On the Pricing Role of Idiosyncratic Risk

Transcription

1 On the Pricing Role of Idiosyncratic Risk Jun (Tony) Ruan a,1,, Qian Sun b, Yexiao Xu c a Xiamen University, Institute for Financial and Accounting Studies, 422 Siming South St., Xiamen , China b Fudan University, School of Management, 670 Guoshun Road, Shanghai , China c The University of Texas at Dallas, School of Management, 800 W. Campbell Rd, Richardson, Texas 75080, the United States. Abstract Total idiosyncratic risk is commonly used to test the pricing role of idiosyncratic risk. However, theory suggests that only the undiversified idiosyncratic risk component should matter. When such a component is economically important in pricing but small in magnitude relative to total idiosyncratic risk, the conventional test has low statistical power to reject the null hypothesis that idiosyncratic risk does not matter due to the large non-priced noise component. We propose a simple regression-based method in the time-series test, which can substantially improve test power while sacrificing little in test size. Based on the new approach, we provide evidence that strongly supports a significantly positive relation between aggregate undiversified idiosyncratic risk and future market excess returns. In fact, at least 4% of the market excess returns can be explained by market risk and the priced component of aggregate total idiosyncratic risk. Extensive robustness study confirms that the pricing effect of aggregate undiversified idiosyncratic risk is pervasive and cannot be attributed to plausible sources of systematic risk, state variables, or microstructure noises. Keywords: Dual predictor, expected return, idiosyncratic risk, predictability. JEL Classification: G12; G14; G17 We are grateful to Jay Cao, Danling Jiang, Fangjian Fu, Qianqiu Liu, Burt Malkiel, Yuhang Xing, Chu Zhang, Harold Zhang, Yuanchen Chang, Daehwan Kim, Joshua Spizman, Laurent Calvet, Sandra Stankiewicz, and to the seminar participants at the Institute for Financial and Accounting Studies (IFAS) and the Wang Yanan Institute for Studies in Economics (WISE) of Xiamen University, Fudan University, Southwestern University of Finance and Economics, Massey University (New Zealand), the 2010 CICF conference, the Fifth International Conference on Asia- Pacific Financial Markets (Seoul, South Korea, Outstanding Paper Award), the 2010 National Taiwan University (NTU) International Conference on Finance, the 2011 Asian Finance Association Conference (Macau, China), and the Paris December 2013 Finance Meeting for helpful comments. We have substantially benefited from an anonymous referee for his/her insightful comments. The early version and a revised version of the paper were circulated under the titles of When Does Idiosyncratic Risk Really Matter? and A New Test for the Detection of the Pricing Role of Aggregate Undiversified Idiosyncratic Risk in the Predictive Regression, respectively. Corresponding author. Address: tony ruan@xmu.edu.cn; Tel: 86-(592) ; Fax: 86-(592) Ruan acknowledges the financial support of both the National Natural Science Foundation of China (NSFC project no , , and ) and the Fundamental Research Funds for the Central Universities in China (project no and ). Preprint submitted to Elsevier May 6, 2014

2 On the Pricing Role of Idiosyncratic Risk Abstract Total idiosyncratic risk is commonly used to test the pricing role of idiosyncratic risk. However, theory suggests that only the undiversified idiosyncratic risk component should matter. When such a component is economically important in pricing but small in magnitude relative to total idiosyncratic risk, the conventional test has low statistical power to reject the null hypothesis that idiosyncratic risk does not matter due to the large non-priced noise component. We propose a simple regression-based method in the time-series test, which can substantially improve test power while sacrificing little in test size. Based on the new approach, we provide evidence that strongly supports a significantly positive relation between aggregate undiversified idiosyncratic risk and future market excess returns. In fact, at least 4% of the market excess returns can be explained by market risk and the priced component of aggregate total idiosyncratic risk. Extensive robustness study confirms that the pricing effect of aggregate undiversified idiosyncratic risk is pervasive and cannot be attributed to plausible sources of systematic risk, state variables, or microstructure noises. Keywords: Dual predictor, expected return, idiosyncratic risk, predictability. JEL Classification: G12; G14; G17

3 1. Introduction Inspired by the upward trend of aggregate total idiosyncratic volatility documented by Campbell, Lettau, Malkiel and Xu (2001), many researchers have taken a close look at the pricing role of idiosyncratic risk. Nevertheless, whether and how idiosyncratic risk is priced remain thorny issues in finance. 1 Existing evidence is almost exclusively based on testing the relationship between stock returns and measures of total idiosyncratic risk. However, theory suggests that only the undiversified part of total idiosyncratic risk should be priced. When such a component is economically important in pricing but relatively small in magnitude, existing tests that use total idiosyncratic risk as a proxy for its priced component lack the statistical power needed to reject the null hypothesis that idiosyncratic risk does not matter. In this paper, we propose a simple regression-based method in the time-series test, which can substantially improve the power of standard tests relative to the conventional predictive regression, while sacrificing little in test size. Based on the proposed method, we provide economically significant and robust time-series evidence on the pricing effect of idiosyncratic risk. Idiosyncratic risk plays no role in the traditional CAPM framework of Sharpe, Lintner, and Black because efficient diversification is achieved in a perfect capital market. Nevertheless, the poor performance of many asset pricing models has led researchers to re-examine the assumption of a perfect capital market. In particular, Merton (1987) shows that investors require a positive risk premium for undiversified idiosyncratic risk when they cannot hold all available securities. When different investors generally hold different subsets of the available securities, the market portfolio is inefficient. Consequently, market risk premium should contain a component that compensates for market-wide aggregate undiversified idiosyncratic risk. 2 Hence, even if investors are mean-variance efficient for their given subsets of all available securities, the theory predicts that idiosyncratic risk should be priced to the extent that it is not fully diversified away. In other words, it is the undiversified (rather than total) idiosyncratic risk that matters. Although Goetzmann and Kumar (2008) find that a vast majority of investors directly hold only a few stocks, Calvet, Campbell and Sodini (2007) show that a large portion of Swedish households do not suffer very significant efficiency loss in diversification when investors cash holdings and indirect 1 For the debate on time-series tests, see, for example, Goyal and Santa-Clara (2003), Bali, Cakici, Yan and Zhang (2005), and Wei and Zhang (2005). For the debate on cross-sectional tests, see, for example, Malkiel and Xu (2002), Ang, Hodrick, Xing and Zhang (2006), Bali and Cakici (2008), Doran, Jiang and Peterson (2008), Huang, Liu, Rhee and Zhang (2010), Fu (2009), Cao and Xu (2009), and Han and Lesmond (2011). 2 Malkiel and Xu (2002) argue a similar point from the supply side of the market. In the presence of market imperfections, the effective supply of securities that investors are able to use to price individual securities can differ greatly from the total published supply that researchers actually observe. As a result, the market portfolio that investors use to price securities is inefficient relative to the would-be market portfolio if investors were able to trade in all the securities. 1

4 stock holdings through mutual funds are taken into account. Calvet, Campbell and Sodini s (2007) results suggest that the undiversified idiosyncratic risk component should be rather small relative to total idiosyncratic risk. 3 Nevertheless, the question remains whether this small component has economically important pricing implications. If investors are reasonably, but not fully, diversified, aggregate measures of total idiosyncratic risk contain not only an undiversified (i.e., a signal) component but also a potentially very large diversified (i.e., noise) component. More importantly, the relative importance of the signal component may vary over time. When a predictive regression of future market excess returns on a measure of aggregate total idiosyncratic risk is used to test the null hypothesis that idiosyncratic risk does not matter, the standard t test generally maintains correct size. However, under the alternative hypothesis that only a small component of total idiosyncratic risk significantly predicts future market excess returns, the power of such a test can greatly deteriorate, making it difficult to reject the null hypothesis. This is caused by the well-known errors-in-variables problem in hypothesis testing. In a time-series test, Goyal and Santa-Clara (2003) first document that equal-weighted aggregate total risk measures (used as proxies for aggregate total idiosyncratic risk) positively predict valueweighted market excess returns, consistent with Merton (1987). However, their main results are subsequently questioned by Bali, Cakici, Yan and Zhang (2005) and Wei and Zhang (2005), all of whom find little evidence for the ability of idiosyncratic risk to predict future market excess returns when using an extended sample period, alternative aggregate measures of total risk (or total idiosyncratic risk), or alternative choices of market indexes to re-examine the relationship. While one can interpret the additional results as evidence against the pricing role of idiosyncratic risk, such a conclusion is prone to type II error unless the test power is adequately assessed for some economically meaningful alternative hypotheses. In other words, the ambiguous evidence based on the same type of regressions can be an indication of low test power. For example, using Monte Carlo experiments, we find that the power of the conventional test is less than 50% for a very noisy predictor, even when its signal component by itself explains 15% of simulated future excess returns. To deal with the potential low-power issue, we propose a unique method that exploits two highly, but not perfectly, correlated measures of aggregate total idiosyncratic risk in one predictive regression. We call such a regression a dual-predictor regression. The measure with a higher (lower) signal-to-noise ratio (defined later as the variation of the undiversified idiosyncratic risk component, also referred to as the signal component, to that of the diversified idiosyncratic risk component, 3 For example, one can reduce total idiosyncratic risk by as much as 60% while only holding an equal-weighted three-stock portfolio. 2

5 also referred to as the noise component) is treated as the signal (noise) predictor. When the signal component is quite small relative to the noise component, the high correlation between the two measures is mostly attributed to the correlation between the two noise components. We show that if the signal component matters, the noise components in the two highly correlated measures tend to cancel out each other in the dual-predictor regression, generally leading to reduced bias in the coefficient estimate of the signal predictor. As a result, the dual-predictor regression can have considerably higher explanatory power than a conventional predictive regression. In addition, we offer guidelines for assessing the economic impact of undiversified idiosyncratic risk on risk premiums. In our empirical analysis, we primarily rely on different weighting schemes to obtain highly correlated aggregate total idiosyncratic risk measures with different signal-to-noise ratios. This is because Merton (1987) shows that a stock s undiversified idiosyncratic risk is inversely related to the stock s proportional investor base (i.e., the fraction of the investors holding the stock), for which the stock s market capitalization can be used as a proxy. Hence, an equal-weighted aggregate measure, for example, should have a higher signal-to-noise ratio than a value-weighted aggregate measure. This reasoning may explain why researchers are able to document a significantly positive relationship between an equal-weighted measure and future market excess returns, and yet fail to find such a relationship when a value-weighted measure is used instead. Using equal- and value-weighted aggregate total idiosyncratic risk measures as a signal predictor and a noise predictor, respectively, in the dual-predictor regression, we strongly reject the null hypothesis that idiosyncratic risk does not matter, regardless of whether market risk and other well-known predictors are controlled for. Consistent with theory and our analysis, we find a significantly positive coefficient for the signal predictor, implying a positive relationship between aggregate undiversified idiosyncratic risk and future market excess returns. 4 Our results suggest that more than 4% of market risk premium can be attributed to every one percent of aggregate total idiosyncratic variance that is not diversified away. Moreover, we find that the dual-predictor regression has substantially improved explanatory power. The dual-predictor regression is designed to alleviate the severe errors-in-variables problem under the alternative hypothesis. However, a regression with highly, albeit not perfectly, correlated 4 Jiang and Lee (2006) use innovations of aggregate total idiosyncratic volatility in a predictive regression to document a significantly positive relation. When the noise component is more persistent than the signal component, innovations should have higher signal-to-noise ratios so that their results can be explained by noise reduction. In a similar context, Guo and Savickas (2006) find that market volatility and total idiosyncratic volatility jointly predict future market excess returns at a quarterly frequency, but neither can do so on a stand-alone basis. This is also consistent with our noise reduction story, because the two volatility measures may contain highly correlated noise components. 3

6 regressors naturally raises a concern about multicollinearity in finite samples. In the absence of the errors-in-variables problem, multicollinearity neither biases the estimation of conditional means nor reduces the explanatory power of the regression. Nevertheless, multicollinearity increases the standard errors of the estimates. In our new tests, the multicollinearity issue is less of a concern for two reasons. First, under the null hypothesis, the noisy predictors can be treated as perfectly measured non-priced noises. Multicollinearity in this case is just like that arising in the absence of the errors-in-variables problem. Therefore, multicollinearity does not affect test size. Second, under the alternative hypothesis, multicollinearity between the noise components introduces a noise reduction mechanism so that the mean coefficient estimate of the signal predictor generally increases, even though the standard error of the coefficient estimate also increase and the common signal component could limit the extent of noise reduction. Nevertheless, our Monte Carlo simulations suggest that as long as the two noisy predictors have quite different signal-tonoise ratios or the two noise components are very highly correlated, the mean coefficient estimate proportionally increases more than the standard error of the estimate, so that standard tests can have substantially improved power. In a nutshell, one may think that multicollinearity combines the two noisy predictors into one predictor with a much higher signal-to-noise ratio. Since total idiosyncratic risk is typically estimated from residual returns with respect to an asset pricing model (e.g., the Carhart four-factor model), it might still contain unknown systematic risk factors. To mitigate this concern, we also estimate total idiosyncratic risk from residual returns relative to a six-factor APT model, which leaves no significant comovement in residual returns. Because variance measures can be accurately estimated from high frequency returns, our main analysis relies on total idiosyncratic variances estimated from daily residual returns. However, it is well recognized that realized daily returns can be affected by market microstructure noises such as bid-ask spreads and zero returns. As a result, bid-ask spreads may inflate, whereas zero returns may deflate, the estimate of total idiosyncratic variance. 5 Even though it is unclear whether and how microstructure noises in the estimated total idiosyncratic variance might affect our results, we check the robustness of our results with respect to the microstructure noises issue by using total idiosyncratic variance estimates constructed from portfolio daily residual returns or monthly residual returns. In addition, we perform extensive robustness analysis against other plausible concerns. For example, we check whether our main results are robust to a wide range of control variables, several 5 See Han and Lesmond (2011) for a discussion about how bid-ask spreads and zero returns can affect the estimate of total idiosyncratic risk. 4

