1

Empirical Tests of Asset Pricing Models with Individual Assets:

Resolving the Errors-in-Variables Bias in Risk Premium Estimation

by

Narasimhan Jegadeesh, Joonki Noh,

Kuntara Pukthuanthong, Richard Roll, and Junbo Wang

September 15, 2015

Abstract

To attenuate an inherent errors-in-variables bias, portfolios are widely employed for risk

premium estimation; but portfolios might diversify away and thus mask relevant risk- or

return-related features of individual assets. We propose a resolution that allows the use of

individual assets while avoiding the bias. It hinges on specific instrumental variables, factor

sensitivities (’s) calculated from alternate observations. Closed-form asymptotics are

provided for large cross-sections and time-series. Simulations indicate that the IV method

delivers unbiased risk premium estimates and well-specified tests with adequate power in

small samples. Empirical implementation finds some evidence of significant risk premiums

for the size and book-to-market. However, when controlling for non-β characteristics,

estimated risk premiums are insignificant for the CAPM, size, book-to-market,

investments, profitability, and the liquidity-adjusted CAPM.

Co-Author Affiliation Voice E-Mail

Jegadeesh Emory University

Atlanta GA 30322 404-727-4821

Jegadeesh@

Emory.Edu

Noh Case Western Reserve University

Cleveland OH 44106 216-368-3737

Joonki.Noh@

Case.Edu

Pukthuanthong University of Missouri

Columbia MO 65211 619-807-6124

PukthuanthongK@

Missouri.Edu

Roll Caltech

Pasadena CA 91125 626-395-3890 RRoll@Caltech.Edu

Wang Louisiana State University

Baton Rouge LA 70808 781-258-8806 junbowang@lsu.edu

Key Words: Risk Premium Estimation, Errors-in-Variables Bias, Instrumental Variables,

Individual Stocks, Asset Pricing Models

2

1. Introduction

A fundamental precept of financial economics is that investors earn higher average

returns by bearing systemic risks. While this idea is well accepted, there is little agreement

about the identity of systematic risks or the magnitude of the supposed rewards. This is not

due to a lack of effort along two lines of enquiry. First, numerous candidates have been

proposed as underlying risk factors. Second, empirical efforts to estimate risk premiums

also have a long and varied history.

Starting with the single-factor CAPM (Sharpe, 1964; Lintner, 1965), and the multi-

factor APT, Ross (1976), the first line of enquiry has brought forth an abundance of risk

factor candidates. Among others, these include the Fama and French size and book-to-

market factors, human capital risk (Jagannathan and Wang, 1996), productivity and capital

investment risk (Cochrane, 1996; Chen, Novy-Marx and Zhang, 2011; Eisfeldt and

Papanikolaou, 2013), different components of consumption risk (Lettau and Ludvigson,

2001; Ait-Sahalia, Parker, and Yogo, 2004; Li, Vassalou, and Xing, 2006), cash flow and

discount rate risks (Campbell and Vuolteenaho, 2004) and illiquidity risk (Pastor and

Stambaugh, 2003; Acharya and Pedersen, 2005). Harvey, Liu and Zhu (2015) survey the

literature and report that more than 300 factors have been proposed.

The second line of enquiry has produced empirical estimates of risk premiums for

many among, what Cochrane (2011) terms as “zoo” of factors. Most estimation methods

have followed those originally introduced by Black, Jensen and Scholes (1972), (BJS), and

refined by Fama and Macbeth (1973), (FM). Their most prominent feature is the use of

portfolios rather than individual assets in testing asset pricing models. This has long been

considered essential because of an error-in-variables (EIV) problem inherent in estimating

risk premiums.

The EIV problem is best appreciated by tracing through the BJS and FM methods. It

involves two-pass regressions: the first pass is a time series regression of individual asset

returns on the proposed factors. This pass provides estimates of factor loadings, widely

called “betas” in the finance literature.1 The second pass regresses asset returns cross-

1 Hereafter, we will adopt the shorthand nomenclature “Beta” to mean “factor sensitivity” or “factor

3

sectionally on the betas obtained from the first pass regression. Since the explanatory

variables in the second pass are estimates, rather than the true betas, the resulting risk

premium estimates are biased and inconsistent; and the direction of the bias is unknown

when there are multiple factors involved in the two-pass regressions.

With a large number (=N) of individual assets, the EIV bias can be reduced by working

with portfolios rather than individual assets. This process begins by forming diversified

portfolios classified by some individual asset characteristic such as a beta estimated over a

preliminary sample period. It then estimates portfolio betas on the factors using data for

second period. Finally it runs the cross-sectional regressions on estimated portfolio betas

using data for a third period. BJS, Blume and Friend (1973), and FM note that portfolios

have less idiosyncratic risk; so the errors-in-variables bias is reduced (and can be entirely

eliminated as N grows without bound).

But using portfolios, rather than individual assets, has its own defects. There is an

immediate issue of test power since dimensioniality is reduced; i.e., there are unavoidably

fewer explanatory variables with portfolios than with individual assets.

Diversification into portfolios can mask cross-sectional phenomena in individual

assets that are unrelated to the portfolio grouping procedure. For example, advocates of

fundamental indexation (Arnott, Hsu and Moore (2005)) argue that high market value

assets are overpriced and vice versa, but any portfolio grouping by an attribute other than

market value itself could diversify away such mispricing, rendering it undetectable.

Another troubling result of portfolio masking involves the cross-sectional relation

between average returns and factor exposures (“betas”). Take the single-factor CAPM as

an illustration (though the same effect is at work for any linear factor model). The cross-

sectional relation between expected returns and betas holds exactly if and only if the market

index used for computing betas is on the mean/variance frontier of the individual asset

universe. Errors from the beta/return line, either positive or negative, imply that the index

is not on the frontier. But if the individual assets are grouped into portfolios sorted by

portfolio beta and the individual errors are not related to beta, the analogous line fitted to

loading.”

4

portfolio returns and betas will display much smaller errors. This could lead to a mistaken

inference that the index is on the efficient frontier.

Test portfolios are typically organized by firm characteristics related to average

returns, e.g., size and book-to-market. Sorting on characteristics that are known to predict

returns helps generate a reasonable variation in average returns across test assets. But

Lewellen, Nagel, and Shanken (2010) point out sorting on characteristics also imparts a

strong factor structure across test portfolios. Lewellen et al. (2010) show that as a result

even factors that are weakly correlated with the sorting characteristics would explain the

differences in average returns across test portfolios regardless of the economic merits of

the theories that underlie the factors.

Finally, the statistical significance and economic magnitudes of risk premiums could

depend critically on the choice of test portfolios. For example, the Fama and French size

and book-to-market risk factors are significantly priced when test portfolios are sorted

based on corresponding characteristics, but they do not command significant risk premiums

when test portfolios are sorted only based on momentum.

In an effort to overcome the deficiencies of portfolio grouping while avoiding the EIV

bias, we develop a new procedure to estimate risk premiums and to test their statistical

significance using individual assets. Our method adopts the instrumental variables

technique, a standard econometric solution to the EIV problem. We define a particular

set of well-behaved instruments and hereafter refer to our approach as the IV method.

To be specific, our IV method first estimates betas for individual assets from a portion

of the observations available in the data sample. These become the “independent” variables

for the second-stage cross-sectional regressions. Then, using completely different sample

observations, it re-estimates the same betas, which become the “instrumental” variables in

the second-stage cross-sectional regressions.

We explore several variants of this basic scheme. One variant estimates betas and beta

instruments using observations from sequential subsamples; betas from the first T

observations and beta instruments from observations T+1 to 2T. A cross-sectional

regression then is run for observations in 2T+1. The entire procedure is rolled forward by

5

one period and repeated up until the last available observation, thereby generating a time

series of risk premium estimates for statistical testing.

Another variant of the basic scheme is to estimate betas from observations in even-

numbered periods, e.g., months 2, 4, …2T, and beta instruments from observations in odd-

numbered periods, e.g., months 1, 3, … 2T-1. The second-stage cross-sectional regressions

are then run using returns in 2T+1. The roles of betas and their instruments are

interchangeable in the second-stage cross-sectional regressions.

The IV method produces N-consistent risk premium estimates for a finite length of

time-series.2 In a more general case that allows both the size of cross-section N and the

length of time-series T to grow without bounds, we prove that the IV method provides NT-

consistent risk premium estimates. We also develop the asymptotic distributions of the IV

estimator for those two different scenarios of N and T.3

While large sample properties can provide some guidance, it is important to examine

the small sample performance of various estimators for practical applications. To do so,

we conduct a number of simulation experiments. We choose simulation parameters

matched to those in the actual data. Simulation results verify that the IV method produces

unbiased risk premium estimates even for relatively short time-series used to estimate

factor sensitivities. In contrast, we find that the standard approach that fits the the second

stage regressions using OLS (hereafter we will refer to this standard approach as the OLS

method) suffers from severe EIV biases. The simulations also show that the root-mean-

squared errors of the IV method are substantially lower than those of the OLS method. For

example, in simulations with a single factor model under time-constant factor sensitivities,

we find that the OLS estimator, if used with individual stocks, is significantly biased

toward zero even when betas are estimated with 2640 time series observations. In contrast,

the IV estimator yields nearly unbiased risk premium estimates when only 264 time-series

observations are available (see Figure 1).

2 According to Shanken (1992), N-consistent risk premium estimator converges to the ex-post risk

premium as N goes to infinity for fixed T. 3 In our simulations and empirical analyses, we employ a truncated version of the IV estimator as an

adjustment for finite N. For detailed discussion, see Section 3.1.

6

In terms of test size (i.e., type I error) and power (i.e., type II error), we find that the

conventional t-tests based on the IV estimator are well specified (under the null hypothesis

that true risk premiums are zero) and they are reasonably powerful (under the alternative

hypothesis that true risk premiums equal the sample means of factor realizations) in small

samples for estimating betas. For the Fama-French three-factor models, similar results are

found.

With actual data, we apply the IV method to estimate the risk premiums for several

risk factors proposed in the literature, which include the CAPM, the three-factor and five-

factor models of Fama and French (1993 and 2014), the q-factor asset pricing model of

Hou, Xue, and Zhang (2014), and the liquidity-adjusted capital asset pricing model

(LCAPM) of Acharya and Pedersen (2005). These risk factors have been empirically

successful when they were tested with portfolios. In contrast to the original papers, when

controlling for corresponding non-β characteristics, we find that none of these factors is

associated with a significant risk premium in the cross-section of individual stock returns.

This failure to find significant risk premiums is not due to the lack of test power of the

IV method. We present simulation evidence that the t-tests based on the IV method provide

reasonably high power under the alternative hypotheses that the true risk premiums equal

the sample means of factor realizations. For example, when the true HML risk premium is

positive, the rejection rates of the null hypothesis (i.e., zero risk premium for HML) are

84.2% and 89.6% under time-constant and time-varying factor sensitivities, respectively.

When analyzing real data, in the absence of non-β characteristics, we find some evidence

that SMB and HML command significant risk premiums in the cross-section of individual

stocks returns. However, this pricing evidence of SMB and HML betas is substantially

weakened when corresponding non-β characteristics are included in the cross-sectional

regressions. This stark difference in pricing evidence without and with non-β

characteristics indicates that insignificant risk premiums are not due to the lack of test

power of the IV method.

Our paper also contributes to a large literature on testing asset pricing models. As the

length of time-series grows indefinitely, Shanken (1992) shows that the EIV bias becomes

7

negligible because the estimation accuracy of betas improves. He also derives an

asymptotic adjustment for the FM standard errors of the OLS method. Jagannathan and

Wang (1998) extend the Shanken’s asymptotic analysis to the case of conditionally

heterogeneous errors in the time series regression. Shanken and Zhou (2007) and Kan,

Robotti and Shanken (2013) extend the result to misspecified models. The evidence and

analyses in those papers mainly focus on test portfoilos. Our paper focuses on individual

stocks as test assets and proposes the IV method to mitigate the EIV bias in testing asset

pricing models, which is likely more severe with individual stocks than with portfolios.

Accordingly, we provide a “double” asymptotic theory of the IV estimator, in which the

size of cross-section and the length of time-series grow simultaneously without bounds.

The double asymptotics reflects a recent development in the econometrics, e.g., Bai (2003)

and it is more appropriate for individual stocks than single asymptotics.4

Using individual stocks in testing asset pricing models is a recent development in the

literature. Kim (1995) corrects the EIV bias using lagged betas to derive a closed-form

solution for the MLE estimator of market risk premium. The solution proposed by Kim is

based on the adjustment by Theil (1971). Other methods proposed by Litzenberger and

Ramaswamy (1979), Kim and Skoulakis (2014), and Chordia et al. (2015) are similar,

producing the EIV correction terms to obtain N-consistent risk premium estimators. In

contrast, the IV method does not require any correction term for the EIV bias. To avoid the

EIV bias, Brennan et al. (1998) advocate risk-adjusted returns as dependent variable in the

second-stage regressions. However, their method does not estimate the risk premiums of

factors. None of these existing papers provides double asymptotic theories as in our paper.

2. Risk-Return Models and IV Estimation

4 Existing papers in the literature analyze “single” asymptotic theories, i.e., one of two dimensions (cross-

section and time) goes to infinity while the other dimension is fixed. A notable exception is Gagliardini

et al. (2011) who show that the EIV bias in the BJS method converges to zero when N and T grow to

infinity.

8

2.1. Multifactor Asset Pricing Models

A number of asset pricing models predict that expected returns on risky assets are

linearly related to their covariances with certain risk factors. A general specification of a

K-factor asset pricing model can be written as:

,γβγ)r( k

K

1k

i

k0

i

E (2.1)

where )r( iE is the expected excess return on stock i,i

kβ is the sensitivity of stock i to

factor k, and kγ is the risk premium on factor k. 0γ is the excess return on the zero-beta

asset. If riskless borrowing and lending are allowed, then the zero-beta asset earns the risk-

free rate and its excess return is zero, i.e. 0γ0 . The CAPM predicts that only the market

risk will be priced in the cross-section. Several multifactor models identify additional risk

factors based on empirical findings or based on variations of models such as ICAPM. APT

also predicts a multifactor pricing model starting with a factor structure for returns.

