Commodity Prices, Convenience Yields, and In�ation

Nikolay Gospodinov� Serena Ng y

February 2010

Abstract

This paper provides a framework for analyzing the determinants of commodity prices andtheir predictive ability for in�ation. We �nd that the common variations in convenience yields,constructed as the principal components from a panel of 23 commodity price futures, havestatistically and quantitatively important predictive power for in�ation even after controllingfor unemployment and oil prices. The results hold up in out-of-sample forecasts, across forecasthorizons, and across G-7 countries. The principal components in convenience yields are shownto be informational variables that are correlated with future U.S. consumption demand and�nancial markets as well as economic conditions of fast growing economies. The recent run-up in commodity prices cannot, however, be explained by convenience yields, interest rates orexchange rates, lending support to the view that speculation is behind the commodity priceincreases. We also propose a bootstrap procedure for conducting inference when the principalcomponents are used as regressors.

Keywords: Convenience yields; Principal components; In�ation; Predictability.

JEL Classi�cation: C53, E37, G12.

�Concordia University and CIREQ, 1455 de Maisonneuve Blvd. West, Montreal, QC H3G 1M8, Canada. Email:nikolay.gospodinov@concordia.ca

yColumbia University, 420 W. 118 St. MC 3308, New York, NY 10027. Email: serena.ng@columbia.eduWe would like to thank Lutz Kilian for helpful comments. We also thank Ibrahim Jamali and Van Hai Nguyenfor research assistance. The second author acknowledges �nancial support from the National Science Foundation(SES-0549978).

1 Introduction

In�ation is a notoriously di¢ cult series to forecast. This paper develops a framework for analyzing

the relation between commodity prices and in�ation. We construct convenience yields from a panel

of 23 commodity price futures. We �nd that while the individual commodity prices and convenience

yields are weak predictors of in�ation, the common variations in convenience yields, estimated by the

method of principal components, have systematic predictive power for in�ation. We attribute this

�nding to the ability of the principal components to aggregate economic information as contained in

the futures market, and such information is apparently not captured by variables being considered

in the Phillips curve.

Monetary authorities seem to hold a long standing view that commodity prices have in�ationary

consequences and thus the ability to predict future commodity price movements can be important

for the time path of economic policies. As the Federal Reserve Chairman Ben Bernanke (2008)

remarked,

Rapidly rising prices for globally traded commodities have been the major source of the

relatively high rates of in�ation we have experienced in recent years, underscoring the

importance for policy of both forecasting commodity price changes and understanding

the factors that drive those changes (Speech at the FRB of Boston, June 9, 2008).

In spite of the general view that commodity price movements have in�ation implications, the

formal link between in�ation and commodity price is not thoroughly understood. Some argue that

commodity prices are leading indicators of in�ation because they respond more quickly to gen-

eral economic conditions. Others believe that idiosyncratic movements in commodity prices work

through the distribution channel and subsequently a¤ect prices in general. While both explanations

are plausible, the magnitude of the commodity price e¤ect on in�ation must necessarily depend on

what triggers changes in commodity prices. In particular, commodity prices can change as a result

of transactions, speculative, or precautionary demand, but these demand components can have a

di¤erent impact on in�ation.

At empirical level, evidence on the relation between commodity prices and in�ation, based

on either a composite commodity price indicator or the price of oil, is far from robust. While

commodity prices appear to improve the in�ation forecasts upon a simple AR benchmark for

certain countries and periods, Stock and Watson (2003) show that these forecasting improvements

are sporadic and unstable. Hooker (2002) �nds empirical evidence of a statistically signi�cant

impact of oil prices on core U.S. in�ation over the period 1962-1980 but documents a breakdown

in the relationship for the post 1981 period. Several other studies also report an improved ability

1

of commodity prices for forecasting in�ation until the mid 1980s but a substantial deterioration in

the predictive power of commodity prices from 1985 onwards.1

Our approach di¤ers from the existing work in the literature in that we decompose the commod-

ity price variations into a convenience yield component, an interest rate component, an exchange

rate component, and a component that is orthogonal to the three terms. The convenience yield

component re�ects, by construction, expectational information incorporated in futures commodity

prices. While the evidence of the predictive power of futures markets for commodity and gen-

eral price movements is somewhat mixed (Alquist and Kilian, 2010), it seems unreasonable �to

ignore the substantial amount of information about the supply and demand conditions that are

aggregated by futures markets� (Bernanke, 2008). We use information in the futures market in

a particular way:- we isolate the common variations in the convenience yields. We �nd that the

predictive power of the principal components in convenience yields for commodity prices is strong,

and is quantitatively and statistically more important than the interest rate and exchange rate.

Put di¤erently, not every determinant of commodity prices has strong predictive power for in�a-

tion. This is broadly in line with the �nding in Kilian (2009) that the macroeconomic e¤ects of

oil price shocks depends on the source of the shock. The predictive power of principal components

in convenience yields is stable over time and holds up even when demand pressure variables such

as unemployment gap and oil price are controlled for. A decomposition of variance reveals that

the predictive power of the principal components in convenience yields are not only statistically

signi�cant, but also quantitatively important. In contrast, the IMF aggregate commodity index

has little predictive power for in�ation. We attribute this �nding to the ability of the principal

components to isolate those variations in commodity prices that have in�ationary consequences.

The principal components for convenience yields also explain in�ation rates of the rest of the G-7

countries. However, the run up in commodity prices in 2007 and 2008 is too large to be consistent

with the prediction of a regression with coe¢ cients on convenience yields, exchange and interest

rates estimated from 1983-2008. Either the regression function exhibits instability over this sample,

or the commodity price changes are speculative.

The rest of the paper proceeds as follows. Section 2 establishes a relationship between commod-

ity prices and convenience yields from which a relation between in�ation and convenience yields

is obtained. The data and the construction of principal components are discussed in Section 3.

Predictive regressions for commodity prices and in�ation are presented in Section 5. Section 6

takes a closer look at the explanatory power of the principal components. We then show that

these principal components in the convenience yields contain information about future economic

conditions in the U.S. and at a global level. We also propose a bootstrap procedure to assess the

1See Blomberg and Harris (1995), Furlong and Ingenito (1996), among others.

2

sampling error from principal components estimation without imposing a factor structure on the

data. This is presented in Section 4 for the interested readers.

2 The Determination of Commodity Prices and In�ation

The dynamics of commodity price movements are of interest for a variety of reasons. For developing

countries that depend heavily on exports of commodities for revenue, �uctuations in commodity

prices are the main cause of income volatility. In other countries, commodity price movements are

often tied to real exchange rate appreciation and depreciation, depending on whether the country is

an exporter or importer of commodities. For small open economies like Canada, New Zealand and

Australia, �uctuations in commodity prices are su¢ ciently important for the design of economic

policies that these central banks produce their own commodity price indices to re�ect the dynamics

of the appropriate commodities that are being produced by the country.

Commodities share similar characteristics with money in that it is held for everyday use, it can

be stored, and it is also an asset. It is thus useful to think of the demand for commodities as arising

from one of three sources: (i) a component tied to current consumption and production; (ii) a

precautionary demand component that re�ects future needs, and (iii) an asset demand component

that depends on its risk over the holding period relative to its potential for capital gains. Williams

(1937) and Samuelson (1957) suggest that the market clearing price for commodities will depend

on availability (new production plus inventories) relative to total demand (current, precautionary,

plus asset demand).

Our analysis focuses on the so-called convenience yield to holding commodities. Kaldor (1939)

uses the concept of a convenience yield to re�ect the bene�t from using the stored commodity

whenever desired. When the timing or the level of consumption are stochastic, holding commodity

inventories could smooth the production process and hedge against unexpected price movements

or demand shifts. Routledge, Seppi and Spatt (2000) extend the rational expectations competitive

storage model of Deaton and Laroque (1992) and show that the existence of convenience yields

can be linked to the probability of a stockout of inventories. According to this precautionary

demand view, convenience yield is a function of the inventory levels and acts as insurance against

a potential supply shortfall. Gorton, Hayashi and Rouwenhorst (2007) provide empirical evidence

that convenience yields are a decreasing and possibly nonlinear function of inventories. The more

recent literature gives the convenience yield an option-like interpretation that arises from the timing

option embedded in the stored commodity (see, for example, Evans and Guthrie, 2008). Alquist

and Kilian (2010) suggest that inventories in crude oil is re�ected in convenience yields and hence

contain information about oil futures.

Pindyck (1993) develops a model of rational commodity pricing in which the convenience yield

3

incorporates all the information about the fundamentals. In his model, convenience yields play the

role of dividends. As the spot price is the expected discounted future �ow of convenience yields,

the latter should anticipate future changes in spot commodity prices. Instead of using the present

value model for derivative pricing as in Gibson and Schwartz (1990), we use the model to motivate

a link between commodity prices, interest rates, exchange rates, and convenience yields. This is

formalized in the next subsection.

2.1 A Model for Commodity Prices

Let Sjt and Fjt;n denote the spot and futures price of commodity j for delivery at time t + n.

Also, let it;n be the nominal interest earned between period t and t+ n. De�ne the basis to be the

di¤erence between the spot and the futures price, Fjt;n � Sjt. There are two views for the role of

the basis. Instead of treating these as competing hypotheses, we will exploit both.

The �rst view due to Kaldor (1939), Working (1949) and Brennan (1958), posits that negative

basis Fjt;n�Sjt consists of two components: (i) an opportunity cost of forgone interest from havingto borrow and buy the commodity and (ii) a (net of insurance and storage costs) convenience yield

Cjt;n with the convention that Cjt;0 = 0: Thus as in Fama and French (1987, 1988),

Fjt;n � Sjt = Sjtit;n � Cjt;n: (1)

Note that Cjt;n is a forward looking variable to the extent that convenience yields anticipates the

future possibility of stockouts.

An alternative view of the basis is provided by the theory of normal backwardation according

to which risk averse investors earn a risk premium for the �uctuations in the future spot price

that commodity pricers want to hedge. The basis is then comprised of a risk premium component

jt;n � EtSjt+n � Fjt;n and an expected price change component EtSjt+n � Sjt so that

Fjt;n � Sjt = EtSjt+n � Sjt �jt;n: (2)

When jt;n is positive, futures price is backwardated (at a discount). In the storage model, low

inventory leads to low basis and subsequently to low returns on owning the commodity.

Whereas (1) implicitly de�nes the convenience yield, (2) implicitly de�nes the risk premium.

Together, (1) and (2) imply

EtSjt+n � Sjt = Sjtit � Cjt;n +jt;n:

Let cjt;n = Cjt;n=Sjt, jt;n = jt;n=Sjt, sjt = log(Sjt), and Et�nsjt+n = Etsjt+n � sjt. Then

Et�nsjt+n = it;n + jt;n � cjt;n: (3)

4

According to (3), expected changes in log commodity prices have three components: (i) a component

it;n related to the opportunity cost of buying and holding inventories, (ii) a risk premium component

jt;n, and (iii) an expected marginal convenience yield component cjt;n. The three components are

not mutually uncorrelated as (1) and (2) are alternative decompositions of the basis. Now de�ne

�jt;n = jt;n � Proj( jt;njit;n; cjt;n)

to be the component of jt;n that is orthogonal to the interest rate and the convenience yield.

Given that this is the risk premium component that is orthogonal to the fundamentals underlying

commodities viewed as assets, �jt;n can be thought of as a pure speculative component. Equation

(3) can be represented as

Et�nsjt+n = �j1cjt;n + �j2it;n + �jt;n: (4)

Because �j1 and �j2 are now �reduced form�coe¢ cients, they are not constrained to be -1 and 1.

In this setup, convenience yield bears information about future demand for both goods and assets.

We view it as an informational variable about future economic conditions as seen by the commodity

markets.

