Mean Reversion, Momentum and Return Predictability∗
Dashan Huang
Fuwei Jiang
Jun Tu
Singapore Management University
Guofu Zhou
Washington University in St. Louis
First draft: May 2012
Current version: December 2013
∗ This
paper was previously circulated under title “Forecasting the Market Risk Premium: The
Role of Market States”. We thank Michael Brennan, Zhi Da, Ohad Kadan, Raymond Kan, Hong Liu,
Roger Loh, Ernst Maug, Massimo Massa, Pavel Savor, Farris Shuggi, Ngoc-Khanh Tran, Joe Zhang
and seminar participants at Beijing University, London Imperial College, Shanghai Advanced Institute
of Finance, Southwestern University of Finance and Economics, Tsinghua University, University of
Houston, University of Texas–Dallas, University of Warwick, and Washington University in St. Louis
for helpful comments. We are very grateful to Ataman Ozyildirim of The Conference Board for making
the vintage data on Leading Economic Indicator available to us, and thank Amit Goyal for providing
his data on his homepage.
Correspondence: Guofu Zhou, Olin Business School, Washington University in St. Louis, St. Louis,
MO 63130. Phone: (314) 935–6384 and e-mail: [email protected] (Zhou).
Mean Reversion, Momentum and Return Predictability
Abstract
We document significant short-term time-series mean reversion in up-market and momentum
in down-market. We find that the market risk premium for one to 12 months can be negatively
predicted in up-market and positively predicted in down-market by a mean reversion indicator
that is defined as the past year cumulative return of market portfolio minus its long term mean
and standardized by its annualized volatility. This asymmetric predictability is significant
in-sample and out-of-sample, and applies to cross-sectional portfolios sorted by size, book-tomarket ratio, industry, momentum, and long- and short-term reversals. The finding of this
paper is consistent with Veronesi (1999) that investors overreact to bad news in up-market and
underreact to good news in down-market if they are uncertain about the market state.
JEL Classification: C53, C58, G12, G14, G17
Keywords: Return predictability; Mean reversion; Momentum; Market risk premium; Leading
economic indicator; 200-day moving average; Business cycle
1
Introduction
Time-series mean reversion and momentum in the stock market study whether future stock
returns can be negatively or positively predicted by their past returns. For holding periods
more than one year, Fama and French (1988) and Poterba and Summers (1988), among others,
show that future stock returns can be negatively predicted by their past returns. For horizons
less than one year, however, the evidence is subject to some controversy. For example, Jegadeesh
(1991) finds that next month stock returns can be negatively predicted by their lagged multiyear
returns. Lewellen (2002) shows the past one year returns negatively predict future monthly
returns for up to 18 months. In contrast, Conrad and Kaul (1989) show that next month
returns can be positively predicted by their past week or month returns since stocks display
positive and significant autocorrelations. Moskowitz, Ooi, and Pedersen (2012) find that the
past 12-month volatility-scaled returns positively predict the future one to 12 month volatilityscaled returns. These seemingly opposite findings raise the question of whether the stock market
follows a time-series momentum or a mean reversion pattern for a short-term horizon.
This paper examines whether both mean reversion and momentum can coexist over time and
explores conditions under which mean reversion is more pronounced than momentum, and verse
visa. Investors have long held the view that the stock market fluctuates around its long-term
mean. For example, John Bogle (2012), the legendary investor and founder of the Vanguard
Group that manages billions of retirement funds for teachers and college professors, says that
the number one rule of investing (out of his ten rules) is “Remember reversion to the mean.”
What is hot today may not be hot tomorrow. The stock market reverts to its long-term mean
over the long run. To capture this idea, we simply define a mean reversion indicator (MRI)
at any given time as the past year cumulative return minus its long term mean (the mean up
to time t) and standardized by its annualized volatility.1 The intuition is that, when we look
at the market this month, if the cumulative return since one year ago has already been 26%,
the stock market will be more likely to go down than to go up next month since the long-term
1
The past year cumulative return smoothes the noise in the monthly realized returns, which is in line
with Campbell and Shiller (1988), Welch and Goyal (2008), and Hong and Yogo (2012) where the past 12month moving sum of dividends/earnings or open interests are used to predict the market risk premium. Also,
Moskowitz, Ooi, and Pedersen (2012) document that using “look-back period” the same as the “holding period”
masks significant predictability of past returns on future returns.
1
mean is less than 13%.2
Investors also have long viewed that the stock market has up- and down-trends.3 The most
popular concept of an up-market is defined as those periods during which the stock market
index level is above its 200-day moving average. Otherwise, a down-market occurs. According
to Siegel (1994), the use of the moving averages goes back at least to the 1930s. In practice,
the 200-day moving average has been widely plotted for years in investment letters, trading
softwares, and newspapers (such as Investor Business Daily). Intuitively, the up-trend is the
period during which the stock prices rise up in general, in response to perhaps a positive
economic outlook. However, when the index breaks down the trend line captured by the 200day moving average, the economy may have fundamentally worsened and is expected to worsen
further. Following the popular practice, we define the market as an up-market if and only if
when the market index is above its 200-day moving average, and a down-market otherwise.
We follow Fama and French (1988) by using the predictive regression to explore the timeseries mean reversion and momentum behavior. When we do not distinguish the market state
(running the state-independent regression of future stock returns on MRI), MRI is weakly
positively associated with future stock returns over one to six month horizons, and negatively
associated with future stock returns over one year horizon. These results are consistent with
Moskowitz, Ooi, and Pedersen (2012) and Fama and French (1988) but not significant.
However, once when the market state, either up- or down-market, is accounted for in the
state-dependent predictive regression as Boyd, Hu, and Jagannathan (2004), MRI reveals a
mean reversion behavior in up-market and a time-series momentum behavior in down-market
in forecasting future stock returns. In particular, MRI is negatively related to future stock
returns in up-market and positively related to future stock returns in down-market for one to
12 month horizons. For example, in the up-market state, one-standard deviation increase in
MRI forecasts a decrease of 0.6% in the next month expected return and a decrease of 1.29%
in the next three month expected return. Instead, in the down-market state, one-standard
2
This is also consistent with various newspapers and investment publications that post 52-week (roughly one
year) high, low and return on individual stocks and the market. George and Hwang (2004) and Li and Yu (2012)
study the anchoring effects of the 52-week high. We focus here on market mean reversion and momentum that
may not necessarily be psychologically based.
3
Zhang (2005) shows that costly reversibility and countercyclical price of risk cause assets in place to be
riskier than growth options in bad times, suggesting that market reactions can be fundamentally different in
up- and down-markets.
2
deviation increase in MRI forecasts a 1.08% increase in the next month expected return and
a 2.30% increase in the next three month expected return. The in-sample R2 increases from
earlier 0.14% and 0.19% to 2.47% and 3.54% in the state-dependent regression for the one and
three month horizons, respectively.
To alleviate the data-mining and data-snooping concerns, we use the stringent Campbell and
2
as the statistical performance measure and the certainty
Thompson (2008) out-of-sample ROS
equivalent return (CER) gain as the economic performance measure. Over the out-of-sample
2
period of 1985:01–2012:12, the ROS
is 2.86% for the one month horizon and 4.39% for the three
2
month horizon, which are significant at the 1% level. The monthly ROS
of 2.86% is much greater
than 1.3% in Ferreira and Santa-Clara (2011) and 1.8% in Neely, Rapach, Tu, and Zhou (2012),
the best levels to date in the literature. With respect to the CER gain criteria, the strategy
using MRI prediction yields 5.37% more annualized return relative to the strategy using the
historical mean as the market risk premium estimate.
How does MRI perform when it is augmented by another predictor? We address this question
using 14 well recognized individual macroeconomic variables in Welch and Goyal (2008) and an
aggregate economic variable, leading economic indicator (LEI), published by The Conference
Board. For one, three or six month horizon, the forecasting ability and patterns of MRI still
hold, negatively in up-market and positively in down-market. The in-sample R2 improves
slightly, and the out-of-sample R-square is a little worse than the prediction with MRI alone.
Two exceptions are inflation (INFL) and LEI, which are complementary to MRI and improve the
forecasting power substantially. With monthly horizon, the in-sample R2 s of MRI with inflation
2
and LEI are 3.84% and 3.10%, and the out-of-sample ROS
s are 3.86% and 3.34%, respectively.
The prominent role of MRI in predicting the market risk premium is confirmed by its low
correlations with the 15 alternative predictors. For example, its correlations with dividendprice ratio, T-bill rate, term spread, and LEI are -0.10, 0.07, 0.03, and -0.09, respectively.
The forecasting pattern of MRI is also apparent for portfolios sorted by size, book-to-market
ratio, industry, momentum, long- and short-term reversals. While the forecasting power is
varying across assets, MRI is always forecasting portfolio returns negatively in up-market and
positively in down-market. The in- and out-of-sample R-squares, with several exceptions, are
all larger than 1%, and significant at the conventional statistical level. Internationally, our
3
MRI, defined by the U.S. market data, also negatively forecasts international stock markets in
up-market (with Japan as an exception), and positively forecasts the markets in down-market.
This result is consistent with the finding in Rapach, Strauss, and Zhou (2013) that the lagged
U.S. return is leading non-U.S. industrialized countries.
Why can MRI forecast the stock market in different patterns? One explanation is due to
investors’ asymmetric response to uncertainty on the state of the economy. Veronesi (1999)
shows that, when the market shifts between two unobservable states, investors overreact to bad
news in good times and underreact to good news in bad times. The economic intuition is as
follows. Suppose that investors believe that good times will almost surely prevail. If a bad
news arrives, their expected future asset values decrease and the risk of the economy shifting
to another state increases. Hence, they require additional premiums for bearing the additional
risk, which drives the current price to drop by more than it would be in a present value model.
On the other hand, if they believe the economic state is bad. Then, although a good news
increases their expectation of future asset values, their uncertainty about the economy state
increases too. As a result, the equilibrium price increases but not as much as it would be in a
present value model. Empirically, Ozoguz (2009) finds support for the Veronesi model.
MRI can be regarded as a proxy of news. With Veronesi’s argument, a negative MRI is a
bad news that drives investors to overreact in up-market, suggesting that the return is expected
to revert in the future, so the prediction of MRI is negative in up-market. More specifically,
since investors have abundant capital due to past rising prices and have less constraints in
borrowing, they buy aggressively when MRI is negative, which drives the price up and lifts the
future return back to its long-term mean or above it. On the other hand, a positive MRI is
a good news that drives investors to underreact in down-market, suggesting that the current
return will continue. There are two intuitive reasons. First, as many investors follow the
down-market indicator, they may not buy aggressively in a down-market or even start selling
to reduce stock exposure. Second, those investors who use leverages are likely forced to sell as
margins relative to asset values are increased. Both explanations contribute to the empirical
fact that selling generates more selling in down-markets, resulting in a positive regression slope
of the market risk premium on MRI.4
4
This time-series momentum behavior is consistent with the “financial accelerator” macroeconomic models
4
Empirically, allowing for the up- and down-market states does reveal some fundamentally
different behaviors of the stock market. For example, in terms of the S&P 500 index over
1959:03–2012:12, the annualized excess return in up-market is 7.03%, far greater than −1.58%,
the annualized excess return in down-market, which is quite interesting since the market states
are defined ex ante or out-of-sample. The standard deviations are 12.97% and 18.51%, respectively.5 This asymmetry supports our findings that the stock market follows a mean reversion
pattern in up-market since the price evolves around its mean and so the volatility is low. In
down-market, however, the stock market departs from its mean and displays a short-term momentum pattern, so the volatility is much higher. In addition, a large daily drop of 5% or more
in the stock market occurs five times more often in down-market than in up-market. These summary statistics suggest that the popular practitioner’s measures of up-market (down-market)
reflect well the true good (bad) times of the stock market. Moreover, the root mean-squared
pricing error of the well-known Fama-French (1993) three-factor model increases by 57% from
up-market to down-market.