7 subsample periods, various subsets of stocks, alternative weighting schemes, alternative equity index returns, and the quantile regression method. In all these cases, we continue rejecting the null hypothesis strongly. In particular, we find that more than 4% of the variation in future monthly CRSP value-weighted index excess returns can be explained by the dual predictor alone. Surprisingly, the dual predictor generally outperforms several measures of market risk. Therefore, the pricing effect of aggregate undiversified idiosyncratic risk cannot be ignored. Nevertheless, we find that different measures of market risk often improve the performance of our dual-predictor regression, suggesting that both market risk and aggregate undiversified idiosyncratic risk contribute to the predictive power of the regression, in line with Merton (1987). Even though we focus on the debate on time-series tests in this study, our approach helps to reconcile the mixed evidence from cross-sectional tests. Malkiel and Xu (2002) find a positive crosssectional relationship between portfolio idiosyncratic risks and portfolio expected returns. Fu (2009) documents a positive cross-sectional relationship based on conditional idiosyncratic volatilities. To the contrary, using quintile portfolios sorted by idiosyncratic volatility, Ang, Hodrick, Xing and Zhang (2006) report a negative cross-sectional relationship between past realized idiosyncratic volatilities and current stock returns. 6 Like the time-series tests, these cross-sectional tests typically rely on some sort of total idiosyncratic risk measure as well. If the diversified idiosyncratic risk component is dominant in total idiosyncratic risk, it is difficult to establish a positive link between the priced component of total idiosyncratic risk and average returns in a cross-sectional study. Cao and Xu (2009) take a decomposition approach and find additional cross-sectional evidence on the pricing role of idiosyncratic risk, consistent with our results. We contribute to the literature on idiosyncratic risk in several aspects. First, we advocate the importance of separating the undiversified idiosyncratic risk component from the diversifiable idiosyncratic risk component in testing the pricing effect of idiosyncratic risk. Our new method provides economically important and statistically robust evidence in support of the pricing effect of undiversified idiosyncratic risk, while offering a much simpler explanation for the extant ambiguous or even conflicting results on the same issue. 7 Our new approach adds to existing remedies to the errors-in-variables problem, which include the instrumental variable approach and the higher-order moments approach (see Erickson and Whited, 2002 and Erickson, Jiang and Whited, 2013). The 6 Bali and Cakici (2008) have questioned robustness of the negative cross-sectional relationship with alternative methodologies, whereas Huang, Liu, Rhee and Zhang (2010) have claimed that the results can be explained away by short-term return reversals. Moreover, Han and Lesmond (2011) argue that the microstructure noise component of estimated total idiosyncratic risk can explain away the negative cross-sectional relationship. 7 Although our method is developed for testing the effect of undiversified idiosyncratic risk, it may also be applied to other tests in empirical research, where reduction of the effect of large measurement noises is critical in testing a theory. 5

8 former approach requires instrumental variables. The latter approach requires nonnormality of the mismeasured regressors and nonzero true coefficients. Neither of the two approaches is applicable to the question we ask, because an appropriate instrument for aggregate undiversified idiosyncratic risk is lacking and because whether idiosyncratic risk matters is still in debate. Second, our time-series evidence for the pricing role of idiosyncratic risk complements the early evidence for a positive relationship between market risk premium and market volatility, as documented by French, Schwert and Stambaugh (1987). 8 In particular, we show that inclusion of the priced component of idiosyncratic risk often improves the performance of market risk measure and thus the performance of a conditional asset pricing model. Consequently, the priced component of idiosyncratic risk should have implications for dynamic asset allocations. Third, broadly speaking, our results on idiosyncratic risk support implications of the theories in Mayers (1976) and Barberis and Huang (2001). In a similar vein, our results complement other evidence for the pricing role of idiosyncratic risk. For instance, Green and Rydqvist (1997) show that the prices of Swedish lottery bonds are affected by idiosyncratic risk. Storesletten, Telmer and Yaron (2001) find that idiosyncratic risk in labor income helps to explain equity returns. In addition, Moskowitz and Vissing-Jørgensen (2002) show that private equity is larger than public equity, so that investors in private equity need to care about idiosyncratic risk. The rest of the paper proceeds as follows. In Section 2, we first revisit the theoretical argument for the pricing role of undiversified idiosyncratic risk and draw important implications from the theory. Subsequently, we develop the dual-predictor regression and use Monte Carlo simulations to compare the size and power of standard tests between the conventional predictive regression and the dual-predictor regression. In Section 3, we describe the data and variable construction. In Section 4, we detail empirical analysis and discuss our main results and extensive robustness checks. In Section 5, we offer concluding remarks. 2. Theory and Methodology There are several channels for idiosyncratic risk to affect stock returns. Apart from possible irrational explanations, several rational models make idiosyncratic risk relevant. 9 In this section, we focus on the Merton (1987) model to motivate our new test methodology. 8 For other related research, see Merton (1980), Kandel and Stambaugh (1996), Campbell (1987), Turner, Startz and Nelson (1989), Chan, Karolyi and Stulz (1992), Glosten, Jangannathan and Runkle (1993), and Whitelaw (1994). 9 In addition to Merton (1987) and Malkiel and Xu (2002), the arbitrage pricing theory (APT) of Ross (1976) can also motivate our test for the pricing effect of undiversified idiosyncratic risk. The APT model requires a sufficiently large number of risky assets to eliminate idiosyncratic risk in a well-diversified portfolio. When this condition is violated in practice, expected returns will be affected by undiversified idiosyncratic risk (see Eq. (10) for a simple example in Ross, 1976). 6

9 2.1. Implications of the Merton (1987) Model Under the assumption of incomplete information, each investor tends to hold a subset of all available securities. In equilibrium, Merton derives E(R k ) R f = δvar( R m )β k + λ k, (1) where E(R k ) is the expected return of stock k; R f is the risk-free rate; δ is a risk aversion parameter; R m is the market return; β k is the conventional beta measure for stock k; and λ k is stock k s aggregate shadow price in terms of risk premium. The shadow price, λ k, (see Merton, 1987, Eq. (23) on p. 493) can be expressed as λ k = δw k (1/q k 1)σ 2 k, (2) where q k is the proportional investor base of stock k (or the fraction of the investors holding stock k); w k is stock k s market value weight in the market portfolio; and σk 2 is the total idiosyncratic variance of stock k. From Eqs. (1) and (2), it is clear that investors in stock k are rewarded with a positive risk premium, λ k, for their exposure to the stock s idiosyncratic risk, when q k is less than unity. Intuitively, such a risk premium rewards these investors for the inability of the rest of the investors to hold the stock. Hence, the term w k (1/q k 1)σk 2 in the shadow price can be interpreted as undiversified idiosyncratic risk. Although in Merton s (1987) theory a stock s investor base is exogenously determined, market imperfections such as information costs, institutional restrictions, liquidity concerns, and costly short sales make plausible the assumption that investors portfolios are tilted towards large stocks. Moreover, most firms set their total number of shares outstanding in such a way that their share prices fall into an affordable trading range for investors to hold a minimum number of shares. Consequently, smaller stocks should have fewer investors than larger stocks do, which implies that a stock s investor base is very likely to be positively related to the stock s market value. If a stock s proportional investor base is equal to its market value weight (for example, q s = w s = 0.10 for a small stock and q l = w l = 0.90 for a large stock in a simple two-stock world), then λ s = δ(0.9σs) 2 and λ l = δ(0.1σl 2). In this case, even if σ2 s = σl 2, the risk premium due to undiversified idiosyncratic risk is eight times greater for the small stock than for the large stock (i.e., λ s = 9λ l ). In reality, a stock s proportional investor base is far greater than the stock s market value weight (i.e., q l w l and q s w s ), and a large stock s market value weight and proportional investor base can be far greater than a small stock s market value weight and proportional investor base, respectively (i.e., w l w s and q l q s ). If w l /w s q l /q s, it is easy to show that w s (1/q s 1) 7

10 w l (1/q l 1), and thus λ s λ l even if σl 2 = σs. 2 If w l /w s > q l /q s, λ s can still be greater than λ l because small stocks typically have much higher total idiosyncratic volatilities than large stocks do (see Malkiel and Xu, 2003 and Table 4 in Fu, 2009). For example, a typical large stock can be 100 times as large in market capitalization as a typical small stock (i.e., w l = 100w s ). If the large (small) stock s investor base is 0.90 (0.10) (i.e., q l = 9q s = 0.90) and the large stock is only 50% as volatile as the small stock, Eq. (2) implies that λ s = 3.25λ l. With these empirical regularities, we can infer from Eq. (2) that the risk premium for undiversified idiosyncratic risk is very likely to be much larger for small stocks than for large stocks. The pricing effect of undiversified idiosyncratic risk on a stock s returns will be reflected in market returns. By aggregating both sides of Eq. (1) across all stocks with market value weights, we have the following result: E(R m ) R f = δvar( R m ) + λ m, (3) where E(R m ) is the expected market return; and λ m = n k=1 w kλ k is the market-wide shadow price of aggregate undiversified idiosyncratic risk (see Merton, 1987, Eq. (24) on p. 494), which is proportional to the value-weighted aggregate undiversified idiosyncratic risk of individual stocks. Eq. (3) says that the market risk premium rewards investors not only for market risk but also for aggregate undiversified idiosyncratic risk. Therefore, it is the aggregate undiversified idiosyncratic risk, rather than aggregate total idiosyncratic risk, that matters at the market level. This important implication has been largely overlooked in time-series tests (see also Malkiel and Xu, 2002), but it forms the theoretical basis for our study. Although the relation in Eq. (3) is static in nature, it may be extended to a dynamic setting in the spirit of a conditional asset pricing model. In practice, most of the idiosyncratic risks may have been diversified away, but the undiversified idiosyncratic risk components do not vanish quickly, due to the nonlinearity of the diversification effect, even when investors hold a large number of stocks. As a result, the risk premium attributed to aggregate undiversified idiosyncratic risk can still be sizable relative to the total market risk premium. For example, if 500 stocks have equal weights of and each stock is held by 10% of all investors, the risk premium for aggregate undiversified idiosyncratic risk is λ m = δ(0.018 σ 2 ), where σ 2 is the average total idiosyncratic risk of all stocks. In this case, even though only about 2% of a typical stock s total idiosyncratic risk matters, it contributes to 6.72% of the total market risk premium if Var( R m ) = 1 4 σ2. When the average investor base is reduced by a half, the reward for aggregate undiversified idiosyncratic risk can now account for 13.19% of the total market risk premium! The lack of detailed information on all investors stock holdings makes it difficult to construct a 8

11 clean measure of aggregate undiversified idiosyncratic risk. In practice, aggregate total idiosyncratic risk is commonly employed as a proxy for its undiversified component in testing the economic significance of the undiversified component. This is reasonable if the undiversified idiosyncratic risk component is proportional to total idiosyncratic risk for all stocks and the proportion is constant over time. However, these requirements are unlikely to be satisfied in reality. Therefore, we argue that the diversified idiosyncratic risk component (i.e., the large noise component) in the conventional measures is the culprit for the less robust relationship between aggregate total idiosyncratic risk and future market excess returns Methodology We propose a novel approach to test the relationship between aggregate undiversified idiosyncratic risk and market risk premium. The new approach exploits the high correlation between the large non-priced noise components in a so-called dual-predictor regression, which uses two different noisy measures of aggregate undiversified idiosyncratic risk as a dual predictor. Let r m = R m R f be the market excess return. Motivated by Eq. (3), we postulate that the conditional expected market excess return can be determined as follows: E t (r m,t+1 ) = κv t + κ zz t, (4) where v t is the aggregate undiversified idiosyncratic variance (referred to as the signal thereafter) at time t with mean µ v and variance σv; 2 Z t is a vector of other known predictors at time t; and κ z is a vector of coefficients for Z t. Theory predicts κ > 0, and thus testing the null hypothesis that idiosyncratic risk does not matter is equivalent to testing κ = 0. In a multivariate predictive regression of r m,t+1 on v t and Z t, the (partial) regression coefficient estimate ˆκ reflects the marginal effect of v t on E t (r m,t+1 ). For ease of discussion, but without loss of generality, we assume that v t is orthogonal to Z t. With this assumption, we develop our method based on the following reduced empirical model of Eq. (4): r m,t+1 = κv t + ε t+1, (5) where ε t+1 is the unexpected market return at time t+1 with mean zero and variance σ 2 ε. We define θ = σ v /σ ε as a relative signal measure, which should be rather small. 10 As a result, the population coefficient of determination, R 2 = κ2 θ 2 1+κ 2 θ 2, may still be low under the alternative hypothesis of κ Even though v t may significantly contribute to the conditional market risk premium, its volatility should be quite small relative to that of the unexpected market return. 9

12 In empirical studies, we assume that researchers employ the following two measures of aggregate total idiosyncratic risk as proxies for the signal v t : x 1,t = v t + s 1,t and (6a) x 2,t = av t + s 2,t, (6b) where s 1,t and s 2,t are the aggregate diversified idiosyncratic risk components (referred to as the noises) with variances σ1 2 and σ2 2, respectively; both s 1,t and s 2,t are uncorrelated with the signal v t and the white noise ε t at all time t, 11 but they have a correlation coefficient of ρ because x 1,t and x 2,t are generally constructed from idiosyncratic risks of the same set of stocks; and a is a scaler with 0 a 1, which allows us to study general cases for the signal and noise structure in x 2,t. We measure the relative importance of the signal component in x 1 and x 2 by defining the signal-to-noise ratios of x 1 and x 2 as θ 1 = σ v /σ 1 and aθ 2 = aσ v /σ 2, respectively. Because the noise component generally dominates the signal component, we assume that both x 1 and x 2 are very noisy and, without loss of generality, that x 1 is less noisy than x 2 (i.e., 0 aθ 2 < θ 1 1). For that reason, we refer to x 1 as the signal predictor and x 2 as the noise predictor in the following sections. The conventional approach to testing whether idiosyncratic risk matters over time is based on a predictive regression of r m,t+1 on either x 1,t or x 2,t (together with other predictors). We call such an approach a univariate-predictor regression test. Under the null hypothesis of κ = 0, the true coefficient of x 1 or x 2 is simply zero. If ε t N(0, σε), 2 the standard t test in the univariate-predictor regression has correct size under the null hypothesis. Under the alternative hypothesis of κ 0 (e.g., κ > 0), however, such a test can have very low power due to the large downward bias in the coefficient estimate that results from the severe errors-in-variables problem. To improve test power, we propose to regress r m,t+1 on both x 1,t and x 2,t (together with other predictors), provided that x 1 and x 2 are not perfectly correlated. Under the null hypothesis, the population coefficients of both x 1 and x 2 in the dual-predictor regression are zero. Therefore, if ε t N(0, σε), 2 the t statistic used to test whether the coefficient of x 1 or x 2 is zero and the F statistic used to test whether a linear restriction on the two coefficients is zero still have correct sizes. Under economically motivated alternative hypotheses, these standard tests in the dual-predictor regression can generally have large improved power relative to the univariate-predictor regression tests. Intuitively, the improved power results from noise reduction because highly correlated noise 11 We note that v t and s 1,t (or s 2,t) may be correlated. However, it can be shown that our basic results do not change. Detailed discussions are available upon request. 10