So far, the two fundamentals of commodity prices are the interest rate, which is not commodity

speci�c, and convenience yield, which is commodity speci�c. Barsky and Kilian (2002) suggest that

lower interest rates will raise commodity prices. Frankel (2006) formalizes the idea and show that

in an open economy, a monetary expansion in country k will lead to a lower interest rate, causing

commodity prices in local currencies to rise until an expected depreciation restores equilibrium.

But irrespective of the number of commodities, there is only one bilateral exchange rate that must

adjust to o¤set the lower real interest rate relative to all convenience yields. This suggests that an

open economy extension of (4) can be written as

Et�nsjt+n = �j1cjt;n + �j2ijt;n + �j3�

nxkt + �jt;n; (5)

where xkt is the nominal exchange rate between the U.S. and country k. Chen, Rogo¤ and Rossi

(2010) �nd exchange rates to forecast commodity prices.

2.2 In�ation and Commodity Prices

The previous subsection provides a framework for understanding the determinants of expected

changes in commodity prices. Accounting for hedging and open economy decisions, expected ap-

preciation in commodity price is now linear in a convenience yield component cjt;n, an interest rate

component, an exchange rate component, and a speculative component �jt that is orthogonal to

the convenience yield, interest rate, and exchange rate. This section provides a link from these

determinants of commodity prices to in�ation.

5

Let pt denote the logarithm of the economy�s general price index and let qkjt be de�ned as the

real price of commodity j in the currency of country k relative to its equilibrium. In other words,

if ~qkjt = sjt� pkt , qkjt = ~qkjt� qk�jt , where q�jt is the equilibrium value of ~qkjt. Our starting point is that

if real commodity prices are mean reverting, we can write

E�npt+n = �jqjt + Et�nsjt+n:

This model for in�ation is essentially Dornbusch�s (1976) overshooting model extended to commod-

ity prices by Frankel (2006). In a closed economy, after substituting for Et�nsjt+n; the expression

in (4) becomes

Et�npt+n = �jqjt + �j1cjt;n + �j2it;n + �jt;n:

If �j1 = ��j2 = �1, rearranging terms implies that the disequilibrium in real commodity price

qjt is inversely proportional to the real carrying cost, which equals real interest rate minus net

convenience yield. The open economy analog is

Et�npt+n = �jqjt + �j1cjt;n + �j2it;n + �j3�

nxkt + �jt;n: (6)

We use this model of Frankel (2006) for real commodity prices in a di¤erent way. Observe that

pt is the general price level, but there is a convenience yield corresponding to every commodity, and

an exchange rate with every trading partner. Let ct denote an aggregate measure of convenience

yields and qt be an aggregate measure of disequilibrium real commodity price. Also let xt be a

weighted average of exchange rates. The aggregate analog to equation (6) is given by

Et�npt+n = �qt + �1ct + �2it;n + �3�

nxt + �t;n: (7)

This in�ation model (7) posits that the expected n period ahead in�ation is a function of the

aggregate disequilibrium in real commodity price, aggregate convenience yield, and interest rate.

Note that the use of aggregate convenience yield for predicting in�ation is new and has not been

previously explored in the literature. To the extent that convenience yield measures excess demand

as seen by participants in the commodities market, it is an alternative to the goods market view

of excess demand.

3 Data

This section discusses the data that will be used in the empirical work. The daily commodity price

data are obtained from the Commodity Research Bureau and are available at daily frequency for the

period March 1983 to July 2008. The data set contains spot and futures prices of 23 commodities

6

from 6 commodity groups: energy (crude oil, heating oil), foodstu¤s (cocoa, co¤ee, orange juice,

sugar), grains and oilseeds (canola, corn, oats, soybeans, soybean oil, wheat), industrials (cotton,

lumber), livestock and meats (cattle feeder, cattle live, hogs lean, pork bellies) and metals (copper,

gold, palladium, platinum, silver). A description of the trading characteristics of these commodities

is provided in Table A.1 in Appendix A. This particular choice of a cross section of commodity

prices and time period is dictated by data availability. The spot price is approximated by the price

of the nearest futures contract and the futures price is the price of the next to the nearest futures

contract. It should be noted that the time separating the nearest and next to the nearest futures

contracts typically di¤ers across commodities and may not be equally spaced over the course of a

year (see Table A.1). As a result, we do not match exactly the contract maturities with the forecast

horizon. It might prove bene�cial to augment further the data set with longer maturity contracts

(for example, three-month or one-year contracts for energy commodities) but we do not pursue

this due to lack of continuous and liquid record of longer maturity futures prices over our sampling

period. Some of the properties of commodity futures as an asset class are summarized in Gorton

and Rouwenhorst (2006).

As consumer price data are observed at no higher than monthly frequency, monthly commod-

ity price series are constructed from daily data by averaging the daily prices in the corresponding

month.2 The real commodity prices qjt are obtained by de�ating the spot prices by the U.S. CPI

(seasonally adjusted) index obtained from the Bureau of Labor Statistics (BLS). The 3-month U.S.

T-bill rate and the exchange rate data (U.S. dollar trade weighted index) are from the Federal

Reserve Economic Database FRED R . We use the IMF commodity non-fuel price index to approx-imate the aggregate behavior of commodity spot prices.3 The detrended real commodity price qjt

is obtained by applying the HP �lter to sjt � pt.4

As noted earlier, convenience yield is a forward looking variable. We proxy this forward looking

information using data on commodity price futures. Speci�cally, the percentage net convenience

yield for commodity j is computed as

cjt;n =(1 + it;n)Sjt � Fjt;n

Sjt; (8)

using the three-month U.S. Treasury bill as it;n, adjusted for the time that separates the two

futures contracts. Note that while commodity spot and futures prices may be volatile and subject

2Empirical results for end-of-period data and quarterly data are also available from the authors upon request.3The data for CPI, interest and exchange rates, and commodity price index can be downloaded from

http://research.stlouisfed.org/fred2/ and http://www.imf.org/external/np/res/commod/index.asp.4The smoothing parameter for the HP �lter throughout the paper is set to its default value for monthly data of

14,400. Results based on data detrended by a one-sided (exponentially weighted) moving average �lter are availablefrom the authors upon request and do not appear to be sensitive to the choice of a detrending method. We implementone-sided �lters in Section 6.3 for the purposes of out-of-sample forecasting.

7

to occasional spikes, the convenience yield (being the di¤erence between the two series) should be

less noisy. Statistically, convenience yield should be stationary even though the spot and future

prices might be non-stationary processes.

The mean, standard deviation and �rst-order autocorrelation coe¢ cient of the convenience

yields and detrended real commodity prices are reported in Table A.2 in Appendix A. The con-

venience yields exhibit relatively high persistence, with most of the �rst-order autocorrelation

coe¢ cients being in the range 0.7-0.9 (the largest autocorrelation is 0.936 for co¤ee). As expected,

the convenience yields in precious metals such as gold and silver are very small with variance that

is orders of magnitude smaller than the convenience yield variance for the other commodities. The

convenience yields of sugar, oats, hogs, pork bellies and industrials appear to be most volatile.

Fama and French (1987) argue that the di¤erences in the variability of convenience yields (basis)

arise from di¤erent degree of seasonality in demand and supply as well as di¤erences in adjustment

to demand and supply shocks. The detrended real commodity prices are also persistent (with an

autocorrelation coe¢ cient close to 0.9) with foods being the most volatile commodity group.

3.1 Constructing ct and qt by Principal Components

Because of the heterogeneous nature of the commodities, one can expect the e¤ects of an aggregate

measure of commodity prices on in�ation to be di¤erent than those of the individual commodity

prices. Boughton and Branson (1991) have argued that an aggregate measure of commodity price

may not be a reliable leading indicator of in�ation when the individual commodities are a¤ected

by idiosyncratic supply shocks that are not accommodated by the monetary authorities. They

conjectured that better predictors for future in�ation movements can be constructed by some

transformations of the commodity price variables that are driven primarily by demand shocks. We

now propose one way of extracting information from the individual commodity prices.

In our model for expected commodity price changes, variations in the interest rate it;n are

common to all commodities. While cjt;n is commodity speci�c by de�nition, its variations are likely

to have a common and an idiosyncratic component. One possible transformation of the individual

commodity prices is to average over the convenience yields to isolate the common variations, but

simple averaging might not yield optimal weights in view of the the heterogeneous nature of the

market-traded commodities. To this end, we extract principal components from our panel of 23

convenience yields. Note that for this exercise, it is not required that the convenience yields have a

factor representation. We are simply using principal components as a dimension reduction device.

If convenience yields are indeed driven by latent common factors, then the principal components

consistently estimate the space spanned by the common factors as the number of commodity prices

tends to in�nity. Pindyck and Rotemberg (1990) �nd excess co-movement in commodity prices,

8

while Tang and Xiong (2009) show that commodity prices have been increasingly exposed to macro

shocks; a common factor representation of commodity prices is defensible though this is not crucial

to the analysis.

The �rst r principal components of the convenience yields are the eigenvectors corresponding

to the largest eigenvalues of the N �N matrix (NT )�1c0ncn, where cn is a T �N matrix of conve-

nience yields, with N = 23. By construction, these principal components are mutually orthogonal.

Similarly, principal components are also extracted from the panel of 23 detrended real commodity

prices qjt for j = 1; :::; N . We denote these by ct and qt, respectively. The �rst two principal

components ct = (c(1)t ; c

(2)t )

0 explain 23% of the variance in convenience yields. Both c(1)t and c(2)tare persistent with �rst-order autocorrelation coe¢ cients of 0.93 and 0.81, respectively. Similarly,

the two principal components for disequilibrium commodity prices qt = (q(1)t ; q

(2)t )

0 explain slightly

more than 31% of the variance of the panel of qjt with �rst-order autocorrelation coe¢ cients on

the estimated principal components of 0.90 and 0.88, respectively.

The marginal R2, obtained from regressing each cjt;n on ct, and qjt on qt are presented in

Table 1. The �rst element of ct loads heavily on corn, co¤ee, cotton, wheat and crude oil while

the second principal component is highly correlated with some metals (silver, copper) as well as

soybeans, cocoa and heating oil. The �rst element of qt is closely associated with the commodity

group of grains and oilseeds while its second element captures the price variation of the metals

group. However, neither of these principal components appears to be strongly correlated with the

energy commodities. We will return to the interpretation of ct subsequently.

4 Bootstrap Inference

Since ct and qt are used as regressors in the commodity price (5) and in�ation (7) models, we need

to account for the sampling uncertainty associated with the estimation of the principal components.

In this paper, we propose a bootstrap method for conducting inference on the parameters in the

predictive regression. Several bootstrap procedures for factor models have been suggested (with

no proof of validity) for idiosyncratic errors that are iid across units and over time, which is too

restrictive for the data being analyzed. As well, we want to allow but does not want to impose

a factor structure on the data. In other words, we want to leave open the possibility that the

principal components are simply weighted averages of the individual convenience yields, which

are meaningful predictors in their own right. Given these considerations, a bootstrap procedure

for estimation using principal components as predictors seems more appropriate. Again, some

procedures have been suggested for iid data without a proof of their asymptotic validity.

Let x be a genericN�T data matrix, where xit (i = 1; :::; N; t = 1; :::; T ) denotes the ith observedseries at time t; N is the total number of variables (convenience yields, detrended real commodity

9

prices) and T is the number of time series observations. The �rst r principal components of matrix

x; denoted by xt, are the eigenvectors corresponding to the largest eigenvalues of the N �N matrix

(NT )�1xx0.

Our interest lies in conducting statistical inference in the predictive regression

yt+h = �0xt + 0wt + "t+h;

where wt denotes a p � 1 vector of other observable predictors (interest rate, exchange rate, realoil price) as well as deterministic terms and lag values of yt; and the errors "t+h are possibly

autocorrelated and heteroskedastic. The predictive regression is estimated by OLS and the inference

procedure on the estimated parameters should potentially take into account that xt are �generated�

regressors.

