On the economic significance of stock return predictability: Evidence from macroeconomic state variables

Transcription

1 On the economic significance of stock return predictability: Evidence from macroeconomic state variables Huacheng Zhang * University of Arizona This draft: 8/31/2012 First draft: 2/28/2012 Abstract We explore the question of whether macroeconomic states are able to predict cross-sectional stock returns from the perspective of asset allocation. We find that conditioning on macroeconomic state variables leads to optimal portfolios with a certainty equivalent return that is 58 basis points per month higher than unconditional optimal portfolios out-of-sample. The conditional portfolio performance during economic recession periods is comparable to that in expansion periods. Unlike unconditional investors, conditional investors underweight value and winner stocks during economic recession periods. Our results reveal that macroeconomic state variables interacted with stock characteristics is able to predict not only cross-sectional stock returns but also the time series variations in the returns. Key words: stock return predictability, economic significance, business cycle, asset allocation * I thank Scott Cederburg and Christopher Lamoureux (my dissertation chair) for numerous discussions and insightful suggestions. I am also grateful for helpful comments by Keisuke Hirano, Richard Sias and workshop participants at the University of Arizona. Corresponding address: Finance department, Eller College of Management, University of Arizona, Tucson, Arizona zhangh@ .arizona.edu.

2 I. Introduction In this study we explore the question of whether macroeconomic states are able to predict crosssectional stock returns. Current studies show that cross-sectional stock returns are related to macroeconomic states. For example, Chen, Petkova and Zhang (2008) show that the premiums for value stock portfolios are time-varying; Chordia and Shivakumar (2002) and Cooper, Gutierrez and Hameed (2004) show that momentum premiums are related to macroeconomic states. Kandel and Stambaugh (1996) and Avramov and Chordia (2006) advocate quantifying the economic, rather than statistical, inference of stock return predictability. The economic significance analysis is appealing because it can avoid the statistical issues in direct predictability tests and is useful for decision makers. 1 We adopt the same logic to examine whether conditioning on macroeconomic state variables is able to improve crosssectional asset allocation decisions. We use the time-varying parametric portfolio selection algorithm proposed by Brandt, Santa-Clara and Valkanov (2009) (BSV) to measure the economic significance of cross-sectional stock allocations conditioning on macroeconomic state variables. Unlike the approach used by Kandel and Stambaugh, and Avramov and Chordia, the BSV algorithm allows us to directly derive the portfolio weights from the investor s optimization problem. In the unconditional approach, portfolio weights depend on the stock characteristics such as size, book-to-market and past 12-months returns. In the conditional approach, the portfolio weights are a function of macroeconomic state variables and stock s characteristics. We use four macroeconomic variables, the short-term interest rate, term spread, default spread and dividend-price ratio and three stock characteristics, market capitalization of equity, book-to-market and momentum. We compare the out-of-sample performances of the optimal portfolios from the conditional approach to those from the unconditional approach. We find evidence that supports the hypothesis that conditioning on macroeconomic states is economically significant for asset allocations. The certainty equivalent return of the conditional portfolios exceeds that of the unconditional portfolios by 58 basis points per month for our base case of a constant relative risk aversion utility (CRRA) function with a risk aversion level of five. We also compare the ex-post performance of the optimal portfolios using CAPM, Fama-French and Carhart alphas. The portfolios selected using the conditional approach provide a CAPM alpha which is 50 basis points higher than portfolios from the unconditional approach. The conditional portfolios perform 1 Stambaugh (1986, 1999), Lewellen (2004) and Campbell and Yogo (2006) discuss the power of predictability tests. 1

3 similarly over both economic expansion and recession periods while the unconditional portfolios perform poorer over bad economic periods. We further find that conditional investors allocate their money differently from unconditional investors. Unconditional investors consistently overweight small, value and winner stocks over the whole out-of-sample period of Conditional investors, however, overweight more in small stocks and less in value stocks over economic recession periods. They overweight more winner stocks in bad times in 1970s-1990s and less in 2000s. We also find that all four state variables are needed to fully describe the state of economy. Conditioning on single macroeconomic state variable leads to portfolios with substantially lower certainty equivalent returns and alphas. We perform two major robustness checks and find similar results. First, we obtain optimal portfolio weights by rolling instead of updating historical predictive information. Second, we test whether our findings hold up across various investment opportunity sets. In both cases, the conditional portfolios deliver higher Carhart alphas and certainty equivalent returns than the unconditional portfolios. Our results show that the conditional portfolios have the following properties. First, the conditional portfolios, in each month, have similar average absolute weights, maximum and minimum weights, aggregated shorted weights, and fractions of shorted stocks to unconditional portfolios. The weight similarities between the two approaches suggest that the conditional gains are not driven by extreme bets on individual stocks. The turnover of conditional portfolios is higher than that of unconditional portfolios, but the gap is mild, indicating that conditioning on macroeconomic variables does not cause extreme large trading activities. Secondly, the conditional optimal portfolios provide higher average return and more positive skewness than the unconditional portfolios, suggesting a link between predictable variations in investment sets and profitable opportunities. Finally, the conditional optimal portfolios have higher standard deviation and kurtosis than the unconditional portfolios, suggesting much more noise introduced by conditioning on macroeconomic state variables. The gains by conditioning on macroeconomic state variables come from short positions, so that the conditional approach yield similar gains to the unconditional approach if short-selling is not allowed. However, both approaches are still able to deliver positive certainty equivalent returns and alphas, indicating that short selling is not the only source for both investment approaches. Our study contributes to the literature on whether stock return predictability is economically important. The study by Kandel and Stambaugh (1996) explores the economic significance of stock return predictability. They focus on allocation between one risky asset and one risk-free asset, and show 2

4 that return predictability impacts investor s asset allocation and investment performance. Avramov (2004) examines whether an investor s priors of stock return predictability impact their asset allocation and investment performance. Avramov and Chordia (2006) examine the economic value and determinants of cross-sectional predictive models. They document that stock return predictability is economically significant if the alpha and beta are allowed to vary according to macroeconomic state variables. Unlike these studies, the BSV algorithm used in this paper allow us to derive cross-sectional portfolio weights directly from investor s maximization process. The study by Brandt, Santa-Clara and Valkanov (2009) is the first paper to use this approach to explore the economic significance of crosssectional return predictability. They provide evidence that stock characteristics are able to predict crosssectional stock returns. Our results support that macroeconomic state variables interacted with stock characteristics is able to predict not only cross-sectional stock returns, but also the time series variations in returns. All studies support Cochrane s (2011) proposition that stock characteristics and macroeconomic state variables may drive both time series and cross sectional stock price movements. The rest of this paper proceeds as follows. We briefly introduce the asset allocation algorithm and empirical methodology, and discuss the data and descriptive statistics in section II. The main empirical results of the economic significance conditioning on macroeconomic state variables and the robustness checks of the results are presented in Section III. We examine whether the economic significance is robust to estimation approach and various investable asset sets. We investigate the properties of conditional portfolios in section IV and the impact of short-sale on the economic significance in section V. Section VI concludes this study. II. Methodology 2.1 The conditional portfolio allocation algorithm The parametric portfolio selection algorithm is proposed by Brandt, Santa-Clara and Valkanov (2009) to circumvent the statistical issues in conventional tests of stock return predictability. Investors are riskaverse and utility-maximizing; they have a constant relative risk aversion (CRRA) preference with a risk aversion level of five. They allocate their capital among a large set of stocks, portfolio selection as:, and parameterize their (1) 3

5 where is the market equilibrium weight defined by the ratio of the market capitalization of stock i to the aggregated market capitalization of all stocks in the portfolio in month t, is the vector of normalized characteristics of stock i in month t, and is the associated parameter vector. The stock characteristics vector is normalized to ensure that it is stationary over time and that the sum of in each month equals one. 2 The constant parameter vector,, reflects investor s beliefs about overweighting featured stocks such as small, value, and winner stocks. If each element in this vector is zero, then investors do not deviate from the market equilibrium weights. If it is a non-zero vector, say (-1, 1, 1), then investors deviate from the benchmark weights and overweight small, value, and winner stocks. Since is constant over t by construction, investors will overweight featured stocks the same level in both economic recession and expansion times. The optimal stock allocations in equation (1) are directly derived from the following utility maximization process: [ ( )] [ ( )] where is the expected utility in month t, and are the portfolio and stock i s return in month t+1. While the base BSV algorithm is able to capture the cross-sectional return predictability of firm s characteristics, it does not capture the time series variations of cross-sectional returns. The later implies that the investor s utility may not be optimized when investors still allocate into featured stocks, say, momentum stocks, when there is no premium for such stocks. To highlight this and incorporate the time-variation of cross-sectional stock predictability, is modified to be time-varying and condition on K macroeconomic state variables in this study as: (2) each reflects the impact of the corresponding macroeconomic state variable on the investor s marginal beliefs about overweighting the associated featured stocks. The unconditional approach, in which theta is constant from any given estimation, sets the coefficients to be zero, i.e.. If at least one of is non-zero, then equation (2) predicts that the conditional theta 2 See Brandt, Santa-Clara and Valkanov (2009) for details. 4