Empirical tests of asset pricing models typically use the Fama-MacBeth (FM) two-

stage regression procedure to estimate factor risk premiums. The first stage estimates factor

sensitivities using the following time-series regressions with T periods of data:

,fβαr i

ttk,

K

1k

i

k

ii

t ε

(2.2)

where tk,f is the realization of factor k in time t, i

tε is the regression residual. We will

assume that the factors and residuals are stationary process, the residuals are cross-sectional

and time-series uncorrelated, and uncorrelated with factors. The time series estimates of

factor sensitivities, say i

kβ̂ are the independent variables in the following second-stage

cross-sectional regressions used to estimate factor risk premiums:

,γβ̂γr i

ttk,

K

1k

i

kt0,

i

t ξ

(2.3)

where excess return i

tr is the dependent variable. The standard FM approach fits OLS

regression to estimate the parameters of regression (2.3). These OLS estimates are biased

due to the EIV problem since i

kβ̂ s are estimated with errors. To mitigate such bias, the

9

literature typically uses selected portfolios as test assets rather than individual stocks since

portfolio betas are estimated more precisely than individual stock betas.

The use of test portfolios, however, presents a different set of problems. The test

portfolios are typically sorted on a few characteristics such as size and book-to-market.

Sorting on characteristics known to predict returns helps generate a reasonable variation in

average returns across test assets. Lewellen, Nagel, and Shanken (2010) point out sorting

on characteristics also imparts a strong factor structure across the test portfolios. They show

that as a result even factors that are weakly correlated with the sorting characteristics would

explain the differences in average returns across the test portfolios regardless of the

economic merits of the theories that underlie the factors.

Moreover, the statistical significance and economic magnitudes of risk premiums

estimated using regression (2.3) could critically depend on the choice of tests portfolios.

For example, the Fama and French size and book-to-market risk factors are significantly

priced when test portfolios are sorted based on these characteristics, but they do not

command significant risk premiums if test portfolios are sorted only based on momentum.

This paper proposes a method that uses individual stocks as test assets, which addresses

these problems. The use of individual assets preserves the dimensionality of the variation

in expected returns that we observe in the stock market. Also, since our tests use all listed

stocks individually (after reasonable criteria for screening individual stocks), the results

are not dependent on subjective choices made to construct the test portfolios.

2.2. Instrumental Variables Estimator

Define ]ˆ1;[1ˆ evene ββ and ]ˆ;1[1ˆ oddo ββ where evenβ̂ and oddβ̂ ( NK

matrix) are estimated factor senstivities (loadings) based on even and odd months,

respectively. “^” indicates an estimate. 1 is N1 vector whose entities are 1 and the

operator “;” is to stack the first row vector on top of second matrix. Similarly, we define

]ˆ 1;[1ˆ IVIV ββ and ]ˆ 1;[1ˆ EVEV ββ where the “IV” subscript indicates the beta

10

instruments and the “EV” subscript denotes the corresponding explanatory variables,

respectively. Rewrite regression (2.3) in matrix form as:

tsamplet 1ˆˆ ξβγr

where tr is the N1 vector of realized excess returns in month t, γ̂ is the 1)(K1

vector of risk premiums, sample1β̂ is the N1)(K matrix that contains the intercept and

K factor loadings, and tξ denotes the N1 vector of return residuals. We then propose

the following IV estimator for risk premiums in month t:

,)'1ˆ()'1ˆ1ˆ(='ˆtIV

1

EVIVt rβββγ

where IV1β̂ ( EV1β̂ ) can be either e1β̂ (even-month betas) or o1β̂ (odd-month betas).

For example, if IV1β̂ contains odd-month betas, EV1β̂ would have even-month betas,

and vice-versa.

In principle, we can use two sets of betas estimated over any non-overlapping periods

as instruments and independent variables. In our empirical tests, we employ even- and odd-

month betas so that they are estimated within the same overall sample period using about

the same number of time-series observations. In addition, we use odd-month beta estimates

as instruments when month t is even and the even-month beta estimates as instruments

when month t is odd.

3. Asymptotics, Adjustment for Finite N, and Standard Errors

The proposed IV estimator in month t is:

'

tIV

1

EVIVt 1ˆ)'1ˆ1ˆ(ˆ rβββγ (3.1)

Let 'γ̂ be the sample average of 'ˆtγ over the sample period. Proposition 1 below shows

the N-consistency and NT-consistency of 'γ̂ . We also provide the asymptotic distributions

of the IV estimator for two different secnarios of N and T interaction.

11

Proposition 1: Suppose that asset returns follow a factor structure described as in

equation (2.1). Assume that (1) The residual process ],,,[ N

s

2

s

1

ss εεε ε is stationary.

The elements in sε are cross-sectionally uncorrelated, and sε and tε are

uncorrelated when s is not equal to t. (2) Factor process also is stationary. If the number

of individual assets in the cross-section is sufficiently large then, under mild regularity

conditions, the sample average of the estimator in equation (3.1), 'γ̂ , is N-consistent for

any finite T and it is also NT-consistent when both N and T go to infinity simultaneously.

The asymptotic distributions for N-consistent and NT-consistent IV estimators can be

derived.5

Proof: See Appendix 1.

In contrast, the standard OLS estimator is biased toward zero as N goes to infinity when

T is fixed. It is straightforward to show that both IV and OLS estimators converge to the

true risk premiums as T goes to infinity when N is fixed.

3.1. Adjustment for Finite Number of Stocks (N)

As the number of assets grows infinitely, we show that the IV estimator is normally

distributed. However, since the cross-product term of IV1β̂ and EV1β̂ in equation (3.1)

might not be positive definite when N is finite, there is tiny but still positive probability

that the IV estimator has an unreasonably large value, which can make its moments not

exist. To avoid this ill-behaved property for finite N, we truncate the IV estimator based

on the sample means and standard deivations of risk factors. Shanken and Zhou (2007) also

employ a truncation for ML estimator of risk premiums when portfolios are employed as

test assets. Specially, suppose that tj,γ̂ is the risk premium estimate of factor j (j=1,…,K)

5 Shanken (1992) defines a risk-premium estimator as N-consistent if for a finite T the estimator converges

to the realized risk premium in the sample as N (the number of stocks in the cross-section) increases

indefinitely.

12

in tγ̂ in month t. We treat tj,γ̂ as a missing value if the deviation of tj,γ̂ from the

sample mean of factor j realizations is greater than six times of their sample standard

deviation. Excluding the extreme values of tj,γ̂ ensures that all moments of the IV

estimator exist even for finite N.6 For all results in our simulations and empirical analyses,

we employ the truncated IV estimator.

3.2. Standard Errors for Risk Premium Estimates

Equation (3.1) gives the IV risk premium estimate for one cross-section at time t. The

rolling beta method pioneered by FM re-estimates betas for a sequence of successive time

period samples and repeats the cross-sectional regression in each time period. The

analogous IV procedure is simply to repeat the regression in (3.1) for a sequence of periods,

and then takes the averages of the cross-sectional coefficients. These averages produce the

final point estimates of the risk premiums. To assess their statistical significance, we need

to compute their standard errors.

In Appendix 1, we derive the asymptotic distributions of the IV estimator and the

corresponding asymptotic covariance matrices for two different scenarios: 1) when N goes

to infinity for a fixed T and 2) when N and T go to infinity together. The first scenario is

the standard in the literature and consistent with the aysmptotic analysis in Shanken (1992).

The double asymptotics in the second scenario is novel in the literature and related to the

recent development in the econometrics literature, e.g., among others, see Bai (2003). One

can employ these theoretical asymptotic covariance matrices to compute the standard errors

of IV risk premium estimates. A reasonable alternative is the sample covariance of the IV

risk premium estimates as in the standard FM procedure. We call them IV-FMSE and OLS-

FMSE, respectively, and they are straightforward to compute.

In our simulations, when tested under the CAPM and Fama-French three-factor model,

6 The 𝑚-th population moment of random variable x is defined as f(x)dx,x)x( mm

E where

f(x) is the probability density function of x . If f(x) has positive values only for a finite interval,

then )x( mE exists for all m.

13

we find that IV-FMSEs are fairly close to the root-mean-squared errors (RMSEs) of the IV

estimator, which capture the variability of risk premium estimates relative to the true risk

premiums. In constrast, it turns out that OLS-FMSEs significantly overestimate the

accuracy of OLS risk premium estimates. This simulation evidence supports that the IV-

FMSEs provide accurate standard error estimates. In our empirical analyses, we thus

employ the FM standard errors (FMSEs) to compute the corresponding t-statistics.

4. Small Sample Properties of the IV Method - Simulation Evidence

To evaluate the small sample properties of the IV method (i.e., when T is fixed and

small relative to N), we conduct a battery of simulations based on real data. We first

investigate the bias and RMSE of the IV estimator and then examine the size and power of

the t-test statistic based on the IV estimator

4.A. Bias and RMSE

We fix the simulation parameters to correspond with actual data during the sample

period for empirical analyses in Section 5: January 1956 through December 2012. For a

single factor model, simulation parameters are matched to the average market risk premium,

the risk-free rate, the cross-sectional distribution of betas, and volatility of firm-specific

returns.

The CRSP value-weighted index is the market return and the short-term T-bill rate

is the risk-free rate. For each stock, a market model regression provides the beta and

residual returns. We conduct simulations with the cross-sectional size of N=2000 stocks,

which is matched to the real data.7 We randomly generate daily returns using the following

procedure:

1) For each stock, randomly generate beta and standard deviation of return

residuals i

εσ from normal distributions with means and standard deviations equal

to the corresponding sample means and standard deviations from the data.8 We

7 In our empirical analyses, an average month has 1934 individual stocks (see Table 3). 8 If the random draw of 𝜎𝜀

𝑖 is negative, we replace it with its absolute value.

14

generate betas and i

εσ s in the beginning of each simulation and keep them constant

across 1000 repetitions.

2) For each day, generate market excess return as a random draw from a

normal distribution with mean and standard deviation equal to the sample mean and

standard deviation from the data.

3) For each stock and each day, generate residual return i

τε from independent

normal distributions with mean zero and standard deviation corresponding to the

value generated in step (1).

For stock i, the excess return on day τ is defined as

,rβ r i

ττMKT,

ii

τ ε (4.1)

where τMKT,r is the market excess returns.

For the first-stage regression in the simulation, we estimate betas using the

following market model regression with daily excess returns for each stock:9

.rβαr i

ττMKT,

i

i

i

τ ε (4.2)

Each “month” in the simulation has 22 trading days and we use two years of daily returns

(T=528) to fit the time-series regression (4.2). For the IV method, we use daily returns

from odd and even months during the two-year estimation period to compute independent

and instrumental variables.

We fit the second stage regression with monthly returns, following the common

practice in the literature. We could have fit the second stage regression with daily returns

as well, but this method will not help us improve the precision of the second stage estimates.

To see this intuitively, compare fitting one cross-sectional regression for month t with

fitting 22 separate daily regressions for the month and averaging the daily regression

estimates over the month. With the same set of firms in both regressions and same betas

9 We use daily returns rather than monthly returns to obtain more precise beta estimates in the first stage

regression.

15

for the month, the slope coefficient of the monthly regression would be exactly 22 times

the average slope coefficient of the daily regressions and the standard error of the monthly

regression would also be 22 times the standard error of average daily regression coefficient.

As a result, both specifications would yield exactly the same t-statistic for the slope

coefficient. There would be some differences between the two specifications if daily

returns are compounded to compute monthly returns but such differences are likely small.

We compound daily stock and factor returns to compute corresponding monthly

returns. We fit the cross-sectional IV regression equation (3.1) for each month t to estimate

t0,γ and t1,γ . We then roll the two-year estimation window forward by one month and

repeat the two-stage IV estimation procedure over 660 months (=55 years).10 Finally, we

take the averages of the monthly slope coefficients across 660 months and then compare

them to their FMSEs.

We conduct the three-factor model simulations analogously, but in addition to

market returns and market betas, additional factors and betas correspond to the Fama-

French SMB and HML factors and betas. We match means and standard deviations of the

simulation parameters to the actual data, then carry out the two-stage IV estimation

procedure to estimate 0γ , MKTγ , SMBγ , and HMLγ . Appendix 2 describes the

simulation parameters and the simulation experiment design in more detail.

For each repetition, we compare the true factor risk premiums used to generate

returns and the corresponding sample estimates. The averages of these differences over the

1000 repetitions are the biases in risk premium estimates relative to the true risk premiums,

which are the EIV-induced ex-ante biases. Since all risk premium estimates within a

sample are conditional on particular factor realizations, we also report the biases relative

to the average realized risk premiums in that particular sample, which are the EIV-induced

ex-post biases (see Shanken, 1992).

Panel A of Table 1 presents the ex-ante and ex-post biases, as a percentage of the true

10 Out of 684 months (=57 years) that correspond to the sample period from Jan. 1956 to Dec. 2012, the

first 24 months are deducted.

16

market premium. The OLS estimates are biased towards zero by about 28%, because of

the errors-in variable problem. In contrast, the differences between average IV estimates

and both ex-ante and ex-post risk premiums are less than 1%, and statistically not different

from zero.

The next two columns in Panel A present the ex-ante and ex-post root-mean squared

errors (RMSEs). The bias and standard error of risk premium estimates contribute to

RMSE. It is well known that OLS standard error tends to be smaller than IV standard error.

Because of the tradeoff between bias and standard error, it is important to examine the

RMSE to assess the overall performances of the OLS and IV estimators. The ex-ante

RMSE for both OLS and IV estimators are about equal. The ex-post RMSE for the OLS

estimator is .156 for the OLS estimator, compared with .088 for the IV estimator. These

results indicate that because of the bias, the overall accuracy of the IV estimator would be

better than the OLS estimator conditional on factor realizations.

Figure 1 plots the biases of the IV and OLS estimators as a function of the number of

time-series observations, with N=2000 stocks. The vertical axis reports the ex-ante and ex-

post biases as percentages of the true market risk premium. The bias of the OLS estimator

is fairly large at -43% when T=264 observations.11 The bias is larger than 5% even when

T=2640 days, or 10 years. In contrast, the bias is fairly close to zero for the IV estimator

even for T=264 days, or 1 year.