13 components tend to cancel out each other, increasing the relative importance of the signal vs. the noise in the regression Large-Sample-Limit Analysis Under the Alternative Hypothesis Before proceeding to analyze the power of the new test, we first study the large sample limits of the estimators in the univariate- and dual-predictor regressions under the alternative hypothesis of κ 0. The large-sample-limit analysis allows us to directly see how the asymptotic downward bias in the population coefficient of x 1 in the univariate-predictor regression may be alleviated in the dual-predictor regression. In a univariate-predictor regression of r m,t+1 on x 1,t or x 2,t, it is straightforward to show that the large-sample limit of the slope coefficient estimator is and that the population R 2 is R1,u 2 = θ θ1 2 β 1,u = θ θ1 2 κ or β 2,u = aθ a 2 θ2 2 κ, κ 2 θ κ 2 θ 2 or R2,u 2 = a2 θ a 2 θ2 2 κ 2 θ κ 2 θ 2. It is clear that measurement noise in x 1 (or x 2 ) asymptotically biases both the slope coefficient and the R 2 towards zero as the signal-to-noise ratio, θ 1 (or aθ 2 ), decreases to zero. In the extreme, for example, if x 2 were pure noise (i.e., a = 0), we would have β 2,u = 0 and R 2 2,u = 0. However, if x 1 were perfectly measured as v (i.e., θ 1 ), its population coefficient and the population R 2 of the regression would be κ (the true coefficient) and κ2 θ 2 1+κ 2 θ 2 (the true explanatory power), respectively. We assess the precision of the conditional forecast with β 1,u by computing the absolute bias of the forecast conditioning on the signal v as below: E(ˆr ˆr u v) = κv E(β 1,u x 1 v) = θ1 2 κv, (7) where E( ) is the expectation taken over s 1. Apparently, the absolute bias solely depends on θ 1 and reduces as θ 1 increases. When x 1 has a large noise component, the absolute bias can be as large as the absolute value of the risk premium attributed to the signal. In contrast, in our dual-predictor regression, we regress r m,t+1 on both x 1,t and x 2,t and the large-sample limits of the slope coefficient estimators for x 1 and x 2 are (θ 1 aρθ 2 )θ 1 β 1,d = (1 ρ 2 ) + (θ 1 aθ 2 ) 2 κ and (8a) + 2aθ 1 θ 2 (1 ρ) (ρθ 1 aθ 2 )θ 2 β 2,d = (1 ρ 2 ) + (θ 1 aθ 2 ) 2 κ, (8b) + 2aθ 1 θ 2 (1 ρ) 11

14 respectively. To gain intuition of the role of ρ in bias reduction, we first examine a special case, in which x 2 is pure noise (i.e., a = 0). In this case, we have β 1,d = θ 2 1 (1 ρ 2 ) + θ 2 1 ρθ 1 θ 2 κ and β 2,d = (1 ρ 2 ) + θ1 2 κ. Obviously, β 1,d should be positive. Even though x 2 in this case has no explanatory power when used alone, its coefficient, β 2,d, should be negative in the limit for positive values of ρ when x 2 is used alongside with x 1 in the dual-predictor regression. This is because x 2 automatically serves as a noise remover in the dual-predictor regression when the correlation between x 2 and the noise component in x 1 is high. Compared to the asymptotic bias in β 1,u, the asymptotic bias in β 1,d will be reduced towards zero as ρ increases to 1. For the general cases, in which 0 < a 1, β 1,d should still be positive because θ 1 > aθ 2. However, β 2,d should be negative only when ρ > aθ 2 /θ 1 (i.e., the positive correlation between the two noise components is sufficiently high). 12 Even though the asymptotic bias in β 1,d is largely reduced relative to that in β 1,u, β 1,d is still generally biased downward from κ. In the next subsection, we compare the asymptotic biases in β 1,u and β 1,d, and discuss the cases, in which an improved lower bound for κ may be constructed from β 1,d and β 2,d. Conditioning on the signal v, the absolute conditional forecast bias of the dual-predictor regression is E(ˆr ˆr d v) = κv E(β 1,d x 1 + β 2,d x 2 v) = 1 ρ 2 (1 ρ 2 ) + (θ 1 aθ 2 ) 2 κv, (9) + 2aθ 1 θ 2 (1 ρ) where E( ) is the expectation taken over both s 1 and s 2. Holding θ 1 and aθ 2 constant, the absolute forecast bias declines to zero as ρ approaches to 1. The difference in the absolute conditional forecast bias between the two types of regressions is E(ˆr ˆr u v) E(ˆr ˆr d v) = (ρθ 1 aθ 2 ) 2 (1 + θ1 2)[(1 ρ2 )(1 + θ1 2) + (ρθ 1 aθ 2 ) 2 κv 0. (10) ] Since the coefficient of κv in Eq. (10) is non-negative, the dual-predictor regression in general reduces the absolute conditional forecast bias. The noise cancellation from the dual predictor naturally improves the explanatory power of the dual-predictor regression. This can be seen from the large-sample limit of the dual-predictor 12 Since x 2 contains a common signal component in the general cases, the noise reduction is tempered by the common signal component if the positive correlation between the two noise components is not sufficiently high. In our empirical study, the positive correlation between the noise components in the dual predictor, as approximated by the positive correlation between the two measures in the dual predictor, is quite high. Moreover, it is possible for β 2,d to be positive if the correlation between the two noise components is negative. We further discuss different cases in Section

15 regression s R 2 : Rd 2 = (θ 1 aθ 2 ) 2 + 2aθ 1 θ 2 (1 ρ) (1 ρ 2 ) + (θ 1 aθ 2 ) 2 + 2aθ 1 θ 2 (1 ρ) κ 2 θ κ 2 θ 2. (11) When x 2 is pure noise (i.e., a = 0), Eq. (11) reduces to R 2 d = θ 2 1 (1 ρ 2 )+θ 2 1 κ 2 θ 2 1+κ 2 θ 2. In contrast to R 2 1,u, the gain in the explanatory power can be substantial. In fact, when ρ approaches to 1, R2 d increases to the population explanatory power of the true model, R 2 = κ2 θ 2 1+κ 2 θ 2, just as if only the signal v were used in the regression. In general, it can be shown that Rd 2 increases (decreases) with ρ if and only if aθ 2 /θ 1 < ρ 1 ( 1 ρ aθ 2 /θ 1 ), holding both θ 1 and aθ 2 constant. 13 Moreover, we can also show that R 2 d increases as aθ 2/θ 1 decreases, holding θ 1 and ρ constant. 14 These results suggest that a large increase in the explanatory power of the dual-predictor regression requires a high correlation between the noise components in the two noisy predictors or a large difference in the signal-to-noise ratios of the two noisy predictors. In comparison to the explanatory power of a univariate-predictor regression, Eq. (11) can be rewritten as [ Rd 2 = (ρθ 1 aθ 2 ) 2 ] θ1 2 (1 ρ 2 )(1 + θ1 2) + (ρθ 1 aθ 2 ) 2 R1,u 2 0. (12) Since the term in the square brackets in Eq. (12) is close to (1 + 1/θ1 2 ) ( 1) if either ρ is close to unity or (ρθ 1 aθ 2 ) 2 (1 ρ 2 )(1 + θ1 2 ), the explanatory power of the dual-predictor regression can be many times larger than that of the univariate-predictor regression. Although we cannot directly measure ρ between the two noise components, we may be able to use the correlation coefficient between x 1 and x 2 as a proxy for ρ, which is corr(x 1, x 2 ) = ρ + aθ 1 θ 2 (1 + θ 2 1 ) (1 + a 2 θ 2 2 ). (13) When x 1 and x 2 are very noisy (i.e., both θ 1 and aθ 2 are close to zero), the correlation coefficient between x 1 and x 2 should be close to ρ. 13 After taking the derivative of R 2 d with respect to ρ and collecting terms, we have R 2 d ρ = ρ(θ 1 aθ 2) 2 aθ 1θ 2(1 ρ) 2 [(1 ρ 2 ) + (θ 1 aθ 2) 2 + 2aθ 1θ 2(1 ρ)] 2 R2. When ρ = 1, the first derivative is positive. When ρ 1, the first derivative is positive if and only if ρ (1 ρ) 2 > aθ 2/θ 1 (1 aθ 2/θ 1) 2. The function, z = w, is increasing in w if and only if 1 < w < 1. Since 0 aθ (1 w) 2/θ 2 1 < 1, the above inequality is satisfied if and only if aθ 2/θ 1 < ρ < 1. Taken together, Rd 2 increases with ρ if and only if aθ 2/θ 1 < ρ 1. Conversely, Rd 2 decreases with ρ if and only if 1 ρ aθ 2/θ It can be shown that the conditional forecast improvement of the dual-predictor regression (i.e., Eq. (10)) depends on ρ and aθ 2/θ 1 in the same manner as does R 2 d. 13

16 2.4. The Lower Bound For the True Parameter κ We are interested in estimating κ, the risk premium per unit of the signal. Due to the dominant noise components, we show in the previous section that 0 β 2,u < β 1,u κ in the univariatepredictor regression. Using the dual-predictor regression, we offer two approaches for constructing improved lower bounds for κ in large samples. For clarity, β L,κ is used to denote the improved lower bound. In the first approach, we make use of the coefficients from the dual-predictor regression to obtain β L,κ. We note that 0 < β 1,d + aβ 2,d = θ 2 1 2aρθ 1θ 2 +a 2 θ 2 2 (1 ρ 2 )+(θ 2 1 2aρθ 1θ 2 +a 2 θ 2 2 ) κ κ. Since β 1,d, β 2,d, and β 1,d + aβ 2,d crucially depend on ρ for given θ 1 and aθ 2, we discuss below the conditions for ρ, under which improved lower bounds for κ, relative to β 1,u, can be obtained. 15 Specifically, under the assumption of θ 1 > aθ 2, we solve for different intervals of ρ to obtain β L,κ in different cases by comparing the differences among β 1,u, β 1,d, and β 1,d + aβ 2,d. Case I: When 1 ρ < aθ 1 θ 2, we set β L,κ = β 1,d + aβ 2,d. In this case, ρ is negative, and we can show β 2,d > 0 and 0 < β 1,u < β 1,d β 1,d + aβ 2,d κ. Case II: When aθ 1 θ 2 ρ aθ 2 /θ 1, we again set β L,κ = β 1,d + aβ 2,d. In this case, ρ may vary from a slightly negative value to a relatively small but positive value, and we can show β 2,d 0 and 0 < β 1,d β 1,u β 1,d + aβ 2,d κ. Case III: When aθ 2 /θ 1 < ρ < ( 1 2 aθ 1θ 2 + 1) 2 aθ 2 (θ 1 aθ 2 ) 1 2 aθ 1θ 2, we set β L,κ = β 1,d. In this case, ρ is positive but strictly less than 1, and we can show β 2,d < 0 and 0 < β 1,u < β 1,d + aβ 2,d β 1,d < κ. Case IV: When ( 1 2 aθ 1θ 2 + 1) 2 aθ 2 (θ 1 aθ 2 ) 1 2 aθ 1θ 2 ρ 1, we set β L,κ = β 1,d +aβ 2,d. In this case, ρ is still positive and may be very close to, or equal to, 1; we can show β 2,d < 0 and 0 < β 1,u < β 1,d + aβ 2,d κ β 1,d. Note that β L,κ = β 1,d in Case III, whereas β L,κ = β 1,d + aβ 2,d in all other cases. We consider Cases III and IV as the relevant cases for our empirical tests for two reasons. First, all measures of aggregate total idiosyncratic risk in the dual predictors used in our tests are highly positively correlated with each other, a condition that most likely satisfies the required range of values for ρ in Cases III and IV but least likely satisfies that in Cases I and II (in Case I ρ is strictly negative, whereas in Case II ρ is slightly negative or, if aθ 2 and θ 1 are quite different, at best moderately positive). Second, the estimate of β 2,d in the dual-predictor regression is always negative, consistent with β 2,d < 0 in both Cases III and IV but inconsistent with β 2,d > 0 in Case I and β 2,d 0 in Case 15 We note that β 1,u, β 2,u, β 1,d, and β 2,d are different scaled versions of κ with scalers being nonlinear functions of the unknown parameters θ 1, θ 2, a, and ρ. However, given β 1,u, β 2,u, β 1,d, and β 2,d, κ cannot be solved for explicitly. 14