Our up-market indicator, the 200-day moving average, is negatively correlated with economic uncertainty proxies, such as the implied variance (VIX2 ), expected realized variance, variance
risk premium (VRP), the Baker, Bloom and Davis (2013) economic policy uncertainty index,
the variance of Chicago Fed National Activity Index (CFNAI), and the variance of industrial
production growth. It is also positive related with the Pastor and Stambaugh (2003) liquidity
factor. The absolute value of these correlations ranges from 0.24 to 0.53, suggesting that when
the 200-day moving average indicates an up-market, the economic uncertainty is low, and verse
visa.
We also examine whether the predictive ability of MRI steps from cash flows or discount
rates, and find that MRI strongly forecasts the future dividend growth, dividend yield, and
stochastic discount factor (SDF), but it only forecasts government bond returns in up-market
and forecasts stock variance in an opposite pattern with the market risk premium. These
results imply that the predictability of MRI may come from both the cash flow channel and
the discount rate channel.
of Brunnermeier, Eisenbach and Sannikov (2012) and references therein that uncertainty over collateral values
reduces lending and exacerbates economic downturns.
5
Over the sample period, the overall annualized excess return and standard deviation are 4.24% and 15.01%.
5
The rest of the paper is organized as follows. Section 2 describes the data and the empirical
methodology. Section 3 provides empirical results on the market risk premium predictability
of MRI. Section 4 investigates cross-sectional and international predictability of MRI. Section
5 explores sources for the predictability of MRI. Section 6 concludes.
2
Data and Method
2.1
Key variables
The data sets span from March 1959 through December 2012 due to the availability of the
leading economic indicator (LEI). We focus on monthly market risk premium predictability
although we do report some results for three, six, and 12 month horizons. One reason is that
this paper attempts to reconcile the seemingly opposite findings in Conrad and Kaul (1989)
and Moskowitz, Ooi, and Pedersen (2012) who explore different stock return behavior with onemonth horizon. Another reason is that Ang and Bekaert (2007) and Boudoukh, Richardson,
and Whitelaw (2008) suggest short horizon predictability to avoid the overestimation issues in
long-horizon regressions.
The market risk premium is the log return on the S&P 500 index (including dividends)
minus the log return on a risk-free bill.
2.1.1
Mean reversion indicator (MRI)
MRI, as the proxy of mean reversion indicator, at any time t is defined by
MRIt =
rt−12→t − µ
,
σt−12→t
(1)
where rt−12→t is the cumulative return of the S&P 500 index over the past year (from month
t − 11 to month t), µ is the long-term mean,6 and σt−12→t is the annualized moving standard
deviation estimator (Mele, 2007), which appears more appropriate to use for investors when
they lookback at the volatility over the past year.7 Intuitively, the greater MRI, the greater
6
The long-term mean is computed with sample up to t as more samples improve mean estimate.
We have also examined some alternative volatility measures such as the realized volatility in Welch and
Goyal (2008) and find that they do not alter the qualitative conclusions.
7
6
the over-valuation of the market at time t relative to its long-term mean. Hence, a reversal in
next period is more likely than otherwise.
The use of the past 12-month cumulative return rather than the past one month is to
smooth the noise in the monthly realized returns. For example, the realized excess returns
could be negative, however, the expected excess returns should be positive. Campbell and
Shiller (1988), Welch and Goyal (2008), and Hong and Yogo (2012) use the past 12-month
moving sum of dividends/earnings and open interest rates to predict the market risk premium.
Also, Jegadeesh (1991) shows that using the past one or multiple year returns to predict the
future one month return can improve the statistical power, which is confirmed by Moskowitz,
Ooi, and Pedersen (2012).
Over our sample period, MRI is positive in 339 months out of 646 and negative in the
remaining 307 months, suggesting that the stock market does evolve around its long-term
mean. The first three-order autocorrelations are 0.91, 0.83, and 0.74, respectively.
The predictive explanation of MRI is different from the volatility-scaled lagged return in
Moskowitz, Ooi, and Pedersen (2012) in two aspects. First, our numerator is the difference
between the past one year cumulative return and its long-term mean, which captures the deviation of current stock price relative to its fundamental value. Even when the lagged return
and MRI simultaneously and positively predict future stock returns, they may have different
interpretations. The positive prediction in Moskowitz, Ooi, and Pedersen (2012) says that a
positive return in this month predicts an increase in the next month expected return (there is
an intercept in the regression). The positive prediction in MRI means that if this month cumulative return is larger than the long-term mean, the next month expected return will increase
too. However, if this month cumulative return is less than the long-term mean but even when
it is positive, MRI predicts a decrease in the next month expected return, which is in contrast
to the prediction of the lagged return in Moskowitz, Ooi, and Pedersen (2012).
Second, the predictive power of MRI is different from the lagged return in that the component of
µ
σt−12→t
in MRI also has predictive power as Guo (2006) shows that stock variance
is a predictor of future stock returns if it is augmented by the consumption-wealth ratio. The
predictive ability of the scaled lagged return in Moskowitz, Ooi, and Pedersen (2002) may be
from the lagged volatility even when the lagged return does not have any predictive ability. In
7
this case, MRI has no forecasting power at all.
2.1.2
Up- and down-market indicator
We define an up-market as time periods when the market index is above its 200-day moving
average, which is given by
Iup,t+1

 1, if P ≥ 1 ∑200 P
t
i=1 t+1−i ;
200
=
 0, otherwise,
(2)
where Pt is the daily price level of the market index.8 Statistically, one may use a different lag,
say 100 days, to re-define the moving average, or use an optimal lag to yield better performance
than what are reported in this paper. However, to mitigate the concerns of data mining and
data snooping (see., e.g., Lo and MacKinlay, 1990), we simply use the 200-day moving average
that had been used by practitioners for decades before our out-of-sample periods (see, e.g.,
Siegel 1994), and that has been widely plotted in investment letters, trading softwares, and
newspapers. Economically, since the 200-day moving average is widely followed, its effect
might be easy to understand. If enough investors believe it, they may herd on this information,
thereby generating impact on the market price (see, e.g., Froot, Schaferstein, and Stein, 1992,
and Bikhchandani, Hirshleifer and Welch, 1992), and making it interesting to study whether it
matters in predicting the market risk premium when it defines the up- and down-market states.
2.1.3
Alternative predictors
For comparison, we use 14 well recognized macroeconomic predictors from Welch and Goyal
(2008).
1. Dividend-price ratio (DP): log of a twelve-month moving sum of dividends paid on the
S&P 500 index minus the log of stock prices (S&P 500 index).
2. Dividend yield (DY): log of a twelve-month moving sum of dividends minus the log of
lagged stock prices.
8
In the monthly market risk premium forecasting, the market state of next month is determined by the last
trading day’s 200-day moving average indicator of current month.
8
3. Earnings-price ratio (EP): log of a twelve-month moving sum of earnings on the S&P 500
index minus the log of stock prices.
4. Dividend-payout ratio (DE): log of a twelve-month moving sum of dividends minus the
log of a twelve-month moving sum of earnings.
5. Book-to-market ratio (BM): book-to-market value ratio for the Dow Jones Industrial Average.
6. Treasury bill rate (TBL): three-month Treasury bill rate (secondary market).
7. Long-term yield (LTY): long-term government bond yield.
8. Long-term return (LTR): return on long-term government bonds.
9. Term spread (TMS): difference between the long-term yield on government bonds and the
Treasury bill rate.
10. Default yield spread (DFY): difference between Moody’s BAA- and AAA-rated corporate
bond yields.
11. Default return spread (DFR): long-term corporate bond return minus the long-term government bond return.
12. Stock variance (SVAR): monthly sum of squared daily returns on the S&P 500 index.
13. Net equity expansion (NTIS): ratio of a twelve-month moving sum of net equity issues by
NYSE-listed stocks to the total end-of-year market capitalization of NYSE stocks.
14. Inflation (INFL): consumer price index (all urban consumers). As in Welch and Goyal
(2008), we use inflation with one month lag to account for the delay in releases.
In addition, we consider an aggregate economic variable, leading economic indicator (LEI),
which is published by The Conference Board on a monthly basis and constructed to predict
economic turning points (peaks and troughs) over business cycles. As a composite index, LEI
consists of ten individual economic leading indicators: 1) average weekly hours (manufacturing),
2) average weekly initial claims for unemployment insurance, 3) manufacturers’ new orders
(consumer goods and materials), 4) vendor performance (slower deliveries diffusion index), 5)
manufacturers’ new orders (nondefense capital goods), 6) building permits (new private housing
units), 7) stock prices (S&P 500 Index), 8) Money supply (M2), 9) interest rate spread (10-year
Treasury bonds less Federal Funds rate), and 10) The Conference Board index of consumer
expectations. All of these indicators have an established tendency to decline before recessions
9
and rise before recoveries. As other macroeconomic indices, the data of LEI creleased today
on the past may include possible revisions and adjustments. We use the vintage data, which
are the actual released data and have no updates (detrending the data month-by-month), for
out-of-sample forecasting to obtain the only practically feasible forecasts over time (the last
vintage is used for in-sample forecasting).9
LEI captures the future state of the overall economy. The classic Merton (1971) model
provides the theoretical basis for the state of the economy or the changing investment opportunity set as the source for the time-varying market risk premium. Surprisingly, in the vast
empirical literature on predictability, no studies have ever used a simple aggregate measure of
the economy to predict the market, though various individual economic variables have been
used. Since each individual economic variable summarizes only one aspect of the economy, it
does not capture the overall state of the economy. To utilize fully the insight from Merton
(1971), an overall measure of the state of the economy is needed; LEI is designed exactly for
this purpose and widely used today from Bloomberg and other news sources.10
2.2
State-independent and dependent regression
The standard state-independent predictive regression model for forecasting the market risk
premium is
rt→t+h = α + β · MRIt + εt→t+h ,
(3)
where rt→t+h is the excess return of a market index, say the S&P 500, from month t + 1 to
month t + h and εt→t+h is the error term. We focus on h = 1 but also consider h = 3, 6 and 12
sometime to show the robustness of MRI. The out-of-sample forecast of next period’s market
risk premium is computed recursively from
r̂t→t+h = α̂t + β̂t · MRIt ,
9
(4)
We are grateful to Ataman Ozyildirim of The Conference Board for providing us with the vintage data.
In the finance literature, although LEI has been used as a measure of economic state in earlier studies by
Perez-quiros and Timmermann (2000), Lamont, Polk, and Saá-Requejo (2001), Ozoguz (2009), and Lee (2012),
it is not used for forecasting the market risk premium.
10
10
where α̂t and β̂t are the ordinary least squares (OLS) estimates of α and β, respectively, based
on data from the start of the available sample through t. The in-sample forecast is computed the
same as above except that α̂t and β̂t are replaced by those estimated by using the entire sample.
In the literature, almost all of the predictability studies prior to Welch and Goyal (2008), such
as Rozeff (1984), Fama and French (1988), and Campbell and Shiller (1988a, 1988b), are based
on in-sample results.
Welch and Goyal (2008) find that almost all the established predictors fail to forecast the
market risk premium out-of-sample. One reason for the failure is that the data-generating
process for stock returns may be subject to regime shifts while regression (3) does not permit.