Panel B of Table 1 presents the results for the Fama-French three-factor model. The

EIV problem always biases OLS slope coefficient estimates towards zero in univariate

regressions, but in theory the direction of the bias is indeterminate in multivariate

regressions. The results in Panel B indicate that the OLS estimates of the slope coefficients

in the case of the Fama-French model are all biased towards zero. For example, the ex-

ante biases of the OLS estimates are -64.7% and -66.3% for SMB and HML, respectively.

In contrast the biases of the IV estimates are all less than 2.1%. The last two columns in

Panel B indicate that the IV method outperforms the OLS method substantially in terms

of ex-ante and ex-post RMSEs.

11 Since the simulation assumes 22 days per month, T=264 corresponds to one year.

17

4.B. Size and Power of t-Test

Our tests follow the Fama-MacBeth approach to test whether the risk premiums

associated with various common factors are reliably different from zero. For example, in

the case of a single factor model, the tests statistic is:

,σ̂

γ̂t

γ

γ (4.3)

where γ̂ is the time-series average of monthly IV risk premium estimates and γσ̂ is the

corresponding Fama-MacBeth standard error (FMSE).

To examine the small sample distribution of the t-statistic in equation (4.3) under the

null hypotheses, we follow the same steps as above to generate simulated data, but we set

all true risk premiums equal to zero. We then examine the percentage of repetitions (out of

1000 total repetitions) when the t-statistics are positively significant at the various levels

(one-sided) using critical values based on the standard normal distribution.

Panels A and B of Table 2 present the test sizes under the CAPM and the Fama-French

three-factor model for N=2000 stocks, respectively. The results indicate that the tests are

well specified when T=528 days (=two years of daily data) are used for rolling beta

estimation. For example, the test sizes for all risk premiums at the 5% significance level

are between 4.7% and 5.3% and those at the 10% significance level are between 9.8% and

10.3%. In untabulated results, we found that the distribution of the test statistic was closer

to the theoretical distribution as we increased T. These results indicates that we can draw

reliable statistical inferences about risk premiums based on conventional t-test statistics

with the IV approach. .

We now investigate the power of the IV tests to reject the null hypotheses when the

alternative hypotheses are true. To evaluate power, we modify the simulation experiments

by adding risk premiums equal to the average risk premiums that we observe in the sample.

All the other simulation parameters are the same as in the simulations under the null

hypotheses. We fix the size of IV tests at the 5% significance level.

Panel C of Table 2 shows that the power of the IV test to reject the null hypothesis

18

under the CAPM is 82.8%. Under the three-factor model (Panel D), we find that the

frequency of rejection of the null of zero market risk premium is 78.1% and that of zero

HML risk premium is 84.2%. The test power is somewhat weaker to detect the positive

SMB premium but it is still greater than 50%. We also find that in 98.2% of the simulations,

at least one of the three factor risk premiums is different from zero. Overall, these results

indicate that our IV tests are reasonably powerful to detect non-zero risk premiums.

4.C. Time-varying Factor Sensitivities

Our simulations so far assume that betas are constant over time. In practice,

however, betas may vary over time. Therefore, we also conduct the simulations to

investigate the small sample properties and power of our tests with time-varying betas.

When we allow betas to follow AR(1) processes, we find that the small sample properties

of IV risk premium estimates and the size and power of the IV tests are similar to what we

report with constant betas in Tables 1 and 2. For brevity, we report the details of this

simulation and the results in Appendix 2.

5. IV Risk Premium Estimates for Selected Asset Pricing Models

The simulations of the previous section indicate that our IV method accurately

assesses risk premiums when they are present and does not falsely indicate their presence

when they are absent. In this section, we apply the IV method to ascertain whether risk

premiums seem to be present for several of the most prominent asset pricing models. We

first describe the data, then carry out a battery of examination, and finally provide evidence

that the weak instruments are not a problem in these applications.

5.A. Data

Stock return and market capitalization data from CRSP and balance sheet data from

COMPUSTAT are compiled from January 1956 through December 2012.12 With respect

12 The sample periods vary depending on asset pricing models tested.

19

to equities, we include only common shares (CRSP share codes of 10 or 11)13 and also

exclude stocks with prices below $1 and market capitalizations less than $500,000 at the

end of a month from the sample in the following month. Since daily returns are employed

to estimate betas, we restrict the sample to stocks with at least 200 daily observations per

year.14

Table 3 presents summary statistics for stocks included in our empirical analyses. A

total of 7508 distinct stocks enter the sample at different points in time; 1934 stocks are

available in an average month.

5.B. The CAPM and the Fama-French Three-Factor Model

This section tests the CAPM and the Fama-French three-factor model. We first test

whether the estimated factor risk premiums under the CAPM and the Fama-French three-

factor models are different from zero using the IV method and individual stocks as test

assets. We then examine whether the risk premiums are present after controlling for stock

characteristics.

Early empirical tests of the CAPM by Fama and MacBeth (1973) and others find strong

support for the CAPM. However several subsequent papers find that market betas are not

priced after controlling for other characteristics. For instance, Jegadeesh (1992) and Fama

and French (1993) conclude that the market risk premium is not significantly different from

zero after controlling for the firm size.

The inability of the CAPM to account for any of the cross-sectional differences in

expected returns reinvigorates the search for alternative asset pricing models. The arbitrage

pricing theory, Ross (1976) provides the general multifactor framework, but the Fama-

French three-factor version of the APT is perhaps the most widely used alternative. This

model identifies size and book-to-market factors in addition to the market factor.

The empirical support for the Fama-French three-factor model is mixed. Fama and

13 Excluded are American depository receipts (ADRs), shares of beneficial interest, Americus Trust

components, close-end funds, preferred stocks, and real estate investment trusts (REITs). 14 We repeat the asset pricing tests with different thresholds for the number of observations per year, i.e.,

100 and 150 observations per year, and find that our conclusions on the asset pricing tests do not change.

20

French (1992) estimate factor risk premiums using the portfolios sorted on size and book-

to-market and show that both premiums are significantly positive. But the loadings on these

risk factors are highly correlated with the size and book-to-market characteristics of the

test portfolios. Therefore, as Lewellen et al. (2010) point out, it is hard to reliably conclude

that these risk premiums are indeed compensation for systematic risks rather than for

portfolio characteristics. This highlights the low dimensionality problem when portfolios

are used as test assets in testing asset pricing models.

The conflicting results of the empirical tests in Daniel and Titman (1997) and Davis,

Fama, and French (2000) further illustrate the difficulty in making reliable inferences with

portfolios as test assets. Daniel and Titman argue that the differences of average returns in

size and book-to-market sorted portfolios are due to their characteristics and are not

necessarily related to factor risks. However, Davis, Fama, and French (2000) extend the

sample period back to 1925 and argue, based on this extended sample period, that the SMB

and HML factor risks are priced significantly. They counter that the differences in average

returns across the test portfolios are due to factor risks and not due to non-risk portfolio

characteristics.

This subsection uses individual stocks in the tests and avoids the low dimensionality

problem inherent in the tests that employ characteristics-sorted portfolios as test assets. We

use daily rolling windows from month t-36 to month t-1 to estimate betas for month t. In

unreported tests, we find similar asset pricing test results when betas are estimated with

60-, 24-, and 12-month rolling windows.

To account for non-synchronous trading effects in daily returns, beta estimation is

supplemented with one-day lead and lag of the independent variables (Dimson, 1979). For

example, the following regression estimates betas for the CAPM: for firm i and day 𝜏,

,rβαr i

τ

1

1k

k-M K T ,

iii

τ M K T . k ε

(5.1)

.β̂β̂β̂β̂ i

M K T , 1

i

M K T , 0

i

M K T , - 1

i

M K T

We estimate odd- and even-month betas separately using returns on days belonging to odd

21

and even months, respectively. Because of the non-synchronous trading adjustment in (5.1),

the first and the last days of each month are excluded to avoid overlap.15 An analogous

multivariate regression estimates the three betas for the Fama-French three-factor model.

For each stock and month, the Size characteristic is the natural logarithm of market

capitalization at the end of the previous month. BM is the book value divided by the market

value where book value is the sum of book equity value plus deferred taxes and credits

minus the book value of preferred stock. We compute correlations between each pair of

firm-specific variables each month and Table 4 presents the average cross-sectional

correlations among betas and characteristics. The CAPM beta estimated using the market

model is negatively correlated with both Size and BM. In the Fama-French model, the

correlation between market betas and the betas on other factors are positive. The correlation

between Size and SMB betas is negative, and the correlation between HML factor and BM

is positive, which reflect the fact that the SMB and HML factors are constructed using

these characteristics.

To see an impact of portfolio formation, Table 4 also presents the average cross-

sectional correlations for 25 Fama-French size and book-to-market sorted portfolios. For

each portfolio, we compute Size and BM each month as the value-weighted averages across

all stocks in the portfolios. The correlation among portfolio betas and characteristics is

much larger; between the SMB betas and Size it is -.97 and between the HML beta and

BM it is .88.

We next estimate factor risk premiums using the IV method. Table 5 presents the factor

risk premium estimates for several different specifications of the second stage regressions.

We first test the CAPM using betas estimated with the univariate regression. The market

risk premium estimate is -.189%, which is not reliably different from zero, (Table 5,

column (1).) Therefore, we do not find support for the CAPM with individual stocks.

For the Fama-French three-factor model, betas come from multivariate time-series

15 We find almost identical results while including the first and last days of each month. Also, the results

are qualitatively similar when there is no adjustment for non-synchronous trading, i.e., .β̂β̂ i

MKT,0

i

MKT

22

regressions with all three factors. The market risk premium estimate is now an insignificant

-.315% and the SMB and HML risk premiums are .311% and .504%. The risk premiums

of SMB and HML are significant at conventional levels (column (2).)

The significance of SMB and HML suggests that these risks are priced, but this

inference might be compromised by the correlation between SMB and HML betas and the

underlying size and book-to-market characteristics documented in Table 4. To examine

this issue, the Size and BM characteristics are included as additional independent variables

in the second stage cross-sectional regressions. With the single-factor (CAPM) model,

column (3), the mean coefficients of the Size and BM characteristics are -.152% and .163%,

respectively, and both are statistically significant at the 1% level. The market’s risk

premium estimate is .010%, which is still not significantly different from zero. In the

regression, column (4), that includes betas and characteristics for the FF three-factor model,

none of the risk premiums is significant at the 5% level, including the previously significant

SMB and HML betas. Both Size and book-to-market characteristics are significant at any

conventional levels.

Table 5 also reports on two roughly equal subperiods. The factor risk premiums are not

significant in any subperiod when Size and book-to-market characteristics are included in

the cross-sectional regressions. The Size characteristic is significant in both subperiods,

while the BM is significant only in the first subperiod at the 5% level.

Overall, using the IV method to correct the EIV problem while still relying on

individual stocks, factor risk premium estimates are not significant for the CAPM or any

of the Fama-French factors. However, the individual firm characteristics, Size and BM, are

significant for the entire 1956-2012 sample and Size is significant in both subsamples.

Given that the IV method works very well with simulated data, there are several

interpretations possible concerning these empirical results. First, something in the real data

compromises the IV method; i.e., something that is missing from the simulated data. For

example, although our simulation evidence indicates that the conventional t-tests based on

the IV method are reasonably powerful, they might not in the real data. This interpretation

does not seem convincing due to the following observations. First, without controlling for

23

characteristics, Panel A finds that the risk premiums of SMB and HML are significant.

This evidence indicates that the test power is not a big issue when an average month has

about 2000 stocks. Second, even with 40% larger cross-sections on average, i.e., with more

powerful t-tests, none of the risk premiums becomes significant in the second subperiod.

Second, the Size and BM characteristics are associated with loadings on risk factors that

are badly captured by the Fama-French SMB and HML portfolios while the CRSP value-

weighted index is a poor proxy for the true aggregate market. Third, Size and BM

characteristics represent anomalies that offered the opportunity to earn sizeable returns

without bearing much risk. For the BM characteristic, this third interpretation is buttressed

by its disappearance from the 1986-2012 data after being strong from 1956 through 1985.

Size, in contrast, is impressively persistent and is even more significant in the second

subperiod than in the first subperiod.

5.C. The Fama-French Five-Factor Model

Novy-Marx (2013) and Aharoni, Grundy, and Zeng (2013) among others find that stock

returns are significantly related to profitability and investment after controlling for Fama-

French three factors. Fama and French (2014) propose the following five-factor model that

adds factors to capture these anomalies as well:

CMA

i

CMARMW

i

RMWHML

i

HMLSMB

i

SMBMKT

i

MKT

i

t γβγβγβγβγβ)(r E (5.2)

where i

RMW

i

HML

i

SMB

i

MKT β ,β ,β ,β and i

CMAβ are the betas with respect to market, size,

book-to-market, profitability, and investment factors, and RMWHMLSMBMKT γ, γ, γ,γ and

CMAγ are the corresponding risk premiums. The RMW factor is the difference between

the returns on diversified portfolios of stocks with robust and weak operating profitability

and the CMA factor is the difference between the returns on diversified portfolios of the

stocks of low and high investment.

We use the same procedure as in Fama and French (2014) and construct daily HML,

RMW, and CMA factors. For example, to construct the construct the RMW factor we first

independently sort of stocks into two Size groups and three operating profitability groups.

24

We compute the value-weighted returns for the six size-profitability portfolios. The

average of the small and big high profitability portfolio return minus the average of the

small and big low profitability portfolio return is the RMW factor.

Following Fama and French (1993 and 2014), we use the annual balance sheet data to

compute the levels of book-to-market, operating profitability and investment and allow six-

month delay when combining with financial variables.16 As in Fama and French (2014),

the sample period for the tests in this subsection is from 1964 through 2012.

