Analyzing the short-term predictability
of the stock market
Alex P. Taylor∗
Alliance Manchester Business School
Jingya Wang†
Alliance Manchester Business School
February 14, 2016
∗
Alex P. Taylor, Accounting and Finance Group, Alliance Manchester Business School,
The University of Manchester, Booth Street West, Manchester, M15 6PB, England, e-mail:
[email protected], Tel: +44(0)161 275 0441, Fax: +44(0)161 275 4023.
†
Corresponding Author: Jingya Wang, a PhD student in finance, Alliance Manchester Business School, The University of Manchester, Booth Street West, Manchester, M15 6PB, England,
e-mail: [email protected]
1
Abstract
This paper evaluates the ability of lagged commodity returns to forecast market returns. In order to exploit the predictive information from a
relatively large amount of commodity returns, we apply the partial-leastsquares (PLS) method pioneered by Kelly and Pruitt (2013). We find that
the commodity returns measured over previous twelve months show strong
predictive power in monthly and three-month forecasts, in-sample and outof-sample. The findings are robust to controlling for risk factors such as
momentum, Fama-French three factors and industry returns previously
identified to be significant predictors of market returns (Hong, Torous and
Valkanov, 2007).
2
1
Introduction
A large array of predictors have been identified in the literature on forecasting
aggregate market returns. Prominent amongst these predictors are valuation
ratios, for example the dividend-price (dp) ratio, the dividend-yield (dy) ratio, the
book-to-market (bm) ratio, the earnings-dividend (ed) ratio, the price-earnings
(pe) ratio.1 Researchers find that these valuation ratios show good performance
in forecasting market returns at long horizons due to their high persistence (e.g.,
Jegadeesh, 1991; Lettau & Ludvigson, 2001; Welch & Goyal, 2008; Cochrane,
2005, pp.391-395). Researchers also have linked the stock market with other
markets, such as the bond market predictors (e.g., Campbell, 1987; Fama &
French, 1989; Asness, 2003; Fama & French, 1989; Campbell, 1987; Hodrick,
1992), the exchange market (Amihud, 1994; Bartov & Bohnar, 1994; Obben,
Pech, & Shakur, 2007).
However, relatively little attention has been paid to whether the the commodity market contains information relevant for predicting stock market returns. The
relationship between commodity returns and stock market returns has mostly
been investigated by examining the contemporaneous correlation of stocks returns and commodity factors (e.g., Brooks, Fernandez-Perez, Miffre, & Nneji,
2014; Baker & Routledge, 2015). The literature primarily concentrates on oil
commodity exposure (e.g., Chen, Roll, and Ross (1986), Hamilton (1983); Jones
and Kaul (1996)), however, other commodities are seldom investigated. More
recently, some studies have examined the role of commodity returns as predictors of stock returns, and find evidence that the commodity risk indeed matters
for future investment opportunities (e.g., Jacobsen, Marshall, & Visaltanachoti,
2009, Hu & Xiong, 2013; Hong & Yogo, 2012; Bakshi, Gao, & Rossi, 2014; Miffre,
1
For dp ratio see Rozeff (1984), Fama and French (1988), Hodrick (1992), Campbell and
Viceira (2002), Campbell and Yogo (2006), Lewellen (2004), Menzly and Veronesi (2004); for
the book-to-market (bm) ratio see Kothari and Shanken (1997), Pontiff and Schall (1998); for
the earnings-dividend (ed) ratio see Campbell and Shiller (1988, 1998), Lamont (1998); for the
price-earnings (pe) ratio see Cole, Helwege, and Laster (1996), Campbell and Shiller (1998),
Lander, Orphanides, and Douvogiannis (1997), Pu (2000), Welch and Goyal (2008).
3
Fuertes, & Fernandez-Perez, 2015).
Among these studies, the work by Jacobsen, Marshall, and Visaltanachoti
(2009), Hu and Xiong (2013) examines the predictive power of commodity returns
for aggregate market returns. These studies both rely on univariate regressions
of market returns on selected commodity returns, and do not evaluate the overall
predictive ability of a collection of commodities. In this paper we consider a
wide cross-section of commodities in order to establish the maximum amount of
information about future returns that can be extracted from commodity prices.
Our study is motivated by the findings of Kelly and Pruitt (2013) that there is a
large amount of information about future stock returns hidden in the cross-section
of disaggregate bm ratios.
The main problem, when faced with the large pool of potential predictor
variables, as is the case when considering a large cross-section of commodities,
is the issue of dimensionality which means that standard approaches, such as an
OLS regression of the market returns on the predictors variables, cannot be used.
However, there are various techniques to overcome this problem by reducing the
dimension of the predictors. The most well-known technique is the method of
principal component analysis (PCA) (e.g., Stock & Watson, 2002; Ludvigson &
Ng, 2007). More recently Kelly and Pruitt (2013, 2015) show that a partial-leastsquares (PLS) methodology can be used to extract the predictive information
from a large group of potential predictor variables. They argue that because
PCA relies on the structure of the covariance between predictors, rather than the
covariance between predictors and the forecast target as in PLS, the estimated
expected return may not be an optimal predictor.2
In this paper we use the PLS methodology to optimally extract forecasts of
market returns from the past commodity returns. We use data on 16 commodities
over the period from January 1950 to January 2015. The lagged returns of the
commodities are used as potential predictor variables and the three-step process
2
See Kelly and Pruitt (2013, 2015) for details.
4
of the PLS methodology is used to extract an estimate of the expected return for
both in-sample and out-of-sample tests. Various horizons of commodity returns
are considered from one to sixteen months so we can identify the horizon over
which the commodity returns show the strongest predictive power.
Our results may be summarized as follows. First, we show that information
contained in commodities prices can predict market returns at short horizons (e.g.,
monthly and three-month). The optimal predictor variable is constructed from
lagged commodity returns measured over last twelve months. The R̄2 values for
the in-sample forecast of monthly and three-month market returns reach 1.79%
and 5.66%, respectively. Also out-of-sample forecasts show that the forecast
performance of the commodity predictor outperforms the historical mean forecast
of market returns. Moreover, the forecast performance holds in subsample tests.
Second, the latent factor extracted from commodity returns retains its significance
in the presence of other short-term predictors such as momentum, SMB, HML,
lagged excess returns and industry returns.3
Our research contributes to the literature in a number of ways. Firstly, we
show that the cross-section of commodity prices contain important information
about future market returns and that this may be extracted using the PLS technique of Kelly and Pruitt (2013, 2015). The resulting estimate of the expected
market return is a strong predictor both in-sample and out-of-sample. Secondly,
compared to previous analysis, based on univariate regressions on individual commodity predictors, we are able to establish the overall predictive information from
the cross-section of available commodities. Thirdly, the finding that the commodity returns remain significant after accounting for commonly used predictors
indicates that the information we identify in commodity prices is novel and provides new evidence that market returns are predictable over the short horizon.
The source of the predictability, whether it could be due to variation in risk or
3
Hong, Torous, and Valkanov (2007) show that various industry returns lead stock market
returns.
5
mispricing, is difficult to analyze and we leave this question to future work.4
The rest of the paper is organized as below, section 2 introduces the PLS
method along with the univariate predictive regression which has been widely used
in conventional studies; section 3 describes the data; section 4 shows empirical
findings and the results of robustness check; section 5 concludes.
2
Methodology
In this section, we introduce the PLS methodology and univariate predictive
regression.
2.1
The univariate predictive regression
We firstly describe the univariate predictive regression conventionally used for
testing the predictive power of commodity returns:
e
rt+M
= β0 + βi ri,t−H:t + ωt+1 .
(1)
e
is stock market returns in excess of the risk-free rate at a horizon
Where rt+M
of M months; ri,t−H:t is the commodity returns measured over last H periods;
e
βi reflects how rt+M
varies with one-unit change in ri,t−H:t .5 The measure-of-fit
(R̄2 ) indicates the amount of expected market returns that is predicted by each
ri,t−H:t .
4
Various explanations of short term predictability have been proposed including the gradualdiffusion-hypothesis (Hong, Torous, & Valkanov, 2007). They show that information from industry returns can pass slowly to the aggregate stock market. Hong, Torous, and Valkanov
(2007) provide empirical findings confirming the hypothesis that some industries show significant performance in forecasting one-month- and two-month-ahead stock market returns. One
explanation for the hypothesis is that investors have to spend some time in reactions as the
limited capacity for understanding all the available information (e.g., Shiller, 2000; Sims, 2001).
5
It appears that current literature about the sign of βi lacks explicit interpretations. However, empirical evidence suggests that the commodity market may be negatively related to the
stock market. Hong, Torous, and Valkanov (2007) show that industry returns associated with
industrial minerals returns are negatively related to market returns. Jacobsen, Marshall, and
Visaltanachoti (2009) observe negative predictive relationship between commodity returns that
are measured over various horizons and the stock market returns.