6 is time-varying because of the associated variation in state variables. Rewriting equation (1) by plugging (2) gives the explicit conditional optimal portfolio selection: ( ) (3) Equation (3) suggests that the optimal portfolio weights should relate to the interaction of state variables and the realized stock characteristics if macroeconomic state variables are able to predict the time series variations in cross-sectional stock returns. Since the coefficient matrix in equation (3) is constant, Brandt, Santa-Clara and Valkanov (2009) show that the investor s conditional maximizing problem can be solved unconditionally. 3 We estimate the coefficients by numerical methods and obtain the asymptotic covariance matrix of the coefficients proposed by Hansen (1982). 4 Finally, the realized conditional and unconditional portfolio returns at time t are: ( ) (4) where is the realized return of stock i in period t+1. We compare the out-of-sample performance between the conditional and unconditional portfolios selected using equation (2). The logic of our study is the follows. If macroeconomic state variables interacted with stock characteristics is able to predict the cross-sectional stock returns and the time series variations in returns, conditioning on state variables should lead to different allocation behavior across the business cycle from the unconditional approach. As a result, conditional investors should be able to perform better than unconditional investors in general and over bad times. For example, in the past decade, the returns to value and past winner stocks become trivial or negative when the macroeconomy is bad, then the conditional approach leads to underweight value and winner stocks in those periods and investors can profit more or lose less from these underweighting allocations Variable construction We use the short-term interest rate, term spread, default spread and aggregate market dividendprice ratio to describe macroeconomic states. Previous studies show that these variables are able to characterize different aspects of macroeconomic states and are related to future stock returns. Fama 3 Brandt, Santa-Clara and Valkanov (2009) provide detailed discussions how to solve this maximizing problem unconditionally. 4 Specifically, the Newton-Raphson algorithm works here, the appendix provides the details of this approach. 5

7 (1981), and Fama and Schwert (1977) document that the short-term interest rate is a proxy of the expectations for future economic activities; Keim and Stambaugh (1988), Campbell and Schiller (1988), and Fama and French (1988) show that dividend-price ratio is associated with the slow mean reversion of stock returns across economic cycles; and Fama and French (1988) show that default and term spreads are related to long-term and short-term business cycles. The short-term interest rate (INT) is defined as the yield on the 3-month Treasury bills (T-bill). Dividend is defined as the 12-month moving aggregated dividend paid on S&P 500 Index; Dividend-price ratio (D/P) is the aggregated dividend divided by current S&P 500 index. The term spread (TERM) is defined as the difference between the average yield of Treasury bonds with more than 10 years to maturity and the average yield of T-Bills that mature in three months and the default spread (DEF) as the difference in the average yields between the BAA corporate bonds and the AAA bonds which are rated by Moody s. Our definitions of these macroeconomic variables are consistent with literature (e.g. Chordia and Shivakumar, 2002; and Goyal and Welch, 2008). We use the three most common stock characteristics in literature: market capitalization of equity, book-to-market and momentum, and define them as Brandt, Santa-Clara and Valkanov (2009). The market capitalization of equity (me) is defined as the log of stock price per share multiplied by number of shares outstanding. The book-to-market (B/M) ratio is log of one plus book value of equity divided by market value of equity. The book value of equity equals total assets minus liabilities and preferred equity value, plus deferred taxes and investment tax credits. We require that the accounting numbers lag the stock returns at least six months. The momentum (mom) is calculated as the compounded return between month t-13 and t-2. To measure portfolio performance, we use certainty equivalent value, Sharpe ratio, and CAPM, Fama-French and Carhart alphas. The certainty equivalent return is derived directly from the expected CRRA utility function with a risk aversion of five; Sharpe ratio is the time series mean of optimal portfolio excess return over the risk-free asset return divided by the corresponding standard deviation; alphas are obtained from the conventional linear asset pricing regressions. All measures are annualized. They are used by Kandel and Stambaugh (1996) and Avramov and Chordia (2006), among others, to examine the economic significance of stock return predictors, and by DeMiguel, Garlappi and Uppal (2009) to compare the 1/N strategy with different sophisticated mean-variance asset allocation strategies. The certainty equivalent value is considered a better measure for CRRA utility-maximizing investors (see Brandt, Santa-Clara and Valkanov, 2009) as it is derived directly from the maximized utility. 6

8 2.3 Data and descriptive statistics The sample data period is from 1964 through Firms annual accounting data are from the COMPUSTAT database and the monthly stock returns from the CRSP database, and are merged by the CRSP-COMPUSTAT linktable file. We screen the merged data and construct the firm s three characteristic variables following the procedure proposed by Brandt, Santa-Clara and Valkanov (2009) after we exclude the 20% smallest stocks from our sample. 5 Finally, we normalize the characteristics to ensure that they are stationary across time and that the sum of overweighting featured stocks in equation (3) equals zero in each month. On average, there are 3516 stocks per month in our sample with the fewest stocks in February 1964 (903 stocks) and the most stocks in September 1997 (5929 stocks) (see Table I). In general, the number of stocks increases over time before 1997 and decreases afterwards. Figure 1 illustrates the time-series distribution of cross-sectional mean of stock characteristics before normalization. The magnitude and time trend of each characteristic variable are comparable to that of Brandt, Santa-Clara and Valkanov (2009). Overall, all characteristics are persistent over time, consistent with Campbell and Yogo (2007). [Figure I is about here] Table I reports the summary statistics of cross-sectional weights and characteristics of the sample dataset (before normalization). The time series averages of the cross-sectional equal-weighted and value-weighted weights over are 0.04% and 3.55%, respectively. In each month, we also pin down the largest and smallest value-weighted weights. Over the whole sample period, the largest weight is 9% on average and the smallest weight is zero. The value- and equal-weighted weights are useful benchmarks to examine whether our strategy causes extreme allocations. The values of the characteristic variables in each month are cross-sectional averages of all qualified stocks in that month. The time series means of the cross-sectional average of me, B/M and mom in each month are 18.33, 0.54, and 0.15, respectively. The largest and smallest cross-sectional average me are in May 2008 and in January 1975; the largest and smallest B/M are 0.99 in January 1975 and 0.38 in January 1969; and the largest and smallest mom are 1.03 in August 1983 and in May The results are similar to that of Brandt, Santa-Clara and Valkanov (2009, Figure 1). [Table I is about here] 5 See the appendix in Brandt, Santa-Clara and Valkanov (2009) for details. 7

9 The data of macroeconomic state variables are from two sources: the Federal Reserve website (the INT, TERM and DEF) and Robert Shiller s website (the D/P). 6 They are also normalized. Figure 2 is the time series distribution of each macroeconomic variable before normalization. All variables have their own patterns across business cycles. INT decreases, and TERM and DEF increase during the economic contractionary periods. D/P does not change monotonically during the economic recession periods, it increases at the beginning and then decreases, indicating that the dividend-price ratio lags the business cycle. The pronounced different and independent evolution patterns of these variables suggest that they complement each other to reflect different aspects of the macroeconomy, consistent with findings in previous studies (e.g. Campbell and Shiller, 1988; Fama and French, 1988; among others). [Figure 2 is about here] The monthly factor returns and risk-free asset return used for portfolio performance evaluation are from Kenneth French s online data library. 7 III. Main empirical results In this section, we first investigate the impacts of macroeconomic state variables on asset allocations. Specifically, we examine the impact of macroeconomic state variables on the time series variation of thetas in equation (2), which are the overall beliefs about over/underweighting featured stocks. We predict that the conditional thetas are more volatile to reflect the impact of macroeconomic states on investors beliefs. We also examine the marginal impact of macroeconomic state variables on investor s beliefs, the deltas in equation (2). We then draw the economic inference for macroeconomic state variables predictability, i.e. we explore the issue of whether the portfolios selected using the conditional approach perform better than that from the unconditional approach. This comparison in economic crisis periods is interesting in particular for predictability testing. If macroeconomic state variables are able to predict the time series variations in cross-sectional returns, we predict that conditional investments perform better than unconditional investments over economic recession periods. 3.1 Impact of macroeconomic states on stock allocations 6 The links are and We thank Robert Shiller for providing this dataset 7 We thank Kenneth French for making these data available. 8

10 Figure 3 illustrates the time series thetas conditioning on the full set of macroeconomic state variables and the unconditional thetas. Before analyzing the plot, we first explain how we find the thetas. For both conditional and unconditional approach, equation (2) implies that there are three linear equations corresponding to firm s characteristics. In each linear equation, the deltas need to be estimated and thetas are then derived by equation (2). Notice that the delta associated with each macroeconomic state variable indicates the marginal impact of the corresponding variable on investor s beliefs about over/underweighting featured stocks. The overall impact of macroeconomic state variables on investor s overall beliefs about over/underweighting a type of featured stocks, theta, is the sum of all terms except the intercept on the right hand side of equation (2). The time series deltas and thetas are estimated as follows. We estimate the deltas for year 1974 from the data over the first 10 years ( ) and derive the thetas for year 1974 from the estimated deltas. We update the data to the end of year 1974 to find the deltas and thetas for the subsequent year (1975) and so forth. This estimation procedure is similar to that by Brandt, Santa-Clara and Valkanov (2009). We end up with a sequence of deltas and thetas from 1974 through In Figure 3, the first graph is the thetas associated with market capitalization of equity, the second with book-to-market, and the third with momentum. Figure 3 shows the unconditional theta in the equation of market capitalization of equity is negative across time, the unconditional thetas in equations of book-to-market and momentum are positive, not volatile over time. These findings suggest that the unconditional approach leads to overweight small, value and winner stocks similarly across time, consistent with that this approach does not take into account the time varying predictability of crosssectional size, book-to-market ratio and momentum. However, not only the magnitudes but also the signs of the thetas conditioning on all macroeconomic variables vary across time, consistent with our prediction that the macroeconomic state variables predictability changes investors beliefs about overweighting featured stocks. For example, the investment returns on value and winner stocks become trivial or negative during the latest economic recession period ( ); conditioning on macroeconomic state variables leads to negative thetas and underweight those stocks over these periods. The volatile conditional thetas are also consistent with the finding by Kandel and Stambaugh (1996) that stock return predictability impacts investor s asset allocation behavior. [Figure 3 is around here] Figure 4 plots the time series of the estimated deltas of equation (2) for each single-variable conditional approach using the same estimation procedure above. Figure 4 shows that the marginal 9