Under suitable regularity conditions (Bai and Ng, 2006), the OLS estimator (�0; 0)0 is root-

T consistent and asymptotically normal. Furthermore, the presence of generated predictors xt

does not require any adjustments to the standard errors of the parameter estimates provided thatpT=N ! 0: Unfortunately, in our analysis the cross-sectional dimension N is relatively small and

the regularity conditions of Bai and Ng (2006) do not seem to hold. As a result, we resort to

bootstrap methods for inference that account for uncertainty associated with the estimation of

principal components. Our proposed bootstrap algorithm is based on a moving block resampling

of the original data.

More speci�cally, we stack the data from both stages, x1t, .., xNt, yt+h and wjt; j = 1; :::; p,

(after truncating the last h observations of xit); into the matrix

Z =

266664x11 ::: xN1 yh+1 w11 ::: wp1x12 ::: xN2 yh+2 w12 ::: wp2::: ::: ::: ::: ::: ::: :::::: ::: ::: ::: ::: ::: :::

x1(T�h) ::: xN(T�h) yT w1(T�h) ::: wp(T�h)

377775 :

The bootstrap samples (x�1t; :::; x�Nt; y

�t+h; w

�1t; :::; w

�pt) for t = 1; :::; T � h are then generated by

drawing with replacement blocks of m = mT 2 N (1 � m < T ) observations from matrix Z. This

ensures that the bootstrap samples preserve possible model misspeci�cation, serial correlation,

heteroskedasticity and cross-sectional dependence in the data: The block size m is allowed to grow,

but at a slower rate, with the time series dimension T .

Let zt be the tth row of the data matrix Z above. Also, let Bt;m = (zt; zt+1; :::; zt+m�1) denote

a block of m consecutive observations of zt; k = [T=m], where [a] signi�es the largest integer

that is less than or equal to a, and T = km. We resample with replacement k blocks from

(B1;m; B2;m; :::; BT�m+1;m) by drawing k iid uniform random variables [u1]; :::; [uk] on (1; k + 1):

10

Then, the bootstrap sample is given by Z� = [(z�1 ; z�2 ; :::; z

�m); (z

�m+1; z

�m+2; :::; z

�2m); :::; (z

�T�m;

z�T�m+1; :::; z

�T)] = (B[u1];m; B[u2];m; :::; B[uk];m):

For each bootstrap sample fx�itg (i = 1; :::; N; t = 1; :::; T ), the r principal components, denotedby x�t ; are re-estimated as the largest eigenvalues of the N �N matrix (NT )�1x�x�0 and plugged

into the predictive regression for the bootstrap data

y�t+h = ��0t x

�t +

�0w�t + "�t+h;

where ��and � are OLS estimates.

Let �l (l = 1; :::; r) denote the estimated coe¢ cient on the lth principal component from the

observed sample and ��l;j denote the estimated coe¢ cient from the jth bootstrap sample with

corresponding standard errors s:e:(�l) and s:e:(��l;j) computed using a HAC estimator. Due to the

sign indeterminacy of the principal components, we set the sign of ��l;j to be the same as the sign of

�l estimated from the original sample. We then construct the sequence t��l;j = (��l;j � �l)=s:e:(�

�l;j);

sort it in ascending order and let v�� and v�(1��) denote the �th and (1 � �)th elements of the

sorted sequence for a pre-speci�ed nominal level �. The 100(1 � �)% equal-tailed percentile-t

bootstrap con�dence interval for �l is obtained as [�l�s:e:(�l)v�(1��=2); �l�s:e:(�l)v��=2]: In addition

to better approximating the small-sample distribution of �l, these bootstrap con�dence intervals

take into account the estimation uncertainty for the generated regressors (principal components).

Con�dence intervals for the remaining coe¢ cients are obtained in a similar manner. Note that

this bootstrap procedure allows for possible asymmetry in the �nite-sample distribution of the

parameter of interest. In empirical analysis, we set m = 4 although the results are similar for other

values of m in the range m 2 [4; 24]. The number of bootstrap replication B is 4,999.

Since this inference procedure is used for both the aggregate commodity price and in�ation equa-

tions, in the practical implementation we stack the two dependent variables, 4hst+h and 4hpt+h,

along with all regressors in the matrix Z which is resampled as described above. We then select

the appropriate columns of the bootstrap matrix Z� to conduct the principal component analysis

and estimate the corresponding predictive regressions. Some Monte Carlo simulation results on the

coverage rates of the proposed bootstrap con�dence intervals are presented in Appendix B.

5 Estimation Results

This section �rst assesses the relative importance of the determinants of individual commodity

prices sjt and the aggregate (IMF) commodity price index sIMFt . We then turn to a more detailed

analysis of the in�ation rates �pht in the U.S. and the other G-7 countries.

11

5.1 Commodity Prices

The key predictors in the commodity price regressions are individual convenience yields cjt;n, the

�rst two principal components of the 23 convenience yields ct = (c(1)t ; c

(2)t )

0, interest rate it, and

the �rst principal component in exchange rate xt. Recall also that the convenience yields are

constructed from futures price data.

Our �rst set of results is based on the predictive regression

�sjt+1 = a+ �j0(L)�sjt + �j1(L)cjt;n + �j2(L)it + �j3(L)�xt + ejt+1;

where we recall that �sjt+1 is the log change in the price for commodity j, cjt;n is the convenience

yield for commodity j, it is the nominal interest rate and�xt is the log change in the trade-weighted

USD exchange rate index. Our preliminary speci�cation analysis suggests that �j1(L) = �j1;

�j2(L) = �j2; �j2(L) = �j3 and �j0(L) is a second-order polynomial in the lag operator. For

each of the three determinants of commodity price changes, we test the null of no predictability

conditional on the presence of the remaining two determinants. This leads to three hypotheses.

The �rst test is for the null of no predictability of convenience yields in the presence of interest

and exchange rates, or H0 : �j1 = 0. The second hypothesis is non-predictability of interest

rate for changes in commodity prices in the presence of convenience yields and exchange rates, or

H0 : �j2 = 0. The third test is for lack of predictability of exchange rates H0 : �j3 = 0.

The estimation results are presented in Table 2 with statistically signi�cant coe¢ cients (at 10%

level) in bold font. Several observations can be made. First, marginal convenience yields have highly

statistically signi�cant predictive power for most commodity prices but the interest and exchange

rates do not. The exceptions are those commodities in the metals group whose components are

somewhat more substitutable. In these cases, the interest and exchange rates exhibit predictive

power. There is evidently substantial heterogeneity in the dynamic response of commodity prices

to convenience yields. The last row of Table 2 reports results from a regression of the aggregate

commodity price changes (IMF index) on a simple average of the convenience yields in addition to

interest rate and exchange rate. Non-predictability of the average convenience yield and exchange

rate is rejected. In contrast, interest rate has no predictive power once we control for the presence

of convenience yields and exchange rates.

As the futures contract maturities for di¤erent commodities vary between 1 month (for energy

commodities) to 5 months (for sugar), the convenience yields and the principal components ex-

tracted from them are expected to have predictive power not only for one-month but also for longer

horizons. For this reason, we also vary the prediction horizon, denoted by h, which may di¤er from

the futures contract maturity n. The h-period predictive regression for aggregate commodity price

12

is then given by

�hsIMFt+h = a+ �0(L)�s

IMFt + �01ct + �2it + �3�xt + et+h: (9)

The speci�cation analysis selects two lags for the dependent variable and reveals no need for extra

lags for the predictors. The results for di¤erent forecasting horizons (h = 1; 3; 6; 12) are presented in

Table 3. Since the cross sectional dimensionN is relatively small and ct are �generated regressors�in

the predictive regressions considered in this paper, we employ the bootstrap procedure described in

Section 4 that (i) re�ects the uncertainty associated with the estimation of the principal components

and (ii) provides a better approximation to the �nite-sample distribution of the parameters of

interest. We are unaware of a bootstrap procedure that considers estimation by the method of

principal components when the data are also weakly dependent. The procedure is thus of interest

in its own right.

Table 3 reports the parameter estimates from (9) along with their corresponding Newey-West

HAC standard errors, bootstrap standard errors and 90% bootstrap con�dence intervals. The

decision on the statistical signi�cance of each coe¢ cient is made based on whether zero is contained

in the bootstrap con�dence interval or not. Table 3 reveals that the �rst principal component of

convenience yields is statistically signi�cant (at 10% level) for all horizons except for h = 12.5 The

lack of predictability at long horizons is perhaps not surprising since the convenience yields are

constructed with futures prices of maturity n = 1 to 5 months. The second principal component

is signi�cant at one-month horizon and borderline insigni�cant at three-month horizon, but the

uncertainty from estimating this principal component at longer horizons appears to o¤set its large

coe¢ cient in the predictive regression and it becomes insigni�cant.

Several features of our bootstrap inference procedure deserve a more detailed discussion. As

expected, the bootstrap makes the coe¢ cients less signi�cant or even insigni�cant due to �nite-

sample corrections that come primarily from two sources: (i) accounting for estimation error in ct

and (ii) �nite-sample corrections of the critical values of the t-statistic. To illustrate the magnitude

of these adjustments, consider the second principal component of convenience yields in the predictive

regression at three-month horizon in Table 3. The estimated coe¢ cient is -3.096 which is rather

large and indicates that 1% increase in c(2)t reduces the commodity price index by 3.096%. The HAC

standard error on this coe¢ cient, treating this regressor as observed and not estimated, is 0.973

and using the conventional asymptotic testing procedure renders this coe¢ cient highly signi�cant.

Incorporating the estimation uncertainty for the �generated regressor�by the bootstrap increases

the standard error to 1.451. Using this bootstrap standard error would still suggest that this

5Since the signs of the principal components cannot be uniquely determined, we use the commodity price regressionto set the signs of the principal components for convenience yields so that they are consistent with the theoreticalmodel (3).

13

coe¢ cient is signi�cant at 10% (and even 5%) level if the standard asymptotic critical value is

employed. Our bootstrap method, however, further adjusts the critical value by computing a

better approximation to the tails and possible skewness in the �nite-sample distribution of the t-

statistic. As a result, the coe¢ cient on the second principal component becomes insigni�cant at 10%

signi�cance level. Similar magnitude and pattern of bootstrap corrections occur for all coe¢ cients

on the principal components in the commodity price and in�ation prediction regressions reported

in this and the subsequent sections. This illustrates that the signi�cant coe¢ cients on c(1)t and

c(2)t pass a very stringent test and the evidence of their predictive power proves to be robust and

convincing.

As for the other determinants of commodity prices, the exchange rate also appears to be signif-

icant for all horizons which lends support to the �ndings in Chen, Rogo¤ and Rossi (2010). The

coe¢ cients on both exchange rate and interest rate have the expected signs. While the interest

rate is not signi�cant at short horizons (one and three months), the magnitude of the interest rate

increases with the horizon and it becomes signi�cant at twelve-month horizon, at which point ct is

no longer signi�cant.

In summary, we �nd that individual and aggregate convenience yields contain statistically and

quantitatively important information for predicting commodity price changes. The documented

predictability could be attributed to a risk premium that measures the exposure of commodity prices

to risk factors (Fama and French, 1987; Gorton, Hayashi and Rouwenhorst, 2006; among others).6

Our explanation is that there is expectational information about future spot prices contained in

futures commodity prices, and such information are captured by the convenience yields.7

5.2 U.S. In�ation

Our approach to forecasting in�ation with commodity prices di¤ers from the previous studies in

two respects. First, we extract information from a panel of commodity prices and convenience

yields and, second, we use expectational information incorporated in futures prices that re�ects

demand/supply conditions in commodity markets. Our key predictors are the individual real de-

trended commodity prices qjt, their �rst two principal components qt = (q(1)t ; q

(2)t )

0, the individual

convenience yields cjt;n, their �rst two principal components ct = (c(1)t ; c

(2)t )

0, and the real detrended

price of oil, qoil;t.