6
2.2
The PLS methodology
Kelly and Pruitt (2013) apply the PLS methodology to extract a unique predictive
factor from portfolio-level bm ratios. This method can be implemented as a threestep OLS regression-based procedure. In their study, the potential predictor
variables are a set of disaggregate bm ratios and they assume that these variables
are differentially exposed to the state variable driving variations in the expected
market returns. In our research, the potential predictors are the past returns of
various commodities and we assume that fluctuations in commodity returns are
also driven by the same state variable. The PLS technique, which consists of
three steps of OLS regressions, is able to isolate the component related to the
expected returns, whilst filtering out factors related to cash-flow shocks.
The first step of PLS procedure is to run a time-series regression of disaggregate commodity returns on excess market returns
e
ri,t−H:t = αi + βi rm,t+1:t+M
+ εi,t−H:t .
(2)
e
Where αi is the intercept of regression; βi is the loading on rm,t+1:t+M
. β̂i de-
scribes the amount of variations in each ri,t−H:t with respect to one-unit change
e
. This regression is the empirical implementation of the folin expected rm,t+1:t+M
e
] + εi,t−H:t , which measures the
lowing relationship: ri,t−H:t = αi + βi Et [rm,t+1:t+M
exposure of each of the potential predictor variables, ri,t−H:t , to expected market
e
returns Et [rm,t+1:t+M
]. The empirical implementation substitutes the expected
returns with an unbiased proxy, the realized market returns.
Using β̂i from the first-step regression we run the second-step cross-sectional
regression at time t
ri,t−H:t = φi + Ft β̂i + i,t−H:t .
(3)
where Ft is the estimated cross-sectional regression coefficient. Since β̂i approximately captures how each potential predictor variable, ri,t−H:t , is affected
e
by variations in Et [rm,t+1:t+M
], the cross-sectional regression of ri,t−H:t provides
7
e
estimation of Ft which asymptotically equals the latent factor Et [rm,t+1:t+M
].6
The cross-sectional regression is repeated for all t to build up a time series of the
expected returns, or the composite predictor variable, Ft .
e
By running the first and second steps of regressions, Et [rm,t+1:t+M
] represents
the predictive factor exploited from ri,t−H:t and is a linear combination of disaggregate commodity returns. The weight of each ri,t−H:t shows contribution of
each ri,t−H:t to the forecast.7
Finally the composite predictor variable, F̂t , is tested in a predictive regression
e
on F̂t
of rm,t+1:t+M
e
rm,t+1:t+M
= c + γt+M F̂t + ξt+1:t+M .
(4)
e
Where γ̂ captures how rm,t+1:t+M
varies with one-unit change in F̂t and γ̂t+M is
asymptotically normally distributed 8 The measure-of-fit of equation (4) indicates
the amount of expected rm,t+1:t+M that is predicted by F̂t .
2.2.1
The out-of-sample implementation
In our analysis, we perform a recursive out-of-sample forecast procedure to examine whether the forecast from ri,t−H:t is reliable in a real-time manner. The
procedure exactly follows the way suggested by Welch and Goyal (2008) in which
the forecast of each time point only uses the information prior to the forecast point
of date. This ensures that the forecast at each time point avoids look-ahead bias.
We describe this procedure as below.
We take the example of forecasting the three-month market returns from come
modity returns measured over last four months.9 For simplicity, we define rm,τ
+3
6
Kelly and Pruitt (2011) provide the matrix algebra in details.
We provide the matrix algebra of calculating the weight in appendix.
8
Kelly and Pruitt (2015) provide statistical proofs with underlying assumptions in details.
9
The monthly forecast would not have look-ahead bias as the data is not overlapped. Regressions on overlapping data may suffer estimation bias. Hansen and Hodrick (1980), Hansen
(1982) show that there may be serial correlation existed in residuals when the data is overlapping. Boudoukh, Richardson, and Whitelaw (2008), Amihud, Hurvich, and Wang (2009) show
that when forecasting long-horizon market returns, which have overlapping observations, using
7
8
as three-month excess return measured over the period from τ + 1 to τ + 3 and
ri,τ as commodity returns over the period from τ − 3 to τ . The full sample having
observations {1, 2, 3, ..., T } is split at ttr and the training sample has observations
{1, 2, 3..., ttr }. In the first step regression, the market returns used as independent
e
e
e
variable are observations {rm,7
, rm,8
, ..., rm,t
}, and the commodity returns used
tr
as dependent variable are observations {ri,4 , ri,5 , ..., ri,ttr−3 }. In the second step,
we run regressions of the cross-sectional commodity returns, which are observations {ri,4 , ri,5 , ..., ri,ttr−3 }, on β̂i estimated from the first-step regression. The
estimated coefficient F̂ttr−3 corresponds to predictive information extracted up to
the time point {4, 5, ..., ttr−3 }. Finally, we run a time-series regression of market
e
e
e
} on F̂ttr−3 . The forecast of rm,ttr+3
, ..., rm,t
, rm,8
returns using observations {rm,7
tr
equals sum of the constant and the product of last observation of F̂ttr−3 and the
predictive loading estimated in the last step regression. We repeat the whole
procedure by expanding the training sample period-by-period until ttr = T − 3.
We compare the forecast with forecast equals historical mean of the market returns. The error of our forecast (errunrestr ) equals the difference between realized
market returns and forecast from commodity returns, and the error of historical
mean model (errrestr ) equals the difference between realized market returns and
historical mean. The pseudo R2 is calculated as
pseudo R2 = 1 −
M SEunrestr
.
M SErestr
Where
M SEunrestr = err2 unrestr
M SErestr = err2 restr .
persistent predictors, the significance of predictive coefficients may be over-estimated and the
R̄2 of predictive model may increase with the forecast horizon. We follow Kelly and Pruitt
(2013) and report Newey-West corrected t-statistics to account for the serial correlation that
may exist in residuals caused by overlapping data.
9
3
Data
〈Insert table 1 here〉
We obtain the daily spot prices of commodities from the Global Financial Data
(GFD).10 We aim to form the dataset with the longest time series and have to omit
commodities, such as energy products, which mostly become available after 1986,
that start late. There are 16 commodities, which may be categorized into four
industries: agriculture, industrial metals, energy and precious metals, included in
the dataset. Table 1 presents the name and start date of each commodity along
with the corresponding industry. We measure commodity price change over last
H month (ri,t−H:t ) as the log-price difference between the last trading day in
month t and the last trading day in month t − H + 1. We let the sample start
from 1950 and form the dataset without missing values. Therefore, our sample
period is January 1950-March 2015. We split the entire sample to investigate
whether the stock market return predictability varies with sample period. The
first subsample is from January 1950 to December 1984 and the second subsample
is from January 1985 to March 2015.
We use market returns available from the Kenneth R. French Data library.
The stock market return equals the value-weighted market return based on all
CRSP firms which become corporations in the US and are traded in NYSE,
AMEX or NASDAQ, and the risk-free rate is one-month treasury bill rate which
is from Ibbotson Associates.11 The three-month, six-month and twelve-month
returns are computed from monthly data.
Apart from the commodity returns, we use the predictors investigated in
Welch and Goyal (2008) as benchmarks and the data file which ends in December
2013 is from Amit Goyal’s personal website. These alternative predictors include
the price-dividend ratio (pd), price-to-earning ratio (pe), book-to-market ratio
10
By comparing GFD and Datastream (DS), we find that the commodity data included in
GFD generally have longer series than the data in DS.
11
Please see Keneth R. French Data library for details.
10
(bm), the t-bill rate (tbl), the net equity expansion ratio (ntis), the long-term
yield (lty), the inflation rate (inf l) the long-term rate of return (ltr) and stock
variance (svar).12
In addition, we perform a robustness check with control variables that are
frequently used as risk factors in multivariate asset pricing models, such as the
momentum factor, SMB, HML, lagged excess returns, and industry returns that
are identified to be significant predictors of market returns (Hong, Torous, &
Valkanov, 2007). The data of risk factors and lagged values of excess returns is
also from Kenneth R. French Data library. Following Hong, Torous, and Valkanov
(2007), the data of industry returns is Fama-French 38 value-weighted industry returns and the data of real estate industry is returns on REIT index from NAREIT
website. Apart from the data of real estate industry, which starts from January
1972, the data of other control variables all starts from January 1950 and ends
in December 2014.
We further test the predictability of industry returns from commodity returns.
For this test, the industry returns are those given by the Fama-French 49 industry
portfolios. We follow the definition of industry classification and use returns of
agriculture industry, mines (regarded as the industrial metals industry), gold
(regarded as the precious metals industry), and oil (regarded as energy industry)
in analysis.
4
Empirical findings
4.1
Forecasting monthly excess returns from commodity
returns
〈Insert table 2 here〉
The findings of Jacobsen, Marshall, and Visaltanachoti (2009) show that
12
We exclude the cross-section premium (csp) from the data file as it is no longer available
from January 2003.
