essays in market integration, and economic forecasting · essays in market integration, and...

Essays in Market Integration,

and Economic Forecasting

Alonso E. Gomez Albert

A thesis submitted in conformity with the requirements

for the degree of Doctor of Philosophy,

Department of Economics,

University of Toronto

c© Copyright by Alonso E. Gomez Albert, 2009

Essays in Market Integration, and Economic Forecasting

Alonso E. Gomez Albert

Doctor of Philosophy

Department of Economics

University of Toronto

2009

Abstract

In this thesis I study two fields of empirical finance: market integration and economic

forecasting. The first two chapters focus on studying regional integration of Mexican and

U.S. equity markets. In the third chapter, I propose the use of the daily term structure of

interest rates to forecast inflation. Each chapter is a free-standing essay that constitutes

a contribution to the field of empirical finance and economic forecasting.

In Chapter 1, I study the ability of multi-factor asset pricing models to explain the

unconditional and conditional cross-section of expected returns in Mexico. Two sets of

factors, local and foreign factors, are evaluated consistent with the hypotheses of segmen-

tation and of integration of the international finance literature. Only one variable, the

Mexican U.S. exchange rate, appears in the list of both foreign and local factors. Empirical

evidence suggests that the foreign factors do a better job explaining the cross-section of

returns in Mexico in both the unconditional and conditional versions of the model. This

evidence provides some suggestive support for the hypothesis of integration of the Mexican

stock exchange to the U.S. market.

In Chapter 2, I study further the integration between Mexico and U.S. equity markets.

ii

Based on the result from chapter 1, I assume that the Fama and French factors are the

mimicking portfolios of the underlying risk factors in both countries. Market integration

implies the same prices of risk in both countries. I evaluate the performance of the asset

pricing model under the hypothesis of segmentation (country dependent risk rewards) and

integration over the 1990-2004 period. The results indicate a higher degree of integration

at the end of the sample period. However, the degree of integration exhibits wide swings

that are related to both local and global events. At the same time, the limitations that

arise in empirical asset pricing methodologies with emerging market data are evident. The

data set is short in length, has missing observations, and includes data from thinly traded

securities.

Finally, Chapter 3, coauthored with John Maheu and Alex Maynard, studies the abil-

ity of daily spreads at different maturities to forecast inflation. Many pricing models

imply that nominal interest rates contain information on inflation expectations. This has

lead to a large empirical literature that investigates the use of interest rates as predictors

of future inflation. Most of these focus on the Fisher hypothesis in which the interest

rate maturity matches the inflation horizon. In general, forecast improvements have been

modest. Rather than use only monthly interest rates that match the maturity of inflation,

this chapter advocates using the whole term structure of daily interest rates and their

lagged values to forecast monthly inflation. Principle component methods are employed

to combine information from interest rates across both the term structure and time se-

ries dimensions. Robust forecasting improvements are found as compared to the Fisher

hypothesis and autoregressive benchmarks.

iii

Acknowledgements

I am greatly indebted to many people whose encouragement, support and guidance

were invaluable in helping me complete this dissertation.

I have been fortunate to be directed by Angelo Melino. His guidance, support and

patience have been decisive in my academic development. His many suggestions and ideas

are reflected in all chapters. I am deeply grateful to John Maheu for his detailed and

constructive comments. I would also like to thank Gordon Anderson for taking the effort

in reading and providing me with valuable comments on earlier versions of this thesis.

I thank Richard Deaves for his support as member of my academic committee. Special

thanks go to Alex Maynard whose support, encouragement and ideas were invaluable in

helping me complete this dissertation.

To Blanca, I acknowledge her as a coauthor of my work. Her being at my side has

been the highlight of my years as a doctoral student.

I am grateful to Juan Manuel Perez Porrua whose mentorship led me to pursue a PhD.

I want to thank my classmates. Specially, Jean Eid and Carlos Rosell for invaluable

times and conversations.

Last but not least I want to thank my family for their love and unconditional support.

To my mother and sister who probably had numerous moments wondering whether I would

manage to finish. To my father, who unfortunately died in the summer of 2006. If only

he could have been here for this moment. I guess he would have been a proud father.

I gratefully acknowledge support from CONACYT.

iv

Contents

1 Determinants of the Cross-Section of Mexican Stock Returns 1

1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Empirical Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.2.1 Linear Factor Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.2.2 Unconditional and Conditional Factor Pricing Models . . . . . . . . 9

1.3 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.3.1 Returns on Mexican Portfolios . . . . . . . . . . . . . . . . . . . . . 11

1.3.2 Risk Factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

1.3.3 Information Variables . . . . . . . . . . . . . . . . . . . . . . . . . . 15

1.4 Empirical Evidence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

1.4.1 Summary Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

1.4.2 Predictability and Description of Stock Portfolio Returns . . . . . . 17

1.4.3 Unconditional Factor Models . . . . . . . . . . . . . . . . . . . . . . 20

1.4.4 Conditional Factor Models . . . . . . . . . . . . . . . . . . . . . . . 29

1.4.5 Pricing Errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

1.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

v

2 Prices of Risk and Integration 39

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

2.2 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

2.2.1 Factor Pricing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

2.2.2 Beta Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

2.2.3 SDF Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

2.2.4 Measuring Integration . . . . . . . . . . . . . . . . . . . . . . . . . . 55

2.3 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

2.3.1 Portfolios Returns . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

2.3.2 Risk Factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

2.4 Empirical Evidence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

2.4.1 Summary Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

2.4.2 Time-Series Regressions . . . . . . . . . . . . . . . . . . . . . . . . . 63

2.4.3 Prices of Risk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

2.4.4 Pricing Errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

2.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

2.5.1 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

3 Improving Forecasts of Inflation using the Term Structure of Interest

Rates 91

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

3.2 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

3.3 Benchmark Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

3.4 Principal Components Models . . . . . . . . . . . . . . . . . . . . . . . . . . 98

3.4.1 Incorporating Term Structure and Time Series Information . . . . . 99

vi

3.4.2 Dimension Reduction via Principal Components . . . . . . . . . . . 101

3.4.3 Selection of Principal Components . . . . . . . . . . . . . . . . . . . 103

3.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

3.5.1 Benchmark . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

3.5.2 Fixed Number of Principal Components . . . . . . . . . . . . . . . . 107

3.5.3 Bayesian Information Criteria . . . . . . . . . . . . . . . . . . . . . . 109

3.5.4 Model Averaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

3.5.5 Significance of Forecast Improvements . . . . . . . . . . . . . . . . . 110

3.5.6 Summary of results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

3.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

vii

List of Tables

1.1 Composition of Industrial Portfolios . . . . . . . . . . . . . . . . . . . . . . 13

1.2 Summary Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

1.3 Predictability of Industrial Portfolios . . . . . . . . . . . . . . . . . . . . . . 19

1.4 Risk Factors Regressions, Local Factors Model . . . . . . . . . . . . . . . . 21

1.5 Risk Factors Regressions, Fama and French Factors . . . . . . . . . . . . . . 22

1.6 Risk Factors Regressions, Fama and French Factors with Exchange rate . . 24

1.7 Unconditional Pricing Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

1.8 Cross-Section Regressions: Unconditional Model . . . . . . . . . . . . . . . 27

1.9 Conditional Beta Regressions; Local Factors . . . . . . . . . . . . . . . . . . 30

1.10 Conditional Beta Regressions; Fama and French . . . . . . . . . . . . . . . . 31

1.11 Conditional Beta Regressions; Fama and French with Exchange . . . . . . . 32

1.12 Tests for Time Varying Betas . . . . . . . . . . . . . . . . . . . . . . . . . . 33

1.13 Cross-Section Regressions: Conditional Model . . . . . . . . . . . . . . . . . 34

1.14 Pricing Errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

2.1 Summary Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

2.2 Risk Factor Regressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

2.3 Prices of Risk: 1990-2004 Weekly Data . . . . . . . . . . . . . . . . . . . . . 67

viii

2.4 Prices of Risk: 1996-2004 Weekly Data . . . . . . . . . . . . . . . . . . . . . 68

2.5 Prices of Risk: 1990-2004 Monthly Data . . . . . . . . . . . . . . . . . . . . 75

2.6 Prices of Risk: 1996-2004 Monthly Data . . . . . . . . . . . . . . . . . . . . 76

2.7 Tests of Integration, Monthly Data . . . . . . . . . . . . . . . . . . . . . . . 80

3.1 Location of Forecast Summary Results for Various Models . . . . . . . . . . 113

3.2 Benchmark Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

3.3 Out-of-Sample Forecasting for Competing Models, Variance Sort. . . . . . . 115

3.4 Out-of-Sample Forecasting for Competing Models, Correlation Sort . . . . . 116

3.5 Out-of-Sample Forecasting using BIC . . . . . . . . . . . . . . . . . . . . . . 117

3.6 Model Averaging: π12t+12 − π1

t+1. . . . . . . . . . . . . . . . . . . . . . . . . . 118

3.7 Model Averaging: π36t+36 − π1

t+1. . . . . . . . . . . . . . . . . . . . . . . . . . 120

3.8 Diebold Mariano Tests on Regression based Model Averages: π12 − π1. . . . 122

3.9 Diebold Mariano Tests on Regression based Model Averages: π36 − π1 . . . 123

ix

List of Figures

1.1 Realized vs. Fitted returns unconditional model . . . . . . . . . . . . . . . 29

1.2 Realized vs. Fitted returns conditional model. . . . . . . . . . . . . . . . . 35

2.1 Growth in Portfolio Investment by Foreigners . . . . . . . . . . . . . . . . . 40

2.2 Holdings of Mexican Stocks by Foreign and National Investors . . . . . . . 42

2.3 Value Traded as ADRs to Value Traded in Mexico . . . . . . . . . . . . . . 43

2.4 Mexican Industrial and US Fama and French Portfolios. Weekly data. Re-

alized returns are on the horizontal axis and predicted are on the vertical

axis. Circles represent Mexican portfolios and triangles U.S. portfolios. . . . 70

2.5 Mexican Industrial and US Industrial. Weekly data. Realized returns are on

the horizontal axis and predicted are on the vertical axis. Circles represent

Mexican portfolios and triangles U.S. portfolios. . . . . . . . . . . . . . . . . 71

2.6 Mexican Industrial and US Small Caps. Weekly data. Realized returns

are on the horizontal axis and predicted are on the vertical axis. Circles

represent Mexican portfolios and triangles U.S. portfolios. . . . . . . . . . . 72

2.7 Mexican Industrial and US Industrial Portfolios: Monthly Returns. Real-

ized returns are on the horizontal axis and predicted are on the vertical


x

2.8 Mexican Industrial and US Fama and French Portfolios: Monthly Returns.

Realized returns are on the horizontal axis and predicted are on the vertical


2.9 Mexican Industrial and US Small Caps Portfolios: Monthly Returns. Re-

alized returns are on the horizontal axis and predicted are on the vertical


2.10 Pricing Errors. Mexican Industrial and US Fama and French Portfolios.

Weekly data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

2.11 Pricing Errors. Mexican Industrial and US Industrial Portfolios. Weekly

data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

2.12 Pricing Errors. Mexican Industrial and US Small Caps Portfolios. Weekly

data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

2.13 Pricing Errors. Mexican Industrial and US Fama and French Portfolios.

Monthly data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

2.14 Pricing Errors. Mexican Industrial and US Industrial Portfolios. Monthly

data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

2.15 Pricing Errors. Mexican Industrial and US Small Caps Portfolios. Monthly

data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

3.1 Inflation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

xi

Chapter 1

Determinants of the Cross-Section

of Mexican Stock Returns

1.1 Introduction

The purpose of this chapter is to study the determinants of equity returns in Mexico.

The pricing performance of two sets of factors, local and global, inspired by the hypotheses

of segmentation and integration of the Mexican Stock Exchange (MSM) to the U.S. stock

market are evaluated. I examine the ability of multi-factor asset pricing models to explain

the unconditional and conditional cross-section of expected returns of industry portfolios

in Mexico. The results provide evidence that U.S. risk factors are priced in the MSM.

Interest in emerging markets has grown rapidly in recent years as a result of liberal-

ization of capital accounts in emerging markets and as investors seek higher returns and

international diversification. The average net capital flows to emerging market economies

from 1995 to 2003 was 103.12 billion U.S. dollars, of which 8 percent was portfolio invest-

1

ment1. Foreign investment can have a significant impact on returns in emerging markets

(EMs) because they are generally small and illiquid compared to more mature interna-

tional markets. For example, Bekaert and Harvey (2000) present evidence of a negative

relation between the cost of capital and the degree of integration with the world market

in emerging markets.

Defining risk in general, and the factors that affect stock returns, has been the focus of

much work in empirical finance. Interest has focused on identifying which factors (mainly

local) are important to explain the cross-section of expected returns in EMs (Bailey and

Chung (1995), Fama and French (1998)). This paper contributes to the international

finance literature in studying the determinants of the cross-section of Mexican returns

using both local and U.S. risk factors.

I examine the cross-section of returns of industry-based portfolios of Mexican equities.

Data and studies of the MSM, indeed of any Latin American capital market, are scarce.

To my knowledge, this is the first paper that examines whether international factors affect

the cross-section of expected returns in Mexico.

Following Bailey and Chung (1995), the local-factor model includes as factors the local

market risk, exchange rate risk and political risk as the only sources of systematic risk in

expected returns in Mexico.

Fama and French U.S. portfolios were selected as the set of foreign factors used to

explain the cross-section of returns in Mexico. In response to the failures of the CAPM

to explain the cross-section of expected returns sorted by size and book-to-market in the

U.S., alternative models have been suggested to explain the pattern of returns. Fama and

French (1993) developed a three-factor model, with factors related to market risk, book-1International Monetary Fund, World Economic Outlook: Growth and Institutions, World Economic

and Financial Surveys, April 2003.

2

to-market and firm size, that has proved to be successful in capturing the cross-section

of average returns in the U.S. and internationally. I compare the power of the Fama and

French factors relative to the local factors for explaining the cross-section of expected

returns. Empirically, integration between the Mexican stock exchange to the U.S. market

can not be rejected if the Fama and French factors summarize better the risk exposures

of the cross-section of returns in Mexico relative to local factors. Based on the empirical

evidence that exchange rate risk (from a U.S. investor’s perspective) should be included

as a source of systematic risk, I also test the power of an augmented Fama and French

model that includes exchange rate risk as an additional factor.

I search for both unconditional and conditional versions of the local-factor model and

Fama and French model. In the unconditional model, risk premia are assumed to be

constant. In the conditional model, factors in the stochastic discount factor are expected

to price assets only conditionally, leading to time-varying rather than fixed linear factor

models. If risk premia are time-varying, the parameters in the stochastic discount factor

will depend (among other conditional moments), on investors’ expectations of future av-

erage returns. To capture this variation, I assume that the parameters of the stochastic

discount factor depend on current-period information variables, as in Cochrane (2001),

Ferson and Harvey (1999) and Lettau and Ludvigson (2001).

Factors are scaled by variables (instruments) that are likely to be important in sum-

marizing variation in expected future returns. A conditional linear factor model can be

expressed as an unconditional multi-factor model on the scaled factors. However, the

choice of conditioning variables is of central importance for this approach. The fact that

expected returns are a function of investors’ conditioning information, which is unob-

servable, represents a practical obstacle in testing conditional factor models. In order to

address this problem, conditioning variables are selected based on their empirical perfor-

3

mance in forecasting future returns.

The empirical results from cross-section regressions suggest that the unconditional

model using the Fama and French factors does a good job explaining the cross-section of

MSM returns (see Figure 1.1). This result suggests that risk factors that have proven to be

successful in explaining the cross-section of U.S. returns are also important in explaining

the cross-section of MSM returns. This results supports the hypothesis of integration

between both markets. Compared to the Fama and French model, the local-factor model

was not able to capture the cross-section of average returns (see upper-right graph of

Figure 1.1). In time-series regressions, results show that portfolio returns appear to be

highly correlated with local factors, specially with the Mexican market return, yielding

high R2s. However, in cross-section regressions these risk exposures have low explanatory

power when compared to the Fama and French risk exposures.

Results for the conditional asset pricing models suggest that risk premiums are signifi-

cantly time-varying in the case of Fama and French factors, whereas in the local-factor case

the hypothesis of time-varying risk exposures is rejected. In both specifications, uncondi-

tional and conditional, Fama and French factors dominates the local-factor specification.

When the exchange rate is included, the conditional version of the Fama and French

model outperforms all of the other specifications by explaining 60 percent of the cross-

section of expected returns compared to 47 percent for the local-factor model. Global

factors, in particular, the Fama and French factors and exchange rate risk appear to be

more important in explaining the cross-section of returns than local factors, suggesting

integration of the MSM to the U.S. market.

The chapter is organized as follows. In section 2, I give a brief summary of factor

pricing models and address the difference between conditional and unconditional asset

pricing. A detailed description of the data used in this paper is given in section 3. Section

4

4 presents the empirical results. Conclusions are presented in the final section.

1.2 Empirical Methodology

1.2.1 Linear Factor Model

In the absence of arbitrage, the fundamental pricing equation is

Pt = Et(mt+1(Pt+1 + Dt+1)) (1.1)

where Pt is a vector of asset prices at time t, Dt+1 represents a vector of interest, dividends

or other payments at t+1, and mt+1 is the stochastic discount factor (SDF)2. Et represents

the conditional expectation with respect to Ωt, the market-wide information set. Since

Ωt is unobservable from a researcher’s perspective, expectations are usually conditioned

on a vector Zt of observable variables (instruments) that are contained in Ωt. Equation

(1.1) can also be written in terms of returns. Strict positivity of mt+1 follows from no

arbitrage opportunities alone, but more structure is needed in order to explore the model

empirically. Multiple factor models for asset pricing follow when mt+1 can be written

as a function of a set of risk factors. The notion that the SDF comes from an investor

optimization problem, and is equal to the intertemporal rate of substitution, suggests

that likely candidates for the factors are variables that can proxy consumption growth

or wealth, or any state variable that affects the marginal rate of substitution along an

optimal consumption-investment path. Equation (1.1) implies

Et(mt+1rt+1,i) = 0 i = 1, ..., N (1.2)

2Also known as the pricing kernel or intertemporal marginal rate of substitution.

5

where rt+1,i are excess returns. Expanding Equation (1.2) in terms of the covariance

Et(rt+1,i) =Covt(rt+1,i,−mt+1)

Et(mt+1)i = 1, ..., N (1.3)

The conditional covariance of the excess return with the SDF is a general measure of

systematic risk. In standard economic models, it measures the component of returns that

is related to fluctuations in the marginal utility of wealth. In terms of returns, investors

are willing to trade off overall performance to improve payoffs in “bad” states of nature.

A linear factor model is of the form: mt+1 = a + b′ft+1, where ft+1 is a k × 1 vector

of risk factors. In the special case when the test assets are all zero wealth portfolios so

that we only use data in excess returns, then Et(mt+1rt+1) = 0 does not identify the mean

of mt+1, and at can be normalized arbitrarily to one: mt+1 = 1 + b′ft+1. In general,

mt+1 can be written as mt+1 = mt+1 + εt+1 where mt+1 is the projection of mt+1 on the

asset space and εt+1 is orthogonal to the asset space, so E(mt+1εt+1) = 0. Any random

variable orthogonal to returns can be added to m, leaving the pricing implications of (1.3)

unchanged.

In the case of conditional factor models, the coefficients at and bt vary over time as a

function of conditioning information, mt+1 = at + b′tft+1. To illustrate this heuristically, I

assume that the factors ft+1 are returns on tradeable assets. Imposing the condition that

the model correctly prices the risk free rate Rft and the factors, ft+1, yields

ιk = Et(mt+1ft+1) and 1 = Et(mt+1Rft ) (1.4)

6

where ιk is a k × 1 vector of ones. Solving for at and bt we obtain

at =1

Rft

−Et(f ′t+1)bt and bt = (V art(ft+1))−1

(ιk − Et(ft+1)

Rft

)(1.5)

Equation (1.5) shows explicitly that both at and bt are functions of the risk free rate,

Rft , and the conditional first and second moments Et(ft+1), and V art(ft+1). Therefore,

time variation in conditional moments of mimicking portfolio returns imply time-variation

in the parameters governing the stochastic discount factor. Following Cochrane (2001),

Ferson and Harvey (1999) and Lettau and Ludvigson (2001), I assume that the denomina-

tor in bt is not likely to be highly variable3. On the other hand, a large body of literature

has documented that excess returns are predictable to some degree using monthly or quar-

terly data. Therefore, in the conditional model, I will restrict attention to instruments

that help predict the equity premia.

Combining Equation (1.3) with the linear specification of the SDF, mt+1 = at +b′tft+1,

we obtain the beta representation of expected returns

Et(rt+1,i) = −Covt(rt+1,i, f′t+1)

Et(mt+1)bt ≡ −β′t,i

V art(ft+1)Et(mt+1)

bt ≡ β′t,iλt (1.6)

where βt,i are the population time-varying regression coefficients of a regression of rt+1,i

on ft+1, and are the loadings or risk exposures to ft+1 risks. The components of λt are

the associated prices of risk for each unit of risk exposure. Following Cochrane (2001),

and Lettau and Ludvigson (2001), the conditional factor pricing model given above is

implemented by explicitly modeling the dependence of the parameters in the stochastic

discount factor, at and bt, on time-t information variables, Zt, that forecast future excess3Predictable movements in volatility may be a source of variation in bt, however they appear to be more

concentrated in high-frequency data (Campbell, Lo, and MacKinlay (1997)) than in monthly, or quarterlyreturns.

7

returns.

Differences in risk factors exposures are measured using time-series regressions of in-

dustrial portfolio excess returns on contemporaneous risk factors.

rt+1,i = ct,i + β′t,ift+1 + εt+1,i, i = 1, ..., N (1.7)

where rt+1,i are excess returns over a one-month government zero coupon bond yield, and

ft+1 is a vector of excess returns of economic risk factors. The vector of coefficients β′t,i

are risk exposures of portfolio excess returns to the factors ft+1. In section 1.2.2 below, I

further describe the scaled factor approach used to estimate βt,i in Equation (1.7). The

property Et(εt,ift+1) = 0 captures the fact that the coefficients βt,i are the conditional

betas of the returns. The idea behind the beta representation (1.6) is to explain the

variation in excess returns across assets where betas are a measure of risk compensation

between assets, and the λs are the premiums per unit of risk. Equation (1.6) is estimated

using cross-section regressions,

Et(rt+1,i) = βt,i′λt + αi,t i = 1, ..., N (1.8)

where the betas are the right-hand variables that come from (1.7), the factor risk premia λ

are the regression coefficients, and αi are the pricing errors (differences between expected

and predicted returns). This method is also known as a two-pass regression estimate. In

applying standard OLS formulas to cross-sectional regressions, it is implicitly assumed that

the right-hand variables (in this case β) are fixed. However, the βs in (1.8) are estimates

from a time-series regression resulting in errors-in-variables bias. Shanken (1992) provides

the corrected asymptotic standard errors for λ and α (see Cochrane (2001)).

8

1.2.2 Unconditional and Conditional Factor Pricing Models

As mentioned above, betas are the variables that explain the variation in average

returns across assets. Therefore, the general model for expected returns should have betas

that vary by asset and time.

To evaluate if expected returns and risks are time varying, I first estimate the uncon-

ditional version of equation (1.7) and (1.8) assuming constant coefficients. However, if

risk exposures vary through time in a predictable manner, for example, with the business

cycle, the unconditional model will be misspecified.

To test for time-varying risk exposures, the unconditional version of the model is

taken as the null hypothesis, and different specifications of the conditional model with

time-varying risk exposures are set as alternatives.

To proceed, first I must specify the risk factors. Two sets of factors are used to explain

the cross-section of returns in Mexico. The first set correspond to the hypothesis of full

segmentation of the MSM to the U.S. stock market. Under this hypothesis, risk exposures

on the MSM are represented only by local factors. The vector of local factors is composed

by the local stock market return, the exchange rate risk and a proxy of political risk. The

second set of factors is consistent with the hypothesis of integration between the MSM

and the U.S. market. Given the ability of Fama and French (1993) factors to explain the

cross-section of expected returns in the U.S., these factors appear as good candidates to

summarize risk exposures in the MSM if markets are integrated. Therefore, not only the

pricing performance of the two sets of factors is evaluated, but also the empirical results

provide an assessment of the hypothesis of integration as measured by the ability of the

Fama and French factors to explain the pattern of returns in Mexico. Consequently, I

9

interpret as supporting evidence of integration if the cross-section of Mexican returns is

better summarized by the Fama and French factors than by the set of local factors.

Scaled Factors Approach

A popular and simple approach to incorporate conditioning information is based on

scaled factors. As shown above (Equation (1.5)), in a conditional setting, the coefficients

associated with the discount factor mt+1 are time-varying and depend on the time-t in-

formation set. A partial solution is to model the dependence of the betas in Equation

(1.8) with a subset of variables that belong to the time-t information set. Furthermore, if

a linear specification is assumed, we can write

βt,i = D′iZt (1.9)

ct,i = c′iZt

where Zt is an L× 1 vector of information variables (including a constant) known at time

t, and the elements of the matrix Di are fixed parameters to be estimated. In choosing

the instruments, Zt, I focus only on variables that can forecast conditional returns4. Con-

versely, the unconditional factor model is characterized by fixed betas and is a special case

of Equation (1.9) when Zt is constant.

Combining Equation (1.7) and Equation (1.9), time series regressions with conditional

betas are

rt+1,i = c′iZt + d′i(Zt ⊗ ft+1) + εt+1,i (1.10)

4As shown in Equation (1.5), at and bt are functions of conditional returns, therefore variables that cansummarize variation in conditional moments are used as instruments Zt.

10

where every factor is scaled by every instrument, and di is given by V ec(Di)5. It is worth

mentioning that the coefficients di in Equation (1.10) are linear and fixed on the scaled

factors (Zt ⊗ ft+1), so the conditional version of the factor model can be viewed as an

unconditional factor model over scaled factors.

In order to evaluate the ability of the scaled-factor model to explain the cross-section

of returns, time-varying betas are recovered using the estimated version of Equation (1.9),

βt,i = D′Zt and cross-section regressions (1.8) of returns on βt,i are estimated.

1.3 Data

The sample is limited to the period following the devaluation suffered by the Mexican

peso at the end of 1994; it runs from May 1995 to October 2003. Monthly Mexican stock

prices and Mexican bond returns were obtained from Infosel Financiero6. The rest of the

variables were obtained from the Central Bank of Mexico, the Board of Governors of the

Federal Reserve System web page, and the Fama and French web page.

The data comprise two types of series: financial and macro variables, and are used to

construct portfolio returns, risk factors, and information variables.

1.3.1 Returns on Mexican Portfolios

To construct monthly returns, log differences of end-of-month closing prices were cal-

culated. If the end-of-month price was not available, the closest quote preceding the

end-of-month was used. There are a total of 101 months in the sample. Stock prices were

adjusted for splits and dividends7. I compute returns for all Mexican stocks that traded5V ec(A) is the operation represented by the vectorization columnwise of matrix A.6Mexican electronic provider of financial information.7In the sample analyzed, very few stocks payed dividends before 2001. However, by the end of the

sample a high proportion of stocks were paying dividends.

11

between 1995 and 2003 and the Mexican stock index. The average number of firms listed

in the MSM during the sample is of 124, peaking in 1998 with 131 stock series8, and the

MSM index. I applied some filtering rules and summarized the stock returns by returns

on industry portfolios.

In order to evaluate the pricing performance of different sets of factors (in a com-

mon currency, and from a U.S. perspective), nominal log returns in Mexican pesos were

converted to U.S. dollar returns. Excess returns of Mexican industrial portfolios were

computed and are defined as the difference between its log U.S. return and the 30-days

T-bill return.

Given the thinness of trading in many of the Mexican stocks in the sample, and in

order to help address potential problems such as survivorship bias9, missing observations

for individual stocks, and noise in individual security returns, I aggregated individual

stocks into industrial portfolios. The industrial categories resemble the official categories

defined by the MSM and are given by: 1) Beverages, Food Products and Tobacco, 2)

Financial Services, 3) Building Products, that includes engineering, construction and the

real state sectors, 4) Conglomerates, 5) Media, entertainment and telecommunications,

6) Chemical and Metal Production, 7) Industrial, an aggregation that contains the paper

and pulp products industry, textiles industry, glass production and tubes production, 8)

Machinery and Equipment, 9) Retail Services and 10) Transportation. Table 1.1 presents

a summary of the number of firms that comprise each portfolio, as well as the relative

annual average liquidity, measured as the value of the transactions of the portfolio to the

value of all transactions of the market. Industrial portfolios are formed using weights based8The average number of series traded daily in the MSM between 2000 and 2003 is around 70 stocks.