Panel A of Table 6 presents the results of asset pricing tests of the Fama-French

five-factor model. Consistent with Table 5 (although the sample periods are different and

an average month has larger cross-section in Table 6), columns (1) to (3) indicate that SMB

and HML risks are priced, while RMW and CMA are not priced in the cross-section of

individual stock returns. In column (6), the pricing evidence of SMB and HML risks

disappears when we control for firm characteristics in the regressions. The slope

coefficients of characteristics, especially for investment/total asset, are highly significant

and reliable. We find similar results in the subperiods as well.

5.D. The q-factor Asset Pricing Model

Cochrane (1991) and Liu, Whited and Zhang (2009) present production-based asset

pricing models in which productivity shocks are tied to the changes in the investment

opportunity set, which is consistent with Merton’s (1973) ICAPM framework. Since

shocks to productivity are difficult to accurately measure, Hou, Xue, and Zhang (2015)

(HXZ) propose an investment factor and an ROE factor to capture productivity shocks.

The q-factor model is specified as:

ROE

i

ROEI/A

i

I/AME

i

MEMKT

i

MKT

i

t γβγβγβγβ)(r E (5.3)

where β ,β ,β i

I/A

i

ME

i

MKT and i

ROEβ are the betas with respect to market, size, investment

16 The investment for June of year t is the change in total assets from the fiscal year ending in year t-2 to the

fiscal year ending in year t-1, divided by total assets in year t-2. The operating profitability for June of year

t is annual revenues minus cost of goods sold, interest expense, and selling, general, and administrative

expenses divided by book equity for the last fiscal year end in year t-1.

25

and ROE factors, respectively, and I/AMEMKT γ, γ,γ and ROEγ are the corresponding

risk premiums.

The investment factor captures the level of investments and the ROE factor captures

the return on investments, i.e., profitability. The investment factor is constructed as the

return difference between firms with low and high levels of investment and the ROE factor

is constructed as the return difference between firms with high and low return on

investment. Following HXZ, we control for size when constructing the investment and

ROE factors. Intuitively, investments and rates of return on investments are likely to reflect

sensitivity to unanticipated productivity shocks, and these factors are supposed to capture

the price impact of such shocks. HXZ argue that their factors better explain cross-sectional

return differences across portfolios constructed based on various firm-level anomalies, e.g.,

book-to-market, size, momentum, and earnings surprise than the Fama-French three-factor

model and the Carhart four-factor model.

The HXZ model is appealing since an underlying theory rather than empirical

regularities suggests the factors. Also, HXZ’s empirical approach employs a variety of

different common factors and test portfolios. For instance, their tests of size and book-to-

market uses the 25 Fama-French size and book-to-market sorted portfolios, the test of

momentum uses 10 portfolios formed based on momentum, and the test of the earning

surprises (SUE) uses 10 SUE sorted portfolios. However, all their tests employ portfolios

and are subject to potential low dimensionality problem.

We examine whether the HXZ factors are priced using individual stocks as test

assets. Here, we follow procedure in HXZ to construct daily market, size, investment and

ROE factors. For example, we first sort firms by size, investment as a fraction of total

assets (I/A), and ROE based on the NYSE breakpoints. We then assign stocks to groups

according to the top and bottom 50% of size and the top and bottom 30% and the middle

40% of I/A and ROE, producing a total of 18 (=2x32) groups. We form value-weighted

portfolios of stocks in each of the 18 groups. The investment factor is the equal-weighted

portfolio that is long the six low I/A portfolios and short the six high I/A portfolios. The

ROE factor is the equal-weighted portfolio that is long the six high ROE portfolios and

26

short the six low ROE portfolios.

We use the last announced quarterly financial statement data to compute the level

of investments and ROE each month.17 The HXZ apply earnings announcement dates to

determine when financial data become available to the market. Since earnings

announcement dates on COMPUSTAT are available only after 1972, as in HXZ (2015),

the sample period for this portion of the study is from 1972 to 2012.

Table 7, Panel A, reports average cross-sectional correlations among estimated

factor sensitivities (betas) and firm characteristics. Sensitivities to I/A and ROE factors are

positively correlated across stocks. I/A beta is negatively correlated with size and positively

correlated with BM, and the ROE beta is positively correlated with size and negatively

correlated with BM. The correlations between these betas and the characteristics are

smaller than those for the SMB and HML factors in Table 4. In Panel B, Table 7 also

reports analogous correlations for the 25 Fama-French size and book-to-market sorted

portfolios. For these portfolios, the correlation between I/A beta and BM is .88 and the

correlation between ROE beta and Size is .74. Such high correlations suggest that the issues

discussed in Lewellen et al. (2010) could influence results of tests that use the 25 Fama-

French portfolios.

Table 8 presents results of asset pricing tests with individual stocks and the IV

method. For comparison, (since the sample period is different,) column (1) reports the

single-factor market risk premium; it is quantitatively similar to the premium reported in

Table 5 and is still insignificant statistically. Column (5) reports the slope coefficients of

HXZ’s q-factor loadings without controlling for characteristics. In this case, both I/A risk

premium and ROE risk premium are negative and the former is insignificant and the latter

is significant at the 5% level. The mean of the ROE factor during the sample is .7% per

month, which is significantly positive, so if the ROE factor reflected risk, its premium

should be positive as well. Columns (6) to (8) in Table 8 indicate that the inclusion of Size

and BM does not change the results of asset pricing tests. The slope coefficients of Size

17 Following HXZ (2015), the investment to total assets is defined as the annual change in total assets

(COMPUSTAT annual item AT) divided by 1-year-lagged total assets. ROE is income before

extraordinary items (COMPUSTAT quarterly item IBQ) divided by book equity lagged by one quarter.

27

and BM are significant and have the signs consistent with existing studies.

5.E. The Liquidity-Adjusted CAPM

This subsection examines the liquidity-adjusted capital asset pricing model (LCAPM)

proposed by Acharya and Pedersen (2005), which accounts for the impact of illiquidity-

based trading frictions on asset pricing.18 According to the LCAPM, the level of illiquidity

and the covariances of return and illiquidity innovation with the market return and

illiquidity innovation vary across assets. The unconditional expected return in excess of the

risk-free rate ( )(r i

tE ) under the LCAPM is:

),βββλ(β)(c)(r i

4

i

3

i

2

i

1

i

t

i

t EE (5.4)

where i

tc is the illiquidity cost, the risk premium is the market excess return minus

aggregate illiquidity cost (i.e., )c-(rλ tMKT,tMKT,E ), and the betas are

,))]c(c[)r(Var(r

))r(r,r(β

tMKT,1ttMKT,tMKT,1ttMKT,

tMKT,1ttMKT,

i

ti

1

EE

ECov (5.5)

,))]c(c[)r(Var(r

))c(c),c(c(β

tMKT,1ttMKT,tMKT,1ttMKT,

tMKT,1ttMKT,

i

t1t

i

ti

2

EE

EECov

,))]c(c[)r(Var(r

))c(c,r(β

tMKT,1ttMKT,tMKT,1ttMKT,

tMKT,1ttMKT,

i

ti

3

EE

ECov

.))]c(c[)r(Var(r

))r(r),c(c(β

tMKT,1ttMKT,tMKT,1ttMKT,

tMKT,1ttMKT,

i

t1t

i

ti

4

EE

EECov

18 Several other papers, e.g., Pastor and Stambaugh (2003), also propose models where a stock’s return

sensitivity to market-wide liquidity is priced. Since we do not have daily Pastor and Stambaugh liquidity

factors, we do not examine this model here.

28

The term )(c i

tE is the reward for firm-specific illiquidity level, which is the compensation

for holding an illiquid asset as in Amihud and Mendelson (1986). Acharya and Pederson

define illiquidity-adjusted net beta as:

.βββββ i

4

i

3

i

2

i

1

i

LMKT (5.6)

The LCAPM implies that the linear relation between risk and return applies for

liquidity-adjusted beta and not for the market beta under the standard CAPM. The LCAPM

also implies that the linearity between risk and return applies to excess returns net of firm

specific illiquidity cost.

Acharya and Pedersen test the LCAPM using two sets of test portfolios formed based

on illiquidity and the standard deviation of illiquidity. They sort stocks based on Amihud

(2002) illiquidity measures during each year and form 25 value-weighted illiquidity test

portfolios for the subsequent year. They also form 25 σ (illiquidity) portfolios similarly

by sorting based on the standard deviation of illiquidity.

We examine the correlations between i

LMKTβ and the value-weighted averages of Size

and BM for these portfolios used by Acharya and Pedersen. Correlations of i

LMKTβ with

Size for illiquidity and σ (illiquidity) portfolios are -.96 and -.97, and those with BM are

.71 and .74, respectively. Such high correlations between liquidity-adjusted betas, i.e.,

i

LMKTβ , and size suggest that it would be particularly hard to determine empirically whether

average returns differ across test portfolios due to size or illiquidity-adjusted betas. This

situation parallels that in Chan and Chen (1988) who use 20 size-sorted portfolios as test

assets and find strong support for the CAPM. The correlations between market betas and

Size for Chan and Chen’s test portfolios range from -.988 to -.909 over different periods,

and the corresponding correlations in the case of illiquidity and σ (illiquidity) portfolios

are within this range. Jegadeesh (1992) shows that when test portfolios are constructed so

that size and market beta have low correlations, the market risk is not priced and that the

significant market risk premium found using size-sorted portfolios is due to a high

correlation between Size and market beta.

29

To avoid such ambiguity, we use the IV method with individual stocks to investigate

whether the liquidity-adjusted market risk i

LMKTβ under the LCAPM is priced in the

cross-section. To facilitate comparability, we follow the same procedure as in Acharya and

Pederson (2005) in all other respects. Because of the differences in the market structures

of the NYSE/AMEX and NASDAQ, the trading volumes reported in these two markets are

not comparable and hence NASDAQ stocks are excluded for this test. In addition to

existing screening criteria, following Acharya and Pederson, we exclude stocks that do not

trade for at least 100 days per year, which can suppress noisy illiquidity measures.19

Acharya and Pederson define illiquidity cost as follows:20

,|r|

ILLIQi

τ

i

τi

τ

(5.7)

),30,PILLIQ3.025.0min(c 1τMKT,

i

τ

i

τ (5.8)

where i

τr is the return on day τ , i

τν is the dollar volume (in millions) and 1τMKT,P is

the day 1-τ value of $1 invested in the market portfolio as of the end of July 1962.

Equation (5.7) is based on Amihud’s (2002) illiquidity measure. Acharya and Pederson use

equation (5.8) as a measure of illiquidity cost where 1τMKT,P is used to adjust for inflation

and the illiquidity cost is capped at 30% to avoid an obviously unreasonable value for it.

Market illiquidity cost τMKT,c is the value-weighted average of the illiquidity costs of the

individual stocks.

As in Acharya and Pederson (2005), we estimate innovations in illiquidity using an AR

model and then estimate each individual component of betas in equation (5.6) using a time-

series GMM approach and Dimson-type corrections.21 We then fit the following cross-

sectional regression each month:

19 We follow Acharya and Pederson and impose the 100 days per year data requirement for inclusion in the

sample. 20 Acharya and Pederson use illiquidity costs at monthly frequency but we use them at daily frequency. 21 Appendix 3 presents the AR models that we use to estimate expected and unexpected components of

30

.εβ̂γcγαr i

t

i

LMKTtLMKT,

i

ttILLIQ,t

i

t (5.9)

where i

tc is the average illiquidity for stock i in month t.22

The IV estimator in month t is:

,'ˆ)ˆˆ('ˆttIV,

1'

tEV,tIV,t rΨΨΨγ

where tIV,Ψ̂ is teven,Ψ̂ when month t is odd and is todd,Ψ̂ when month t is even,

and

teven,Ψ̂ N3 matrix of independent variables with unit vector as the first row,

i

tc , and estimated even-month LMKT betas for N stocks as the second and

third rows, respectively. We estimate the even-month LMKT betas using

daily data in even months in the period of month t-36 to month t-1.

todd,Ψ̂ Analogous to teven,Ψ̂ estimated using all daily data in odd months.

We use the FMSEs to compute the standards errors of IV risk premium estimates.

Table 9 presents the regression estimates with individual stocks. The slope coefficient

on the Amihud illiquidity measure is .184%, and it is significantly positive at the 1% level.

However, the liquidity-adjusted market risk premium (the risk premium for LMKTβ ) is

.140%, which is not reliably different from zero. These results suggest that firm-specific

illiquidity, a firm characteristic, is positively related to average returns, but a stock’s

liquidity-adjusted beta, supposedly a systematic risk, does not command a risk premium.

Table 9 also shows that illiquidity risk is not priced in either subperiods and that the

Amihud illiquidity characteristic is not reliably associated with average returns at the 5%

level in the second subperiod.

In comparison, Acharya and Pederson report a liquidity-adjusted market risk premium

estimate of about 2.5% per month using the value-weighted index (see Panel B of Table 5

in AP), which is about 30% per year.23 The equity risk premium puzzle literature argues

illiquidity.

22 As in Acharya and Pedersen (2005), 30% capping is applied after taking monthly average. 23 The liquidity-adjusted market risk premium equals market risk premium minus expected illiquidity costs

31

that even an annual risk premium of about 6% observed in the data is hard to justify with

realistic levels of risk aversion, and larger risk premiums would be harder to justify. The

large risk premium estimate obtained with portfolios seems likely to be the result of

correlation between LMKTβ and portfolios characteristics rather than a true depiction of

the rewards to systematic risk.

These findings further illustrate the problems that arise when portfolios are used as test

assets in asset pricing tests. In the earlier size versus beta debate in the literature, portfolios

were formed based on size ranks and hence it would be fairly natural to check the

correlation between size and beta and to uncover the problem. In the case of illiquidity-

sorted portfolios, size was not explicitly used as a sorting variable to form portfolios and

hence it is not readily apparent that one should check the correlation with this variable, but

such correlations could lead to mistaken statistical inferences. Our tests with individual

stocks avoid such confounding issues.