11
the predictive power of commodity returns may vary according to the horizon
over which the commodity return is measured. Consequently we apply the PLS
method separately to commodity returns measured at various horizons. We let
the horizon be one, two, four, and up to sixteen months (ri,t−1:t , ri,t−2:t , ri,t−4:t ,...,
ri,t−16:t ). Table 2 shows the prediction results for equation (4). The in-sample
test provides estimation of predictive loading on the latent factor (γ̂t+M ), the
measure-of-fit (R̄2 ) of the third-step regression, the t-statistic (t-statKP ) suggested by Kelly and Pruitt (2013, 2015) and Newey-West t-statistic (t-statN W ).13
The pseudo R2 of the out-of-sample forecast (Welch & Goyal, 2008),
pseudo R2 = 1 −
M SEunrestricted
M SErestricted
Where M SErestricted is the mean-squared-error (MSE) of the forecast which equals
the average value of historical excess returns; M SEunrestricted is the MSE of the
forecast from predictive model. A positive pseudo R2 means the forecast from
the predictive model is more accurate than the forecast equals historical mean
of excess returns. Since the value of pseudo R2 is sensitive to the size of training sample (e.g., Brennan & Xia, 2005), we report pseudo R2 s generated from
two training samples: January 1950-December 1984 and January 1950-December
1994. Also, we perform the ENC-NEW test (e.g., Clark & MacCracken, 2001)
under the null hypothesis that the historical mean of forecast target encompasses
the predictive information carried by commodity returns.
In panel A of table 2, we report the results estimated from the entire sample
which covers a period over 65 years. Columns 2-10 correspond to regressions using
ri,t−1:t , ri,t−2:t ,..., ri,t−16:t as predictors, respectively. Throughout these results,
all predictors have t-statKP and t-statN W values consistently greater than 2.58,
13
Kelly and Pruitt (2013, 2015) provide statistical evidence that the predictive loading of the
third-step regression is asymptotically normally distributed when the number of observations
and predictors are large. Please see their paper for details. As the forecasts from commodity
returns at horizons over two months consist of overlapping data, we use t-statistics with NeweyWest corrections to overcome heteroskedasticity and serial correlations existed in residuals.
12
suggesting that the predictors are significantly different from zero at the 1%
significance level, excluding ri,t−1:t and ri,t−2:t of which values of t-statN W are
slightly smaller than 2.58. We notice that the value of R̄2 generally follows a
hump shape, starting from 0.86% (using ri,t−1:t as predictor), rising to the peak
value 1.98% (using ri,t−14:t as predictor) and falling to 1.51% (using ri,t−16:t as
predictor). The value of estimated predictive loading (γ̂t+M ), also shows the
same trend that it reaches the peak point 2.10 (in percentage) when using ri,t−14:t
as predictor, indicating the greatest impact of the predictor.
However, in the last two rows of panel A, the out-of-sample forecast results
provide little evidence that the commodity returns outperform the historical mean
of market returns. Only the forecast from ri,t−12:t performs well, regardless of the
choice of training sample. When the training sample period is January 1950December 1984, the pseudo R2 of forecast from rt−12:t is 0.52% with ENC-NEW
statistic significantly rejects the null hypothesis at the 10% significance level;
when the training sample expands to December 1994, the pseudo R2 increases to
1.74%, which is higher than values generated by other predictors, and the ENCNEW statistic significantly rejects the null hypothesis at the 1% level. Although
ri,t−14:t shows the strongest predictive power in-sample, it fails to show convincing
evidence that the out-of-sample forecast outperforms the forecast which equals
average value of historical market returns with pseudo R2 s of -0.93% and 1.28%
(the ENC-NEW statistic significantly rejects the null hypothesis at the 1% level)
corresponding to the two training samples.
In addition, we examine whether the significant predictive performance of
commodity returns still holds in subsamples. The entire sample is split evenly
into two parts: January 1950-December 1984 and January 1985-March 2015.
Panels B and C report the results. Indeed, the prediction results from subsample
tests show that all commodity returns are significant predictors with values of
t-statKP and t-statN W that are at least greater than 1.65 (significant at the 10%
level), and the value of R̄2 is somewhat higher than the value identified in full
13
sample test. The R̄2 ranges from 1.34% (using ri,t−1:t as predictor) to 2.59% (using
ri,t−14:t as predictor) and from 2.44% (using ri,t−1:t as predictor) to 5.46% (using
ri,t−2:t as predictor) when testing on the first and second subsamples, respectively.
4.2
Forecasting the stock market returns at three-month
horizon
〈Insert table 3 here〉
Given that ri,t−12:t appears to be the most appropriate predictor of monthly
market returns, in this section, we propose examining whether the ability of
the strongest predictor still holds in three-month forecast. To achieve this, we
regress three-month market returns on the same predictors studied in section 4.1
in-sample and out-of-sample, and compare the measure-of-fit of regressions.
Table 3 presents the results. The results show that ri,t−12:t has the greatest
ability in forecasting three-month market returns in-sample and out-of-sample.
We begin with the forecast on entire sample (panel A), the value of R̄2 keeps the
hump shape and ranges from 0.56% (using ri,t−1:t as predictor), peaks at 5.66%
(using ri,t−12:t as predictor) and falls to 4.74% (using ri,t−16:t as predictor). As for
the significance of predictors, only the coefficients estimated from regressions on
ri,t−1:t and ri,t−2:t have values of t-statKP and t-statN W that are greater than 1.96
(significant at the 5% level), the coefficients on other commodity returns all have
values of t-statKP and t-statN W that are greater than 2.58 (significant at the 1%
level).
Compared with the out-of-sample performance in monthly forecast, the performance in three-month forecast slightly improves. The pseudo R2 s are consistently
positive with forecasts starting from January 1985 and January 1995 when using
ri,t−10:t , ri,t−12:t and ri,t−14:t as predictors. Moreover, the ENC-NEW statistics are
primarily significant at the 1% level, suggesting that the forecast performance
from these predictors are significantly better than the performance of historical
14
mean.
The findings in subsample tests (panels B and C) are similar with findings in
monthly forecast: commodity returns measured over various horizons all predict
market returns significantly and ri,t−12:t forecasts three-month market returns substantially with R̄2 s values of 6.63% and 11.01% in the first and second subsample
tests.
4.3
Forecasting market returns at six-month and twelvemonth horizons
We examine whether the predictive power continues to hold over longer horizons
of six and twelve months in tables 4 and 5.
〈Insert table 4 here〉
Table 4 shows the results of forecasting six-month market returns. The findings in entire sample forecast (panel A) are somewhat similar with findings in
monthly and three-month forecasts that, the commodity returns all show significant predictive impact, the value of R̄2 remains the hump shape and peaks at
10.02% when using ri,t−10:t as predictor; the out-of-sample forecasts from ri,t−6:t ,
ri,t−8:t , ri,t−10:t generate positive pseudo R2 s with forecasts starting from January
1985 and January 1995, and the ENC-NEW statistics are significant at 1% level.
Interestingly, ri,t−12:t , the most appropriate predictor in monthly and three-month
forecast, no longer has the strongest ability in six-month forecast. The measureof-fit (R̄2 ) of the model approximates to 9.69% which is slightly lower than the
highest value generated from the model using ri,t−10:t as predictor. Moreover,
the results of out-of-sample forecasts indicate that the forecast from ri,t−12:t is
no longer reliable in a real-time manner, with inconsistent signs of pseudo R2 s:
-1.32% (forecast starting from January 1985) and 0.33% (forecast starting from
January 1995).
15
As reported in panels B and C, the significant predictive impact of commodity
returns exists in subsample tests, except for ri,t−4:t of which t-statN W has a value
of 1.45 that fails to reject the null hypothesis that the coefficient of predictor is
zero.
〈Insert table 5 here〉
Table 5 presents results of forecasting twelve-month market returns. In panel
A, the in-sample results estimated from the entire sample show that commodity returns measured over various horizons are significant predictors, excluding
ri,t−16:t of which t-statN W is 1.49 which fails to reject the null hypothesis that
the coefficient of predictor is zero, and that the value of R̄2 follows a clear hump
shape, with R̄2 starting from 0.97% (using ri,t−1:t as predictor), peaking at 14.09%
(using ri,t−12:t as predictor) and gradually diminishing when commodity returns
are measured over 14 months or longer horizons. The steadily diminishing predictive ability is in line with changes in γ̂t+M and t-statistics. For instance, γ̂t+M
estimated from regressions on ri,t−12:t has a value of 14.21 (in percentage) which
is higher than the values estimated from regressions on other commodity returns,
suggesting the greatest predictive impact; and the findings that t-statKP has a
value of 9.50 (significant at the 1% level), t-statN W has a value of 1.87 (significant
at the 5% level) consistently confirm the significance of the predictor. However,
as the horizon of commodity returns extends to 14 months, γ̂t+M slightly reduces
to 13.67 (in percentage) with smaller t-statKP (9.05) and t-statN W (1.72); as
the horizon extends to 16 months, γ̂t+M reduces to 12.04 (in percentage) with
even smaller t-statKP (8.38) and t-statN W (1.49), the inconsistency in t-statistics
makes us believe that ri,t−16:t no longer has significant predictive impact on the
twelve-month market returns.