17 II. When θ 1 is relatively small, it can be shown that (1 1 8 θ2 1 ) θ2 1 (aθ 2/θ 1 )(1 aθ 2 /θ 1 ) < ( 1 2 aθ 1θ 2 + 1) 2 aθ 2 (θ 1 aθ 2 ) 1 2 aθ 1θ 2. For a given θ 1, the above inequality provides a simple conservative estimate of the minimum value of ρ in Case IV. For low values of θ 1, Case IV requires extremely high values of ρ. For example, when θ 1 = 0.20, Case IV requires ρ to be above Because our primary measures in the dual predictor only have a correlation coefficient of about 0.80 (see Table 3), Case III applies to most of our tests and thus β L,κ = β 1,d in most of our empirical analysis. When Case IV becomes relevant, estimating the improved lower bound of β 1,d + aβ 2,d requires an estimate of a. Since Var(x 1 )/Var(x 2 ) = σ2 v+σ1 2 = 1+θ2 a 2 σv 2 1 θ2 +σ2 2 1+a 2 θ2 2 2, a can be estimated from θ1 2 Var(x 1 )/Var(x 2 ) along with β 1,u and β 2,u derived in the previous subsection as: a = Var(x 2) Var(x 1 ) β2,u β 1,u. (14) However, Eq. (14) breaks down when the estimate of β 1,u or β 2,u is negative. In most of our tests, we do find the estimate of β 2,u to be negative and insignificant. To deal with this situation, we make two additional assumptions about θ 1 and aθ 2 /θ 1. Specifically, we assume θ and aθ 2 /θ These assumptions may appear to be ad hoc, but they accommodate a wide range of parameter values for which standard tests have low power in the univariate-predictor regression but have much improved power in the dual-predictor regression, as we show in the subsection for the finite-sample analysis. With these assumptions, we have β 2,u β 1,u < 1 a ( 2 3 ) Using Eq. (14), we obtain a < σ(x 2) σ(x 1 ). Because β 2,d < 0 in Case IV, we have β 1,d σ(x 2) σ(x 1 ) β 2,d < β 1,d + aβ 2,d. Therefore, we can set β L,κ = β 1,d σ(x 2) σ(x 1 ) β 2,d in Case IV. 16 In the second approach, we make use of the explanatory power of the dual-predictor regression to construct improved lower bounds for κ. According to Eqs. (11) and (12), R 2 1,u R2 d κ2 θ 2 1+κ 2 θ 2 or R 2 1,u /(1 R2 1,u )/θ R 2 d /(1 R2 d )/θ κ. If we are willing to assume an upper bound for θ, say Rd 2/(1 R2 d )/θ Max. θ Max, the improved lower bound for κ can be computed explicitly as β L,κ = Hence, we can use the estimate of bound for κ Large-Sample-Limit Analysis For Multiple Noisy Predictors R 2 d /(1 R2 d )/θ Max as an alternative estimate of the lower In our previous analysis, we have assumed that all other predictors in the vector Z are perfectly measured and are orthogonal to v. Because some predictors in Z can also contain substantial 16 We could also set β L,κ = β 1,d + β 2,d in Case IV, but β 1,d + β 2,d can be too conservative and even turn negative when a is small. 17 Because in general Rd 2 is still low in magnitude, we find that the estimate of Rd 2/(1 R2 d ) has large standard errors in our preliminary simulation analysis. Therefore, we only use Rd 2/(1 R2 d ) for point estimation rather than for testing. 15

18 measurement noises, we now briefly discuss how our noise reduction mechanism can still work when there are large correlated noise components in multiple predictors. For simplicity, we consider the case of two noisy predictors. Consider the following true predictive model: r m,t+1 = κ 1 z 1,t + κ 2 z 2,t + ε t+1, (15) where z 1,t and z 2,t are unobserved return predictors with mean zeros; and, without loss of generality, we assume that both z 1,t and z 2,t have the same variance σ 2 z and have a correlation coefficient of ρ z. Suppose that researchers only observe the following two very noisy measures of z 1,t and z 2,t : x 1,t = z 1,t + s 1,t and (16a) x 2,t = z 2,t + s 2,t, (16b) respectively, where s 1,t and s 2,t are measurement noises with mean zeros and respective variances σ 2 1 and σ2 2 ; and both s 1,t and s 2,t are uncorrelated not only with the unobserved predictors z 1,t and z 2,t but also with the unexpected return ε t at all time t, yet they have a correlation coefficient of ρ s. As in Section 2.2, we define the signal-to-noise ratios of x 1 and x 2 as θ 1 = σ z /σ 1 and θ 2 = σ z /σ 2, respectively, and assume θ 2 < θ 1. In a regression of r m,t+1 on both x 1,t and x 2,t, the large-sample limits of the slope coefficient estimators can be shown to be γ 1 = κ 1 κ 1(1 ρ 2 s) + κ 1 θ 2 2 κ 2ρ z θ ρ s(κ 2 κ 1 ρ z )θ 1 θ 2 (1 ρ 2 s) + θ θ2 2 2ρ sρ z θ 1 θ 2 + (1 ρ 2 z)(θ 1 θ 2 ) 2 and γ 2 = κ 2 κ 2(1 ρ 2 s) + κ 2 θ 2 1 κ 1ρ z θ ρ s(κ 1 κ 2 ρ z )θ 1 θ 2 (1 ρ 2 s) + θ θ2 2 2ρ sρ z θ 1 θ 2 + (1 ρ 2 z)(θ 1 θ 2 ) 2. Our previous analysis corresponds to the case of ρ z = 1 and κ 2 = 0. Under the assumption that the noise component dominates the signal component in both x 1 and x 2 (i.e., θ 1, θ 2 1), we have (1 ρ 2 z)(θ 1 θ 2 ) 2 0. The above two equations can be simplified to: γ 1 κ 1 κ 1(1 ρ 2 s) + κ 1 θ 2 2 κ 2ρ z θ ρ s(κ 2 κ 1 ρ z )θ 1 θ 2 (1 ρ 2 s) + θ θ2 2 2ρ sρ z θ 1 θ 2 and (17) γ 2 κ 2 κ 2(1 ρ 2 s) + κ 2 θ 2 1 κ 1ρ z θ ρ s(κ 1 κ 2 ρ z )θ 1 θ 2 (1 ρ 2 s) + θ θ2 2 2ρ sρ z θ 1 θ 2. (18) If there is no correlation between the two noise components (i.e., ρ s = 0), the two coefficients θ1 become γ 1 = 2 θ 1+θ1 2 2 (κ 1 + ρ v κ 2 ) 0 and γ 2 = 2 (κ +θ2 2 1+θ ρ v κ 1 ) 0 and are thus severely biased +θ2 2 towards zero. If there is a perfect correlation between the two noise components (i.e., ρ s = 1) but no correlation between the two signal components (i.e., ρ z = 0), the coefficient of the less 16

19 noisy predictor is γ 1 = 1 (κ 2/κ 1 )(θ 2 /θ 1 ) 1+(θ 2 /θ 1 ) 2 κ 1, where 1 (κ 2/κ 1 )(θ 2 /θ 1 ) 1+(θ 2 /θ 1 ) 2 decreases with θ 2 /θ 1, holding κ 2 /κ 1 constant, and decreases with κ 2 /κ 1, holding θ 2 /θ 1 constant. by changing the subscripts of κs and θs. γ 2 can be symmetrically obtained If 0 < κ 2 κ 1 and θ 2 θ 1, the bias in γ 1 can be greatly reduced. In other words, the correlation between the noise components helps to restore the coefficient of the strong predictor that is measured with less noise. Nevertheless, this restoration may be at the cost of a distortion of the coefficient of the weak predictor that is measured with more noise. Even though the noise reduction is seen more clearly for the case of ρ s = 1 and ρ z = 0, we expect it to hold in a more general sense. For example, when θ 1 = 2θ 2, ρ s = 1, and ρ z = 0.50, we will have γ 1 = κ 1 and γ 2 0.5κ 1 in a large sample. In a more realistic case, in which κ 1 = κ 2 = 1, θ 1 = 2θ 2 = 0.30, ρ s = 0.80, and ρ z = 0.50, we will have γ 1 = and γ 2 = 0.043, compared to γ 1 = and γ 2 = if ρ s is reset to zero. This might explain why the aggregate total variance measure has a significantly positive coefficient estimate, whereas the value-weighted market volatility has a significantly negative coefficient estimate when both measures are used in the same predictive regression (see the last regression of Table 5 in Goyal and Santa-Clara, 2003 and our regressions in Panel A of Table 4), whereas the market volatility often has an insignificant coefficient estimate when it is used alone Finite-Sample Analysis Under the Alternative Hypothesis Our analysis of the univariate- and dual-predictor regressions in their large-sample limits allows us to demonstrate how the asymptotic biases from the univariate-predictor regressions can be reduced to a certain degree in the dual-predictor regression under the alternative hypothesis. In particular, we show that the asymptotic bias in the coefficient estimate of a noisy predictor in the univariate-predictor regression decreases with the signal-to-noise ratio of the predictor, whereas the ability of the dual predictor to reduce the noise effect depends on both the correlation between the two noise components and the relative signal-to-noise ratio of the two noisy predictors. finite samples, reduction in the noise effect is expected to increase test power under the alternative hypothesis. However, aggregate total idiosyncratic risk measures are quite persistent over time, which implies that the dominant noise component (and possibly the signal component as well) in the measure should be quite persistent. Using highly correlated and persistent predictors in the predictive regression naturally raises concerns about how multicollinearity and persistent regressors may affect the tests in finite samples. In this subsection, we use Monte Carlo simulations to assess the performance of the dual-predictor regression in finite samples. We note that, in the absence of the errors-in-variables problem, multicollinearity does not bias the estimation of the conditional means and the explanatory power of the regression, but merely In 17

20 increases the standard errors of the coefficient estimates. In our new tests, multicollinearity is less of a concern for two reasons. First, under the null hypothesis, the two highly correlated noisy predictors can be treated as perfectly measured non-priced noises. Therefore, multicollinearity under the null hypothesis is the same as that in the absence of the errors-in-variables problem, and it will not affect the sizes of standard tests for normally distributed regression residuals. Second, under the alternative hypothesis, multicollinearity between the two noise components creates a noise reduction mechanism so that the mean coefficient estimate of the signal predictor increases, even though the standard errors of the coefficient estimate also increase and the common signal components could limit the extent of noise reduction. Nevertheless, as long as the mean coefficient estimate of the signal predictor or the mean of a particular linear combinations of the coefficient estimates of the signal and noise predictors increases proportionally faster than the standard errors of the respective estimates, we will more likely reject the null hypothesis by the conventional t or F critical values. 18 In our simulations, we specify the stochastic processes of v t, s 1,t, and s t,2 as the first-order autoregressive processes, and φ v, φ 1, and φ 2 are used to denote the first-order autocorrelation coefficients of v t, s 1,t, and s t,2, respectively. Details of the simulations are provided in the Appendix. In the first set of simulations, we assume that residual returns are normally distributed, and thus the sizes of standard tests are correct. For this reason, our discussions are focused on the test power of the t test and the estimation of the coefficient of the signal predictor in the univariateand dual-predictor regressions. Figure 1 offers a simple comparison of the test power of the t test between the univariateand dual-predictor regressions for various sample sizes and different values of ρ. The sample size increases from 100 to 600 by an increment of 100. For the alternative hypotheses of κ 0, we set κ = 1, R 2 = 0.10, θ 1 = 0.20, aθ 2 /θ 1 = 0.20, ρ {0.75, 0.80, 0.85, 0.90, 0.95, 0.975, 0.99, 0.999}, and φ v = φ 1 = φ 2 = The choice of parameter values not only reflects our sample characteristics but also highlights an economically meaningful alternative hypothesis. Panel (a) of the figure shows that the test power in the univariate-predictor regression increases slowly from 0.13 to 0.35 as the sample size increases from 100 to 600, but it does not depend on ρ. In contrast, Panel (b) of the figure shows that the test power in the dual-predictor regression increases not only with the sample size but also with ρ. For a sample of 100, the test power is about 0.15 when ρ = 0.80 and rises to 0.48 when ρ = For a sample of 600, the test power increases sharply to 0.53 when ρ = Because the estimates of β 1,d and β 2,d should be negatively correlated in Cases III and IV, the standard errors of β 1,d + aβ 2,d and β 1,d + âβ 2,d (e.g., â = 2 10 ˆσ(x 2) 9 ˆσ(x 1 ) ) should not increase greatly. 18

21 and to 1.00 when ρ = Therefore, the test in the dual-predictor regression can have large power improvement even for quite small samples and a moderate true R 2. In the presence of the errors-in-variables problem, multicollinearity seems to improve, rather than reduce, the test power. Insert Figures 1 and 2 Approximately Here Under the same assumptions of parameter values, Figure 2 displays a comparison of the average coefficient estimates of the signal predictor between the two types of regressions. For the univariatepredictor regression, the average coefficient estimate is almost flat at 0.04 (compared to the true value of 1) across different sample sizes. In contrast, the average coefficient estimate of the signal predictor in the dual-predictor regression increases dramatically with ρ and slightly with sample size. For example, for a sample of 300, the average coefficient estimate increases from 0.08 when ρ = 0.80 to 0.63 when ρ = We note that when the correlation coefficient ρ is very high (more than ) and the sample size is over 400, the coefficient estimate of the signal predictor in the dual-predictor regression exceeds the true coefficient, consistent with the analysis in Case IV of Section 2.4. Despite the remaining downward bias in almost all cases, the average coefficient estimates from the dual-predictor regression are generally much closer to the true coefficient than are those from the univariate-predictor regression. Insert Figures 3 and 4 Approximately Here Since R 2 s tend to be biased upward in small samples (see Cramer, 1987) and the dual-predictor regression loses one degree of freedom relative to the univariate-predictor regression, we further compare the average adjusted R 2 s of the above univariate- and dual-predictor regressions in Figure 3. Interestingly, the patterns of the average adjusted R 2 s closely resemble those of the average coefficient estimates in Figure 2. The univariate-predictor regression explains only about 0.50% of the variation in simulated future excess returns when the R 2 of the true model is 10%. In contrast, the adjusted R 2 of the dual-predictor regression is about 1% for relatively low values of ρ and about 8% for very high values of ρ. The results in Figure 3 suggest that even small incremental explanatory power of the dual-predictor regression can be used to establish an economically significant pricing effect. The power of the t test in the dual-predictor regression also depends on the relative signal-tonoise ratio of the two noisy predictors. Using aθ 2 /θ 1 as a measure of the relative signal-to-noise ratio, we illustrate its effect on test power in Figure 4. In Panels (a) and (b) of the figure, ρ is set to 0.85 and 0.95, respectively. In both panels, the test power declines with aθ 2 /θ 1. However, for relatively large samples the percentage drop in test power is smaller when ρ = 0.95 than when 19