5.F. On the Strength of Instrumental Variables

An important issue to consider in instrumental variable regressions is the correlation

between the instrumental variables and the corresponding independent variables. The

cross-product matrix of instrumental variables and independent variables could be close to

singular if the correlation is too low. Nelson and Startz (1990) show that if the instruments

are sufficiently weak then the expected value of the IV estimator may not exist. The

intuition behind this result can be seen in a univariate regression with weak instruments. If

the covariance between the independent variable and the instrument is close to zero then

the sample covariance could be small and be either negative or positive, resulting in large

variations in both the sign and magnitude of the slope coefficient estimates in finite samples.

However, if the covariance and the sample size are sufficiently large, then the likelihood

that the sample estimate of the covariance is close to zero becomes negligibly small, and

the IV estimator is well behaved.

and hence it is smaller than the unadjusted market risk premium.

32

Nelson and Startz (1990) show that weak instruments would be a concern if

N,ρ̂

12

xz

(5.10)

where xzρ̂ is the correlation between the independent variable and the corresponding

instrument (which in our method is the correlation between betas estimated from different

observations) and N is the number of observations in the cross-sectional regression, i.e.,

the number of individual stocks. For example, in the CAPM and Fama-French model tests

of section 5.3, there are 1934 stocks per month on average and the minimum number of

stocks is 305. From (5.10), there would be a weak instrument concern based on the

minimum (average) number of stocks, if the correlation were less than 0.057 (0.023) in

absolute value.

Table 10 presents average correlations between the odd and even month beta estimates.

The correlation for market beta under the CAPM is .67. The market beta of Fama-French

three-factor (five-factor) model is less precisely estimated and the correlation is smaller

at .52 (.42). The market beta in the q-factor asset pricing model and the LCAPM betas also

exhibit similar levels of correlation as the Fama-French three-factor market betas. The

average correlations for SMB, HML, RMW, CMA, I/A, and ROE betas range from .14

to .44. Although these correlations are smaller than those for market betas, they are all

comfortably above the Nelson and Startz (1990) critical value.

Nelson and Startz (1990) and Staiger and Stock (1997) also show that the conventional

IV standard error estimator based on asymptotic theory is not reliable in small samples if

the instruments are weak. However, this concern is not relevant in our application because

we use the Fama-MacBeth approach to estimate standard errors and do not use the

asymptotic estimator in our empirical analyses in the previous subsections. Nevertheless,

we find, in the tests proposed by Staiger and Stock (1997), that the instruments give no

cause for concern.24

24

Staiger and Stock (1997) regress the independent variable against the instrumental variable and develop

a test based on the goodness of fit for this regression. In unreported results, we find that their test statistics

in our applications were well above critical values for all instruments in all months.

33

To provide further insights into the strength of the instruments, we also estimate the

correlation between the instruments that we use and the corresponding true but

unobservable factor betas. Although the true beta is unobservable, we can estimate this

correlation based on the correlation between the odd- and even-month betas as we show in

the following proposition:

Proposition 2: Let i

kβ be stock i’s true unobservable sensitivity to factor k and let

i

kodd,β̂ and i

keven,β̂ be the odd and even month estimates of the corresponding

betas, respectively. Then:

.)β̂,β̂(n correlatio

)β̂,(βn correlatio )β̂,(βn correlatio

i

keven,

i

kodd,

i

kodd,

i

k

i

keven,

i

k

Proof: See Appendix 4.

Table 10 also presents the mean correlation between estimated betas and true betas.25

The average correlation between even- and odd-month market betas is .67 and the average

correlation between estimated market beta and the unobserved true market beta is .82. We

find smaller correlations for SMB, HML, RMW, and CMA betas, but even for CMA the

average correlation between estimated beta and unobservable true beta is .38. The

correlations for the I/A and ROE betas are about the same as that for the HML beta. All

these estimates are significantly above the cutoff prescribed by Nelson and Startz (1990).

6. Conclusion

We propose a method for estimating risk premiums using individual stocks as test

25 To compute the mean correlation between estimated betas and true betas, we first compute the square root

of the correlation between odd- and even-month betas each month and then compute the average across

months. Because the variability of correlation between odd- and even-month betas is relatively small, the

square root of average correlation is about the same as the mean of the square root of the correlation.

34

assets. It sidesteps concerns about risk premium estimated with portfolios, which have been

employed in almost all previous research to mitigate an inherent errors-in-variables

problem. Estimated β s from different observations can serve as effective instruments for

estimated β s from other observations that serve as the explanatory variables in second-

stage cross-sectional regressions. We prove the consistency and provide the asymptotic

theory of the proposed risk premium estimator when the size of cross-section and the length

of time-series grow simultaneously without bounds. In simulations, our instrumental

variables (IV) method estimates risk premiums accurately even for relatively short time-

series and also provides valid tests of statistical significance. Our simulations also indicate

that our tests are reliable under time-varying betas.

We use the new IV method to test whether risk premiums suggested by several popular

factor models are different from zero. These models include the CAPM, the Fama-French

three- and five-factor models, the q-factor asset pricing model proposed by Hou, Xue, and

Zhang (2015), and the liquidity-adjusted CAPM proposed by Acharya and Pedersen

(2005). Previous empirical examinations, employing portfolios as tests assets, find strong

support for these models, but Lewellen, Nagel and Shanken (2010) suggest caution about

the low dimensionality issue when portfolios are used. We find that none of the factor risks

in these asset pricing models commands a significant risk premium in the cross-section of

individual stock returns after controlling for firm characteristics. Simulations results

indicate that this failure cannot be attributable to a lack of test power, so it represents a

puzzle that calls for further research.

35

References

Acharya Viral and Lasse Heje Pedersen, 2005, Asset Pricing with Liquidity Risk,

Journal of Financial Economics 77, 375-410.

Aharoni Gil, Bruce Grundy, and Qi Zeng, 2013, Stock returns and the Miller

Modigliani valuation formula: Revisiting the Fama French analysis, Journal of Financial

Economics 110(2), 347-357.

Ait-Sahalia Yacia, Jonathan A. Parker, and Motohiro Yogo, 2004, Luxury Goods

and the Equity Premium, Journal of Finance 59, 2959-3004.

Amihud Yakov, 2002, Illiquidity and Stock Returns: Cross-section and Time-series

Effects, Journal of Financial Markets 5, 31-56.

Amihud Yakov and Haim Mendelson, 1986, Asset Pricing and the Bid-ask Spread,

Journal of Financial Economics 17, 223-249, Dec. 1986.

Arnott Robert, Jason Hsu, and Philip Moore, 2005, Fundamental Indexation,

Financial Analysts Journal 61, 83-99.

Bai Jushan, 2003, Inferential theory for factor models of large dimensions,

Econometrica 71(1), 135-171.

Black Fisher, Michael C. Jensen, and Myron Scholes, 1972, The capital asset

pricing model: Some empirical tests, Michael C. Jensen, ed: Studies in the Theory of

Capital Markets, 79–121.

Blume Marshall, and Irwin Friend, 1973, A New Look at the Capital Asset Pricing

Model, Journal of Finance 28, 19-34.

Brennan Michael, Tarun Chordia, and Avanidhar Subrahmanyam, 1998,

Alternative Factor Specifications, Security Characteristics, and the Cross-Section of

Expected Returns, Journal of Financial Economics 49, 345-373.

Campbell John and Tuomo Vuolteenaho, 2004, Bad Beta, Good Beta, American

Economic Review 94, 1249-1275.

Chan K.C., and Nai-Fu Chen, 1988, An Unconditional Asset Pricing Test and the

Role of Firm Size as an Instrumental Variable for Risk, Journal of Finance 43, 309-325.

Chen Long, Robert Novy-Marx, and Lu Zhang, 2011, An Alternative Three-Factors

36

Model, working paper, Ohio State University.

Chordia Tarun, Amit Goyal, and Jay Shanken, 2015, Cross-Sectional Asset Pricing

with Individual Stocks: Betas versus Characteristics, Working paper, Emory University.

Cochrane John, 1991, Production-based Asset Pricing and the Link Between Stock

Returns and Economic Fluctuations, Journal of Finance 46, 209-237.

Cochrane John, 1996, A Cross-Sectional Test of an Investment-Based Asset Pricing

Model, Journal of Political Economy 104, 572-621.

Cochrane, John , 2011, Presidential Address: Discount Rates, Journal of Finance

66, 1047–1108.

Daniel Kent, and Sheridan Titman, 1997, Evidence on the Characteristics of Cross

Sectional Variation in Stock Returns, Journal of Finance 52 (1), 1-33.

Davis James, Eugene Fama, and Kenneth French, 2000, Characteristics,

Covariances, and Average Returns: 1929-1997, Journal of Finance 55, 389–406.

Dimson Elroy, 1979, Risk Measurement When Shares Are Subject to Infrequent

Trading, Journal of Financial Economics 7, 197-226.

Eisfeldt Andrea, and Dimitris Papanikolaou, 2013, Organization Capital and the

Cross-Section of Expected Returns, Journal of Finance 68, 1365-1406.

Fama Eugene, and Kenneth R. French, 1992, The Cross-Section of Expected Stock

Returns, Journal of Finance 47, 427-465.

Fama Eugene and Kenneth French, 1993, Common Risk Factors in the Returns on

Stocks and Bonds, Journal of Financial Economics 33, 3–56.

Fama, Eugene and Kenneth French, 2014, A Five-factor Asset Pricing Model,

Journal of Financial Economics 116, 1-22.

Fama Eugene F, and James D. MacBeth, 1973, Risk, Return and Equilibrium:

Empirical Tests, Journal of Political Economy 81, 607-636.

Gagliardini Patrick, Elisa Ossola and O. Scaillet, 2011, Time-Varying Risk

Premium in Large Cross-Sectional Equidity Datasets, Swiss Finance Institute, working

paper.

Harvey Campbell, Yan Liu, and Heqing Zhu, 2015, …and the Cross-Section of

37

Expected returns, forthcoming Review of Financial Studies.

Hou Kewei, Chen Xue, and Lu Zhang, 2014, A comparison of new factor models.

Ohio State University, working paper.

Jagannathan Ravi, and Zhenyu Wang, 1996, The conditional capm and the cross-

section of expected returns, Journal of Finance 51, 3-53.

Jagannathan Ravi, and Zhenyu Wang, 1998, An asymptotic theory for estimating

beta-pricing models using cross-sectional regression, Journal of Finance 53, 1285-1309.

Jegadeesh Narasimhan, 1992, Does Market Risk Really Explain the Size Effect?,

Journal of Financial and Quantitative Analysis 27, 337-351.

Kan Raymond, Cesare Robotti and Jay Shanken, 2013, Pricing model performance

and the two-pass cross-sectional regression methodology, Journal of Finance 68, 2617–

2649.

Kim Dongcheol, 1995, The Errors-In-Variables Problem in the Cross-Section of

Expected Stock Returns, Journal of Finance 50, 1605-1634.

Kim Soohun and Georgios Skoulakis, 2014, Estimating and Testing Linear Factor

Models using Large Cross Sections: The Regression-Calibration Approach, Working

paper, Georgia Institute of Technology

Lettau Martin and Sydney Ludvigson, 2001, Resurrectng the (C)CAPM: A

Cross-sectional test when risk premia are time-varying, Journal of Political Economy 109,

1238-1287.

Lewellen Jonathan, Stefan Nagel and Jay Shanken, 2010, A skeptical appraisal of

asset pricing tests, Journal of Financial Economics 96, 175-194.

Li Qing, Maria Vassalou, and Yuhang Xing, 2006, Sector investment growth rates

and the cross section of equity returns, Journal of Business 89, 1637-1665.

Lintner John, 1965, The valuation of risk assets and the selection of risky

investments in stock portfolios and capital budgets, Review of Economics and Statistics 47

(1), 13–37.

Litzenberger Robert H, and Krishna Ramaswamy, 1979, The effect of personal

taxes and dividends of capital asset prices: The theory and evidence, Journal of Financial

38

Economics 7, 163-196.

Liu Laura Xiaolei, Toni Whited, and Lu Zhang, 2009, Investment-based Expected

Stock Returns, Journal of Political Economy 117, 1105-1139.

Merton Robert, 1973, An intertemporal asset pricing model, Econometrica 41, 867-

888.

Nelson Charles and Richard Startz, 1990, The Distribution of the Instrumental

Variable Estimator and its t ratio When the Instrument is a Poor One, Journal of Business

63, S125-S140

Novy-Marx, Robert, 2013, The other side of value: The gross profitability

premium, Journal of Financial Economics 108, 1–28.

Pastor Lubos and Robert Stambaugh, 2003, Liquidity risk and expected stock

returns, Journal of Political Economy 111, 642-685.

Ross Stephen A., 1976, The arbitrage theory of capital asset pricing, Journal of

Economic Theory 13, 341-360.

Shanken Jay, 1992, On the estimation of beta-pricing models, Review of Financial

Studies 5, 1-33.

Shanken Jay, and Guofu Zhou, 2007, Estimating and testing beta pricing models:

alternative methods and their perfor- mance in simulations, Journal of Financial Economics

84, 40-86.

Sharpe William, 1964, Capital asset prices: A theory of market equilibrium under

conditions of risk, Journal of Finance 19 (3), 425–442.

Staiger Douglas and James Stock, 1997, Instrumental Variables Regression with

Weak Instruments, Econometrica 65, 557–586.

Theil Henri 1971, Principles of Econometrics, 1st Edition, John-Wiley & Sons, New

York.

39

Appendix 1: Asymptotic Theorems and Proofs

A1.1. The N-consistency and asymptotic distribution of the IV method

In this section, we show the consistency of the estimator and obtain its asymptotic

distribution. To simplify the exposition, we make following definitions and assumptions:

As in section 2.2, define ]ˆ;1[1ˆ evene ββ and ]ˆ;1[1ˆ oddo ββ where evenβ̂ and oddβ̂ are

estimated factor loadings for even and odd periods. Similarly, define ]ˆ;1[1ˆ samplesample ββ ,

]ˆ;1[1ˆ EVEV ββ , and ]ˆ;1[1ˆ IVIV ββ where sample, EV or IV=odd or even. In this paper,

the column vectors are used to define time-series variables (such as returns for one stock),

and row vectors represent cross-sectional variables (returns and factors in one period).

Moreover, risk premium is a row vector.