However, the total number of firms listed in 1995 is 185, reaching its maximum of 195 listed firms in 1998and falling to 158 for 2003. Only about 60 percent of these stocks trades at least once per week.

9Only successful companies are included in the sample.

12

Table 1.1: Composition of Industrial PortfoliosComposition of industrial portfolios in the sample. The column labelled Firms gives themaximum number of firms used to construct each portfolio. The average relative liquiditymeasures the share of the value of the total trading within the sample of firms.

Industrial Firms Average Relative Liquidity of Industrial PortfoliosSector Year

1995 1996 1997 1998 1999 2000 2001 2002Beverages, Food 22 7.97 6.56 8.08 6.82 9.41 16.42 8.46 6.71

and TobaccoFinancial Ser- 18 7.51 2.85 3.23 3.30 3.47 3.88 6.89 9.61

vicesBuilding 21 31.26 19.34 26.04 20.02 17.88 15.70 14.22 9.38

Conglomerates 18 15.69 8.88 10.50 10.59 9.76 9.73 9.03 4.70

Media & Telecoms 10 24.53 24.10 16.90 12.52 13.47 17.83 24.58 27.86

Chemical & Metal 11 1.28 8.29 11.44 6.38 4.80 2.42 3.15 5.03

Industrial 13 2.53 2.08 3.64 5.09 3.14 3.82 2.42 3.29

Mach. & Equip. 5 0.00 0.22 0.01 0.01 0.03 0.02 0.00 3.50

Retailing 17 8.88 27.51 19.69 32.95 37.78 29.98 31.21 29.62

Transportation 3 0.35 0.16 0.48 2.32 0.27 0.20 0.05 0.31

on the previous year’s annual average liquidity and are re-balanced each January. The

weights for each stock in each industrial portfolio are given by the relative annual average

volume of the stock to the annual average volume of the portfolio. The cross-section of

the sample includes many industries and all of the components of the IPC index10.

10The IPC is the most important market index and is comprised of 35 stocks (see next section).

13

1.3.2 Risk Factors

I specify two sets of factors, ft+1, that represent potential sources of rewarded risk in

the MSM. The choice of each set of factors is based on different assumptions concerning

the degree of integration of the Mexican market to the North American market. In what

follows, the factors will be divided into two categories: a) Local Factors and b) Foreign

Risk Factors.

Local Factors: IPC is the monthly log-difference of the Mexican market index ex-

pressed in U.S. dollars, and in excess of the 30-day T-bill. The IPC is the most important

index of the MSM and is computed as the market capitalization weighted average price of

35 of the most liquid stocks listed on the MSM. It represents a broad sample of industries.

Exchange rate risk, Exch, is computed as the log-difference of the “fix” exchange rate

(in terms of U.S. dollars/ Mexican pesos). The “fix” rate is determined on a daily basis

by the central bank and is computed as the interbank market exchange rate at the close.

As a proxy of political risk, the spread between the 5 year yield of the UMS and

the matching maturity of a U.S. Treasury note, Diff , was computed. To obtain this

spread, I calibrated a time series of a zero-coupon term structure at fixed terms from the

observed prices of Mexican government bond issued in US dollars (UMS)11. Diff reflects

perceived national credit risk, and is assumed to be highly correlated with political risk.

Changes in sovereign yield spreads, like credit ratings, generally reflect changes in bond

markets’ perceptions of an indebted country’s credit worthiness. Sudden increases are

usually followed by a drying up of liquidity and a flow out of national equity markets.

Alder and Qi (2003) interpret sovereign default risk as a measure of relative segmentation.

Their rationale is that when sovereign default risk cannot be completely diversified, and11These bonds pay a fixed semi-annual coupon. The maturity of the bonds that I used to estimate a

zero coupon structure are: 06-Apr-05 15-Jan-07 12-Mar-08 17-Feb-09 15-Sep-16 15-May-26

hence is a priced factor, international investors will respond to an unexpected increase in

default risk by liquidating their positions of assets subject to default risk. The same effect,

they argue, will occur if the market becomes suddenly segmented.

Foreign Factors:

If the MSM is integrated to the U.S. stock markets, risk factors that have been success-

ful in explaining the cross-section of different sorts of U.S. portfolios should also explain

the cross-section of Mexican portfolios12. Therefore foreign risk factors are assumed to

be summarized by the U.S. Fama and French factors13. The Fama and French mimicking

portfolios related to market, size and book-to-market equity ratios are: a) Market risk,

Mkt, that is the monthly return of the U.S. market portfolio in excess of the 30-days

T-bill, b) Size factor: SMB (Small Minus Big) is the average return on three small port-

folios minus the average return on three big portfolios, and c) Value factor: HML (High

Minus Low) is the average return on two value portfolios (stocks with high book-to-market

ratio) minus the average return on two growth portfolios (stocks with low book-to-market

ratio)14.

1.3.3 Information Variables

To evaluate the scaled factor model, information variables Zt that track variation in risk

exposures in time t+1 are assumed to be known by investors in time t. Following previous

studies (see Campbell, Lo, and MacKinlay (1997); Ferson and Harvey (1993)), I explored12One of the most questionable issues in the empirical international finance literature seeking to mea-

sure integration of national stock markets, is the use of the CAPM or ICAPM to explain internationalreturns. Given the documented empirical failure of the CAPM in a domestic environment (see for exampleBanz (1981); Fama and French (1993)), a multi-factor approach appears to be more appropriate in aninternational setting.

13Given the differences in size between U.S. stock markets and the MSM, the U.S. Fama and Frenchfactors are a good proxy of a weighted portfolio of Mexican and U.S. factors, where the weights areproportional to the capitalization of the U.S. and Mexican markets.

14See Fama and French (1993) for a detailed explanation of these portfolios.

15

various candidates that include: the monthly growth rate of labor income, the monthly

growth of financial assets holdings, the spread between the one year Cetes(Certificados

de la Tesoreria) and the 28 days Cetes15, the ex-post real return of the 28-days Cetes,

the lagged exchange rate, lagged Diff , lagged Fama and French factors, lagged U.S.

Momentum factor, the U.S. term premium, measured by the spread between the five-year

and one-month Treasuries rates, and the short term spread between T-bills and Cetes.

Of these variables, only three appear to have forecasting power for future Mexican

returns. The first one is the monthly growth rate of labor income, ∆y; the second variable

is the monthly growth of financial asset holdings, ∆FA; and the last one is the short-term

premium of the Mexican government term structure measured by the spread between the

one year and the 28 days Cetes, CetSp.

15The Cetes is a zero coupon bond auctioned weekly by the Mexican Treasury that represents the leadinginterest rate in Mexico. Typically, the term structure is composed of bonds with maturities of 28, 91, 182,364 and occasionally of 724 days

16

1.4 Empirical Evidence

1.4.1 Summary Statistics

Table 1.2 presents summary statistics for the portfolios’ excess returns, risk factors and

information variables; the means and standard deviations for returns are annualized. The

Media & Telecoms portfolio is not only the most liquid portfolio, but also the portfolio

with highest average excess return in our sample, with an annualized average excess return

of 10.52 percent in U.S. dollars. This portfolio is dominated by the telephone company

monopoly, privatized at the beginning of the 90’s. It represents the most active Mexican

stock in both the MSM and NYSE, and is the leading stock in the composition of the

IPC.

Diff , the risk premium of Mexican sovereign debt measured by the spread between

the 5 years yield of the UMS and the U.S. Treasury note of the same maturity, has an

average of 3.08 percent. The political risk premium is highly persistent.16.

Cross-correlations are presented in panel D of Table 1.2. The IPC is highly correlated

with both Mkt and Exch17.

1.4.2 Predictability and Description of Stock Portfolio Returns

To implement the conditional asset pricing model, the set of instrument variables Zt that

capture the dependence of at and bt on the information set Ωt has to be defined. Since only

variables that forecast returns and/or the stochastic discount factor, mt+1, are relevant

to the pricing problem (see equation (5)), I concentrate on a small set of variables that

have the ability to forecast future returns. Table 1.3 summarizes the results of regressing16Autocorrelation coefficients for this variable suggest that Diff follows an AR(1). The first order

autocorrelation is 0.81 while the second autocorrelation is of 0.64.17Remember that Exch is measured as the price of Mexican Pesos in U.S. dollars

17

Table 1.2: Summary StatisticsReturns are in U.S. dollars and measured in excess of the 30 days T-bill. All sample meansand standard deviations are annualized. The sample period is March 1995 to October 2003.In panels A-C, the sample autocorrelations, ρj , are presented in the first row, and p-values ofthe Ljung-Box statistic for testing the joint significance of the autocorrelation coefficient up tothe corresponding lag are presented in the second row. Panel D presents the sample correlationmatrix of selected factors.

Mean Std. Dev. ρ1 ρ2 ρ3 ρ4 ρ12 ρ24

Panel A. Industrial PortfoliosBeverage, Food 1.96 33.48 -0.19 0.04 0.05 -0.17 0.11 -0.09and Tobacco 0.03 0.47 0.29 0.06 0.32 0.11Financial Services 7.55 53.38 -0.05 0.03 0.10 -0.26 0.05 -0.13

0.30 0.40 0.15 0.01 0.39 0.41Building -7.01 39.86 0.01 -0.08 0.06 -0.07 0.00 -0.02

0.46 0.23 0.28 0.23 0.35 0.28Conglomerates -8.84 42.10 -0.10 0.08 0.11 -0.14 0.06 -0.11

0.16 0.24 0.12 0.10 0.44 0.09Media & Telecoms 10.52 38.08 -0.12 -0.02 0.09 -0.11 0.02 -0.06

0.13 0.36 0.21 0.19 0.34 0.20Chemical & Metal -24.87 45.17 -0.03 0.09 0.09 -0.06 0.01 -0.12

0.40 0.19 0.19 0.25 0.45 0.08Industrial -7.20 32.77 -0.08 -0.07 0.13 -0.23 0.10 0.12

0.23 0.22 0.12 0.01 0.32 0.27Machinery & Equipment -21.72 54.43 0.19 0.10 0.05 -0.15 -0.06 0.07

0.03 0.26 0.44 0.04 0.45 0.38Retailing 6.71 54.52 0.08 -0.01 0.05 0.00 0.03 -0.04

0.22 0.42 0.29 0.47 0.42 0.25Transportation -6.77 43.50 -0.14 0.00 -0.06 -0.05 0.10 -0.10

0.09 0.41 0.27 0.24 0.13 0.20Panel B. Risk FactorsIPC 5.03 34.39 -0.09 -0.05 0.10 -0.14 0.00 -0.06

0.18 0.29 0.19 0.12 0.44 0.10Exch -4.55 8.92 -0.05 -0.15 0.01 0.10 0.09 0.10

0.31 0.06 0.48 0.23 0.41 0.48Diff 3.08 0.43 0.81 0.64 0.48 0.37 -0.16 -0.07

0.00 0.26 0.26 0.32 0.28 0.45Mkt 5.72 17.72 0.04 -0.08 0.01 -0.08 0.02 0.04

0.34 0.23 0.44 0.20 0.36 0.43SMB -5.05 16.62 0.27 0.13 0.00 0.15 -0.02 -0.01

0.00 0.29 0.32 0.06 0.36 0.23HML 10.28 15.11 0.27 0.11 0.20 0.13 0.05 -0.11

0.00 0.35 0.04 0.38 0.43 0.27Panel C. Information Variables∆y 6.24 9.77 -0.57 0.02 0.33 -0.33 0.40 -0.11

0.00 0.00 0.05 0.38 0.03 0.44∆FA 6.82 3.69 -0.01 -0.08 -0.01 -0.09 0.33 0.20

0.46 0.22 0.44 0.16 0.00 0.20CetSp 2.26 1.70 0.61 0.27 0.15 0.03 0.10 0.23

0.00 0.06 0.16 0.10 0.48 0.26Panel D. Cross-Correlations

IPC Exch Diff Mkt SMBExch 0.67Diff -0.14 -0.11Mkt 0.72 0.38 -0.12SMB 0.11 0.05 -0.08 0.09HML -0.31 -0.16 -0.17 -0.51 -0.52

18

Table 1.3: Predictability of Industrial PortfoliosMonthly excess returns are regressed on a set of lagged instruments. The instrumentalvariables are “∆yt−1” the lagged real growth in labor income, “∆FAt−1” the laggedreal growth in asset holdings, and CetSp is the lagged spread between the one year andone month cetes. HAC consistent t-ratios are on the second line below the coefficients.R2 is the coefficient of determination, with the adjusted R2 on the second line. ρis the first order autocorrelation of the regression residual, with its t-value in thesecond column. F is the F -statistic of testing the hypothesis of zero coefficients ineach regression.

Const ∆yt−1 ∆FAt−1 CetSp R2 ρ F

Beverage, Food 1.05 -1.65 1.85 -0.55 0.22 -0.51 12.67and Tobacco 0.69 -5.04 2.24 -1.06 0.19 -5.88Financial Service 3.20 -2.49 1.96 -1.22 0.20 -0.49 11.35

1.33 -4.76 1.49 -1.46 0.17 -5.26Building 2.76 -1.67 1.98 -1.69 0.20 -0.23 11.18

1.52 -4.24 1.99 -2.68 0.17 -2.20Conglomerates 1.25 -1.90 1.13 -0.80 0.18 -0.48 10.05

0.64 -4.50 1.06 -1.18 0.15 -5.27Media & Telecoms 2.29 -1.33 1.11 -0.66 0.11 -0.44 5.59

1.24 -3.34 1.10 -1.04 0.08 -4.41Chemical & Metal 1.54 -1.88 1.66 -1.60 0.17 -0.17 9.53

0.72 -4.10 1.44 -2.18 0.15 -1.70Industrial 0.22 -0.99 1.77 -0.64 0.10 -0.29 4.98

0.14 -2.86 2.03 -1.16 0.07 -3.28Machinery & Equipment -0.64 -0.75 1.61 -0.66 0.02 0.11 1.14

-0.23 -1.26 1.07 -0.69 -0.01 1.23Retailing 2.78 -1.81 2.11 -1.30 0.12 -0.39 5.87

1.09 -3.27 1.51 -1.46 0.09 -3.97Transportation 0.54 -0.97 0.11 -0.33 0.05 -0.46 2.17

0.25 -2.04 0.09 -0.43 0.01 -5.05IPC 2.17 -1.44 1.28 -0.83 0.16 -0.46 8.75

1.34 -4.13 1.45 -1.48 0.13 -4.70

future excess returns on the 10 industrial portfolios and the IPC on lagged information

variables Zt. The regressions produce significant t-statistics in many cases. The R2 for

the IPC is of 16 percent.

The F -statistic for the joint hypothesis of zero coefficients is rejected in 10 of the 11

portfolios. In addition, the F -test associated with the joint hypothesis of zero coefficients

in all portfolios was rejected with a p-value of 3.5×10−3. To further evaluate the ability

of these information variables Zt to forecast returns, and to mitigate possible problems

concerning data mining, I conducted the forecasting exercise with out-of-sample returns on

19

the IPC using a sample from January of 1982 to August of 2004. An R2 of 5 percent was

obtained for the whole sample and of 10 percent using a subsample from January of 1982

to January of 1991. Despite the structural transformation experienced in Mexico during

the last 20 years, my choice for Zt appears to have forecasting power on stock returns over

these years. The hypothesis of a change in the value of the coefficients associated with

the forecasting variables between 1982-1995 and 1995-2003 was conducted. Parameter

constancy between samples was rejected.

1.4.3 Unconditional Factor Models

Time-Series Evidence of the Factor Model

Results for time-series regressions of realized returns on contemporaneous factors ft+1, as

described in Equation (1.7), assuming that both at,i and βt,i are constant, are presented

from Table 1.4 to Table 1.6 for the the local-factor model, the Fama and French model and

the augmented version of the Fama and French that includes exchange rate, respectively.

The objective of regressions on contemporaneous factors is to measure risk exposures to

the proposed factors. In other words, we are trying to measure if risk factors can account

for the variability in the cross-section of returns. In the next section I evaluate if these

risks are priced.

Table 1.4 presents results for the local-factor model18. Excluding the transportation

sector19, the R2 coefficients for the local-factor model range from 51 percent in the In-18I included the local market stochastic volatility (that is a measure of market idiosyncratic risk, mea-

sured as both the squared sum and absolute value of daily returns in both U.S. dollars and Mexican pesos)as an additional local risk factor. This was motivated by the international finance literature that exploresintegration with a weighted average of the ICAPM and CAPM. Under this specification, systematic riskunder the hypothesis of segmentation, is quantified by the variance of the local market. However, riskexposures for this measure of idiosyncratic risk were not significant for any of the industrial portfolios.

19As noted in Table 1.1, the transportation sector accounts for less than 2.4 percent of total transactionsin the MSM.

20

Table 1.4: Risk Factors Regressions, Local Factors ModelMonthly data from March 1995 to October 2003. Excess returns in U.S. dol-lars are regressed on the excess return on the the MSM index “IPC”, exchangerate “Exch” and Diff is the UMS spread. HAC consistent t-ratios are on thesecond line below the coefficients. R2 is the coefficient of determination, withthe adjusted R2 on the second line. ρ is the first order autocorrelation of theregression residuals, with its t-value in the second column. F is the F -statisticfor the hypothesis of zero coefficients in each regression.

Const IPC Exch Diff R2 ρ FBeverage, Food -0.28 0.84 0.39 0.08 0.87 -0.09 202.59and Tobacco -0.34 17.18 2.06 0.32 0.87 -0.95Financial Services 0.51 1.21 0.86 -0.02 0.78 -0.08 107.07

0.29 11.93 2.20 -0.04 0.77 -0.94Building 0.94 0.86 0.85 -0.51 0.79 -0.08 115.11

0.75 11.63 3.00 -1.38 0.79 -1.00Conglomerates 0.29 0.97 0.65 -0.38 0.80 -0.08 121.84

0.22 12.70 2.23 -1.02 0.80 -0.99Media & Telecoms 0.60 1.15 0.53 -0.13 0.92 -0.09 351.92

0.82 26.44 3.18 -0.62 0.92 -0.88Chemical & Metal 1.56 0.72 0.96 -1.16 0.52 0.00 32.32

0.72 5.64 1.97 -1.84 0.50 -0.05Industrial -1.17 0.47 0.83 0.22 0.44 -0.06 23.38

-0.69 4.71 2.15 0.45 0.42 -0.76Machinery & Equipment 2.78 0.76 -0.57 -1.67 0.22 0.05 8.57

0.84 3.90 -0.76 -1.72 0.20 0.65Retailing 2.14 1.21 0.66 -0.60 0.72 -0.08 77.72

1.07 10.34 1.47 -1.03 0.71 -0.99Transportation 3.75 0.25 1.29 -1.28 0.22 0.03 8.53

1.41 1.61 2.16 -1.65 0.20 0.36

21

Table 1.5: Risk Factors Regressions, Fama and French FactorsMonthly data from March 1995 to October 2003. Excess returns in U.S. dollars areregressed on the excess return of the the standard’s and Poor Index “S&P”, thesmall minus big factor “SMB”, and high minus low factor“HML” of Fama andFrench. HAC consistent t-ratios are on the second line below the coefficients. R2 isthe coefficient of determination, with the adjusted R2 on the second line. ρ is thefirst order autocorrelation of the regression residual, with its t-value in the secondcolumn. F is the F -statistic for the hypothesis of zero coefficients in each regression.

Const Mkt SMB HML R2 ρ FBeverage, Food -0.75 1.41 0.32 0.44 0.46 0.11 25.50and Tobacco -0.99 8.22 1.73 1.88 0.44 0.98Financial Services -0.56 1.87 0.12 0.40 0.33 0.06 14.64

-0.41 6.14 0.36 0.97 0.31 0.56Building -1.66 1.59 0.71 0.71 0.42 0.18 21.62

-1.76 7.50 3.11 2.46 0.40 1.63Conglomerates -1.63 1.59 0.70 0.50 0.42 0.18 21.60

-1.63 7.12 2.92 1.62 0.40 1.60Media & Telecoms -0.13 1.69 0.06 0.26 0.55 0.06 36.68

-0.16 9.51 0.32 1.07 0.54 0.54Chemical & Metal -3.09 1.48 0.87 0.79 0.30 0.29 12.89

-2.64 5.64 3.09 2.20 0.28 2.40Industrial -1.47 0.99 0.39 0.65 0.21 0.19 7.99

-1.62 4.89 1.78 2.33 0.18 1.71Machinery & Equipment -2.94 1.56 0.62 0.76 0.20 0.22 7.69

-1.95 4.60 1.71 1.64 0.18 1.90Retailing -1.08 2.25 0.43 0.87 0.41 0.10 21.14

-0.83 7.72 1.38 2.19 0.39 0.93Transportation -1.26 0.89 0.62 0.63 0.12 0.27 3.92

-0.99 3.11 2.03 1.61 0.09 2.34

dustrial sector to 87 percent for the Beverage, Food and Tobacco sector. The two most

important factors in the local-factor model are the market return IPC and the exchange

rate (Exch). For all portfolios, the constant term appears not significant.

Table 1.5 shows the same results for the Fama and French factor model. The slope on

the U.S. market factor, Mkt, appears uniformly significant and positive for all industrial

portfolios. Risk exposures to SMB and HML, are also significant in several industrial

portfolios. An interpretation for HML, not universally accepted, is provided by Fama and

French (1998). They suggest that HML acts as a proxy for relative distress. Weak firms,

with low earnings, tend to have high book-to-market ratios and positive loadings on HML,

whereas the contrary effect is observed for strong firms. Therefore, high positive exposures

22

to HML can be interpreted as a measure of financial distress. With this interpretation,

the results suggest that Mexican portfolios have a positive exposure to financial distress.

Mexican portfolios also have a positive exposure to the size factor, SMB. The SMB

factor captures the common variation of small stocks. Given the size of Mexican stocks

relative to U.S. firms, it is not surprising that Mexican portfolios have a positive exposure

to SMB. Compared to the local-factor model, R2s for the Fama and French model are

lower, however, the constant terms appear insignificant for almost all portfolios. The fact

that R2s for the local model are higher than for the Fama and French model is a result of

the high correlation between the IPC and portfolio returns20.

Finally, and following the international finance literature where the exchange rate has

proven to be an important risk factor within an international setting, I extended the Fama

and French model by including exchange rate risk. In general, the coefficients associated

with the Fama and French factors are very similar when Exch is included (Table 1.5

and Table 1.6). Exch appears significant and positive in all portfolios. From a time-

series perspective, it appears that the local-factor model measured by R2s, does a better

job in explaining the pattern of industrial returns in Mexico. In the following section,

however, cross-section regressions give evidence that despite the high R2s from time-series

regression in the local-factor model, betas from the Fama and French model do a better

job explaining the cross-section of returns in Mexico.

To complement these results and to assess the relative importance of each risk factor,

I performed F -tests21 to test the joint significance of each risk factor in all industrial20Given the high weight that some of the portfolios (such as Telecom) have in the IPC, some of this

hight correlation is most likely “spurious”21For testing linear restrictions in a SURE representation, the analogous F -statistic under GLS assump-

tions is: F = (Rβ−r)′[RV ar(β)R′]−1(Rβ−r)/q

e′V e/(N−K), where V = Σ ⊗ I; Σ is the FGLS estimate of the covariance

matrix. N is the number of observations of each equation times the number of equations and K standsfor the number of parameters estimated in the system. An alternative test statistic (Wald test), under the

23

Table 1.6: Risk Factors Regressions, Fama and French Factors with Exchange rateMonthly data from March 1995 to October 2003. Excess returns in U.S. dollars are regressedon the excess return of the “S&P” the standard’s and Poor Index, the small minus bigfactor “SMB”, and high minus low factor “HML” of Fama and French and the exchangerate Exch. HAC consistent t-ratios are on the second line below the coefficients. R2 is thecoefficient of determination, with the adjusted R2 on the second line. ρ is the first orderautocorrelation of the regression residual, with its t-value in the second column. F is theF -statistic of testing the hypothesis of zero coefficients in each regression.

Const Mkt SMB HML Exch R2 ρ FBeverage, Food 0.19 1.02 0.27 0.35 1.85 0.67 0.06 44.40and Tobacco 0.31 7.06 1.83 1.91 7.63 0.65 0.67Financial Services 1.04 1.21 0.03 0.25 3.15 0.56 0.01 28.63

0.93 4.58 0.11 0.75 7.11 0.54 0.13Building -0.44 1.08 0.64 0.60 2.38 0.66 0.10 43.29

-0.60 6.26 3.67 2.70 8.18 0.65 1.13Conglomerates -0.41 1.08 0.63 0.38 2.40 0.64 0.09 39.20

-0.51 5.74 3.34 1.58 7.55 0.62 1.04Media & Telecoms 0.61 1.39 0.02 0.19 1.44 0.65 0.04 40.84

0.85 8.22 0.11 0.88 5.09 0.63 0.43Chemical & Metal -1.95 1.01 0.81 0.68 2.24 0.47 0.17 19.52

-1.87 4.10 3.28 2.17 5.42 0.44 1.88Industrial -0.62 0.64 0.34 0.57 1.67 0.39 0.11 13.93

-0.76 3.34 1.77 2.32 5.17 0.36 1.31Machinery & Equipment -2.66 1.44 0.61 0.73 0.56 0.21 0.22 5.95

-1.74 3.99 1.67 1.59 0.92 0.18 1.90Retailing 0.31 1.67 0.35 0.74 2.73 0.58 0.07 31.11

0.28 6.35 1.34 2.20 6.18 0.56 0.75Transportation -0.38 0.52 0.57 0.54 1.73 0.22 0.12 6.36

-0.31 1.82 1.99 1.48 3.59 0.19 1.46

24

portfolios. Table 1.7 presents results of the different specifications (local factor, Fama

and French and Fama and French with Exchange rate). In the local-factor model, and

as observed in the time-series regressions (Table 1.4), Diff is statistically insignificant.

These results are consistent with the results of Bailey and Chung (1995). Using a sample

that spans from 1988-1994 Bailey and Chung (1995) report that sovereign default risk

is not significant for explaining portfolio returns in Mexico. For both Fama and French

models, all factors appear jointly significant. Panel B of Table 1.7 tests the hypothesis of

zero intercept (omitted risk factors). Interesting and surprising, the test of zero intercept

is not rejected for any of the three specifications.

Cross-Section of Expected Returns

The performance of the different unconditional models (i.e. local-factor vs. Fama and

French) is measured using cross-section regressions of the form

rt+1,i = β′iλt+1 + αt+1,i i = 1, ..., N,

where λt+1 is the vector of risk prices, and αt+1,i is the pricing error. The βi are the

“first pass” time-series betas using information up to time t. Risk premia λ = E(ft+1), is

estimated using sample means.

Table 1.8 summarizes the results of cross-sectional regressions for the unscaled model.

Time-series averages of the cross-sectional regressions coefficients λt+1, Fama-MacBeth

t-statistics, and time-series average of R2s are presented. Betas are estimated using ex-

panding samples and 36-month rolling windows. Results of the domestic CAPM are also

presented.

hypothesis that e′V e/(N −K) converges to one, that measures the distance between Rβ and r is given by

25

Table 1.7: Unconditional Pricing TestsPanel A presents the results from testing the joint hypothesis of zerocoefficients on all portfolio for the different factors. The first two columnspresents results for the local-factor models, the next two for the Famaand French and the last two for the Fama and French that includes theexchange rate. p-values for the F -tests and Wald tests are presentedbelow the value of the test statistic. Panel B presents the results fromtesting the joint hypothesis of zero coefficients for the omitted factors ineach model. The Fama and French factors for the local-factor model, thelocal-factors in the Fama and French model and IPC and UMS in theFama and French with exchange. p-values are presented in the secondline.