It is widely documented that U.S. in�ation is persistent and its persistence changes over time.

We therefore start with an AR(2) model for in�ation �hpt+h = pt+h � pt and ask what variables6 In an earlier version of the paper, we also proxied the unobserved risk premium by principal components extracted

from realized volatilities of the 23 individual commodity price changes. The volatility components indeed show someincremental predictive power.

7Note that unlike tests of the e¢ cient market hypothesis, we estimate models changes in commodity prices andnot commodity returns.

14

have additional predictive power beyond lags of in�ation. Because di¤erent commodities have

di¤erent e¤ects on the components of CPI, we consider six measures of in�ation based on: (i) CPI

all items, (ii) CPI less food and energy, (iii) CPI less food, (iv) CPI less energy, (v) CPI food only

(vi) CPI energy only. All CPI series are seasonally adjusted (source: BLS).

According to the model of Frankel (2006), disequilibrium in commodity prices should predict

in�ation. Our model for commodities suggest that commodity prices should itself be determined

by convenience yields. We start with a baseline speci�cation that does not include interest and

exchange rates. Thus, for �hpt+1 = pt+h�pt being one of the six measures of in�ation, we considerthe predictive regression:

4hpt+h = b+ �0(L)4 pt + �j1cjt;n + �j2qjt + vjt+h:

We test whether (i) the convenience yield cjt;n for commodity j has predictive power in the presence

of the detrended real price of commodity j, and (ii) the detrended real price of commodity j has

predictive power in the presence of the convenience yield for commodity j. Table 4 presents the

t-tests of predictability with 2 lags for the dependent variable for h = 1 (the results for h = 12 are

similar).

While the individual convenience yields predict changes in commodity prices, these same vari-

ables have limited predictive power for the di¤erent measures of U.S. in�ation. The only commodi-

ties for which convenience yield exhibits some predictive power are orange juice, soybeans, oats,

copper and silver. On the other hand, the real commodity prices of crude and heating oil, platinum

and sugar appear to be most useful for predicting in�ation. Since the real energy prices are not

highly correlated with the �rst and second principal components of real commodity prices (see

Table 1) and given their importance for predicting various in�ation measures, in the subsequent

analysis we include the detrended real price of crude oil as a separate variable in the aggregate

in�ation regressions.

A predictive equation for h-period in�ation using the principal components instead of the indi-

vidual convenience yields and real commodity prices is speci�ed as

4hpt+h = b+ �0(L)4 pt + �1(L)0ct + �2(L)

0qt + �3(L)zt + vt+h: (10)

In the base case, zt is empty. As in the case of commodity prices, the lag length for all variables is

selected based on a preliminary speci�cation analysis.

Table 5 reports the estimation results from model (10) for h = 1; 3; 6 and 12. The second

principal components of convenience yields c(2)t and real commodity prices q(2)t tend to be strongly

signi�cant at short horizons. Furthermore, c(2)t remains signi�cant at longer horizons. Recall that

most of the individual convenience yields are not signi�cant in the regressions reported in Table

15

4. Yet, as seen in Table 5, c(2)t is strongly signi�cant. We attribute this result to the fact that the

principal components isolate those variations common across individual convenience yields and real

commodity prices that are relevant for predicting in�ation. In contrast, the individual commodity

prices may contain a large idiosyncratic component (such as production disruptions or unfavorable

weather conditions) that are not relevant for forecasting aggregate in�ation of goods and services.

Table 5 also presents results from an augmented predictive model of in�ation with zt being

the detrended real price of crude oil, and other determinants (interest and exchange rates) that

proved useful for predicting future commodity prices. As in the baseline speci�cation, c(2)t appears

to contain important information for predicting in�ation at all forecasting horizons. Including the

real oil price picks up some of the predictive power of ct. The real oil price is particularly helpful

in capturing some sharp movements in the in�ation rate for all goods and services. While the

interest rate is insigni�cant in a model with only autoregressive dynamics, it is highly signi�cant

in the augmented predictive regression. It indicates that there may be some crucial interactions

between the interest rate and the other predictors included in the model. Finally, exchange rate is

insigni�cant at all horizons despite its predictive power for commodity prices reported in Table 3.

It is instructive to compare our in�ation model to the forward looking form of the Phillips curve.

Following Gordon (1977), let the forward looking Phillips curve be formulated as

�hpt = Et�hpt+h + 1(ut � u�) + 2zt + &t;

where zt is a vector of supply shocks and ut � u� is a measure of output gap (typically proxied bythe unemployment rate minus an estimate of NAIRU). If zt = st� st�h, we can rewrite the Phillipscurve as

�hpt+h = � 2(qt � qt�h)� (1� 2)�hpt � 1(ut � u�) + "t+h; (11)

where qt = st � pt and "t+h is a composite error term. This would suggest to specify in�ation

as an autoregressive distributed lag model with lags of real commodity prices and output gap as

regressors. While this in�ation equation bears some similarities to model (7), it is based entirely

on past information and does not incorporate any forward looking variables such as cjt;n:

Given the popularity of the Phillips curve as a forecasting equation for in�ation, we include in our

augmented model of in�ation the variable ut� u� proxied by the deviations of U.S. unemploymentrate (source: BLS) from its HP trend. The estimated coe¢ cients on this variable are reported

in Table 5 and are insigni�cant at all forecast horizons. We note that ut � u� is signi�cant in

the simple version of the Phillips curve that excludes ct. This suggests that ct and the other

determinants incorporated into our model play a key role in capturing the information about the

economy�s fundamentals that is typically re�ected in the output gap. Table 6 reports results using

other measures of in�ation. As expected, the detrended real oil price proves useful for predicting

16

in�ation rates that include energy items (in�ation rates of energy and less food). However, ct

continue to be successful at predicting other in�ation measures especially those based on the CPI

less food and energy, and CPI less energy.

While our results suggest a relation between in�ation and the commodity prices or its determi-

nants, it remains to see if the relationships are stable. Hooker (2002) identi�es a structural break in

the relationship between commodity prices and in�ation after 1980. Cogley, Primiceri and Sargent

(2009) �nd a sharp reduction in the in�ation persistence in the post 1980 period. In contrast, the

recursive estimates from our model of 4pt+1 on ct, real commodity prices, detrended real price ofoil, interest rate and two autoregressive terms of in�ation (not reported to preserve space) exhibit

very little time variation and are almost constant from the beginning of 1990s until the end of the

sample period.

5.3 G-7 In�ation Rates

In order to see if the predictability of in�ation by the convenience yields and real commodity prices

also holds for other countries, we use data on CPI (all items) from Source OECD to construct

in�ation rates for the rest of the G-7 countries: Canada, Japan, Germany, France, Italy, and UK.

We also use interest and exchange rate data for these countries from Source OECD.

For each of these countries, the convenience yields and real commodity prices are computed in

local currency. In particular, we convert all commodity price variables in domestic currency using

the market exchange rate and use the country�s interest rate in equation (8). Similarly, the real

commodity prices are converted in local currency and de�ated by the corresponding CPI index.

For all countries, we estimate a model that includes two principal components of convenience yields

and detrended real commodity prices as well as two lags of the detrended real oil price and the

dependent variable.

The estimation results for each country at horizons h = 1 and 12 are presented in Table 7. At

one-month horizon, c(1)t and q(1)t as well as the real oil price appear to be statistically signi�cant for

almost all countries. For example, the �rst principal component of convenience yields is signi�cant

at 10% level (for all countries except for Germany for which it is marginally insigni�cant) even

after incorporating (by bootstrap methods) its estimation uncertainty in the predictive regression.

At twelve-month horizon, the predictive power of c(1)t remains signi�cant for all countries (except

Germany) but the e¤ects of the other determinants (real commodity and oil prices) are substantially

diminished. These results are consistent with our �ndings for the U.S. The robust high predictive

power of c(1)t across countries and forecast horizons is impressive given the very noisy and volatile

dynamics of some international in�ation rates at monthly frequency.

17

6 Assessing the Importance of Aggregate Convenience Yields

The previous section shows that the principal components of the convenience yields have statistically

signi�cant predictive power for commodity prices and in�ation. In this section, we demonstrate

that the predictive power is quantitatively important by looking at decomposition of variances,

counterfactual analysis focusing on the U.S. data, and out-of-sample forecast evaluation. We also

relate ct to some global and U.S. economic variables.

6.1 Decomposition of Variances

We perform a variance decomposition of changes in commodity prices based on a �ve-variable VAR

with �hsIMFt ordered �rst, followed by ct (of dimension two), �xt, and then it. The bootstrap

standard error is computed as the standard deviation of the variance decomposition estimates

over the bootstrap samples. To obtain bootstrap samples, we follow the block resampling method

described in Section 4 but with a larger block size of m = 18 since we do not impose the VAR

structure in generating the data.

The results for up to 36 months ahead variance decomposition are presented in Table 8. The c(2)tappears to be more important at short prediction horizons whereas c(1)t explains a larger portion of

commodity price changes at longer horizons. After three years, the vector ct collectively explains

from 11% of the variations in the monthly change in commodity prices �sIMFt , to 27% of the

variations in the six month change in commodity prices, �6sIMFt . The exchange rate �xt explains

6-9% of the variance of commodity price changes and is dominated by the principal components in

convenience yields at all horizons. The contribution of interest rate increases from 2% for monthly

commodity price changes to 5% for annual changes and is consistent with the regression results

in Table 3. These �ndings are in line with the results in Hong and Yogo (2009) who document

that the aggregate adjusted basis explains a large portion of the variance of expected commodity

returns as its importance increased substantially after 1986.

We also use a six-variable VAR model to obtain a decomposition of variations in h period ahead

in�ation (for all items and less food and energy). This is reported in Table 9. In�ation is ordered

�rst, followed by qt (q(1)t and q

(2)t ), real oil price, and ct (c

(1)t and c

(2)t ). For the in�ation rate

based on CPI all items, these factors explain collectively between 22% and 33% of the variation in

in�ation within a 3-year period. While the real oil price explains the largest portion of the in�ation

variations for monthly (11%) and quarterly (17%) in�ation, ct dominates the other determinants for

semi-annual and annual in�ation. For the core in�ation rate (less food and energy), the combined

contribution of c(1)t and c(2)t varies between 9% and 20% and often exceeds the combined contribution

of q(1)t and q(2)t . The contributions of these principal components increase further at quarterly

18

frequency as the in�ation rates at lower frequencies are less noisy.

6.2 Counterfactual Analysis

Although commodity prices are expected to �uctuate as supply and demand conditions change,

increases in commodity prices in recent years have been unprecedented. In particular, the IMF

commodity price index rose by 30 percent between 2006 and 2007, and by another 10 percent

in 2007. Heilbling and Blackman (2008) summarize many explanations that have been suggested

for the recent surge in commodity prices. To shed some light on the source of commodity price

increases in recent years, we use the estimates for the baseline model of �sIMFt+1 (Table 3). We

then ask what would have been the level of the IMF commodity price index if (i) the convenience

yields from 2007:M1 onwards were held at the level of 2006:M12 but with it and �xt at their actual

levels, (ii) it from 2007:M1 onwards was held at the level of 2006:M12 with ct and �xt at their

actual levels, (iii) �xt from 2007:M1 onwards was held at the level of 2006:M12 with ct and it at

their actual levels, and (iv) all these three variables were held �xed at their 2006:M12 level. These

counterfactual commodity prices, denoted sIMFt+1jc; s

IMFt+1ji ; s

IMFt+1jx and s

IMFt+1jc;i;x, are plotted in Figure 1.