Let 1-T1 ,, ff be factors for each period. They are row vectors (K vector). The

estimation error in the first pass is d

sample

d

sample

1d

sample

d

samplesample ''ˆ ΩFFFββ

, where

];;[ d

1-T

d

1

d

o

d

sample ffFF when sample contains odd periods and

];;[ d

T

d

2

d

e

d

sample ffFF when sample contains even periods. The superscript d

indicates the demeaned factor or residual (constructed by subtracting the factor or residual

from their sample average). Thus, d

1-T

d

1 ,, ff are demeaned factors for each period.

Moreover, operator “;” is to stack the first row vector on top of second row vector.

The dependent variable in the second pass cross-sectional regression could be any

return vector rs for a disjoint period s not in the sample periods in the first pass. This

regression can be written as ssamples 1ˆˆ ξβγr . Since the true model is sss )( εβγfr ,

the cross-sectional residuals are

ssampless )ˆ)(( εββγfξ .

We show the T-consistency and convergent rate in section 3 with relative weak

assumptions. In order to show the N-consistency and asymptotic distribution, we need to

40

make the following assumptions:

Assumptions: (1) The residual process ],,,[ N

s

2

s

1

ss εεε ε is a stationary. The elements

in sε are cross-sectionally uncorrelated, and sε and tε are uncorrelated when s is not

equal to t. Let Σ be the covariance matrix for the residuals, then the above assumption

implies that it is a diagonal matrix. (2) Factor process sf is a stationary. With these

assumptions and several regularity conditions (shown in the Theorem), N-consistent and

asymptotic distribution can be shown through the following Theroem.

Theorem A1 (a) Assume that N

t

1

N

1

t

1

1 ξβ,,ξβ (where 1=]β,,β[ 1

N

1

1 β and

t

N

t

1

t ξ]ξ,,[ξ ) have finite variances, and when N∞, N/11 ββ and N/11 Σββ

converge to invertible matrices (denote the matrices by 'bb and 'bbΣ ), then the estimated

risk premiums 'ˆtγ converges to )',0( tfγ an when N converges to infinity.

Thus, 'γ̂ converges to )',0( fγ in probability when N converges to infinity (where

'γ̂ and 'f are sample average of 'ˆtγ and 'tf , respectively) .

Define i

s

id,

s

d

s

1

d

EV

d

EV

EVt

t ]''T

2)[

T

2( εεFFFγf

i

s for any s in EV. Assume that

N

T

1

N

N

1

1

N

1

1

1

1 β,,β,,β have finite variances, then 'γ̂ converges to )',0( fγ in

probability when both N and T converge to infinity.

(b) In addition to the assumptions in (a), we further assume that (1) 2N

1i

i

tN

))ε(var(N

1lim

L

exists, (2) ]ξβ,,ξβ[ N

t

1

N

1

t

1

1 satisfies a Lindeberg condition, 26 and (3)

26 Assume that the covariance matrices for N

t

1

N

1

t

1

1 ξβ,,ξβ are tNt1 ,, VV , let

j

tjVV , then the Lindeberg

condition is that 0)1)ξβ((limj

})|ξ(β{|

2j

t

1

j

1

njt

1j1

1

VV E for any 0 where 1 is the indicator function.

41

]')ξ)(''(,,')ξ)(''[( N

t

Nd,

IV

d

IV

1d

IV

d

IV

1

t

d,1

IV

d

IV

1d

IV

d

IV εFFFεFFF

for any t not in IV, satisfies a

Lindeberg condition, then (A) the asympotic distribution of the estimated risk premium

conditional on F is

,)(0,)),('ˆ(N 11

tt

BAAfγ0γ N

where '= bbA , ,))~

'(= IV0,0 LbbΣB c where

)')'(()')'(

0=

~d

IV

1d

IV

d

IV

dd

IV

1d

IV

d

IV1k

k1

IV0,FFFIFFFL0

0L

with

)')')((T

2T2)')')((()')')((1= t

d

EV

1d

EV

d

EVt

d

EV

1d

EV

d

EVt

dd

EV

1d

EV

d

EVt0 lFFFfγFFFfγIFFFfγ

c

where )T

2,,

T

21,,

T

2,

T

2(=t

l with the termT

21 the t'th entry of the vector,

and

T

2T

T

2

T

2

T

2

T

2T

T

2T

2

T

2

T

2T

=d

I

;

Similarly, ,)(0,)),('ˆ(N 11 ABAfγ0γ N where

,)))~

'())~

'((4

1= o0,ee0,o LbbΣLbbΣB cc and

))')')((()')')((T

2= d

e

1d

e

ddd

e

1d

e

d

ee

FFFfγIFFFfγ ec ,

))')')((()')')((T

2= d

o

1d

o

ddd

o

1d

o

d

oo

FFFfγIFFFfγ oc ;

(B) If ]β,,β,,β[ N

T

1

N

N

1

1

N

1

1

1

1 satisfies a Lindeberg condition and other conditions in

(A) are satisfied, then if both T and N converges to infinity,

,)(0,)),('ˆ(TN 11 DAAfγ0γ N where

,')1(= -1 bbΣγ'γD F and F is the covariance matrix of factors.

This condition implies that N

t

1

N

1

t

1

1 ξβ,,ξβ have similar variances.

42

As from the theorem, both asympotitc standard deviations have sandwitch forms.

Following Shanken (1992), the formulas can be decomposed as follows:

In case (A), 1

IV0,00

11111 )~

')1(()'( ALbbΣAAbbΣABAA cc .

In case (B), 1-111111 )')(()'( AbbΣγ'γAAbbΣADAA F .

In above decompostion, the first term is the asymptotic standard deviation of an OLS

estimator when there is no error in factor loadings, i.e. β is known. The second term is

the EIV adjustment on standard deviation. In particular, the standard deviation in case (B)

takes the same formula as Shanken (1992), although the rate of convergence for IV

estimator is faster. Gagliardini, Ossola and Scaillet (2011) show that when both T and N

large, the estimated risk premiums in the BJS method converge to their true values at the

speed of )NT

1O( 27 only if )O(TN when 3 . The rate of convergence of IV

estimator is )NT

1O( , and convergent rate does not depend on the relative size of T and

N.

Next, we will prove Theorem A1:

Proof:

We first prove the consistentency, note that:

.)'1ˆN

1()'1ˆ1ˆ

N

1(=)')(0,'ˆ

tIV

1

EVIVtt ξβββfγγ

The consistency is established based on Markov’s Law of Large Number: since (1) N

t

1

N

1

t

1

1 ξβ,,ξβ have finite variances; (2) for any i, 0=)ξβ( i

t

1

iE ; (3) regression residuals

i

tε are not cross-sectional and time-sereies correlated; (4) it is clear that

')ξ)(''(,,')ξ)(''( N

t

Nd,

IV

d

IV

1d

IV

d

IV

1

t

d,1

IV

d

IV

1d

IV

d

IV εFFFεFFF

have finite variances (i

IVε is the

residuals vector for stock i); then,

27Here, for any real number X, O(X) is defined as follows: there exist two positive numbers M and N, such that MX<O(X)<NX.

43

N

1i

i

t

id,

IV

d

IV

1d

IV

d

IV

1

itIV 0')ξ)(''(βN

1='1ˆ

N

1εFFFξβ , and

.')'1ˆ1ˆ(N

1EVIV bbββ This implies that the estimator is an N-consistent estimator to zero.

When T is finite, the sample average of these estimators is an N-consistent estimator to

zero. Hence, )',0('ˆ fγγ is also an N-consistent estimator to zero.

For the same reason, when (1)N

T

1

N

N

1

1

N

1

1

1

1 β,,β,,β have finite variances, (2) for any

i and t, 0=)β( N

t

1

iE , (3) regression residuals ],,,[ N

s

2

s

1

ss εεε ε are not cross-sectional

and time-sereies correlated, and (4) 0'' d

sample

d

sample

1d

sample

d

sample

ΩFFF as T ,

))'T

21ˆ

N

1()'1ˆ1ˆ

N

1()'

T

21ˆ

N

1()'1ˆ1ˆ

N

1((

2

1=),('ˆ

odd ist

te

1

oe

even ist

to

1

eo ξβββξβββγf0γ

converges to 0 when both N and T goes to infinity (i.e. the estimator is NT-consistent)

since

N

1i even ist

i

t

id,

o

d

o

1d

o

d

o

1

i

even ist

to 0)''(βNT

2)'

T

21ˆ(

N

1εFFFξβ , by Markov’s Law

of the large number,

N

1i odd ist

i

t

id,

e

d

e

1d

e

d

e

1

i

odd ist

te ,0)''(βNT

2)'

T

21ˆ(

N

1εFFFξβ by Markov’s Law of

the large number, and 1

eo ))'1ˆ1ˆ(N

1(

ββ and 1

oe ))'1ˆ1ˆ(N

1(

ββ are bounded given

,')'1ˆ1ˆ(N

1eo bbββ and '.)'1ˆ1ˆ(

N

1oe bbββ

Next, we show the asymptotic distribution of )',0('ˆtt fγγ . Since (1)

converges to when N∞, and (2) ]ξβ,,ξ[β N

t

1

N

1

t

1

1 and

]'))(''(,,'))(''[( N

t

Nd,

IV

d

IV

1d

IV

d

IV

1

t

d,1

IV

d

IV

1d

IV

d

IV ξεFFFξεFFF

both satisfy the Lindeberg

condition, one can apply the Lindeberg-Feller Central Limit Theorem to show the

normality.

It remains to calculate the aympotitic covariance of the estimator. We first define

d

sample

d

sample

1d

sample

d

samplesamplet ''== ΩFFFuu

for any samplet where sample can be

either IV or EV. To obtain the asymptotic covariance, notice that as N∞,

,')'1ˆ1ˆ(N1/ EVIV bbββ

and for EVt , the numerator of the variance can be writte as:

)|)'1ˆ'1ˆ(N(1/ IVttIV FβξξβE

N/ββ

'bb

44

)|)1))(())(1((N

1(= tttttt Fβεufγεufγβ E

)|)'))(())(((N

1( IVttttttIV Fuεufγεufγu E

1)|))(())(((1N

1tttttt

βFεufγεufγβ E

.)|'~)|))(())(((~(N

1IVttttttIVIV FuFεufγεufγu EE

)|(IV FE is expected value of a random variable conditioning on all the information F

and residuals in IV periods. Moreover, ],'[='~IVN1IV u0u

. One can show that

)|))(())((( ttttttIV Fεufγεufγ E

ΣFεufγεufγ 0tttttt =)|))(())(((= cE

hence,

)|)'1ˆ'1ˆ(1/N( IVttIV FβξξβE

.~

' IV0,00 LbbΣ cc

Then the asymptotic covariance matrix can be written as

.)')(~

'()'(=)|'))(0,('ˆ(Acov 1

IV0,

1

0tt

bbLbbΣbbFfγγ c

Note the key step is to show

ΣFεufγεufγ 0tttttt =)|))(())((( cE .

This result follows the proof in Shanken (1992). The details are shown below:

Since d

EV

d

EV

1d

EV

d

EVt ')'(= FFFu ,

,)()')')(((=)( d

EV

d

EV

1d

EV

d

EVttt VecFFFfγIfγu

where )( d

EVVec reshapes the N2

T matrix d

EV into the a N2

T column vector, i.e

,),,,,,,,,,(=)( Nd,

1-T

Nd,

1

d,2

1-T

d,2

1

d,1

1-T

d,1

1

d

EV εεεεεεVec when EV are estimated using

data in odd periods, and ,),,,,,,,,,(=)( Nd,

T

Nd,

2

d,2

T

d,2

2

d,1

T

d,1

2

d

EV εεεεεεVec when

EV are estimated using data in even periods.

Applying this formula, one has )|)()('( tttt FE ufγfγu

)')')(()()(')')(((= d

EV

1d

EV

d

EVt

dd

EV

1d

EV

d

EVt

FFFfγIIΣFFFfγI

.)')')((()')')((= d

EV

1d

EV

d

EVt

dd

EV

1d

EV

d

EVt ΣFFFfγIFFFfγ

Using the same method, one can show that:

.'=)|1))(())(1(N

1( 0tttttt bbΣFβεufγεufγβ cE

Similarly, one can show that

45

.~

=

)|'~)|))(())(((~

N

1(

IV0,0

IVttttttIVIV

L

FuFεufγεufγu

c

EE

When T is finite, the asymptotic distribution of '),('ˆ fγ0γ is similar to that

of '),('ˆtt fγ0γ following Lindeberg-Feller Central Limit Theorem.

The asymptotic covariance is calculated following the same logic, notice that

)')'T

21ˆ

N

1)('

T

21ˆ

N

1(NT(

EVt

tIV

EVt

tIV

ξβξβE

)|)1))(())(1((N

1(= EVEV Fβεufγεufγβ E

)|)'))(())(((N

1( IVEVEVIV Fuεufγεufγu E

1)|))(())(((1N

1EVEV

βFεufγεufγβ E

.)|'~)|))(())(((~(N

1IVEVEVIVIV FuFεufγεufγu EE

Here, ε and f are the sample average of the residual and factors for t in EV. Following

the same derivation as before, we can show that the above expression converges to

IV0,EVEV

~' LbbΣ cc . Here EV=o when IV=e and EV=e when IV=o. Thus, the asymptotic

convariance takes the form in Theorem A1.

Finally, we prove the asymptotic distribution of '),('ˆ fγ0γ when both T and

N are large. Since (1) converges to when N∞, (2)

]β,,β,,β[ N

T

1

N

N

1

1

N

1

1

1

1 satisfies a Lindeberg condition, and (3)

0'' d

sample

d

sample

1d

sample

d

sample

ΩFFF as T , one can apply the Lindeberg-Feller

Central Limit Theorem to show the normality.