The out-of-sample forecast shows poor performance of commodity returns
that forecasts from commodity returns over various horizons never generate positive pseudo R2 s when the forecast starts from either January 1985 or January
1995. The poor out-of-sample performance stands in contrast to the performance
16
identified in monthly, three-month and six-month forecasts that strong predictors
in in-sample forecast always show good performance in out-of-sample forecast.
Not surprisingly, the results in subsample tests show that the market returns
are predictable by commodity returns, although the significant predictive performance may not exist in the first subsample test (e.g., values of t-statN W estimated
from regressions on ri,t−1:t , ri,t−2:t , ri,t−4:t and ri,t−6:t are smaller than 1.65).
〈Insert figure 1 here〉
Given the superior performance of the predictor created from lagged 12-month
commodity returns (ri,t−12:t ), we focus on this estimate of the expected return.
We plot out the weight of each commodity i in the composite predictor in Figure 1.14 The figure clearly provides us clues about the contribution from each
commodity returns to the forecast. In monthly forecast, apart from returns of
aluminum, high grade copper, live hog and zinc special high grade, returns of
other commodities all contribute to the predictability. Specifically, returns of
soybean oil, Brazil Santos Arabicas, Chicago yellow corn, tin, sugar, soybeans,
West Texas intermediate oil, Wheat and silver make great contributions. In
three-month forecast, in addition to the commodity returns that have little to
contribute to predictability, returns of soybean meal, gold and gold bullion (New
York) also make small contributions. We note that, among the 16 commodities, returns of soybean oil, Chicago yellow corn and West Texas intermediate oil
always contribute substantially to monthly and three-month forecasts.
4.4
Forecasts from other predictors
In this section, we start with investigating predictive ability of individual commodity returns, and then examine predictive ability of the benchmark: predictor
variables studied in Welch and Goyal (2008).
14
We show the matrix logarithm of calculating the coefficient B in Appendix.
17
4.4.1
Univariate regressions on individual commodity returns
〈Insert table 6 here〉
Table 6 presents forecast results from each ri,t−12:t . We report in-sample results
including the estimated predictive loadings (β̂i ), t-statistics with Newey-West corrections in brackets along with the measure-of-fit (R̄2 ) of the predictive model,
and the out-of-sample pseudo R2 of which the training sample is January 1950December 1984. We also perform the ENC-NEW test (e.g., Clark & MacCracken,
2001) to examine the significance of out-of-sample performance. Panels A and
B report forecast results of monthly and three-month market returns, respectively. We begin with the monthly forecast (panel A). Combining the in-sample
and out-of-sample forecast results together, when forecasting monthly market
returns, only returns of soybean oil and West Texas intermediate oil show consistently significant predictive performance. The returns of a few commodities show
significant in-sample predictive impact, but fail to outperform historical mean of
market returns in out-of-sample forecast. For instance, returns of Chicago yellow corn, soybeans, have t-statistics of -3.28 (significant at 1% level) and -2.12
(significant at 5% level), however, the out-of-sample forecasts fail to outperform
forecasts from historical mean significantly, with pseudo R2 s of -0.22% and 0.17%,
respectively.
The results in panel B show that returns of some commodities have stronger
ability in forecasting three-month market returns. Commodity returns, such as
soybean oil, Chicago yellow corn, soybeans, West Texas intermediate oil, exhibit
significant forecast performance in-sample and out-of-sample. This finding is
largely consistent with findings in figure 1, potentially suggesting that the PLS
procedure efficiently extracts the relevant information from a cross-section of
potential predictors.
4.4.2
A comparison with the benchmark
〈Insert table 7 here〉
18
Table 7 shows forecast results of benchmark predictors. Among the predictors
that have been examined in Welch and Goyal (2008), only the long-term yield
(lty) shows significant performance in forecasting three-month market returns insample and out-of-sample for which the first forecast starts from January 1985.15
Other predictors fail to show consistently significant performance in-sample and
out-of-sample.
4.5
Robustness checks
〈Insert tables 8 here〉
The information about future market returns extracted from commodities
may be correlated with existing predictor variables. To address this concern,
we conduct a robustness check by adding control variables into the predictive
regression. We let the well-known risk factors, the momentum factor and FamaFrench three factors, along with the industry returns that are identified to be
significant predictors of monthly market returns (Hong, Torous, & Valkanov,
2007) be control variables. All factors, excluding the market returns and industry
returns that are lagged values, are contemporaneous.
Table 8 presents the results. For consistency, we test on the entire sample
along with the two subsamples.16 Panels A and B report results from regressing
on monthly and three-month market returns, respectively. The t-statistics with
Newey-West corrections (Newey & West, 1987) are reported in brackets. We focus
on the composite predictor created from lagged 12-month commodity returns
(ri,t−12:t ).
With findings in panels A and B, we find statistically strong evidence that the
significant predictive impact of the factor (Fcommodity,t−1 ) extracted from ri,t−12:t
always exists in tests on entire sample, the first and second subsamples, controlling
15
This finding is consistent with the monthly forecast results reported in Kelly and Pruitt
(2013).
16
Since the commercial real estate returns are available from January 1972, we have to include
the return (rRLEST,t−1 ) in the second subsample test.
19
for momentum factor, Fama-French three factors and industry returns. We notice
that the industry returns provide no significant predictive power. This finding
is not against our expectation as these industry returns are inherently correlated
with fundamentals of stock market which the size and book-to-market stocks are
proxying for (e.g., Hong, Torous, & Valkanov, 2007; Fama & French, 1995).
4.6
Forecasting industry returns from commodities
〈Insert table 9 here〉
As an extension of our research we examine whether commodities can predict
industry returns, and report the results in table 9. As we show in table 1, the
commodities are related to agriculture, industrial metals, energy and precious
metals, returns of these industries may be predictable from commodity returns,
ri,t−12:t . Therefore, we conduct regressions of monthly returns of related industry
on the composite predictor extracted from ri,t−12:t .
In in-sample tests, all industry returns are significantly predictable by the
composite predictor with values of t-statKP and t-statN W that are consistently
greater than 2.58 (significant at the 1% level), except that the values of t-statKP
and t-statN W of the regression on agriculture industry returns are 2.35 and 2.43
(both significant at the 5% level). Specifically, the predictor exhibits the strongest
power in forecasting industry returns of industrial metals with an R̄2 2.35% that
is higher than R̄2 s generated from other forecasts, and the magnitude of estimated
coefficient γ̂t+M , which is also the greatest, reaches 2.48 (in percentage). Indeed,
the strong and significant predictive impact of ri,t−12:t is only robust for outof-sample forecast of industrial metal industry. The pseudo R2 s with forecasts
starting from January 1985 and January 1995 are 0.39% and 1.15% with ENCNEW statistics significant at the 5% and 1% levels, respectively. Findings in
forecasting other industry returns lack convincing evidence that the forecasts are
reliable in a real-time manner. For example, the forecast of agriculture industry
returns generates an extremely low pseudo R2 0.04% with ENC-NEW statistic
20
significant at the 10% level, whereas the pseudo R2 obtained from a smaller
training sample (January 1950-December 1984) becomes negative (-1.30%). This
inconsistency makes us believe that the positive pseudo R2 is insufficient for
reliably forecasting agriculture industry returns. The situation is even worse in
forecasting returns of energy and precious metals industries, the pseudo R2 s are
never positive for any training sample, suggesting greater errors of forecasts from
the composite predictor extracted from ri,t−12:t than forecasts from average value
of historical industry returns.
5
Conclusion
In this paper, we propose evaluating predictive power of disaggregate commodity
returns that are measured over various horizons. By assuming that variations
in commodity returns are driven by the state variable that determines expected
market returns, we employ the PLS methodology (Kelly & Pruitt, 2013) to extract
the latent factor from a cross-section of commodity returns.
The main findings may be summarized as following: i) the amount of predictive information carried by commodity returns may be different, depending on the
horizon over which the commodity returns are measured; ii) the predictabilities
of monthly and three-month stock market returns are significantly improved by
using commodity returns measured over a horizon of twelve months; iii) significant predictive impact of commodity returns measured over last twelve months
holds in subsample tests; iv) in the forecasts of monthly and three-month market
returns, returns of soybeans, Chicago yellow corn and West Texas intermediate oil
typically make substantial contributions; v) the strong predictive ability of commodity returns measured over last twelve months appears to exist in monthly and
three-month forecasts, rather than six-month and twelve-month forecasts; vi) the
returns of industrial metals portfolio are significantly predictable by commodity
returns measured over a horizon of twelve months.
21
To confirm that the success we have achieved is not by chance we examine
the performance of out-of-sample forecasts by reporting pseudo R2 s obtained at
different forecast starting points to lessen the possibility of accidentally reporting
positive pseudo R2 . Finally, we perform a robustness check with control variables
and find that our predictor is not proxying for any determinants of expected
market returns captured by common predictors from the literature.
22
References
Amihud, Y. (1994). Exchange rates and the valuation of equity shares. In
Y. Amihud & R. Levich (Eds.), Exchange rates and corporate performance
(p. 49-50). Irwin.
Amihud, Y., Hurvich, C. M., & Wang, Y. (2009). Multiple-predictor regressions:
hypothesis testing. Review of Financial Studies, 22 (1), 410 - 434.