Panel A: Tests on significance of risk factorsLocal Factors Fama and French Fama and French

and ExchangeFactor F -test Wald test F -test Wald test F -test Wald testIPC 244.33 2551.93

0.00 0.00Exch 2.50 26.09 9.64 101.80

0.01 0.00 0.00 0.00Diff 1.39 14.52

0.18 0.15Mkt 11.69 122.08 9.42 99.51

0.00 0.00 0.00 0.00SMB 2.49 26.04 2.94 31.02

0.01 0.00 0.00 0.00HML 1.44 15.00 1.65 17.47

0.16 0.13 0.09 0.06Panel B: Tests on omitted risk factorsConst 0.75 7.85 1.20 12.49 0.86 9.13

0.68 0.64 0.29 0.25 0.57 0.52

26

Table 1.8: Cross-Section Regressions: Unconditional ModelResults for average λ estimates from monthly cross-sectional regressions for industrialportfolios: Rt+1,i = β′λ. The betas come from time-series regressions using informa-tion up to time t of industrial portfolios excess returns on the factors excess returns.Individual λi estimates for the beta of the factor listed are presented. “IPC” is theexcess return in U.S. dollars of the MSM Index over the 30 day T-Bill, Mkt is the ex-cess return of U.S. market over the 30 day T-Bill, “Exch” is the US. dollar/Mexicanpeso exchange rate growth, “Diff” is the spread betweem UMS bond and a T-Noteof 5 years, “SMB” and “HML” are the Fama-French mimicking portfolios relatedto size and book-to-market equity ratios. The table reports cross-sectional regressionusing expanding sample (es) and rolling windows of 36-month (rw) coefficients.Fama-MacBeth t-statistics are presented below the coefficients in parenthesis.

Risk Factors R2

Model IPC Mkt Exch Diff SMB HMLCAPM λes 2.34 0.21

(1.29)λrw 1.76 0.19

(1.15)Local λes 2.97 0.31 -0.48 0.47

Factor (1.58) (0.42) (-0.62)λrw 1.36 -0.16 -1.13 0.50

(0.82) (-0.25) (-1.80)Fama and λes 0.50 -2.33 0.25 0.42

French (0.42) (-1.12) (0.12)λrw -0.77 -2.87 2.16 0.42

(-0.73) (-1.47) (0.95)Fama and λes -0.74 0.79 -3.95 1.88 0.54

French with (-0.40) (1.10) -(1.27) (0.76)Exchange λrw -1.12 0.26 -4.01 3.51 0.54

(-0.79) (0.37) (-1.74) (1.18)

27

The first four rows of Table 1.8 present results for the CAPM using the IPC as a

proxy of the local market return. The poor performance of the CAPM in explaining the

cross-section of returns is summarized by the low average R2s.

A significant improvement in the pricing model is observed by the inclusion of exchange

rate risk, Exch, and political risk Diff as additional factors. On average the local-factor

model explains around 50 percent of the cross-sectional variation in returns. Meanwhile,

the Fama and French factors explain on average 42 percent of the cross-sectional variation

in returns. Finally, the last rows correspond to the results of the augmented Fama and

French model.

Figure 1.1 summarizes the above results for the different factor models. In particular,

cross- section regressions of the form:

E(rt+1,i) = β′iλ i = 1, ..., N,

were computed, where E(rt+1,i) is the sample average of industrial returns in excess of

the risk free rate, βi are the betas of time-series regressions using the whole sample. If the

proposed model fits perfectly expected returns, all the points in the figure would lie along

the 45-degree line. The figure shows clearly that few do, and that both local factor models

(CAPM and local-factor) have small power in explaining returns in Mexico22. The results

show that the augmented Fama and French factor model does a better job in capturing

the pattern of average returns in Mexico than the local CAPM and the local factor model

specifications. The results suggest that risk exposures that have been proven to be priced

in the U.S. are also important in the MSM. In the context of linear pricing methodology,

qF . This test statistic has a limiting χ2(q) distribution.22Fama-French model performs better (in terms of R2s) than the local-factor model when betas are

estimated using the whole sample.

28

a linear pricing kernel with fixed coefficient that is approximated by the Fama and French

factors and exchange rate, does a better job in pricing the cross-section of returns in Mex-

ico than a specification that uses exclusively local risk factors.

Figure 1.1: Realized vs. Fitted returns unconditional model

−2 −1 0 1 2

−2

−1

0

1

2

CAPM

Fitted

Rea

lized

−2 −1 0 1 2

−2

−1

0

1

2

Local Factor

Fitted

Rea

lized

−2 −1 0 1 2

−2

−1

0

1

2

Fama and French

Fitted

Rea

lized

−2 −1 0 1 2

−2

−1

0

1

2

Augmented Fama and French

Fitted

Rea

lized

1.4.4 Conditional Factor Models

Time-Series Evidence of the Factor Model

Conditional asset pricing presumes the existence of a set of instruments Zt that track

variation in expected returns, E(rt+1|Zt) or E(mt+1|Zt).

29

Table 1.9: Conditional Beta Regressions; Local FactorsExcess returns on 10 industrial portfolios are regressed on lagged instruments, “IPC” Mexican stock marketindex multiplied by the instruments and a constant, “Exch” the exchange rate multiplied by the instrumentsand a constant and, “Diff” political risk multiplied by the instruments and a constant. R2 of this regression ispresented in the second column. R2 of the restricted model (constant betas), where excess returns are regressedonly on instruments and risk factors is presented in the first column. The p-value of an F -test that comparesthe two models is presented in the third column. The last three columns present similar results assuming afixed constant. In the fourth column the R2 when excess returns are regressed on a constant and the local riskfactors are multiplied by the instruments and the constant. The p-value of F -test that tests the significance oftime varying betas is presented in the last column.

Panel A: Time-varying constant Panel B: Fixed constant

R2 R2 R2 R2

Time-varying Constant F -test Time-varying Constant F -testBetas Betas (p-value) Betas Betas (p-value)

Beverage, Food & Tobacco 0.8809 0.8798 (0.3757) 0.8824 0.8667 (0.0131)Financial Services 0.7878 0.7762 (0.1283) 0.7948 0.7738 (0.0325)Building 0.7858 0.7956 (0.8636) 0.7778 0.7864 (0.8145)Conglomerates 0.7830 0.7967 (0.9672) 0.7848 0.7958 (0.9140)Media & Telecoms 0.9225 0.9185 (0.1406) 0.9202 0.9188 (0.3049)Chemical & Metal 0.4953 0.5051 (0.6262) 0.4809 0.5026 (0.8423)Industrial 0.3690 0.4136 (0.9876) 0.3722 0.4193 (0.9966)Machinery & Equipment 0.1963 0.1825 (0.3134) 0.2214 0.1962 (0.2145)Retailing 0.7189 0.7157 (0.3535) 0.7192 0.7122 (0.2579)Transportation 0.1917 0.2131 (0.6930) 0.2010 0.1955 (0.3839)

The fact that industry portfolios wander between growth and distress suggest that risk

exposures for industry portfolios are time-varying. Time-variation in conditional betas was

achieved by allowing betas to depend linearly on a set of instruments Zt with forecasting

power on future portfolio returns.

Tables 1.9 to 1.11 present results from testing the hypothesis of time varying betas for

the local-factor model and both versions of Fama and French model. These tests sum-

marize the power of the instruments Zt to track variation in risk exposures. I performed

F -tests for the hypothesis of time-varying betas. Under the null, the coefficients asso-

ciated with the scaled factors, (Zt ⊗ ft+1) in equation (1.9), are restricted to be jointly

equal to zero. Panel A of Tables 1.9 to 1.11 present results from testing the hypothesis of

time-varying betas when the constant is allowed to be time-varying. R2 of the unrestricted

and restricted models are presented in the first two columns, together with the p-values of

30

Table 1.10: Conditional Beta Regressions; Fama and FrenchExcess returns on 10 industrial portfolios are regressed on lagged instruments and Fama and French factorsmultiplied by the instrumental variables and a constant. R2 of this regression is presented in the second column.R2 of the restricted model (constant betas), where excess returns are regressed only on instruments and thefactors are presented in the first column. The p-value of an F -test that compares the two models is presentedin the third column. The rest of the columns presents similar results when the constant is assumed to be fixed.The fourth column presents the R2 when excess returns are regressed on a constant and the factors multipliedby the instruments and the constant. The restricted version of this model is the unconditional model. Thep-value that tests the hypothesis of constant betas is presented in the last column.


R2 R2 R2 R2



the F -tests that compares both models (restricted and unrestricted) in the third column.

The hypothesis of fixed betas, conditional on time-varying intercepts, is not rejected in the

local-factor model. In contrast, with Fama and French factors we obtain strong evidence

on time-varying betas. Panel B presents results on the hypothesis of time-varying betas

conditional on a fixed intercept. Again the results suggest that risk exposures in the case

of Fama Frenc factors are time-varying. Time-variation in risk exposures is consistent

with the results from Fama and French (1997). They observe that risk exposures to the

value factor are time-varying using U.S. industry portfolios.

Table 1.12 extend the above results by testing the joint hypothesis of zero coefficients

associated with scaled factors. Results for the local factor model are consistent with those

obtained in Table 1.9. The joint hypothesis of fixed risk exposures for the local factor model

is not rejected. In contrast, for Fama French factors the hypothesis of zero coefficients

31

Table 1.11: Conditional Beta Regressions; Fama and French with ExchangeExcess returns on 10 industrial portfolios are regressed on lagged instruments, Fama and French factors multi-plied by the instrumental variables and a constant, and “Exch” the exchange rate multiplied by the instrumentsand the constant. R2 of this regression is presented in the second column. R2 of the restricted model (constantbetas), where excess returns are regressed only on instruments and the factors are presented in the first column.The p-value of an F -test that compares the two models is presented in the third column. The last three columnspresents the results when the constant is assumed to be fixed. In the fourth column, the R2 when excess returnsare regressed on a constant and the factors multiplied by the instruments and the constant is presented. Therestricted version of this model is the unconditional model. The p-value that tests the hypothesis of constantbetas is presented in the last column.


R2 R2 R2 R2



associated with scaled factors is rejected, suggesting that risk exposures for these factors

are time-varying. Notice that for the augmented Fama French model time-variation for

exchange rate risk exposures is rejected.

Cross-Section of Expected Returns

To evaluate the performance of the conditional model, cross-section regressions with time-

varying betas are estimated

Rt+1,i = β′t,iλt + αt+1,i i = 1, ..., N.

Table 1.13 summarizes results from the different cross-section regressions. Time-series

averages of coefficients along with their Fama-MacBeth t-ratios are shown. Betas are

32

Table 1.12: Tests for Time Varying BetasPanel A presents the results from testing the joint hypothesis of zero coefficients onall portfolio for the scaled factors ft ⊗ Zt. The first two columns present results forthe local-factor models, the following two for the Fama and French and the last twocolumns for the Fama and French that includes the exchange rate. p-values for the F -tests and Wald tests are presented below the value of the test statistic in parenthesis.Panel B presents the results from testing the joint hypothesis of constant alphas andzero alphas, p-values are presented in parenthesis.

Panel A: Tests on significance of scaled factorsLocal Factors Fama and French Fama and French

and ExchangeFactor F -test Wald test F -test Wald test F -test Wald testIPC 0.9185 11.0930

(0.5938) (0.9993)Exch 0.3177 3.8375 0.8773 11.1765

(0.9998) (0.9544) (0.6576) (0.3439)Diff 0.4594 5.5491

(0.9947) (0.8516)Mkt 1.7791 21.4879 1.7428 22.2024

(0.0064) (0.0179) (0.0083) (0.0141)SMB 3.2200 38.8906 2.4226 30.8634

(0.0000) (0.0000) (0.0000) (0.0006)HML 2.4939 30.1211 3.2043 40.8219

(0.0000) (0.0008) (0.0000) (0.0000)Panel B: Tests on alphasconstant 0.7184 8.6773 0.4063 4.9069 0.3581 4.5617alpha (0.8675) (0.5630) (0.9983) (0.8973) (0.9995) (0.9185)zero 0.7968 9.6234 1.0893 13.1569 0.8340 10.6250alpha (0.8134) (0.4741) (0.3264) (0.2150) (0.7589) (0.3875)

33

Table 1.13: Cross-Section Regressions: Conditional ModelResults for average λ estimates from monthly cross-sectional regressions for industrialportfolios: Rt+1,i = β′λ. The betas come from time-series regressions using informa-tion up to time t of industrial portfolios excess returns on the factors excess returns.Individual λi estimates for the beta of the factor listed are presented. “IPC” is theexcess return in U.S. dollars of the MSM Index over the 30 day T-Bill, Mkt is the ex-cess return of U.S. market over the 30 day T-Bill, “Exch” is the US. dollar/Mexicanpeso exchange rate growth, “Diff” is the spread betweem UMS bond and a T-Noteof 5 years, “SMB” and “HML” are the Fama-French mimicking portfolios relatedto size and book-to-market equity ratios. The table reports cross-sectional regressionusing expanding sample (es) and rolling windows of 36 months (rw) coefficients.Fama-MacBeth t-statistics are presented below the coefficients in parenthesis.

Risk Factors R2

Model IPC Mkt Exch Diff SMB HMLCAPM λes 1.48 0.20

(0.86)λrw 0.22 0.20

(0.16)Local λes 2.60 1.08 -0.16 0.51

Factor (1.24) (1.40) -(0.23)λrw 1.12 0.66 -0.78 0.47

(0.76) (1.60) -(1.81)Fama and λes -1.38 -0.53 1.00 0.46

French (-1.06) (-0.36) (0.84)λrw 0.11 -1.36 0.44 0.51

(0.10) -(1.15) (0.38)Fama and λes -1.74 -0.07 -1.20 2.09 0.55

French with (-1.47) (-0.11) (-0.76) (1.60)Exchange λrw -0.60 0.04 0.36 -0.81 0.59

(-0.54) (0.09) (0.33) (-0.72)

estimated using an expanding sample and a rolling window of 36-month prior estimation.

Average R2s are presented. Results show again that the augmented Fama-French model

performs the best in pricing the cross-section of returns in Mexico.

No significant differences are observed between the conditional and unconditional ver-

sion for the local factor model, i.e., average R2 are very similar for both representations.

However, for Fama-French factors we observe a significant improvement in the performance

of the conditional model relative to its unconditional version.

34

1.4.5 Pricing Errors

The theoretical content of the factor model relies on whether the alphas or pricing errors

are jointly equal to zero. Table 1.14 presents portfolios’ pricing errors. Figures 1.1 and 1.2

provide a visual representation of the relative empirical performance of the unconditional

and conditional versions of each model. Two measures of the performance are shown in

the last two rows of Panel A and Panel B of Table 1.14. Average, is the average norm of

the pricing error vector, and χ2 is the result of an asymptotic Wald test with the null of

zero pricing errors. Under this metric the Fama and French model performs the best for

the unconditional version. However, when using conditional information the augmented

Fama and French model has the lowest average pricing error.

Figure 1.2: Realized vs. Fitted returns conditional model.

−2 −1 0 1 2

−2

−1

0

1

2

CAPM

Fitted

Rea

lized

−2 −1 0 1 2

−2

−1

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Local Factor

Fitted

Rea

lized

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Fitted

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lized

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−2

−1

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2

Augmented Fama and French

Fitted

Rea

lized

35

Table 1.14: Pricing ErrorsMonthly pricing errors for the cross sectional regressions are reported. In each column, the average price for theunscaled and scaled versions for different models are compared: CAPM, Local Factors, Fama and French andFama and French with Exchange are reported. The last two rows reports the square root of the average squaredpricing errors across all portfolios and a χ2 statistic for the test that the pricing errors are zero.

CAPM Local Factors Fama and French Fama and Frenchwith Exch

Scaled Scaled Scaled ScaledPanel A. Expanding Sample

Beverage, Food & Tobacco 0.03 0.07 0.15 0.57 0.04 0.28 0.54 0.04Financial Services 0.77 1.51 0.41 0.67 0.38 0.60 -0.15 0.48Building 0.11 0.18 0.26 -0.24 0.70 -0.43 0.55 -0.57Conglomerates -0.79 -0.58 -0.51 -0.26 -0.08 0.31 -0.03 0.07Media & Telecoms 0.74 0.78 0.46 1.27 0.25 0.63 0.56 0.93Chemical & Metal -1.43 -1.26 -0.95 -1.01 -0.61 -0.91 -0.67 -0.80Industrial 0.05 -0.12 0.34 -0.63 0.24 -0.05 0.58 -0.08Machinery & Equipment -0.86 -1.49 -1.47 -0.51 -0.89 -0.51 -0.80 -0.33Retailing 0.14 0.26 0.10 -0.44 -0.25 0.09 -0.33 0.22Transportation 1.25 0.64 1.21 0.59 0.22 -0.01 -0.26 0.04Average 0.62 0.69 0.59 0.62 0.37 0.38 0.45 0.36χ2 3.31 5.75 2.91 6.22 2.99 6.08 3.57 9.03

Panel B. Rolling WindowsBeverage, Food & Tobacco 0.08 0.01 -0.06 0.33 -0.04 0.51 0.16 -0.10Financial Services 1.31 1.95 1.27 1.31 0.88 0.66 0.82 0.59Building 0.08 0.09 0.04 -0.90 0.62 -0.39 0.35 -0.02Conglomerates -0.87 -0.60 -0.17 -0.29 -0.22 -0.63 0.17 -0.40Media & Telecoms 0.84 0.88 0.61 1.19 0.85 0.35 0.68 0.80Chemical & Metal -1.43 -1.34 -0.99 -0.91 -0.53 -0.08 -0.24 -0.60Industrial 0.09 0.11 1.02 -0.07 -0.16 0.36 0.19 0.08Machinery & Equipment -1.04 -1.55 -1.18 -1.12 -0.18 -0.69 -0.89 0.22Retailing 0.11 0.25 -0.52 0.07 -0.95 -0.08 -0.66 -0.37Transportation 0.83 0.21 -0.03 0.40 -0.28 -0.01 -0.57 -0.20Average 0.67 0.70 0.59 0.66 0.47 0.37 0.47 0.34χ2 3.25 7.81 6.97 11.47 9.71 5.01 8.48 5.47

Results for the unconditional model do not reject the hypothesis of zero pricing errors

when using expanding sample betas. However, this result should be interpreted carefully.

Lettau and Ludvigson (2001), Burnside and Eichenbaum (1996) and Hansen, Heaton, and

Yaron (1996)) show that statistical tests that rely on the variance-covariance of pricing

errors have very poor small-sample properties.

Results from average pricing errors confirm that Fama and French factors do a better

job in explaining the cross-section of returns in Mexico than local factors. Notice that

based on pricing errors, the augmented Fama and French model only outperforms the

36

Fama and French specification in its conditional version. Consistent with results above,

the local factor model pricing errors are smaller for the unscaled version than when betas

are allowed to be time-varying. In contrast, allowing for time-variation in risk exposures to

Fama and French factors result in smaller pricing errors in general. The only exception is

for the case of expanding sample risk exposures where the Fama and French model shows

no improvement from scaled factors.

1.5 Conclusions

After the failure of the CAPM to explain the cross-section of expected returns sorted

by size and book-to-market in the U.S. stock market, researchers have sought alternative

models to explain the pattern of returns. The Fama and French (1993) three factor model,

despite the controversy of whether these factors truly capture nondiversifiable risk, has

proved to be successful in capturing the cross-section of returns in the U.S..

This chapter investigates which factors explain the cross-section of returns in a partic-

ular emerging market, Mexico. Much of the work in empirical asset pricing has focused on

developed markets, in particular, the U.S. stock market. Few studies have concentrated

in studying the cross-section in a developing market or, the degree of integration of an

emerging market taking into account the pattern of cross-section returns.

To test the factor model, two sets of factors were evaluated. The first set corresponded

to local factors. Under this specification, the underlying hypothesis was of segmentation

of the MSM to the North American market. The local factors were chosen based on

a previous study by Bailey and Chung (1995), and following much of the international

finance literature that has concentrated on explaining returns in developing countries. In

this literature, factors such as exchange rate risk, political risk, and local market risk

37

are used frequently to explain returns in national markets, and to evaluate the degree of

integration of national markets to the world market. Assuming segmentation of MSM, a

local-factor model that represents an extension of the CAPM that includes exchange rate

risk and political risk is evaluated. Meanwhile, supporting the hypothesis of integration

between the MSM and U.S. market, foreign factors that appeared to be successful in

explaining the cross-section of returns in the U.S. are evaluated in the cross-section of

returns in Mexico. Fama and French factors appeared as natural candidates to support

the hypothesis of integration.

Tests allow time-varying risk premia using conditional or scaled versions of the model.

In the conditional version, factors were scaled with instruments that incorporate in-

vestors’expectation of future expected returns. In particular, the instruments were the

lagged growth in labor income, the lagged growth in holdings of financial instruments and

the lagged term spread of Mexican government zero coupon bonds, Cetes.

The empirical evidence suggests that the Fama and French factors can explain a sub-

stantial fraction of the cross-sectional variation in average returns sorted by industry. The

hypothesis of time-varying risk exposures for the local-factor model was rejected but for

the Fama-French factor specifications the evidence was supportive of time-varying risk

exposures. Evidence for both unconditional and scaled factor models reveals that the

augmented Fama and French with exchange rate does a better job in explaining the cross-

section of returns than the local-factor model. These results seem to be supportive of the

hypothesis of integration of MSM to U.S. markets.

38

Chapter 2

Prices of Risk and Integration

2.1 Introduction

The purpose of this chapter is to study market integration between an emerging market

and a developed market. I study integration between the Mexican Stock Exchange (MSM)

and U.S. capital markets.

Financial markets have become steadily more open to foreign investors. During the

late 80’s and beginning of the 90’s, Emerging Markets (EMs) underwent a series of reforms

aimed to trigger foreign investment. These reforms lead to an impressive increase in capital

flows from developed markets making EMs more integrated to global markets. Intuitively,

market integration should be understood as a gradual process. The speed and degree of

integration is determined by the particular institutional and economic situations prevail-

ing in each country, for example, effectiveness of economic reforms, the country’s legal

system, and contract enforceability (which help mitigate moral hazard). A big challenge

falls in dating liberalization, and its direct effect on market integration. In general, market

integration is broadly defined as an increase in foreign investment activity (see Bekaert,

39

Harvey, and Lundblad (2003)), and has been measured using three different indicators:

1) the introduction of American deposit receipts (ADRs) and country funds; 2) lifting of

investment restrictions; and 3) capital flows into emerging equity markets. Bekaert and

Harvey (1995) recognizes three major categories of investment barriers to EMs; 1) legal

barriers, e.g., restrictions on foreign investment; 2) indirect barriers that arise from infor-

mation asymmetries, accounting standards; and 3) risks exclusive to EMs, e.g., liquidity

risk and political risk.

Figure 2.1: Growth in Portfolio Investment by Foreigners

Jan89 Jul91 Jan94 Jul96 Jan99 Jul01 Jan04 Jul06−20

−10

0

10

20

30

40

50

60

70

Beginning in 1989 Mexico experienced a transformation from a closed and protected

economy to one of the most open economies in Latin America (see Bekaert, Harvey, and

40

Lundblad, 2003). Figure 2.1 presents the growth in portfolio investment by foreigners from

the end of 1989 to the end of 2007. By 1995, after the Mexican Peso’s devaluation of almost

70%, control of the exchange rate was eliminated. At this time, domestic companies listed

in MSM sought to broaden their shareholder base by raising capital abroad. An increasing

number of firms started listing in foreign equity markets, in particular, in the U.S.1.

Foreign investors accounted for over 30 percent of Mexican holdings2 and up to 80 percent

of the trading of Mexican stocks since 1990. Figure 2.2 and Figure 2.3 present the ratio of

the value of holdings of Mexican stocks by foreign investors to domestic investors, and the

ratio of the value of volume traded in ADRs to their Mexican counterpart respectively.

These figures reveal that foreign investors holdings were mainly through Mexican ADRs.

The large role play by foreigners in Mexican stocks, specially through ADRs, provides

prima facie support for the hypothesis of integration.

Recent empirical studies on market integration have explored integration between mar-

kets by using a linear combination between the domestic and international versions of

the conditional CAPM (Bekaert and Harvey (1995, 2000), Carrieri, Errunza, and Hogan

(2007), and Alder and Qi (2003)). Under the CAPM, excess returns on a national index

are equal to the price of local risk times the variance of the local market when markets

are segmented; and equal to the price of global risk times the covariance of the local

market and global market when markets are integrated. These studies, in an attempt

to capture the more realistic case of mild segmentation, have modelled excess returns as

a time-varying linear combination of these two extreme cases. With this approach, the

time-varying weights represent a measure of the degree of integration. In general, results1A striking increase of firms have undertaken ADRs programs, passing from 8 firms in 1992 to 71 in

2001. Many of the ADRs are traded over the counter, but by 2001 there were 28 different series traded onmajor exchanges.

2Banco de Mexico, Development of Equity Markets, 2003.

41

Figure 2.2: Holdings of Mexican Stocks by Foreign and National Investors

Jan90 Jul92 Jan95 Jul97 Jan00 Jul02 Jan050

10

20

30

40

50

60

70

80Ratio of Value of Mexican Stocks Holdings Foreign/Domestic

on integration using different versions of the CAPM have been inconclusive. However,

there is widespread agreement that market integration is a time-varying process where the

degree of integration exhibits wide swings related to global and local events.

The large body of empirical research on multi-factor models relies on the fact that

market return covariances are not the only measure of systematic risk. Therefore, cor-

relations between proxies of a country market portfolio return and the world portfolio

market return are an incomplete measure of risk in an international framework. Further,

the use of national indices as a measure of market risk, and a measure of the cross-section

of national stock has been widely criticized. The way national indices are constructed in

terms of scope, coverage, and industrial composition has raised serious concerns about the

42

Figure 2.3: Value Traded as ADRs to Value Traded in Mexico

Jan90 Jul92 Jan95 Jul97 Jan00 Jul02 Jan050.5

1

1.5

2

2.5

3

3.5

4

4.5

5

5.5Ratio of Trading Value ADR/Mex

use of cross-country correlations to measure integration.

This chapter contributes to the growing literature on emerging markets by studying

regional integration between a developed market and an emerging market using a multi-

factor asset pricing model and the cross-section of returns from both countries. Most of

the empirical work in EMs concentrates in country level aggregate indices. Little research3

has focused on the effect of liberalization and its implications on the integration process

of financial markets using cross-sectional data. Specifically, I examine market integration

using industrial portfolio returns of all securities listed in the MSM and different U.S.

portfolio returns from 1990 to 2004.3None to my knowledge.

43

When two markets are integrated, one stochastic discount factor (SDF) prices assets

in both national markets. Given the empirical success of the three factor model of Fama

and French (1993), market risk, small minus big portfolios (SMB) and high minus low

portfolios (HML), to explain the cross-section of expected returns on different sorts of

portfolios in the U.S.4; and the results from Chapter 1 where I found strong covariation

between the cross-section of returns in the MSM and U.S. Fama and French factors, the

Fama and French factors are assumed to be the mimicking portfolios of the underlying

risk factors within an international asset pricing model.

From a theoretical perspective, financial markets are said to be integrated when similar

assets have the same price, regardless of their nationality and trading location. On the

contrary, if asset prices differ depending on their trading location, markets are said to

be segmented. In the context of asset-pricing models, markets are integrated when risk

factors and their prices are the same between markets. Hence, differences in prices of risk

are a result of investment barriers that could take the form of differences in transaction

costs, liquidity constraints, and capital flows restrictions, among others.

Even if the Mexican market is integrated to the world capital market, theoretical

and empirical evidence suggests that exchange rate risk is priced and should be included

as a source of systematic risk. Whenever a domestic investor holds a foreign asset, her

return in domestic currency depends on the exchange rate and therefore bears exchange

rate risk. Ferson and Harvey (1993), Ferson and Harvey (1994), Brown and Toshiyuki

(1993), Bekaert and Harvey (1995), Dumas and Solnik (1995), DeSantis and Gerard (1998),

Karolyi and Stulz (2003) and references therein, find that the price of currency risk, from

the U.S. perspective, is significantly different from zero. Hence, I include exchange rate

as an additional source of risk.4See Fama and French (1993, 1997)

44

However, liberalization is a necessary, but by no means sufficient condition for a market

to become integrated to the world markets. The structural change from local to global

pricing takes place only if, after liberalization, foreign investors begin to hold stocks of the

liberalizing country; this is known as Merton’s (1987) recognition hypothesis. Therefore,

we can argue that integration in general is influenced by global as much as by local events

and does not necessarily start immediately after local markets are liberalized.