Intuitively, predictors may have in general different predictive abilities in different time periods
or market states.
In the spirit of Boyd, Hu, and Jagannathan (2005), we consider an extension of the standard
state-independent predictive regression model (1) by allowing for two states: up- and downmarkets. That is, we run the following state-dependent predictive regression as
rt→t+h = α + βup · Iup,t · MRIt + βdown · (1 − Iup,t ) · MRIt + εt→t+h .
(5)
Apparently, (5) nests regression (3) as a special case when the up- and down-market reactions
are the same.11
To the best of our knowledge, Cooper, Gutierrez, and Hameed (2004) appear the first to
use the up- and down-market regression in their study of the momentum strategy and find
that the profitability depends on the state of the market.12 However, their definition of the
up- and down-market is based on lagged three-year returns that are not directly related to the
moving averages. Econometrically, our modified predictive regression model with the up- and
down-markets may capture some common regime effects of sophisticated econometric models
such as Hamilton (1989), Perez-Quiros and Timmermann (2000), Lettau and Van Nieuwerburgh
(2008), and Tu (2010). We do not use these models for two reasons. First, it is well known
11
We also confirm that the predictability of MRI is robust when the lagged two month MRIt−1 and Iup,t−1
are used to predict rr→t+h .
12
To capture the leptokurtosis of the momentum strategy, Daniel, Jagannathan, and Kim (2012) identify the
state of the stock market as “calm” or “turbulent” with a two-state hidden Markov Chain model, and find that
severe losses mainly occur in the “turbulent” state.
11
that complex models can be counter-productive in out-of-sample forecasting due to estimation
errors, which is why the simple predictive regression model is the primary model used in the
predictability literature. Second, our definition of state is easy to interpret since it is what many
investors are actually using to assess the market state. Therefore, it is of economic interest to
see how it works in practice.
Are the concepts of the up- and down-markets defined above useful? Before showing in the
next section that they have a huge impact on predictability, we illustrate here that they capture
two fundamental characteristics of stock returns. First, large daily drops of the stock market
occur much often in down-market than up-market. Table 1 reports the numbers of daily drops
for the Dow Jones Industrial Average (DJIA) and the S&P 500 index when the daily drop is
larger than 3%, 5%, and 10%, respectively, where the drop is measured by the daily arithmetic
return without dividends. The market state, up or down, on day t + 1 is determined by the
corresponding index’s 200-day moving average on date t. The data on DJIA is from May 26,
1896 to December 30, 2012.13 During this period, there are 75 days in which the returns drop
more than 5%. Of these drops, 54 occur in down-market and the remaining 21 happen in upmarket. Moreover, all the four daily drops larger than 10% occur only in down-market. During
the sample period of 1959 to 2012 (the period for our return predictability study), DJIA has 3
daily drops larger than 5% in up-market and 16 such drops in down-market. Accordingly, the
S&P 500 index has 3 drops larger than 5% in up-market and 19 such drops in down-market, five
times larger than the state of up-market. These results show that there is a difference on large
return drops over up- and down-markets, suggesting that the stock market behaves differently
over different market states.
[Insert Table 1 about here]
The second characteristic is that the pricing errors of the Fama-French three-factor model
are much greater in down-market than up-market. Table 2 presents the pricing errors on their
25 portfolios sorted by firm size and book-to-market ratio. The portfolio betas are estimated
with all data from March 1959 to December 2012, but the pricing errors, alphas, are evaluated
in each of the up- and down-market states, respectively. One may interpret that a portfolio
is underpriced if its pricing error is positive, and is overpriced if its pricing error is negative.
13
The data are downloaded from http://www.djaverages.com/.
12
With this interpretation, 12 out of 25 portfolios are overpriced in up-market (Panel A), and 9
are overpriced in down-market (Panel B). In particular, except for the smallest size and lowest
book-to-market ratio portfolio, all the remaining smallest size portfolios are underpriced in
up-market and overpriced in down-market.
[Insert Table 2 about here]
To measure the aggregate market pricing error, we compute the root mean-squared pricing
error (RMSE),
√
∑25
i=1
RMSE =
25
αi2
,
(6)
where αi is the pricing error for portfolio i. Table 2 shows that the root mean-squared error
increases from 0.14% in up-market to 0.22% in down-market, implying that the aggregate
pricing error increases by 57% around in down-market. Both Tables 1 and 2 suggest that it is
important to divide the market into up- and down-markets.
3
Empirical Results
In this section, we present the in- and out-of-sample results for predicting the market risk
premium with MRI. We also examine the performance over business cycles.
3.1
Forecasting performance
In assessing the forecasting ability of MRI, we use the in-sample R2 and the Campbell and
2
Thompson (2008) out-of-sample ROS
,
∑T
2
ROS
= 1 − ∑t=1
T
(r̂t − rt )2
t=1 (r̄t
− rt ) 2
,
(7)
where T is the number of out-of-sample observations, r̂t is the excess return forecast estimated
from regression (5), and r̄t is the historical average return, both of which are estimated using
2
data up to month t − 1. If MRIt is viable, ROS
will be positive, which implies a lower mean-
squared forecast error (hereafter MSFE) relative to the forecast based on the historical average
13
2
return. Campbell and Thompson (2008) show that a monthly ROS
of 0.5% is economically
significant.
2
If ROS
is positive, the forecast outperforms the historical average return. The null hypothesis
2
2
of interest is therefore ROS
≤ 0 against the alternative hypothesis that ROS
> 0. We test this
hypothesis by using the Clark and West (2007) MSFE-adjusted statistic. Define
ft+1 = (rt+1 − r̄t+1 )2 − [(rt+1 − r̂t+1 )2 − (r̄t+1 − r̂t+1 )2 ].
(8)
Then, the Clark and West (2007) MSFE-adjusted statistic is the t-statistic from the regression
of ft+1 on a constant.
All the forecasts below are estimated recursively with the expanding windows approach as
Welch and Goyal (2008). We use the first 26 years of data for in-sample training and the
remaining 28 years of data for the out-of-sample evaluation. That is, our out-of-sample period
starts in January 1985 and ends in December 2012. Since the vintage data of leading economic
indicator (LEI) is only available to us until December 2011, the forecasting results involved
with LEI will end in that month.14
Table 3 reports the key results of this paper, including the regression slopes, Newey-West
t-statistics with h − 1 lag correction (in the brackets), in- and out-of-sample R-squares in
forecasting the market risk premium. To explore the predictive ability of MRI over different
holding periods, we regress future one, three, six month or one year returns on current MRI.
Panel A provides the results using the state-independent predictive regression (3). In general,
MRI positively forecasts future one, three, and six month risk premiums and negatively forecasts
future one year risk premium, although all the predictions are not significant. The in-sample
2
R2 s are less than 0.2% for any horizon and the out-of-sample ROS
s are negative or almost zero,
confirming Fama (1965) that monthly returns are somewhat predictable from past returns since
returns tend to be positively autocorrelated, but the predictability is negligible and cannot
survive trading costs. The negative forecasting with one year horizon is consistent with Fama
and French (1988) who find that returns tend to be negatively autocorrelated over one year
14
The data on LEI starts from January 1959 but the first vintage is only available in December 1968 (the
first forecast is April 1959 since LEI was initially released with two month delay). So, we have effectively used
16 years of vintage data for the in-sample training.
14
horizon since stock returns consist of a mean reverting permanent component.
[Insert Table 3 about here]
Penal B shows another picture on the predictive ability of MRI. In particular, when the
market state is controlled by running the state-dependent regression (5), MRI forecasts future
risk premiums negatively in up-market and positively in down-market, and the regression slopes
are significant for the one and three month horizons, but not significant for the six month and
one year horizons, which suggests that MRI captures only short-term movements of the market
risk premium. One-standard deviation increase in MRI forecasts a 0.6% or a 1.29% decrease
in the next one or three month risk premiums if the market is in the up state. Instead, if
the market is in the down state, one-standard deviation increase in MRI forecasts a 1.08% or
a 2.30% increase in the next one or three month risk premiums. As the forecasting horizon
increases to six months or one year, the forecasting pattern still holds but the forecasting power
is diminishing. This trend is directly captured by the in-sample R2 , which increases from 2.47%
for the one month horizon to 3.54% for the three month horizon, then decreases to 2.56% for
the six month horizon and drops further to 0.30% for the one year horizon.
2
The out-of-sample ROS
shows the same forecasting pattern as in-sample. With monthly
2
horizon, the ROS
is as large as 2.86%, much larger than 1.3% in Ferreira and Santa-Clara
(2011) and 1.8% in Neely, Rapach, Tu, and Zhou (2012), the best performance so far. Re2
garding the three, six month and one year horizons, the ROS
s are 4.39%, 2.10%, and -4.72%,
respectively.15 Again the predictability of MRI concentrates on short-term horizons. As Camp2
bell and Thompson (2008) show that even an ROS
of 0.5% can yields significant economic value,
the predictability of MRI is obviously substantial and convincing.
To make the results of Panel B more apparent, Panel C shows the differences of the in- and
out-of-sample R-squares between the state-independent forecasting (3) and the state-dependent
forecasting (5). Except for the one year horizon result, the improvement is substantial, either
in-sample or out-of-sample.
The result in Table 3 is in line with Moskowitz, Ooi and Pedersen (2012) that the past one
year return can predict the future one month return although our MRI and their volatility-scaled
15
2
It should be mentioned that in general, the in-sample R2 is larger than the out-of-sample ROS
although
they capture different aspects of stock returns and do not have a theoretical connection. This relationship still
holds with respect to MRI if we focus on the same forecasting horizon.
15
return have different forecasting patterns. While Moskowitz, Ooi, and Pedersen (2012) concentrate on the existence of predictability of the past returns, we explicitly show the magnitude of
predictability and the out-of-sample performance.
Fama and French (1988) conjecture that stock returns consist of two components, a temporal
component that follows a random walk and a permanent component that is mean reverting.
With horizons more than one year, the permanent component is dominant and hence the future
returns are negatively predicted. With short-term horizon, the temporal component is more
pronounced and so the future returns are almost unpredicted. Our results in Table 3 are
complementary with Fama and French (1988) that the temporal component may follow two
different stochastic processes, mean reversion in up-market, and momentum in down-market.
3.2
Forecasting with alternative predictors
One interesting question is whether MRI is highly correlated with known macroeconomic predictors and its predictive ability still exists when it is augmented by alternatives. To answer
this question, Table 4 exhibits the correlations between the 14 predictors in Welch and Goyal
(2008), LEI and MRI. While the correlations between the 14 predictors are well documented,
we focus on LEI and MRI. First, LEI, as an aggregate measure of the whole economy, has
substantial correlations with most economic variables. For example, the highest correlation
(in absolute value) is −0.72 with LTY, suggesting that the long-term government bond yield
should be lower with better economic outlook. The next highest correlation is with DFY. The
better the outlook of the whole economy, the less the chance of default.
In comparison with LEI, MRI has a correlation no more than 0.1 (in absolute value) with
any of the existing predictors, except DE, DFY, and SVAR, which are −0.18, −0.13, and −0.24,
respectively. The low correlations between the known economic predictors and MRI imply that
they capture different fundamental sources in predicting stock returns.
[Insert Table 4 about here]
To examine the predictive ability of MRI when an alternative predictor is included, we run
16
the following regression,
rt→t+h = α + βMRI,up · Iup,t · MRIt + βMRI,down · (1 − Iup,t ) · MRIt
+βZ,up · Iup,t · Zt + βZ,down · (1 − Iup,t ) · Zt + εt→t+h ,
(9)
where Zt is one of the 14 predictors in Welch and Goyal (2008) or LEI. We do not include all
the predictors together to run a kitchen sink regression since Welch and Goyal (2008) show this
approach performs much worse than individual predictors in general. An alternative approach
is to extract the principal component from the individual predictors, which has been proved
not promising either by Neely et al. (2013).