22 ρ = For example, for a sample of 600, when aθ 2 /θ 1 increases from 0 to 0.50, the percentage drop in test power is about 48% (a decline of 0.36 from 0.74) and 34% (a decline of 0.33 from 0.97) for ρ = 0.85 and 0.95, respectively. Holding aθ 2 /θ 1 constant, we find that the test power increases with sample size quickly. The results in Figure 4 suggest that in order to improve test power, two noisy predictors with very different signal-to-noise ratios are desired for given ρ and sample size. Because stock returns have a leptokurtic distribution, we assume that residual returns in the predictive model follow a Student s t distribution with four degrees of freedom in the second set of simulations. 19 Because the effect of sample size on test power is presented in Figure 1, we fix the sample size to 600 in this exercise. We implement similar simulations for alternative R 2 s, different values of θ 1 and aθ 2 /θ 1, and different levels of persistence for the signal and noise components in the two types of regressions. As expected, the test power increases with both R 2 and θ 1 for both types of regressions, but decreases with aθ 2 /θ 1 for the dual-predictor regression. We present in Table 1 selected results from the second set of simulations with a focus on the effects of persistent signal and noise components on test power. Insert Table 1 Approximately Here We fix θ 1 = 0.20 in all the panels of Table 1, but set aθ 2 /θ 1 = 0 (the pure noise case for the noise predictor) in Panels A through C and set aθ 2 /θ 1 = 0.20 in Panels D through F. Comparing Panels A and B of the table (i.e., φ 1 = φ 2 = φ v = 0 vs. φ 1 = φ 2 = φ v = 0.80), we find that, holding other things constant, persistence in both signal and noise components only slightly decreases (increases) the power of the t test in the dual-predictor (univariate-predictor) regression. Nevertheless, the power improvement from the dual-predictor regression is still substantial even for low values of ρ. When the persistence in the signal component increases from 0.80 to 0.90 (Panel B vs. Panel C of the table), the test power further declines slightly in the dual-predictor regression, even though it does not seem to change in the univariate-predictor regression. When the noise predictor also contains a signal component, the effects of moderately or highly persistent signal and noise components on test power in the dual-predictor regression, as shown in Panels D and E of the table, remain similar to those found in Panels B and C of the table. Finally, when the signal component is extremely persistent (i.e., φ v = 0.99), we find considerable power reduction in both types of regressions in Panel F of the table. However, holding other things constant, the percentage decline in test power from the case of φ v = 0.90 to the case of φ v = 0.99 is limited to about 25% in the dual-predictor regression when R 2 = 15% and ρ More importantly, in the presence 19 The simulation results for normally distributed residual returns are quantitatively similar. 20

23 of an extremely persistent signal component, the tests in the dual-predictor regression can still be more powerful than the test in the univariate-predictor regression even for low values of ρ. When ρ is extremely high, say ρ 0.99, so that Case IV is invoked, 20 we also conduct the F tests of β 1,d +aβ 2,d = 0 and β 1,d +âβ 2,d = 0 in Panels D through F of the table, where â = ˆσ(x 2) ˆσ(x 1 ) (see Section 2.4). It turns out that the power of the F tests for the lower bound for κ in this case is only slightly less than that of the t test of β 1,d = 0 for the same value of ρ. Moreover, it is also interesting to find that the adjusted R 2 of the dual-predictor regression increases from that of the univariate-predictor regression by about 70% for ρ = 0.75 or 0.80 and by more than 9 times for ρ = 0.99 or (not reported in the paper). Overall, these simulation results suggest that the t test of β 1,d = 0 in the dual-predictor regression is generally much more powerful than that of β 1,u = 0 in the univariate-predictor regression for a wide range of parameter values that are likely to capture the main features of the noisy predictors and the potential and economically meaningful pricing effect of a small component in these measures. Therefore, in our empirical study, we mostly focus on the t-test for the pricing effect of aggregate undiversified idiosyncratic risk. We supplement the t-test with the F tests in a few special cases, where the correlation between the two noisy predictors is quite close to unity. In addition, unreported simulation results suggest that the marginal improvement in the adjusted explanatory power from the dual predictor can be used to assess the true population explanatory power of the priced component Alternative Interpretations Although our new method of testing the pricing role of aggregate undiversified idiosyncratic risk is motivated by theory, rejecting the null hypothesis empirically might still be open to alternative explanations for three reasons. 21 First, time-varying market risk premiums can be determined by some underlying economic state variables, for which business cycle variables are noisy proxies. Because Campbell, Lettau, Malkiel and Xu (2001) show that average total idiosyncratic volatility tends to increase in business cycle troughs, it may also be a proxy for a state variable. We will make two attempts to check whether this is the case. In our first attempt, we assume that all control variables (to be discussed later) as a whole are very noisy proxies for state variables and that the signal predictor of the dual predictor is a relatively less noisy proxy for some state variable, and then regress future market excess returns on the signal predictor and all control variables. If aggregate total idiosyncratic 20 For θ 1 = 0.20 and aθ 2/θ 1 = 0.20, Case IV is invoked when ρ > We appreciate the referee s suggestions for these plausible alternative interpretations of our main results. 21

24 variance measures indeed represent a state variable, the significance of the coefficient estimate of the signal predictor should increase due to noise reduction (see Section 2.5). In our second attempt, we assume that all control variables are collectively less noisy proxies for state variables than are aggregate total idiosyncratic variance measures, and orthogonalize market excess returns and aggregate total idiosyncratic variance measures with respect to these control variables. If our measures are indeed noisier proxies for a state variable, the dual predictor consisting of residual average total idiosyncratic variances should have no or very weak predictive power for residual market excess returns. Second, systematic risk factors in stock returns may also be related to business cycles through their correlations with state variables. If we do not adequately remove systematic risk factors from our aggregate total idiosyncratic variance measures, these measures might still reflect some state variables. In addition to the standard factor model, we will use a multifactor APT model to further account for pervasive systematic risk factors. If some state variable drives our results, the measures with pervasive systematic risk factors removed should have no or very weak predictive power for future market excess returns. If we still find that our measures have strong predictive power, it is unlikely that some state variable drives our results. Despite our effort of exhaustively estimating total idiosyncratic risk in a less model-dependent way, one may still insist on the imperfect nature of any empirical study. In this case, we can at least say that we are testing for a significantly priced component in residual risk, no matter whether it is truly undiversified idiosyncratic risk in nature, or it is some factors yet to be discovered by researchers. Lastly, because both market risk and idiosyncratic risk can be positively correlated, the pricing effect of idiosyncratic risk may arise from the correlation between market risk and idiosyncratic risk if market risk is not adequately controlled for. To check against this alternative interpretation, it is important to accurately estimate market variance. In our empirical study, in addition to the popular realized market variance measure, we will also use both the CBOE Volatility Index (VIX) variance measure and the Mixed Data Sampling (MIDAS) estimator of the conditional monthly market variance (see Ghysels, Santa-Clara and Valkanov, 2005 and Ghysels, Santa-Clara and Valkanov, 2006) as better proxies for market risk. After carefully addressing these alternative explanations, we should be able to conclude more definitively on the empirical relevance and importance of the pricing effect of aggregate undiversified idiosyncratic risk. The pricing role of undiversified idiosyncratic risk should have implications for investors asset allocation decisions, mutual funds performance evaluation, and cost of capital, just to name a few. 22

25 3. Data and Variable Construction The success of our dual-predictor approach hinges on the construction of highly correlated aggregate total idiosyncratic risk measures. In this section, we discuss various methods of constructing these measures, and briefly describe control variables Data We employ monthly returns on various broad equity indexes from the Center for Research in Security Prices (CRSP) index file in excess of the risk-free rate as the dependent variable over the period of July 1963 to December Aggregate total idiosyncratic risk measures are constructed from daily or monthly stock returns over the same period. 22 The choice of this period is consistent with other recent time-series studies (e.g., Goyal and Santa-Clara, 2003; Bali, Cakici, Yan and Zhang, 2005; and Wei and Zhang, 2005) on the pricing role of idiosyncratic risk. 23 As further described in Section 3.4, we use Goyal and Santa-Clara (2003) and Goyal and Welch (2006) as references for all other commonly used predictors for monthly stock returns. In addition, we also employ the implied monthly market variance constructed from the daily CBOE Volatility Index (VIX) and the institutional holdings data from Thomson Financial Institutional (13f) Holdings Measuring Aggregate Total Idiosyncratic Risk Estimating Idiosyncratic Returns Total idiosyncratic risk potentially accounts for a large portion of an individual stock s total risk and is typically measured by the variance or volatility of idiosyncratic returns. However, idiosyncratic returns are not directly observed and are thus commonly estimated by residual returns from an asset pricing model. In this study, our main measures of total idiosyncratic risk are estimated with respect to the Carhart four-factor model. 24 The Carhart model can be estimated with daily returns in each month or with monthly returns in pre-determined estimation windows. The estimation windows can be consecutive 60-month subperiods from July 1963 to June 2003 and a 78-month subperiod from July 2003 to December 2009 or rolling 60-month subperiods that begin 22 To reduce the market microstructure effects on daily returns, we exclude stocks with an average monthly closing price of less than $5. We impose the same requirement on stocks for monthly returns. In any given month, the sample size may vary due to estimation requirements that are detailed in subsequent subsections. 23 Since we primarily use within-month daily stock residual returns from the Carhart four-factor model estimated in each month to construct various measures of aggregate total idiosyncratic risk, the starting point of our sample period corresponds to the starting date of the standard daily factors. 24 Campbell, Lettau, Malkiel and Xu (2001) offer a model-free approach for estimating total idiosyncratic volatility, but we adopt a model-based approach that directly accounts for the well-known sources of systematic variations in stock returns. Both daily and monthly factor returns are obtained from Professor Kenneth French s data library on his website: library.html. 23

26 in July For convenience, we refer to these three procedures as the within-month Carhart model, the fixed-period Carhart model, and the rolling-window Carhart model, respectively. We address the concern about omitted factors by also using a multifactor APT model to further account for unknown systematic risks. model. 25 Due to space limitation, we focus on a six-factor APT The six factors of the APT model are the first six principal components of daily or monthly stock returns and are extracted from the return correlation matrix of all stocks that have complete returns over the aforementioned fixed periods or rolling windows. We refer to the procedure that is based on daily (monthly) returns and fixed periods/rolling windows as the daily (monthly) fixed-period/rolling-window six-factor APT model Total Idiosyncratic Risk Measures Based on Daily Stock Returns We consider several methods of estimating total idiosyncratic risk from residual returns. To be consistent with Merton s (1987) theory, we focus exclusively on total idiosyncratic variance (rather than volatility) measures in this study. 26 For comparison, our primary measure is a modified version of the total idiosyncratic variance measure employed by Goyal and Santa-Clara (2003). Following Campbell, Lettau, Malkiel and Xu (2001), Goyal and Santa-Clara (2003) use a stock s within-month daily returns to estimate the stock s monthly total variance, which is then used as a proxy for the stock s monthly total idiosyncratic variance. Specifically, the total idiosyncratic variance is computed as follows: σ 2,GS k,t = D t d=1 D t rk,d,t r k,d,t r k,d 1,t, (19) where r k,d,t is stock k s realized return in day d of month t, and D t is the number of trading days in month t. 27 Eq. (19) is consistent with Merton s argument that realized volatility estimates from high-frequency returns offer better precision when volatilities are time-varying. However, Goyal and Santa-Clara s (2003) total idiosyncratic variance measure apparently contains systematic risk components. In our modified measures, we use residual returns from the within-month Carhart model, the daily fixed-period APT model, or the daily rolling-window APT model in place of realized returns in Eq. (19) to estimate a stock s monthly total idiosyncratic variance. d=2 25 In our preliminary analysis, we find that the average marginal explanatory power is less than 1% for every additional factor, from the seventh to the twelfth factor, added to the six-factor APT model. Our main results are qualitative similar and remain statistically and economically strong even if we use the twelve-factor APT model. 26 Our preliminary analysis shows that the variance measures mostly outperform the volatility measures. 27 The second term on the right-hand side of Eq. (19) adjusts for serial correlations in daily returns, as proposed by French, Schwert and Stambaugh (1987). 24

27 Total Idiosyncratic Risk Measures Based on Daily Portfolio Returns The total idiosyncratic variance measures estimated from individual stocks daily residual returns are subject to market microstructure noises such as bid-ask spreads and zero returns (see Han and Lesmond, 2011). A practical solution is to use portfolio residual returns to mitigate such effects. Our second type of total idiosyncratic variance measure is estimated by replacing a stock s daily returns in Eq. (19) with a portfolio s daily residual returns from a factor model. If the factor model is adequate, the total idiosyncratic variance of a portfolio can be estimated more accurately, but will be smaller, than that of an individual stock because of more precise estimates of the factor loadings. Due to space limitation and for computational ease, we focus on daily portfolio residual returns from the daily fixed-period six-factor APT model. 28 Since a very large number of portfolios can be formed out of all available stocks at any given point of time, we limit our attention to a set of non-overlapping portfolios with an equal number of stocks chosen from nearly all available stocks. Specifically, our procedure consists of the following two steps. First, in the beginning month of our sample period, we randomly select four stocks without replacement to form each non-overlapping portfolio until there are fewer than four stocks left. Second, in subsequent months, we retain the four-stock portfolios formed in the previous month if all stocks in the portfolio have survived to the current month, and then randomly form additional non-overlapping four-stock portfolios in the same manner from stocks that have just appeared, or were discarded in the previous month, or have just been released from a ceased portfolio. 29 We repeat the procedure for 100 replications Total Idiosyncratic Risk Measures Based on Monthly Stock Returns Another way to deal with the market microstructure effects is to use monthly stock returns. Hence, our third type of total idiosyncratic variance measures is constructed from monthly residual returns of individual stocks. As described in Section 3.2.1, we use the monthly Carhart four-factor model factor or the monthly six-factor APT model to estimate monthly residual returns over either fixed periods or rolling windows. When the factor model is estimated in fixed periods, we simply treat a stock s squared monthly residual return as the stock s monthly total idiosyncratic variance. When the factor model is estimated in rolling windows, we use a weighted average of a stock s squared monthly residual returns over the rolling window as the stock s monthly total idiosyncratic variance for the last month of the rolling window. Weights for the squared monthly residual returns 28 Our results are robust to aggregate total idiosyncratic variance measures constructed from daily portfolio residual returns from the within-month Carhart model. 29 We have also tried two-stock, eight-stock, sixteen-stock, and thirty-two-stock random portfolios. Results are qualitatively similar. 25