11 impact of macroeconomic state variables on investor s over/underweighting beliefs varies across time. Take conditioning on the dividend-price ratio as an example. The time series delta in the equation of market capitalization of equity varies from 1.0 to -4.2 with a mean of It is negative before year 2000 and positive after that point, indicating that investors believe that they should overweight more in small stocks over 1980s and 1990s but overweight less over 2000s. Figure 4 shows similar variation patterns of deltas in B/M and MOM equations conditioning on dividend-price ratio. Figure 4 also shows that deltas conditioning on other single macroeconomic variables vary over time. The sharp changes in conditional investor s belief about over/underweighting featured stocks are consistent with the hypothesis that macroeconomics state variables impact investor s cross-sectional asset allocation and this impact is time-varying. It is interesting and important to examine whether investors strongly believe that they should over/underweight a specific type of stocks at any point of time. For this purpose, we investigate the statistical significance of deltas in equation (2) in each month. [Figure 3 is around here] Table II illustrates the estimated deltas and the associated asymptotic standard deviations for year Models 1-4 each conditions on one macroeconomic variable. Models 1 to 4 help us understand whether each macroeconomic variable alone is sufficient to characterize the macroeconomic states. The base case is Model 5, which conditions on all four macroeconomic state variables. Let us start with strategies conditioning on single macroeconomic variables. In Model 1, the estimated coefficient on the short-term interest rate is 1.5 in the equation of market capitalization of equity, 2.1 in the book-tomarket equation, and 1.7 in the momentum equation. These coefficients suggest that this conditional strategy leads to overweight more in small stocks and less in value and momentum stocks than the unconditional approach in January, April, May, August 2010 during which the short-term interest rate decreases. The associated t-statistics are significant, suggesting that investors strongly believe that they should follow these over/underweighting strategies. Similarly, the estimated slopes in Model 2 suggest that conditioning on term spread will lead to overweight more in small, value and loser stocks than the unconditional approach whenever the term spread increases in The negative estimated slope in the book-to-market equation in Model 3 implies that conditioning on default spread leads to less overweight value stocks when the economy is becoming bad in The positive slope signs in Model 4 suggest that conditioning on dividend-price ratio leads to overweight less in small stocks and more in value and winner stocks if dividend-price ratio decreases. 10

12 The signs and magnitudes of the coefficients conditioning on the full set of macroeconomic state variables are different from that conditioning on single variables as the state variables interact with each other during the predicting process. For example, Model 5 shows that the coefficients on all macro variables in the equations of market capitalization of equity are larger than the counterparts in Models 1-4. The coefficients on dividend-price ratio in the equations of market capitalization of equity and book-to-market are negative while they are positive in Models 2 and 3. As mentioned above, however, the overall impact in investment opportunity set is the aggregated marginal impacts of all state variables. These differences of coefficients in signs and magnitudes suggest that the portfolios conditioning on the full set of macroeconomic state variables may deliver better returns than that conditioning on subsets of state variables as the full set variables better describe the macroeconomic states. [Table II is around here] To summarize, the results above support that macroeconomic state variables have impacts on investor s asset allocation decisions. When the macroeconomic states change, the conditional approach leads to different loading behavior in featured stocks from the unconditional approach. The full set of state variables may be able to describe the macroeconomic states from different aspects while single state variable describes one specific attribute Economic significance In this subsection, we start to explore the economic significance of the impacts of macroeconomic state variables on investor s stock allocation decisions. Figure 5 shows the time series portfolio returns using the base conditional approach and the unconditional approach out-of-sample. The portfolio returns in each month are obtained by plugging the coefficients estimated in the previous section in equation (4). Figure 5 shows that the conditional approach delivers higher portfolio returns in most times than the unconditional portfolios. The conditional portfolio returns are more volatile than that of the unconditional, suggesting that conditioning on macroeconomic state variables may produce portfolios with higher standard deviations than the unconditional approach. The underperformance of conditional portfolios in some periods suggests that investor s utility is maximized by the higher returns in good times, which dominate their loss in bad times. The underperformance may also suggest that even the full set of macroeconomic state variables do not efficiently describe the macroeconomy. [Figure 5 is around here] 11

13 Table III illustrates the out-of-sample asset allocation performance of the conditional and unconditional portfolios over the whole out-of-sample period (Panel A), and economic expansion (Panel B) and recession (Panel C) periods. The economic expansion and recession periods are defined by the National Bureau of Economic Research (NBER). 8 In each panel, we report the certainty equivalent return, Sharpe ratio, and CAPM, Fama-French and Carhart alphas of the optimal portfolios selected conditioning on the full set and subsets of macroeconomic state variables and the portfolio from the unconditional approach. The differences in the performance measures between each conditional portfolio and the unconditional portfolio are used to gauge the economic significance of conditioning on the associated macroeconomic state variables. Consistent with Brandt, Santa-Clara and Valkanov (2009), Panel A of Table III provides evidence that stock characteristics are able to predict cross-sectional stock returns, i.e. the unconditional approach can produce portfolios with high certainty equivalent return and Sharpe ratio, and positive CAPM alpha. The alpha, Sharpe ratio and certainty equivalent return are 15%, 0.96 and 14%, respectively, which are comparable to the 18%, 0.94 and 12% in Brandt, Santa-Clara and Valkanov (2009). More interestingly, Panel A of Table III provides evidence that conditioning on the full set of macroeconomic state variables (Model 5) produces portfolios with higher certainty equivalent returns and alphas than the unconditional portfolio. The conditional portfolio provides an annualized certainty equivalent return as high as 21%, about 7% per year or 58 basis points per month higher than the unconditional portfolio. The annualized CAPM, Fama-French and Carhart alphas of the conditional portfolio are 21%, 13% and 9%, which are around 4-6% higher than that of the unconditional portfolio and the increases in alphas are statistically significant. The conditional portfolio, however, delivers a Sharpe ratio that is slightly lower than that of the unconditional portfolio. This is consistent with Figure 5 that the conditional portfolio returns are more volatile than the unconditional portfolio returns. 9 One possible reason for this low Sharpe ratio may be that the full set of macroeconomic state variables do not sufficiently characterize the business conditions and introduce additional noise into the parameterization process. Another reason may be that the investor s utility is maximized by earning more in good time while losing more in bad times. It may also be because the stale information used in estimation process misleads allocations. 10 Nevertheless, the higher certainty equivalent value, the better measure of economic value, in Panel A of Table III supports the conclusion that portfolios conditioning on the full set of macroeconomic state variables outperform the unconditional portfolio. Panel A of This is also verified by the lower adjusted R-squares of the factor model regressions on the unconditional portfolio returns. 10 When we use rolling estimating approach, which excludes distant historical information, to obtain estimates in later section, we obtain higher Sharpe ratios. 12

14 Table III also shows that the economic value conditioning on single macroeconomic state variables becomes trivial. For example, the portfolio formed conditioning on short-term interest rate delivers a portfolio with an annualized certainty equivalent return of 15.5%, which is only 1% higher than that of the unconditional portfolio. Its CAPM, Fama-French and Carhart alphas are equivalent to that of the unconditional portfolio. Our weak results for single-variable conditional investments are stronger than but consistent with the findings by Brandt, Santa-Clara and Valkanov (2009) that portfolio selected conditioning on the slope of yield curve provides a certainty equivalent return that is only 0.02% per year higher and a Sharpe ratio slightly lower than the unconditional portfolio. Our results also suggest that large sets of macroeconomic state variables describe the macroeconomy and the time-varying premiums of cross-sectional stock return predictors better than small sets. Panels B and C in Table III illustrate the economic values conditioning on macroeconomic state variables over economic expansion and recession periods. Since the unconditional approach ignores the time-varying properties of stock return predictor premiums, we may predict higher economic values of conditional portfolios than that of the unconditional portfolio over recession periods. Panel B shows comparable economic values to that in Panel A. The difference in certainty equivalent value between the unconditional and the base conditional portfolios over economic expansion periods equals that over the whole sample period. The difference in Carhart alpha between the two portfolios over economic expansion periods is statistically significant but smaller than that over the whole out-of-sample period. Conditioning on subsets of state variables in Panel B provides performance improvements which are as trivial as that in Panel A. Panel C of Table III shows stronger evidence than Panel A that conditioning on macroeconomic state variables is economically significant. The base conditional portfolio provides much higher certainty equivalent return, alphas and Sharpe ratio than the unconditional portfolio. The certainty equivalent, Carhart alpha and Sharpe ratio of the base conditional portfolio are 21%, 13% and 0.53, which are 9%, 10% and 0.48 higher than that of the unconditional portfolio. These improvements are larger than that in Panels A and B and shows that conditioning on macroeconomic state variables can effectively capture the time-varying returns of investment in featured stocks. The economic significances conditioning on single macroeconomic state variables, however, is as trivial as that in Panels A and B. The findings in Panels B and C suggest that the economic values conditioning on macroeconomic state variables are robust to the business cycle. To summarize, Table III shows that conditioning on macroeconomic state variables are economically significant for cross-sectional stock allocations, supporting that macroeconomy state variable interacted with stock characteristics is able to predict cross-sectional stock returns and time series variation in 13