Asness, C. S. (2003). Fight the fed model. Journal of Portfolio Management,
30 (1), 11 - 24.
Baker, S. D., & Routledge, B. R. (2015). The price of oil risk. Working paper.
Bakshi, G., Gao, X., & Rossi, A. (2014). A better specified asset pricing model to
explain the cross-section and time-series of commodity returns? Working
paper.
Bartov, E., & Bohnar, G. (1994). Firm valuations, earnings expectations and the
exchange rate exposure effect. Journal of Finance, 49 , 1755 - 1785.
Boudoukh, J., Richardson, M., & Whitelaw, R. F. (2008). The myth of longhorizon predictability. Review of Financial Studies, 21 (4), 1577 - 1605.
Brennan, M. J., & Xia, Y. (2005). tay’s as good as cay. Finance Research Letters,
2 , 1 - 14.
Brooks, C., Fernandez-Perez, A., Miffre, J., & Nneji, O. (2014). Commodity risk
factors and the cross-section of equity returns. ICMA Centre.
Campbell, J. Y. (1987). Stock returns and the term structure. Journal of
Financial Economics, 18 (2), 373 - 399.
Campbell, J. Y., & Shiller, R. J. (1988). Stock prices, earnings, and expected
dividends. Journal of Finance, 43 (3), 661 - 676.
Campbell, J. Y., & Shiller, R. J. (1998). Valuation ratios and the long-run stock
market outlook. Journal of Portfolio Managements, 24 (2), 11 - 26.
Campbell, J. Y., & Viceira, L. M. (2002). Strategic asset allocation: portfolio
choice for long-term investors. Oxford: Oxford University Press.
Campbell, J. Y., & Yogo, M. (2006). Efficient tests of stock return predictability.
23
Journal of Financial Economics, 81 (1), 27 - 60.
Chen, N., Roll, R., & Ross, S. A. (1986). Economic forces and the stock market.
Journal of Business, 59 , 383 - 403.
Clark, T. E., & MacCracken, M. W. (2001). Tests of equal forecast accuracy and
encompassing for nested models. Journal of Econometrics, 105 , 85 - 110.
Cochrane, J. H. (2005). Asset pricing (Rev ed.). Princeton, NJ: Princeton
University Press.
Cole, K., Helwege, J., & Laster, D. (1996). Stock market valuation indicator: is
this time different? Financial analysts Journal, 52 (3), 56 - 64.
Fama, E. F., & French, K. R. (1988). Dividend yields and expected stock returns.
Journal of Financial Economics, 22 (1), 3 - 25.
Fama, E. F., & French, K. R. (1989). Business conditions and expected returns
on stock and bonds. Journal of Financial Economics, 25 (1), 23 - 49.
Fama, E. F., & French, K. R. (1995). Size and book-to-market factors in earnings
and returns. Journal of Financie, 50 , 131 - 155.
Hamilton, J. (1983). Oil and the macroeconomy since world war II. Journal of
Political Economy, 91 , 228 - 248.
Hansen, L. (1982). Large sample properties of generalized method of moments
estimators. Economitrica, 1029-1054.
Hansen, L., & Hodrick, R. (1980). Forward exchange rates as optimal predictors
of future spot rates. Journal of Political Economy, 829-853.
Hodrick, R. J. (1992). Dividend yields and expected stock returns: alternative
procedures for inference and measurement. Review of Financial Studies,
5 (3), 257 - 286.
Hong, H., Torous, W., & Valkanov, R. (2007). Do industries lead stock markets?
Journal of Financial Economics, 83 , 367 - 396.
Hong, H., & Yogo, M. (2012). What does futures market interest tell us abou
tthe macroeconomy and asset prices?
Journal of Financial Economics,
105 , 473-490.
24
Hu, C., & Xiong, W. (2013). The informational role of commodity futures
prices. Working paper. Retrieved from http://www8.gsb.columbia.edu/
rtfiles/finance/misc/Information Commodity4 c.pdf
Jacobsen, B., Marshall, B., & Visaltanachoti, N. (2009, May). Return predictability revisited. FMA Asian Conference.
Jegadeesh, N. (1991). Seasonality in stock price mean reversion: evidence from
the u.s. and u.k. Journal of Finance, 46 , 1427 - 1444.
Jones, C. M., & Kaul, G. (1996). Oil and the stock markets. Journal of Finance,
51 , 463 - 491.
Kelly, B., & Pruitt, S. (2011). Market expectations in the cross-section of present
values. Working Paper.
Kelly, B., & Pruitt, S. (2013). Market expectations in the cross-section of present
values. Journal of Finance, 68 (5), 1721 - 1756.
Kelly, B., & Pruitt, S. (2015). The three-pass regression filter: A new approach
to forecasting using many predictors. Journal of Econometrics, 186 , 294 316.
Kothari, S., & Shanken, J. (1997). Book-to-market, dividend yield, and expected
market returns: a time-series analysis. Journal of Financial Economicss,
44 (2), 169 - 203.
Lamont, O. (1998). Earnings and expected returns. Journal of Finance, 53 (5),
1563 - 1587.
Lander, J., Orphanides, A., & Douvogiannis, M. (1997). Earnings forecasts and
the predictability of stock returns: evidence from trading the s&p. Journal
of Portfolio Management, 23 (4), 24 - 35.
Lettau, M., & Ludvigson, S. (2001). Consumption, aggregate wealth and expected
stock returns. Journal of Finance, 56 (3), 815 - 849.
Lewellen, J. (2004). Predicting returns with financial ratios. Journal of Financial
Economicss, 74 (2), 209 - 235.
Ludvigson, S. C., & Ng, S. (2007). The empirical risk-return relation: a factor
25
analysis approach. Journal of Financial Economics, 83 , 171-222.
Menzly, L. T., & Veronesi, P. (2004). Understanding predictability. Journal of
Political Economy, 112 (1), 1 - 47.
Miffre, J., Fuertes, A.-M., & Fernandez-Perez, A. (2015). Commodity risk factors
and intertemporal asset pricing? Working paper. Retrieved from http://
papers.ssrn.com/sol3/papers.cfm?abstract id=2432884
Newey, W. K., & West, K. D. (1987). A simple, positive semi-definite, heteroskedasticity and autocorrelation consistent covariance matrix. Econometrica, 55 , 703 - 708.
Obben, J., Pech, A., & Shakur, S. (2007). Analysis of the relationship between
the share market performance and exchange rates in new zealand: a cointegrating var approach. Working paper.
Pontiff, J., & Schall, L. D. (1998). Book-to-market ratios as predictors of market
returns. Journal of Financial Economicss, 49 (2), 141 - 160.
Pu, S. (2000). The p/e ratio and stock market performance. Economic Review
of the Federal Reserve Bank of Kansas City, 4th Quarter, 23 - 36.
Rozeff, M. S. (1984). Dividend yields are equity risk premiums. Journal of
Portfolio Management, 11 (1), 68 - 75.
Shiller, R. J. (2000). Irrational exuberance. New York: Broadway Books.
Sims, C. A. (2001). Rational inattention: beyond the linear-quadratic case.
American Economic Review, 96 (2), 158 - 163.
Stock, J. H., & Watson, M. W. (2002). Forecasting using principal components
from a large number of predictors. Journal of the American Statistical
Association, 97 , 1167-1179.
Welch, I., & Goyal, A. (2008). A comprehensive look at the empirical performance
of equity premium prediction. Review of Financial Studies, 21 , 1455 - 1508.
26
Tables
Table 1: Monthly forecast from commodity returns
row
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
category
agriculture
industrial metals
energy
precious metals
ticker
BO1599D
COFSAD
C US2D
SU1599D
SYB TD
SYM 4D
W USSD
IHXD
CMALSD
CU NYD
MZN2MD
SN NYD
WTC D
XAG HD
XAU BD
XAU D
start date
31/12/1948
31/12/1946
02/01/1900
31/12/1948
31/12/1936
01/12/1947
02/01/1900
02/01/1900
31/12/1919
02/01/1900
31/12/1903
31/12/1911
31/12/1874
31/12/1919
08/09/1933
08/09/1933
name
Soybean Oil
Brazil Santos Arabicas
Chicago Yellow Corn
Sugar
Soybeans
Soybean Meal
Wheat
Live Hog
Aluminum
High Grade Copper
Zinc Special High Grade
Tin
West Texas Intermediate Oil
Silver
Gold Spot Price-London PM Fixing
Gold Bullion Price-New York
Notes: This table lists the names of commodities which include commodities in agricultures,
industrial metals, energy, and precious metals. The data is obtained from Global Financial
Data. The start date of the commodity varies from each other but all the data ends in March
2015 at daily frequency.