28 gradually increase from the first month to the last month of the rolling window. Specifically, we estimate stock k s total idiosyncratic variance at month t, σ 2 k,t, as 59 ˆσ k,t 2 = w j ˆε 2 k,t j, (20) j=0 where w j = g j / 59 i=0 gi and g is set to 0.95; and ˆε k,t j is stock k s residual return at month t j, estimated from a factor model with monthly excess returns from month t 59 to month t. Alternatively, as in Fu (2009), in addition to the Carhart model, a stock s monthly total idiosyncratic variance can be directly estimated from the Exponential Generalized Autoregressive Conditional Heteroscedasticity (EGARCH) model with monthly excess returns. In general, total idiosyncratic variance estimates from low-frequency returns can be less precise, despite the advantage of avoiding some issues with estimates from high-frequency returns. Consequently, these estimates may be less capable of capturing the time-varying undiversified idiosyncratic risk component, and are thus primarily used in our robustness checks Aggregate Total Idiosyncratic Risk Measures At the market level, aggregate total idiosyncratic variances of individual stocks or portfolios can be equal- or value-weighted. As discussed earlier, the equal-weighted measures should have a higher signal-to-noise ratio than the value-weighted measures. Therefore, the equal- and valueweighted measures can be used as the signal and noise predictors in the dual-predictor regression, respectively. If a stock s investor base is positively related to the stock s institutional ownership (IO), idiosyncratic risk of a stock with high (low) IO should have a small (large) undiversified idiosyncratic risk component. To link different levels of undiversified idiosyncratic risk to investor base, we can also aggregate individual stocks total idiosyncratic variances by their IO measures. We use institutional holdings data from Thomson Financial Institutional (13f) Holdings to compute a stock s quarterly aggregate IO from December 1979 to December Assuming that stocks quarterly IO information does not change in the subsequent three months, we match stocks quarterly IOs to their monthly total idiosyncratic variances, which are estimated from residual returns relative to the within-month Carhart model. We then sort these stocks into IO tercile portfolios in each month by their IO measures. 30 Lastly, we use total idiosyncratic variances of individual stocks in 30 Stocks with no reported institutional ownership are assumed to have zero institutional ownership. Because a high proportion of stocks have no reported institutional ownership in early years, sorting is implemented for stocks with positive institutional ownership. Stocks with no reported institutional ownership are re-grouped with the stocks in the lowest institutional ownership tercile. We have also tried to sort stocks into institutional ownership quintiles or deciles. Results are qualitatively similar. 26

29 the lowest or the highest IO terciles to compute monthly equal- and value-weighted aggregate total idiosyncratic variances. The equal-weighted aggregate total idiosyncratic variance for the lowest IO tercile should have a higher signal-to-noise ratio than the equal- or value-weighted aggregate total idiosyncratic variance for the highest IO tercile. If total idiosyncratic risks of large stocks have lower signal-to-noise ratios, shifting more weights towards these stocks in the aggregate measure should produce an aggregate total idiosyncratic variance measure with an even lower signal-to-noise ratio than obtained with the standard valueweighted aggregate total idiosyncratic variance measure. Following this logic, we construct a supervalue-weighted measure, in which weights on individual stocks total idiosyncratic variances are computed based on an increasing nonlinear function of the stocks market capitalizations. In particular, if we let V k be stock k s market capitalization in thousands of dollars at the beginning of each month, we can use (V k ) 1.01 to compute the super-market-value weight for the same stock. 31 Relative to the super-value-weighted measure, the standard value-weighted measure can now be used as the signal predictor in the dual-predictor regression. Small shifts in weights will produce very highly correlated aggregate total idiosyncratic variance measures. If the bias reduction in the coefficient estimate from the very high correlation between the two measures outweighs the increase in the standard error of the coefficient estimate from multicollinearity, our simulation results suggest that the test power and the incremental adjusted R 2 will be further improved in the dual-predictor regression. We explore the robustness of our results to this simple super-value weighting scheme Measuring Market Risk In our large-sample-limit and Monte Carlo simulation analysis, we have assumed that both the signal and noise components of aggregate total idiosyncratic variance measures are uncorrelated with systematic risk factors. However, as we will show in Table 3, market variance measures are typically highly correlated with aggregate total idiosyncratic variance measures, indicating that a common component exists in these measures. Therefore, it is important to control for market risk in testing the pricing effect of aggregate undiversified idiosyncratic risk (see discussions in Section 2.7). In theory, market variance should be a strong predictor for future market excess returns. We employ three measures of market risk. The first measure is the simple monthly realized market variance computed according to Eq. (19) from daily CRSP value- or equal-weighted index returns. The simple market variance is our primary measure of market risk since it is available for the full sample period. For other choices of equity indexes, we compute their corresponding monthly variances in the same manner. Both the ex-ante market volatility measure and the conditional 31 We have also tried 1.02, 1.04, 1.06, and 1.08 for the exponent. Results are qualitatively similar. 27

30 market volatility measure may be better proxies for market risk. 32 Hence, our second measure is the implied monthly market variance constructed from the daily CBOE Volatility Index (VIX), which is available from February 1990 to December Our third measure is the Mixed Data Sampling (MIDAS) estimator of the conditional monthly market variance, advocated by Ghysels, Santa- Clara and Valkanov (2005) and Ghysels, Santa-Clara and Valkanov (2006). The MIDAS estimator is postulated to be a weighted average of past squared daily market returns with the weight function optimally determined. Ghysels, Santa-Clara and Valkanov (2005) show that the MIDAS estimator has a significantly positive relationship with market excess returns. We investigate whether the predictive power of the dual predictor can be subsumed by the three measures of market risk Existing Predictors and Other Systematic Risk Factors The return predictability literature offers quite a few predictors that have been shown to predict future market excess returns. Like Goyal and Santa-Clara (2003), we employ the following predictors as control variables: relative Treasury bill rate (RTB), term spread (TS), default spread (DS), and dividend yield (DY). 33 Moreover, we augment the above predictors with additional predictors used by Goyal and Welch (2006). These predictors include earnings-price ratio (E/P), book-to-market ratio (B/M), net equity expansion (NTIS), and the rate of inflation (Infl). 34 In addition, recent studies find that the liquidity risk factor is useful in explaining the crosssectional average returns. We employ two existing measures of the liquidity risk factor. The first measure is the aggregate liquidity innovation measure constructed by Pástor and Stambaugh (2003) from daily returns and volumes. The second measure is the aggregate permanent variable component of price impacts estimated by Sadka (2006) from intraday trades. These liquidity risk factors are downloaded from WRDS. We note that the second liquidity risk measure only covers a subperiod of our full sample period. 32 We thank the anonymous referee for these suggestions. 33 Relative Treasury bill rate is the relative three-month Treasury bill (T-bill) rate, and is calculated as the difference between the T-bill rate and its 12-month moving average. Term spread is the difference between the yield on longterm government bonds and the T-bill rate. Default spread is the difference between yields on BAA- and AAA-rated corporate bonds. Dividend yield is the ratio of the last 12-month aggregate dividends to the current index value of the CRSP value-weighted index. We are grateful to Professor Amid Goyal for making available the data used in Goyal and Santa-Clara (2003), and we have cross checked the consistency of the variables used in our study. 34 The earnings-price ratio is the past 12-month earnings of S&P 500 index divided by the level of the index in the current month. The book-to-market ratio is the ratio of book value to market value for the Dow Jones Industrial Average. Net equity expansion is the ratio of past 12-month net issues by NYSE listed stocks divided by the total end-of-year market capitalization of NYSE stocks. Inflation is the percentage change in the Consumer Price Index. These variables are downloaded from the RFS website. We have also experimented with the cross-sectional risk premium measure and find that this variable typically substantially strengthens our results in the related periods. Because Goyal and Welch (2006) find it difficult to replicate this variable and the variable is available only up to December 2002, we decided to exclude it from our study. 28

31 4. Empirical Analysis To ensure that the data in our study are consistent with prior studies, we begin our empirical analysis by first describing summary statistics of our main variables. Subsequently, we compare the results from our preliminary tests to those in recent studies. We then build our case for the pricing role of aggregate undiversified idiosyncratic risk in time-series tests Summary Statistics Summary statistics of our main variables are reported in Table 2. As shown in Panel A of Table 2, the average CRSP equal- and value-weighted index excess returns (i.e., r ew and r vw ) are slightly lower but the average CRSP equal- and value-weighted index variances (i.e., σew 2 and σvw) 2 are slightly higher than those reported by Goyal and Santa-Clara (2003). This is likely due to market downturns and market swings in 2007 and A comparison of the three market variance measures shows that the VIX measure (i.e., σvw,vix 2 ) and the MIDAS estimator (i.e., σ2 vw,midas ) have much lower skewness and excess kurtosis, but much higher autocorrelations, than the realized market variances do. 35 Differences in the characteristics of these market variance measures may capture different aspects of time-varying market risk. Insert Table 2 Approximately Here Panel B of Table 2 presents the characteristics of aggregate equal- and value-weighted total variances (i.e., σ 2,GS 1,ew and σ2,gs 1,vw, referred to as the GS measures) and our three types of aggregate total idiosyncratic variance measures. For convenience, the aggregate total idiosyncratic variance measure of individual stocks is referred to as σ 2,M,P,F 1,w, where M refers to either the Carhart fourfactor model (Carhart), the six-factor APT model (APT), or the EGARCH model (EGARCH); P refers to within-month estimation periods (Within), fixed estimation periods (Fixed), or rolling estimation periods (Rolling); F refers to either daily return frequency (Daily) or monthly return frequency (Mon); and w denotes the weighting scheme used to aggregate total idiosyncratic variances of individual stocks, and takes ew (vw) for the equal-weighting (value-weighting) scheme. Similarly, the aggregate total idiosyncratic variance measure of four-stock portfolios is referred to as σ 2,APT,Fixed,Daily 4,w 1,w 2, where w 1 denotes the weighting scheme used to form portfolios and w 2 denotes the weighting scheme used to aggregate portfolio total idiosyncratic variances; and each takes ew (vw) for the equal-weighting (value-weighting) scheme. 35 Excluding the market crash periods of October 1987 and October 2008 will substantially reduce both skewness and excess kurtosis of the equal- and value-weighted market variances. 29

32 The summary statistics of the GS measures in our sample period are largely consistent with those reported for Goyal and Santa-Clara s (2003) period, with two minor exceptions. First, both the skewness and excess kurtosis are somewhat higher in our sample. Again, these differences can be attributed to the recent financial crisis period of Second, due to the exclusion of lowpriced stocks, the equal-weighted GS measure is lower on average in our sample period. Due to removal of the systematic risk components from total risk, our aggregate total idiosyncratic variance measures are much smaller than the GS measures. Moreover, our measures typically have much lower skewness and excess kurtosis than the GS measures do, regardless of the factor models, the estimation periods, the use of individual stocks or portfolios, the weighting schemes, or the frequency of returns, suggesting that our measures are less susceptible to extreme values. Nevertheless, our measures generally have higher autocorrelations (largely over 0.80 for measures constructed from daily residual returns and largely over 0.90 for measures constructed from monthly residual returns) than the GS measures do. Our simulation results in Section 2.6 suggest that highly persistent signal and noise components of the noisy predictors will reduce test power in the dual-predictor regression, which should only bias us finding the pricing effect of the signal component. According to the weighting schemes, the frequency of returns, and the use of individual stocks or portfolios, we classify different versions of aggregate total idiosyncratic variance measures into several major groups. Within each major group, the characteristics of these measures could differ significantly, depending on the factor models and the estimation periods. First, the equal-weighted measures are almost twice larger, but typically much less skewed and leptokurtic, than are the corresponding value-weighted measures. Second, the measures constructed from within-month daily residual returns from the daily six-factor APT model are larger than those constructed from within-month daily residual returns from the within-month Carhart model because the former is estimated over a 60-month period whereas the latter is estimated over a one-month period. In contrast, when both types of measures are estimated from monthly residual returns over the same 60-month period, the pattern is reversed. Third, as expected, the measures constructed from within-month daily portfolio residual returns from the daily fixed-period six-factor APT model are smaller than those constructed from within-month daily stock residual returns from the same model. Fourth, the measures constructed from monthly residual returns estimated in rolling windows are typically much more persistent than those constructed from monthly residual returns estimated in fixed periods and those constructed from daily residual returns estimated in fixed periods or rolling windows, because they are simply a weighted moving average of squared monthly residual returns. Fifth, the equal-weighted measures constructed from monthly residual returns from the fixed-period EGARCH model are less skewed and leptokurtic than those 30

33 constructed from monthly residual returns from the fixed-period Carhart model. However, this difference is absent for the corresponding value-weighted measures. Nevertheless, the measures based on the fixed-period EGARCH model are more persistent than those based on the fixed-period Carhart model because the former is also essentially a weighted moving average measure. Panel C of Table 2 reports summary statistics of control variables. Most of the control variables are obtained from authors of several well-cited papers. All control variables are very persistent (even more so than our aggregate total idiosyncratic variance measures), except for the rate of inflation, Pástor and Stambaugh s (2003) aggregate liquidity innovation measure (denoted as PSi in the table) and Sadka s (2006) aggregate permanent variable component of price impacts (denoted as Spv in the table). As discussed in Section 2.5, if these control variables are much less noisy and contain signals positively correlated with aggregate undiversified idiosyncratic risk, including these variables in the regression will likely reduce the significance of our idiosyncratic risk measures. Table 3 reports the correlation matrix among market excess returns, market variances, and aggregate total idiosyncratic variance measures. Because market excess returns and the three types of market variance measures have similar correlations with various aggregate total idiosyncratic variance measures, we choose to present the correlations with a few representative aggregate total idiosyncratic variance measures for simplicity. Moreover, according to the use of the equal- or valueweighting scheme in aggregation, we roughly classify the aggregate total idiosyncratic variance measures into high and low signal-to-noise ratio groups. All correlations in the table are highly significant at the 5% significance level or better, except a few ones printed in italics. Insert Table 3 Approximately Here Like Glosten, Jangannathan and Runkle (1993) and Adrian and Rosenberg (2008), we find that market excess returns are negatively correlated with the realized market variances. In contrast, market excess returns are mostly uncorrelated with the VIX measure, the MIDAS estimate, and various aggregate total idiosyncratic variance measures. In particular, the value-weighted market excess returns have very low correlations with most aggregate total idiosyncratic variance measures in the high signal-to-noise group. Moreover, quite high correlations are found between market variances and various aggregate total idiosyncratic variances (mostly about 0.50), suggesting the importance of controlling for market risk in testing for the pricing effect of undiversified idiosyncratic risk. The correlations among aggregate total idiosyncratic variances are in the range of 0.29 to 0.99, and most of them are above According to Eq. (13), the differences in correlations may reflect the differences in signal-to-noise ratios and in the correlations between unobserved noise 31