15 returns. However, the predictability of single macroeconomic state variables is not as strong as that of the full set of macroeconomic variables. [Table III is around here] 3.3 Robustness checks There are two major concerns about the results above. The first is whether the conditional approach is effective if investors allocate their capital based on different historical information sets of underlying stocks and business conditions. We test this by applying rolling approach to obtain the estimated coefficients. The second is whether the economic significance of conditioning on macroeconomic state variables is robust to different investable sets of stocks. We test this by changing the data screening criteria to generating various sets of stocks Rolling estimation approach It is difficult to conjecture how much historical information investors use for their portfolio allocation decisions. Moreover, the facts that the conventional state variables do not sufficiently characterize the time-varying premiums of cross-sectional stock return predictors suggest that shorter estimation periods may introduce more noise during the estimation process. We find that the estimates and economic values are less impacted when 10 or more years of historical information of stock predictors are used. Table IV reports the portfolio performances and economic significances conditioning on rolling information of macroeconomic states and stock characteristics over the very past 15 years. The estimation procedure is similar to that in previous sections except that the estimation period is not expanded but rolled each year. Panel A in Table IV shows stronger evidence than Panel A of Table III that conditioning on macroeconomic state variables can significantly improve cross-sectional stock allocations. The base conditional approach (Model 5) leads to a portfolio with not only higher certainty equivalent returns and alphas, but also higher Sharpe ratio than the portfolio from the unconditional approach. The base conditional portfolio delivers certainty equivalent return, Carhart alpha and Sharpe ratio that are 18%, 19% and 0.12 than the unconditional portfolio. However, the adjusted R-squares in the factor model regressions are only half of the counterparts in Table III, suggesting that more historical information used in coefficient estimation helps to reduce the noise in portfolio returns. The economic values by Models 1-4 are much smaller than Model 5 and none of them earn higher Sharpe ratio than the base conditional approach. 14

16 Panel B of Table IV shows similar pattern of economic significance conditioning on macroeconomic state variables over economic expansion periods to Panel A except that the base conditional approach fails to produce higher Sharpe ratio than the unconditional strategy. Panel C, however, shows that conditioning on multiple macroeconomic variables produces portfolios with higher certainty equivalent returns, alphas and Sharpe ratios than the unconditional strategy. For example, conditioning on the full set of state variables leads to a portfolio with certainty equivalent return, Carhart alpha and Sharpe ratio that are 15%, 24% and 0.78 higher than the unconditional approach and the improvements in alphas are statistically significant. 11 These results are stronger than that in Table III. However, the low adjusted R- squares for factor model regressions of Models 5, which are lower than that in Panel A, suggest that much noise is introduced by macroeconomic state variables in economic recession periods. Similar to Table III, conditioning on short-term interest rate or term spread provides portfolios with higher certainty equivalent returns, alphas and Sharpe ratios than the unconditional approach but conditioning on default spread or dividend-price ratio does not. [Table IV is around here] In short, the results above are similar to that in Table III and suggest that the economic significance of conditioning on macroeconomic state variables is robust to estimation approaches Size of investment opportunity set The investment opportunity set above is the database that 20% of the smallest qualified stocks are excluded. In this sub-section, we generate four alternative opportunity sets from four cutoffs of the smallest qualified stocks: 0%, 10%, 40% and 60%. The portfolio performances of the conditional and unconditional approaches of each dataset are reported in in Table V. The investment sets in Panels A and B are larger than the base case. Panel A illustrates the economic significance conditioning on macroeconomic state variables with all qualified stocks. Both conditional and unconditional approaches lead to portfolios with higher certainty equivalent returns, alphas and Sharpe ratios than the base case (Table III). For example, the certainty equivalent return, Carhart alpha and Sharpe ratio of the unconditional portfolio are 17%, 8% and 1.15, respectively, which are 2%, 2% and 0.18 higher than the counterparts of the base dataset (Table III). More interestingly, the base conditional strategy (Model 5) produces portfolios with higher certainty equivalent return and alphas 11 The Carhart alpha of Model 5 is higher than the corresponding Fama-French alpha because the coefficient on momentum factor returns is in Carhart regression. Similarly, the coefficient on momentum in Model 3 is , which drives the Carhart alpha slightly higher than the Fama-French alpha. 15

17 than the unconditional strategy. Their economic significances are comparable to the counterparts in Table III. The economic significance conditioning on single macroeconomic variable is smaller than that from Model 5, but their economic significances are comparable to that in Table III (the base dataset). The higher Sharpe ratios delivered by conditioning on single macroeconomic state variables and the lower adjusted R-squares in Model 5 suggest that conditioning on multiple variables introduces extra noise in portfolio decision. Panel B reports the performances of portfolios selected using the conditional and unconditional strategies based on the dataset that 10% of the smallest stocks are deleted. The results are similar to that in Panel A. The investment opportunity sets in Panels C and D are smaller than the base dataset. In Panel C, 40% of the smallest stocks are eliminated and 60% in Panel D. The magnitudes of the performance measures in Panels C and D are smaller than the counterparts in Panels A and B, and Table IV, indicating that the outstanding performance of portfolios from both conditional and unconditional approaches are partially from overweighting small stocks. More interestingly, the performance improvements in certainty equivalent returns by conditional strategies are not impacted by the further eliminations of small stocks. For example, the improvement in certainty equivalent returns by the base conditional strategy in Panels C and D are 7% and 9%, respectively. However, the improvements in Carhart alphas are smaller than the base case in Panel A of Table III while Model 5 still leads to portfolios outperforming that from the unconditional approach. [Table V is around here] In words, the economic significance conditioning on macroeconomic state variables is robust to both alternative estimation approaches and investment opportunity sets. IV. Properties of the conditional portfolios From the investment perspective, investors can enjoy the economic value conditioning on macroeconomic state variables only if the conditional approach leads to smart portfolio choices. If this is the case, we should expect that conditioning on macroeconomic variables produces portfolios with higher returns, and/or more positive skewnesses, and lower risks (standard deviations and/or kurtoses) than that from the unconditional approach. We explore this by examining all moments of the conditional and unconditional portfolio return distributions. However, extreme bets on a few stocks or extreme large trading volumes may also lead to portfolios with outstanding performance (see Brandt, Santa-Clara and Valkanov, 2009; and Goetzmann, Ingersoll, Spiegel and Welch, 2007). We address the 16

18 concern by comparing the weight distributions of the conditional portfolios with that of the unconditional portfolio. We first report the weight distributions of the conditional portfolios and then their moments Portfolio weight distribution Table VI reports the time series means of cross-sectional weights of the conditional and unconditional portfolios. We report the turnovers, maximum and minimum weights, and cross-sectional average absolute stock weights of the optimal portfolios. We also report the fraction of the shorted stocks and the aggregated shorted weights of each strategy. The first column contains the portfolio weight properties of the unconditional strategy. Columns 2-6 are the properties of the conditional portfolios defined in Table II. The first few rows are the weights during the whole out-of-sample period. These rows show that there are no significant differences in time series means of maximum and minimum weights, aggregated short positions, and cross-sectional average absolute stock weights between the conditional and unconditional portfolios. The differences in turnover between the two types of portfolios are small and mild. For example, the time series means of the cross-sectional average absolute stock weights for the unconditional portfolio and the base conditional portfolio are 0.15% and 0.17%. Their aggregated shorted weights are 125% and 145%, respectively. The turnover of the base conditional portfolio is 1.99, which is not extremely high but doubles that of the unconditional portfolio. The high turnover may suggest that conditioning on macroeconomic state variables leads to more frequent trades to capture good investment opportunities over the business cycle. The increase in turnover also suggests that the portfolio outperformance conditioning on multiple state variables may be partially attributed to transaction costs. The turnovers of portfolios selected conditioning on single state variables, however, are close to that of the unconditional portfolio, suggesting that their economic values are not driven by transaction costs while their economic values are small. The next few rows illustrate the portfolio weights of the conditional and unconditional portfolios over the economic expansion periods and the last few rows over the economic recession periods. The results for both cases are similar to that over the whole period. [Table VI is around here] 4.2 Moments of portfolio return distributions In the last section, we documented that the economic values conditioning on macroeconomic state variables cannot be explained away by transaction costs or extreme bets on individual stocks. In this section, we examine whether such economic values are related to smart stock picking and trading. 17

19 Specifically, we examine the first to fourth moments of the conditional and unconditional portfolio returns to investigate whether conditioning on macroeconomic state variables leads to portfolios with higher returns and/or lower risks. Table VII reports the first to fourth moments of the monthly return distributions of the ex-post conditional and unconditional portfolio returns over the whole out-of-sample period, and economic expansion and recession periods. Let us start with the whole sample period. The time series average return of the base conditional portfolio (Model 5) is 26.5%, 6% higher than that of the unconditional portfolio. The base optimal portfolio returns have a positive skewness of 2.2 but the skewness of the unconditional portfolio returns is The kurtosis is 26.3 for the conditional portfolios and 3.6 for the unconditional one. The positively skewed conditional portfolio returns suggest that the associated higher kurtosis is driven by high positive, rather than negative, realized returns, which is preferable. The higher average return, more positive skewness and higher kurtosis of the conditional portfolios suggest that conditioning on macroeconomic state variables helps to identify good stocks and trading. However, the unconditional portfolio returns have a standard deviation of 15.5% but the base conditional portfolio s is 24.6%. The increase in standard deviation is greater than the increase in average return, which causes the conditional portfolio s lower Sharpe ratios. The moments of portfolios selected conditioning on one macroeconomic variable, however, are different from that of the base conditional approach. The increases in average returns and skewnesses by these conditional portfolios are trivial and much smaller than the increase in standard deviations, consistent with that their economic significances are small and their Sharpe ratios are lower than that of the unconditional approach. Take the investment conditioning on short-term interest rate as an example. The average return of this conditional optimal portfolio is 20%, which equals that of the unconditional strategy. The skewness of this conditional portfolio is -0.44, which is only 0.29 higher than that of the unconditional portfolio. The net increase in portfolio skewness conditioning on short-term interest rate is also much smaller than that of the base conditional approach. Moreover, the portfolio selected conditioning on short-term interest rate provides higher standard deviation and kurtosis than that from the unconditional approach. The mixed facts of all these moments suggest that the economic significance conditioning on short-term interest rate is small. The moments of the portfolio return distributions of the conditional strategies over economic expansion periods show similar patterns to that over the whole sample period while the magnitudes of increases in average return and skewness are smaller. However, the difference in average return between the base conditional portfolio returns and the unconditional returns over the economic 18