Extending Chapter 1, in this chapter I evaluate integration by comparing the prices

of risk between Mexico and the U.S. using a linear factor model. Under the hypothisis of

integration, the prices of risk between Mexico and the U.S. should be the same. Hence,

not only should the same risk factors explain the cross-section of returns in both markets,

but the rewards of holding those risks should be equal in both, Mexico and the U.S..

I explore the ability of Fama and French factors to price different sorts of U.S. port-

folios: Fama and French, Industrial and Small Caps; and Mexican Industrial portfolios

assuming that the markets are segmented and integrated. The restricted model that in-

corporates the hypothesis of integration assumes the same prices of risk in both markets.

Empirical results suggest that the degree of integration is a time-varying process. In

particular, when using pricing errors as a metric to assess the integration of both markets,

we observe that market integration (or segmentation) is related to global and local events.

This result provides evidence that investors’ risk perceptions of the future investment

opportunity set, which need not reflect only domestic regime changes, affect the degree of

integration of Mexico to the U.S. market. In particular, the Mexican market shows strong

signs of disassociation, or segmentation, linked to international events (Russian crisis of

1998 and Argentinean crisis of 2001) that lead international investors to liquidate their

positions in EMs, drying out liquidity in Mexico.

The chapter is organized as follows. In Section 2, I describe the relation between the

45

SDF representation and Beta representation and describe the methodology used to test

integration and the inclusion of exchange rate risk. In section 3, I describe the data used

to test integration and discuss empirical results.

2.2 Methodology

The objective in this paper is to measure the degree of integration between the MSM and

U.S. markets and not Mexico’s integration to the world market. In particular, the preva-

lence of regional events in the early part of the sample period, including the liberalization

of the equity market, signing of the NAFTA, change in the exchange rate regime from a

fixed parity to a floating parity, supports the hypothesis of regional market integration

between Mexico and the U.S..

When markets are integrated, the same pricing kernel or stochastic discount factor

(SDF) should price assets in both countries. In the context of linear factor models, the

SDF is defined as an affine transformation of a set of risk factors. The criteria for choosing

the relevant risk factors is that the true SDF lies in the subspace spanned by the risk

factors. The empirical success of Fama and French three-factor model over different sample

periods, and with different portfolio sorts in the U.S., suggests that the U.S. pricing kernel

lies in or close to the subspace span by the Fama and French factors: market, HML, SMB.

Therefore, in order to evaluate the integration between MSM and U.S. markets, I use the

parsimonious factor representation of Fama and French to capture style exposures in an

international context.

If we define the regional Fama and French factors as the value weighted Fama and

French factors in both countries, the U.S. Fama and French factors act as reasonable

proxies of regional factors given the size differences between both markets.

46

2.2.1 Factor Pricing

Asset-pricing models specify what the expected return on a financial asset should be in

terms of observable variables and model parameters at each point in time. Most of the

models start by studying the first order conditions (FOC) for the optimal consumption,

investment and portfolio choice problem faced by a model investor. From the FOC the

SDF representation is obtained.

The degree of market integration between markets is related to both the kind of risk

faced by investors and the level of reward for given levels of risk. Therefore, I proceed from

an asset-pricing model to characterize these kinds of relations and ultimately to measure

the degree of integration between Mexico and U.S. equity markets.

In an international asset-pricing context, the fundamental pricing relation for country

j = 1, ..., J is given by,

Et(mjt+1r

ji,t+1) = 0, i = 1, ..., Nj , (2.1)

where mjt+1 is the stochastic discount factor for country j, rj

i,t+1 are excess returns for

asset i = 1, ..., Nj in country j, and Et is the conditional expectation given information

up to time t. The price assigned by the model to a financial asset is equal to the model

specific discounted future payoff.

Empirical work on asset-pricing models relies on the hypothesis of rational expecta-

tions. Errors made by investors, namely the difference between observed and expected

returns are uncorrelated with the information set on which expectations are formed. In an

informationally efficient market, where the econometrician has less information than the

model investor, it should not be possible to explain differences between market prices and

the price assigned by the model based on the information available to the econometrician.

47

Asset-pricing models identify the SDF, that together with (2.1) provide the economet-

ric specification to evaluate a particular model. Empirical tests on asset-pricing models

assumes a particular structure on mt+1. Multiple factor models follow when the SDF

can be written as a function of a set of risk factors. In particular, linear factor models

arise when the SDF is specified as an affine transformation of a set of risk factors f jt+1,

observable or latent. In this case, the country specific discount factor, mjt+1, is given by

mjt+1 = 1− bjf j

t+1, j = 1, ..., J (2.2)

With separated markets, investors face only country-specific risks, f jt+1, and only do-

mestic risks are priced in the asset-pricing model all of which are summarized by mjt+1 for

j = 1, ..., J .

If, on the contrary, Mexico and U.S. markets are fully integrated, the opportunity set

available to investors in both countries includes the stocks traded in both markets. In this

case one SDF, mt+1 = 1 − b′ft+1, would price assets in both countries. Therefore, with

integrated markets, the asset-pricing relation (2.1) can be written as

Et(mt+1rji,t+1) = 0, j = 1, ..., J i = 1, ..., Nj (2.3)

In an international asset-pricing framework with linear factor models, the hypothesis

of integration (equality in the SDF) implies not only that the same risk factors should span

the set of returns in both countries, but the parameters that define the SDF, b, should

be the same for both countries. In the language of expected returns, when markets are

integrated the prices of risk are the same irrespective of the nationality of the assets and

the place of trading. Asset return differences among national assets are driven exclusively

by differences in risk exposures to the relevant risk factors.

48

In the empirical finance literature, two estimation methods are widely used. The first

is the classical beta method which relies on the beta representation of the factor model,

where expected returns are decomposed into a non-diversifiable risk component that is

equal to the sum of the product of risk exposures times prices of risk and an idiosyncratic

or residual component. This method relies on the covariance decomposition of the SDF

representation. The second method, involves estimating the asset-pricing model using its

SDF representation by the generalized method of moments (GMM).

2.2.2 Beta Representation

The most popular method for estimating linear factor models is the beta representation.

Its popularity comes from the appealing decomposition of risk into risk reward and risk

exposure. The classic two-step cross-sectional regression (CSR) approach proposed by

Fama and MacBeth (1973) is widely used for estimation of the beta representation.

The beta representation can easily be obtain using the covariance decomposition for

the SDF representation, (2.1). Expanding the left hand side of (2.1), E(mjt+1, r

ji,t+1) =

cov(mjt+1, r

ji,t+1) + E(mj

t+1)E(rji,t+1). This, together with the definition of mj

t+1, (2.2),

yields

Et(rji,t+1) =

Covt(rji,t+1, f

jt+1)b

j

Et(mjt+1)

(2.4)

Equation (2.7) can be expressed in terms of betas, or risk exposures as

Et(rji,t+1) = βj

i

V ar(f jt+1)b

j

E(mjt+1)

= βji λ

j (2.5)

Therefore, when assuming a linear factor model, the beta representation, (2.5), yields

a data generating process for asset i = 1, ..., Nj excess return in country j = 1, ..., J of the

49

form

rji,t+1 = Et(r

ji,t+1) +

K∑

k=1

βji,k

(f j

k,t+1 − λjk

)+ εj

i,t+1, (2.6)

and

Et(rji,t+1) = αj

i +K∑

k=1

λjkβ

ji,k. (2.7)

Where f jk,t is the k risk factor for country j, βj

i =[βj

i,1, ..., βji,K

]are the risk exposures of

excess returns for asset i in country j to the set of risk factors f jk,t+1

5.

Betas are defined as: Σ−1F E

[(f j

t − E(f jt ))(Ri

t − E(Rit))

′], where the variance-covariance

matrix of factors is given by, ΣF = E [(ft − E(ft))(ft −E(ft))′]. The set of beta vectors

βji , i = 1, ..., N can be consistently estimated using time-series regressions of excess returns

on risk factors: rji,t+1 = aj

i + f ′t+1,jβji + εj

t+1,i.

When markets are integrated, the return generating process for asset i = 1, ..., N is

given by

rji,t+1 = Et(r

ji,t+1) +

K∑

k=1

βji,k

(f j

k,t+1 − λk

)+ εj

i,t+1, (2.8)

and

Et(rji,t+1) = αj

i +K∑

k=1

λkβji,k. (2.9)

In this case, the restriction that prices of risk λk among different markets are equal is

imposed.

2.2.3 SDF Representation

In order to tie the appealing empirical interpretation of the beta representation (2.7) with

the robustness of the SDF methodolgy, in this section I present the relation between both5Fama and French (1997) suggest that when tested portfolios transit between growth and distress, i.e.

industrial portfolios, it is important to allow time variation in factor loadings.

50

representations.

To have results that are robust to the presence of conditional heteroskedasticity and

time independence, I estimate (2.1) and (2.3) using the GMM method. Following Jagan-

nathan and Wang (2002), I expand the moment conditions implied by (2.1) and (2.3) and

include the first two moments of the vector of factor returns to estimate prices of risk, λ,

and Jensen’s measure of pricing errors, α.

The moment restrictions for the GMM under segmentation are obtained by combining

(2.1) with equation (2.2). In this case, the Mexican stochastic discount factor is given by

1− ft+1bmex and 1− ft+1b

us for the U.S.. The SDF representation yields

E[rmt+1(1− f ′t+1b

mex)] = 0Nmex×1, (2.10)

E[rust+1(1− f ′t+1b

us)] = 0Nus×1,

where the first set of moments are for Mexican returns, and the second set of moments

are for U.S. returns. Nmex and Nus are the number of portfolios in Mexico and the U.S.

respectively.

When markets are integrated, the same SDF price assets in both countries. In this

case, the moment restrictions are

E[rmt (1− f ′tb)] = 0Nm×1, (2.11)

E[rust (1− f ′tb)] = 0Nus×1

From the moment restrictions (2.10) and (2.11) we obtain the GMM estimate under

segmentation bmex and bus for Mexico and the U.S. respectively. In the case markets are

integrated, only one SDF prices assets in both countries therefore obtaining a common b

51

for both markets.

Jagannathan and Wang (2002) present the equivalence between the SDF method and

the classical beta model. In specific, they show that the SDF representation is as efficient

as the beta representation for estimating risk premiums, and specification tests based on

the SDF method are as powerful as those based on the beta model. To maintain the intu-

itive interpretation of the beta representation, I compute prices of risk and Jensen’s pricing

errors using the SDF. Further, the flexibility of the SDF representation allows to incor-

porate exchange rate risk as an additional risk factor affecting only Mexican portfolios.

Hence, under market integration prices of risk for Fama and French factors are determined

by the covariance between Mexican and U.S. portfolios with Fama and French factors, and

exchange rate risk is determined exclusively by the covariance between Mexican portfolio

returns and exchange rate risk.

The link between prices of risk and estimates from (2.10) and (2.11) are obtained from

the covariance decomposition (2.6) and (2.5). From these expressions we observe that the

vector of risk premiums for country j, λj , is given by

λj =V ar(f j

t+1)bj

E(mjt+1)

, (2.12)

together with the definition of the stochastic discount factor mt+1 in (2.2), yield

λj =V ar(f j

t+1)bj

1− µ′bj, (2.13)

where the expectation of the vector of risk factors is E(ft+1) = µ. Inspection of (2.13)

reveals that prices of risk, commonly used in the linear factor pricing literature, are a

one-to-one transformation of the parameters in the SDF representation (2.2). However,

to obtain prices of risk from the SDF representation we need to compute the estimates

52

for the mean µ and variance-covariance of the factors, V ar(ft+1). In addition, standard

errors of prices of risk λ depend on estimation errors of µ and V ar(ft+1).

As mentioned above,equality of SDFs between Mexico and U.S. (2.1) imply the same

vector of risk prices, λj , in both countries. Therefore, market integration is related to

both, the type of risks faced by investors, and the rewards for the given levels of risk.

Exchange Rate Risk

Under the hypothesis that purchasing power parity is violated, international investors are

exposed to exchange risk exposure, Solnik (1983). Dumas and Solnik (1995), DeSantis

and Gerard (1997, 1998) among other authors include exchange rate risk as an additional

risk factor in an international setting.

In order to test for integration of Mexican and U.S. market I take the perspective

of an U.S. investor who is concerned with obtaining returns in U.S. dollars. Therefore,

excess returns for Mexican and U.S. portfolios in (2.10) and (2.11) are measured in U.S.

dollars. Following the international asset pricing literature, I include exchange rate risk

as an additional risk factor for Mexican portfolios6. Hence, expected excess returns for

Mexican portfolios use regional factors, fk,t, and exchange rate risk, fex, as the relevant

or priced risk factors. The return generating process for Mexican returns that includes

exchange rate risk is given by

rmi,t+1 = E(rm

i,t+1|Ωt) +K∑

k=1

βmi,k (fk,t+1 − λk) + βi,ex (fex,t+1 − λex) + εj

i,t+1, (2.14)

6See section 3.2 where time-series regression of Mexican and U.S. portfolios are regressed on Famaand French factors and Mexican peso-U.S. dollar exchange rate. Risk exposures to exchange rate risk fordifferent sorts of U.S. portfolios are zero.

53

and,

E(rmi,t+1|Ωt) = αm

i +K∑

k=1

λk(Ωt)βmi,k + λex(Ωt)βi,ex. (2.15)

where λe is the reward from bearing exchange rate risk ft+1,ex, and βex,t represents an

additional source of risk induced by PPP deviations.

Next, consider the SDF representation. Market integration is characterized by the

same SDF for all portfolio returns. In the context of a linear SDF, the moment condition

in (2.1) is given by (2.3).

In order to characterize PPP deviations, exchange rate risk is included as an additional

risk factor for Mexican portfolio returns. Therefore, the SDF for Mexican portfolios is

given by: mmt+1 = 1 − ft+1b − ft+1,exbex; and for U.S. portfolios only Fama and French

factors are priced therefore the SDF is given by: mut+1 = 1− ft+1b.

When exchange risk is included as a risk factor that is priced only for Mexican port-

folios, integration is characterized by having the same b associated with Fama and French

factors for both, Mexican and U.S. portfolios. However, from (2.13) we observe that

prices of risk for Fama and French factors are a function of exchange rate risk through

the estimate of the factor variance-covariance matrix. This dependence violates the as-

sumption that U.S. portfolios are independent of the Mexican peso/U.S. dollar exchange

rate (that have zero Mexican peso exchange rate risk). Hence, in order to keep the rela-

tion between prices of risk for both countries and the SDF representation that includes

exchange rate risk only for Mexican portfolios, two conditions need to be satisfied. First,

the price of the risk free rate in U.S. dollars needs to be the same in both countries, that

is E(mmt+1) = E(mu

t+1), secondly, exchange rate risk has to be orthogonal to the set of

common risk factors, in this case Fama and French factors, cov(fk,t+1, fex,t+1) = 0 for

k = 1, ..., K.

54

Therefore, defining exchange rate between Mexican peso and U.S. dollar as:

fex = fex + uex, (2.16)

where fex is the projection of fex on the sub-space span by Fama and French factors and

uex is the component of fex orthogonal to Fama and French factors. With this definition

of fex, the modified Mexican SDF is given by: mmt+1 = 1− ft+1b− ut+1,exbex.

The exchange rate risk factor for Mexican portfolios is constructed to be orthogonal

to the rest of the risk factors and with zero mean. Therefore, equation (2.17) including

exchange rate risk is given by:

λ

λex

=

1

E(mjt+1)

V ar(ft+1) 01×1

01×K V ar(ft+1,ex)

b

bex

. (2.17)

2.2.4 Measuring Integration

Two tests for measuring regional integration between Mexico and the U.S. are performed

based on the asymptotic distribution of GMM estimates.

The first one is based on prices of risk between Mexico and U.S.. When markets

are segmented, investors face country-specific risks, f jt+1, and only domestic risks are

priced. However, in the case that the same risk factors are priced in Mexico and the U.S.,

differences in prices of risk can arise from capital controls or some other forces that restrict

investors’ possibilities to allocate their wealth freely across national markets. Therefore,

under the hypothesis of market integration the reward for bearing risks should be the

55

same.

The second test for market integrations is based on pricing errors. The difference

between observed expected returns and expected returns implied by the pricing model,

αji , in the beta representation (2.6) is the pricing error of asset i in country j. Since the

model under market integration is a “restricted” version of the model under segmentation,

we can use the GMM distance statistics to compare the value of the objective function

under the null of integration against the alternative of segmentation.

Prices of Risk

The beta representation of the linear factor model under integration (2.14) implies that

differences in expected realized returns are driven exclusively by risk exposures βji . There-

fore, a natural way of testing integration is by comparing the price of risk between the

unrestricted model (2.7) and the restricted model (2.14).

However, in order to estimate λk from (2.17) the moment restrictions for the GMM

need to incorporate estimates of the factor mean, µ and variance V ar(ft+1). Following

Jagannathan and Wang (2002), I expand the moment conditions implied by (2.1) and

included the definition of the mean and variance of the risk factors, necessary to obtain

the prices of risk. The expanded moment conditions are:

E(rjt+1(1− f ′t+1b

j)) = 0n×1 (2.18)

E(ft+1 − µ) = 0k×1 (2.19)

E((ft+1 − µ)(ft+1 − µ)′) = Σk×k (2.20)

56

The GMM estimates is the set of parameters θ = bj , µ, Σ. Hansen (1982) shows that

the asymptotic distribution of the GMM estimate, θ is

√T (θ − θ) → N [0, (D′S−1D)−1], (2.21)

where D = ∂g(θ,rt,ft)∂θ′ where g(θ, rt, ft) are the moment conditions (2.18)-(2.20) and S is

the variance-covariance matrix of the moments.

Under the hypothesis of integration between the MSM and U.S. markets, bj for Mexico

and U.S. is the same for both countries, b. The vector estimate of the prices of risk, λj , is

given by:

λj =Σbj

1− µ′bj, (2.22)

. The asymptotic standard error of (2.22) is obtained from the distribution of θ, (2.21),

applying the delta-method7, and is given by

Σλ = D(θ)′Avar(θ)D(θ) (2.23)

where D(θ) = [D(θ)′1, ..., D(θ)′K ]′ and D(θ)i = ∂λi∂θ .

We can obtain a test for the equality of prices of risk between Mexico and U.S. using the

asymptotic distribution of λj . Under the null of market integration, λmex = λus. Defining

λseg = [λmex, λus]′, the null for market integration can be expressed as Rλseg = 0K , where

R = [IK − IK ] and the associated Wald test for testing the hypothesis of integration is

given by

T (Rλseg)′(RΣλR′)−1(Rλseg) → χ2K (2.24)

7See appendix 1 for the derivation of the variance of λk

57

Pricing Errors

The most popular way to assess the pricing performance of an asset pricing model is

through pricing errors. The theoretical content of factor models rely on whether the vec-

tor of pricing errors or alphas from (2.7) are jointly equal to zero. Specification tests

of asset pricing models are given by the difference between realized average returns and

the corresponding expected return according to the pricing model. Hence, I use pricing

errors to measure integration between Mexico and the U.S.. The GMM distance statistics

(that resembles the likelihood ratio test) provides a test statistics that enables to com-

pare the performance of the asset pricing model under both hypotheses: integration and

segmentation.

Defining JT = TgT (b)′S−1gT (b), where gT (b) is the vector of moment conditions (2.18)-

(2.20), from Hansen (1982) we know that JT → χ2N−K it follows that:

TJT (integration)− TJT (segmentation) → χ2K (2.25)

Therefore, integration can be tested by comparing pricing errors of U.S. and Mexican

markets under integration against the pricing errors under the hypothesis of segmentation.

However, the weakness of this approach is that results are sensitive to the particular choice

of the asset pricing model, in particular, assuming that the true pricing model is linear

and that the Fama and French factors mimic the underlying risk factors in both markets.

The vector of pricing errors in the SDF representations is given by the difference be-

tween observed excess returns and expected excess returns implied by the pricing model:

ωj = E(rjt+1) − E(rj

t+1, f′t+1)b

j , where the sample analog is obtained by replacing expec-

tations with the corresponding sample means, ω = rjt − rj

t f′t b.

The analog of the pricing errors vector in the beta representation is given by Jensen’s

58

alpha. In general ωj and αj are not equal. Solving for bj in equation (2.5) and substituting

into the expression for ωj , we obtain the relation between pricing errors in the beta and

SDF representations:

ωj = αjE(mjt+1). (2.26)

As in the case of prices of risk, pricing errors in the SDF representation, ωj are related

to their beta representation counterpart, αj , by a one-to-one transformation.

2.3 Data

The sample covers the period that spans from 1990 to 2004. It is composed of weekly

and monthly closing prices from the first week of January 1991 to the end of December

of 2004, totalling 783 observations for weekly data. The data set comprises all securi-

ties in the MSM, U.S. industrial portfolios, Fama and French risk factors: Market that

is the CRSP equally weighted index that includes NYSE, AMEX and NASDAQ firms;

HML portfolio that is the difference between high book to market (value stocks) minus

low book to market (growth); and SMB portfolio, that is the difference between small mi-

nus big firms; the Mexican Peso/U.S. dollar exchange rate, and U.S. Treasuries. Mexican

data were obtained from Infosel Financiero8. All series were adjusted to account for splits

and dividends. U.S. data were obtained from the Center for Research in Security Prices

(CRSP) at the University of Chicago and Kenneth French web page9.

8Financial electronic information provider in Mexico.9http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/index.html

59

2.3.1 Portfolios Returns

The sample for Mexican securities includes all firms listed in the MSM. A total of 206

different series have been listed during the 1990 to 2004 period, of these, 163 are from

different firms. As of December of 2004, there were a total of 158 firms listed. The major

problem encountered when constructing the dataset for Mexican securities is the fact that

many companies changed their social name and/or series designations. An other problem

is the large proportion of firms delisted during the 1990-1995 period, especially after the

Peso crisis.

Returns were computed as the natural logarithm difference of the current period minus

the last period price. All returns for Mexican securities are expressed in U.S. dollars.

Excess returns are defined as the difference between the continuously compounded return

minus the equivalent weekly or monthly interest rate of the one-month term of US Treasury

bill.

In order to address potential problems such as survivorship bias, missing observations

and thin trading, individual stocks of the MSM were aggregated into 10 industrial port-

folios. The industrial categories are given by: 1) Beverages, Food Procucts and Tobacco,

2) Financial Services, 3) Building Products, 4) Conglomerates, 5) Media, entertainment

and telecommunications, 6) Chemical and Metal Production, 7) Industrial, an aggregate

of the paper and pulp products industry, the textiles industry, glass production and tubes

production, 8) Machinery and Equipment, 9) Retail Services and 10) Transportation. Two

types of Industrial portfolios are constructed. The first type uses previous year’s annual

liquidity weights, while the second consists of equally weighted portfolios. Re-balancing

takes place annually in January.

For U.S. portfolios, Fama and French industrial portfolios are formed based on its

60

four-digit SIC codes from stocks traded in the NYSE, AMEX and NASDAQ exchanges.

Portfolio re-balancing occurs in an annual basis.

2.3.2 Risk Factors

The Fama and French factors are market, HML and SMB. Market is the value-weighted

return on all NYSE, AMEX and NASDAQ stocks from CRSP in excess of the one-month

Treasury bill rate. HML (High minus Low) is the average return on the two value portfolios

minus the average return on the two growth portfolios. SMB (Small minus Big) is the

average return on the three small portfolios minus the average return on the three big

portfolios10. Exchange rate risk is constructed as the log difference in the exchange rate

between the Mexican peso and U.S. dollar. Weekly exchange rates for the sample were

obtained from the Mexican Central Bank. Finally, to construct excess returns, weekly

U.S. T-bill rates were obtained from the Federal Reserve Bank of St. Louis.

2.4 Empirical Evidence

2.4.1 Summary Statistics

Table 2.1 presents summary statistics for Mexican and U.S. portfolio returns using the

whole sample 1990-2004, and for the sub-samples that span from 1990-1995 and from

1996-2004. Means and standard deviations are annualized. We observe big differences in

Mexican portfolio mean returns between sub-samples. For example the Financial Services

industry presents a return of -10.42 percent during the 1990-1995 period, and of 12.72

percent from 1996-2004. These differences in average returns and standard deviations are

largely due to the Mexican pesos crisis of 1995. For example, average Industrial returns10See Fama and French, 1993, for a complete description on the construction of factor portfolios.

61

in U.S. dollars were of 16.20 percent from 1990-1994 and of -30.00 percent during 1995.

In contrast, U.S. portfolio average returns appear more stable between sub-samples.

However, there is a significant increase in volatility in Industry returns and Fama and

French portfolio returns during the second sub-sample.

62

2.4.2 Time-Series Regressions

Table 2.2 presents results from time-series regressions of Mexican and U.S. portfolio re-

turns on contemporaneous Fama and French and exchange rate risk factors. Although

integration was tested using the SDF representation via GMM, time-series regressions

provide a consistent measure of the magnitude of risk exposures of the tested portfolios

to the Fama French risk factors.

Mexican portfolios are more exposed to the size factor, SMB, then to the value factor,

HML. It is no surprise that the coefficients associated with SMB are positive and significant

for almost every Mexican portfolio. Mexican’s market capitalization is much smaller than

the U.S.. Meanwhile, the value factor is both significant and positive only for chemicals,

industrial, and transportation portfolios. In the case of the market and exchange rate

factor, all of Mexican portfolios have a positive and significant exposure. With exception of

the transportation sector (that represents less then 3 percent of the market capitalization),

time-series regressions R2s range between 30 and 50 percent.

63

Table 2.1: Summary StatisticsAnnualized means and annualized standard deviations are presented for excess returns of industrial portfolios inMexico and the U.S., the exchange rate risk return, and returns for Fama and French risk factors. All the statisticsare presented for the whole sample, and for the sub-samples that span from 1990-1995 and 1996-2004. Excessreturns are measured in U.S. dollars and calculated with respect to the 30 days U.S. T-bill. The column labelled ρpresents the first order sample autocorrelation of returns.