Over these 19 months, the IMF commodity price index increased by more than 27%. The

hypothetical commodity price sIMFt+1jc follows closely the trend of the actual index, while s

IMFt+1ji ;

sIMFt+1jx and s

IMFt+1jc;i;x exhibit much slower growth. Some of the increases in 2007 and 2008 appear to

be driven by interest rate and exchange rate changes, as suggested by Hamilton (2008) and Frankel

(2006, 2008). However, the counterfactual commodity price is inconsistent with the one implied

by the predictive regression with coe¢ cients on convenience yield, interest rate, and exchange rate

estimated over the full sample. The predictive regression could well be unstable in the sense that

larger or smaller weights on these fundamentals are warranted. However, it is also possible that

speculation plays a non-trivial role in commodity price changes between 2007 and 2008.8

6.3 Out-of-Sample Forecast Performance

The prediction models for commodity prices and in�ation were evaluated in the previous sections

only based on their in-sample �t and performance. This section performs a recursive out-of-sample

forecasting exercise by estimating the models using T1 observations (T1 = 150; 151;...; T � h) and

producing h period ahead forecasts for h = 1; 3; 6 and 12. The principal components are also

computed with information only up to time T1: We use a one-sided two-year moving average �lter

with exponentially decreasing weights �(1 � �)i for � = 0:15 and i = 1; :::; 24 to detrend real

commodity prices and unemployment. We forecast changes in the IMF commodity price index

8A similar counterfactual analysis for U.S. in�ation reveals that keeping the ct unchanged for the last 19 monthsof the sample underpredicts the actual price level but incorporating the rise in commodity prices over this periodbrings the predicted CPI closer to its actual level.

19

(�sIMFt+h ) and in�ation (�p

ht+h), both for CPI all items and CPI less food and energy. Root mean

squared errors (RMSEs) from three models with forecast horizons of up to one year are reported

in Table 10.

The three forecasting models for �sIMFt+h include as predictors (i) two principal components

of convenience yields ct and two lags �sIMFt and �sIMF

t�1 (M1 in Table 10), (ii) interest rate it,

exchange rate changes�xt and two lags�sIMFt and�sIMF

t�1 (M2 in Table 10), and (iii) only two lags

�sIMFt and �sIMF

t�1 (M3 in Table 10). We separate the determinants with good predictive power

(ct; it and �xt) into two models in order to assess their contribution for out-of-sample forecasting.

The three forecasting models for in�ation use as predictors (i) c(1)t ; c(2)t , q

(1)t , q

(2)t , qoil;t, qoil;t�1,

�pt and �pt�1 (M1 in Table 10) ; (ii) same as in M1 with c(1)t and c(2)t replaced by (ut � u�)

(M2 in Table 10) ; and (iii) only �pt and �pt�1 (M3 in Table 10). Notably, M1 incorporates

forward-looking (expectational) variables c(1)t and c(2)t whereas M2 does not.

Several observations emerge from the results in Table 10. For both commodity prices and

in�ation, model M1 that incorporates ct dominates in terms of RMSE, supporting the �ndings

of the in-sample analysis. The largest forecasting gains for commodity prices occur for one- to

six-month forecast horizons, likely because the convenience yields are constructed using futures

prices with one to �ve months to maturity. The interest and exchange rates also possess some

out-of-sample forecast ability and deliver lower forecast errors than the autoregressive speci�cation

for h = 1; 6 and 12. The in�ation model with ct (M1) clearly has smaller forecast errors than the

Phillips curve model (M2) and the AR model (M3). For core in�ation, the forecast error reduction

compared to the best performing competitor is as large as 18%. In summary, ct contains valuable

information for forecasting the future dynamics of commodity prices and in�ation.

6.4 Relation between ct and Economic Variables

The �nding that ct = (c(1)t ; c

(2)t )

0 have robust predictive power for in�ation in- and out-of-sample,

even in the presence of oil price and the usual variables in the Phillips curve, can arise only

if the variations in c(1)t and c

(2)t are largely independent of the oil price and unemployment of

the output gap. A similar observation was made by Hong and Yogo (2009), who report that a

simple average of the individual convenience yields is uncorrelated with stock and bond returns

and their common predictors such as short interest rate, yields spread and dividend yield. This

has led Hong and Yogo (2009) to interpret the aggregate convenience yield as a predictor that is

local to commodity market. Our analysis reaches a similar conclusion, though we use the method

of principal components instead of simple averaging to aggregate information in the convenience

yields, and our focus here is in�ation forecasts.

To better understand the information content of c(1)t and c(2)t , we study the cross-correlation

20

between a number of macroeconomic variables yt and ct+j . A signi�cant correlation at j < 0

indicates that ct has predictive power for future yt. A signi�cant correlation at j > 0 indicates

yt leads ct. Figure 2 presents these cross correlation plots for several U.S. variables: the annual

growth rates of real personal consumer expenditures, real industrial production, value-weighted

stock returns, as well as the spread between the 10-year government bond and 3-month T-bill.

The one and two standard error bands of � 1pTand � 2p

Tare also given. The contemporaneous

correlation between c(1)t or c(2)t and the U.S. macro aggregates is evidently weak, consistent with

the �ndings in Hong and Yogo (2009). However, lags of c(1)t seems to lead consumption and stock

returns, suggesting c(1)t as an information variable about expected demand. On the other hand,

industrial production seems to lead c(2)t . One explanation is that strong production in the past

draws down inventories, making future stockouts more likely, with the result of increasing c(2)t .9

While these results suggest that ct carry information about future demand for goods, assets,

and raw materials in the U.S., this explanation is incomplete. As seen earlier in Table 1, c(1)t loads

heavily on corn, co¤ee, cotton, wheat and crude oil while c(2)t is highly correlated with some metals

(silver, copper) as well as soybeans, cocoa and heating oil. Given that there is a world demand

for these commodities, the possibility remains that ct capture economic conditions outside the U.S.

For example, it is believed that the recent run up in commodity prices has been driven primarily

by the strong economic growth and commodity demand in the emerging and developing countries

(Heilbling and Blackman, 2008) and especially in China and India.

To investigate this possibility, we use annual GDP growth rates of the world economy, 34

advanced economies (as classi�ed by IMF excluding China and India), China and India, provided

by the IMF. The cross-correlation plots in Figure 3 show that ct is not strongly correlated with

world GDP growth. However, GDP growth of China is signi�cantly correlated with future c(1)t ,

and GDP growth of India is signi�cantly correlated with future c(2)t . Furthermore, c(1)t leads the

GDP growth in the advanced economies and India. These results suggest that c(1)t and c(2)t bear

information about future consumption demand and the higher possibility of stockout for production

materials that such future demand might induce. Such information about U.S. and world economic

conditions is apparently not contained in variables in the Phillips curve and has predictive power

for in�ation.

The foregoing results suggest that the common variation in futures prices as captured by the

principal components in convenience yields, contain important information about future economic

conditions relevant for in�ation. It should be stressed that the predictive power comes from the

common variations in convenience yields (or the commodity price futures), after idiosyncratic noise

from the individual convenience yields has been washed out.

9Predictive regression results at monthly frequency are also available from the authors upon request.

21

7 Conclusions

This paper proposes a new data reduction approach to predicting future commodity prices and

in�ation. We estimate principal components from a panel of individual convenience yields and

real commodity prices that are used for short- and medium-term predictions. A premise of our

analysis is that a series aggregated over variables with heterogeneous properties masks important

explanatory power provided by the individual components. The estimated principal components

for the convenience yields appear to capture potentially important information about future de-

mand. On the other hand, factors like interest and exchange rates, while also useful for prediction,

are associated with asset allocation motives and their predictive power appears to be more elusive

and fragile. Our results provide convincing evidence of the ability of the extracted principal com-

ponents of convenience yields to predict the future commodity price dynamics. We also evaluate

the relevance of aggregated commodity price determinants for forecasting in�ation in the U.S. and

other G-7 countries. The documented improved prediction of future in�ation is expected to play a

promising role for estimation of monetary policy reaction function and determination of the policy

rate (Bernanke, 2010).

22

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24

Table 1. Regression: yjt = a0 + a1y(1)t + a2y

(2)t + et.

R2 (� 0:2 in bold).

yjt cjtn qjt

j yt c(1)t c

(2)t q

(1)t q

(2)t

Cocoa 0.073 0.208 0.150 0.019Co¤ee 0.453 0.072 0.006 0.017Orange Juice 0.127 0.056 0.193 0.040Sugar 0.026 0.095 0.043 0.086Canola 0.153 0.081 0.651 0.046Corn 0.580 0.004 0.527 0.117Oats 0.028 0.023 0.378 0.001Soybeans 0.141 0.271 0.317 0.341Soybean Oil 0.106 0.041 0.504 0.137Wheat 0.202 0.137 0.338 0.034Cotton 0.261 0.038 0.000 0.259Lumber 0.052 0.001 0.028 0.014Cattle, Feeder 0.000 0.189 0.174 0.034Cattle, Live 0.058 0.180 0.052 0.091Hogs 0.036 0.006 0.137 0.137Pork Bellies 0.002 0.041 0.065 0.097Copper 0.037 0.251 0.017 0.133Gold 0.102 0.004 0.000 0.224Palladium 0.155 0.169 0.029 0.312Platinum 0.022 0.093 0.076 0.497Silver 0.000 0.305 0.033 0.241Crude Oil 0.201 0.092 0.160 0.149Heating Oil 0.086 0.200 0.177 0.148

25

Table 2. Estimates and Standard Errors:

�sjt+1 = a+

1Xk=0

�j0k�sjt�k + �j1cjt;n + �j2it + �j3�xt + ejt+1:

j �j1 �j2 �j3Cocoa -0.323 (0.186) -0.092 (0.198) -0.348 (0.149)Co¤ee -0.167 (0.189) -0.239 (0.268) -0.053 (0.286)Orange Juice -0.545 (0.197) 0.303 (0.261) -0.276 (0.146)Sugar -0.797 (0.141) -0.669 (0.379) -0.315 (0.210)Canola -0.673 (0.197) 0.039 (0.226) 0.133 (0.138)Corn -0.784 (0.137) 0.224 (0.185) 0.088 (0.124)Oats -0.368 (0.108) -0.364 (0.217) -0.078 (0.195)Soybeans -1.159 (0.294) -0.275 (0.161) -0.050 (0.136)Soybean Oil -0.570 (0.355) -0.105 (0.242) -0.039 (0.157)Wheat -0.444 (0.095) 0.095 (0.167) -0.298 (0.200)Cotton -0.737 (0.123) 0.395 (0.183) 0.078 (0.226)Lumber -0.572 (0.145) -0.371 (0.240) 0.066 (0.187)Cattle, Feeder -0.597 (0.152) 0.005 (0.107) -0.211 (0.090)Cattle, Live -0.658 (0.062) 0.098 (0.135) -0.163 (0.092)Hogs -0.586 (0.054) 0.296 (0.181) -0.264 (0.181)Pork Bellies -0.921 (0.099) -0.017 (0.216) 0.084 (0.275)Copper -0.148 (0.185) -0.234 (0.142) -0.386 (0.157)Gold 0.709 (0.779) -0.238 (0.058) -0.431 (0.075)Palladium -0.568 (0.331) 0.186 (0.266) -0.073 (0.247)Platinum 0.332 (0.237) -0.230 (0.110) -0.488 (0.152)Silver 0.195 (0.980) -0.509 (0.113) -0.175 (0.170)Crude Oil -0.604 (0.408) -0.130 (0.225) -0.185 (0.239)Heating Oil -0.948 (0.196) 0.021 (0.221) -0.277 (0.203)

ct it �xtIMF commod. index -0.994 (0.401) -0.050 (0.050) -0.206 (0.056)

Notes: Newey-West HAC standard errors in parentheses. Bold font indicates statistical signi�cance

at 10% level. ct =PNj=1 cjt;n.