20 recession periods is much higher than that over the whole sample period. The base conditional portfolio delivers an average return of 28%, three times larger than that of the unconditional portfolio. Its standard deviation almost doubles that of the unconditional portfolio. The skewness of the base conditional portfolio returns is 2.7 while that of the unconditional portfolio returns is These findings strongly suggest that conditioning on macroeconomic state variables leads to preferable variations in investment opportunity sets. Similar to the findings over the economic expansion and the whole sample periods, the improvements in the average returns and skewness of portfolio returns conditioning on single macroeconomic variables are small, which is consistent with their trivial economic significances. [Table VII is around here] V. Impact of short-sale constraint Miller (1977), Shleifer and Vishny (1997), Stambaugh, Yu and Yuan (2012) argue that short-sales are the resources of the stock anomalies. Consistently, Ali and Trombley (2006), and Lee and Swaminathan (2000) provide evidence that the profits of characteristic-based strategies are mainly driven by shortselling activities, which implies that the economic value conditioning on macroeconomic state variables may also come from short sales. 12 On the other hand, the U.S. investors, however, are not allowed to short stocks unlimitedly in practice, according to the Federal Reserve Board Regulation T. Most institutional investors, such as mutual fund or pension fund managers, are not allowed to short stocks. As a result, these investors may not able to collect the gains conditioning on macroeconomic state variables if the benefits are due to short-sales. Furthermore, Brandt, Santa-Clara and Valkanov (2009) also show that the short-sale constraints impact the performance of the unconditional portfolio significantly. The Sharpe ratio, CAPM alpha and certainty equivalent return are reduced by one-third to one half (see Brandt, Santa-Clara and Valkanov, 2009, Table 3). It is interesting to examine whether conditional approaches can achieve better asset allocation than the unconditional approach when shortsales are not allowed. There are several possible approaches to impose short-sale constraints. To simplify the computing task, we impose short-selling constraints following Brandt, Santa-Clara and Valkanov (2009). We truncate the portfolio weights at zero and renormalize them during the parameterization process. The parametric weight of a stock in the realized portfolio with short sale constraint becomes: 12 We abuse the terminology to some extent here. In their papers, they focus on momentum strategy. However, there is evidence that momentum premium is related to firm size (see, Hong, Lim and Stein, 2000; and Lesmond, Schill and Zhou, 2004). 19

21 However, unlike equation (2), the individual weight is now a non-linear function of the stock return predictors and the portfolio is not first-order optimized. Table VIII reports the conditional and unconditional portfolio performances under short-sale constraints over the whole out-of-sample period, and economic expansion and recession periods. The magnitudes of the portfolio performance measures under short sale constraints are smaller than that without short-sale constraints, consistent with the findings by Brandt, Santa-Clara and Valkanov (2009) that short-sales play an important role in the profitability of characteristics-based investment strategies. However, the positive certainty equivalent returns and alphas suggest that short-selling is not the only source of the BSV investment strategy, which is not consistent with Ali and Trombley (2006), and Stambaugh, Yu and Yuan (2012). More interestingly, Panels A and B of Table VIII show that conditioning on macroeconomic state variables is not economically significant anymore over the whole out-of-sample period and economic expansion periods when short sales are not allowed. The conditional portfolios deliver similar certainty equivalent values, CAPM alphas and Sharpe ratios to that from the unconditional portfolio. The annualized certainty equivalent return and Sharpe ratio of the portfolio selected conditioning the full set of macroeconomic state variables are 8% and 0.6 under short-sale constraints, which equal that of the unconditional approach. The conditional approach also produces a portfolio with CAPM, Fama-French and Carhart alphas of 10.4%, 2.4% and 1.5%, which are only 0.5%, 0.8% and 0.1% higher than that from the unconditional approach and not statistically significant. The economic values conditioning on single macroeconomic state variables are smaller than that of the base case and similar to the unconditional approach. The findings in Panel A suggest that the benefits from conditioning on macroeconomic state variables are mostly driven by short sale activities. Given the fact in the last section that there is no significant difference in aggregated short position between the conditional and the unconditional portfolios, the findings in Panel A Table VIII may also suggest that the conditional strategies help investors to buy and sell stocks more effectively than the unconditional approach. Panel B shows that none of the conditional strategies beat the unconditional strategy during economic expansion periods as none of them produce portfolios with higher certainty equivalent returns, or alphas than the unconditional strategy. The certainty equivalent return and Sharpe ratio of the base conditional portfolio are 11% and 0.8, which are 1% and 0.5 lower than that of the 20

22 unconditional portfolio. The annualized CAPM and Carhart alphas are 9.5% and 1.3%, which are also slightly lower than that of the unconditional portfolio but not statistically significant. However, Panel C shows that the base conditional approach can lead to portfolios with higher certainty equivalent returns, alphas and Sharpe ratios than that of the unconditional approach while conditioning on single macroeconomic variables does not. The annualized certainty equivalent return and Sharpe ratio delivered by the base conditional portfolio are and 0.21 while that by the unconditional portfolio are and The base conditional portfolio provides annualized CAPM, Fama-French and Carhart alphas of 11%, 6% and 3%, which are 6%, 4% and 2% significantly higher than that of the unconditional portfolio. The economic significance of the base conditional approach suggests that macroeconomic variables are able to predict the time-varying returns of investment in size, value and winner stocks even under short-sale constraints. This economic significance also supports the finding by Chen, Petkova and Zhang (2008) that cross-sectional predictor premiums are time-varying. Conditioning on single state variable, however, fails to produce portfolios with higher certainty equivalent returns, Sharpe ratio and alphas than the unconditional approach. Overall, Table VIII provides evidence that short-sale activities are the main sources of the economic values conditioning on macroeconomic state variables in good times. Asset allocation conditioning on macroeconomic states is able to profit more than that independent of business conditions in bad times. These facts suggest that, for investors who are not allowed to short-sell stocks, the stock return predictability of macroeconomic state variables may be useful in bad times but not helpful on average. [Table VIII is around here] VI. Conclusion The question whether stock return is predictable is interesting to both academicians and practitioners. Direct tests, however, suffer severe statistical issues because of the persistence of predictive variables and the correlation between the innovations of predictive variables and stock returns (e.g. Stambaugh, 1986, 1999; Lewellen, 2004; and Campbell and Yogo, 2006). Switching the metric of analysis to asset allocation is a novel way to tackle these issues. The traditional Bayesian approach (Kandel and Stambaugh, 1996; Avramov, 2004; and Avramov and Chordia, 2006), however, is not free of statistical issues (Roskelley, 2008). The alternative algorithm proposed by Brandt Santa-Clara and Valkanov (2009) can ideally deal with these issues by obtaining portfolio weights directly from maximizing investor s utility function. 21

23 We use the BSV algorithm and document that cross-sectional asset allocation conditioning on macroeconomic state variables are economically significant in terms of certainty equivalent returns and alphas. The conditional portfolios provide higher average returns and more positive skewness than the unconditional portfolios. While conditioning on state variables may not be helpful in good times if shortsales are not allowed, it is still helpful when the macroeconomy is bad. Our findings suggest that macroeconomic state variables are able to predict the time series variations in cross-sectional stock returns. Moreover, conventional macroeconomic state variables used in this study do not sufficiently describe macroeconomic states. This insufficiency introduces additional noise and causes the conditional portfolios delivers lower Sharpe ratios and adjust R-squares in linear performance evaluation regressions. Accordingly, the economic values will be stronger for better business condition proxies. An alternative way, but not a perfect way, to improve this problem is to include more macroeconomic variables and assign each variable a weight according to their ability to characterize the business conditions. Another concern in this study is the impact of transaction costs. In section IV, we found that the turnover of the base conditional strategy, which performs better than any other single-variable strategy, almost doubles that of the unconditional strategy. This high turnover may cause higher transaction costs and partially reduce the performance improvement of the base conditional strategy. It is interesting to investigate how much of the performance improvement of conditional strategies can be explained by transaction costs. Third, the portfolios under short-sale constraints in last section are not the first-order optimized portfolio while this approach simplifies the cumbersome maximizing calculations. It is worth examining whether it is robust to conventional Kuhn-Tuck nonlinear optimizations that the economic values of conditioning on macroeconomic variables are driven by short-selling trades. 22

24 Appendix The nonlinear estimation procedure For a case of C characteristics variables and K macroeconomic variables, then equation (2) can be written explicitly as: [ ] [ ] [ ] (A1) The second term of the right hand side of equation (1) without the scale term,, becomes: [ ] [ ] [ ] ( ) ( ) Rewrite the optimizing problem with state-dependent parameters by plugging (A1) into a general CRRA utility function: The first and second order derivatives of (A2) are: (A2) { } (A3) where { } (A4) 23

25 where is a vector, is a matrix. To apply the Newton-Raphson method, let, the utility maximizing problem can be solved by minimizing EU and the Newton-Raphson algorithm can be applied as: (A5) By plugging the (A3) and (A4) into (A5), and solve deltas recursively, the optimal deltas will be obtained when preferable convergence is reached. 24