Industry 1990-2004 1990-1995 1996-2004Mean Std. Dev. Mean Std. Dev. Mean Std. Dev. ρ

Panel A. Mexican PortfoliosFood Products 4.94 36.81 5.10 43.59 4.84 31.80 -0.14Financial 3.54 46.39 -10.42 39.71 12.72 50.30 -0.03Building 3.44 46.54 10.01 57.66 -0.88 37.74 0.12Conglomerates -5.17 41.58 -11.29 43.13 -1.15 40.70 -0.04Media & Telecoms -0.64 43.75 -13.89 34.99 8.07 48.65 -0.08Chemical & Metal -5.19 43.71 2.05 40.00 -9.94 46.11 0.00Industrial 2.77 33.63 8.55 39.42 -1.03 29.37 0.12Machinery & Equipment 11.87 51.90 16.04 53.56 9.14 51.01 -0.06Retailing 18.22 36.87 18.61 41.88 17.96 33.38 -0.02Transportation 4.90 41.84 7.90 44.19 2.93 40.42 -0.05Panel B. U.S. PortfoliosFood 7.85 14.32 10.01 13.14 6.41 15.10 0.20Beer 10.50 19.03 18.81 15.71 4.99 20.86 -0.03Smoke 15.82 29.12 17.69 23.55 14.58 32.40 -0.04Games 9.54 26.24 11.81 18.83 8.03 30.25 0.35Books 5.79 15.89 5.20 13.60 6.18 17.31 -0.16Consumer Goods 9.76 15.78 12.18 13.89 8.16 16.96 -0.15Apparel 8.90 22.58 7.82 21.80 9.61 23.18 0.08Health 10.38 16.30 15.10 16.75 7.25 16.01 -0.17Chemicals 7.06 18.50 10.04 14.57 5.08 20.75 0.04Textiles 3.74 21.11 2.26 19.78 4.71 22.04 0.04Construction 8.15 17.45 7.95 15.53 8.29 18.69 -0.10Steel 5.91 24.52 6.06 17.07 5.81 28.48 -0.26Machinery 10.26 21.82 10.85 16.86 9.86 24.64 -0.07Electrical Equip 7.98 18.10 11.73 14.16 5.48 20.34 -0.04Auto and Trucks 6.71 23.54 7.20 19.74 6.38 25.84 -0.31Carry Equip. 10.94 21.09 13.45 15.53 9.28 24.14 0.16Mines 2.96 25.21 2.21 23.50 3.46 26.38 -0.11Coal 9.26 36.12 -10.74 21.45 22.52 42.87 -0.10Oil 8.44 16.27 4.63 12.72 10.96 18.27 -0.15Utilities 6.31 14.30 6.41 10.90 6.25 16.21 0.21Telecoms 4.28 18.74 8.54 12.31 1.46 22.01 -0.16Services 10.03 24.67 10.66 15.13 9.61 29.41 -0.15Business Equip. 13.47 31.84 17.20 20.20 11.00 37.72 0.12Paper 8.16 15.24 7.66 13.12 8.48 16.56 0.12Transportation 8.28 17.25 9.98 16.50 7.16 17.79 0.02Wholesale 5.49 15.59 8.39 12.31 3.57 17.46 0.03Retail 9.82 19.01 7.47 17.63 11.39 19.94 0.04Meals 8.77 17.95 12.39 17.82 6.37 18.08 -0.12Financial 12.85 17.99 14.86 16.35 11.51 19.06 -0.14Other 9.30 19.17 9.22 16.18 9.36 20.99 -0.14Panel C. Risk FactorsMarket 7.90 14.87 9.38 11.40 6.93 16.80 0.01SMB 2.67 13.25 0.67 8.22 3.99 15.73 -0.04HML 4.07 12.23 2.30 8.09 5.24 14.34 0.32Exchange Rate -9.65 13.77 -18.05 18.99 -4.12 8.54 0.19

64

For U.S. industrial portfolio returns, risk exposures to Fama and French factors are

significant for almost all industries. U.S. portfolios exposures to SMB and HML are more

heterogenous for U.S. returns than for Mexican returns. Finally, U.S. returns have no

exposure to Mexican exchange rate risk.

In general, we observe that Fama and French factor returns are significant in both

Mexican and U.S. portfolio returns and exchange rate risk is the only significant local

factor (Mexican) that will be included in testing integration. In order to incorporate

exchange rate risk as a Mexican local factor, two different stochastic discount factors are

defined. In both cases Fama French factors will be included, however in the case of Mexico

we will also include exchange rate. With the presence of a pure local factor, integration

results when the prices of risk for common factors, in this case the Fama and French

factors, are the same in both countries, and the price of risk for exchange rate risk is

exclusively determined by the cross-section of Mexican returns.

2.4.3 Prices of Risk

Table 2.3 to Table 2.6 present results from estimating equation (2.18) and equation (2.22)

using weekly and monthly data.

Weekly Data

Table 2.3 and Table 2.4 present results for weekly data using the sample that spans from

1990 to 2004 and the sub-sample that spans from 1996 to 2004. Panel A presents the

results from estimating the moment conditions using U.S. Industrial portfolios as the

testing portfolios for the U.S. and in Panel B U.S. Fama and French portfolios are tested.

Estimates under the hypothesis of integration are presented first, followed by the results

65

Table 2.2: Risk Factor RegressionsExcess returns in U.S. dollars of Mexican and U.S. industrial portfolios are regressed on the excess returns of themarket portfolio (Mkt), the small minus big portfolio (SMB), the high minus low portfolio (HML) of Famaand French, and the exchange rate return (Exch). HAC consistent t-stats are on the second line following thecoefficients. The columns labelled R2 present the R-square on the first row and the adjusted R2 on the second andρ present the sample autocorrelation with t-value on the second row.

Industry Const Mkt SMB HML Exch R2 ρPanel A. Mexican PortfoliosFood 0.63 1.04 0.36 0.22 1.31 0.50 0.15

1.05 6.67 2.15 1.06 9.06 0.48 2.60Financial 1.05 1.03 0.26 -0.01 1.86 0.50 0.14

1.39 5.30 1.23 -0.03 10.22 0.49 2.00Building 0.76 1.13 0.53 0.40 1.84 0.50 0.17

1.01 5.82 2.53 1.57 10.13 0.49 2.80Conglomerates -0.21 1.17 0.47 0.37 1.51 0.50 0.16

-0.31 6.69 2.50 1.58 9.32 0.49 2.55Media & -0.04 1.34 0.73 0.10 1.36 0.54 0.15Telecoms -0.06 7.63 3.85 0.44 8.28 0.53 2.63Chemicals -0.71 1.17 0.46 0.84 1.09 0.30 0.20

-0.84 5.38 1.97 2.92 5.39 0.28 3.63Industrial 0.65 0.78 0.11 0.35 1.34 0.45 0.15

1.13 5.28 0.68 1.81 9.69 0.44 2.21Machinery 1.53 1.20 0.69 0.02 1.86 0.49 0.16

1.79 5.45 2.91 0.05 9.03 0.48 2.89Retail 1.44 1.14 0.24 -0.27 0.79 0.43 0.13

2.26 6.93 1.37 -1.26 5.15 0.42 2.66Panel B. U.S. PortfoliosFood 0.13 0.61 -0.22 0.32 -0.07 0.34 0.00

0.47 8.87 -2.94 3.56 -1.07 0.32 0.04Health 0.53 0.65 -0.33 -0.24 -0.05 0.45 0.00

1.93 9.05 -4.34 -2.48 -0.75 0.44 0.06Chemicals -0.31 1.10 0.00 0.64 0.04 0.59 0.09

-1.15 15.72 0.04 6.91 0.67 0.58 1.59Textiles -0.83 1.07 0.58 0.95 0.00 0.51 0.28

-2.45 12.22 6.23 8.20 0.05 0.50 5.95Construction -0.29 1.13 0.09 0.58 -0.02 0.70 0.12

-1.31 20.11 1.41 7.81 -0.32 0.70 2.03Steel -0.78 1.44 0.44 0.82 0.05 0.63 0.18

-2.28 16.28 4.67 7.01 0.63 0.62 3.50Machinery -0.18 1.30 0.43 0.38 0.03 0.75 0.16

-0.74 20.19 6.25 4.47 0.42 0.75 2.80Electrical Equip -0.29 1.10 0.18 0.41 -0.07 0.66 0.13

-1.21 17.49 2.62 4.91 -1.14 0.65 2.18Auto and Trucks -0.71 1.39 0.20 0.99 0.00 0.58 0.16

-2.03 15.36 2.04 8.29 -0.05 0.57 2.93Mines -0.67 0.85 0.45 0.74 0.00 0.22 0.28

-1.31 6.42 3.20 4.23 -0.02 0.20 5.84Oil 0.15 0.72 -0.09 0.48 0.07 0.35 0.09

0.50 9.24 -1.06 4.70 0.98 0.33 1.44Utilities -0.13 0.61 -0.05 0.70 -0.03 0.39 0.13

-0.50 9.22 -0.72 8.01 -0.42 0.38 1.83Telecoms -0.26 0.99 -0.24 -0.08 -0.05 0.63 -0.02

-0.98 14.61 -3.35 -0.95 -0.77 0.62 -0.38Services 0.37 1.09 0.04 -0.73 -0.01 0.82 0.06

1.55 17.82 0.63 -8.95 -0.13 0.82 1.14Transportation -0.15 1.06 0.04 0.53 0.03 0.64 0.11

-0.63 17.33 0.60 6.56 0.53 0.63 1.84Retail 0.17 1.00 -0.04 0.13 0.06 0.57 0.06

0.61 13.49 -0.49 1.27 0.82 0.56 1.08Financial Serv 0.12 1.23 -0.18 0.59 0.02 0.81 0.05

0.65 26.22 -3.65 9.43 0.35 0.80 0.80Other 0.15 1.05 -0.32 0.01 0.01 0.64 -0.02

0.58 15.45 -4.33 0.13 0.11 0.63 -0.33

66

Table 2.3: Prices of Risk: 1990-2004 Weekly DataThe table presents the estimation results of the pricing model for Mexican industrial portfolios and U.S. portfolios.

E(rjt+1(1− ft+1bj)) = 0 and λj = Σbj/(1− µ′bj)

The columns labelled bj present estimates, t-stat and p-value for bj for the different factors under integration andsegmentation. Columns labelled λj present the same results for prices of risk. The model is estimated using weeklyexcess returns of Mexican and U.S. portfolios from January 1990 to December 2004. Results with U.S. IndustrialPortfolios and Fama and French Portfolios are presented in Panel A and Panel B respectively. The stochasticdiscount factor is linear in the Fama and French factors and the exchange rate between the U.S. dollar and MexicanPeso. In the case of integration, the stochastic discount factor is restricted to be the same for Mexico and the U.S..In contrast, under segmentation the model is estimated using a stochastic discount factor for each country.

Panel A. Industrial Portfolios1. Integration

bj λj

Mkt SMB HML Exch Mkt SMB HML ExchEstimate 0.04 -0.10 -0.06 -0.02 0.27 -0.15 -0.12 -0.08

t-ratio 1.71 -2.18 -1.22 -0.87 2.99 -1.98 -1.72 -0.81p-value 0.09 0.03 0.22 0.39 0.00 0.05 0.09 0.42

2. Segmentation

bj λj

Mexico Mkt SMB HML Exch Mkt SMB HML ExchEstimate -0.16 0.14 -0.38 0.03 -0.12 0.44 -0.54 0.09

t-ratio -0.85 0.47 -1.18 0.38 -0.28 0.77 -1.41 0.35p-value 0.40 0.64 0.24 0.70 0.78 0.44 0.16 0.72

U.S. Mkt SMB HML Exch Mkt SMB HML ExchEstimate 0.04 -0.11 -0.04 0.22 -0.16 -0.07

t-ratio 1.54 -2.28 -0.79 2.44 -2.25 -0.99p-value 0.12 0.02 0.43 0.01 0.02 0.32

Panel B. Fama and French Portfolios1. Integration

bj λj

Mkt SMB HML Exch Mkt SMB HML ExchEstimate 0.13 0.08 0.22 -0.05 0.22 0.03 0.13 -0.21

t-ratio 6.09 2.91 6.72 -1.93 2.65 0.50 2.67 -1.63p-value 0.00 0.00 0.00 0.05 0.01 0.62 0.01 0.10

2. Segmentation

bj λj

Mexico Mkt SMB HML Exch Mkt SMB HML ExchEstimate -0.16 0.14 -0.38 0.03 -0.12 0.44 -0.54 0.09

t-ratio -0.85 0.47 -1.18 0.38 -0.28 0.77 -1.41 0.35p-value 0.40 0.64 0.24 0.70 0.78 0.44 0.16 0.72

U.S. Mkt SMB HML Exch Mkt SMB HML ExchEstimate 0.12 0.08 0.20 0.23 0.03 0.15

t-ratio 5.42 2.74 6.08 2.75 0.59 3.01p-value 0.00 0.01 0.00 0.01 0.55 0.00

67

Table 2.4: Prices of Risk: 1996-2004 Weekly DataThe table presents the estimation results of the pricing model for Mexican industrial portfolios and U.S. portfolios.


The columns labelled bj present estimates, t-stat and p-value for bj for the different factors under integration andsegmentation. Columns labelled λj present the same results for prices of risk. The model is estimated using weeklyexcess returns of Mexican and U.S. portfolios from January 1996 to December 2004. Results with U.S. IndustrialPortfolios and Fama and French Portfolios are presented in Panel A and Panel B respectively. The stochasticdiscount factor is linear in the Fama and French factors and the exchange rate between the U.S. dollar and MexicanPeso. In the case of integration, the stochastic discount factor is restricted to be the same for Mexico and the U.S..In contrast, under segmentation the model is estimated using a stochastic discount factor for each country.


bj λj

Mkt SMB HML Exch Mkt SMB HML ExchEstimate 0.03 -0.08 -0.04 -0.05 0.26 -0.17 -0.10 -0.06

t-ratio 1.16 -1.57 -0.76 -0.72 2.00 -1.53 -1.06 -0.62p-value 0.25 0.12 0.45 0.47 0.05 0.12 0.29 0.53

2. Segmentation

bj λj

Mexico Mkt SMB HML Exch Mkt SMB HML ExchEstimate 0.16 0.22 -0.42 -0.09 0.06 0.86 -0.88 -0.09

t-ratio 0.83 0.78 -1.34 -0.38 0.10 1.14 -1.92 -0.31p-value 0.41 0.43 0.18 0.70 0.92 0.25 0.06 0.76

U.S. Mkt SMB HML Exch Mkt SMB HML ExchEstimate 0.03 -0.06 -0.03 0.22 -0.12 -0.08

t-ratio 1.00 -1.20 -0.50 1.68 -1.16 -0.79p-value 0.32 0.23 0.62 0.09 0.25 0.43


bj λj


t-ratio 5.02 2.90 5.82 -2.02 1.63 0.53 2.53 -1.59p-value 0.00 0.00 0.00 0.04 0.10 0.60 0.01 0.11

2. Segmentation

bj λj


t-ratio 0.83 0.78 -1.34 -0.38 0.10 1.14 -1.92 -0.31p-value 0.41 0.43 0.18 0.70 0.92 0.25 0.06 0.76


t-ratio 4.26 2.71 5.50 1.63 0.59 3.08p-value 0.00 0.01 0.00 0.10 0.55 0.00

68

when markets are assumed to be segmented. The first four columns labelled bj present

coefficients for the SDF and the columns labelled λj show the prices of risk.

Tables 2.3 and 2.4 show that when markets are assumed to be integrated the results

are highly driven by U.S. portfolios. For example, the market price of risk is of 0.27 under

integration, where under segmentation the market premium is of 0.22 for U.S. and of -0.12

for Mexican portfolios. When using weekly data, Fama and French risk factors appear

not statistically significant under segmentation for Mexican Portfolios in both samples

(1990-2004 and 1994-2004).

For U.S. portfolios, we observe that results are robust to sample selection. Estimates

using 1990 to 2004 returns are close to those when using the sub-sample from 1996 to

2004. When the tested portfolios are the 25 Fama and French portfolios, we observe

that Mkt, HML and SMB are all significant. However, and consistent with evidence

on the performance of Fama and French factors on Industrial Portfolios, see Fama and

French (1997), prices of risk are not significant for U.S. Industrial portfolios. Despite low

significance levels in prices of risk for Mexican portfolios, big differences differences in the

magnitude of the estimates between sub-samples are observed. This can be interpreted

as evidence of a possible structural change in the rewards of risk and/or risk factors in

Mexican market after the peso crisis and the adoption of a flexible exchange rate regime.

Figure 2.4 to 2.6 provide a visual representation of the empirical performance of the

pricing model under segmentation and integration. These figures show average return

against the expected return implied by the factor model, E(rjt+1) = E(rj

t+1ft+1bj). If the

proposed model perfectly fit expected returns, all points in the figure would lie along the

45-degree line. Realized returns are on the horizontal axis, and predicted returns on the

vertical axis. Circles represent Mexican portfolios and triangles U.S. portfolios. Figure

2.4 presents results for US Fama and French portfolios. In Figure 2.5 results for U.S.

69

Figure 2.4: Mexican Industrial and US Fama and French Portfolios. Weekly data. Realizedreturns are on the horizontal axis and predicted are on the vertical axis. Circles representMexican portfolios and triangles U.S. portfolios.

−2 −1 0 1 2−2

−1

0

1

2Integration. Sample 1990−2004

−2 −1 0 1 2−2

−1

0

1

2Segmentation. Sample 1990−2004

−2 −1 0 1 2−2

−1

0

1


−2 −1 0 1 2−2

−1

0

1


−2 −1 0 1 2−2

−1

0

1


−2 −1 0 1 2−2

−1

0

1


70

Figure 2.5: Mexican Industrial and US Industrial. Weekly data. Realized returns areon the horizontal axis and predicted are on the vertical axis. Circles represent Mexicanportfolios and triangles U.S. portfolios.

−2 −1 0 1 2−2

−1

0

1


−2 −1 0 1 2−2

−1

0

1


−2 −1 0 1 2−2

−1

0

1


−2 −1 0 1 2−2

−1

0

1


−2 −1 0 1 2−2

−1

0

1


−2 −1 0 1 2−2

−1

0

1


71

Figure 2.6: Mexican Industrial and US Small Caps. Weekly data. Realized returns areon the horizontal axis and predicted are on the vertical axis. Circles represent Mexicanportfolios and triangles U.S. portfolios.

−2 −1 0 1 2−2

−1

0

1


−2 −1 0 1 2−2

−1

0

1


−2 −1 0 1 2−2

−1

0

1


−2 −1 0 1 2−2

−1

0

1


−2 −1 0 1 2−2

−1

0

1


−2 −1 0 1 2−2

−1

0

1


72

Industrial portfolios are presented. These figures clearly illustrate the fact that Fama

and French factors do a much better job in explaining the cross-section of different sorts

of U.S. portfolios than for Mexican portfolios, especially in the sub-sample that spans

from 1990 to 1995. The power of Fama and French portfolios to explain the cross-section

of Mexican returns is much stronger after the peso crisis than in the post-liberalization

period. This could be interpreted as evidence that integration is a gradual process that

started after the liberalization of capital accounts, and that was interrupted after the

default on its Tesobonos in January of 1995. In contrast, U.S. portfolios present small

differences between the restricted and unrestricted model specifications. Consistent with

the results on the first chapter, Fama-French factors do a good job in explaining the cross-

section of returns for the sample that spans from 1996-2004. The unrestricted version of

the model performs better that restricted, suggesting that the rewards of risk are different

in both markets.

Monthly Data

Tables 2.5 to 2.6 present the same results than those in tables 2.3 to 2.4 with the difference

that monthly data are used instead of daily data.

In Panel A, results from estimating the moment conditions using U.S. Industrial portfolios

and, in Panel B, results for U.S. Fama and French portfolios are presented. Estimates for

the restricted model are shown first, followed by the estimates assuming markets are

segmented. In the first four columns results for the coefficient associated with the SDF

are presented. Prices of risk are in the last four columns.

Again we observe that results when rewards of risk are resctricted to be the same

across markets are driven by U.S. data. For example, this can be illustrated by the fact

that the price of risk associated with the value factor, HML, under integration is -0.16

73

, while under segmentation the value HML price of risk is -0.24 for U.S. and -0.81 for

Mexican portfolios.In contrast to the results obtained with daily data, Fama and French risk factors appear

significant for Mexican portfolios in both samples (1990-2004 and 1995-2004).

The size factor appears significant only when using Fama and French U.S. portfolios asthe tested portfolios. We observe that risk rewards for SMB and HML factors undersegmentation have the same sign and are close in magnitude between U.S. industrialportfolios and Mexican portfolios.

Visual representation of monthly results are presented in Figure 2.7 to Figure 2.9. As

in the case with weekly data, the unrestricted model performs better for Mexican portfolios

for all samples. That is, we observe a better fit of the unrestricted pricing model than

from the restricted model.

Comparing Figure 2.4 to Figure 2.6 with their monthly counterparts (Figure 2.7 to

Figure 2.9) we observe that, on average, the factor model performs much better when

using monthly data. This phenomenon could be a result of thin trading and market

microstructure in particular for Mexican returns, e.g., bid-ask bounce, and lack of liquidity

in the Mexican market.

Statistical tests to measure integration are presented in Table 2.7. Panel A present

results for the equality of prices of risk between the unrestricted version of the models

in Mexico and the U.S.. Results suggest that equality in the reward of market risk and

SMB risk cannot be rejected using monthly data in both sub-samples and for Industrial

and Fama and French portfolios. However, the joint hypothesis of equality of prices of risk

is rejected for both Industrial and Fama and French portfolios. In Panel B results for the

GMM distance statistics (2.25) are presented. The results again reject the hypothesis of

market integration between Mexico and the U.S. using Industrial and Fama and French

portfolios.

74

Table 2.5: Prices of Risk: 1990-2004 Monthly DataThe table presents the estimation results of the pricing model for Mexican industrial portfolios and U.S. portfolios.


The columns labelled bj present estimates, t-stat and p-value for bj for the different factors under integrationand segmentation. Columns labelled λj present the same results for prices of risk. The model is estimated usingmonthly excess returns of Mexican and U.S. portfolios from January 1990 to December 2004. Results with U.S.Industrial Portfolios and Fama and French Portfolios are presented in Panel A and Panel B respectively. Thestochastic discount factor is linear in the Fama and French factors and the exchange rate between the U.S. dollarand Mexican Peso. In the case of integration, the stochastic discount factor is restricted to be the same for Mexicoand the U.S.. In contrast, under segmentation the model is estimated using a stochastic discount factor for eachcountry.


bj λj

Mkt SMB HML Exch Mkt SMB HML ExchEstimate 0.07 -0.02 0.02 -0.03 1.11 -0.21 -0.21 -0.33

t-ratio 3.49 -0.66 0.58 -1.38 2.90 -0.56 -0.68 -0.85p-value 0.00 0.51 0.56 0.17 0.00 0.58 0.50 0.39

2. Segmentation

bj λj

Mexico Mkt SMB HML Exch Mkt SMB HML ExchEstimate 0.22 0.13 0.15 -0.03 6.00 3.04 -1.11 0.34

t-ratio 8.78 7.18 8.44 -0.48 6.08 5.63 0.97 1.08p-value 0.00 0.06 0.05 0.28 0.00 0.06 0.26 0.68

U.S. Mkt SMB HML Exch Mkt SMB HML ExchEstimate 0.07 -0.03 0.01 1.17 -0.24 -0.27

t-ratio 3.22 -0.90 0.25 2.85 -0.61 -0.87p-value 0.00 0.37 0.80 0.00 0.54 0.38


bj λj

Mkt SMB HML Exch Mkt SMB HML ExchEstimate 0.19 0.12 0.19 -0.01 3.08 1.50 0.26 0.53

t-ratio 8.64 3.80 8.33 -4.88 4.09 0.89 2.69 -3.48p-value 0.00 0.00 0.00 0.63 0.00 0.00 0.33 0.28

2. Segmentation

bj λj

Mexico Mkt SMB HML Exch Mkt SMB HML ExchEstimate 0.22 0.13 0.15 -0.03 6.00 3.04 -1.11 0.34

t-ratio 8.78 7.18 8.44 -0.48 6.08 5.63 0.97 1.08p-value 0.00 0.06 0.05 0.28 0.00 0.06 0.26 0.68


t-ratio 8.89 6.50 7.37 4.20 3.82 0.14p-value 0.00 0.00 0.00 0.00 0.00 0.89

75

Table 2.6: Prices of Risk: 1996-2004 Monthly DataThe table presents the estimation results of the pricing model for Mexican industrial portfolios and U.S. portfolios.


The columns labelled bj present estimates, t-stat and p-value for bj for the different factors under integrationand segmentation. Columns labelled λj present the same results for prices of risk. The model is estimated usingmonthly excess returns of Mexican and U.S. portfolios from January 1996 to December 2004. Results with U.S.Industrial Portfolios and Fama and French Portfolios are presented in Panel A and Panel B respectively. Thestochastic discount factor is linear in the Fama and French factors and the exchange rate between the U.S. dollarand Mexican Peso. When markets are assumed to be integrated, the stochastic discount factor is restricted to bethe same in Mexico and the U.S.. In contrast, under segmentation a stochastic discount factor is estimated for eachcountry.


bj λj

Mkt SMB HML Exch Mkt SMB HML ExchEstimate 0.08 0.03 0.05 -0.16 0.90 0.26 -0.05 -0.75

t-ratio 3.72 0.98 1.71 -3.81 1.55 0.48 -0.11 -2.35p-value 0.00 0.33 0.09 0.00 0.12 0.63 0.91 0.02

2. Segmentation

bj λj


t-ratio 1.98 0.27 -0.33 -2.92 2.63 0.88 -2.21 -1.89p-value 0.05 0.79 0.74 0.00 0.01 0.38 0.03 0.06

U.S. Mkt SMB HML Exch Mkt SMB HML ExchEstimate 0.06 0.01 0.03 1.26 0.09 -0.22

t-ratio 2.43 0.20 0.95 1.88 0.15 -0.47p-value 0.02 0.84 0.34 0.06 0.88 0.64


bj λj


t-ratio 4.19 3.80 10.69 -2.00 1.76 0.07 3.63 -1.17p-value 0.00 0.00 0.00 0.05 0.08 0.95 0.00 0.24

2. Segmentation

bj λj


t-ratio 1.98 0.27 -0.33 -2.92 2.63 0.88 -2.21 -1.89p-value 0.05 0.79 0.74 0.00 0.01 0.38 0.03 0.06


t-ratio 4.34 8.56 9.53 1.65 4.32 2.60p-value 0.00 0.00 0.00 0.10 0.00 0.01

76

Figure 2.7: Mexican Industrial and US Industrial Portfolios: Monthly Returns. Realizedreturns are on the horizontal axis and predicted are on the vertical axis. Circles representMexican portfolios and triangles U.S. portfolios.

−2 −1 0 1 2−2

−1

0

1


−2 −1 0 1 2−2

−1

0

1


−2 −1 0 1 2−2

−1

0

1


−2 −1 0 1 2−2

0

2


−2 −1 0 1 2−2

−1

0

1


−2 −1 0 1 2−2

−1

0

1


77

Figure 2.8: Mexican Industrial and US Fama and French Portfolios: Monthly Returns.Realized returns are on the horizontal axis and predicted are on the vertical axis. Circlesrepresent Mexican portfolios and triangles U.S. portfolios.

−2 −1 0 1 2−2

−1

0

1


−2 −1 0 1 2−2

−1

0

1


−2 −1 0 1 2−2

−1

0

1


−2 −1 0 1 2−2

0

2


−2 −1 0 1 2−2

−1

0

1


−2 −1 0 1 2−2

−1

0

1


78

Figure 2.9: Mexican Industrial and US Small Caps Portfolios: Monthly Returns. Realizedreturns are on the horizontal axis and predicted are on the vertical axis. Circles representMexican portfolios and triangles U.S. portfolios.

−2 −1 0 1 2−2

−1

0

1


−2 −1 0 1 2−2

−1

0

1


−2 −1 0 1 2−2

−1

0

1


−2 −1 0 1 2−2

0

2


−2 −1 0 1 2−2

−1

0

1


−1 0 1 2 3−1

0

1

2

3

79

Table 2.7: Tests of Integration, Monthly DataThis table presents results of testing equality in prices of risk between Mexican and U.S. estimates under segmen-tation equation (2.24) and; testing the hypothesis of integration using equation (2.25). Panel A present results forprices of risk. The column labelled ALL presents the results of testing the joint hypothesis of equality of all pricesof risk between Mexico and the U.S.. The columns labelled Mkt, SMB and HML present the results of testingequality of each risk factor respectively. In Panel B results for testing the hypothesis of integration using (2.24)

are presented. The columns labelled Int and Seg present standardized pricing errors Jt = gT (b)′S−1gT (b) for themodel under the hypothesis of integration and segmentation respectively. J-test presents the result of the GMMdistance statistics given by J-test=TJT segmentation−TJT integration. p-values are presented in the last column.

Panel A. Prices of RiskIndustrial Portfolios

ALL Mkt SMB HML1990-2004 5.63 1.04 0.34 0.80

p-value 0.23 0.31 0.56 0.371996-2004 13.08 3.74 1.15 1.37

p-value 0.01 0.05 0.28 0.24Fama and French

ALL Mkt SMB HML1990-2004 14.11 1.84 0.38 1.39

p-value 0.01 0.18 0.54 0.241996-2004 52.95 2.57 0.64 12.56

p-value 0.00 0.11 0.42 0.00Panel B. Pricing ErrorsIndustrial Portfolios

Int Seg J-test p-value1990-2004 0.33 0.24 15.36 0.001996-2004 0.29 0.21 8.65 0.03Fama and French1990-2004 0.62 0.55 11.79 0.011996-2004 0.50 0.39 11.30 0.01

2.4.4 Pricing Errors

To complement the above results, and in an attempt to capture the dynamic performance

of the pricing model, figures (7) to (12) present the time series of the magnitude of pricing

errors of the different model specifications. The magnitude of the vector of pricing errors

for U.S. and Mexican Portfolios is computed using `2 norm: ‖ ωjt+1 ‖=

√∑Ji ωj

i,t+1 where

ωji,t+1 = ri,t+1 − ri,t+1ft+1b

j .