26

Table 3. Estimates and Standard Errors:

�hsIMFt+h = a+

1Xk=0

�10�sIMFt�k + �11c

(1)t + �12c

(2)t + �2it + �3�xt + et+h:

h = 1 h = 3 h = 6 h = 12

c(1)t -0.877

(0.344){0.493}

[-1.560, -0.035]

-3.181(1.034){1.442}

[-5.992, -0.410]

-6.227(1.942){2.753}

[-12.888, -0.069]

-5.665(3.452){3.449}

[-13.315, 3.080]

c(2)t -1.175

(0.323){0.483}

[2.223, -0.155]

-3.096(0.973){1.426}

[-6.218, 0.026]

-3.603(1.907){2.535}

[-8.506, 1.875]

-3.893(2.964){3.288}

[-11.180, 3.079]

it -0.058(0.060){0.075}

[-0.177, 0.052]

-0.232(0.174){0.212}

[-0.601, 0.154]

-0.568(0.316){0.385}

[-1.228, 0.227]

-1.775(0.464){0.581}

[-2.731, -0.469]

�xt -0.195(0.054){0.059}

[-0.292, -0.103]

-0.311(0.119){0.126}

[-0.547, -0.107]

-0.522(0.181){0.197}

[-0.8948 -0.237]

-0.647(0.254){0.279}

[-1.176, -0.266]

�sIMFt 0.174

(0.062){0.068}

[0.063, 0.282]

0.269(0.109){0.125}

[0.073, 0.476]

0.197(0.185){0.201}

[-0.113, 0.593]

0.252(0.264){0.275}

[-0.210, 0.735]

�sIMFt�1 0.044

(0.045){0.051}

[-0.035, 0.118]

0.001(0.109){0.124}

[-0.189, 0.207]

-0.028(0.164){0.179}

[-0.324, 0.277]

0.133(0.283){0.295}

[-0.333, 0.725]

R2 0.199 0.250 0.286 0.326

Notes: Newey-West HAC asymptotic and bootstrap standard errors, and 90% bootstrap con�dence

intervals are reported below the parameter estimates in parentheses, squiggly brackets and square

brackets, respectively. Bold font indicates statistical signi�cance at 10% level.

27

Table 4. Robust t Tests for Predictability:

�pt+1 = b+1Xk=0

b0k�pt + �j1cjt;n + �j2qjt + vt+1:

j�p all items less f & e less food less energy food energycjt;n qjt cjt;n qjt cjt;n qjt cjt;n qjt cjt;n qjt cjt;n qjt

Cocoa 1.78 -0.79 1.42 -0.92 1.57 -0.72 1.19 -0.62 1.35 -0.42 1.13 -0.35Co¤ee 0.29 -0.82 1.48 -1.37 0.22 -0.97 1.22 -0.82 0.73 0.22 -1.27 -0.38O. J. 3.56 -0.30 5.51 -1.62 3.27 -0.06 5.87 -2.30 2.01 -1.13 0.32 0.41Sugar -0.94 0.54 -2.68 2.07 -0.52 -0.16 -3.51 2.79 -1.79 2.12 0.84 -1.87Canola -1.10 1.43 1.04 -0.74 -0.62 0.70 0.24 0.03 -2.53 2.50 -1.36 1.01Corn -1.01 1.59 1.93 -1.31 -0.88 1.06 1.66 -0.46 0.03 1.84 -2.63 2.34Oats -2.89 3.15 -1.58 0.39 -2.66 2.58 -2.01 0.87 -1.78 1.47 -1.85 1.89Soyb. -3.35 2.17 -0.66 -0.41 -2.85 -1.59 -1.41 0.44 -2.11 2.84 -1.59 1.96Soy O 0.25 0.91 2.11 -1.05 0.81 0.52 1.46 -0.59 -1.74 1.63 -1.04 1.18Wheat -0.97 1.49 1.80 -1.10 -0.99 1.23 1.77 -0.54 -0.40 1.73 -2.61 2.48Cott. -0.22 0.90 2.39 -0.36 -0.49 0.55 3.15 0.32 0.78 1.22 -4.07 1.03Lumb. -0.74 1.17 0.44 0.82 -0.52 1.10 -0.04 0.73 -1.03 0.88 -0.87 1.04Cat.F. 0.16 0.47 1.21 0.67 0.29 0.54 1.11 0.31 -0.10 0.12 -0.84 0.17Cat.L. 0.57 1.01 1.89 -0.21 0.46 0.79 2.02 -0.09 0.29 1.73 -0.90 1.10Hogs 0.12 1.01 1.60 -0.14 0.17 0.81 1.29 0.02 0.22 1.17 -1.08 1.26Pork -0.72 0.36 -0.67 0.96 -0.67 0.25 -0.73 0.79 -1.76 1.12 0.34 -0.50Copp. 3.41 -0.60 2.02 -0.06 3.05 -0.57 2.50 -0.44 2.91 -0.52 0.70 -0.08Gold -0.00 0.23 1.54 0.21 0.33 0.26 0.78 0.21 -1.36 0.41 -1.30 0.08Pallad. -0.59 1.38 0.98 0.62 -0.68 1.12 0.89 1.44 0.03 2.30 -2.48 0.38Plat. -0.88 2.11 -3.04 1.31 -0.90 1.55 -2.44 2.28 -1.28 3.88 1.36 0.71Silver -1.95 0.68 -2.31 0.19 -1.97 0.70 -2.02 0.46 -1.95 0.47 0.33 0.37Oil -0.30 2.28 0.59 0.60 -0.77 2.96 0.89 0.09 2.14 -1.30 -1.31 2.82Heat O -0.04 2.38 -0.01 0.86 -0.46 2.88 0.70 0.60 1.62 -0.35 -0.79 2.50

Notes: Bold font indicates statistical signi�cance at 10% level using asymptotic critical values.

28

Table 5. U.S. Estimates and 90% Bootstrap Con�dence Intervals:

�hpt+h = b+1Xk=0

�0k�pt�k+2Xk=1

�1kc(k)t +

2Xk=1

�2kq(k)t +

1Xk=0

�3kqoil;t�k+�4it+�5�xt+�6(ut�u�)+vt+h:

h = 1 h = 3 h = 6 h = 12

(1) (2) (1) (2) (1) (2) (1) (2)

c(1)t 0.010

[-0.09,0.08]-0.086

[-0.17,-0.00]0.064

[-0.21,0.25]-0.151[-0.31,0.11]

0.177[-0.36,0.53]

-0.154[-0.59,0.54]

0.337[-1.02,1.51]

-0.170[-1.18,1.34]

c(2)t -0.074

[-0.14,-0.00]-0.080

[-0.14,-0.00]-0.262

[-0.51,-0.03]-0.280

[-0.50,-0.06]-0.538[-1.12,0.01]

-0.544[-1.08,-0.01]

-1.153[-2.22,-0.12]

-1.131[-2.11,-0.18]

q(1)t -0.009

[-0.04,0.03]-0.052

[-0.09,-0.01]-0.089[-0.17,0.06]

-0.109[-0.20,0.06]

-0.204[-0.44,0.07]

-0.111[-0.22,0.17]

-0.228[-0.49,0.42]

0.122[-0.39,0.50]

q(2)t -0.071

[-0.12,-0.02]-0.044[-0.08,0.02]

-0.178[-0.32,-0.03]

-0.145[-0.26,0.03]

-0.125[-0.25,0.14]

-0.172[-0.34,0.15]

-0.084[-0.67,0.61]

-0.341[-0.70,0.23]

4pt 0.372[0.28,0.59]

0.171[0.11,0.34]

0.165[-0.04,0.44]

-0.185[-0.35,0.01]

0.261[-0.02,0.63]

0.022[-0.18,0.32]

0.695[0.31,1.12]

0.597[0.27,0.98]

4pt�1 -0.217[-0.36,-0.08]

-0.139[-0.30,-0.01]

-0.176[-0.41,0.04]

-0.002[-0.27,0.24]

0.060[-0.25,0.41]

0.279[-0.06,0.69]

-0.029[-0.55,0.54]

0.411[-0.12,1.06]

qoil;t - 1.165[0.85,1.63]

- 1.838[0.99,3.43]

- 1.003[-0.13,2.72]

- 0.044[-1.38,1.80]

qoil;t�1 - -1.069[-1.40,-0.77]

- -2.186[-3.60,-1.40]

- -2.317[-4.13,-1.32]

- -3.478[-5.54,-2.35]

it - 0.022[0.01,0.03]

- 0.053[0.02,0.09]

- 0.092[0.01,0.16]

- 0.151[0.03,0.26]

�xt - -0.003[-0.01,0.01]

- 0.001[-0.02,0.02]

- 0.011[-0.02,0.05]

- 0.013[-0.03,0.05]

ut-u� - -0.010[-0.05,0.12]

- -0.153[-0.41,0.06]

- -0.224[-0.73,0.22]

- -0.264[-1.06,0.37]

R2 0.176 0.324 0.084 0.248 0.132 0.304 0.192 0.430

Notes: Bold font indicates statistical signi�cance at 10% level.

29

Table 6. U.S. Estimates and 90% Bootstrap Con�dence Intervals:

�hpt+h = b+1Xk=0

�0k�pt�k +2Xk=1

�1kc(k)t +

2Xk=1

�2kq(k)t +

1Xk=0

�3kqoi;t�k + vt+h:

less f & e less food less energy food energyh = 1

c(1)t 0.079

[0.013, 0.153]-0.011

[-0.109, 0.082]0.070

[0.011, 0.133]0.041

[-0.083, 0.112]-1.141

[-2.081, -0.553]

c(2)t -0.053

[-0.116, -0.000]-0.087

[-0.158, -0.017]-0.058

[-0.120, -0.004]-0.101

[-0.192, -0.011]0.021

[-0.533, 0.895]

q(1)t 0.022

[-0.008, 0.040]-0.037

[-0.074, 0.012]0.019

[-0.017, 0.034]-0.011

[-0.123, 0.095]-0.646

[-1.087 -0.259]

q(2)t -0.004

[-0.034, 0.027]-0.025

[-0.051, 0.040]-0.022

[-0.043, 0.006]-0.149

[-0.255, -0.065]-0.228

[-0.510, 0.516]

qoil;t -0.068[-0.174, 0.063]

1.451[1.092, 1.942]

-0.065[-0.168, 0.080]

-0.016[-0.509, 0.471]

14.750[11.953, 18.367]

qoil;t�1 0.080[-0.034, 0.189]

-1.282[-1.626, -0.946]

0.040[-0.067, 0.137]

-0.196[-0.523, 0.161]

-12.941[-15.793, -9.990]

4pt 0.211[0.140, 0.327]

0.139[0.064, 0.275]

0.258[0.180, 0.379]

0.127[0.033, 0.261]

0.156[0.060, 0.294]

4pt�1 0.222[0.111, 0.350]

-0.099[-0.291, 0.040]

0.145[0.046, 0.274]

-0.063[-0.148, 0.083]

-0.198[-0.355, -0.062]

R2 0.319 0.292 0.308 0.095 0.382h = 3

c(1)t 0.213

[0.029, 0.440]0.034

[-0.321, 0.305]0.213

[0.035, 0.426]0.176

[-0.118, 0.388]-2.978

[-5.963, -0.806]

c(2)t -0.155

[-0.331, 0.007]-0.259

[-0.505, -0.023]-0.181

[-0.357, -0.009]-0.347

[-0.619, -0.077]0.144

[-2.015, 2.902]

q(1)t 0.064

[-0.023, 0.122]-0.066

[-0.158, 0.128]0.052

[-0.041, 0.099]-0.029

[-0.292, 0.285]-1.663

[-2.957, -0.376]

q(2)t -0.022

[-0.091, 0.065]-0.149

[-0.285, 0.024]-0.065

[-0.134, 0.033]-0.398

[-0.716 -0.120]-1.169

[-2.130, 0.663]

qoil;t -0.036[-0.428, 0.347]

2.160[1.242, 3.792]

-0.105[-0.469, 0.247]

-0.262[-1.052, 0.467]

23.948[17.848, 34.362]

qoil;t�1 0.036[-0.307, 0.380]

-2.408[-3.942 -1.531]

-0.009[-0.306, 0.284]

-0.574[-1.161, 0.071]

-25.608[-35.570, -17.151]

4pt 0.692[0.483, 0.961]

-0.145[-0.327, 0.093]

0.612[0.411, 0.876]

0.030[-0.154, 0.246]

-0.287[-0.429, -0.116]

4pt�1 0.734[0.506, 0.996]

0.121[-0.182, 0.427]

0.599[0.399, 0.821]

-0.108[-0.314, 0.096]

0.030[-0.271, 0.308]

R2 0.538 0.166 0.495 0.176 0.227

Notes: Bold font indicates statistical signi�cance at 10% level.