34 components. Because our previous analysis indicates that the performance of our dual-predictor regression depends largely on the correlation coefficient between the two noise components of the dual predictor, we are most interested in the inter-group correlations and especially in those between the measures that differ only in the weighting scheme. Because the measures that differ only in the weighting scheme are used as the signal and noise predictors of the dual predictor in the subsequent analysis, we print in boldface the correlations between these measures. These correlations range from 0.71 to 0.97, which should lead to moderate or very good noise reduction in the dual-predictor regression, according to our simulation results The New Empirical Test Based on our analysis in the previous sections, we propose the following predictive regression to test for the pricing effect of aggregate undiversified idiosyncratic risk at the market level: R m,t+1 R f,t+1 = β 0 + β 1,d x 1,t + β 2,d x 2,t + γ z Z t + ν t+1, (21) where R m,t+1 R f,t+1 is the market excess return in month t+1; x 1,t and x 2,t are the dual predictor consisting of two very noisy measures of aggregate undiversified idiosyncratic risk in month t, and Z t is a vector of control variables in month t and includes market variance σ 2 m,t in particular. To be consistent with Section 2.2, we assume that x 1,t has a higher signal-to-noise ratio than x 2,t. Under the null hypothesis of κ = 0, the aggregate undiversified idiosyncratic risk has no predictive power for future market excess returns. Hence, we should have β 0 = β 1,d = β 2,d = 0 and γ z = κ z (see Eq. (4)). Since the correlations between the noisy predictors in most of the dual predictors used in our study are in the range of 0.71 to 0.97 (see the boldface numbers in Table 3), β 1,d can directly serve as an improved lower bound for the true underlying parameter κ under the alternative hypothesis (see Case III in Section 2.4). Our simulation analysis has shown that the t test of β 1,d = 0 carries substantially large power, yet it only tests against a lower bound for the underlying parameter under the alternative hypothesis. Therefore, we will primarily focus on this test in our empirical analysis. When the noisy predictors in the dual predictor are extremely positively correlated, as is in the case of the dual predictor consisting of the value- and super-value-weighted aggregate total idiosyncratic variance measures, we can test β 1,d + âβ 2,d = 0 (e.g., â = ˆσ(x 2) ˆσ(x 1 ) ; see the discussion for Case IV in Section 2.4 and the simulation results in Table 1), in addition to β 1,d = 0. In our new tests, we expect to find a positive coefficient estimate for the signal predictor x 1 and a negative coefficient estimate for the noise predictor x 2 (predictions from Cases III and IV in Section 2.4). Moreover, if the null hypothesis is rejected, we also expect to find that the dual- 32

35 predictor regression improves the adjusted explanatory power relative to the regression using only the signal predictor Preliminary Results In Table 4, we first replicate the main time-series results in the literature on idiosyncratic risk. These results are then compared to those obtained from our dual-predictor regression. The dependent variable is the monthly CRSP value-weighted index excess return. Panel A of the table displays the results of Goyal and Santa-Clara s (2003) main regression (see Model 3 of Table II in p. 986) for their sample period. The conventional univariate-predictor regression uses the lagged realized market variance to control for market risk, but employs only one noisy proxy for aggregate undiversified idiosyncratic risk. measures (denoted as σ 2,GS 1,ew The noisy predictors include the equal- and value-weighted GS and σ2,gs 1,vw, respectively, in the table) and one set of our primary equaland value-weighted measures (denoted as σ 2,Carhart,Within,Daily 1,ew and σ 2,Carhart,Within,Daily 1,vw, respectively in the table), which is constructed from within-month daily stock residual returns relative to the within-month Carhart model. Insert Table 4 Approximately Here The coefficient estimate for the equal-weighted GS measure is positive and highly significant. Both the significance level for the coefficient estimate and the adjusted R 2 of the regression are very close to those reported in Goyal and Santa-Clara s (2003) study (a t-statistic of 3.85 and an adjusted R 2 of 2.23% in our regression vs. a t-statistic of 3.86 and an adjusted R 2 of 2.12% in their study). 36 When the value-weighted GS measure is used instead, the significance for the coefficient estimate declines to the 10% level (a t-statistic of 1.70). The significance levels for the coefficient estimates and the R 2 s of the regressions increase by a moderate amount when idiosyncratic variance measures replace total variance measures. For example, the adjusted R 2 increases from 2.23% to 2.57% for the equal-weighted aggregate total idiosyncratic variance measure. Moreover, the coefficient estimate for our value-weighted measure is significant at the 5% level, but the regression has a lower adjusted R 2 than that using our equal-weighted measure, consistent with our argument that value-weighted measures are noisier than equal-weighted measures. In Panel B of Table 4, we extend the sample period to December 2009 and repeat the above analysis, along with our dual-predictor regression. Two distinctive results emerge. First, despite the fact that our equal- and value-weighted measures perform better than the respective GS measures, 36 The magnitude of the coefficient estimate from our regression is almost twice as large as that in Goyal and Santa-Clara s (2003) study. This may be due to exclusion of low-priced stocks in our sample, as noted in summary statistics. 33

36 none of the measures alone significantly predicts future market excess returns. These results confirm the findings of Bali, Cakici, Yan and Zhang (2005) and Wei and Zhang (2005) that Goyal and Santa- Clara s (2003) results are not robust to different sample periods and to the value-weighting scheme. Second, consistent with our simulations, the coefficient estimate for our equal-weighted (valueweighted) measure is positive (negative) and highly significant in the baseline dual-predictor regression. This initial result highlights the importance of test power in testing the null hypothesis that idiosyncratic risk does not matter. Similar to the finding in prior research, our results show that the lagged realized market variance is loaded with a significantly negative coefficient in many cases, especially when used together with an equal-weighted total or total idiosyncratic variance measure. Current literature is silent on this perplexing fact. Our discussion in Section 2.5 offers one possible explanation, i.e., both the realized market variance measure and the aggregate total idiosyncratic variance measure may contain correlated noise components, and the sign and significance for the coefficient estimate of each variable depend on the relative importance of the signal components and the relative signal-to-noise ratio of the predictors. Interestingly, the lagged realized market variance has an insignificant and less negative coefficient estimate in our dual-predictor regression, a result that moves in the direction towards a sensible risk-return relation. To alleviate the concern about the model-dependent estimation of idiosyncratic variance, we repeat the analysis in Panel B of Table 4 with the aggregate total idiosyncratic variance measures constructed from within-month daily stock residual returns relative to the six-factor APT model estimated over fixed periods and rolling windows in Panels C and D of the table, respectively. Hence, the signal and noise predictors of the dual predictor are denoted as σ 2,APT,Fixed,Daily 1,ew σ 2,APT,Fixed,Daily 1,vw in Panel C of the table and as σ 2,APT,Rolling,Daily 1,ew and σ 2,APT,Rolling,Daily 1,vw and in Panel D of the table. The results for the univariate-predictor regressions are fairly weak and quite similar across different measures. In contrast, the results for the dual-predictor regressions in both panels remain quite strong and are qualitatively similar to the result in Panel B of the table. 37 For the dual-predictor regressions in Panels B through D of the table, the differences in the magnitude of the coefficient estimates and in the adjusted R 2 s may be attributed to the differences in the variation of the individual noisy predictors, in the correlations between the signal and noise predictors in the dual predictors, 38 and in the signal-to-noise ratios of the two measures in the dual predictors. Even 37 We note that idiosyncratic variance measures used in Panel C of the table are not strictly based on past information, which may induce a forward-looking bias that might partially explain the particularly strong results. Nevertheless, if the forward-looking bias does not fully offset the bias induced by measurement noise, the explanatory power of the regression is still biased downward. 38 For example, as shown in Table 3, the correlation coefficient between σ 2,Carhart,Within,Daily 1,ew σ 2,Carhart,Within,Daily 1,vw is 0.79, and the one between σ 2,APT,Rolling,Daily 1,ew and σ 2,APT,Rolling,Daily 1,vw is and 34

37 though our preliminary results from the dual-predictor regressions lend support to the pricing effect of undiversified idiosyncratic risk, these results might also be consistent with a few other interpretations, as discussed in Section 2.7. In the subsequent analysis, we will try to rule out the alternative interpretations Main Results In Table 5, we control for commonly used predictors in the dual-predictor regressions. The signal and noise predictors in this table are the equal- and value-weighted aggregate total idiosyncratic variances constructed from within-month stock residual returns from the Carhart model. For ease of interpretation, we classify lagged market excess returns, relative T-bill rate (RTB), term spread (TS), default spread (DS), lagged dividend yield (DY) as the first set of control variables and lagged earnings-price ratio (E/P), lagged book-to-market ratio (B/M), lagged net equity expansion (NTIS), and the lagged rate of inflation (Infl) as the second set of control variables. In Panel A of Table 5, the dependent variable is the monthly CRSP value-weighted index excess return, and we control for the first set of predictors in the first regression and for both sets of predictors in the second regression. In both regressions, the coefficient estimates for the dual predictors remain highly significant and are comparable to those from the last regression in Panel B of Table 4. For example, the coefficient estimate of the signal predictor from the second regression is 1.26, compared to 1.42 from the last regression in Panel B of Table 4. This estimate can serve as an estimate of the improved lower bound for the risk premium per unit of aggregate undiversified idiosyncratic risk (see Case III in Section 2.4). Over the sample period, the monthly value-weighted market risk premium and the average signal predictor are and , respectively (see Table 2). Thus, even if 1% of the aggregate total idiosyncratic variance is undiversified, more than 4% (= 1.26 ( )/0.0042) of the value-weighted market risk premium can be attributed to the aggregate undiversified idiosyncratic variance. In contrast, the only significant control variable in the first and second regressions is the relative T-bill rate, consistent with the Ang and Bekaert s (2009) finding. We note that, in the absence of the noise predictor, the signal predictor neither has a significant coefficient estimate nor changes the adjusted R 2 of the regression with all control variables, inconsistent with the state variable interpretation. Insert Table 5 Approximately Here Comparing the regressions with and without the dual predictor, we find that the dual predictor by itself adds 1.35 (1.10) percentage points in the adjusted R 2 to the first (second) regression of Panel A of Table 5. Given that the correlation coefficient between the two noise components is 35

38 about 0.80, 39 our simulation results for the adjusted R 2 s (not reported) indicate that these estimates can only account for a small fraction of the population marginal explanatory power of the priced component. For example, when the population R 2 is 15%, the average adjusted R 2 is only 1.25% for the case of ρ = 0.80, θ 1 = 0.20, aθ 2 /θ 1 = 0.20, and φ v = φ 1 = φ 2 = In other words, even when the estimated marginal adjusted R 2 is only 1.10%, the population marginal R 2 can be as high as 13%! If we assume 0.04 θ Max and use R 2 d /(1 R2 d )/θ Max as a conservative estimate for the underlying parameter κ (see the discussion in Section 2.4), the estimate is between 1.05 and 2.63, in line with the coefficient estimate of 1.26 from the regression. These estimates are not only internally consistent but also suggest that the pricing effect of aggregate undiversified idiosyncratic risk is economically important. In a predictive regression with a highly persistent predictor, when shocks to the predictor are negatively correlated with contemporaneous return innovations, the coefficient estimate of the predictor will be biased upward in small samples (e.g., see Stambaugh, 1999 and Lewellen, 2004). Even though this source of bias applies to dividend yield, the book-to-market ratio, and the earnings-price ratio, it is unlikely to affect our dual-predictor estimates for two reasons. First and foremost, both the signal and noise predictors of the dual predictor are at best very weakly negatively correlated with market excess returns, as shown in Table 3. In particular, we have confirmed that no contemporaneous correlation exists between the residuals from the second regression in Panel A of Table 5 and those of an autoregressive process of the signal predictor. Second, the first-order autocorrelations of our main aggregate total idiosyncratic variance measures are in the range of 0.77 to 0.89 (see the summary statistics in Table 2), and are much lower than the first-order autocorrelation of 0.95 or above for the well-known financial predictors. In Panel B of Table 5, we repeat the regression analysis in a similar fashion but use the monthly CRSP equal-weighted index excess return as the dependent variable. If the prices of small stocks are affected by undiversified idiosyncratic risk more than those of large stocks, we should find even stronger results for the equal-weighted index excess return. This is indeed the case. For example, in the first regression, which includes only the signal predictor and the lagged realized market variance, we find that the coefficient estimate of the signal predictor is 1.12 with a t-statistic of 39 We assume that the correlation coefficient between the two noise components can be approximated by the correlation coefficient between the signal and noise predictors of the dual predictor. 40 Because r vw = κv + κ zz + ε, where v is the market-wide undiversified idiosyncratic variance and Z is a vector of other predictors, θ = σ v/σ ε σ v/σ(r vw). If σ v σ(5% σ 2,Carhart,Within,Daily 1,ew ) = (assuming that at least 5% of the equal-weighted aggregate total idiosyncratic variance is undiversified) and σ(r vw) = , θ / = See Table 2 for the values of σ 2,Carhart,Within,Daily 1,ew and σ(r vw). 36