Figure 2.10 to Figure 2.12 show pricing errors for U.S. Fama and French, U.S. Industrial

and U.S. Small Caps against Mexican pricing errors using weekly data. The first row of

figures present pricing errors when prices of risk are estimated using the entire sample.

80

Figure 2.10: Pricing Errors. Mexican Industrial and US Fama and French Portfolios.Weekly data.

Feb90Jun91Oct92Mar94Jul95Dec96Apr98Sep99Jan01May02Oct030

0.5

1

1.5

2

2.5Integration. Sample 1990−2004


1

2

3


Feb90Aug90Mar91Sep91Apr92Oct92May93Dec93Jun94Jan95Jul950

0.5

1

1.5

2



0.5

1

1.5

2

2.5Segmentation. Sample 1990−1995

Nov96 Mar98 Aug99 Dec00 Apr02 Sep030

0.5

1

1.5

2



0.5

1

1.5

2


MexUS

MexUS

MexUS

MexUS

MexUS

MexUS

81

Figure 2.11: Pricing Errors. Mexican Industrial and US Industrial Portfolios. Weeklydata.


0.5

1

1.5

2



1

2



0.5

1

1.5

2



0.5

1

1.5

2



0.5

1

1.5

2



0.5

1

1.5

2


MexUS

MexUS

MexUS

MexUS

MexUS

MexUS

82

Figure 2.12: Pricing Errors. Mexican Industrial and US Small Caps Portfolios. Weeklydata.


0.5

1

1.5

2



1

2

3



0.5

1

1.5

2



0.5

1

1.5

2



0.5

1

1.5

2



0.5

1

1.5

2


MexUS

MexUS

MexUS

MexUS

MexUS

MexUS

83

The first columns present results in the case of segmentation and in the second column

results under integration are presented. The second and third row show the results for the

sub-samples 1990-1995 and 1996-2004 respectively. Figure 2.13 to Figure 2.15 present the

same results but using monthly data.

From Figures 2.13 to Figure 2.15 we observe that the magnitude of pricing errors is

much smaller and more stable for U.S. portfolios than for Mexican portfolios. In the case

of Mexican portfolios, pricing errors are very similar when using either U.S. Fama and

French, U.S. Industrial or U.S. small caps as tested portfolios. In all cases, huge pricing

errors for Mexican portfolios are observed during the episodes of 1994, 1998 and 2001.

These periods coincide with three major events that affected all emerging markets. The

first one is the Mexican peso crisis in 1994, the other two were not originated in Mexico

(Russian crisis of 1998 and Argentinean crisis of 2001) but had a strong impact in Mexico

that is reflected by the poor performance during this periods of the pricing model.

Crises that are local in nature are followed by liquidity shocks in financial markets

of developing countries. This contagion effect is a result of an increase in the perception

of risk in developing markets from global investors. Therefore, we observe that liquidity

shocks, not necessarily generated in Mexico, segmented the MSM from U.S.markets.

A comparison of the model performance in both sub-samples, excluding the local and

global events discussed above, shows an improvement in the magnitude of pricing errors of

Mexican portfolios in the sub-sample covering 1995-2004. This suggests that integration

is an evolving phenomenom that in the case of Mexico was triggered by liberalization of

capital accounts in the end of the 80’s and beginning of the 90’s. Moreover, the results

suggest that integration is not only a matter of regime shifts within an individual market

that is independent of global events. Instead it expands and contracts in association with

global markets depending only partially on local developments.

84

Figure 2.13: Pricing Errors. Mexican Industrial and US Fama and French Portfolios.Monthly data.

Sep92 Jun95 Mar98 Dec00 Sep030

5



5


May91 Sep92 Feb94 Jun950

2

4



2

4


Mar98 Dec00 Sep030

2

4

6


Mar98 Dec00 Sep030

2

4


85

Figure 2.14: Pricing Errors. Mexican Industrial and US Industrial Portfolios. Monthlydata.


5



5



2

4



2

4


Mar98 Dec00 Sep030

2

4

6


Mar98 Dec00 Sep030

2

4


86

Figure 2.15: Pricing Errors. Mexican Industrial and US Small Caps Portfolios. Monthlydata.


5



5



2

4



2

4


Mar98 Dec00 Sep030

2

4

6


Mar98 Dec00 Sep030

2

4

6

Segmentation. Sample 1996−2004

87

2.5 Conclusions

This paper investigates the degree of integration between Mexican and U.S. equity mar-

kets between 1990 and 2004. I test the power of Fama and French factors to explain the

cross-section of expected returns in both markets. Theoretically, integration of financial

markets between the U.S. and Mexico implies that not only the same risk factors should

be priced in both countries, but the rewards for holding these risks should be the same

across countries.

Fama and French factors are assumed to be the mimicking portfolios of the true underly-

ing risk factors in both markets and exchange rate was included as the only local factor

affecting exclusively the cross-section of Mexican returns. In this multi-factor approach,

integration implies that one stochastic discount factor price portfolio returns in Mexico

and the U.S..

Overall, the empirical results suggest that the degree of integration is a time-varying pro-

cess related to global markets events as to particular conditions within the Mexican mar-

ket. Liberalization of capital accounts is necessary but does not causes market integration.

Despite the MSM being fully investible by May 1989 our model rejects market integra-

tion using data from 1990-2004. However, testing market integration using a multi-factor

pricing model is problematic. For example, empirical evidence on multi-factor models of

expected returns has found significant local factor premiums in developing markets that

appear not to be segmented by any barrier to capital flows. For example asymmetric tax-

ation and differences in transaction costs (see Stulz (1981)) can affect investors decisions

when they decide to invest either locally or globally; home bias as in Lewis (1999) can ac-

count for a large local factor even without segmentation. Therefore, the results presented

in this chapters should not be interpreted as evidence of true segmentation.

88

Further, we observe a significant improvement in the performance of the factor model

in the sub-sample that spans from 1995-2004 relative to the results obtained in periods fol-

lowing the liberalization of Mexican market 1990-1995. This evidence suggests that even

after rejecting full integration of the MSM we observe that these markets are relatively

more integrated in the most recent periods than in the past.

2.5.1 Appendix

Derivation of standard errors for λ.

To obtain standard errors for (2.22) using the delta-method, it is convenient to re-

express the prices of risk (2.13) for k = 1, ..., K as:

λji =

∑Kk=1 σikb

jk

1− µ′bj(2.27)

Taking the derivative of (2.27) with respect to the vector of estimates, θ = [b′, µ′, vec(Σf )′]′,

vector of dimension 2K + K2. vec(Σf ) = [σ′.1, ..., σ′.K ]′ and σ.i is the i-th column of the

variance-covariance of risk factors, Σf ,

D(θ)i =∂λj

i

∂θ=

1(1− µ′b)2

(1− µ′b)σ.i + (σi.b)µ

(σi.b)b

0(i−1)K

(1− µ′b)b

0(K−i)K

(2.28)

89

Applying the delta method, the asymptotic variance of λji is given by:

Avar(λji ) = D(θ)′iAvar(θ)D(θ)i (2.29)

In the same way it is possible to obtain the asymptotic covariances between λi and λh,

Acov(λji , λ

jh) = D(θ)′iAvar(θ)D(θ)h (2.30)

90

Chapter 3

Improving Forecasts of Inflation

using the Term Structure of

Interest Rates

3.1 Introduction

There has been a growing empirical literature that finds a statistical relationship between

interest rates and future inflation. Papers by Mishkin 1990a; 1990b; 1991, Jorion and

Mishkin (1991), Estrella and Mishkin (1997), and Kozicki (1997) document the predictive

power of interest rates for future inflation. In general, these papers investigate the statisti-

cal relationship between a yield spread that matches an inflation spread. This is motivated

by the Fisher equation which states that the nominal interest rate is composed of the real

interest rate and expected inflation. The stability of this relationship is investigated in

Estrella, Rodrigues, and Schich (2003). For the U.S. they find evidence of a structural

91

break in 1979 and 1982 for short horizon regressions. Estrella (2005) discusses the reasons

why the yield curve contains predictive content for both real and nominal variables, and

argues that monetary policy plays a large role in this relationship.

Nevertheless, the practical forecast improvements that come from employing the Fisher

equation sometimes appear modest relative its importance as a theoretical relationship.

The literature finds poor predictive power at the short end of the yield curve but improved

prediction around 12 months and the best performance from 3 to 5 years out. Stock and

Watson (2003) also note that the Fisher equation often loses its predictive content after

controlling for lagged inflation.

The failure of the Fisher equation to consistently beat autoregressive benchmarks has

two possible implications for inflation forecasting. One possible implication, as argued by

Stock and Watson (2003), is that, despite their theoretical appeal, interest rates may have

limited practical value for inflation forecasting. An alternative interpretation is that the

performance of the Fisher equation is hindered by its rigid restrictions, in which only the

information in a single interest rate spread with maturities matching the inflation spread is

employed in the forecast. While theoretically appealing, using only one source of informa-

tion goes against some of the more recent trends in forecasting, in which improvements are

found from combing information from multiple predictors or forecasts (Bernanke, Boivin,

and Eliasz (2005), Stock and Watson (2002), Favero, Marcellino, and Neglia (2005)).

Recent research on macro-finance models, including Ang and Piazzesi (2003), Chernov

and Biokbov (2006), Dewachter, Lyrio, and Maes (2006), and Diebold, Rudebusch, and

Aruoba (2006), demonstrate the importance that macro aggregates and monetary policy

play in determining the shape of the term structure and its dynamics.

In this paper we contribute to the literature on inflation forecasting in several ways.

First, we investigate whether, for a given inflation spread, additional forecasting informa-

92

tion can be extracted from the entire interest rate term structure. We also investigate if

information in daily lags of the term structure are useful for prediction. In contrast to

previous work, we focus exclusively on the out-of-sample predictive performance relative

to autoregressive type models, and we work with readily available interest rates on gov-

ernment T-bill and bonds.1 Our focus is on the empirical forecasting content of interest

rates.

One reason why multiple interest rates may improve inflation forecasts is that we face a

signal extraction problem. For example, suppose that the Fisher equation holds as in ikt =

rkt +Etπ

kt+k, where ikt is the k period nominal interest rate, rk

t is the matching real interest

rate, Etπkt+k is the expected inflation over this period, and πk

t+k = log(pt+k/pt), where pt

denotes the price level. If real interest rates are unobserved and stochastic then ikt provides

a noisy measure of Etπkt . However, the interest rate ilt, l > k, also contains information

on Etπkt+k, since we can decompose the longer inflation rate as πl

t+l = πkt+k + πl−k

t+l to get

ilt = rlt + Etπ

kt+k + Etπ

l−kt+l . Therefore, ikt and ilt have Etπ

kt+k as a common factor. This

suggests additional information on Etπkt+k may be available in other interest rates and the

term structure as a whole. If inflation expectations are persistent then similar reasoning

suggest that there may be benefits to using lags of the term structure.

By employing multiple interest rates we may also allow for a more robust predictive

relationship with inflation. For instance, the information content of the term structure

may be constant, but the location of inflation expectations may be easier to extract at

different maturities, depending on the stance and expectation of monetary policy.

The use of the whole term structure and lags of the term structure results in a di-

mensionality problem. To extract the predictive content but retain parsimony we follow1The majority of research in this area uses zero coupon yields. This requires estimation of the yield

curve at each point in time.

93

the diffusion index method of Stock and Watson (2002). We extract principle components

from both current term structures and lagged term structures. We select the principle

components using a variance criteria and a correlation criteria. The latter is based on the

correlation between the principle components and regression residuals from an AR model

of inflation. Therefore, they pick up remaining structure not captured by the AR model.

We investigate several variable selection methods and report results on a model average

over different principle component specifications.

To gauge the success of our methods we include, as benchmarks, an autoregressive type

model, a Fisher equation model augmented with lags of inflation and a Phillips curve. To

allow for changes in the model relationship we use a rolling window for estimation.

In general, our results show that an AR type benchmark is better than an augmented

Fisher model. Our principle component methods improve upon these results. The forecast-

ing improvements come from two dimensions. The term structure (current term structure)

dimension and the time (lagged term structures) dimension. For the 12 month ahead pre-

dictive regressions, only the term structure dimension improves forecasts, while for the 36

month regressions both the term structure and time dimension are useful. The relative

performance of the variance and correlation criteria used to select principle components

criteria is mixed, and depends on the subsample and the forecast horizon. Our recom-

mended approach is a model average over specifications that have between 3 and 5 principle

components from both the term structure and time dimensions. This approach provides

good results for all time periods and forecast horizons.

Our results are supportive of the broad implication of the Fisher equation that interest

rates provide a useful tool in inflation forecasting. At the same time, we find that em-

ploying the whole term structure and its recent lags provides clear improvements over the

restrictions of the Fisher equation. This is consistent with recent developments in fore-

94

casting which emphasize the value of combining information and forecasts from multiple

sources.

While this paper focuses on predictive content of the term structure, it is not our

intention to suggest that interest rates be used in isolation to forecast inflation. There is

a large literature on inflation forecasting and many useful techniques have been developed

using information other than interest rates that could potentially be combined with term

and lagged term structure variables considered here. Examples, include methods based on

structural relationships Fisher, Liu, and Zhou (2002); Orphanides and van Norden (2005);

Stock and Watson (1999), the use of asset prices, Ang, Bekaert, and Wei (2005); Stock

and Watson (2003), and factor models Banerjee and Marcellino (2006); Camba-Mendez

and Kapetanios (2005); Stock and Watson (2002).2.

This paper is organized as follows. Section 3.2 reviews the data, Section 3.3 discusses

the benchmark models, while the new principle components models are introduced in

Section 3.4. Results are found in Section 3.5 and conclusions in Section 3.6.

3.2 Data

Our data set comprise daily interest rates and monthly data on consumer price indexes,

and the total civilian unemployment rate. The daily interest rates included in our panel

are U.S. Treasury yields with maturities of 3-months, 6-months, 1-year, 3-years, 5-years,

and 10-years. Inflation was computed as the log difference of the seasonally adjusted

consumer price index for all urban items. Interest rates and consumer price indices were

obtained from the Federal Reserve of St. Louis FRED data base. The unemployment

rate was obtained from the Citibase data base. The sample period spans from March2Recent work on time series models of inflation emphasize the changing inflation dynamics Cogley and

Sargent (2002); Dossche and Everaert (2005); Pivetta and Reis (2006); Stock and Watson (2006).

95

1962 to December 2004. This yields a sample size of 514 for the monthly inflation rate

data and a sample size of 11,175 for the daily interest rate data. Out-of-sample forecasts

are computed from September 1974 to December 2004, giving a total of 364 monthly

out-of-sample forecasts, with the proceeding 150 months reserved for the training sample.

The top panel of Figure 3.1 presents the annual inflation rate for the whole sample. Two

major episodes are observed during the post-war U.S. history. The first one is characterized

by a period of high inflation from the early 1960s through the mid 1980s, and a second

period of low inflation runs from the mid 1980s to the present.

We define the annualized j-th term inflation rate observed at time t as:

πjt = (1200/j) ln(pt/pt−j).

Following Stock and Watson (1999), we will focus on forecasting annualized inflation

percentage spreads of the form:

πkt+k − π1

t+1 = (1200/k) ln(pt+k/pt)− 1200 ln(pt+1/pt)

where k = 12, 36.

Finally, note that in the following we use interest rates for T-bills and bonds which

may have coupon payments. This has two attractions for our forecasting exercises. First,

it avoids the need to estimate a zero-coupon term structure and the associated estimation

errors that can arise form this. Second, the interest rates we use are readily available,

for example from the St. Louis FRED data base. Throughout the paper we use the

terminology Fisher equation even though our rates may not be derived from a zero-coupon

instrument.

96

3.3 Benchmark Models

In order to obtain the benchmark with the best performance in our sample, a set of

competing models were estimated.

1. Pure autoregression-type benchmark (AR-type) Our base-line benchmark

model for inflation forecasting is the simple autoregressive-type specification given

by:

πkt+k − π1

t+1 = γ +P−1∑

p=0

φpπ1t−p + εt+k, (3.1)

where εt+k is an error term. We also consider two other standard benchmark models

in which this AR-type specification is augmented by other predictors suggested by

economic models.

2. Phillips Curve (PhCu)

The unemployment-based Phillips curve specification used in this paper is

πkt+k − π1

t+1 = γ +P−1∑

p=0

φpπ1t−p +

Q−1∑

q=0

βqwt−q + εt+k, (3.2)

where wt is total civilian unemployment rate (LHUR) in month t.

3. Augmented-Fisher Equation

Finally, a benchmark that appears as the natural counterpart to the model that

includes daily spreads as predictors, is a model that incorporates the Fisher hypoth-

esis. In this case, we extend the autoregressive model by including as an additional

predictor the monthly yield spread with the same maturity as the target inflation

forecast. The empirical specification of the augmented Fisher equation hypothesis

97

is given by

πkt+k − π1

t+1 = γ +P−1∑

p=0

φpπ1t−p + β(ikt − i1t ) + εt+k, (3.3)

where ikt is the observed treasury yield with maturity k at time t.

All three benchmark models are estimated by OLS, with the number of lagged inflation

rates (1 ≤ P ≤ 12) and unemployment rates (1 ≤ Q ≤ 12, PhCu model only) determined

recursively using the Bayesian information criterion (BIC). For the Phillips curve, P is

selected first and then conditional on that value Q is selected. This process of lag selection

is repeated recursively as new information arrives.

The benchmark models above were estimated under the assumption of an I(1) price

index. We also compared these to the equivalent benchmark models that hold under the

assumption that prices are I(2), in which case the terms involving π1t−p are replaced by

∆π1t−p. As in Stock and Watson (1999; 2002) we find little difference between the I(1) and

I(2) specifications with respect to the accuracy of the inflation forecasts. Thus we present

only the results that hold when prices are modelled as I(1) variables. Generally speaking,

we found that the forecasts using the pure AR-type model with prices modelled as I(1)

provided the best benchmark forecasts.

3.4 Principal Components Models

Models that use interest rate spreads as predictors of future inflation are based on a ver-

sion of the Fisher’s hypothesis. The empirical specification of Fisher’s hypothesis includes

only monthly spreads with the same maturity as the target inflation rate. Thus, Fisher’s

specification discards any other potential information contained in the term structure of

98

interest rates or its daily lags. In this paper, we propose forecasting inflation by incor-

porating information contained in both dimensions of the interest rate data: the term

structure dimension and the time dimension. The time series dimension is incorporated

via the use of daily interest rate observations.

The transition from Fisher’s specification that includes just one spread of interest rates

as predictors to the multivariate case that incorporates different terms for different dates

raises the important problem of parsimony. In order to mitigate overfitting and poor

forecast performance we proceed by constructing a small number of factors from the large

set of daily spreads using principal components. The methodology is based on the premise

that the most useful information for forecasting purposes can be summarized by the first

few principal components.

3.4.1 Incorporating Term Structure and Time Series Information

We denote by Xt the full vector of interest rates used as inflation predictors. In our

main forecast results the interest rates included in Xt span both the maturity and time

series dimensions. However, in order to better understand the source of the forecast

improvements, we first consider the two dimensions separately.

Let spk,jt = ikt − ijt denote the spread at time t between yields with a maturity of k

and j respectively. To simplify notation, the length of every month is assumed to be of 22

business days. We use integer subindices to denote monthly frequencies, whereas for daily

data subindices we use the conventional notation in which t+h/22 for h = 0, ..., 22−1 refers

to business day h in month t. Time t refers to the first business day of month t. Inflation

rates are assumed to be observed the first business day of every month. Following the

above notation, and to avoid the use of information not available to the econometrician,

the information set available at time t is given by inflation rates observed up to time t,

99

and interest rate spreads observed up to time t− 1/22.

Term Structure Dimension

In this case, we extend the augmented Fisher equation in the term structure dimension

by including the full set of interest interest rate spreads as predictors for future inflation.

The matrix of predictors, Xt, is therefore given by

X ′t =

[sp12,3

t−1/22, sp36,3t−1/22, sp

36,12t−1/22, sp

60,3t−1/22, sp

60,12t−1/22, sp

60,36t−1/22, sp

120,3t−1/22,

sp120,12t−1/22, sp

120,36t−1/22, sp

120,60t−1/22

]

such that only the interest rate spreads observed on the last business day of month t− 1

are included in the information set It.

Time Series Dimension

This case extends the augmented Fisher specification in the time series dimension by

including m daily lags of the interest rate spread with the same maturity as the target

inflation spread as predictors. In other words, for the target inflation forecast period

πkt+k − πj

t+j , k > j, Xt is given by,

X ′t =

[spk,j

t−1/22, spk,jt−2/22, ..., sp

k,jt−m/22

].

Combined term structure and time series information

In this case both dimensions, the term structure and time dimension, are pooled together

to construct predictors to forecast future inflation. The predictors’ matrix, Xt, is formed

100

as

X ′t =

[x1t

′, x′2t, . . . , x′mt

]

xht =[sp12,3

t−h/22, sp36,3t−h/22, sp

36,12t−h/22, sp

60,3t−h/22, sp

60,12t−h/22,

sp60,36t−h/22, sp

120,3t−h/22, sp

120,12t−h/22, sp

120,36t−h/22, sp

120,60t−h/22

], h ∈ 1, ..., m.

Unfortunately, there is little guidance as to the number of days, m, that should be

included in Xt. To measure the relative information content in daily data, we constructed

three matrices of predictors by setting different values for m. The first matrix sets m

equal to 20, including four weeks of daily data. In the second matrix m is set equal to 10,

and finally, the last case includes only a week of daily data setting m equal to 5. In the

following let n denote the number of columns in Xt.

From the above definitions the information set available for forecasting at time t, It,

is given by lag values of inflation rates and Xt,

It = πt, πt−1, πt−2, ..., Xt .

3.4.2 Dimension Reduction via Principal Components

Following Stock and Watson (2002), forecasting is performed in a two-step procedure.

First, the principal components of the set of interest spreads are computed; second, the

estimated factors are used as predictors to forecast future inflation.

As discussed above, we denote by Xt the vector of n daily interest rate predictors,

potentially varying across both the time series and maturity dimensions, that are observed

up until the last day of time t− 1. Because the dimension of Xt is large, it is impractical

101

to use all n predictors. Instead, we employ the principle component decomposition

Xt = AZt + AcZct = AZt + vt, vt ≡ AcZc

t (3.4)

where Z ′t =(zt,(1), zt,(2), . . . , zt,(F )

)denotes the first F < n (sorted) principal components,

and Zct contains the remaining n−F components. The factor loading matrices for Zt and

Zct are denoted by A and Ac respectively. This allows us to achieve dimension reduction by

extracting only the first F factors Zt for inclusion as regressors in our forecasting equation

for inflation, together with the lagged values of inflation. Thus our forecasting model

becomes

πkt+k − π1

t+1 = γ +P−1∑

p=0

φpπ1t−p +

F∑

f=1

αfzt,(f) + εt+k, (3.5)

All of the regression coefficients in (3.5), including the coefficients αf on the first F

principal components are estimated by OLS. Since the lagged values of inflation already

enter (3.5) parsimoniously, we do not include them in the principal component analysis.

As in the benchmark models, the number of lagged inflation rates is chosen using the

Bayesian Information Criteria (BIC), with a maximum of 12 (1 ≤ P ≤ 12). We discuss

the choice of F in Section 3.4.3 below.

The advantage of this approach is that we have reduced the number of term structure

regressors from n down to F , while allowing the factor decomposition to pick out what

we expect to be the most important components in Zt for forecasting inflation. This

potentially allows us to incorporate substantially more information on inflation than in

the augmented Fisher regressions, while still maintaining parsimony.

102

3.4.3 Selection of Principal Components

In order to determine which principal components to employ in a principal components

model two choices are required. First, one needs a rule for ordering the components from

first to last. This involves taking a stand on which components will be most useful for

forecasting inflation. Secondly, one needs to decide how many components to use. We

discuss these questions separately in the two subsections below.

Ordering the principal components

Two different criteria are employed in the selection of factors. The first criterion, Vari-

ance Sort , consists in selecting the principal components that explain the highest per-

centage of the second moment of the predictors matrix Xt. Regardless of this being

the most common practice in the selection of principal components, there is no guarantee

that this methodology will result in the set of factors that maximize forecasting power for

inflation. Hence, we also propose and compare to a second criterion for selecting factors,

Correlation Sort , in which we sort principal components according to their in-sample

correlation with the residual from the AR-type benchmark model in (3.1). The idea con-

sists in selecting the principal components that have the highest in-sample correlation with

the residual term from the AR model to capture the component of inflation not explained

by the AR model.

Selecting the number of principal components

Once we have decided on a rule for ordering the components, we must next choose the

number of components (F) to include. Here we compare three approaches. The first is

simply to fix F, the second is to select F by applying a model selection criterion to the

103

forecasting equation, and the third is to model average across forecasts using a range of

different choices for F.

Because our primary interest is in forecasting inflation, we apply BIC to the forecasting

equation in (3.5), while treating the principle components zt(f) as regressors. Thus we

measure the (penalized) fit in terms of the explanatory power of the principle components

of zt(f) for inflation πkt+k − π1

t+1. This is similar to the approach of Stock and Watson

(1999). An alternative approach, not pursued here, is to measure the penalized fit based

on the explanatory power of the principle components zt(f) for Xt itself in (3.4). This is

the approach pursued by Bai and Ng (2002) in a more general context, where the primary

focus is often on the modeling of Xt itself, rather than on the use of the components of

Xt in the forecast of another variable, say yt. Bai and Ng (2002) provide consistent model

selection procedures for the choice of F in (3.4).

For the model averaging approach, we let fpct+k,F denote the out-of-sample forecast of

the principal component model using F factors. The model average forecasts are then

given by

fa,t+k =u∑

F=l

ωF,t+kfpcF,t+k, (3.6)

where l and u are the lower and upper bounds for the number of principal components used

to forecast inflation. We compare two simple approaches to choosing the weights ωF,t. In

the first approach, we simply weight all forecasts equally, setting ωF,t ≡ 1/(u− l+1), so

that fa,t+k is just the simple average of all date t+k forecasts. In our second approach, we

choose weights using an in-sample regression of the realized inflation rate on the inflation

predictions from each model. In other words, the weights are given by the estimated

coefficients in the following regression:

104

πkt+k − π1

t+1 =u∑

F=l

ωF,t+kfpcF,t+k + εt+k. (3.7)

3.5 Results

The ability of the different models, benchmarks and factor models, to forecast inflation is

summarized by the mean-squared-error (MSE) and mean-absolute-error (MAE) of their

forecasts relative to the MSE and MAE of forecasts based on the autoregressive type model

(AR-type) respectively. Table 3.1 summarizes the results from different tables while results

for the competing benchmark models are found in Table 3.2.

Tables 3.3-3.7 present results for the different specifications of the factor models; Term

Structure Dimension, Time Series Dimension, and All Information (using both the term

structure and time series dimension). For the latter two models results are shown for

5, 10, and 20 days of lagged daily interest rate spreads. As mentioned above, principal

components were selected using two different methods: 1) the variance sort criteria and 2)

the correlation sort criteria. We also considered three competing approaches to chose the

number of principal components: a) the specification of parsimonious models with fixed

numbers of principal components, b) selection by BIC, and c) model averaging across

forecasts with different numbers of principal components. Results for the different models

are summarized in Tables 3.3-3.7 following the organization described in Table 3.1.

3.5.1 Benchmark

Table 3.2 summarizes results for the different benchmark models: the autoregressive-type

model (AR-type), the augmented Fisher model, and the Phillips Curve (PhCu) specifica-

tion. The columns labelled I(1) and I(2) refer to cases where price indices are modelled as

105

I(1) and I(2) respectively. Panel A shows results when the parameters are estimated using

recursive least squares3 and in Panel B rolling windows estimates are presented. In order

to make comparisons between the different estimates, the MSE and MAE are computed

relative to the recursive least squares AR type model. Results are shown for the 12-month

and 36-month inflation forecasts, and are divided in two forecast sub-samples: 1974-1983,

1984-2004, and for the whole period.