30

Table 7. G7 Estimates and 90% Bootstrap Con�dence Intervals:

�hpt+h = b+1Xk=0

�0k�pt�k +2Xk=1

�1kc(k)t +

2Xk=1

�2kq(k)t +

1Xk=0

�3kqoi;t�k + vt+h:

Canada Japan France Germany Italy UKh = 1

c(1)t -0.136

[-0.251, -0.000]-0.157

[-0.242, -0.061]-0.143

[-0.229, -0.054]-0.123

[-0.268, 0.014]-0.116

[-0.156, -0.039]-0.234

[-0.366, -0.086]

c(2)t 0.057

[-0.147, 0.220]-0.001

[-0.188, 0.178]-0.064

[-0.171, 0.116]0.000

[-0.222, 0.200]-0.052

[-0.121, 0.082]-0.091

[-0.260, 0.136]

q(1)t -0.078

[-0.140, -0.001]-0.109

[-0.173, -0.049]-0.058

[-0.096, -0.014]-0.078

[-0.133, -0.011]-0.013

[-0.025, 0.036]-0.158

[-0.238, -0.087]

q(2)t -0.117

[-0.215, -0.016]0.027

[-0.107, 0.085]0.046

[-0.046, 0.089]-0.007

[-0.155, 0.103]0.019

[-0.063, 0.068]-0.090

[-0.049, 0.157]

qoil;t 0.855[0.376, 1.373]

0.564[0.106, 1.161]

0.518[0.129, 0.837]

0.465[0.124, 0.861]

0.270[0.087, 0.498]

0.626[0.097, 1.170]

qoil;t�1 -0.687[-1.267, 0.025]

-0.344[-0.778, 0.069]

-0.756[-1.041, -0.423]

-0.526[-0.825, -0.187]

-0.249[-0.426 -0.098]

-0.968[-1.544, -0.413]

4pt 0.004[-0.103, 0.149]

0.091[-0.040, 0.224]

0.187[0.124, 0.347]

-0.052[-0.165, 0.091]

0.316[0.258, 0.443]

0.065[0.033, 0.169]

4pt�1 0.052[-0.025, 0.165]

-0.308[-0.381, -0.190]

-0.061[-0.160, 0.100]

0.054[-0.013, 0.172]

0.194[0.111, 0.350]

-0.054[-0.115, 0.020]

R2 0.108 0.166 0.210 0.057 0.426 0.125h = 12

c(1)t -1.851

[-2.517, -1.057]-1.725

[-2.678, -0.753]-1.013

[-1.582, -0.166]-1.176

[-2.711, 0.247]-1.470

[-2.070, -0.528]-2.602

[-3.682, -1.008]

c(2)t 0.232

[-0.877, 1.256]0.253

[-1.588, 2.124]-0.596

[-1.483, 0.648]-0.817

[-2.211, 0.665]0.597

[-0.720, 2.102]-0.794

[-2.070, 3.202]

q(1)t -0.167

[-0.545, 0.351]-0.570

[-1.161, -0.183]-0.058

[-0.234, 0.333]-0.037

[-0.406, 0.595]-0.392

[-0.788, 0.008]-0.560

[-1.074, 0.281]

q(2)t -0.197

[-0.783, 0.521]-0.089

[-0.814, 0.750]0.543

[-0.182, 1.197]0.105

[-0.805, 0.670]0.390

[-0.705, 0.908]0.880

[-0.183, 1.755]

qoil;t 1.198[-0.690, 3.345]

1.450[-0.293, 4.300]

-2.219[-4.166, -0.540]

0.632[-1.688, 3.763]

-0.439[-2.387, 1.629]

-1.877[-4.027, 0.434]

qoil;t�1 -2.365[-4.281, -1.042]

-1.333[-4.186, 0.122]

-0.169[-1.571, 1.166]

-0.665[-3.409, 1.480]

-1.773[-3.372, -0.672]

0.197[-2.040, 2.069]

4pt 0.242[-0.244, 0.756]

0.084[-0.040, 0.335]

1.408[0.795, 2.356]

0.633[0.254, 1.155]

2.699[2.362, 3.578]

0.598[0.321, 1.075]

4pt�1 0.252[-0.177, 0.748]

0.218[0.119, 0.469]

1.572[0.934, 3.176]

0.607[0.173, 1.091]

2.640[2.121, 3.799]

0.557[0.241, 1.222]

R2 0.345 0.391 0.438 0.255 0.697 0.465

Notes: Bold font indicates statistical signi�cance at 10% level.

31

Table 8. Variance Decomposition of �hsIMFt .

4sIMFt 43sIMF

t 46sIMFt 412sIMF

t

Horizon 3 18 36 3 18 36 3 18 36 3 18 364hsIMF

t 90:8(3:7)

82:2(4:3)

81:0(4:3)

94:0(2:4)

74:9(5:9)

71:9(6:1)

95:8(2:2)

63:2(8:9)

59:7(9:1)

97:4(1:7)

78:6(11:2)

70:5(11:3)

c(1)t 0:4

(1:8)5:6(3:5)

5:6(3:5)

0:5(1:6)

11:9(5:1)

11:9(5:0)

1:2(1:4)

16:2(7:0)

16:4(6:9)

0:3(0:7)

8:2(8:8)

11:1(8:8)

c(2)t 3:5

(2:2)4:5(3:7)

5:0(3:6)

2:0(1:6)

6:3(5:1)

7:4(5:1)

1:1(1:4)

9:7(6:1)

10:8(6:0)

0:1(0:7)

2:6(7:2)

4:5(7:2)

�xt 4:3(2:0)

6:6(2:3)

6:5(2:3)

3:5(1:4)

6:3(2:4)

6:0(2:4)

1:6(1:3)

9:6(3:3)

9:1(3:3)

1:6(1:1)

9:7(4:9)

9:0(4:8)

it 0:9(1:2)

1:1(1:7)

1:9(1:7)

0:1(0:7)

0:6(2:5)

2:7(2:8)

0:3(0:6)

1:3(3:8)

3:9(4:1)

0:5(0:5)

1:0(4:7)

5:1(5:1)

Notes: The variance decomposition at forecasting horizons 3, 18 and 36 months is obtained from a

VAR(4) model with a Choleski decomposition using the ordering (4hsIMFt ; c

(1)t ; c

(2)t ;�xt; it). 90%

bootstrap con�dence intervals are reported in parentheses.

Table 9. Variance Decomposition of �hpt (U.S.).

4pt 43pt 46pt 412ptPeriod 3 18 36 3 18 36 3 18 36 3 18 36

CPI all items4hpt 85:0

(3:2)78:6(4:8)

78:6(4:9)

80:9(4:9)

67:6(7:5)

67:4(7:7)

89:1(4:9)

73:4(9:6)

72:5(9:9)

93:4(3:0)

79:7(8:1)

78:8(8:8)

q(1)t 0:8

(1:6)1:8(1:8)

1:8(1:8)

0:8(1:8)

3:0(2:6)

3:0(2:6)

0:1(1:6)

2:8(3:6)

2:9(3:6)

0:3(0:8)

3:6(3:7)

4:1(4:0)

q(2)t 2:8

(2:2)3:7(2:1)

3:7(2:1)

4:5(2:3)

5:4(3:4)

5:4(3:5)

3:7(2:0)

6:4(4:4)

6:4(4:4)

3:0(1:6)

6:0(3:3)

5:1(3:4)

qoil;t 10:3(2:6)

11:4(2:7)

11:4(2:7)

13:1(4:8)

17:3(5:5)

17:4(5:5)

5:7(3:8)

6:8(4:9)

7:1(4:9)

2:8(2:4)

2:7(3:3)

2:5(3:3)

c(1)t 0:8

(0:8)1:3(2:0)

1:3(2:1)

0:7(1:3)

2:9(3:4)

2:9(3:5)

1:4(1:3)

4:1(4:4)

4:2(4:5)

0:4(0:7)

4:0(3:8)

5:8(4:3)

c(2)t 0:3

(0:8)3:1(1:9)

3:1(1:9)

0:0(1:1)

3:9(3:7)

3:9(3:8)

0:0(1:3)

6:5(4:6)

7:0(4:8)

0:2(0:9)

4:1(4:0)

3:7(4:4)

CPI less food and energy4hpt 95:9

(2:8)84:7(7:3)

81:0(7:9)

99:1(1:9)

90:1(8:0)

87:1(8:9)

98:4(1:5)

86:4(7:9)

83:5(8:8)

97:3(1:4)

65:0(8:3)

53:6(9:4)

q(1)t 1:8

(1:0)1:6(1:6)

1:6(1:8)

0:4(0:9)

2:0(2:6)

1:7(2:9)

0:4(0:4)

2:9(3:0)

2:4(3:3)

0:1(0:4)

2:9(4:4)

3:2(4:7)

q(2)t 0:5

(0:7)1:0(2:0)

1:0(2:2)

0:0(0:6)

1:8(3:1)

2:0(3:3)

0:5(0:6)

3:8(4:4)

3:5(4:7)

2:0(1:0)

19:4(4:4)

23:1(4:8)

qoil;t 0:4(0:7)

0:5(1:9)

0:7(1:9)

0:1(0:7)

0:3(4:1)

0:5(4:2)

0:1(0:5)

0:3(4:0)

0:9(4:2)

0:1(0:5)

0:2(3:3)

0:4(3:5)

c(1)t 0:9

(2:2)2:4(6:0)

2:5(6:4)

0:2(1:1)

0:6(4:9)

0:6(5:4)

0:2(0:9)

0:3(3:1)

0:3(3:8)

0:1(0:4)

0:4(4:0)

0:9(4:7)

c(2)t 0:5

(1:4)9:9(4:5)

13:3(4:8)

0:3(0:6)

5:3(3:7)

8:1(4:3)

0:4(0:7)

6:3(2:9)

9:4(3:5)

0:3(0:4)

12:1(2:7)

18:8(3:4)

Notes: The variance decomposition is obtained from a VAR(4) model with a Choleski decomposition

using the ordering (4hpt; q(1)t ; q

(2)t ; qoil;t; c

(1)t ; c

(2)t ). 90% bootstrap con�dence intervals are reported

in parentheses.

32

Table 10. Recursive Out-of-Sample Forecasts:

Root Mean Squared Error.

commodity price changes in�ation (all items) in�ation (less food & energy)h = 1 h = 3 h = 6 h = 12 h = 1 h = 3 h = 6 h = 12 h = 1 h = 3 h = 6 h = 12

M1 1.928 4.098 6.597 10.068 0.230 0.469 0.649 1.091 0.095 0.189 0.302 0.557M2 1.982 4.497 7.220 10.255 0.236 0.505 0.722 1.201 0.101 0.218 0.373 0.735M3 2.001 4.496 7.553 11.310 0.245 0.513 0.676 1.058 0.098 0.206 0.353 0.679

Notes: The initial sample for recursive estimation of the models is 150 observations. For commodity

prices �hsIMFt+h , M1 denotes the model that includes c

(1)t ; c

(2)t ; �sIMF

t and �sIMFt�1 ; M2 denotes the

model with regressors it;�xt; �sIMFt and �sIMF

t�1 ; and M3 is a pure autoregressive model with

AR terms �sIMFt and �sIMF

t�1 : For in�ation �hpt+h (both all items and less food and energy), M1

includes c(1)t ; c(2)t , q

(1)t , q

(2)t , qoil;t, qoil;t�1, �pt and �pt�1; M2 contains (ut � u�), q(1)t , q

(2)t , qoil;t,

qoil;t�1, �pt and �pt�1; and M3 is an AR model with �pt and �pt�1 as regressors.