Panel A of Table 5 reports the forecasting results with monthly horizon. As expected, the
slope of MRI is negative in up-market and positive in down-market. The forecasting magnitude
is similar with the case without augments. All the slope coefficients are significantly different
from zero. With respect to the macroeconomic variables, their predictability is very weak and
depends on the market state. For example, DP, DY and LTR can significantly predict the
market risk premium in down-market at the 10% level, while DFY, SVAR, and NTIS, and LEI
show significant predictive power in up-market. One exception is INFL whose predictability is
significant in both the up- and down-market states. The inclusion of an alternative predictor
does improve the forecasting power of MRI in-sample. All R2 s except LTY are larger than
2
2.47%, the level with MRI alone. The out-of-sample ROS
s, however, decrease. The reason is
simple. The inclusion of an alternative predictor does improve the forecasting power of MRI
in sample, however, this inclusion increases more estimation error in the regression coefficients.
Hence, the out-of-sample performance becomes worse. Two exceptions are INFL and LEI,
2
whose ROS
s are 3.86% and 3.34%, respectively.
[Insert Table 5 about here]
Panels B and C of Table 5 report the results with forecasting horizons of three and six
months. The forecasting pattern of MRI is the same as the one month horizon. The market
risk premium is negatively predicted in up-market and positively predicted in down-market.
2
The in-sample R2 increases and the ROS
decreases relative to the case without alternative
predictors, which is especially pronounced for the six month horizon. Summarizing Table 5, we
17
can conclude with high confidence that the predictive ability of MRI is different from the wellrecognized predictors and concentrates on short-term horizons, such as one to three months.
3.3
Performance over business cycle
This section investigates whether MRI can forecast the market risk premium during expansions,
as well as recessions. This is of interest since recent studies, such as Rapach, Strauss, and Zhou
(2010), Henkel, Martin, and Nardari (2011), and Dangl and Halling (2012), find that the
forecasting power of traditional macroeconomic variables is significant during recessions but
insignificant during expansions. Since the 14 predictors in Welch and Goyal (2008) have been
exhaustively explored in the literature, to save space, we focus on MRI and LEI and report
their individual and joint forecasting powers over business cycles.
2
We calculate the ROS
s during expansions and recessions over the NBER business cycle dates
separately. Since the data of LEI ends in December 2011. The out-of-sample period is over
1985:01–2011:12. Panel A of Table 6 reports the results. When the only predictor is MRI, the
2
ROS
is 2.12% in expansions, significant at the 1% level, and 5.26% in recessions, insignificant.16
2
When LEI is the unique predictor in regression (3), the ROS
s are 0.00% and 1.77% during
expansions and recessions, respectively, in which 1.77% is significant at the 1% level. These
results indicate that LEI performs relatively better in recessions while MRI works better in
2
expansions. When they are put together, the ROS
s are 2.32% and 6.89% during expansions
2
and recessions, and are significant at the 5% level and the 10% level, respectively. The ROS
s
2
from the joint prediction are almost equal to the sum of individual ROS
s, implying that MRI
and LEI are complementary with each other in the market risk premium forecasting.
[Insert Table 6 about here]
Why is the performance of MRI and LEI, as well as other recognized variables, in predicting
the market risk premium weaker during economic expansions than during recessions? One
explanation is the countercyclical pattern of the market risk premium. In expansions when
consumption, output, and investment are strong, investors are less risk-averse and require a
16
The insignificance is likely due to the small sample size, i.e., there are only 34 months during our out-ofsample period identified as economic recessions. However, this small sample size does not alter the significance
when LEI and MRI are jointly used.
18
lower premium for risk taking. In addition, they are less constrained and have ample capital
to eliminate any arbitrage opportunity. As a result, any information/news will be incorporated
quickly into the market, and the predictability is weaker. On the contrary, in recessions when
consumption, output and investment are weak, investors are more risk-averse and require a
higher risk premium (Campbell and Cochrane, 1999; Cochrane, 2011). In this case, investors
suffer from both leverage and capital constraints (He and Krishnamurthy, 2012). Hence, the
lack of sufficient arbitrage capital may limit the speed of news diffusion, resulting in stronger
prediction in recessions.
The above explanation can yield some analytical insights via a simple model. Suppose that
the market risk premium is governed by the following process
rt+1 = ζst µt + σst εt+1 ,
(10)
µt+1 = (1 − ρ)µ0 + ρµt + ut+1 ,
(11)
where st = G, B is the business cycle barometer of expansion or recession, and is independent
of εt+1 and ut+1 . When ζst and σst are constant, the assumed return process reduces to Pastor
and Stambaugh (2009) where the risk premium is time-varying and follows an AR(1) process.
When µt is constant, it reduces to the simplest regime shifting process (Ang and Timmermann,
2012). Our assumption says that the expected return is time-varying and may shift from a
high-growth state to a low-growth state at random times (Veronesi, 1999; Ozoguz, 2009).
The Sharpe ratio conditional on business cycle is
[
Shst = Est
]
ζs µ0
ζst µt
= t .
stdt (rt+1 )
σst
(12)
According to Lettau and Ludvigson (2010) and Lustig and Verdelhan (2012), the Sharpe ratio
is higher in economic recessions. This implies that
ζB
σB
>
ζG
σG
(the unconditional expected return
µ0 > 0 is self-evident). Moreover, the predictive R-square over business cycles is
Rs2t
Obviously, if
ζB
σB
>
ζG
,
σG
ζs2t Var(µt )
Var[Et (rt+1 )|st ]
.
=
= 2
Var(rt+1 |st )
ζst Var(µt ) + σs2t
(13)
then the R-square during recessions must be larger than that during
19
expansions, that is,
2
2
RB
> RG
.
(14)
This helps to understand why stock returns are more predictable in recessions than expansions.
3.4
Economic value of MRI
Following Campbell and Thompson (2008) and Ferreira and Santa-Clara (2011), among others,
we use the certainty equivalent return (CER) gain as an economic performance measure for
market risk premium forecasting.
Suppose a mean-variance investor invests his wealth between one risky asset and one riskfree bill. At the start of each month, he allocates a proportion of wt to the risky asset to
maximize his next month’ expected utility
γ
U (Rp ) = E(Rp ) − V ar(Rp ),
2
(15)
where Rp is the (simple) return on the investor’s portfolio, E(RP ) and V ar(Rp ) are the mean
and variance of the portfolio return, and γ is the investor’s coefficient of relative risk aversion.
Let rt+1 and Rf,t+1 be the excess return and risk-free rate. The investor’s portfolio return
at the end of each month is
Rp,t+1 = wt rt+1 + Rf,t+1 ,
(16)
where Rf,t+1 is known at t. With a simple calculation, the optimal portfolio is
wt =
1 r̂t+1
,
2
γ σ̂t+1
(17)
2
are the investor’s estimates on the mean and variance of the risky asset
where r̂t+1 and σ̂t+1
based on information up to time t.
20
The CER of the portfolio is
γ
CER = µ̂p − σ̂p2 ,
2
(18)
where µ̂p and σ̂p2 are the mean and variance of the investor’s portfolio over the out-of-sample
evaluation period. The CER can be interpreted as the compensation to the investor for taking
the risky asset. The difference between the CERs for the investor using the predictive regression
and the historical mean as the forecast of the equity risk premium is naturally an economic
measure of predictability significance.
Panel B of Table 6 presents the welfare benefits generated by optimally trading on each
predictor for the investor with a relative risk aversion of three. That is, we report the CER
difference between the strategy using predictability and the strategy using historical mean of the
equity premium. We annualize the CER by multiplying 1200 so that the CER difference denotes
the percentage gain for the investor to use the predictive regression forecast instead of the
historical mean forecast. In contrast to Campbell and Thompson (2008), we do not impose any
restrictions to the intercept, the slope, and the sign of risk premium on the predictive regression.
Also, in the calculation of CER, we do not impose portfolio constraints to prevent the investor
from shorting stocks or taking high leverage. We believe that these unconstrained results are
more convincing to support the existence of out-of-sample predictability (in an unreported
table, we do find that restrictions can significantly improve the out-of-sample performance),
and make our paper comparable to Welch and Goyal (2008). We assume that the investor
estimates variance using a rolling five-year window of monthly data.
When MRI is a single predictor, its CER gains in expansions, recessions, and overall are
4.63%, 10.75%, and 5.37%, respectively, which means that an investor can generate 4.63%,
10.75%, and 5.37% more annual returns with MRI prediction relative to the historical return
average. Instead, the CER gain of LEI only exists in recessions, which is 7.25%. The overall
gain is 0.95% per annum. When both MRI and LEI are jointly used to predict the market risk
premium, the CER gain is again almost equal to the sum of individual gains. In particular, an
investor with a risk aversion of three can get 6.06% more returns over the whole period.
21
4
Predictability of MRI on Other Assets
In this section, we show that MRI can also predict risk premiums on cross-sectional portfolios
sorted by size, book-to-market ratio, industry, momentum, long- and short-term reversals, and
on international portfolios. To save space, we only report the results on monthly horizon
forecasting. The results with other horizons are available upon request.
4.1
Forecasting cross-sectional portfolios
Return predictability on size portfolios has been extensively investigated in the literature (Ferson and Harvey, 1991; Ferson and Korajczyk, 1995; Kirby, 1998) and the basic characteristic is
that smaller size portfolios are more predictable than larger size portfolios.
Panel A of Table 7 presents the in- and out-of-sample forecasting of MRI on the ten size
portfolios. The regression slopes are negative in up-market and positive in down-market, and
all are significantly different from zero. The in- and out-of-sample R-squares are increasing
in general. This finding is inconsistent with Ferson and Harvey (1991) and Kirby (1998) that
small firms are more positively affected by improving economic fundamentals but more vulnerable during economic downturns (Perez-Quiros and Timmermann, 2000). The reason for
this increasing predictability is the increasing correlation between the size portfolios and the
market portfolio, which ranges from 0.55 for the smallest size portfolio to 0.99 for the largest
size portfolio. The returns we use in this paper are value-weighted, which implies that a larger
size portfolio should have a higher correlation with the market portfolio than a smaller size
portfolio. As shown in the previous section, MRI should exhibit stronger predictive ability
for portfolios with a larger weight in the market portfolio, since it can significantly forecast
the market risk premium. The smallest size portfolio predicted by MRI significantly since its
correlation with the market portfolio is still high.
[Insert Table 7 about here]
Whether value premium is predictable has drawn considerable attention in the past two
decades. Janannathan and Wang (1996), Pontiff and Schall (1999), and Chen, Petkova, and
Zhang (2008) document positive evidence, while Lewellen and Nagel (2006) find that the covariance between the value-minus-growth risk and the aggregate risk premium is small and
22
therefore value premium is unpredictable. We revisit this problem by considering ten book-tomarket portfolio risk premium forecast, and present the results in Panel B of Table 7. Without
any surprise, the ten value portfolios can be negatively predicted in up-market and positively
2
predicted in down-market. While the R2 and ROS
s do not show a monotonic pattern, they are
significant at least at the conventional level.
Studies on industry portfolio risk premium are relatively limited. Ferson and Harvey (1991)
and Ferson and Korajczyk (1995) consider this problem on a small set of economic variables
that serve as predictors. Cohen and Frazzini (2008) and Menzly and Ozbas (2010) provide
supporting evidence that some industry portfolios are predictable while others are not. Panel
C of Table 7 shows strong evidence that industry portfolios can be significantly predicted.