There are important differences in the forecasting performance between recursive least

squares estimates and rolling windows estimates. Both, MSE and MAE deteriorate dra-

matically when parameters are estimated using rolling windows, especially the 36-month

inflation rate forecasts. The MSE for the 12-month inflation forecasts over the period

1974-2004 using the AR model when prices are modelled as I(1) increases by 14 percent

when estimated by rolling windows (1.144, panel B.1, AR) compared to the least square

estimation (1.000, panel A.1). For the 36-month inflation forecasts, least squares estima-

tion outperforms rolling windows forecasts by 68 percent measured by MSE (1.684, panel

B.2, AR) and by 20 percent in terms of the MAE (1.206, panel B.2, AR). The augmented

Fisher representation and Phillips Curve benchmarks exhibit the same pattern. In the

case of the Fisher’s representation, the MSE for the 12-month inflation rate goes from

0.847 (panel A.2, Fisher) in the least square case to 1.480 (panel B.2, Fisher) in the rolling

windows case. Therefore, in the results that follow we concentrate exclusively on recursive

least squares estimates.

The AR type model and the augmented Fisher specification outperform the rest of the

benchmarks. In the case of the 12-month inflation period, the AR type model with an I(1)

specification performs the best in both sub-samples.4 For the 36-months inflation rate,3In other words the model is re-estimated and a forecast computed as each new observation arrives.4We compared I(1) and I(2) specifications in which the left-hand side variable is identical annualized

inflation differences, but the right hand side inflation variables where I(1) (π1t−p) or I(2) (first differences

106

forecasts using Fisher’s model outperform the rest of the benchmarks with a relative MSE

of 0.847 (panel A.2, Fisher) compared to 1.000 and 1.203 for the AR model and Phillips

Curve model respectively. Modelling price indexes as I(1) result in better forecasts than

I(2) specifications in the 1974-1983 sub-sample for both the 12 and 36-month inflation

terms. However, for the 36-period inflation, the AR-type and Fisher’s representation with

price indices modelled as I(2) outperform the I(1) specification in the 1984-2004 sub-

sample. Despite this result, benchmarks using the I(1) specification have a smaller MSE

and MAE than the I(2) specification for the whole sample.

Although no-single benchmark model dominated in all cases, we concluded that the

AR-type model provided the best overall benchmark model. The comparisons in the pro-

ceeding tables (Tables 3.3-3.7) all employ this benchmark when presenting the relative

MSEs and MAEs. Since the Fisher model provided the starting point for the term struc-

tures models proposed here, Tables 3.3-3.4 also include a column with the results from the

augmented Fisher model as an additional point of comparison.

3.5.2 Fixed Number of Principal Components

Tables 3.3 and 3.4 present the results for different specifications of the factor model using

a fixed number of principal components (from one to five) for the variance and correlation

sort criteria respectively. In Panel A the results for the 12-period inflation period are shown

and in Panel B results for the 36-period inflation are presented. The columns labelled

“Term Structure” refer to the factor model that includes as predictors the current term

structure. The columns labelled “Time” present results for the model that incorporates

lags of daily spreads with the same maturity as the target inflation. Finally, both are

combined in the columns labelled “5 days”, “10 days” and “20 days” for a factor model

of π1t−p).

107

that respectively include 5, 10 and 20 lags of the daily term structure as predictors.

Several findings come out from these tables. All factor models, except the time dimen-

sion model, outperform the AR type benchmark uniformly in the 1974-1983 sub-sample.

Important differences in forecasting performance between models are observed depending

on the target inflation rate. Forecast errors are smaller when factors are selected using the

correlation approach than the variance approach in the 1974-1983 sub-sample. However,

in the 1984-2004 sub-sample, the variance sort produce smaller MSE for the 12-month

inflation period. For example, the relative MSE for the Term Structure model using three

principal components is 0.820 (Table 3, panel A.3, Term Structure) in the case of the vari-

ance sort and 0.805 (Table 4, panel A.3, Term Structure) in the correlation sort for the

1974-1983 subsample, however for the 1984-2004 subsample the results for the variance

and correlation sort are 0.933 and 1.300 respectively.

The term structure model outperforms the rest of the factor models when forecasting

the 12-month inflation rate for the whole sample. However, there is an improvement in

forecasts during the 1984-2004 sub-sample when including the time dimension into the

model. Forecasts that use 5 or 10 lags of the daily term structure of interest rates with

three principal components perform the best, with a relative MSE of 0.930 and 0.899

(Table 3, panel A.3, 5 and 10 days) respectively. In general, models with three or more

principal components improve upon the more parsimonious representations with only one

or two principal components.

For the 36-month inflation rate, the correlation sort criteria produces superior forecasts

than the variance sort. In contrast with the 12-month inflation, we observe that the time

dimension contains information on future inflation beyond the AR-type specification, and

uniformly outperforms the AR-type model in all sub-samples. Forecasts with one principal

component outperform higher dimension models that include more than one principal

108

component.

Different numbers of lags of daily spreads, m, in the predictors matrix, X for the

All Information model are tested. In general, forecasts using 5 daily lags of interest rate

spreads perform better than when 10 or 20 lags are included.

3.5.3 Bayesian Information Criteria

Table 3.5 shows the results when the number of principal components is selected recur-

sively using BIC. The results are qualitatively similar to those presented in Tables 3.3

and 3.4. Overall, the term structure dimension appears more informative than the time

dimension, specially for the 12-month inflation rate. The time dimension of interest rates

adds information relative to the AR-type model for the 36-month inflation, but not for the

12-month inflation. In contrast to the results obtained in Tables 3.2 and 3.3, we obtain

mixed results regarding the forecast performance between the correlation and variance

sorts. The specification that performs the best for the 12-month inflation includes only

the term structure dimension and has a relative MSE of 0.908 (panel A, Term Structure)

and 0.907 (panel B, Term Structure) for the correlation and variance sort respectively.

3.5.4 Model Averaging

Finally, Tables 3.6 and 3.7 present results for model averaging. Important improvements

in forecasting performance are observed from model averaging. Two different methods are

employed to compute forecast averages. The first method is a simple average across models

with different number of principal components. The second method is a weighted average

across models where weights are obtained by regressing in-sample inflation realizations

on forecasts with different numbers of principal components. OLS averaging outperform

simple averages. Consistent with the results observed in Tables 3.3 and 3.4, averages of

109

models with 3 to 5 principal components perform the best when forecasting the 12-period

inflation; and for the 36-period inflation, averages from 1 to 3, and from 1 to 5 principal

components are the best. The variance criteria outperforms the correlation criteria for the

12-month inflation rate. The opposite is true for the 36-period inflation rate.

3.5.5 Significance of Forecast Improvements

Tables 3.8 and 3.9 show the results of the Diebold and Mariano (1995) tests used to assess

the significance of the out-of-sample improvements of the forecasts using the regression

based model average relative to the autoregressive benchmark. In order to generate a

sufficient sample size for reliable out-of-sample testing, the tests are based on the full

sample period.

Since the forecast horizons, (k = 12 and k = 36 months) exceed the monthly sampling

period, the errors from even an unbiased forecast would follow a moving average process

of order k− 1 and it would therefore be unrealistic to expect uncorrelated errors in either

forecast model. Thus to implement the Diebold and Mariano (1995) test we require a

kernel estimator for the long-run variance of the difference in the squared forecast errors.

To this end, we employ the Newey-West (Barttlet) kernel, with a baseline bandwidth

choice of k − 1. While k − 1 seems a natural choice in the context of a k-period forecast,

our relatively large values of k may result in a noisy variance estimator. For this reason,

and to assess the robustness of our findings, we also provide test results using several

smaller bandwidth values.

The results from the Diebold and Mariano (1995) tests confirm the statistical sig-

nificance of the forecast improvements relative to our baseline model. In fact, all the

t-statistics shown in Tables 3.8 and 3.9 exceed the standard critical values by a substan-

tial margin. This strong significance appears robust across the bandwidth choice, forecast

110

horizon (k =12,36), the collection of principal component models included in the average

(1 to 3, 3 to 5, 1 to 5), and the choice of term structure and lag information used for the

principle components (columns 3-7). On the other hand, the tests do not adjust for the

effects of parameter estimation West (1996), model encompassing McCracken (2007), or

potential data snooping concerns White (2000). This would be of particular concern in

the case of marginally significant results. However, any resulting distortions would have

to be quite large in order to overturn the very highly significant results shown in Tables

3.8 and 3.9.

3.5.6 Summary of results

In summary, the AR type model outperforms all benchmarks for the 12-month inflation

rate, and the augmented Fisher model outperforms all benchmarks when forecasting the

36-period inflation. Forecasts improvements over the benchmark model for the 12-month

inflation rate come from the term structure dimension, with better results obtained when

using the variance sort criteria than the correlation sort.

For the 36-period inflation both dimensions, the term structure and time dimension

improve upon the benchmark model, with the correlation criteria performing better than

the variance criteria. However, model averaging provide robust forecasting improvements

as compared with both the Fisher model and autoregressive type models.

3.6 Conclusion

This paper proposes a new approach to forecasting inflation using daily interest rate data.

We consider a large number of potential interest rates predictors and organize them along

a term structure and time series dimension. Principal component methods are used to

111

extract useful predictors. For 12 month predictive regressions, only the term structure

dimension improves forecasts, while for the 36 month regressions both the term structure

and time dimension are useful. We find robust forecasting improvements in general as

compared to the augmented Fisher equation and autoregressive benchmarks.

The performance of variance and correlation criteria used to select principle compo-

nents criteria is mixed, and depends on the subsample and the forecast horizon. Our

recommended approach is the model average across models using between 3 to 5 principle

components from both term structure and time dimensions. This approach provides good

results for all time periods and forecast horizons.

112

Table 3.1: Location of Forecast Summary Results for Various Models

Number ofPrincipal Components

Selection of Principal ComponentsVariance Sort Correlation Sort

Fixed from 3.2 to 3.6 Table 3.3 Table 3.4BIC Table 3.5 Table 3.5Model Averaging Tables 3.6 and 3.7 Tables 3.6 and 3.7

113

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1.0

28

1.0

14

1.1

12

1.0

37

All

1.0

00

1.0

50

1.0

49

1.0

87

1.0

71

1.1

27

1.0

00

0.9

73

1.0

23

1.0

05

1.0

61

1.0

33

2.

π36

t+36−

π1 t+

1

1974-1

983

1.0

00

1.9

99

0.8

27

1.9

31

0.8

25

1.8

86

1.0

00

1.3

95

0.9

03

1.3

05

0.8

72

1.3

53

1984-2

004

1.0

00

0.6

90

0.8

82

0.7

37

1.8

48

1.1

75

1.0

00

0.6

72

0.9

36

0.7

02

1.3

62

0.9

27

All

1.0

00

1.5

16

0.8

47

1.4

90

1.2

03

1.6

24

1.0

00

1.0

27

0.9

20

0.9

98

1.1

22

1.1

36

Pan

elB

.R

ollin

gW

indo

ws

Sam

ple

Rel

ati

ve

MSE

MA

EA

RFis

her

PhC

uA

RFis

her

PhC

uI(

1)

I(2)

I(1)

I(2)

I(1)

I(2)

I(1)

I(2)

I(1)

I(2)

I(1)

I(2)

1.

π12

t+12−

π1 t+

1

1974-1

983

1.0

78

1.1

59

1.0

60

1.1

11

1.1

19

1.1

01

1.0

30

1.0

53

1.0

37

1.0

24

1.0

31

1.0

31

1984-2

004

1.2

79

1.0

07

1.3

95

1.0

49

1.5

00

1.3

91

1.1

37

0.9

46

1.1

51

0.9

71

1.2

47

1.0

70

All

1.1

44

1.1

08

1.1

70

1.0

91

1.2

45

1.1

97

1.0

82

1.0

01

1.0

93

0.9

98

1.1

36

1.0

50

2.

π36

t+36−

π1 t+

1

1974-1

983

1.1

67

2.0

41

0.9

18

2.0

59

1.0

56

1.6

58

1.0

94

1.4

07

0.9

32

1.2

74

1.0

10

1.2

58

1984-2

004

2.5

67

0.7

62

2.4

40

0.7

64

3.7

56

2.3

22

1.3

14

0.7

19

1.4

03

0.7

00

1.7

41

1.2

85

All

1.6

84

1.5

69

1.4

80

1.5

81

2.0

53

1.9

03

1.2

06

1.0

56

1.1

72

0.9

81

1.3

83

1.2

72

114

Tab

le3.

3:O

ut-o

f-Sa

mpl

eFo

reca

stin

gfo

rC

ompe

ting

Mod

els,

Var

ianc

eSo

rt.

Rel

ativ

eM

SE

and

MA

Efo

rdi

ffere

ntsp

ecifi

cati

ons

are

pres

ente

d.A

llre

sult

sar

ere

lati

veto

the

AR

-typ

ebe

nchm

ark

mod

el.

The

colu

mns

labe

lled

“Fis

her”

pres

ent

resu

lts

for

the

mod

elth

atin

corp

orat

esas

pred

icto

rsth

em

onth

lyyi

eld

spre

adw

ith

the

sam

em

atur

ity

asth

efo

reca

sted

infla

tion

diffe

renc

e.T

heco

lum

nsla

belle

d“T

erm

Stru

ctur

e”pr

esen

tre

sult

sfo

rth

em

odel

that

inco

rpor

ates

aspr

edic

tors

the

prin

cipa

lco

mpo

nent

sof

the

mon

thly

term

stru

ctur

esp

read

s.In

the

colu

mns

labe

lled

“Tim

e,”

resu

lts

for

the

mod

elth

atus

esda

ilysp

read

sw

ith

the

sam

em

atur

ity

asin

flati

onar

epr

esen

ted.

The

last

thre

eco

lum

nssh

owth

ere

sult

spr

inci

palco

mpo

nent

mod

elin

corp

orat

ing

both

the

tim

ean

dte

rmst

ruct

ure

dim

ensi

ons,

wit

hei

ther

5,10

,or

20da

ysof

daily

inte

rest

rate

lags

.N

um

.Sam

ple

Augm

ente

dTer

mStr

uct

ure

Tim

eB

oth

Dim

ensi

ons

(Tim

e&

Ter

mStr

uct

ure

)of

Per

iod

Fis

her

Dim

ensi

on

Dim

ensi

on

5D

ays

10

Days

20

Days

PC

sM

SE

MA

EM

SE

MA

EM

SE

MA

EM

SE

MA

EM

SE

MA

EM

SE

MA

EPanel

A.π

12

t+12−

π1 t+

1

11974-1

983

1.0

36

1.0

19

0.9

16

0.8

41

1.0

34

1.0

20

0.9

21

0.8

43

0.9

60

0.8

55

0.9

59

0.8

65

1984-2

004

1.0

75

1.0

28

1.1

74

1.0

58

1.0

93

1.0

40

1.1

56

1.0

54

1.1

29

1.0

44

1.1

29

1.0

46

All

1.0

49

1.0

23

1.0

01

0.9

46

1.0

53

1.0

30

0.9

99

0.9

46

1.0

16

0.9

46

1.0

15

0.9

52

21974-1

983

0.9

54

0.8

83

1.0

51

1.0

18

0.9

64

0.8

89

1.0

02

0.9

03

1.0

32

0.9

28

1984-2

004

1.1

36

1.0

68

1.0

72

1.0

36

1.1

24

1.0

64

1.1

10

1.0

58

1.0

91

1.0

57

All

1.0

14

0.9

73

1.0

58

1.0

26

1.0

17

0.9

74

1.0

38

0.9

78

1.0

51

0.9

91

31974-1

983

0.8

20

0.7

98

1.0

60

1.0

19

0.8

27

0.8

01

0.8

60

0.8

20

0.9

60

0.9

01

1984-2

004

0.9

33

0.9

60

1.0

51

1.0

22

0.9

30

0.9

66

0.8

99

0.9

41

1.0

89

1.0

45

All

0.8

57

0.8

77

1.0

57

1.0

20

0.8

61

0.8

81

0.8

73

0.8

79

1.0

03

0.9

71

41974-1

983

0.8

10

0.8

10

1.0

72

1.0

26

0.8

33

0.8

08

0.8

57

0.8

17

0.9

00

0.8

51

1984-2

004

1.1

02

1.0

28

1.0

53

1.0

22

1.1

87

1.0

57

1.0

53

1.0

19

0.9

18

0.9

58

All

0.9

06

0.9

16

1.0

66

1.0

24

0.9

50

0.9

29

0.9

22

0.9

15

0.9

06

0.9

03

51974-1

983

0.7

73

0.7

90

1.0

96

1.0

27

0.8

44

0.8

08

0.8

86

0.8

29

0.8

91

0.8

29

1984-2

004

1.0

81

1.0

17

1.0

55

1.0

23

1.1

34

1.0

41

1.1

36

1.0

43

1.2

57

1.0

76

All

0.8

75

0.9

00

1.0

83

1.0

25

0.9

40

0.9

21

0.9

69

0.9

33

1.0

12

0.9

49

Panel

B.π

36

t+36−

π1 t+

1

11974-1

983

0.8

27

0.9

03

0.8

31

0.9

04

0.8

69

0.9

22

0.8

47

0.8

43

0.9

60

0.8

55

0.9

21

0.8

43

1984-2

004

0.8

82

0.9

36

0.7

50

0.8

45

0.8

96

0.9

43

0.7

59

1.0

54

1.1

29

1.0

44

1.1

56

1.0

54

All

0.8

47

0.9

20

0.8

01

0.8

74

0.8

79

0.9

33

0.8

15

0.9

46

1.0

16

0.9

46

0.9

99

0.9

46

21974-1

983

1.0

61

1.0

00

0.8

42

0.9

03

1.0

81

0.8

89

1.0

02

0.9

03

0.9

64

0.8

89

1984-2

004

0.8

50

0.8

43

0.8

93

0.9

41

0.8

70

1.0

64

1.1

10

1.0

58

1.1

24

1.0

64

All

0.9

83

0.9

20

0.8

61

0.9

23

1.0

03

0.9

74

1.0

38

0.9

78

1.0

17

0.9

74

31974-1

983

0.7

23

0.7

93

0.8

80

0.9

24

0.7

63

0.8

01

0.8

60

0.8

20

0.8

27

0.8

01

1984-2

004

1.1

76

1.0

49

0.8

95

0.9

42

1.1

94

0.9

66

0.8

99

0.9

41

0.9

30

0.9

66

All

0.8

90

0.9

24

0.8

86

0.9

33

0.9

22

0.8

81

0.8

73

0.8

79

0.8

61

0.8

81

41974-1

983

0.8

22

0.8

53

0.8

98

0.9

32

0.8

87

0.8

08

0.8

57

0.8

17

0.8

33

0.8

08

1984-2

004

0.8

48

0.8

78

0.8

88

0.9

38

0.8

48

1.0

57

1.0

53

1.0

19

1.1

87

1.0

57

All

0.8

32

0.8

65

0.8

95

0.9

35

0.8

72

0.9

29

0.9

22

0.9

15

0.9

50

0.9

29

51974-1

983

0.7

97

0.8

44

0.8

89

0.9

21

0.8

65

0.8

08

0.8

86

0.8

29

0.8

44

0.8

08

1984-2

004

0.9

32

0.9

19

0.9

04

0.9

43

0.8

56

1.0

41

1.1

36

1.0

43

1.1

34

1.0

41

All

0.8

47

0.8

82

0.8

95

0.9

33

0.8

62

0.9

21

0.9

69

0.9

33

0.9

40

0.9

21

115

Tab

le3.

4:O

ut-o

f-Sa

mpl

eFo

reca

stin

gfo

rC

ompe

ting

Mod

els,

Cor

rela

tion

Sort

Res

ults

for

the

MS

Ean

dM

AE

for

diffe

rent

spec

ifica

tion

sar

epr

esen

ted.

All

resu

lts

are

rela

tive

toth

eA

Rbe

nchm

ark

mod

el.

The

colu

mns

labe

lled

“Fis

her”

pres

ent

resu

lts

for

the

mod

elth

atin

corp

orat

esas

pred

icto

rth

em

onth

lyyi

eld

spre

adw

ith

the

sam

em

atur

ity

than

the

fore

cast

edin

flati

ondi

ffere

nce.

The

colu

mns

labe

lled

“Ter

mSt

ruct

ure”

pres

ent

resu

lts

for

the

mod

elth

atin

corp

orat

esas

pred

icto

rsth

epr

inci

palco

mpo

nent

sof

the

mon

thly

term

stru

ctur

esp

read

s.In

the

colu

mns

labe

lled

“Tim

e”,re

sult

sfo

rth

em

odel

that

uses

daily

spre

ads

wit

hth

esa

me

mat

urity

asin

flati

onar

epr

esen

ted.

The

last

thre

eco

lum

nssh

owth

ere

sult

spr

inci

palco

mpo

nent

mod

elin

corp

orat

ing

both

the

tim

ean

dte

rmst

ruct

ure

dim

ensi

ons,

wit

hei

ther

5,10

,or

20da

ysof

daily

inte

rest

rate

lags

.N

um

.Sam

ple

Augm

ente

dTer

mStr

uct

ure

Tim

eB

oth

Dim

ensi

ons

(Tim

e&

Ter

mStr

uct

ure

)of

Per

iod

Fis

her

Dim

ensi

on

Dim

ensi

on

5D

ays

10

Days

20

Days

PC

sM

SE

MA

EM

SE

MA

EM

SE

MA

EM

SE

MA

EM

SE

MA

EM

SE

MA

EPanel

A.π

12

t+12−

π1 t+

1

11974-1

983

1.0

36

1.0

19

0.9

16

0.8

41

1.0

70

1.0

19

0.9

21

0.8

43

0.9

60

0.8

55

0.9

59

0.8

65

1984-2

004

1.0

75

1.0

28

1.1

74

1.0

58

1.0

93

1.0

40

1.1

56

1.0

54

1.1

29

1.0

44

1.1

29

1.0

46

All

1.0

49

1.0

23

1.0

01

0.9

46

1.0

78

1.0

29

0.9

99

0.9

46

1.0

16

0.9

46

1.0

15

0.9

52

21974-1

983

0.8

77

0.8

14

1.0

78

1.0

28

0.8

87

0.8

20

0.9

24

0.8

27

0.9

35

0.8

47

1984-2

004

1.5

72

1.1

97

1.0

79

1.0

36

1.6

82

1.2

45

1.5

63

1.2

11

1.6

53

1.2

01

All

1.1

07

1.0

00

1.0

78

1.0

32

1.1

49

1.0

27

1.1

35

1.0

13

1.1

72

1.0

19

31974-1

983

0.8

05

0.7

81

1.0

91

1.0

28

0.8

14

0.7

82

0.8

31

0.7

95

0.8

83

0.8

28

1984-2

004

1.3

00

1.0

88

1.0

78

1.0

33

1.3

70

1.1

15

1.3

31

1.1

01

1.4

44

1.1

26

All

0.9

69

0.9

30

1.0

87

1.0

30

0.9

98

0.9

44

0.9

96

0.9

43

1.0

68

0.9

72

41974-1

983

0.7

94

0.7

81

1.0

98

1.0

42

0.8

20

0.7

90

0.8

13

0.7

97

0.8

43

0.7

97

1984-2

004

1.1

02

1.0

28

1.0

64

1.0

34

1.2

16

1.0

61

1.0

71

1.0

31

1.2

45

1.0

78

All

0.8

96

0.9

01

1.0

87

1.0

38

0.9

50

0.9

22

0.8

98

0.9

11

0.9

76

0.9

33

51974-1

983

0.7

73

0.7

90

1.1

38

1.0

53

0.8

07

0.7

83

0.8

08

0.8

08

0.8

06

0.7

83

1984-2

004

1.0

81

1.0

17

1.0

45

1.0

26

1.1

71

1.0

26

1.1

13

1.0

22

1.1

99

1.0

46

All

0.8

75

0.9

00

1.1

07

1.0

40

0.9

27

0.9

01

0.9

09

0.9

12

0.9

36

0.9

11

Panel

B.π

36

t+36−

π1 t+

1

11974-1

983

0.8

27

0.9

03

0.6

37

0.7

68

0.8

70

0.9

21

0.6

92

0.8

15

0.6

77

0.7

91

0.6

56

0.7

70

1984-2

004

0.8

82

0.9

36

0.6

77

0.8

03

0.9

29

0.9

59

0.6

87

0.7

98

0.7

73

0.8

49

0.7

23

0.8

15

All

0.8

47

0.9

20

0.6

52

0.7

86

0.8

92

0.9

40

0.6

90

0.8

06

0.7

12

0.8

21

0.6

81

0.7

93

21974-1

983

0.6

62

0.7

56

0.9

36

0.9

60

0.6

84

0.7

94

0.7

25

0.7

92

0.6

28

0.7

43

1984-2

004

1.3

25

1.0

81

0.9

32

0.9

56

1.2

61

1.0

29

0.8

30

0.8

51

1.2

85

1.0

24

All

0.9

07

0.9

21

0.9

35

0.9

58

0.8

97

0.9

14

0.7

64

0.8

22

0.8

71

0.8

86

31974-1

983

0.6

91

0.7

65

0.9

43

0.9

81

0.8

04

0.8

31

0.8

54

0.8

83

0.7

32

0.7

65

1984-2

004

0.9

11

0.8

86

0.9

52

0.9

70

0.9

15

0.8

80

0.6

09

0.7

49

1.3

08

1.0

57

All

0.7

72

0.8

27

0.9

47

0.9

75

0.8

45

0.8

56

0.7

63

0.8

15

0.9

44

0.9

14

41974-1

983

0.7

33

0.7

96

0.9

09

0.9

33

0.7

59

0.8

09

0.8

80

0.8

83

0.8

03

0.7

98

1984-2

004

0.9

96

0.9

31

0.9

42

0.9

55

0.9

46

0.9

00

0.8

43

0.8

68

0.8

88

0.8

69

All

0.8

30

0.8

65

0.9

21

0.9

44

0.8

28

0.8

55

0.8

67

0.8

76

0.8

34

0.8

34

51974-1

983

0.7

97

0.8

44

0.9

56

0.9

56

0.8

01

0.8

30

0.7

69

0.7

99

0.7

58

0.7

76

1984-2

004

0.9

32

0.9

19

0.9

40

0.9

48

0.9

64

0.9

08

0.7

70

0.8

15

0.9

24

0.8

82

All

0.8

47

0.8

82

0.9

50

0.9

52

0.8

61

0.8

70

0.7

69

0.8

07

0.8

19

0.8

30

116

Tab

le3.

5:O

ut-o

f-Sa

mpl

eFo

reca

stin

gus

ing

BIC

BIC

was

used

tode

term

ine

the

num

ber

ofpr

inci

palco

mpo

nent

sfo

rπ

12

t+12−

π1 t+

1.

Rel

ativ

eM

SE

and

MA

Efo

rth

edi

ffere

ntsp

ecifi

cati

ons

are

pres

ente

d.M

SE

and

MA

Efo

rea

chsp

ecifi

cati

onis

calc

ulat

edre

lati

veto

the

AR

-typ

em

odel

.T

heco

lum

nsla

belle

d“T

erm

Stru

ctur

e”pr

esen

tre

sult

sfo

rth

em

odel

that

inco

rpor

ates

aspr

edic

tors

the

prin

cipa

lco

mpo

nent

sof

the

mon

thly

term

stru

ctur

esp

read

s.In

the

colu

mns

labe

lled

“Tim

e”,re

sult

sfo

rth

esp

ecifi

cati

onth

atin

corp

orat

esas

pred

icto

rspr

inci

palco

mpo

nent

sof

daily

mat

ched

spre

ads

are

show

n.T

hela

stth

ree

colu

mns

show

the

resu

lts

prin

cipa

lco

mpo

nent

mod

elin

corp

orat

ing

both

the

tim

ean

dte

rmst

ruct

ure

dim

ensi

ons,

wit

hei

ther

5,10

,15

,or

20da

ysof

daily

inte

rest

rate

lags

.Pan

elA

pres

ents

the

resu

lts

for

the

case

whe

reth

epr

inci

palco

mpo

nent

sw

ith

the

larg

est

in-s

ampl

eco

rrel

atio

nsbe

twee

nth

eA

Rre

sidu

als

and

the

esti

mat

edpr

inci

palc

ompo

nent

sar

ese

lect

ed.