It remains to calculate the aympotitic covariance of the estimator. The asymptotic

covariance for the numerator of the estimator can be written as:

)')'T

21ˆ

N

1)('

T

21ˆ

N

1(NT(

even ist

to

even ist

to ξβξβE

)))''(β())''(β(NT

4(

N

1i even ist

i

t

id,

o

d

o

1d

o

d

o

1

i

N

1i even ist

i

t

id,

o

d

o

1d

o

d

o

1

i

εFFFεFFFE

N/ββ 'bb

46

))'β()β(NT

4(

N

1i even ist

i

t

1

i

N

1i even ist

i

t

1

i

E

)))']''T

2)[

T

2((β(

))]''T

2)[

T

2((β(

NT

4(

N

1i even ist

i

t

id,

t

d

t

1

d

EV

d

EV

EVt

t

1

i

N

1i even ist

i

t

id,

t

d

t

1

d

EV

d

EV

EVt

t

1

i

εεFFFγf

εεFFFγfE

).)'β()']''T

2[(

)]''T

2[(β

NT

4(

1

i

i

t

id,

t

d

t

1

d

EV

d

EV

N

1i even ist

i

t

id,

t

d

t

1

d

EV

d

EV

1

i

εεFFFγ

εεFFFγ

E

The last equation holds because factors are assumed to have zero means, and regression

residuals are both cross-sectional and time-series uncorrelated. Since

,)](1[

))(())(''T

2[(

)']'T

2)(''

T

2[(

))|)']''T

2[)((]''

T

2[(((

))']''T

2[)((]''

T

2[((

2i

t

1-

2i

t

2i

t

d

t

1

d

EV

d

EV

1

d

EV

d

EV

d

t

2i

t

d

t

1

d

EV

d

EV

i

t

id,

t

d

t

1

d

EV

d

EV

i

t

id,

t

d

t

1

d

EV

d

EV

i

t

id,

t

d

t

1

d

EV

d

EV

i

t

id,

t

d

t

1

d

EV

d

EV

γ'γ

FFFγ

γFFFFFFγ

FεεFFFγεεFFFγ

εεFFFγεεFFFγ

F

EE

E

EE

E

we can show that:

.')1(2

))'β())(1((βNT

4

).)'β()']''T

2[(

)]''T

2[(β

NT

4(

1-

1

i

2i

t

1-N

1i even ist

1

i

1

i

i

t

id,

t

d

t

1

d

EV

d

EV

N

1i even ist

i

t

id,

t

d

t

1

d

EV

d

EV

1

i

bbΣγ'γ

γ'γ

εεFFFγ

εεFFFγ

F

F

E

For the same reason, we can show that

47

.')1(2

)')'T

21ˆ

N

1)('

T

21ˆ

N

1(NT(

1-

odd ist

te

odd ist

te

bbΣγ'γ

ξβξβ

F

E

Finally, notice that as N∞,

.')'1ˆ1ˆ(N1/ EVIV bbββ

With these results, we can show that the final asymptotic distribution:

.)(0,)'),('ˆ(TN 11 DAAfγ0γ N

48

Appendix 2: Details of Simulation Experiments

A2.1. Simulation Parameters

Table A.1 presents the parameters that we use with constant betas in sections 4.A

and 4.B. We set these parameters equal to the mean risk premiums of the common factors

and their covariance structure during the 1956 to 2012 sample period (Panel A). We

determine the cross-sectional means and standard deviations of betas and the volatility

of firm-specific returns by running time-series regressions during this sample period and

shrinking the betas with a simple adjustment rule: adjusted beta = 2/3×beta estimate +

1/3 (Panel B). All simulations use a risk-free rate of 0.9996% per annum.

TABLE A.1

Simulation Parameters

Panel A: Time-series means and standard deviations of common factors

Single Factor Model

Fama-French

Three-Factor Model

Mean (%) StdDev (%) Mean (%) StdDev (%)

Factors MKT 5.80 15.69 5.80 15.69

(per annum) SMB 2.64 7.89

HML 4.36 7.56

Panel B: Cross-sectional means and standard deviations of constant betas

Single Factor Model

Fama-French

Three-Factor Model

Mean

StdDe

v Mean StdDev

Betas MKT 0.95 0.42 0.95 0.42

SMB 0.80 0.50

HML 0.19 0.51

Idiosyncratic

Volatility

(per day)

0.036 0.015 0.037 0.015

49

A2.2. Simulations with Time-varying Betas

This section describes the procedure that we use for the simulations with time-

varying betas discussed in section 4.C. We assume that i

tβ , the beta of stock i in month t,

follows an AR(1) process. Specifically:

i

t

ii

1-t

ii

t e)βρ(βββ

where i

te is the shock to beta, and iβ is the mean of beta. We set ρ to equal the

average autocorrelation coefficient during our sample period. To estimate the AR(1)

coefficients, we estimate the three-year rolling betas for each stock. We then trim these

estimates at the 2.5% and 97.5% levels, and shrink them by applying a simple adjustment

rule: adjusted beta = 2/3×beta estimate + 1/3. We compute the average autocorrelation of

the betas across stocks, which equals .96. Table A.2 also presents the average time-series

standard deviations for single-factor and three-factor betas.

To generate time-varying betas, we first randomly generate the time-series mean of

each beta as we did for the constant-beta simulations. We next draw i

te from a normal

distribution with mean zero and standard deviation equal to 2ρ-1 times the average

time-series standard deviation of the corresponding beta. We then compute i

tβ through

the AR(1) specification above. We assume that i

tβ stays constant for 22 trading days for

a given month. Finally, using this time-varying factor sensitivity, we generate daily returns

by following the same simulation procedure described in section 4.A. We conduct the same

IV estimation procedure for risk premiums as in the simulations with constant betas. This

simulation procedure is used for the single factor CAPM and the Fama-French three-factor

model. Table A.3 presents the biases and RMSEs of IV risk premium estimates with time-

varying betas and Table A.4 presents the size and power of IV tests with time-varying betas.

The results here are similar to the corresponding results in Tables 1 and 2.28

28 Back et al. (2015) report somewhat different results for the small sample distribution of the test statistic

compared to the results in Table A.4. We are not able to replicate their results.

50

Table A.2

Time-varying Beta Parameters

Average time-series standard deviations of time-varying betas and their AR(1)

Coefficients (=ρ )

Single Factor Model

Fama-French

Three-Factor Model

ρ StdDe

v ρ StdDev

Betas MKT 0.96 0.15 0.96 0.15

SMB 0.96 0.19

HML 0.96 0.21

TABLE A.3

Small Sample Properties of IV Risk Premium Estimates

with Time-varying Betas

Panel A: Single-factor CAPM

Risk

Factor Estimator

Ex-ante

Bias (%)

Ex-post

Bias (%)

Ex-ante

RMSE

Ex-post

RMSE

MKT OLS -25.8 -25.4 0.191 0.139

IV -2.98 -2.56 0.180 0.079

Panel B: Fama-French Three-factor Model

Risk

Factor Estimator

Ex-ante

Bias (%)

Ex-post

Bias (%)

Ex-ante

RMSE

Ex-post

RMSE

MKT OLS -39.4 -39.3 0.234 0.204

IV -3.76 -3.71 0.194 0.090

SMB OLS -59.6 -62.4 0.144 0.148

IV 0.83 -1.96 0.132 0.096

HML OLS -61.9 -62.9 0.232 0.234

51

IV -3.29 -4.21 0.125 0.097

TABLE A.4

Size and Power of the IV Tests

with Time-varying Betas

Risk Theoretical Percentiles

Factor 1% 2.5% 5% 7.5% 10%

Panel A: Size for Single-factor Model CAPM

MKT 1.2% 2.9% 4.7% 7.3% 11.4%

Panel B: Size for Fama-French Three-factor Model

MKT 0.9% 2.5% 5.3% 7.8% 10.7%

SMB 0.8% 2.8% 5.1% 7.6% 10.4%

HML 1.3% 2.0% 4.2% 8.0% 9.5%

Risk

Factor

Test

Power

Panel C: Power for Single-factor Model CAPM

MKT 84.3%

Panel D: Power for Fama-French Three-factor Model

MKT 80.4%

SMB 56.3%

HML 89.6%

MKT or SMB or

HML 99.4%

52

Appendix 3: Innovations in Illiquidity Costs

We follow Acharya and Pedersen (2005) and fit the following time-series

regression to estimate expected and unexpected components of market-wide illiquidity

cost :])c[cc~( MKT,1τMKT,τMKT, E

,~)PILLIQ3.025(.ααPILLIQ3.025.0 τMKT,1τMKT,1-τMKT,

L

1l

l01τMKT,τMKT, c

where τMKT,ILLIQ is the value-weighted average of ),P30.0

25.030,min(ILLIQ

1τMKT,

i

τ

which

Acharya and Pederson define as un-normalized illiquidity, truncated for outliers. We cannot reject

the hypothesis that the residuals are white noise based on the Durbin-Watson tests for L=2. The

results we report are based on the application of the AR(2) model to estimate expected and

unexpected components of illiquidity for the market as well as for individual stocks. We repeat the

tests with L ranging from 2 to 6 and find that the results are not sensitive to the choice of L.

53

Appendix 4: Proof of Proposition 2

For expositional convenience, assume that the even-month beta is the independent

variable and odd-month beta is its instrument. We need to show that the correlation of true

beta ( x ) and estimated beta ( *x ) from even months is the square root of the correlation of

estimated beta ( *x ) and its instrument ( z ), i.e.,

)z,(xn correlatio)x(x,n correlatio **

where

even

* uxx odduxz

and, evenu x, and oddu are mutually independent and .2

u

2

u

2

u oddeven

By the definition of correlation,

2

u

2

x

2

x

*

**

)var(z)var(x

)z,xcov()z,(xn correlatio

.)z,(xn correlatio

)()xvar(x)var(

)xx,cov()x(x,n correlatio

*

2

u

2

x

2

x

2

x

*

**

54

Figure 1

Percentage Bias versus Time-series Length T

Single Risk Factor

This figure presents the ex-ante and ex-post biases in the estimated market risk premiums

as percentages of the true market risk premium from two sets of simulations, one when risk

premiums estimated using ordinary least squares (=OLS) regression and the other using

Instrumental Variables (=IV) regressions. All simulations are based on a risk-free rate of

0.9996%, a true market risk premium of 5.8008% per annum, The simulations use 2000

individual stocks in the cross-section. Appendix 2 describes the details of the simulation

experiments. The horizontal axis is the number of days (T) used to estimate betas. We run

1,000 repetitions for each T.

0 500 1000 1500 2000 2500 3000-45

-40

-35

-30

-25

-20

-15

-10

-5

0

5

Length of Firm Time-series (T in days)

Bia

s /

Tru

e M

KT

Pre

miu

m [

%]

Percentage Bias vs Time-series Length T

OLS (ex-ante)

IV (ex-ante)

OLS (ex-post)

IV (ex-post)

55

Table 1

Small Sample Properties of Risk Premium Estimates

Panel A presents the bias in the slope coefficients when the second stage regressions are fitted

using the OLS and Instrumental Variable (=IV) methods under the single-factor CAPM and

Panel B presents the results for the Fama-French three-factor model. Appendix 2 describes more

details of the simulations. There are 2000 stocks in the cross-section, and the results are based

on 1,000 repetitions. The sample period for the simulations is 660 months. Rolling betas are

estimated each month using daily return data over the previous 24 months. The IV method uses

data over 12 months to estimate the independent variables (betas) and data other the other 12

months to estimate the instrumental variables. Ex-ante bias is the difference between the mean

risk premium estimate and the corresponding true parameter. Ex-post bias is the difference

between the mean risk premium estimate and the sample mean of the corresponding risk factor

realizations. Ex-ante and ex-post biases are expressed as percentages of the true parameters.

Panel A: Single-factor CAPM

Risk

Factor Estimator

Ex-ante

Bias (%)

Ex-post

Bias (%)

Ex-ante

RMSE

Ex-post

RMSE

MKT OLS -28.4 -28.8 0.194 0.156

IV -0.30 -0.70 0.193 0.088

Panel B: Fama-French Three-factor Model

Risk

Factor Estimator

Ex-ante

Bias (%)

Ex-post

Bias (%)

Ex-ante

RMSE

Ex-post

RMSE

MKT OLS -42.7 -40.4 0.241 0.208

IV -2.10 0.20 0.191 0.095

SMB OLS -64.7 -66.1 0.152 0.157

IV 0.80 -0.70 0.131 0.102

HML OLS -66.3 -65.9 0.246 0.246

IV -0.40 0.00 0.134 0.109

56

Table 2

Size and Power of the IV Test

Panels A and B of this table present the test sizes under the null hypotheses that the risk

premiums equal zero using the t-statistics of the corresponding slope coefficients. The

slope coefficients are obtained from the IV estimator and t-statistics are based on their

Fama-MacBeth standard errors. Panels A and B present the results for the CAPM and for

the Fama-French three-factor model, respectively. Appendix 2 describes the details of the

simulation experiments. The number of stocks in the cross-section is set to N=2000 stocks,

and the results are based on 1,000 repetitions. The sample period for the simulations is 660

months. Rolling betas are estimated each month using daily return data over the previous

24 months, with data over 12 months to estimate the independent variables (betas) and data

other the other 12 months to estimate the instrumental variables. The simulations in Panels

C (CAPM) set market risk premium (MKT) equal to 5.8%, and that in D (Fama-French

three-factor model) sets MKT, SMB and HML equal to 5.8%, 2.64% and 4.36%,

respectively. The panels present the percentage of simulations that reject the null

hypothesis that the respective factor risk premiums less than or equal to zero at the 5%

significance level. The row labeled “MKT or SMB or HML” presents the percentage of

simulations that reject the null hypothesis that at least one of the risk premiums is less than

or equal to zero at the 5% significance level.

Risk Theoretical Percentiles

Factor 1% 2.5% 5% 7.5% 10%

Panel A: Size for Single-factor Model CAPM

MKT 0.9% 2.4% 5.1% 7.7% 9.8%

Panel B: Size for Fama-French Three-factor Model

MKT 1.1% 2.3% 4.7% 7.9% 9.9%

SMB 1.0% 2.4% 4.9% 7.6% 9.8%

HML 0.9% 2.7% 5.3% 7.3% 10.3%

Risk

Factor

Test

Power

Panel C: Power for Single-factor Model CAPM

MKT 82.8%

Panel D: Power for Fama-French Three-factor Model

MKT 78.1%

SMB 50.1%

HML 84.2%

MKT or SMB or

HML 98.2%

57

Table 3

Summary Statistics for Stock Test Data

Summary statistics include the mean, median, standard deviation, and first and third

quartiles. All other rows contain based on the time-series of the corresponding statistics.