2
Except Telcm whose R2 is 0.90%, all the remaining nine R2 s are larger than 1%. The ROS
ranges from 0.70% for Durbl to 3.23% for the NoDur. Nine out of ten are significant at least
at the 10% level.
Cooper, Gutierrez JR., and Hameed (2004) find that the cross-sectional momentum profit is
only from up-market market. Daniel and Moskowitz (2013) document that momentum crashes
are more often in down-market. These two papers, among others, suggest the necessity to
predict the winner and loser’s future returns to avoid big losses. Panel D of Table 7 shows
positive results for the ten momentum portfolios. In- and out-of-sample, all the ten portfolios
are forecasted, but winners are more predictable than losers.
Finally, for robustness, we investigate further on ten long-term reversal and ten short-term
reversal portfolios whose data are readily available from Ken French Library. The long-term
reversal portfolios at month t are constructed based on prior returns from month t − 60 to
month t − 13 while the short-term reversal portfolios are based on the previous month’s return.
The results are reported in Panels E and F of Table 7. The overall patterns can be summarized
as 1) all the portfolios are significantly predicted in either up-market or down-market, and 2)
winner’s portfolios are more predictable than loser’s portfolios.
In summary, MRI is a powerful predictor for cross-sectional portfolios, in addition to the
market portfolio. It forecasts future stock returns negatively in up-market and positively in
down-market.
23
4.2
Forecasting international stock markets
Rapach, Strauss, and Zhou (2012) show that lagged U.S. returns significantly predict returns
in numerous non-U.S. industrialized countries in- and out-of-sample. So one natural question
is whether MRI can predict international stock markets as well. Table 8 documents the results.
Data are from Global Financial Data and are available from David Rapach’s web site. The
period is from 1980:02–2010:12. For comparison, we still start the out-of-sample forecasting
from 1985:01. In general, MRI can predict the international markets, but the forecasting power
is mainly from down-market. Except for the Japanese market where MRI positively predicts
the stock returns in both up- and down-markets. For the other markets, MRI shows negative
prediction in up-market and positive prediction in down-market. All the in-sample R2 are larger
2
than 2%, ranging from 2.04% for Australia to 5.84% for Switzerland. The out-of-sample ROS
s
are positive and significant for all countries except for Australia.
5
Potential Explanation and Forecasting Channel
In Introduction, we explain the forecasting pattern of MRI as investor’s asymmetric response
to uncertainty on the state of the economy. To justify this interpretation, we define a good
news as MRI is positive, i.e.,
IMRI,t

 1, if MRI > 0;
t
=
 0, otherwise,
and run the following regression
rt+1 = α + βup · Iup,t · (1 − IMRI,t ) · MRIt + βdown · (1 − Iup,t ) · IMRI,t · MRIt + εt+1 .
−1.68 [−2.68]
2.39 [1.45]
Apparently, βup measures the impact of bad news in up-market and βdown measures the impact
of good news in down-market. Their negative and positive signs provide direct support for
Veronesi (1999) whose model pertains to a single risky asset although Ozoguz (2009) finds
cross-sectional empirical support.
24
In addition, Veronesi (1999) implies that the uncertainty is low in up-market and high
in down-market, implying that our market state indicator, Iup,t , should be highly correlated
with economic uncertainty. Table 9 presents the correlations of Iup,t with several financial
and economic uncertainty proxies, which include the implied variance of the S&P 500 index
(proxied by the end-of-month Chicago Board of Options Exchange (CBOE) volatility index
(VIX2 /12)), the expected realized variance, the Bollerslev, Tauchen and Zhou (2009) variance
risk premium, the Pastor and Stambaugh (2003) aggregate liquidity factor, the Baker, Bloom
and Davis (2013) economic policy uncertainty index, the conditional variance of the Chicago
Fed National Activity Index (CFNAI), and the conditional variance of the industrial production
growth rate. Among which, the conditional variances of the CFNAI index and the industrial
production growth are calculated by the GARCH(1, 1) model.
As expected, Iup,t is highly correlated with these uncertainty measures, ranging from 0.24
(in absolute value) with the conditional variance of the industrial production growth to 0.53
with the implied variance. If we sort the market risk premiums into up- and down-market, the
annualized excess return is 7.03% in up-market and -1.58% in down-market. The corresponding
volatilities are 12.97% and 18.51%, respectively. This asymmetric return distribution supports
investors’ concern that whether the market state has shifted from up-market to down-market.
In up-market, if there is a bad news, i.e., MRI is negative, investors may overreact since the
future market return could drop down further if the market state has changed. If the market
state is expected to be a down-market, even there is a good news (MRI is positive), investors
may underreact since the expected excess return in down-market is negative.
5.1
Predictability of discount rate and cash flow
Stocks are priced by discounting their cash flows at a discount rate and the source of predictability is obviously from two possible channels: cash flow or discount rate. This section
investigates the source of MRI in predicting market risk premium.
According to the following decomposition as Cochrane (2011),
dt − pt ≈ E(rt+1 |It ) − E(△dt+1 |It ) + ρE(dt+1 − pt+1 |It ),
25
(19)
where ρ is the log-linearization constant and It is the information set up to time t. If a variable
can predict future returns, it must predict either the cash flow (△dt+1 ) or the discount rate
(dt+1 − pt+1 ), or the both.
Table 10 runs the regression of △dt+1 or dt+1 − pt+1 on MRI and LEI. Panel A reports
the predictability on the dividend-price ratio (proxy of discount rate). When MRI is the
single predictor, it negatively predicts the future dividend-price ratio but is only significant in
down-market. The insignificance in up-market may be due to the omitted variables. When
LEI is augmented, the predictability of MRI is significant in both the up- and down-markets.
In this case, both MRI and LEI jointly explain 27.19% variation of the dividend-price ratio.
The negative prediction of MRI on the dividend-price ratio indicates that if the market price
positively deviates from its fundamental, the discount rate of next period will decrease since
investors request a low risk premium regardless of the market state. However, if the market
price negatively deviates from its fundamental, the next period discount will increase since
investors request a high risk premium.
[Insert Table 10 about here]
Panel B of Table 10 shows that MRI positively forecasts the dividend growth no matter the
market is in an up or down state, which implies that a positive deviation of the market price
suggests good cash flows and a negative deviation suggests bad cash flows in the future. MRI
and LEI jointly explain 6.91% variation of the dividend growth rate.
5.2
Predictability of stochastic discount factor
According to Cochrane (2005), risk premium is negatively related to the covariance between
the stochastic discount factor (SDF) and the excess return.
Et (rt+1 ) = −Rf,t Covt (SDFt+1 , rt+1 ),
where SDFt+1 is a measure of the discount rate and Rf,t is the gross risk-free rate at t + 1.
Projecting rt+1 on SDFt+1 , we have
rt+1 = a + b · SDFt+1 + εt+1 .
26
If a variable can predict rt+1 , it must predict SDFt+1 or εt+1 , and SDFt+1 captures discount
rate and εt+1 measures cash flows. Suppose there exist a risk free asset and N risky assets in
the market. A default and observable SDFt+1 is
SDFt+1 =
1
+ (1N − µ/Rf,t )′ Σ(Rt+1 − µ),
Rf,t
(20)
where µ and Σ are the mean and the covariance matrix of the N risky asset return Rt+1 .
Table 11 reports the results of forecasting SDF with MRI, where SDF is constructed by
decile portfolios sorted by size, value, industry, momentum, long-term or short-term reversal.
In general, MRI predicts SDF positively in up-market and negatively in down-market. Since
SDF is contemporaneously negatively correlated with stock returns in general, the forecasting
pattern of MRI on SDF is consistent with that on stock returns. In up-market, a positive
MRI predicts an increase in the SDF and an decrease in the risk premium of next period. In
contrast, in down-market, a positive MRI predicts an decrease in the SDF and an increase in
the risk premium. All the in-sample R2 s are larger than 1% except the case of SDF formed
by long-term reversal portfolios and the out-of-sample R-squares are all positive. Overall, the
predictability of MRI may be from discount rates.
[Insert Table 11 about here]
5.3
Predictability of bond returns
Chen and Zhao (2009) and Campbell, Polk, and Vuolteenaho (2010) argue that the variation
in Treasury bond returns can be solely attributed to news about discount rates since their cash
flows are known in advance. If a variable can predict the excess returns of government bonds,
its predicting power must be from the discount rate channel.
Table 12 reports the results of predicting bond excess returns with MRI. We focus on three
horizons, one, three and six months. The general pattern is that MRI predicts positively the
government bond returns in up-market and negatively in down-market, but the predictability
is only significant in up-market. From the previous section, MRI forecasts the market risk
premium in both the up- and down-markets. Its insignificant ability in down-market suggests
that the predictability source of MRI must come from the cash flow channel in down-market.
27
[Insert Table 12 about here]
5.4
Predictability of stock variance
Rational asset pricing models imply a positive relationship between risk premium and its variance, and Guo (2006) confirms this implication and finds significant predictability of stock
variance on risk premium. Risk premium is compensated for taking risk, and has the same
moving direction with its variance, suggesting that if the predictability of MRI is from discount
rates, MRI must forecast the variance of risk premium in the same pattern.
Since stock variance is persistent, we run the following regression:
SVARt+1 = α + βup · Iup,t · MRIt + βdown · (1 − Iup,t ) · MRIt + β · SVARt + εt+1 ,
0.22 [2.11]
− 0.77 [−3.97]
0.61 [12.58]
where SVARt+1 is the realized variance of the S&P 500 index. Surprisingly, MRI forecasts the
future stock variance positively in up-market and negativly in down-market, which is in stark
contrast to its forecasting pattern on the risk premium. Hence, the predictability of MRI is
inconsistent with the risk-return based hypothesis and the source of predictability may come
from the cash flow channel.
In summary, while the forecasting source of MRI is more likely to come from the cash flow
channel, we cannot exclude the discount rate channel since MRI does predict the discount rate
to some extent.
6
Conclusion
In this paper, we extend the traditional predictive regression model to a state-dependent one
where a state variable indicates an up- or down-market. We find that stock returns for one to
12 months can be predicted negatively in up-market and positively in down-market by a mean
reversion indicator that is defined as the past year cumulative return of the market portfolio
minus its long term mean and standardized by its annualized volatility. This predictive pattern
is robust to cross-sectional portfolios sorted by size, book-to-market ratio, industry, momentum,
28
and long- and short-term reversals. Our findings of time series mean reversion and momentum
is consistent with Veronesi’s (1999) theoretical model that investors overreact to bad news in
up-market and underreact to good news in up-market if they are uncertain about the market
state.
Our study focuses on the stock market. It will be of interest to investigate the predictive
ability and pattern of MRI in other markets, such as commodity market and currency market.
Since the pricing errors of factor models and predictability vary substantially over the up- and
down-markets, our study also calls for state-dependent factor models and new asset pricing
models that quantify the risk premiums on MRI. Because stock returns are state-dependent, it
will also be interesting to examine what corporate decisions are state driven.
29
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Table 1
Large Market Daily Drops in Up- and Down-Markets.
This table reports the numbers of daily big drops in up- and down-markets. The up-market state
on day t + 1 is defined by the market index above its 200-day moving average on day t. The market
return, r, is the daily simple arithmetic return (excluding dividends).