InPan

elB

show

sth

ere

sult

sw

here

the

prin

cipa

lco

mpo

nent

sw

ith

the

high

est

vari

ance

are

sele

cted

are

pres

ente

d.Sam

ple

Ter

mStr

uct

ure

Tim

eB

oth

Dim

ensi

ons

(Tim

e&

Ter

mStr

uct

ure

)Per

iod

Dim

ensi

on

Dim

ensi

on

5D

ays

10

Days

15

Days

20

Days

MSE

MA

EM

SE

MA

EM

SE

MA

EM

SE

MA

EM

SE

MA

EM

SE

MA

EPanelA

.C

orrela

tion

Sort

π12

t+12−

π1 t+

1

1974-1

983

0.7

99

0.7

86

1.0

47

1.0

00

0.8

34

0.8

09

0.8

27

0.7

99

0.9

19

0.8

46

0.8

17

0.7

93

1984-1

990

1.4

94

1.1

97

1.1

56

1.0

45

1.6

55

1.2

43

1.5

07

1.2

20

1.8

20

1.3

51

1.9

79

1.3

41

1991-2

004

0.8

39

0.9

37

1.1

64

1.0

95

0.8

10

0.9

29

0.8

11

0.9

24

0.8

49

0.9

48

0.8

54

0.9

51

All

0.9

08

0.9

09

1.0

83

1.0

35

0.9

50

0.9

28

0.9

24

0.9

17

1.0

39

0.9

74

0.9

93

0.9

45

π36

t+12−

π1 t+

1

1974-1

983

0.7

07

0.7

68

0.8

72

0.9

12

0.7

42

0.8

04

0.8

33

0.8

46

0.7

85

0.8

21

0.7

22

0.7

61

1984-1

990

0.7

81

0.8

57

0.9

67

1.0

08

0.7

70

0.8

34

0.6

43

0.8

03

0.5

47

0.7

10

0.4

54

0.6

32

1991-2

004

1.0

60

0.9

80

0.9

60

0.9

61

1.0

69

0.9

66

1.1

52

0.9

94

1.1

72

1.0

11

1.1

24

0.9

92

All

0.7

97

0.8

51

0.9

05

0.9

45

0.8

20

0.8

60

0.8

79

0.8

84

0.8

39

0.8

60

0.7

76

0.8

09

PanelB

.V

aria

nce

Sort

π12

t+12−

π1 t+

1

1974-1

983

0.7

98

0.7

85

1.0

18

1.0

04

0.8

51

0.8

28

0.8

80

0.8

40

0.9

34

0.8

78

0.8

82

0.8

31

1984-1

990

1.4

94

1.1

97

1.1

53

1.0

35

1.7

92

1.3

03

1.5

00

1.2

25

1.9

09

1.3

49

1.9

50

1.3

44

1991-2

004

0.8

39

0.9

37

1.1

55

1.0

96

0.8

43

0.9

52

0.8

31

0.9

39

0.8

57

0.9

63

0.8

49

0.9

50

All

0.9

07

0.9

09

1.0

61

1.0

36

0.9

88

0.9

56

0.9

62

0.9

43

1.0

63

0.9

95

1.0

33

0.9

65

π36

t+12−

π1 t+

1

1974-1

983

0.7

25

0.7

85

0.8

80

0.9

17

0.8

11

0.8

25

0.7

74

0.8

03

0.7

39

0.7

66

0.7

55

0.7

69

1984-1

990

0.7

86

0.8

59

0.8

98

0.9

76

0.7

71

0.8

38

1.0

57

0.9

72

0.7

88

0.8

53

0.7

45

0.8

27

1991-2

004

0.9

47

0.9

27

0.9

60

0.9

61

0.9

63

0.9

33

0.9

64

0.9

38

0.9

62

0.9

39

0.9

45

0.9

31

All

0.7

84

0.8

43

0.9

00

0.9

42

0.8

40

0.8

62

0.8

55

0.8

77

0.7

96

0.8

37

0.7

97

0.8

30

117

Tab

le3.

6:M

odel

Ave

ragi

ng:

π12

t+12−

π1 t+

1.

See

tabl

eno

teon

the

next

page

.N

um

.Sam

ple

Ter

mStr

uct

ure

Tim

eB

oth

Dim

ensi

ons

(Tim

e&

Ter

mStr

uct

ure

)of

Per

iod

Dim

ensi

on

Dim

ensi

on

5D

ays

10

Days

20

Days

PC

sM

SE

MA

EM

SE

MA

EM

SE

MA

EM

SE

MA

EM

SE

MA

EPanel

A.C

orr

elati

on

Sort

Sim

ple

Aver

age

1to

31974-1

983

0.8

44

0.7

95

1.0

59

1.0

16

0.8

57

0.8

04

0.8

83

0.8

11

0.8

99

0.8

24

1984-1

990

1.9

05

1.3

49

1.1

16

1.0

22

2.0

10

1.3

84

1.8

68

1.3

47

2.0

81

1.4

18

1991-2

004

0.8

61

0.9

42

1.1

53

1.0

87

0.9

01

0.9

69

0.8

95

0.9

56

0.8

61

0.9

30

All

1.0

02

0.9

45

1.0

83

1.0

37

1.0

33

0.9

65

1.0

29

0.9

57

1.0

65

0.9

70

3to

51974-1

983

0.7

81

0.7

80

1.0

97

1.0

36

0.8

13

0.7

92

0.8

09

0.7

97

0.8

34

0.7

98

1984-1

990

1.6

52

1.2

68

1.0

68

1.0

05

1.8

11

1.3

00

1.6

01

1.2

45

1.8

89

1.3

20

1991-2

004

0.7

97

0.9

22

1.1

45

1.0

86

0.8

39

0.9

38

0.8

42

0.9

41

0.8

57

0.9

31

All

0.9

11

0.9

16

1.1

01

1.0

44

0.9

64

0.9

33

0.9

31

0.9

26

0.9

92

0.9

38

1to

51974-1

983

0.8

08

0.7

81

1.0

68

1.0

20

0.8

28

0.7

85

0.8

42

0.8

00

0.8

57

0.8

05

1984-1

990

1.7

32

1.2

93

1.0

85

1.0

10

1.8

62

1.3

26

1.6

96

1.2

81

1.9

23

1.3

46

1991-2

004

0.8

22

0.9

26

1.1

42

1.0

84

0.8

44

0.9

41

0.8

44

0.9

36

0.8

32

0.9

12

All

0.9

46

0.9

23

1.0

83

1.0

36

0.9

82

0.9

35

0.9

67

0.9

33

1.0

09

0.9

41

Reg

ress

ion

base

d1

to3

1974-1

983

0.7

71

0.7

77

1.0

37

1.0

01

0.8

27

0.8

11

0.7

89

0.7

92

0.8

87

0.8

31

1984-1

990

1.5

22

1.1

72

1.1

89

1.0

27

1.3

32

1.1

20

1.0

54

1.1

07

1.3

73

1.1

20

1991-2

004

0.8

54

0.9

58

1.1

70

1.0

68

1.0

27

1.0

16

1.1

76

1.1

32

1.0

73

1.0

68

All

0.8

96

0.9

06

1.0

82

1.0

25

0.9

35

0.9

29

0.8

94

0.8

97

0.9

90

0.9

54

3to

51974-1

983

0.7

44

0.7

57

1.0

60

1.0

11

0.7

70

0.7

89

0.8

20

0.7

80

0.7

99

0.8

12

1984-1

990

1.3

75

1.1

22

1.1

82

1.0

26

1.4

63

1.1

43

1.3

22

1.2

34

1.3

29

1.1

30

1991-2

004

0.9

36

0.9

67

1.1

96

1.0

75

0.8

19

0.9

27

0.8

62

0.8

63

0.9

87

0.9

98

All

0.8

69

0.8

88

1.1

02

1.0

32

0.8

80

0.8

97

0.9

01

0.8

61

0.9

09

0.9

27

1to

51974-1

983

0.8

02

0.7

95

1.0

78

1.0

13

0.8

72

0.8

47

0.8

75

0.8

21

0.8

98

0.8

61

1984-1

990

1.2

87

1.0

85

1.1

87

1.0

27

1.2

87

1.0

87

0.9

21

1.1

71

1.1

41

1.0

34

1991-2

004

0.7

68

0.9

08

1.1

78

1.0

68

0.8

29

0.9

32

0.8

87

1.1

38

0.9

56

1.0

07

All

0.8

67

0.8

84

1.1

11

1.0

32

0.9

25

0.9

18

0.8

84

0.9

27

0.9

44

0.9

36

Panel

B.V

ari

ance

Sort

Sim

ple

Aver

age

1to

31974-1

983

0.8

79

0.8

28

1.0

41

1.0

16

0.8

90

0.8

32

0.9

23

0.8

46

0.9

71

0.8

94

1984-1

990

1.3

82

1.1

48

1.1

10

1.0

18

1.3

55

1.1

39

1.2

84

1.1

03

1.4

21

1.1

74

1991-2

004

0.8

16

0.9

30

1.1

42

1.0

86

0.8

12

0.9

24

0.8

25

0.9

34

0.8

86

0.9

86

All

0.9

42

0.9

19

1.0

69

1.0

36

0.9

45

0.9

18

0.9

59

0.9

21

1.0

22

0.9

75

3to

51974-1

983

0.7

93

0.7

96

1.0

69

1.0

22

0.8

26

0.7

99

0.8

57

0.8

16

0.8

91

0.8

37

1984-1

990

1.3

74

1.1

62

1.0

96

1.0

04

1.5

17

1.2

22

1.3

16

1.1

45

1.3

60

1.1

60

1991-2

004

0.7

97

0.9

22

1.1

24

1.0

81

0.8

28

0.9

43

0.8

30

0.9

42

0.8

35

0.9

47

All

0.8

79

0.9

04

1.0

82

1.0

35

0.9

28

0.9

23

0.9

20

0.9

16

0.9

50

0.9

32

1to

51974-1

983

0.8

26

0.7

95

1.0

51

1.0

19

0.8

52

0.8

00

0.8

85

0.8

16

0.9

18

0.8

55

1984-1

990

1.4

09

1.1

68

1.1

04

1.0

11

1.4

73

1.1

98

1.3

42

1.1

46

1.3

74

1.1

57

1991-2

004

0.7

62

0.8

97

1.1

34

1.0

84

0.7

77

0.9

05

0.7

83

0.9

07

0.8

15

0.9

35

All

0.9

00

0.8

97

1.0

73

1.0

36

0.9

30

0.9

08

0.9

34

0.9

07

0.9

67

0.9

37

Reg

ress

ion

base

d1

to3

1974-1

983

0.7

39

0.7

63

1.0

28

1.0

00

0.7

42

0.7

51

0.7

92

0.7

79

0.8

45

0.8

45

1984-1

990

1.1

65

1.0

37

1.2

22

1.0

42

1.2

02

1.0

94

1.1

07

1.0

14

1.6

70

1.2

48

1991-2

004

1.0

20

1.0

44

1.1

65

1.0

70

1.1

37

1.1

00

1.1

32

1.0

90

1.0

56

1.0

37

All

0.8

50

0.8

96

1.0

80

1.0

28

0.8

77

0.9

17

0.8

97

0.9

13

1.0

02

0.9

78

3to

51974-1

983

0.7

10

0.7

30

1.0

27

0.9

85

0.7

50

0.7

59

0.7

80

0.7

84

0.8

05

0.7

91

1984-1

990

1.4

25

1.1

20

1.1

77

1.0

20

1.2

69

1.0

71

1.2

34

1.0

27

1.2

73

1.0

68

1991-2

004

0.9

39

0.9

68

1.1

43

1.0

60

0.8

52

0.9

50

0.8

63

0.9

52

0.9

09

0.9

81

All

0.8

54

0.8

74

1.0

69

1.0

13

0.8

43

0.8

74

0.8

61

0.8

79

0.8

92

0.8

99

1to

51974-1

983

0.7

69

0.7

70

1.0

76

1.0

14

0.7

66

0.7

60

0.8

21

0.8

07

0.8

04

0.8

16

1984-1

990

1.2

10

1.0

52

1.2

16

1.0

34

1.2

77

1.1

16

1.1

71

1.0

42

1.4

51

1.1

48

1991-2

004

1.0

91

1.0

57

1.1

48

1.0

59

1.1

72

1.1

13

1.1

38

1.0

95

1.1

24

1.0

79

All

0.8

89

0.9

06

1.1

09

1.0

30

0.9

10

0.9

30

0.9

27

0.9

34

0.9

54

0.9

55

118

Model Averaging: π12t+12 − π1

t+1.Relative MSE and MAE for equally weighted forecasts

using 1 to 3 principal components, 3 to 5 principal components, and 1 to 5 principal components.

All results are relative to the AR benchmark model. The columns labelled “Term Structure”

present results for the model that incorporates as predictors the principal components of the

monthly term structure spreads. In the columns labelled “Time”, results for the model that uses

daily spreads with the same maturity as inflation are presented. The last three columns show the

results principal component model incorporating both the time and term structure dimensions,

with either 5, 10, or 20 days of daily interest rate lags.

119

Tab

le3.

7:M

odel

Ave

ragi

ng:

π36

t+36−

π1 t+

1.

See

tabl

eno

teon

the

next

page

.N

um

.Sam

ple

Ter

mStr

uct

ure

Tim

eB

oth

Dim

ensi

ons

(Tim

e&

Ter

mStr

uct

ure

)of

Per

iod

Dim

ensi

on

Dim

ensi

on

5D

ays

10

Days

20

Days

PC

sM

SE

MA

EM

SE

MA

EM

SE

MA

EM

SE

MA

EM

SE

MA

EPanel

A.C

orr

elati

on

Sort

Sim

ple

Aver

age

1to

31974-1

983

0.6

23

0.7

39

0.9

16

0.9

47

0.7

79

0.8

28

0.7

08

0.7

97

0.6

35

0.7

34

1984-1

990

0.5

22

0.7

45

0.9

96

1.0

29

0.4

98

0.7

15

0.6

76

0.7

88

0.5

37

0.7

26

1991-2

004

1.1

82

1.0

41

0.9

59

0.9

58

1.0

49

0.9

74

0.7

15

0.8

13

1.4

19

1.1

27

All

0.7

35

0.8

35

0.9

37

0.9

66

0.8

01

0.8

53

0.7

05

0.8

01

0.7

98

0.8

56

3to

51974-1

983

0.7

40

0.7

98

0.9

38

0.9

48

0.7

66

0.8

06

0.8

10

0.8

34

0.7

51

0.7

67

1984-1

990

0.6

67

0.7

84

1.0

14

1.0

33

0.6

40

0.7

53

0.5

51

0.7

56

0.5

79

0.7

28

1991-2

004

1.1

46

1.0

09

0.9

60

0.9

47

1.1

00

0.9

71

0.8

38

0.8

63

1.3

11

1.0

73

All

0.8

22

0.8

62

0.9

53

0.9

64

0.8

24

0.8

48

0.7

81

0.8

28

0.8

54

0.8

56

1to

51974-1

983

0.6

65

0.7

65

0.9

17

0.9

39

0.7

39

0.8

06

0.7

20

0.8

00

0.6

67

0.7

42

1984-1

990

0.5

78

0.7

70

0.9

95

1.0

26

0.5

29

0.7

23

0.5

76

0.7

56

0.4

87

0.6

80

1991-2

004

1.1

49

1.0

25

0.9

56

0.9

52

1.0

39

0.9

61

0.8

11

0.8

61

1.2

93

1.0

77

All

0.7

62

0.8

48

0.9

37

0.9

59

0.7

78

0.8

39

0.7

21

0.8

11

0.7

84

0.8

36

Reg

ress

ion

base

d1

to3

1974-1

983

0.5

84

0.7

08

0.9

06

0.9

26

0.8

16

0.7

79

0.6

56

0.5

69

0.5

92

0.6

92

1984-1

990

0.4

75

0.6

89

1.0

50

1.0

42

0.3

94

0.6

05

0.7

71

1.5

68

0.5

47

0.7

28

1991-2

004

0.7

85

0.8

57

0.9

17

0.9

33

0.9

76

0.9

47

0.9

13

1.0

54

1.2

21

1.0

52

All

0.6

14

0.7

51

0.9

28

0.9

50

0.7

94

0.7

99

0.7

29

0.8

15

0.7

28

0.8

12

3to

51974-1

983

0.6

56

0.7

23

0.9

49

0.9

34

0.7

89

0.7

89

0.8

14

0.7

75

0.7

38

0.7

38

1984-1

990

0.7

70

0.8

60

1.0

97

1.0

55

0.6

76

0.7

55

0.5

02

1.1

54

0.7

79

0.8

78

1991-2

004

1.2

50

1.0

19

0.9

08

0.9

12

1.0

91

0.9

60

0.6

40

0.9

80

1.2

14

1.0

19

All

0.8

06

0.8

42

0.9

60

0.9

50

0.8

42

0.8

36

0.7

32

0.8

73

0.8

51

0.8

53

1to

51974-1

983

0.5

64

0.6

83

0.8

89

0.9

04

0.7

84

0.7

42

0.6

21

0.5

48

0.6

19

0.7

03

1984-1

990

0.4

48

0.6

62

1.0

97

1.0

44

0.4

03

0.5

96

0.5

60

0.9

92

0.4

92

0.6

85

1991-2

004

0.8

04

0.8

61

0.9

17

0.9

23

0.9

41

0.9

31

0.8

31

1.1

43

1.0

54

0.9

84

All

0.6

02

0.7

35

0.9

24

0.9

36

0.7

67

0.7

74

0.6

60

0.7

43

0.7

00

0.7

88

Panel

A.Vari

ance

Sort

Sim

ple

Aver

age

1to

31974-1

983

0.8

29

0.8

80

0.8

69

0.9

11

0.8

66

0.9

03

0.8

67

0.8

97

0.9

62

0.9

41

1984-1

990

0.9

71

0.9

73

0.8

88

0.9

76

1.0

28

0.9

91

1.1

23

1.0

21

1.0

92

0.9

64

1991-2

004

0.8

28

0.8

74

0.9

60

0.9

59

0.8

29

0.8

76

0.8

24

0.8

74

0.7

60

0.8

10

All

0.8

48

0.8

96

0.8

92

0.9

38

0.8

80

0.9

11

0.8

92

0.9

13

0.9

34

0.9

04

3to

51974-1

983

0.7

87

0.8

25

0.8

97

0.9

22

0.8

30

0.8

38

0.7

82

0.8

11

0.7

89

0.8

07

1984-1

990

0.9

68

0.9

40

0.8

87

0.9

79

0.9

70

0.9

37

1.1

87

1.0

38

1.1

90

1.0

40

1991-2

004

1.0

20

0.9

68

0.9

62

0.9

56

0.9

79

0.9

49

0.9

62

0.9

45

0.8

31

0.8

83

All

0.8

64

0.8

92

0.9

10

0.9

43

0.8

82

0.8

92

0.8

78

0.8

96

0.8

54

0.8

75

1to

51974-1

983

0.8

12

0.8

61

0.8

78

0.9

14

0.8

40

0.8

72

0.8

07

0.8

52

0.8

21

0.8

49

1984-1

990

0.8

32

0.8

98

0.8

86

0.9

78

0.8

59

0.9

02

0.9

91

0.9

57

1.0

52

0.9

75

1991-2

004

0.8

67

0.9

03

0.9

60

0.9

57

0.8

48

0.8

92

0.8

43

0.8

90

0.7

94

0.8

60

All

0.8

27

0.8

81

0.8

97

0.9

40

0.8

44

0.8

84

0.8

40

0.8

84

0.8

47

0.8

76

Reg

ress

ion

base

d1

to3

1974-1

983

0.5

53

0.6

47

0.8

79

0.9

11

0.5

55

0.6

67

0.5

69

0.6

72

0.8

22

0.8

45

1984-1

990

1.3

14

1.1

69

0.9

72

1.0

00

1.4

87

1.2

32

1.5

68

1.2

83

0.7

64

0.8

79

1991-2

004

1.1

16

1.0

18

0.9

30

0.9

36

1.1

05

1.0

14

1.0

54

0.9

98

0.8

63

0.8

88

All

0.7

84

0.8

62

0.9

04

0.9

36

0.8

06

0.8

83

0.8

15

0.8

90

0.8

23

0.8

65

3to

51974-1

983

0.6

18

0.7

09

0.9

45

0.9

20

0.7

87

0.7

98

0.7

75

0.7

74

0.7

79

0.7

35

1984-1

990

1.7

36

1.1

92

0.9

39

1.0

01

1.3

01

1.0

12

1.1

54

0.9

75

0.7

36

0.7

77

1991-2

004

1.1

26

1.0

00

0.9

52

0.9

48

0.9

21

0.9

10

0.9

80

0.9

22

0.8

91

0.8

90

All

0.8

85

0.8

91

0.9

46

0.9

44

0.8

87

0.8

73

0.8

73

0.8

58

0.7

98

0.7

91

1to

51974-1

983

0.4

92

0.6

21

0.9

15

0.9

09

0.5

43

0.6

42

0.5

48

0.6

67

0.4

88

0.6

09

1984-1

990

1.4

26

1.2

03

0.9

97

1.0

09

1.2

56

1.1

09

0.9

92

0.9

73

1.0

28

0.9

64

1991-2

004

1.2

21

1.0

41

0.9

46

0.9

42

1.1

44

1.0

00

1.1

43

0.9

88

1.1

39

1.0

08

All

0.7

84

0.8

63

0.9

33

0.9

38

0.7

76

0.8

43

0.7

43

0.8

25

0.7

09

0.8

01

120

Model Averaging: π36t+36 − π1

t+1. Relative MSE and MAE for equally weighted forecasts using

1 to 3 principal components, 3 to 5 principal components, and 1 to 5 principal components. All results

are relative to the AR benchmark model. The columns labelled “Term Structure” present results for the

model that incorporates as predictors the principal components of the monthly term structure spreads.

In the columns labelled “Time”, results for the model that uses daily spreads with the same maturity as

inflation are presented. The last three columns show the results principal component model incorporating

both the time and term structure dimensions, with either 5, 10, or 20 days of daily interest rate lags.

121

Res

ult

sfo

rth

eD

iebold

Mari

ano

test

stati

stic

for

regre

ssio

nbase

dav

erage

fore

cast

susi

ng

1to

3pri

nci

palco

mponen

ts,3

to5

pri

nci

pal

com

ponen

ts,and

1to

5pri

nci

palco

mponen

tsare

pre

sente

d.

All

resu

lts

are

rela

tive

toth

eA

Rben

chm

ark

model

.T

he

colu

mns

label

led

“Ter

mStr

uct

ure

”pre

sent

resu

lts

for

the

model

that

inco

rpora

tes

as

pre

dic

tors

the

pri

nci

palco

mponen

tsofth

em

onth

lyte

rmst

ruct

ure

spre

ads.

Inth

eco

lum

ns

label

led

“T

ime”

,re

sult

sfo

rth

em

odel

that

use

sdaily

spre

ads

wit

hth

esa

me

matu

rity

as

inflati

on

are

pre

sente

d.

Fin

ally,

the

colu

mns

label

led

“5

day

s”,“10

day

s”and

“20

day

s”pre

sent

resu

lts

for

the

model

that

inco

rpora

tes

pri

nci

palco

mponen

tsofpre

dic

tors

that

inco

rpora

teboth

dim

ensi

ons:

Ter

m-S

truct

ure

dim

ensi

on

and

Tim

edim

ensi

on.

Tab

le3.

8:D

iebo

ldM

aria

noTes

tson

Reg

ress

ion

base

dM

odel

Ave

rage

s:π

12−

π1.

Lags

Num

.Ter

mT

ime

5D

ays

10

Days

20

Days

NW

est

ofP

CStr

uct

ure

Panel

A.π

12−

π1

Corr

elati

on

Sort

31

to3

7.0

87.3

27.4

46.7

17.2

13

to5

7.2

37.4

37.3

37.2

47.6

01

to5

7.1

87.3

47.2

86.4

17.3

96

1to

35.9

26.4

16.3

45.7

06.0

73

to5

6.1

76.4

76.2

96.1

36.4

01

to5

6.0

96.4

16.2

55.4

86.2

111

1to

35.0

95.7

45.5

54.9

95.2

63

to5

5.3

35.7

35.5

25.3

15.4

91

to5

5.2

55.7

25.5

04.8

15.3

5Vari

ance

Sort

31

to3

6.6

77.2

46.8

16.6

67.4

83

to5

7.2

77.2

46.9

56.7

36.5

81

to5

6.9

37.1

96.7

36.3

56.8

46

1to

35.1

55.9

15.2

65.1

35.9

33

to5

5.6

75.8

85.3

55.1

04.9

81

to5

5.4

45.8

45.2

34.8

85.2

311

1to

34.3

95.2

24.4

84.3

55.0

43

to5

4.8

55.1

84.5

84.3

54.2

51

to5

4.6

55.1

64.4

54.1

34.4

3

122

Results for the Diebold Mariano test statistic for regression based average forecasts using 1 to 3 principalcomponents, 3 to 5 principal components, and 1 to 5 principal components are presented. All results are

relative to the AR benchmark model. The columns labelled “Term Structure” present results for themodel that incorporates as predictors the principal components of the monthly term structure spreads.In the columns labelled “Time”, results for the model that uses daily spreads with the same maturity asinflation are presented. Finally, the columns labelled “5 days”, “10 days” and “20 days” present resultsfor the model that incorporates principal components of predictors that incorporate both dimensions:

Term-Structure dimension and Time dimension.

Table 3.9: Diebold Mariano Tests on Regression based Model Averages: π36 − π1

Lags Num. Term Time 5 Days 10 Days 20 DaysNWest of PC StructurePanel A. π36 − π1

Correlation Sort3 1 to 3 8.27 7.93 8.14 8.55 9.10

3 to 5 8.50 8.30 7.77 7.71 8.181 to 5 8.27 8.14 7.15 7.65 8.79

9 1 to 3 5.79 5.55 5.85 5.91 6.343 to 5 6.10 5.82 5.65 5.47 5.691 to 5 5.88 5.68 5.20 5.44 6.05

15 1 to 3 4.91 4.72 4.94 4.97 5.343 to 5 5.26 4.93 4.86 4.58 4.801 to 5 5.02 4.81 4.42 4.57 5.05

20 1 to 3 4.49 4.36 4.52 4.53 4.873 to 5 4.86 4.52 4.46 4.13 4.411 to 5 4.60 4.43 4.04 4.11 4.59

30 1 to 3 4.05 4.03 4.07 4.06 4.413 to 5 4.44 4.12 4.00 3.64 4.011 to 5 4.13 4.05 3.60 3.61 4.11

35 1 to 3 3.91 3.93 3.91 3.90 4.263 to 5 4.27 3.99 3.83 3.47 3.871 to 5 3.96 3.94 3.45 3.45 3.96

Variance Sort3 1 to 3 6.86 8.21 6.92 6.99 8.08

3 to 5 7.82 8.12 8.22 7.27 6.881 to 5 7.04 8.23 6.77 6.74 6.63

9 1 to 3 4.78 5.77 4.79 4.84 5.623 to 5 5.64 5.69 5.76 5.08 4.791 to 5 4.96 5.77 4.70 4.71 4.61

15 1 to 3 4.03 4.91 4.05 4.08 4.773 to 5 4.86 4.84 4.85 4.33 4.061 to 5 4.20 4.91 3.98 4.01 3.90

20 1 to 3 3.67 4.52 3.71 3.72 4.413 to 5 4.46 4.46 4.44 3.97 3.721 to 5 3.82 4.53 3.64 3.69 3.57

30 1 to 3 3.28 4.16 3.36 3.35 4.043 to 5 4.01 4.09 3.99 3.59 3.381 to 5 3.41 4.16 3.29 3.36 3.24

35 1 to 3 3.16 4.05 3.25 3.24 3.913 to 5 3.84 3.97 3.84 3.48 3.281 to 5 3.26 4.05 3.19 3.26 3.15

123

Figure 3.1: Inflation

Apr61 Feb66 Dec70 Oct75 Aug80 Jun85 Apr90 Feb95 Dec99 Oct040

5

10

15Annualized 12−month Inflation

Apr61 Jun65 Aug69 Oct73 Dec77 Feb82 Apr86 Jun90 Aug94 Oct98 Dec020

5

10

15Annualized 36−month Inflation

124

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