Market capitalization is price multiplied by the number of shares outstanding. We compute

book-to-market ratios as in in Davis et al. (2000). Excess return is relative to the one-month

T-bill rate. Return volatility is the standard deviation of daily returns. The sample period

is from January 1956 through December 2012.

Mean Median Standard Deviation Q1 Q3

Number of Stocks each month 1934 1980 900 1368 2697

Time-series length 176 136 134 76 231

Capitalization, $ billion 1.498 0.191 6.509 0.052 0.751

Book-to-market ratio 0.904 0.750 0.674 0.463 1.151

Excess Return (%) 0.888 0.044 11.202 -5.391 5.996

Return Volatility (%) 2.711 2.336 1.652 1.628 3.351

58

Table 4

Correlations Among Estimated Factor Sensitivities (Betas)

and Size and Book-to-Market Characteristics

This table presents the average cross-sectional correlations among betas and size and book-

to-market ratios. Betas are estimated for each month using daily returns data from the

previous 36 months. SIZE is the natural logarithm of market capitalization and BM is the

book-to-market ratio. Panel A reports correlations for the CAPM and Panels B and C report

analogous correlations for the Fama-French three-factor model. The sample period is from

January 1956 to December 2012.

Panel A: CAPM

SIZE BM

Individual Stocks MKT -0.18 -0.20

25 Fama-French

portfolios MKT -0.56 -0.44

Panel B: Fama-French three-factor model: Individual stocks

MKT SMB HML SIZE BM

MKT 1

SMB 0.35 1

HML 0.14 0.13 1

SIZE 0.15 -0.44 -0.15 1

BM -0.12 0.06 0.28 -0.35 1

Panel C: Fama-French three-factor model: 25 size and BM sorted portfolios

MKT SMB HML SIZE BM

MKT 1

SMB -0.08 1

HML -0.08 -0.15 1

SIZE 0.19 -0.97 -0.01 1

BM 0.07 -0.03 0.88 -0.08 1

59

Table 5

Risk Premium Estimates with Individual Stocks

CAPM and Fama-French Three-Factor Model

The IV method estimates risk premiums, in percent per month, using individual stocks as

test assets. Rows labelled MKT, SMB and HML are risk premiums for the market, SMB

and HML factors, respectively and the corresponding t-statistics are in parentheses (bold if

significant at the 5% level). SIZE is the natural logarithm of market capitalization and BM

is the book-to-market ratio at the end of the previous month. Betas for each month are

estimated using daily returns data over the previous 36 months and cross-sectional

regressions are fitted using the IV method. The sample period is from January 1956 through

December 2012. N is the mean number of stocks in the monthly cross-sections.

(1) (2) (3) (4)

Panel A: 1956-2012, N=1936

Const 1.050 0.784 3.544 3.512

(7.80) (5.66) (5.07) (5.77)

MKT -0.189 -0.315 0.010 0.113

(-1.00) (-1.65) (0.05) (0.62)

SMB 0.311 -0.077

(2.09) (-0.71)

HML 0.504 0.259

(3.22) (1.77)

SIZE -0.152 -0.161

(-4.31) (-5.19)

BM 0.163 0.134

(3.50) (3.13)

Panel B: 1956-1985, N=1239

Const 1.171 0.769 3.350 3.663

(6.39) (4.25) (3.41) (3.94)

MKT -0.386 -0.394 -0.163 0.061

(-1.67) (-1.55) (-0.70) (0.24)

SMB 0.358 -0.082

(1.75) (-0.60)

HML 0.594 0.317

(2.64) (1.56)

SIZE -0.144 -0.167

(-2.80) (-3.52)

BM 0.204 0.171

(3.00) (2.65)

60

Panel C: 1986 to 2012, N=2710

Const 0.871 0.759 3.687 3.462

(4.57) (3.96) (3.77) (4.24)

MKT 0.030 -0.254 0.201 0.301

(0.10) (-0.85) (0.62) (1.07)

SMB 0.322 -0.155

(1.37) (-0.84)

HML 0.321 0.248

(1.47) (1.19)

SIZE -0.160 -0.157

(-3.32) (-3.91)

BM 0.116 0.092

(1.84) (1.67)

61

Table 6

Risk Premium Estimates with Individual Stocks

Fama-French Five-Factor Model

This table reports the risk premiums estimated using the IV method, in percent per month,

using individual stocks as test assets and the corresponding t-statistics in parentheses (bold

if significant at the 5% level). Rows labelled MKT, SMB, HML, RMW, and CMA are

risk premiums for the market, SMB, HML, RMW, and CMA factors, respectively. SIZE is

the natural logarithm of market capitalization and BM is the book-to-market ratio at the

end of the previous month. OP and INV are the operating profitability and investment/total

asset, respectively. Betas for each month are estimated using daily returns data over the

previous 36 months. Panels A, B and C report results with the IV-method. The sample

period is from January 1964 through December 2012. N is the mean number of stocks in

the cross-sections.

(1) (2) (3) (4) (5) (6)

Panel A: 1964-2012, N=2256

Const 0.891 0.886 0.901 0.891 0.954 2.753

(3.38) (3.40) (4.87) (3.50) (3.89) (4.07)

MKT -0.736 -0.125

(-2.81) (-0.52)

SMB 0.508 0.034

(2.20) (0.20)

HML 0.522 0.231

(2.12) (0.99)

RMW 0.247 -0.071 0.183 0.067

(1.45) (-0.32) (1.11) (0.33)

CMA 0.121 0.445 -0.032 0.275

(0.52) (1.60) (-0.14) (1.01)

Size -0.127

(-3.53)

BM 0.167

(3.79)

OP -0.172 0.242

(-1.16) (2.55)

INV -0.659 -0.592

(-5.42) (-7.32)

62

Panel B: 1964-1988, N=1747

Const 0.765 0.457 0.869 0.871 0.653 1.520

(2.02) (1.10) (2.81) (2.11) (1.49) (1.39)

MKT -0.741 -0.373

(-1.66) (-0.86)

SMB 0.316 0.147

(0.82) (0.66)

HML 0.464 0.270

(1.18) (0.79)

RMW 0.233 -0.437 0.17 -0.082

(1.10) (-1.48) (0.80) (-0.29)

CMA 0.794 0.619 0.583 0.062

(2.34) (1.53) (1.71) (0.16)

Size -0.08

(-1.12)

BM 0.298

(3.86)

OP -0.160 0.612

(-0.53) (3.02)

INV -0.991 -1.006

(-3.56) (-5.28)

Panel C: 1989-2012, N=2741

Const 0.983 1.053 0.884 0.957 1.081 3.453

(3.08) (3.36) (3.67) (3.13) (3.46) (4.07)

MKT -0.567 0.190

(-1.70) (0.65)

SMB 0.502 -0.046

(1.68) (-0.19)

HML 0.856 0.429

(2.68) (1.35)

RMW 0.218 -0.052 0.204 0.068

(0.84) (-0.18) (0.87) (0.26)

CMA -0.337 0.102 -0.439 0.033

(-1.16) (0.28) (-1.54) (0.09)

Size -0.152

(-3.79)

BM 0.070

(1.25)

OP -0.202 0.086

(-1.48) (0.93)

INV -0.453 -0.351

(-5.23) (-5.56)

63

Table 7

Correlations Among Estimated Betas

and SIZE and Book-to-market Ratios: The q-factor Asset Pricing Model

This table reports time series averages of cross-sectional correlations among I/A and ROE

betas and size and book-to-market ratios. Betas are estimated for each month using daily

returns data from the previous 36 months. SIZE is the natural logarithm of market

capitalization and BM is the book-to-market ratio. Panel A reports the results for individual

stocks and Panel B reports the results for 25 Fama-French size and book-to-market sorted

portfolios. The sample period is January 1972 to December 2012.

Panel A: Individual stocks

MKT I/A ROE SIZE BM

MKT 1

I/A 0.04 1

ROE -0.03 0.33 1

SIZE 0.24 -0.05 0.12 1

BM -0.17 0.09 -0.07 -0.32 1

Panel B: 25 Fama-French size and BM sorted portfolios

MKT I/A ROE SIZE BM

MKT 1

I/A -0.70 1

ROE -0.69 0.52 1

SIZE -0.44 0.04 0.74 1

BM -0.48 0.88 0.29 -0.08 1

64

Table 8

Risk Premium Estimates and Characteristics

The q-factor Asset Pricing Model

This table reports the risk premiums estimated using the IV method, in percent per month,

using individual stocks as test assets and the corresponding t-statistics in parentheses (bold

if significant at the 5% level). Rows labeled MKT, ME, I/A, and ROE report factor risk

premium estimates. Rows labeled SIZE and BM contain mean cross-sectional slope

coefficients on the natural logarithm of market capitalization and the book-to-market ratio

at the end of the previous month. Betas for each month are estimated using daily returns

over the previous 36 months. The sample period is January 1972 through December 2012.

N is the average number of stocks.

(1) (2) (3) (4) (5) (6) (7) (8)

Panel A: 1972-2012, N=2431

Const 1.157 0.764 0.958 0.881 1.153 0.563 4.253 3.465

(7.11) (4.45) (3.65) (3.65) (6.17) (2.35) (5.23) (4.45)

MKT -0.150 -0.639 -0.064

(-0.56) (-2.30) (-0.27)

ME 0.263 0.252 -0.157

(1.24) (1.12) (-0.97)

I/A 0.320 -0.074 0.246 -0.114

(1.73) (-0.34) (1.36) (-0.54)

ROE -0.313 -0.796 -0.170 -0.508

(-1.33) (-2.80) (-0.75) (-1.86)

Size -0.173 -0.143

(-4.66) (-3.75)

BM 0.286 0.199

(4.41) (4.07)

Panel B: 1972-1992, N=2082

Const 1.081 0.794 0.991 0.812 1.051 0.575 4.270 3.152

(4.21) (2.91) (2.67) (2.30) (3.99) (1.56) (3.22) (2.61)

MKT 0.070 -0.412 0.114

(0.21) (-1.19) (0.35)

ME 0.390 0.430 -0.083

(1.44) (1.34) (-0.41)

I/A 0.831 -0.205 0.526 -0.214

(3.32) (-0.72) (2.19) (-0.76)

ROE -0.207 -0.832 -0.249 -0.561

(-0.76) (-2.35) (-0.94) (-1.61)

Size -0.178 -0.138

(-2.84) (-2.19)

BM 0.289 0.178

(3.07) (2.60)

65

Panel C: 1993-2012, N=2768

Const 1.351 0.875 0.853 0.921 1.250 0.601 4.055 3.601

(5.86) (3.43) (2.47) (2.79) (4.62) (1.56) (4.41) (3.59)

MKT -0.352 -0.848 -0.200

(-0.84) (-2.00) (-0.57)

ME 0.235 0.196 -0.323

(0.70) (0.62) (-1.30)

I/A 0.056 0.105 0.066 -0.073

(0.22) (0.31) (0.26) (-0.23)

ROE -0.206 -0.517 -0.064 -0.476

(-0.54) (-1.11) (-0.17) (-1.12)

Size -0.165 -0.144

(-4.01) (-3.12)

BM 0.335 0.228

(3.74) (3.31)

66

Table 9

Risk Premium Estimates with Individual Stocks: Liquidity-adjusted CAPM

This table reports the risk premiums estimated using the IV method, in percent per month,

using individual stocks as test assets and the corresponding t-statistics in parentheses (bold

if significant at the 5% level). The row labeled LMKT reports estimated illiquidity risk

premiums under the liquidity-adjusted CAPM (LCAPM), and the row labeled Amihud

illiquidity reports the slope coefficient on firm-specific Amihud illiquidity. The slope

coefficients are in percent per month, and the corresponding t-statistics are in parentheses

(bold at the 5% level). N is the average number of stocks.

Sample Period

1956-2012, N=1265 1956-1985, N=1192 1986-2012, N=1344

Constant 0.641 0.551 0.811 0.657 0.431 0.358

(4.64) (4.07) (4.38) (3.84) (1.83) (1.92)

LMKT

0.140 0.075 -0.086 -0.136 0.462 0.300

(0.63) (0.34) (-0.27) (-0.44) (1.47) (0.97)

Amihud

Illiquidity

0.184 0.310 0.040

(3.89) (3.53) (1.91)

67

Table 10

Strength of Instruments

This table presents the average correlations between odd- and even-month factor loading

estimates (betas) under the models indicated in panel headings. The critical value for the

weak instruments tests proposed by Nelson and Startz (1990) is .06, based on the smallest

number of stocks in the sample in any month. The square root of the odd- and even-month

correlation is the correlation between the unobservable “true” betas and the corresponding

beta estimates; (See section 5.F and Appendix 4.)

Panel A: CAPM

Sample period Corr(Odd, Even) Corr(True Beta, Beta Est.)

MKT MKT

1956-2012 0.67 0.82

Panel B: Fama-French Three-factor Model

Sample period Corr(Odd, Even) Corr(True Beta, Beta Est.)

MKT SMB HML MKT SMB HML

1956-2012 0.52 0.44 0.30 0.71 0.66 0.54

Panel C: Fama-French Five-factor Model

Sample

period

Corr(Odd, Even) Corr(True Beta, Beta Est.)

MKT SMB HML RMW CMA MKT SMB HML RMW CMA

1964-2012 0.42 0.35 0.19 0.18 0.14 0.65 0.59 0.44 0.43 0.38

Panel D: q-factor Asset Pricing Model

Sample

period

Corr(Odd, Even) Corr(True Beta, Beta Est.)

MKT ME I/A ROE MKT ME I/A ROE

1972-2012 0.48 0.38 0.20 0.21 0.69 0.62 0.45 0.46

Panel E: Liquidity-adjusted CAPM (LCAPM)

Sample period Corr(Odd, Even) Corr(True Beta, Beta Est.)

LMKT LMKT

1956-2012 0.58 0.76