Sample period
5/26/1896–12/31/2012
5/26/1896–12/31/1958
1/2/1959–12/31/2012
1/2/1959–12/31/2012
up
104
85
19
20
r < −3%
down overall
up
r < −5%
down overall
up
r < −10%
down overall
242
178
64
Panel A: DJIA
346
21
54
263
18
38
83
3
16
75
56
19
0
0
0
4
3
1
4
3
1
71
Panel B: S&P 500
91
3
19
22
0
1
1
34
Table 2
Pricing Errors in Up- and Down-Markets.
This table reports the pricing errors (alphas) of the Fama-French three-factor model in up- and
down-markets on the the Fama-French 25 portfolios formed on size and book-to-market ratio:
Rt − Rtf = α + β1 · (Rmt − Rtf ) + β2 · SMBt + β3 · HMLt + εt .
The betas are estimated with the entire sample (March 1959 to December 2012), and are then used
to compute the pricing errors in up- and
down-markets, respectively. The root mean-squared pricing
√∑
25
∗∗∗ , ∗∗ , and ∗ indicate that the pricing errors
2
error (RMSE) is defined as RMSE =
i=1 αi /25.
significantly differ for the up- and down-markets at the 1%, 5%, and 10% level, respectively.
Low
2
3
4
High
Panel A: Pricing errors (alphas) in up-markets
Small
2
3
4
Big
− 0.35∗∗∗
−0.27∗
−0.11∗
0.07
0.19
0.06∗∗
− 0.13∗∗
0.03
− 0.21∗∗
0.00
0.02
0.03∗
−0.08∗
−0.11∗
− 0.13∗∗
0.20∗∗
0.01∗
0.04
0.04
−0.13
0.23∗∗∗
−0.09∗
0.01
−0.10
− 0.08∗
Root mean-squared error (RMSE)
0.14
Panel B: Pricing errors (alphas) in down-markets
Small
2
3
4
Big
− 0.82∗∗∗
−0.03∗
0.07∗
0.24
0.10
−0.19∗∗
0.14∗∗
0.15
0.16∗∗
0.04
−0.10
0.21∗
0.14∗
0.12∗
0.16∗∗
Root mean-squared error (RMSE)
−0.03∗∗
0.19∗
0.07
0.10
−0.18
− 0.10∗∗∗
0.09∗
0.20
−0.14
−0.37∗
0.22
35
Table 3
Predictability of MRI on Market Risk Premium.
This table reports the results of forecasting the market risk premium with MRI over different
forecasting horizons. Panel A runs the state-independent predictive regression
rt→t+h = α + β · MRIt + εt+1 ,
and Panel B distinguishes the market state by running the state-dependent regression
rt→t+h = α + βup · Iup,t · MRIt + βdown · (1 − Iup,t ) · MRIt + εt+1 ,
where rt→t+h is the market risk premium over month t + 1 to t + h, MRIt is the mean reversion
indicator, and Iup,t is the dummy indicator of up-market, which is defined as to whether or not the
S&P 500 index is above its 200-day moving average. R2 is the in-sample R-square over the period of
2 is the Campbell and Thompson (2008) out-of-sample R-square over 1985:01–
1959:03–2012:12 and ROS
2012:12. Panel C reports the improvements of in- and out-of-sample R-squares. The values in brackets
2
are the Newey-West t-statistics with h − 1 lag correction. Statistical significance for ROS
is based on
2
the p-value for the Clark and West (2007) MSPE-adjusted statistic for testing H0 : ROS ≤ 0 against
2 > 0. ∗∗∗ , ∗∗ and ∗ indicate significance at the 1%, 5% and 10% level, respectively.
HA : ROS
2
In unreported results, when R2 and ROS
are calculated over the same horizon, R2 is in general
2
larger than ROS
although they do not have a theoretical relationship.
Horizon
1-Month
Panel A: state-independent
R2
2
ROS
βup
βdown
R2
2
ROS
△R2
2
△ROS
0.16
0.14
-0.13
-0.60
1.08
2.47
2.86∗∗∗
2.33
2.99
[-2.50]
[2.70]
-1.29
2.30
3.54
4.39∗∗∗
3.35
4.37
[-2.61]
[2.34]
-1.72
2.81
2.56
2.10∗∗
2.46
2.97
[-2.21]
[1.42]
-1.21
0.95
0.30
-4.72
0.27
0.03
[-0.69]
[0.34]
0.33
0.19
0.02
[0.64]
6-Month
0.33
0.09
-0.89
[0.32]
1-Year
Panel C: improvement
β
[0.78]
3-Month
Panel B: state-dependent
-0.22
[-0.13]
0.02
-4.73
36
Table 4
Correlations.
This table reports the correlations of predictors over 1959:03–2011:12. The first 14 variables are
from Welch and Goyal (2008), LEI is the leading economic indicator, and MRI is the mean reversion
indicator, defined as the past year cumulative return minus its long term mean and standardized by
its annualized moving standard deviation.
DP
DY
EP
DE
BM
TBL
DP
1.00
DY
0.99
1.00
EP
0.72
0.72
1.00
DE
0.25
0.25
-0.49
1.00
BM
0.90
0.89
0.81
0.00
1.00
TBL
0.60
0.60
0.69
-0.21
0.65
1.00
LTY
0.60
0.60
0.62
-0.11
0.60
0.86
LTY
LTR
TMS
DFY
DFR
SVAR
NTIS
INFL
LEI
MRI
1.00
LTR
0.01
0.03
0.03
-0.02
0.00
0.00
0.00
1.00
TMS
-0.18
-0.18
-0.32
0.23
-0.27
-0.50
-0.02
-0.09
1.00
DFY
0.37
0.38
0.13
0.29
0.39
0.28
0.48
0.13
0.19
1.00
DFR
0.02
0.04
-0.08
0.14
0.01
-0.03
0.02
-0.45
0.10
0.08
1.00
SVAR
-0.06
-0.10
-0.18
0.17
-0.09
-0.12
-0.05
0.13
0.11
0.32
-0.13
1.00
NTIS
0.08
0.08
0.08
-0.01
0.16
-0.01
-0.06
-0.09
-0.04
-0.35
0.04
-0.25
1.00
INFL
0.35
0.34
0.46
-0.20
0.46
0.48
0.40
-0.08
-0.26
0.07
-0.04
-0.11
0.08
1.00
LEI
-0.51
-0.52
-0.31
-0.21
-0.42
-0.48
-0.72
-0.10
-0.20
-0.64
-0.04
-0.05
0.11
-0.12
1.00
MRI
-0.10
-0.06
0.04
-0.18
-0.12
0.07
0.09
-0.01
0.03
-0.13
0.01
-0.24
0.00
-0.08
-0.09
37
1.00
Table 5
Predictability of MRI on Market Risk Premium with Augmented Predictors.
This table reports the results of forecasting the market risk premium with MRI over different
forecasting horizons
rt→t+h = α + βMRI,up · Iup,t · MRIt + βMRI,down · (1 − Iup,t ) · MRIt
+βZ,up · Iup,t · Zt + βZ,down · (1 − Iup,t ) · Zt + εt+1 , h = 1, 3, 6
where rt→t+h is the market risk premium over month t + 1 to t + h, MRIt is the mean reversion
indicator, Zt is one of the 14 predictors in Welch and Goyal (2008) or the leading economic indicator
(LEI), and Iup,t is the dummy indicator of up-market, which is defined as to whether or not the S&P
2 is the Campbell
500 index is above its 200-day moving average. R2 is the in-sample R-square and ROS
and Thompson (2008) out-of-sample R-square. Panel C reports the improvements of in- and out-ofsample R-squares. The values in brackets are the Newey-West t-statistics with h − 1 lag correction.
2 is based on the p-value for the Clark and West (2007) MSPE-adjusted
Statistical significance for ROS
2
2 > 0. ∗∗∗ , ∗∗ and ∗ indicate significance at the 1%,
statistic for testing H0 : ROS ≤ 0 against HA : ROS
5% and 10% level, respectively.
2
In unreported results, when R2 and ROS
are calculated over the same horizon, R2 is in general
2
larger than ROS
although they do not have a theoretical relationship.
Z
βMRI,up
βMRI,down
βZ,up
βZ,down
R2
2
ROS
Panel A: Predictability with one-month horizon (h = 1)
DP
-0.58 [-2.35]
1.16 [2.90]
0.08 [0.37]
0.58 [1.69]
3.13
1.61∗∗
DY
-0.58 [-2.37]
1.14 [2.85]
0.07 [0.36]
0.61 [1.79]
3.20
1.71∗∗
EP
-0.58 [-2.37]
1.07 [2.66]
0.01 [0.04]
0.15 [0.44]
2.50
1.82∗∗
DE
-0.59 [-2.41]
1.21 [2.93]
0.10 [0.54]
0.41 [1.02]
2.83
1.46∗∗
BM
-0.58 [-2.37]
1.12 [2.79]
-0.02 [-0.09]
0.29 [0.87]
2.63
2.05∗∗
TBL
-0.55 [-2.26]
1.08 [2.66]
-0.24 [-1.17]
-0.08 [-0.25]
2.62
2.14∗∗
LTY
-0.59 [-2.44]
1.09 [2.70]
-0.03 [-0.15]
-0.03 [-0.07]
2.44
2.36∗∗
LTR
-0.59 [-2.41]
1.10 [2.78]
0.30 [1.35]
0.56 [1.76]
3.45
1.91∗∗
TMS
-0.59 [-2.41]
1.08 [2.68]
0.20 [1.04]
0.04 [0.11]
2.57
1.60∗∗
DFY
-0.59 [-2.43]
1.13 [2.84]
0.48 [2.32]
0.04 [0.11]
3.08
2.14∗∗
DFR
-0.56 [-2.33]
1.06 [2.68]
0.26 [1.07]
0.14 [0.36]
2.64
1.61∗∗
SVAR
-0.57 [-2.29]
1.07 [2.71]
0.58 [1.67]
-0.21 [-0.58]
2.88
1.09∗∗
NTIS
-0.61 [-2.50]
1.07 [2.73]
-0.41 [-2.20]
0.22 [0.63]
3.02
1.99∗∗
INFL
-0.59 [-2.42]
1.11 [2.84]
-0.42 [-1.96]
0.60 [1.92]
3.84
3.86∗∗∗
LEI
-0.66 [-2.68]
1.10 [2.72]
-0.38 [-2.32]
-0.92 [-0.31]
3.10
3.34∗∗∗
38
Table 5 (continued)
Z
βMRI,up
βMRI,down
βZ,up
βZ,down
R2
2
ROS
Panel B: Predictability with three-month horizon (h = 3)
DP
-1.23 [-2.42]
2.52 [2.57]
0.33 [0.77]
1.58 [2.00]
5.19
2.66∗∗
DY
-1.24 [-2.45]
2.45 [2.52]
0.34 [0.79]
1.52 [1.91]
5.10
2.87∗∗
EP
-1.28 [-2.50]
2.31 [2.39]
0.24 [0.47]
0.16 [0.17]
3.58
3.04∗∗
DE
-1.30 [-2.57]
2.79 [2.90]
0.17 [0.41]
1.57 [2.11]
5.23
3.45∗∗∗
BM
-1.24 [-2.46]
2.41 [2.40]
0.07 [0.14]
0.74 [0.90]
3.90
3.10∗∗
TBL
-1.23 [-2.42]
2.32 [2.37]
-0.35 [-0.78]
-0.37 [-0.45]
3.73
3.34∗∗∗
LTY
-1.31 [-2.65]
2.34 [2.40]
0.18 [0.37]
-0.12 [-0.16]
3.55
3.63∗∗∗
LTR
-1.28 [-2.58]
2.33 [2.36]
0.82 [1.90]
0.06 [0.10]
4.11
3.88∗∗∗
TMS
-1.29 [-2.52]
2.28 [2.28]
0.48 [1.08]
0.81 [0.91]
4.17
2.93∗∗∗
DFY
-1.27 [-2.52]
2.47 [2.62]
1.16 [2.41]
0.37 [0.36]
4.79
3.07∗∗∗
DFR
-1.24 [-2.42]
2.25 [2.39]
0.32 [0.80]
0.53 [0.81]
3.84
3.53∗∗∗
SVAR
-1.18 [-2.32]
2.82 [2.94]
1.92 [2.56]
0.43 [0.59]
4.76
1.45∗∗∗
NTIS
-1.31 [-2.61]
2.25 [2.44]
-0.94 [-2.16]
0.74 [0.72]
4.71
2.30∗∗∗
INFL
-1.30 [-2.60]
2.30 [2.34]
-0.83 [-2.31]
0.22 [0.30]
4.13
4.60∗∗∗
LEI
-1.47 [-2.64]
2.34 [2.41]
-1.09 [-2.56]
-1.11 [-1.07]
5.44
5.78∗∗∗
Panel C: Predictability with six-month horizon (h = 6)
DP
-1.56 [-1.63]
3.27 [1.70]
0.35 [0.43]
3.77 [2.55]
6.69
0.01
DY
-1.59 [-1.97]
3.10 [1.63]
0.40 [0.49]
3.69 [2.51]
6.54
-0.04
EP
-1.66 [-2.06]
2.77 [1.41]
0.28 [0.31]
0.56 [0.29]
2.62
0.09∗
DE
-1.75 [-2.20]
3.84 [1.91]
0.16 [0.22]
3.46 [3.42]
6.28
3.02∗∗∗
BM
-1.59 [-2.00]
3.07 [1.55]
-0.33 [-0.37]
2.28 [1.36]
4.24
0.29
TBL
-1.62 [-2.06]
2.83 [1.43]
-0.44 [-0.50]
-0.61 [-0.34]
2.68
0.79∗
LTY
-1.74 [-2.22]
2.83 [1.44]
0.42 [0.47]
0.13 [0.08]
2.56
1.46∗∗
LTR
-1.68 [-2.19]
2.87 [1.44]
1.65 [2.87]
1.39 [1.68]
4.29
2.12∗∗∗
TMS
-1.70 [-2.14]
2.74 [1.37]
0.90 [1.16]
1.52 [0.88]
3.56
1.04∗∗∗
DFY
-1.66 [-2.05]
3.40 [1.74]
1.28 [1.30]
2.35 [1.50]
5.02
1.57∗∗∗
DFR
-1.68 [-2.10]
2.71 [1.41]
-0.10 [-0.20]
0.99 [1.07]
2.91
1.13∗
SVAR
-1.58 [-1.99]
4.07 [1.99]
2.06 [1.39]
1.71 [1.60]
4.50
-1.02
NTIS
-1.73 [-2.18]
2.67 [1.43]
-1.69 [-2.12]
1.47 [0.64]
4.42
-4.31
INFL
-1.76 [-2.25]
2.68 [1.40]
-1.08 [-1.62]
-1.55 [-1.18]
3.84
2.74∗∗
LEI
-1.96 [-2.48]
2.80 [1.44]
-1.64 [-2.38]
-3.27 [-1.71]
6.56
4.02∗∗∗
39
Table 6
Predictability on Market Risk Premium over Business Cycles with Monthly Horizon.