26 References Ali, A. and M. A. Trombley, Short sales constraints and momentum in stock returns. Journal of Business Finance and Accounting, 33: Avramov, D., Stock return predictability and asset pricing models. Review of Financial Studies, 17: Avramov, D. and T. Chordia, Predicting stock returns. Journal of Financial Economics, 82: Banz, R. W., The relationship between return and market value of common stocks. Journal of Financial Economics, 9: Bernardo, A. E. and O. Ledoit, Gain, loss and asset pricing. Journal of Political Economy, 108: Blanchard, O. J., R. Shiller and J. J. Siegel, Movements in the equity premium. Brookings Papers on Economic Activity, 1993: Brandt, M. W., Estimating portfolio and consumption choice: A conditional Euler equation approach. Journal of Finance, 54: Brandt, M. W., Portfolio choice problems in Y. Ait-Sahalia and L. P. Hansen (eds), Handbook of Financial Econometrics, Volume 1: Tools and Techniques. North Holland, 2010: Brandt, M. W. and P. Santa-Clara, Dynamic portfolio selection by augmenting the asset space. Journal of Finance, 61: Brandt, M. W., P. Santa-Clara and R. Valkanov, Parametric portfolio policies: Exploiting characteristics in the cross-section of equity returns. The Review of Financial Studies, 22: Breen, W., L. R. Glosten and R. Jagannathan, Economic Significance of predictable variation of stock index returns. Journal of Finance, 44: Campbell, J. Y. and R. J. Shiller, The dividend-price ratio and expectations of future dividends and discount factors. Review of Financial Studies, 1: Campbell, J. Y. and S. B. Thompson, Predicting excess stock returns out of sample: Can anything beat the historical average? Review of Financial Studies, 21: Campbell, J. Y. and M. Yogo, Efficient tests of stock return predictability. Journal of Financial Economics, 81: Carhart, M. M., On persistence in mutual fund performance. Journal of Finance, 52: Chavez-Bedoya, L. and J. R. Birge, Index tracking and enhanced indexation using a parametric approach. Working paper, University of Chicago. 25

27 Chen, L., R. Petkova, and L. Zhang, The expected value premium. Journal of Financial Economics, 87: Chordia, T. and L. Shivakumar, Momentum, business cycle and time-varying expected returns. Journal of Finance, 57: Cochrane, J. H., The dog that did not bark: A defense of return predictability. Review of Financial Studies, 21: Cochrane, J. H Presidential address: Discount rate. Journal of Finance, 66: Cooper, M. J., H. Gulen and M. Vassalou, Investing in size and book-to-market portfolio using information about the macroeconomy: some new trading rules. Working paper, Columbia University Cooper, M. J., R. C. Gutierrez, and A. Hameed, Market states and momentum. Journal of Finance, 59: Daniel, K. and S. Titman, Evidence on the characteristics of cross-sectional variation in stock returns. Journal of Finance, 52:1-33. DeMiguel, V., L. Garlappi and R. Uppal, Optimal versus naïve diversification: How inefficient is the 1/N portfolio strategy? Review of Financial Studies, 22: Dybvig, P. H. and S. A. Ross, Differential Information and Performance Measurement Using a Security Market Line. Journal of Finance, 40: Fama, E. F., Stock returns, real activity, inflation and money. American Economic Review, 71: Fama, E. F., Stock returns, expected returns and real activity. Journal of Finance, 45: Fama, E. F. and K. R. French, Dividend yields and expected stock returns. Journal of Financial Economics, 22:3-25. Fama, E. F. and K. R. French, Business conditions and expected returns on stocks and bonds. Journal of Financial Economics, 25: Fama, E. F. and K. R. French, Common risk factors in the returns of stock and bonds. Journal of Financial Economics, 33: 3-56 Fama, E. F. and G. W. Schwert, Asset returns and inflation. Journal of Financial Economics, 5: Griffin, J. M., X. Ji and J. S. Martin, Momentum investing and business cycle risk: evidence from pole to pole. Journal of Finance, 58: Goetzmann, W., J. Ingersoll, M. Spiegel and I. Welch, Portfolio performance manipulation and manipulation-proof performance measures. Review of Financial Studies, 20: Goyal, A. and I. Welch, A comprehensive look at the empirical performance of equity premium prediction. Review of Financial Studies, 21:

28 Hamilton, J. D. and G. Lin, Stock market volatility and the business cycle. Journal of Applied Econometrics, 11: Hansen, L. P Large sample properties of generalized methods of moments estimators. Econometrica, 50: Hjalmarsson, E. and P. Manchev, Characteristic-based mean-variance portfolio choice. Working Paper, Federal Reserve Board, Washington DC. Hong, H., T. Lim and J. Stein, Bad news travels slowly: Size, analyst coverage, and the profitability of American options. Journal of Finance, 55: Jagannathan, R. and R. A. Korajczyk, Assessing the market training performance of managed portfolios. Journal of Business, 59: Jegadeesh, N., Evidence of predictable behavior of security returns. Journal of Finance, 45: Jegadeesh, N. and S. Titman, Returns to buying winners and selling losers: Implications for stock market efficiency. Journal of Finance, 48: Joenvaara, J. and H. Kahra, Investing in hedge funds when the fund s characteristics are exploitable. Working paper, University of Oulu. Kacperczyk, M., S. V. Nieuwerburg, and L. Veldkamp, Rational attention allocation over the business cycle. Working Paper, New York University. Kandel, S. and R. F. Stambaugh, On the predictability of stock returns: An asset-allocation perspective. Journal of Finance, 51: Keim, D. B. and R. F. Stambaugh, Predicting returns in the stock and bond markets. Journal of Financial Economics, 17: Lee, C. M. C. and B. Swaminathan, Price momentum and trading volume. Journal of Finance, 55: Leland, H. E., Beyond mean-variance: Performance measurement in a nonsymmetrical world. Financial Analysts Journal, 55: Lesmond, D. A., M. J. Schill and C. Zhou, The illusory nature of momentum profits. Journal of Financial Economics, 71: Lewellen, J., Predicting returns with financial ratios. Journal of Financial Economics, 74:

29 Mankiw, N. G. and M. D. Shapiro, Do we reject too often? Small sample properties of tests of rational expectations models. Economics Letters, 20: Miller, E. M., Risk, uncertainty, and divergence of opinion. Journal of Finance, 32: Nelson, C. R. and M. J. Kim, Predictable stock returns: The role of small sample bias. Journal of Finance, 48: Plazzi, A., Torous, W. and Valknov, R., Exploiting property characteristics in commercial real estate portfolio allocation. Working paper, UCLA. Roskelley, K. D., Cromwell s rule and the role of the prior in the economic metric. Journal of Business & Economic Statistics, 26: Shleifer, A. and R. Vishny, The limits of arbitrage. Journal of Finance, 52: Solnik, B., The performance of international asset allocation strategies using conditioning information. Journal of Empirical Finance, 1: Stambaugh, R., Bias in regressions with lagged stochastic regressors. Working paper, University of Chicago. Stambaugh, R., Predictive regressions. Journal of Financial Economics, 54: Stambaugh, R. F., J. Yu and Y. Yuan, The short of it: Investors sentiment and anomalies. Journal of Financial Economics, 104: Torous, W., R. Valkanov and S. Yan, On predicting stock returns with nearly integrated explanatory variables. Journal of Business, 77: Zakamouline, V. and S. Koekebakker, Portfolio performance evaluation with generalized Sharpe ratios: Beyond the mean and variance. Journal of Banking and Finance, 33:

30 Table I Summary statistics of stock characteristics This table illustrates the summary statistics of the sample data. Data are from CRSP monthly database and COMPUSTAT annual database from 1964 through The 1 st (2 nd ) column is the maximum (minimum) value over the whole sample period. The 3 rd column is the time series mean. The 4 th column is the standard deviation of time series numbers. The 2 nd (3 rd ) row is the cross-sectional maximum (minimum) value-weighted weights over The market capitalization of equity is defined as log of number of common stock outstanding multiplied by the associate stock price. Book to market ratio is defined as log one plus book value of equity divided by market equity. Momentum is the compounded return in the past 12 months between t-2 and t-13. The values of market cap of equity, book-to-market and momentum in each month are cross-sectional average of all stocks. Variable Max Min Mean Standard Deviation Number of firms VW weights(max*100) VW weights(min*100) EW weights (*100) Market cap of equity Book to market Momentum

31 Table II Point coefficient estimation This table reports coefficients in for year The coefficients are estimated by the Newton-Raphson numerical methods. The observations over are used to obtain the estimates. The associated asymptotic standard deviations are reported in Models 1-4 condition on single state variable. Model 5 conditions on four macroeconomics state variables. me, B/M and mom are market value of equity, book-to-market ratio, and momentum. *, **, *** indicate statistical significance at 10%, 5% and 1%, respectively. Intercept Short term interest rate Term Spread Default Spread Dividend/Price Model1 size *** (0.073) 1.459*** (0.015) B/M 4.794*** (0.067) 2.130*** (0.013) mom 2.658*** (0.046) 1.676*** (0.010) Model2 size *** (0.041) *** (0.046) B/M 5.252*** (0.039) 0.891*** (0.043) mom 2.160*** (0.034) *** (0.036) Model3 size *** (0.071) *** (0.030) B/M 4.998*** (0.073) *** (0.028) mom 2.278*** (0.050) *** (0.018) Model4 size *** (0.068) 0.129** (0.063) B/M 5.502*** (0.065) 0.897*** (0.065) mom 2.433*** (0.051) 0.619*** (0.051) Model7 size *** (0.101) *** (0.065) *** (0.109) *** (0.315) 1.053*** (0.111) B/M 2.278*** (0.104) 3.208*** (0.223) 2.325*** (0.378) *** (0.961) *** (0.043) mom 1.462*** (0.091) 0.987*** (0.086) 0.497*** (0.149) *** (0.306) *** (0.159) 30