27
Table 2: Monthly forecast from commodity returns
ri,t−1:t
β̂
(t-statKP )
(t-statN W )
R̄2
2
pseudo R198501
2
pseudo R199501
0.98∗∗
(2.95)
(2.53)
0.86
−2.95
−3.53
β̂
(t-statKP )
(t-statN W )
R̄2
1.58∗∗
(2.23)
(1.99)
1.34
β̂
(t-statKP )
(t-statN W )
R̄2
2.71∗∗
(2.41)
(2.37)
2.44
ri,t−2:t ri,t−4:t ri,t−6:t ri,t−8:t ri,t−10:t ri,t−12:t ri,t−14:t ri,t−16:t
Panel A: Entire sample: 195001-201503
1.01∗∗
1.04∗∗∗ 1.09∗∗∗ 1.80∗∗∗ 2.07∗∗∗
1.91∗∗∗
2.10∗∗∗
1.63∗∗∗
(2.78)
(2.84)
(2.84)
(3.10)
(3.93)
(3.15)
(3.95)
(3.66)
(2.55)
(2.98)
(2.86)
(3.14)
(3.99)
(3.25)
(3.89)
(3.34)
0.89
0.92
0.97
1.67
1.95
1.79
1.98
1.51
∗
−1.44
−1.27
0.02
−1.93
−1.07
0.52
−0.93
0.07
−0.32
−1.70
−0.74
−0.91
−0.20
1.74∗∗∗
1.28∗∗∗
1.12∗∗∗
Panel B: Subsample1 : 195001-198412
∗∗∗
2.09
2.03∗∗
1.22∗∗
2.19∗∗
2.43∗∗∗
2.18∗∗
2.82∗∗∗
2.01∗∗∗
(2.82)
(2.80)
(2.21)
(2.91)
(3.16)
(2.49)
(2.82)
(2.71)
(2.59)
(2.34)
(1.96)
(2.14)
(2.85)
(2.30)
(2.74)
(2.75)
1.86
1.79
0.98
1.95
2.20
1.95
2.59
1.78
Panel C: Subsample2 : 198501-201503
∗∗∗
5.73
3.81∗∗∗ 4.02∗∗∗ 4.22∗∗∗ 4.42∗∗∗
4.38∗∗∗
4.04∗∗∗
2.77∗∗∗
(4.39)
(3.79)
(3.27)
(3.65)
(4.07)
(3.43)
(3.43)
(3.16)
(3.60)
(3.57)
(2.47)
(3.23)
(3.53)
(3.25)
(3.07)
(2.87)
5.46
3.55
3.75
3.96
4.15
4.12
3.78
2.50
Notes: This table reports results of forecasting monthly excess returns. The dependent variable is the return
(in percentage) on CRSP value-weighted index for the US stock market in excess of the one-month Treasury
bill rate, the independent variable is the latent factor (F̂t−H:t ) extracted by the PLS procedure (equation
(3)) from commodity returns that are measured over horizons from last one to sixteen months (ri,t−1:t ,
ri,t−2:t ,..., ri,t−16:t ) and columns 2-10 show the results. Panels A-C report the forecast results on the entire
sample January 1950-March 2015, the first subsample January 1950-December 1984 and the second subsample
January 1985-March 2015, respectively. Each panel includes in-sample results which are the coefficients (β̂)
of the latent factor (F̂t−H:t ), the t-statistic of the predictive loading in the third step regression (equation (4))
suggested by Kelly and Pruitt (2013, 2015) (t-statKP ) and t-statistic with Newey-West correction (Newey
and West (1987)), and the R̄2 (in percentage). The last two rows in panel A show the pseudo R2 s generated
from out-of-sample forecast, with the training sample January 1950-December 1984 and January 1985-March
2015, respectively. The ENC-NEW test (Clark and MacCracken (2001)) is performed with the null hypothesis
that the forecast from historical mean model encompasses the forecast from commodity returns. *, **, ***
denote significance at 10%, 5% and 1% levels.
28
Table 3: Three-month forecast from commodity returns
ri,t−1:t
β̂
(t-statKP )
(t-statN W )
R̄2
2
pseudo R198501
2
pseudo R199501
0.69∗∗
(2.06)
(2.12)
0.56
−1.43
−1.58
β̂
(t-statKP )
(t-statN W )
R̄2
2.08∗∗
(2.64)
(2.36)
1.84
β̂
(t-statKP )
(t-statN W )
R̄2
3.55∗
(2.83)
(1.84)
3.28
ri,t−2:t ri,t−4:t ri,t−6:t ri,t−8:t ri,t−10:t ri,t−12:t ri,t−14:t ri,t−16:t
Panel A: Entire sample: 195001-201503
1.57∗∗
2.36∗∗∗ 3.55∗∗∗ 4.64∗∗∗ 5.14∗∗∗
5.78∗∗∗
5.07∗∗∗
4.86∗∗∗
(2.68)
(3.95)
(4.42)
(4.86)
(5.30)
(6.16)
(6.26)
(6.22)
(2.40)
(2.82)
(3.10)
(3.11)
(3.14)
(2.99)
(3.06)
(3.04)
1.45
2.24
3.42
4.51
5.02
5.66
4.95
4.74
∗∗∗
∗∗∗
∗∗
−0.33
−1.57
−2.66
−0.58
2.67
1.44
0.69
−1.16
−1.28
−0.51
0.86∗∗
1.82∗∗∗ 5.38∗∗∗
4.83∗∗∗
1.32∗∗∗ −1.24
Panel B: Subsample1 : 195001-198412
∗∗
3.70
3.54∗
4.32∗
5.89∗∗
6.62∗∗
6.86∗∗
6.98∗∗∗
5.98∗∗∗
(3.42)
(3.43)
(3.53)
(4.35)
(4.62)
(4.49)
(5.51)
(5.23)
(2.30)
(1.82)
(1.75)
(2.06)
(2.45)
(2.24)
(2.64)
(2.76)
3.47
3.31
4.09
5.66
6.40
6.63
6.76
5.75
Panel C: Subsample2 : 198501-201501
∗∗∗
7.20
7.96∗∗∗ 9.56∗∗∗ 10.17∗∗∗ 10.43∗∗∗ 11.25∗∗∗
8.45∗∗∗
7.28∗∗∗
(4.81)
(4.75)
(4.65)
(5.37)
(5.29)
(5.42)
(5.12)
(5.10)
(4.41)
(3.64)
(3.17)
(3.79)
(5.20)
(6.20)
(5.73)
(5.15)
6.95
7.70
9.31
9.92
10.18
11.01
8.19
7.02
Notes: This table reports forecast results of the three-month excess returns. The forecast target is the excess
returns (in percentage) at three-month horizon that is transformed from the monthly data, the independent
variable is the latent factor (F̂t−H:t ) extracted by the PLS procedure (equation (3)) from commodity returns
that are measured over horizons from last one to sixteen months (ri,t−1:t , ri,t−2:t ,..., ri,t−16:t ) and columns
2-10 show the results. Results in panels A-C correspond to forecasts on the entire sample January 1950-March
2015, the first subsample January 1950-December 1984 and the second subsample January 1985-March 2015.
In each panel, the reported in-sample test results include the coefficients (β̂) of the latent factor (F̂t−H:t ),
the t-statistic of the predictive loading in the third step regression (equation (4)) suggested by Kelly and
Pruitt (2013, 2015) (t-statKP ) and t-statistic with Newey-West correction (Newey and West (1987)), and the
R̄2 (in percentage). The last two rows in panel A show the pseudo R2 s generated from the out-of-sample
forecast, with the training sample January 1950-December 1984 and January 1985-March 2015, respectively.
The ENC-NEW test is performed with the null hypothesis that the forecast from historical mean model
encompasses the forecast from commodity returns. *, **, *** denote significance at 10%, 5% and 1% levels.