2 s in predicting market
Panel A reports the Campbell and Thompson (2008) out-of-sample ROS
2
risk premium with LEI, MRI, or both. The ROS s are calculated based on the NBER-dated periods
of expansions and recessions, respectively. The results in each column follows the state-dependent
regression (9), where rt+1 is the log return (including dividends) on the S&P 500 index minus the
log return on a risk-free bill. LEI is the leading economic indicator, and MRI is the mean reversion
indicator (the past year cumulative return minus its long term mean and standardized by its annualized
2 is based on the p-value for the Clark and
moving standard deviation). Statistical significance for ROS
2
2
West (2007) MSPE-adjusted statistic for testing H0 : ROS
≤ 0 against HA : ROS
> 0. ∗∗∗ , ∗∗ , and ∗
denote significance at the 1%, 5%, and 10% level, respectively.
Panel B reports the certainty equivalent return gains over NBER business cycles from forecasting the
market risk premium instead of using the historical mean. All numbers are annualized by multiplying
the monthly numbers by 1200 and so are in percentage. The representative investor is assumed to have
a mean-variance preference with a relative risk aversion coefficient of three. The results are calculated
over the period 1985:01–2011:12.
MRI
LEI
MRI+LEI
2
Panel A: ROS
Expansion
2.12∗∗∗
0.00
2.32∗∗
Recession
5.26
1.77∗∗∗
6.89∗
Overall
2.82∗∗∗
0.39
3.34∗∗∗
Panel B: CER gain
Expansion
4.63
0.18
4.32
Recession
10.75
7.25
20.12
Overall
5.37
0.95
6.06
40
Table 7
Predictability of MRI on Cross-Sectional Portfolios with Monthly Horizon.
This table reports the results of forecasting cross-sectional portfolio risk premiums with the statedependent predictive regression
i
i
i
rt+1
= αi + βMRI,up
· Iup,t · MRIt + βMRI,down
· (1 − Iup,t ) · MRIt + εt+1 ,
i
where rt+1
is the excess return of the i-th size, book-to-market ratio, industry, momentum, longand short-term reversal portfolio in month t + 1, MRI is the mean reversion indicator, and Iup,t is the
dummy indicator of up-market, defined by the 200-day moving average indicator of the S&P 500 index.
2
is the Campbell and Thompson (2008)
R2 is the in-sample R-square over 1959:03–2012:12 and ROS
2
out-of-sample ROS s over 1985:01–2012:12. The values in brackets are the heteroskedasticity-consistent
2
t-statistics of the regression coefficients. Statistical significance for ROS
is based on the p-value for
2
the Clark and West (2007) out-of-sample MSPE-adjusted statistic for testing H0 : ROS
≤ 0 against
2
∗∗∗
∗∗
∗
HA : ROS > 0.
, , and denote significance at the 1%, 5%, and 10% level, respectively.
2
In unreported results, when R2 and ROS
are calculated over the same horizon, R2 is in general
2
larger than ROS
although they do not have a theoretical relationship.
Portfolio
βMRI,up
βMRI,down
R2
2
ROS
Panel A: Size portfolios
Small
-0.70 [-2.02]
1.06 [2.26]
1.23
0.82∗
2
-0.69 [-2.02]
1.00 [2.21]
1.13
0.87∗
3
-0.89 [-2.83]
1.02 [2.28]
1.53
1.60∗∗
4
-0.72 [-2.32]
0.93 [2.20]
1.24
1.00∗
5
-0.78 [-2.66]
1.00 [2.31]
1.54
1.53∗∗
6
-0.69 [-2.44]
0.83 [2.10]
1.27
1.26∗∗
7
-0.73 [-2.75]
1.07 [2.48]
1.87
2.09∗∗
8
-0.91 [-3.57]
1.08 [2.67]
2.39
2.71∗∗∗
9
-0.79 [-3.45]
1.06 [2.56]
2.45
3.06∗∗∗
Big
-0.53 [-2.49]
1.09 [3.28]
2.47
2.82∗∗∗
Panel B: Book-to-market portfolios
Low
-0.63 [-2.50]
1.14 [2.92]
1.93
1.96∗∗
2
-0.67 [-2.89]
1.01 [2.94]
1.99
2.21∗∗
3
-0.81 [-3.20]
0.97 [3.05]
2.29
2.66∗∗∗
4
-0.69 [-2.70]
1.17 [2.82]
2.49
3.06∗∗∗
5
-0.60 [-2.56]
0.89 [2.53]
1.75
2.26∗∗
6
-0.63 [-2.70]
0.91 [2.11]
1.84
2.23∗∗
7
-0.57 [-2.55]
1.00 [2.57]
2.05
2.81∗∗∗
8
-0.64 [-2.65]
1.10 [2.29]
2.36
3.06∗∗
9
-0.74 [-2.87]
1.07 [2.74]
2.23
2.79∗∗
High
-0.57 [-1.73]
1.18 [2.27]
1.56
1.79∗
41
Table 7 (continued)
Industry
βMRI,up
βMRI,down
R2
2
ROS
Panel C: Industry portfolios
NoDur
-0.73 [-3.00]
0.93 [2.50]
2.27
3.23∗∗∗
Durbl
-0.32 [-0.91]
1.20 [2.15]
1.27
0.70
Manuf
-0.69 [-2.51]
1.10 [2.59]
2.03
2.23∗∗
Enrgy
-0.68 [-2.53]
1.25 [3.32]
2.17
2.35∗∗∗
HiTec
-0.76 [-2.55]
1.12 [2.47]
1.28
0.82∗
Telcm
-0.30 [-1.31]
0.74 [2.00]
0.91
0.78∗
Shops
-0.60 [-2.09]
0.94 [2.61]
1.40
1.37∗∗
Hlth
-0.63 [-2.29]
0.84 [2.44]
1.36
1.57∗∗
Utils
-0.49 [-2.71]
0.89 [2.56]
1.99
3.00∗∗∗
Other
-0.78 [-2.97]
1.18 [2.34]
2.13
2.51∗∗
Panel D: Momentum portfolios
Low
-0.81 [-2.10]
1.16 [1.75]
0.98
0.42
2
-0.70 [-2.37]
1.12 [2.17]
1.39
1.23
3
-0.50 [-2.08]
0.82 [1.97]
1.01
0.82
4
-0.52 [-2.11]
0.77 [2.16]
1.14
1.21∗
5
-0.66 [-2.83]
0.93 [2.72]
1.98
2.13∗∗
6
-0.63 [-2.70]
0.88 [2.27]
1.73
2.10∗∗
7
-0.49 [-2.09]
0.77 [2.35]
1.28
1.20∗∗
8
-0.62 [-2.49]
1.08 [3.20]
2.34
3.21∗∗∗
9
-0.88 [-3.33]
1.10 [2.65]
2.57
2.96∗∗∗
High
-1.02 [-3.32]
1.41 [3.13]
2.38
2.35∗∗∗
42
Table 7 (continued)
Portfolio
βMRI,up
βMRI,down
R2
2
ROS
Panel E: Long-term reversal portfolios
Low
-1.01 [-2.63]
0.86 [1.95]
1.34
1.17∗∗
2
-0.75 [-2.48]
0.72 [1.75]
1.26
1.25∗∗
3
-0.75 [-2.71]
0.69 [1.71]
1.42
1.48∗∗
4
-0.59 [-2.39]
0.93 [2.39]
1.80
1.65∗∗
5
-0.57 [-2.61]
0.92 [2.61]
1.78
1.97∗∗
6
-0.67 [-3.08]
0.94 [2.47]
2.17
3.00∗∗∗
7
-0.54 [-2.57]
0.92 [2.84]
1.81
2.26∗∗
8
-0.57 [-2.53]
0.99 [3.07]
2.01
2.64∗∗
9
-0.67 [-2.54]
1.23 [3.18]
2.55
3.09∗∗∗
High
-0.73 [-2.58]
1.48 [3.05]
2.29
2.89∗∗∗
Panel F: Short-term reversal portfolios
Low
-1.01 [-3.02]
1.06 [1.49]
1.24
0.81∗
2
-0.73 [-2.69]
0.81 [1.49]
1.10
0.80∗
3
-0.70 [-2.63]
1.16 [2.29]
2.11
2.37∗∗
4
-0.71 [-2.92]
1.15 [3.02]
2.47
2.63∗∗∗
5
-0.53 [-2.36]
0.98 [2.79]
1.83
2.35∗∗
6
-0.53 [-2.44]
0.96 [2.59]
1.91
2.36∗∗∗
7
-0.51 [-2.35]
0.94 [2.91]
1.87
1.77∗∗
8
-0.49 [-2.38]
1.23 [3.50]
2.70
3.14∗∗∗
9
-0.62 [-2.55]
1.12 [3.06]
2.16
2.61∗∗∗
High
-0.82 [-2.92]
1.34 [3.23]
2.42
3.10∗∗∗
43
Table 8
Predictability of MRI on International Market Equity Premiums.