32 Table III Portfolio performance This table reports the performance of the conditional and unconditional portfolios over the whole sample, and economic expansion and recession periods. The expansion and recession periods are defined by NBER. The first row is the performance of unconditional policy. Models 1-5 are defined in table II. CER (certainty equivalent return) is derived from the expected utility. The CAPM, Fama-French and Carhart alpha are reported in column 2, 5 and 8 respectively. The associated standard deviations and adjusted R-squares are in parentheses and brackets, respectively. The Sharpe ratios of portfolio returns are in column 11. The row Econsig is the economic significance defined as the performance difference between each conditional portfolio and the unconditional portfolio. The associated t-stats are in parentheses. *, **, *** indicate statistical significance at 10%, 5% and 1%, respectively. CER CAPM Fama-French CARHART Sharpe Ratio Panel A. whole period Uncond *** (0.004) [0.763] 0.062*** (0.002) [0.900] 0.054*** (0.002) [0.903] Model *** (0.004) [0.765] 0.061*** (0.003) [0.859] 0.048*** (0.003) [0.865] Econsig (-0.968) (-0.266) (-1.562) Model *** (0.004) [0.713] 0.065*** (0.003) [0.844] 0.054*** (0.003) [0.850] Econsig (0.916) (0.765) (0.102) Model *** (0.004) [0.711] 0.081*** (0.004) [0.777] 0.067*** (0.004) [0.785] Econsig (1.265) 0.019*** (4.223) 0.013*** (2.839) Model *** (0.005) [0.694] 0.057*** (0.004) [0.815] 0.043*** (0.004) [0.822] Econsig (-0.171) (-1.134) ** (-2.494) Model *** (0.010) [0.290] 0.126*** (0.010) [0.342] 0.092*** (0.010) [0.362] Econsig *** (5.886) 0.064*** (6.282) 0.048*** (4.712) Panel B Expansion Uncond *** (0.004) [0.714] 0.068*** (0.003) [0.869] 0.059*** (0.003) [0.872] Model *** (0.004) [0.725] 0.060*** (0.003) [0.819] 0.050*** (0.003) [0.823] Econsig *** (-2.791) * (-1.940) ** (-2.111) Model *** (0.004) [0.672] 0.070** (0.003) [0.805] 0.061** (0.003) [0.809] Econsig (0.353) (0.476) (0.461) Model *** (0.005) [0.646] 0.085*** (0.004) [0.749] 0.064*** (0.004) [0.767] Econsig (1.528) *** (3.627) (1.054)

33 Model *** (0.005) [0.657] 0.076*** (0.004) [0.780] 0.062*** (0.004) [0.788] Econsig (1.188) 0.008* (1.769) (0.644) Model *** (0.009) [0.273] 0.119*** (0.009) [0.345] 0.078*** (0.009) [0.384] Econsig *** (4.003) 0.051*** (5.737) 0.019*** (2.108) Panel C Recession Uncond *** (0.009) [0.870] 0.040*** (0.006) [0.951] 0.025*** (0.006) [0.954] Model *** (0.011) [0.845] 0.082*** (0.008) [0.919] 0.056*** (0.008) [0.930] Econsig *** (3.522) 0.042*** (4.134) 0.031*** (3.100) Model ** (0.012) [0.794] 0.049*** (0.009) [0.903] 0.025*** (0.009) [0.917] Econsig (1.114) (0.858) (0.038) Model *** (0.011) [0.834] 0.042*** (0.011) [0.839] 0.022** (0.016) [0.844] Econsig (-0.069) (0.157) (-0.231) Model ** (0.015) [0.766] (0.012) [0.871] *** (0.012) [0.879] Econsig (-1.173) *** (-4.031) *** (-4.910) Model *** (0.039) [0.325] 0.174*** (0.038) [0.424] 0.125*** (0.039) [0.429] Econsig *** (5.361) 0.134*** (3.528) 0.100*** (2.547)

34 Table IV Robustness check: rolling estimation approach This table reports the performance of the conditional and unconditional portfolios selected using rolling estimation approach over the whole sample, and economic expansion and recession periods. The expansion and recession periods are defined by NBER. The first row is the performance of state-independent policy. Models 1-5 are defined in table II. CER (Certainty equivalent return) is derived from expected utility. The CAPM, Fama-French and Carhart alpha are reported in column 2, 5 and 8 respectively. The associated standard deviations and adjusted R-squares are in parentheses and brackets separately. The Sharpe ratios of portfolio returns are in column 11. The row Econsig is the economic significance defined as the performance difference between each conditional portfolio and the unconditional portfolio. The associated t-stats are in parentheses. *, **, *** indicate statistical significance at 10%, 5% and 1%, respectively. CER CAPM Fama-French CARHART Sharpe Ratio Panel A. whole period Uncond *** (0.004) [0.710] 0.071*** (0.004) [0.807] 0.059*** (0.004) [0.813] Model *** (0.006) [0.573] 0.080*** (0.005) [0.691] 0.062*** (0.005) [0.703] Econsig ** (2.216) (1.461) (0.476) Model *** (0.006) [0.553] 0.093*** (0.006) [0.600] 0.055*** (0.006) [0.642] Econsig ** (2.075) 0.022*** (3.069) (-0.561) Model *** (0.006) [0.556] 0.105*** (0.006) [0.571] 0.088*** (0.006) [0.579] Econsig ** (2.360) 0.034*** (4.686) 0.029*** (3.923) Model *** (0.006) [0.609] 0.094*** (0.006) [0.649] 0.066*** (0.006) [0.674] Econsig * (1.821) 0.023*** (3.499) (1.056) Model *** (0.014) [0.127] 0.270*** (0.014) [0.132] 0.246*** (0.015) [0.137] Econsig *** (12.302) 0.199*** (15.763) 0.187*** (12.496) Panel B Expansion Uncond *** (0.005) [0.619] 0.077*** (0.004) [0.720] 0.065*** (0.004) [0.725] Model *** (0.006) [0.491] 0.083*** (0.006) [0.592] 0.064*** (0.006) [0.603] Econsig (1.337) (0.874) (-0.140) Model *** (0.007) [0.499] 0.108*** (0.007) [0.534] 0.069*** (0.007) [0.574] Econsig (1.477) 0.031*** (3.946) (0.502) Model *** (0.006) [0.567] 0.104*** (0.006) [0.611] 0.068*** (0.006) [0.651] Econsig ** (2.457) 0.027*** (3.890) (0.429)

35 Model *** (0.006) [0.492] 0.119*** (0.006) [0.528] 0.082*** (0.006) [0.567] Econsig *** (3.846) 0.042*** (5.588) 0.017*** (2.229) Model *** (0.014) [0.159] 0.221*** (0.015) [0.180] 0.144*** (0.015) [0.236] Econsig *** (9.312) 0.144*** (9.556) 0.079*** (5.158) Panel C Recession Uncond *** (0.010) [0.879] 0.044*** (0.007) [0.944] 0.028*** (0.007) [0.949] Model *** (0.017) [0.679] 0.063*** (0.013) [0.829] 0.043*** (0.014) [0.834] Econsig * (1.660) (1.261) (0.987) Model *** (0.019) [0.622] 0.036** (0.016) [0.757] (0.016) [0.780] Econsig (-0.095) (-0.464) * (-1.713) Model ** (0.025) [0.461] (0.026) [0.477] (0.027) [0.468] Econsig (-0.186) (-0.409) (0.429) Model (0.010) [0.895] (0.008) [0.929] *** (0.008) [0.937] Econsig *** (3.904) *** (-4.354) *** (-5.768) Model *** (0.042) [0.064] 0.296*** (0.044) [0.074] 0.387*** (0.044) [0.098] Econsig *** (5.651) 0.252*** (5.658) 0.359*** (8.099)

36 Table V Robustness check: data screening criteria This table reports the performances of the conditional and unconditional portfolios across alternative data screening criteria of the smallest qualified stocks. The first row is the performance of the unconditional portfolio. Models 1-5 are defined in table II. CER (Certainty equivalent return) is derived from expected utility. The CAPM, Fama-French and Carhart alpha are reported in column 2, 5 and 8 respectively. The associated standard deviations and adjusted R-squares are in parentheses and brackets separately. The Sharpe ratios of portfolio returns are in column 11. The row Econsig is the economic significance defined as the portfolio difference between each conditional portfolio and the unconditional portfolio. The associated t-stats are in parentheses. *, **, *** indicate statistical significance at 10%, 5% and 1%, respectively. CER CAPM Fama-French CARHART Sharpe Ratio Panel A. no exclusion Uncond *** (0.004) [0.718] 0.088*** (0.003) [0.850] 0.077*** (0.003) [0.855] Model *** (0.004) [0.705] 0.090*** (0.004) [0.784] 0.072*** (0.004) [0.797] Econsig (-0.523) (0.425) (-1.046) Model *** (0.004) [0.660] 0.095*** (0.004) [0.791] 0.083*** (0.004) [0.797] Econsig (1.376) (1.540) (1.298) Model *** (0.004) [0.645] 0.112*** (0.004) [0.691] 0.104*** (0.004) [0.694] Econsig (1.219) 0.024*** (4.779) 0.027*** (5.206) Model *** (0.005) [0.640] 0.097*** (0.004) [0.771] 0.087*** (0.004) [0.775] Econsig * (1.820) 0.009* (1.883) 0.010** (2.058) Model *** (0.010) [0.243] 0.169*** (0.010) [0.290] 0.130*** (0.010) [0.316] Econsig *** (7.198) 0.081*** (7.852) 0.053*** (5.044) Panel B 10% exclusion Uncond *** (0.004) [0.749] 0.066*** (0.002) [0.894] 0.059*** (0.003) [0.896] Model *** (0.004) [0.746] 0.064*** (0.003) [0.839] 0.052*** (0.003) [0.844] Econsig (1.302) (0.500) (-1.724) Model *** (0.004) [0.700] 0.069*** (0.003) [0.840] 0.059*** (0.003) [0.843] Econsig (0.724) (0.765) (0.075) Model *** (0.004) [0.674] 0.088*** (0.004) [0.735] 0.079*** (0.004) [0.738] Econsig (1.392) 0.022*** (4.716) 0.020*** (4.165)