29
Table 4: Six-month forecast from commodity returns
ri,t−1:t
β̂
(t-statKP )
(t-statN W )
R̄2
2
pseudo R198501
2
pseudo R199501
0.61∗
(1.89)
(1.98)
0.48
−2.28
−0.65
β̂
(t-statKP )
(t-statN W )
R̄2
1.22∗
(1.81)
(1.71)
0.98
β̂
(t-statKP )
(t-statN W )
R̄2
8.31∗∗∗
(4.23)
(2.73)
8.06
ri,t−2:t ri,t−4:t ri,t−6:t ri,t−8:t ri,t−10:t ri,t−12:t ri,t−14:t ri,t−16:t
Panel A: Entire sample: 195001-201503
1.68∗∗
3.90∗∗∗ 6.65∗∗∗ 8.37∗∗∗ 10.14∗∗∗
9.81∗∗∗
9.13∗∗
8.45∗∗
(2.89)
(4.02)
(5.43)
(6.26)
(8.46)
(8.38)
(8.22)
(7.83)
(2.54)
(2.86)
(3.06)
(3.38)
(3.04)
(2.73)
(2.47)
(2.19)
1.56
3.77
6.53
8.25
10.02
9.69
9.01
8.34
∗∗∗
∗∗∗
∗∗∗
−2.84
−1.65
2.29
4.30
2.77
−1.32
−6.85
−10.33
−1.07
−1.15
1.96∗∗∗ 4.07∗∗∗ 4.25∗∗∗
0.33
−4.57
−8.65
Panel B: Subsample1 : 195001-198412
∗
2.36
4.43
6.70
8.99∗∗ 11.05∗∗
9.39∗∗
10.93∗∗∗ 11.46∗∗∗
(2.78)
(3.29)
(4.31)
(5.26)
(6.19)
(6.15)
(7.75)
(7.73)
(1.91)
(1.45)
(1.64)
(2.02)
(2.40)
(2.11)
(2.66)
(2.87)
2.12
4.20
6.47
8.77
10.83
9.17
10.71
11.24
Panel C: Subsample2 : 198501-201503
∗∗∗
16.01
18.38∗∗ 16.33∗∗ 15.53∗∗ 16.83∗∗∗ 19.26∗∗
18.92∗∗
16.34∗∗
(6.74)
(6.89)
(6.36)
(6.53)
(7.09)
(7.78)
(7.62)
(7.06)
(3.03)
(2.46)
(2.19)
(2.21)
(2.73)
(2.31)
(2.41)
(2.10)
15.77
18.15
16.10
15.29
16.60
19.04
18.69
16.10
Notes: This table reports forecast results of the six-month excess returns. The forecast target is the excess
returns (in percentage) at six-month horizon that is transformed from the monthly data, the independent
variable is the latent factor (F̂t−H:t ) extracted by the PLS procedure (equation (3)) from commodity returns
that are measured over horizons from last one to sixteen months (ri,t−1:t , ri,t−2:t ,..., ri,t−16:t ) and columns
2-10 show the results. Panels A-C report forecast results based on three testable samples: the entire sample
January 1950-March 2015, the first subsample January 1950-December 1984 and the second subsample January
1985-March 2015. Each panel shows in-sample test results including the coefficients (β̂) of the latent factor
(F̂t−H:t ), the t-statistic of the predictive loading in the third step regression (equation (4)) suggested by Kelly
and Pruitt (2013, 2015) (t-statKP ) and t-statistic with Newey-West correction (Newey and West (1987)), and
the R̄2 (in percentage). In addition, the last two rows in panel A show the pseudo R2 s generated from the
out-of-sample forecast, with the training sample January 1950-December 1984 and January 1985-March 2015,
respectively. The ENC-NEW test is performed with the null hypothesis that the forecast from historical mean
model encompasses the forecast from commodity returns. *, **, *** denote significance at 10%, 5% and 1%
levels.
30
Table 5: Twelve-month forecast from commodity returns
ri,t−1:t
β̂
(t-statKP )
(t-statN W )
R̄2
2
pseudo R198501
2
pseudo R199501
1.10∗∗
(2.36)
(2.05)
0.97
−3.62
−2.47
β̂
(t-statKP )
(t-statN W )
R̄2
1.22
(2.00)
(1.32)
0.97
β̂
(t-statKP )
(t-statN W )
R̄2
4.78∗
(2.57)
(1.82)
4.52
ri,t−2:t ri,t−4:t ri,t−6:t ri,t−8:t ri,t−10:t
Panel A: Entire sample: 195001-201503
2.71∗∗
6.63∗∗
9.52∗∗ 12.07∗∗ 14.17∗∗
(3.34)
(6.84)
(7.97)
(8.90)
(9.65)
(2.14)
(2.46)
(2.33)
(2.28)
(2.07)
2.58
6.51
9.41
11.95
14.05
−6.31 −13.08 −18.52 −23.53 −27.74
−3.43
−6.49 −10.97 −16.54 −21.99
Panel B: Subsample1 : 195001-198412
2.12
5.53
8.39
10.37∗
13.58∗∗
(2.18)
(3.65)
(5.02)
(6.16)
(7.62)
(1.26)
(1.18)
(1.41)
(1.65)
(2.11)
1.88
5.30
8.16
10.15
13.37
Panel C: Subsample2 : 198501-201503
∗∗
11.35
18.77∗∗ 25.25∗∗∗ 29.85∗∗∗ 31.13∗∗∗
(5.04)
(7.38)
(9.25) (10.38) (10.49)
(2.26)
(2.13)
(2.60)
(3.03)
(3.10)
11.10
18.54
25.05
29.66
30.94
ri,t−12:t
ri,t−14:t
ri,t−16:t
14.21∗
(9.50)
(1.87)
14.09
−29.10
−24.14
13.67∗
(9.05)
(1.72)
13.56
−34.57
−29.10
12.04
(8.38)
(1.49)
11.92
−34.30
−29.78
18.37∗∗
(9.07)
(2.52)
18.17
29.91∗∗
(10.48)
(2.33)
29.72
23.88∗∗∗ 25.06∗∗∗
(10.84)
(11.45)
(3.23)
(3.30)
23.69
24.87
28.51∗∗
(10.29)
(2.19)
28.31
28.10∗∗
(10.17)
(2.01)
27.90
Notes: This table reports forecast results of the twelve-month excess returns. The forecast target is the excess
returns (in percentage) at twelve-month horizon that is transformed from the monthly data, the independent
variable is the latent factor (F̂t−H:t ) extracted by the PLS procedure (equation (3)) from commodity returns
that are measured over horizons from last one to sixteen months (ri,t−1:t , ri,t−2:t ,..., ri,t−16:t ) and columns 2-10
show the results. Panels A-C report forecast results based on three testable samples: the entire sample January
1950-March 2015, the first subsample January 1950-December 1984 and the second subsample January 1985March 2015. Each panel includes the in-sample test results, the coefficients (β̂) of the latent factor (F̂t−H:t ),
the t-statistic of the predictive loading in the third step regression (equation (4)) suggested by Kelly and Pruitt
(2013, 2015) (t-statKP ) and t-statistic with Newey-West correction (Newey and West (1987)), and the R̄2 (in
percentage), and the last two rows in panel A show the pseudo R2 s generated from the out-of-sample forecast,
with the training sample January 1950-December 1984 and January 1985-March 2015, respectively. The ENCNEW test is performed with the null hypothesis that the forecast from historical mean model encompasses the
forecast from commodity returns. *, **, *** denote significance at 10%, 5% and 1% levels.
31
Table 6: Prediction results from individual commodity returns
Panel A: monthly forecast
β̂
R̄2
pseudo R2
∗∗
soybean oil
−1.28
0.78
1.47∗∗∗
(−2.20)
aluminum
−0.79
−0.02
−0.29
(−0.84)
Brazil Santos Arabicas
0.38
−0.05
−0.43
(0.85)
high grade copper
−0.55
−0.05
1.63∗∗∗
(−0.73)
Chicago yellow corn
−1.99∗∗∗ 1.20
−0.22
(−3.28)
live hog
−0.43
−0.07
−1.02
(−0.60)
zinc special high grade
0.33
−0.09
−0.63
(0.40)
tin
−1.11
0.26
0.20
(−1.51)
sugar
−0.46
0.09
0.05
(−1.16)
soybeans
−1.33∗∗
0.37
0.17
(−2.12)
soybean meal
−0.52
−0.02
−0.06
(−0.70)
West Texas intermediate oil −1.65∗∗
0.92
0.79∗∗
(−2.33)
wheat
−1.06
0.23
0.36
(−1.21)
silver
−0.83
0.16
0.24
(−1.15)
gold
−0.89
0.04
0.19
(−1.00)
gold bullion NY
−0.81
0.01
0.23
(−0.92)
predictor
Panel B: three-month forecast
β̂
R̄2
pseudo R2
∗
−4.14
2.70
2.40∗∗∗
(−1.80)
−2.12
0.11
−1.09
(−0.81)
1.44
0.23
−0.98
(1.01)
−1.58
0.08
−0.93
(−0.91)
−6.18∗∗∗ 3.72
0.69∗∗∗
(−2.89)
−1.45
0.07
0.31∗
(−0.68)
1.04
−0.01
−1.76
(0.29)
−4.09
1.45
1.34∗∗∗
(−1.58)
−1.19
0.31
0.52∗∗
(−0.84)
−3.89∗
1.16
2.10∗∗∗
(−1.74)
−1.97
0.31
0.88∗∗
(−0.73)
−5.06∗∗
2.84
3.04∗∗∗
(−2.33)
−4.80∗∗
2.10
3.24∗∗∗
(−2.00)
−2.55
0.70
1.50∗∗∗
(−1.08)
−3.21
0.52
1.07∗∗∗
(−0.99)
−2.86
0.39
0.91∗∗
(−0.90)
Notes: This table presents forecast results of monthly and three-month market returns from individual commodity returns measured over last twelve months (ri,t−12:t ). The estimation results are from the univariate
regression of market returns on each commodity returns (equation (1)). The details of the commodities
are listed in table 1. Panels A and B show results of forecasting monthly and three-month market returns,
respectively. In each panel, the in-sample results include the estimated coefficient on the predictor (β̂), the
measure-of-fit (R̄2 ) of the mode, and pseudo R2 of the out-of-sample forecast with the training sample January
1950-December 1984. The Newey-West corrected t-statistics are reported in parentheses. The ENC-NEW
test is performed to examine significance of the out-of-sample forecast performance under the null hypothesis
that the performance of the historical mean model encompasses the performance of predictive model. The
sample period is January 1950-March 2015. The three-month excess returns are calculated from monthly
data. *, **, *** denote significant at 10%, 5% and 1% levels.