33

Jan 2007 June 2007 Dec 2007 June 2008130

135

140

145

150

155

160

165

170actual values of commodity price indexcounterfactual: fixed convenience yieldcounterfactual: fixed interest ratecounterfactual: fixed exchange ratecounterfactual: all fixed

Figure 1. Actual and counterfactual values of the IMF commodity price index for 2007:M1 -

2008:M7. The counterfactuals hold (i) the two principal components of convenience yields, (ii)

interest rate, (iii) changes of USD trade-weighted index and (iv) all of these variables �xed at their

2006:M12 values, respectively.

34

c(1)t+j c

(2)t+j

Personal Consumer Expenditures

­5 0 5­0.6

­0.4

­0.2

0

0.2

0.4

0.6

Lag

Sa

mp

le C

ross

 Co

rre

latio

n

­5 0 5­0.6

­0.4

­0.2

0

0.2

0.4

0.6

Lag

Sa

mp

le C

ross

 Co

rre

latio

n

Industrial Production

­5 0 5­0.6

­0.4

­0.2

0

0.2

0.4

0.6

Lag

Sa

mp

le C

ross

 Co

rre

latio

n

­5 0 5­0.6

­0.4

­0.2

0

0.2

0.4

0.6

Lag

Sa

mp

le C

ross

 Co

rre

latio

n

Stock Returns

­5 0 5­0.6

­0.4

­0.2

0

0.2

0.4

0.6

Lag

Sa

mp

le C

ross

 Co

rre

latio

n

­5 0 5­0.6

­0.4

­0.2

0

0.2

0.4

0.6

Lag

Sa

mp

le C

ross

 Co

rre

latio

n

Yield Spread (10-year bond - 3-month bill)

­5 0 5­0.6

­0.4

­0.2

0

0.2

0.4

0.6

Lag

Sa

mp

le C

ross

 Co

rre

latio

n

­5 0 5­0.6

­0.4

­0.2

0

0.2

0.4

0.6

Lag

Sa

mp

le C

ross

 Co

rre

latio

n

Figure 2. Cross correlation functions Corr(yt; ct+j), j = �l; :::;�1; 0; 1; :::; l, where yt are theannualized growth rates of U.S. real personal consumer expenditures, real industrial production,value-weighted stock returns and the spread between 10-year government bond and 3-month T-bill,and ct is the �rst (left graphs) or second (right graphs) principal component of convenience yields.The dotted and dashed lines denote one and two standard error bands, respectively.

35

c(1)t+j c

(2)t+j

World

­5 0 5­0.6

­0.4

­0.2

0

0.2

0.4

0.6

Lag

Sa

mp

le C

ross

 Co

rre

latio

n

­5 0 5­0.6

­0.4

­0.2

0

0.2

0.4

0.6

Lag

Sa

mp

le C

ross

 Co

rre

latio

n

Advanced Economies

­5 0 5­0.6

­0.4

­0.2

0

0.2

0.4

0.6

Lag

Sa

mp

le C

ross

 Co

rre

latio

n

­5 0 5­0.6

­0.4

­0.2

0

0.2

0.4

0.6

Lag

Sa

mp

le C

ross

 Co

rre

latio

n

China

­5 0 5­0.6

­0.4

­0.2

0

0.2

0.4

0.6

Lag

Sa

mp

le C

ross

 Co

rre

latio

n

­5 0 5­0.6

­0.4

­0.2

0

0.2

0.4

0.6

Lag

Sa

mp

le C

ross

 Co

rre

latio

n

India

­5 0 5­0.6

­0.4

­0.2

0

0.2

0.4

0.6

Lag

Sa

mp

le C

ross

 Co

rre

latio

n

­5 0 5­0.6

­0.4

­0.2

0

0.2

0.4

0.6

Lag

Sa

mp

le C

ross

 Co

rre

latio

n

Figure 3. Cross correlation functions Corr(yt; ct+j), j = �l; :::;�1; 0; 1; :::; l, where yt are theannual GDP growth rates (source: IFS) of the world, advanced economies, China and India, andct is the �rst (left graphs) or second (right graphs) principal component of convenience yields. Thedotted and dashed lines denote one and two standard error bands, respectively.

36

A Appendix: Data Description

Table A.1 lists information about each commodity used in the analysis. It contains the symbol,

description, futures exchange where the commodity is traded, contract size, sample price and

contract months. The notation for the futures exchanges is CBOT - Chicago Board of Trade,

CME - Chicago Mercantile Exchange, NYBOT - New York Board of Trade, NYMEX - New York

Mercantile Exchange and WCE - Winnipeg Commodity Exchange and the symbols for futures

contract months are F = January, G = February, H = March, J = April, K = May, M = June, N

= July, Q = August, U = September, V = October, X = November and Z = December. The data

source is Commodity Exchange Bureau.

Table A.1. Commodity Description.

description exchange contract size price contract monthFoodstu¤sCC Cocoa/Ivory Coast NYBOT 10 metric tons 1216 H,K,N,U,ZKC Co¤ee �C�/Columbian NYBOT 37,500 lbs. 78.25 H,K,N,U,ZJO Orange Juice, Frozen Concentr. NYBOT 15,000 lbs. 100.50 F,H,K,N,U,XSB Sugar #11/World raw NYBOT 112,000 lbs. 8.32 H,K,N,VGrains and OilseedsWC Canola/No.1 WCE 20 tonnes 326.30 F,H,K,N,XC- Corn/No.2 Yellow CBOT 5,000 bu. 203.50 F,H,K,N,U,X,ZO- Oats/No.2 White heavy CBOT 5,000 bu. 140.75 H,K,N,U,ZS- Soybeans/No.1 Yellow CBOT 5,000 bu. 559.75 F,H,K,N,Q,U,XBO Soybean Oil/Crude CBOT 60,000 lbs. 20.34 F,H,K,N,Q,U,V,ZW- Wheat/No.2 Soft Red CBOT 5,000 bu. 283.50 H,K,N,U,ZIndustrialsCT Cotton/1-1/16" NYBOT 50,000 lbs. 51.60 H,K,N,V,ZLB Lumber/Spruce-Pine Fir 2�4 CME 110,000 brd. feet 245 F,H,K,N,U,XLivestock and MeatsFC Cattle, Feeder/Average CME 50,000 lbs. 90.38 F,H,J,K,Q,U,V,XLC Cattle, Live/Choice Average CME 40,000 lbs. 77.25 G,J,M,Q,V,ZLH Hogs, Lean/Average Iowa/S M CME 40,000 lbs. 41.90 G,J,M,N,Q,V,ZPB Pork Bellies, Frozen 12-14 lbs. CME 40,000 lbs. 30.25 G,H,K,N,QMetalsHG Copper High Grade/Scrap No.2 NYMEX 25,000 lbs. 81.75 H,K,N,U,ZGC Gold NYMEX 100 troy ounces 334.65 G,J,M,Q,V,ZPA Palladium NYMEX 100 troy ounces 95.75 H,M,U,ZPL Platinum NYMEX 50 troy ounces 362.00 F,J,N,VSI Silver NYMEX 5,000 troy ounces 532.30 H,K,N,U,ZEnergyCL Crude Oil, WTI/Global Spot NYMEX 1,000 barrels 19.50 F-ZHO Heating Oil #2/Fuel Oil NYMEX 42,000 gallons .5474 F-Z

37

Table A.2. Mean, Standard Deviation and First-Order Autocorrelations.

cjt;n qjtmedian std.dev. AR(1) median std.dev. AR(1)

Cocoa -0.721 2.206 0.830 -0.008 0.120 0.860Co¤ee -1.211 3.526 0.936 -0.013 0.172 0.885Orange Juice 0.371 2.855 0.882 -0.006 0.158 0.912Sugar 0.158 5.165 0.798 -0.004 0.174 0.871Canola -0.517 3.171 0.754 -0.004 0.110 0.870Corn -1.777 3.421 0.832 -0.012 0.124 0.880Oats -1.768 4.718 0.826 0.000 0.145 0.873Soybeans -0.213 1.780 0.758 -0.022 0.116 0.889Soybean Oil -0.259 1.526 0.909 -0.019 0.126 0.880Wheat -1.521 3.839 0.866 -0.011 0.113 0.860Cotton -0.441 5.155 0.723 -0.005 0.127 0.860Lumber -0.798 4.256 0.818 -0.002 0.126 0.823Cattle, Feeder 0.966 1.637 0.617 0.003 0.062 0.856Cattle, Live 0.882 3.507 0.679 0.004 0.053 0.743Hogs 1.050 7.133 0.677 0.003 0.130 0.841Pork Bellies 0.604 5.279 0.236 -0.002 0.172 0.830Copper 0.612 2.690 0.892 -0.007 0.116 0.865Gold 0.075 0.202 0.246 0.003 0.055 0.828Palladium 0.765 1.716 0.755 -0.007 0.155 0.886Platinum 0.809 1.085 0.770 -0.001 0.097 0.881Silver -0.119 0.323 0.233 -0.008 0.087 0.791Crude Oil 0.845 1.972 0.843 0.003 0.152 0.868Heating Oil -0.283 2.758 0.759 0.001 0.151 0.870

Notes: cjt;n is convenience yield and qjt is detrended real price of commodity j.

38

B Appendix: Monte Carlo Simulation

This appendix reports some simulation evidence on the coverage properties of the bootstrap con-

�dence intervals that we propose in Section 4. The data generating process for the Monte Carlo

experiment is similar to the one considered by Bai and Ng (2006). More speci�cally, the simulated

data for i = 1; :::; N; t = 1; :::; T and j = 1; 2 are generated as

xit = �0iFt + eit

Fjt = �jFjt�1 + (1� �2j )1=2ujt

yt = 1 + "t;

where �i are drawn from the uniform distribution on [0,1], �j = 0:8j and ujt � iidN(0; 1) for

j = 1; 2; vt is an N � 1 vector of standard normal variables (uncorrelated with ujt) with variance�(i)2 (i = 1; :::; N) that is uniformly distributed on [0.5, 1.5], et = vt(0:5)

1=2 with (0:5)1=2

denoting the Choleski decomposition of the N �N Toeplitz matrix (0:5) whose jth main diagonal

is 0:5j for j � 10 and 0 otherwise, and "t = 0:5"t�1 + �t with �t � iidN(0; 1).

For each simulated sample, we compute two principal components from xit, denoted by x(1)t

and x(2)t , and estimate the parameters of the regression

yt = �0 + �1x(1)t + �2x

(2)t + "t;

where the true values of �1 and �2 are zero. It is important to note that the factor structure of the

model from which the data are generated is not imposed in our bootstrap inference procedure. Table

B.1 reports the coverage rates of the 90% asymptotic (based on the standard normal critical values)

and bootstrap con�dence intervals for �1 and �2: The results reveal the asymptotic con�dence

intervals tend to undercover by 3.8% to 8.6% while the coverage rates of the bootstrap con�dence

intervals are very close to the nominal level even for small sample sizes.

Table B.1. Monte Carlo Coverage Rates of 90% Con�dence Intervals.

N T asymptotic bootstrap�1 �2 �1 �2

20 100 0.814 0.837 0.884 0.902300 0.847 0.862 0.893 0.890600 0.856 0.862 0.901 0.884

40 100 0.816 0.831 0.895 0.912300 0.844 0.855 0.893 0.871600 0.859 0.854 0.906 0.870

39