37 Model *** (0.004) [0.694] 0.065*** (0.003) [0.834] 0.056*** (0.005) [0.826] Econsig (0.174) (-0.240) (-0.583) Model *** (0.010) [0.292] 0.142*** (0.010) [0.339] 0.111*** (0.010) [0.354] Econsig *** (6.357) 0.076*** (7.680) 0.052*** (5.093) Panel C 40% Uncond *** (0.003) [0.802] 0.055*** (0.002) [0.912] 0.046*** (0.002) [0.916] Model *** (0.004) [0.783] 0.053*** (0.003) [0.871] 0.044*** (0.003) [0.874] Econsig (0.832) (-0.552) (-0.552) Model *** (0.004) [0.720] 0.065*** (0.003) [0.827] 0.050*** (0.003) [0.837] Econsig (2.280) 0.010** (2.486) (0.994) Model *** (0.004) [0.764] 0.064*** (0.003) [0.834] 0.050*** (0.003) [0.841] Econsig (0.587) 0.009** (2.237) (0.994) Model *** (0.005) [0.722] 0.045*** (0.004) [0.817] 0.030*** (0.004) [0.824] Econsig (-1.075) (-2.295) (3.602) Model *** (0.011) [0.297] 0.110*** (0.010) [0.363] 0.072*** (0.011) [0.384] Econsig *** (5.945) 0.055*** (5.211) 0.026** (2.419) Panel C 60% Uncond *** (0.003) [0.836] 0.041*** (0.002) [0.923] 0.032*** (0.002) [0.926] Model *** (0.003) [0.802] 0.043*** (0.003) [0.861] 0.036*** (0.003) [0.863] Econsig (0.035) (0.571) (0.117) Model *** (0.004) [0.721] 0.054*** (0.003) [0.814] 0.035*** (0.003) [0.827] Econsig *** (3.346) 0.013*** (3.253) (0.751) Model *** (0.003) [0.808] 0.047*** (0.003) [0.864] 0.031*** (0.003) [0.874] Econsig (0.046) 0.006* (1.675) (-0.279) Model *** (0.004) [0.764] 0.030*** (0.004) [0.831] 0.017*** (0.004) [0.837] Econsig ** (2.000) *** (-2.694) *** (-3.599) Model *** (0.012) [0.294] 0.085*** (0.012) [0.338] 0.042*** (0.012) [0.357] Econsig *** (4.537) 0.044*** (3.612) (0.801)

38 Table VI Weights distribution across business cycle This table illustrates the weight distributions of the conditional and unconditional portfolio over the whole sample, and economic expansion and recession periods. The expansion and recession periods are defined by NBER. The first column is the weight of the state-independent strategy. Column 2-7 are the weight of the statedependent strategies. The definition of each model is in table II. The numbers reported are time series means outof-sample. In each panel, the first row is the cross-sectional average absolute value of monthly stock weights; the second and third rows are the monthly maximum and minimum weights; the fourth row is the monthly aggregated short-selling positions and the fifth row is the fraction of short-sold stocks; the last row is the portfolio turnover. Out-of- Sample Period Weight Uncond Model1 Model2 Model3 Model4 Model5 Whole periods ω i x Max ω i x Min ω i x Σ ω i I(ω i <0) ΣI(ω i <0)/Nt Σ ω i,t - ω i,t Booming ω i x Max ω i x Min ω i x Σ ω i I(ω i <0) ΣI(ω i <0)/Nt Σ ω i,t - ω i,t Recession ω i x Max ω i x Min ω i x Σ ω i I(ω i <0) ΣI(ω i <0)/Nt Σ ω i,t - ω i,t

39 Table VII Moments of the conditional portfolio return distributions This table contains the time series average of moments of the conditional and unconditional portfolio returns out-of--sample over the whole sample, and expansion and recession periods. The expansion and recession periods are defined by NBER. The first column is the moments of the unconditional portfolio returns. Column 2-7 are moments of conditional portfolio returns. Conditional strategies are defined in table II. The first to fourth moments are calculated and reported. The first and second moments are annualized. Model Period Moment Uncond Model1 Model2 Model3 Model4 Model5 Whole periods Ret Std Skew Kurtosis Expansion Ret Std Skew Kurtosis Recession Ret std Skew Kurtosis

40 Table VIII Impacts of short-selling constraint on economic significance This table reports the portfolio performances of the conditional and unconditional portfolio returns out-ofsample over the whole sample, and economic expansion and recession periods when sort-sale is not allowed. The expansion and recession periods are defined by NBER. The first row is the performance of the unconditional portfolios. Models 1-5 are conditional portfolios and defined in table II. CER (Certainty equivalent return) is derived from expected utility. The CAPM, Fama-French and Carhart alpha are reported in column 2, 5, and 8 respectively. The associated standard deviations and adjusted R-squares are in parentheses and brackets separately. The Sharpe ratios of portfolio returns of each model are in column 11. The row Econsig is the economic significance defined as performance difference between each state-dependent portfolio and the state-independent portfolio. The associated t-stats are in parentheses. *, **, *** indicate statistical significance at 10%, 5% and 1%, respectively. Model CER CAPM Fama-French CARHART SR Panel A Whole period Uncond *** (0.003) [0.892] 0.016*** (0.002) [0.969] 0.014*** (0.002) [0.969] Model *** (0.003) [0.909] 0.018*** (0.002) [0.963] 0.014*** (0.002) [0.964] Econsig (-0.815) (-0.912) (-0.043) Model *** (0.003) [0.893] 0.017*** (0.002) [0.967] 0.014*** (0.002) [0.967] Econsig (0.061) (0.456) (0.057) Model *** (0.003) [0.907] 0.021*** (0.002) [0.961] 0.017*** (0.002) [0.961] Econsig (-0.272) 0.005** (2.205) (1.323) Model *** (0.003) [0.889] 0.012*** (0.002) [0.957] 0.005*** (0.002) [0.959] Econsig * (-1.780) * (-1.707) *** (-3.841) Model *** (0.003) [0.851] 0.024*** (0.003) [0.922] 0.015*** (0.003) [0.925] Econsig (1.152) 0.008*** (2.744) (0.343) Panel B Expansion Uncond *** (0.003) [0.872] 0.018*** (0.002) [0.962] 0.018*** (0.002) [0.962] Model *** (0.003) [0.892] 0.019*** (0.002) [0.955] 0.018*** (0.002) [0.955] Econsig (1.570) (0.059) (0.027) Model *** (0.003) [0.885] 0.019*** (0.002) [0.961] 0.019*** (0.002) [0.961] Econsig (-0.514) (0.442) (0.416) Model *** (0.003) [0.891] 0.023*** (0.002) [0.954] 0.017*** (0.002) [0.955] Econsig (-0.523) 0.005** (2.076) (-0.404)

41 Model *** (0.003) [0.893] 0.018*** (0.002) [0.963] 0.015*** (0.002) [0.963] Econsig ** (2.094) (0.042) (-1.285) Model *** (0.003) [0.860] 0.019*** (0.002) [0.937] 0.014*** (0.002) [0.938] Econsig (-1.462) (0.379) (-1.439) Panel C Recession Uncond ** (0.008) [0.932] 0.016*** (0.004) [0.981] (0.004) [0.983] Model *** (0.007) [0.942] 0.028*** (0.005) [0.978] 0.015*** (0.005) [0.980] Econsig (0.565) 0.012* (1.926) 0.012** (1.971) Model *** (0.009) [0.920] 0.023*** (0.005) [0.977] (0.005) [0.980] Econsig (0.861) (1.111) (0.649) Model *** (0.007) [0.939] 0.020*** (0.005) [0.973] (0.005) [0.974] Econsig (0.279) (0.605) (0.906) Model *** (0.010) [0.892] (0.007) [0.955] ** (0.007) [0.960] Econsig (-0.077) (-1.238) ** (2.269) Model *** (0.013) [0.849] 0.058*** (0.010) [0.912] 0.026*** (0.010) [0.923] Econsig *** (4.003) 0.042*** (3.840) 0.023** (2.165)

42 Figure 1 Time series stock characteristics me is the cross-sectional average of market capitalization of equity defined as log of number of shares (in million) outstanding multiplied by stock price. btm is the cross-sectional average of book-to-market ratio defined as book value of equity divided by market value of equity in current month. Book value lags market values at least six months. mom is the cross-sectional average of momentum defined as cumulative monthly returns between t- 2 to t me btm mom Figure 2 Time series macroeconomic state variables INT is the one-month T-bill rate. D/P is the dividend-price ratio of S&P index. DEF is the default spread defined as the average yield rate difference between the BAA corporate bonds and the AAA bonds rated by MOODY. TERM is the term spread, defined as the average yield difference between government bonds with more than 10 years to maturity and T-bills mature in three months D/P TERM DEF INT