32
Table 7: Prediction results from alternative predictors
predictor
dp
pe
bm
tbl
ntis
lty
inf l
ltr
svar
Panel A: monthly forecast
β̂
R̄2
pseudo R2
−0.00
0.08
−0.67
(−1.11)
0.00
0.12
−0.62
(1.26)
0.01
0.05
−0.39
(1.00)
−0.28
0.00
−0.59
(−0.42)
0.02
−0.12
−1.71
(0.17)
−0.07
0.05
−0.41
(−1.11)
−0.76
0.21
−0.55
(−0.24)
−0.02
−0.11
0.05
(−0.33)
0.04
−0.13
−0.50
(0.06)
Panel B: three-month forecast
β̂
R̄2
pseudo R2
−0.00
0.52
−2.00
(−1.00)
0.00
0.63
−1.87
(1.11)
0.02
0.46
−0.94
(1.00)
−1.22
0.77
−1.01
(−1.34)
0.09
−0.09
−5.66
(0.14)
−0.22
0.46
−1.38
(−1.41)
−2.40
0.87
−1.86
(−1.02)
0.23∗∗∗
0.51
0.71∗∗∗
(2.33)
0.72
0.01
−2.49
(0.79)
Notes: This table reports results of examining predictive power of other alternative predictors
including the dividend-price ratio (dp), earning-price ratio (ep), book-to-market ratio (bm),
treasury bill rate (tbl), net equity expansion ratio (ntis), long-term yield (lty), inflation rate
(inf l), return of the long-term rate (ltr) and stock variance (svar). The results include the insample test results which are the estimated predictive loading (β̂), the Newey-West corrected
t-statistic (in parentheses) and R̄2 (in percentage) along with out-of-sample pseudo R2 (in
percentage) for which the training sample is January 1950-December 1984. The NEW-ENC
test is performed under the null hypothesis that the performance of historical mean model
encompasses the performance of the predictive model. *, **, *** denote significance at 10%,
5% and 1% levels, respectively. The data for alternative predictors is obtained from Amit
Goyal’s personal website over the period January 1950-February 2014.
33
Table 8: Robustness check with control variables
e
independent
Panel A: dependent variable: rm,t+1
variable
entire sample subsample1 subsample2
Fcommodity,t−1
(3.16)
(2.25)
(2.10)
M omt
(−2.32)
(0.08)
(0.34)
SM Bt
(4.14)
(4.88)
(−1.43)
HM Lt
(−3.82)
(−1.98)
(−0.40)
e
rm,t−1
(0.66)
(0.49)
(−1.34)
rmines,t−1
(−0.70)
(−0.42)
(0.41)
rstone,t−1
(−1.52)
(−0.65)
(0.15)
rapprl,t−1
(0.14)
(−0.38)
(1.55)
rprint,t−1
(0.33)
(0.72)
(0.45)
rptrlm,t−1
(−1.90)
(−1.55)
(−0.66)
rlethr,t−1
(0.89)
(0.25)
(−1.34)
rmetal,t−1
(−0.71)
(−1.28)
(2.20)
rtrans,t−1
(0.50)
(0.61)
(−1.68)
rtv,t−1
(0.89)
(0.59)
(1.62)
rutils,t−1
(0.12)
(−0.06)
(3.01)
rrtail,t−1
(−1.24)
(−0.88)
(0.88)
rmoney,t−1
(0.81)
(1.75)
(0.64)
rsrv,t−1
(−0.22)
(−1.35)
(−0.34)
rreit,t−1
(−1.55)
R̄2
14.25
13.42
2.65
e
Panel B: dependent variable: rm,t+3
entire sample subsample1 subsample2
(3.04)
(2.04)
(1.93)
(−1.87)
(−0.17)
(1.36)
(2.43)
(2.84)
(−0.83)
(−3.87)
(−1.81)
(−0.42)
(1.10)
(2.34)
(−0.91)
(1.09)
(−0.57)
(0.29)
(−1.16)
(−0.16)
(−0.05)
(0.21)
(0.85)
(−0.44)
(0.39)
(−0.66)
(0.23)
(−2.06)
(−2.32)
(−1.50)
(0.02)
(0.99)
(−1.18)
(−1.33)
(−2.29)
(1.32)
(0.30)
(−0.14)
(0.25)
(0.50)
(−0.65)
(0.56)
(0.33)
(−0.54)
(1.14)
(−1.85)
(−1.74)
(0.70)
(−0.16)
(1.05)
(1.02)
(0.43)
(−0.84)
(0.79)
(−0.22)
10.00
9.22
5.19
Notes: This table shows prediction results of monthly and three-month market returns from the latent factor
(Fcommodity,t−1:t ) extracted from commodity returns measured over last twelve months controlled by the
momentum factor (M omt ), SM Bt , HM Lt , lagged values of excess returns and industry returns. The results
are estimated from the regression,
e
e
rm,t+h
= α0 + β1 Fcommodity,t−1:t + β2 M omt + β3 SM Bt + β4 HM Lt + rm,t−1
+ β5 rmines,t−1 + β6 rstone,t−1
+ β7 rapprl,t−1 + β8 rprint,t−1 + β9 rptrlm,t−1 + β10 rlethr,t−1 + β11 rmetal,t−1 + β12 rtrans,t−1
+ β13 rtv,t−1 + β14 rutils,t−1 + β15 rrtail,t−1 + β16 rmoney,t−1 + β17 rsrv,t−1 + β18 rreit,t−1
The regression is performed on three samples: the entire sample January 1950-December 2014, the first
subsample January 1950-December 1984 and the second subsample January 1985-March 2015. The data
of industry returns that are identified to be significant predictor in Hong, Torous, and Valkanov (2007) is
obtained from Fama-French 38 value-weighted industry portfolios which covers the period from January 1950
to December 2014, and these industry returns include mines, stone, apparel, print, petroleum, leather, metal,
transportation, TV, utilities, retail, money and services. The data of commercial real estate is available from
January 1972, therefore, we exclude the return when testing on the entire sample and the first subsample.
Rows 1-19 report t-statistics with Newey-West corrections of the predictive loadings and the bottom row
reports R̄2 . For the test on commodity returns (ri,t−12:t ), we report the t-statistics suggested in Kelly and
Pruitt (2013) in brackets.
34
Table 9: Forecasting monthly industry returns from commodity returns
β̂
(t-statKP )
(t-statN W )
R̄2 (%)
pseudo R2 (%)198501
pseudo R2 (%)199501
agriculture
0.87
(2.35)
(2.43)
0.75
−1.30
0.04∗
industrial metals
2.48
(3.94)
(3.62)
2.35
0.39∗∗
1.15∗∗∗
energy
2.16
(3.53)
(4.06)
2.03
−1.96
−1.18
precious metals
1.70
(2.75)
(4.36)
1.62
−4.20
−4.56
Notes: This table shows results of forecasting monthly industry returns from the compositive
predictor extracted from commodity returns measured over last twelve months (ri,t−12:t ). The
estimation results are obtained by implementing the PLS procedure. Columns 2-5 report the
results of forecasting returns on four industries, agriculture, industrial metals, energy and precious metals, respectively. The results from the in-sample test include the predictive loading
(β̂) estimated from equation (4), the t-statistic suggested by Kelly and Pruitt (2013, 2015)
and Newey-West t-statistic (Newey and West (1987)) and the measure-of-fit (R̄2 ) of the model.
The last two rows show pseudo R2 s of out-of-sample forecasts, with the training sample January 1950-December 1984 and January 1950-December 1994, respectively. *, **, *** denote
significant at 10%, 5% and 1% levels, respectively.
35
Figures
Predictive loadings on individual commodity returns
2
predictive loadings(%)
●
●
0
●
●
Forecast horizon
●
●
●
●
●
1 month
3 months
●
●
−2
●
●
●
●
●
gold bullion NY
gold
silver
wheat
WT intermediate oil
soybean meal
soybeans
sugar
tin
zinc special high grade
live hog
Chicago yellow corn
high grade copper
Brazil Santos Arabicas
aluminum
●
soybean oil
−4
commodities
Figure 1: This figure plots contribution of each commodity returns to monthly
and three-month market return predictability.
36
Appendix
We follow Kelly and Pruitt (2011, 2015) and show the matrix algebra of calculating the weight of each predictor as below
ŷ = ιȳ + (JT XJN X0 JT y(y0 JT XJN X0 JT XJN X0 JT y)−1 y0 JT XJN X0 JT y) (A.1)
Equation A.1 can be rewritten as
ŷ = ιȳ + JT XJN B
Where ŷ is the expected market return; X is the commodity returns used as
predictors with the dimension (T × N ); ι is a T -vector of ones; B is the predictive
loading on X which is also the weight of each commodity returns for forming the
linear combination.
Therefore,
B = X0 JT y(y0 JT XJN X0 JT XJN X0 JT y)−1 y0 JT XJN X0 JT y
Where JT = IT − T−1 ιT ι0T , IT is T -dimensional identity matrix.
37