This table reports the results of forecasting international market risk premiums with the statedependent predictive regression
i
i
i
rt+1
= αi + βMRI,up
· Iup,t · MRIt + βMRI,down
· (1 − Iup,t ) · MRIt + εt+1 ,
i
where rt+1
is the excess return of country i in month t + 1, MRI is the mean reversion indicator,
and Iup,t is the dummy indicator of up-market, defined by the 200-day moving average indicator of
2
the S&P 500 index. R2 is the in-sample R-square over 1980:02–2010:12 and ROS
is the Campbell
2
and Thompson (2008) out-of-sample ROS s over 1985:01–2010:12. The values in brackets are the
2
heteroskedasticity-consistent t-statistics of the regression coefficients. Statistical significance for ROS
is based on the p-value for the Clark and West (2007) out-of-sample MSPE-adjusted statistic for
2
2
testing H0 : ROS
≤ 0 against HA : ROS
> 0. ∗∗∗ , ∗∗ , and ∗ denote significance at the 1%, 5%, and
10% level, respectively.
2
In unreported results, when R2 and ROS
are calculated over the same horizon, R2 is in general
2
larger than ROS although they do not have a theoretical relationship.
βMRI,up
βMRI,down
R2
2
ROS
Australia
-0.39 [-0.86]
1.24 [2.56]
2.04
-0.14
Canada
-0.57 [-1.85]
1.18 [2.30]
2.39
1.38∗∗
France
-0.41 [-1.02]
1.81 [2.91]
3.32
2.34∗∗
Germany
-0.19 [-0.47]
1.96 [2.81]
3.94
2.42∗∗
Italy
-0.19 [-0.39]
2.24 [3.63]
3.46
2.47∗∗∗
Japan
0.42 [1.17]
1.22 [2.19]
2.35
0.51∗
Netherlands
-0.31 [-0.83]
2.03 [3.09]
4.73
3.87∗∗
Sweden
-0.27 [-0.61]
1.93 [2.66]
2.73
0.76∗
Switzerland
-0.62 [-1.83]
1.91 [4.28]
5.84
4.89∗∗∗
United Kingdom
-0.77 [-2.25]
1.36 [2.74]
3.48
2.30∗∗
United States
-0.65 [-2.19]
1.27 [2.34]
3.14
2.20∗∗
Asset
44
Table 9
Correlation of MRI with Economic Uncertainty Meaures.
This table reports the correlations of the mean reversion index (MRI) with known economic uncertainty measures. IV is the implied variance of the S&P 500 index (proxied by the end-of-month
Chicago Board of Options Exchange (CBOE) volatility index on a monthly basis (VIX2 /12)), RV is the
realized variance, VRP is the Bollerslev, Tauchen and Zhou (2009) variance risk premium, Liquidity
is the Pastor and Stambaugh (2003) aggregate liquidity factor, Uncertainty is the Baker, Bloom and
Davis (2013) economic policy uncertainty index, CFNAI is the Chicago Fed National Activity Index,
and IP is the growth rate of industrial production, which the conditional variances of CFNAI and IP
are calculated with the GARCH(1,1) model.
IVt
Et (RVt+1 )
VRPt
Liquidityt
Uncertaintyt
vart (CFNAIt+1 )
vart (IPt+1 )
IVt
1.00
Et (RVt+1 )
0.79
1.00
VRPt
0.71
0.14
1.00
Liquidityt
-0.33
-0.22
-0.28
1.00
Uncertaintyt
0.44
0.36
0.31
-0.08
1.00
vart (CFNAIt+1 )
0.62
0.59
0.33
-0.20
0.31
1.00
vart (IPt+1 )
0.66
0.45
0.55
-0.22
0.26
0.74
1.00
Iup,t
-0.53
-0.44
-0.35
0.31
-0.31
-0.32
-0.24
45
Iup,t
1.00
Table 10
Predictability of Dividend-Price Ratio and Dividend Growth.
This table reports the results of forecasting dividend-price ratio and dividend growth of S&P 500
with MRI and LEI
yt+1 = α + βMRI,up · Iup,t · MRIt + βMRI,down · (1 − Iup,t ) · MRIt
+βLEI,up · Iup,t · LEIt + βLEI,down · (1 − Iup,t ) · LEIt + εt+1 ,
where yt+1 is either the dividend-price ratio or the dividend growth rate over month t + 1, MRIt is
the mean reversion indicator, and Iup,t is the dummy indicator of up-market, which is defined as to
whether or not the S&P 500 index is above its 200-day moving average. R2 is the in-sample R-square
2
over 1959:03–2011:12 and ROS
is the Campbell and Thompson (2008) out-of-sample R-square over
1985:01–2011:12. The values in brackets are the heteroskedasticity-consistent t-statistics. Statistical
2
significance for ROS
is based on the p-value for the Clark and West (2007) MSPE-adjusted statistic
2 ≤ 0 against H : R2 > 0. ∗∗∗ , ∗∗ and ∗ indicate significance at the 1%, 5% and
for testing H0 : ROS
A
OS
10% level, respectively.
2
In unreported results, when R2 and ROS
are calculated over the same horizon, R2 is in general
2
larger than ROS although they do not have a theoretical relationship.
The rationale of this table is that if MRI can predict stock returns, it must predict either the
dividend growth or the dividend-price ratio, or both.
dt − pt = E(rt+1 |It ) − E(△dt+1 |It ) + ρE(dt+1 − pt+1 |It ).
βMRI,up
βMRI,down
βLEI,up
βLEI,down
R2
2
ROS
1.57
0.23
Panel A: Predictability of dividend-price ratio
MRI
-0.01 [-0.30]
-0.08 [-2.90]
LEI
MRI+LEI
-0.05 [-3.03]
-0.07 [-2.83]
-0.21 [17.8]
-0.16 [-7.49]
24.84
9.04∗∗∗
-0.22 [17.6]
-0.16 [-9.11]
27.19
8.54∗∗
3.01
3.29∗∗∗
Panel B: Predictability of dividend growth
MRI
0.94 [2.16]
1.39 [2.56]
LEI
MRI+LEI
1.24 [2.85]
1.33 [2.47]
1.39 [6.99]
0.72 [2.28]
3.26
4.92∗∗∗
1.55 [7.26]
0.76 [2.21]
6.91
8.83∗∗∗
46
Table 11
Predictability of MRI on SDF.
This table reports the results of forecasting the stochastic discount factor (SDF) with MRI:
SDFt+1 = α + βup · Iup,t · MRIt + βdown · (1 − Iup,t ) · MRIt + εt+1 ,
where SDFt+1 is the model-free and minimum variance stochastic discount factor formed by 10 size,
value, industry, momentum, long- or short-term reversal portfolios. Iup,t is the dummy indicator of upmarket, which is defined as to whether or not the S&P 500 index is above its 200-day moving average.
2
R2 is the in-sample R-square over 1959:03–2012:12 and ROS
is the Campbell and Thompson (2008)
2
out-of-sample ROS s over 1985:01–2012:12. The values in brackets are the heteroskedasticity-consistent
2
t-statistics of the regression coefficients. Statistical significance for ROS
is based on the p-value for
2
the Clark and West (2007) out-of-sample MSPE-adjusted statistic for testing H0 : ROS
≤ 0 against
2
∗∗∗
∗∗
∗
HA : ROS > 0.
, , and denote significance at the 1%, 5%, and 10% level, respectively.
βup
βdown
R2
SDF formed by 10 size portfolios
3.40 [3.07]
-4.92 [-2.34]
2.27
2.92∗∗∗
SDF formed by 10 value portfolios
3.47 [2.52]
-4.56 [-1.98]
1.59
2.21∗∗
SDF formed by 10 industry portfolios
3.71 [2.16]
-6.25 [-2.50]
1.83
2.53∗∗∗
SDF formed by 10 momentum portfolios
3.17 [1.55]
-7.29 [-2.14]
1.47
1.02∗
SDF formed by 10 long-term reversal portfolios
3.10 [2.09]
-2.61 [-1.11]
0.74
0.77∗
SDF formed by 10 short-term reversal portfolios
0.92 [0.60]
-5.74 [-2.23]
1.35
0.72
SDF
47
2
ROS
Table 12
Predictability of MRI on Bond Returns.
This table reports the results of forecasting the government bond excess returns with MRI:
(n)
rt→t+h = α + βup · Iup,t · MRIt + βdown · (1 − Iup,t ) · MRIt + εt+h ,
(n)
where rt→t+h is the log bond return on the n-year Treasury bond over month t + 1 to t + h, rxt→t+h
is the average of excess returns across maturity, and Iup,t is the dummy indicator of up-market, which
is defined as to whether or not the S&P 500 index is above its 200-day moving average. R2 is the in2 is the Campbell and Thompson (2008) out-of-sample
sample R-square over 1964:01–2012:12 and ROS
2 s over 1985:01–2012:12. The values in brackets are the heteroskedasticity-consistent t-statistics of
ROS
2
the regression coefficients. Statistical significance for ROS
is based on the p-value for the Clark and
2 ≤ 0 against H : R2 > 0.
West (2007) out-of-sample MSPE-adjusted statistic for testing H0 : ROS
A
OS
∗∗∗ , ∗∗ , and ∗ denote significance at the 1%, 5%, and 10% level, respectively.
Asset
βup
R2
βdown
2
ROS
Panel A: Predictability with one-month horizon (h = 1)
(2)
rt→t+1
(3)
rt→t+1
(4)
rt→t+1
(5)
rt→t+1
0.56 [3.05]
-0.03 [-0.11]
1.45
-0.22
0.56 [3.06]
-0.13 [-0.62]
1.43
0.10
0.51 [2.71]
-0.06 [-0.30]
1.18
-0.19
0.44 [2.20]
-0.05 [-0.22]
0.84
-1.11
rt→t+1
0.52 [2.80]
-0.07 [-0.32]
1.26
-0.31
Panel B: Predictability with three-month horizon (h = 3)
(2)
rt→t+3
(3)
rt→t+3
(4)
rt→t+3
(5)
rt→t+3
1.93 [2.31]
-0.04 [0.04]
1.97
-0.39
1.97 [2.47]
-0.42 [-0.44]
2.17
-0.01
2.04 [2.55]
-0.40 [-0.42]
2.40
-0.55
1.86 [2.34]
-0.44 [-0.50]
2.13
-1.86
rt→t+3
1.95 [2.44]
-0.32 [-0.34]
2.21
-0.62
Panel C: Predictability with six-month horizon (h = 6)
(2)
rt→t+6
(3)
rt→t+6
(4)
rt→t+6
(5)
rt→t+6
4.05 [2.03]
0.26 [0.10]
2.32
-0.28
4.11 [2.17]
-0.52 [-0.22]
2.50
-0.07
4.32 [2.25]
-0.58 [-0.25]
2.92
-0.64
3.99 [2.12]
-0.72 [-0.34]
2.74
-1.87
rt→t+6
4.12 [2.16]
-0.39 [-0.17]
2.65
-0.62
48