39 2.93 and that the regression has an adjusted R 2 of 1.66%. When we add the noise predictor to the second regression, the adjusted R 2 increases to 5.79%, and the coefficient estimate of the signal predictor increases to 2.89, whereas that of the noise predictor is significantly negative. Including other predictors further enhances return predictability but does not alter the predictive power of our dual predictor. In fact, in the dual-predictor regressions with the two sets of control variables, the coefficient estimate of the signal predictor is 2.70 and the dual predictor has 2.99 percentage points incremental adjusted explanatory power. Therefore, even if 1% of the aggregate total idiosyncratic variance is undiversified, more than 4.8% (= 2.70 ( )/0.0075) of the equal-weighted market risk premium can be attributed to the aggregate undiversified idiosyncratic variance. Under the assumption that the adjusted explanatory power is 8.4% of the true population explanatory power (unreported simulation results for the case of R 2 = 15%, ρ = 0.80, θ 1 = 0.20, aθ 2 /θ 1 = 0.20, and φ v = φ 1 = φ 2 = 0.80), the population marginal explanatory power of the signal component can be as high as 36%! These results demonstrate that the pricing effect of undiversified idiosyncratic risk is remarkably strong among small stocks. When adding control variables in the third and fourth regressions in Panel B of Table 5, we find that the lagged realized market variance measure has a positive, albeit insignificant, coefficient estimate, consistent with the positive risk-return relation, whereas the lagged equal-weighted market excess return has a significantly positive coefficient, consistent with the well-known autocorrelation in equal-weighted index excess returns. Moreover, as in the case of the CRSP value-weighted index excess return, the relative T-bill rate is a very significant predictor. In addition, the earnings-price ratio and the inflation rate show some marginal predictive power. We investigate the pervasiveness of the pricing effect of undiversified idiosyncratic risk by using value-weighted equity index excess returns as the dependent variable in the following analysis, which is also in line with the literature on idiosyncratic risk. The pricing effect of aggregate undiversified idiosyncratic variance could be an artifact of unknown systematic risks. In Table 6, we change the Carhart model to the six-factor APT model in constructing the dual predictors. Panel A of the table extends the results for the dual predictors constructed from within-month daily stock residual returns relative to the daily fixed-period or rolling-window six-factor APT model (see the last regressions in Panels C and D of Table 4) by controlling for the well-known predictors. The results for the dual predictors are very similar to those shown in Panels C and D of Table 4 and appear to be statistically stronger than those shown in Panel A of Table 5. However, the estimated economic effect is very close to that from the dual predictor in Panel A of Table 5. For example, the coefficient estimate of the signal predictor in the regression with the two sets of control variables suggests that if 1% of the aggregate portfolio total 37

40 idiosyncratic variance is undiversified, more than 4% of the value-weighted market risk premium can be attributed to aggregate undiversified idiosyncratic risk. Insert Table 6 Approximately Here Panel B of Table 6 employ two different dual predictors constructed from within-month daily portfolio residual returns relative to the daily fixed-period six-factor APT model. The signal predictors of these dual predictors are σ 2,APT,Fixed,Daily 4,vw,ew and σ 2,APT,Fixed,Daily 4,ew,ew, respectively, and we choose to use σ 2,APT,Fixed,Daily 4,vw,vw as a common noise predictor. 41 The reported coefficients, the t-statistics, and the adjusted R 2 in Panel B of Table 6 are the averages from 100 replications. 42 Using different dual predictors and the averages from many replications should avoid the concerns about data snooping. The results in Panel B of Table 6 remain quite strong after controlling for all other predictors. In particular, the adjusted explanatory power further increases from 4.54% (for the second regression in Panel A of Table 5) to 5.78% for the complete regression that employ the dual predictor consisting of σ 2,APT,Fixed,Daily 4,vw,ew and σ 2,APT,Fixed,Daily 4,vw,vw, and the dual predictor by itself contributes to 2.34 percentage points of the adjusted R 2. The large incremental explanatory power can be attributed to the very high correlation coefficient of 0.97 between the two measures in the dual predictor (see Table 3). Because the aggregate portfolio total idiosyncratic variance measures are essentially scaled versions of the aggregate total idiosyncratic variance measures of individual stocks, the coefficient estimate of the signal predictor should be multiplied by a scaler to obtain an estimate of the improved lower bound for κ in Case III of Section 2.4. The adjusted estimate will be comparable to estimates from other regressions. For example, to make the coefficient estimate of σ 2,APT,Fixed,Daily 4,vw,ew comparable to that of σ 2,APT,Fixed,Daily 1,ew (i.e., 1.86) in the second regression of Panel A of the table, we use the summary statistics in Table 2 to calculate the scaler as the ratio of the standard deviation of σ 2,APT,Fixed,Daily 4,vw,ew to that of σ 2,APT,Fixed,Daily 1,ew (i.e., / = 0.44), and the adjusted coefficient estimate is Similarly, the adjusted coefficient estimate of σ 2,APT,Fixed,Daily 4,ew,ew is Our simulation results suggest that the difference in the two adjusted coefficient estimates can be attributed to the differences in the signal-to-noise ratios of the signal predictors, the differences in the relative signal-to-noise ratios of the signal and noise predictors, and the correlations between the signal and noise predictors. Collectively, the results in Panels A and B of the table suggest 41 We have also tried to use σ 2,APT,Fixed,Daily 1,vw as the common noise predictor and the results are qualitatively similar. 42 In unreported results, we find that the 95% confidence interval of the t-statistic of the coefficient estimate for the signal predictor in the second regression in Panel B of Table 6 is [2.69, 3.95]. The 95% confidence interval of the adjusted R 2 of the same regression is [4.83%, 6.63%]. Both the 95% confidence intervals are quite tight. Therefore, the averages should be very informative. 38

41 that the pricing effect we discover does not likely arise due to unknown systematic risks or market microstructure noises Controlling for Alternative Measures of Market Risk In our main results, the lagged realized market variance typically has a negative but insignificant coefficient estimate, as opposed to a positive coefficient predicted by theory. As discussed in Section 2.7, one possibility is that the market risk measure is also too noisy, so that we may have mistakenly attributed its pricing effect to idiosyncratic risk. Therefore, we repeat our main regression analysis in Table 7 by controlling for two other measures of market risk during the period of February 1990 to December 2009, for which the VIX index data are available. 43 To be consistent with our main results in Table 5, we employ the same dual predictor (i.e., σ 2,Carhart,Within,Daily 1,ew σ 2,Carhart,Within,Daily 1,vw ) in all regressions. and Insert Table 7 Approximately Here Because the above sample period is a subperiod of the full sample period, we use the lagged realized market variance measure in Panel A of Table 7 for comparison. The dual predictor, the market variance measure, and the two sets of other predictors jointly explain 6.01% of the excess return variation over this sample period, compared to only 0.92% for the regression that excludes the dual predictor. Hence, the dual predictor on its own contributes about 85% of the adjusted explanatory power from all predictors. Interestingly, the lagged realized market variance measure has a significantly positive coefficient estimate only after the dual predictor is included in the regression. In Panel B of 7, we find that the predictive power is further strengthened when the forwardlooking VIX measure is used instead. For example, the VIX measure and the two sets of other predictors can now jointly explain 4.80% of the return variation, an increase of 3.88 percentage points from the adjusted explanatory power of a similar regression that uses the realized market variance measure instead. The dual predictor adds another 4.63 percentage points to the adjusted R 2. Similar patterns are observed when the MIDAS estimator of market risk is used in Panel C of the table. Consistent with Ghysels, Santa-Clara and Valkanov (2005), the MIDAS estimator significantly predicts future market excess returns when the two sets of other predictors are included 43 We note that the VIX index has been converted into a monthly time-series of implied market variance in the regression analysis. According to Figure 1 of Han and Lesmond (2011), both bid-ask spreads and the percentage of zero returns decline during this period. Therefore, the market microstructure noise effects on daily returns should be less of a concern for this period. 39

42 in the regression. Nevertheless, the results for our dual predictor remain very strong in the presence of the MIDAS estimator, regardless of how we control for the two sets of other predictors. 44 Since the correlation between the signal and noise predictors of the dual predictor is moderately high, the significant and positive coefficient estimate of the signal predictor provides a lower bound estimate for the risk premium due to undiversified idiosyncratic risk. It varies modestly from 1.49 to 1.98, depending on which measure of market risk is used. These results suggest that the dual predictor contains a strong pricing factor that is distinct from market risk and the existing predictors. In addition, the fact that the proxies for market risk often perform better when all other predictors are included in the regression suggests that these proxies may also be noisy and that the same noise cancellation mechanism may be at work when a relatively complete set of correlated noisy control variables are included (see the discussion in Section 2.5) Orthogonalizing to Non-Market-Variance Systematic Risk Factors and Well-Known Predictors We further check whether the pricing effect of the dual predictor arises from state variables or other systematic risks unaccounted for by conducting a two-stage regression analysis. In the first stage, we regress monthly CRSP value-weighted index excess returns on all known non-marketvariance systematic risk factors, which include the lagged factor returns, such as MKT, SMB, HML, MOM, and a liquidity risk factor, and all other existing predictors used in our previous analysis. We use two measures of the liquidity risk factor: Pástor and Stambaugh s (2003) aggregate liquidity innovation measure (denoted as PSi) and Sadka s (2006) aggregate permanent variable component of price impacts (denoted as Spv). the second-stage regression analysis. The residual returns are used as the dependent variable in This orthogonalization procedure allows us to remove the predictive power attributed to non-market-variance systematic risk factors and existing predictors from the excess returns. In addition, we separately regress each measure in the dual predictor and a measure of market risk, but not the MIDAS estimator, on the same set of variables to obtain residuals for the dual predictor and market risk measures, which are used as residual predictors in the second-stage regression. Again, we focus on the dual predictor consisting of σ 2,Carhart,Within,Daily 1,ew and σ 2,Carhart,Within,Daily 1,vw. Because the MIDAS estimator is not directly observed, it is jointly estimated along with other parameters in the second-stage regression. The sample period varies due to the availability of the liquidity risk factor measures and the VIX index data. The secondstage regression results are reported in Table 8. Insert Table 8 Approximately Here 44 We have re-estimated the same type of the MIDAS regressions over several subsample periods as well as the full sample period. Our dual predictor has very strong performance in all these sample periods. Results are available upon request. 40

43 When PSi is used as the liquidity risk factor in Panel A of the table, we show that the residual dual predictor retains its predictive power over the whole sample period for the residual excess returns after controlling for the orthogonalized realized market variance or the MIDAS estimator. Neither of the two market risk measures has a statistically significant coefficient estimate. When Spv is used instead in Panel B of the table, the results are qualitatively the same. In Panel C of the table, we fix the sample period to be from February 1990 to January 2009, for which all variables are available, and compare the performance of our residual dual predictor in the presence of three different measures of market risk. Overall, the results for the residual dual predictor are qualitatively unchanged and remain statistically strong in almost all regressions, inconsistent with the state variable interpretation. Nevertheless, the significance level of the coefficient estimate for the residual signal predictor is somewhat reduced in the presence of the residual VIX measure. This could be due to either the specific short sample period or the residual VIX measure being a cleaner proxy for market risk. Of the three market risk measures, only the residual VIX measure has a highly significant coefficient estimate, even though the coefficient estimates of the three market risk measures are all positive. Therefore, we conclude that the pricing effect of the dual predictor is independent of all known sources of systematic risk and is unlikely due to state variables Dual Predictors Based on Institutional Ownership Terciles We use institutional ownership (IO) as an alternative proxy for investor base to test two finer implications of the negative relationship between investor base and undiversified idiosyncratic risk. First, returns from stocks in the bottom IO tercile should be more sensitive to undiversified idiosyncratic risk than those in the top IO tercile. Second, total idiosyncratic risks of stocks in the bottom IO tercile should contain a larger priced component than those of stocks in the top IO tercile. Results are reported in Table 9. Insert Table 9 Approximately Here In the left (right) part of Panel A of the table, we use the equal-weighted portfolio returns of stocks in the bottom (top) tercile as the dependent variable. In both parts of the panel, we employ the signal and noise predictors of our primary dual predictor, which are constructed from all stocks within-month daily residual returns relative to the within-month Carhart model, in the predictive regressions. In the left part of Panel A, even without the noise predictor, the signal predictor quite strongly predicts future excess returns of the bottom IO tercile portfolio, regardless of how other predictors are controlled for. Specifically, in the absence of the noise predictor, the signal predictor along with all other predictors jointly explain about 10% of the variation in excess 41

44 returns, and about 24% of the predictive power comes from the signal predictor and the realized market variance. Using the dual predictor further increases the predictive power of the regression to about 13%. Nevertheless, removing the dual predictor from the regression decreases the predictive power of the regression to 9.43%. In contrast, in the right part of the panel, we find that without the noise predictor the signal predictor only weakly predicts future excess returns of the top IO tercile portfolio. The signal and noise predictors together increase the predictability impressively, but the overall predictive power is about 5%, much smaller than that for the case of the bottom IO tercile portfolio. The dual predictor on its own contributes to about 1 percentage point of the adjusted explanatory power of the regression. Because we employ the common signal predictor and the common dual predictor in the same types of regressions, these differential effects are consistent with the first implication and suggest that there exists a negative relation between investor base and the pricing effect of undiversified idiosyncratic risk. In both the left and right parts of Panel B of the table, we use the monthly CRSP value-weighted index excess return as the common dependent variable. Because the aggregate total idiosyncratic variance measures of stocks in the bottom IO tercile are hypothesized to have a larger priced component than those of stocks in the top IO tercile, we use the residuals of a time-series regression of the equal-weighted aggregate total idiosyncratic variance measure of stocks in the bottom IO tercile on the value-weighted (equal-weighted) aggregate total idiosyncratic variance measure of stocks in the top IO tercile as the differential signal (noise) predictor in the left part of the panel. For comparison, we use our primary signal and noise predictors in the same types of regressions in the right part of the panel. As shown in the left part of the panel, the differential signal predictor significantly predicts monthly CRSP value-weighted index excess returns when used alongside with the realized market variance. The level of significance for the coefficient estimate of the differential signal predictor declines when we control for the two sets of other predictors. Nevertheless, adding the differential noise predictor restores the level of significance for the coefficient estimate of the differential signal predictor and increases the adjusted explanatory power by about 1.5 percentage points. In contrast, in the right part of the panel, we find that the signal predictor of our primary dual predictor has a positive but insignificant coefficient estimate in the absence of the noise predictor. Nevertheless, the dual predictor has again very significant predictive power with the incremental explanatory power being about 1.2 percentage points. These results are consistent with the second implication and suggest that the priced component in idiosyncratic risk is indeed negatively related to investor base. 42

45 4.8. Subsample Tests To further examine whether our main results are robust to different sample periods, we divide the whole sample period into two equal subsample periods: from July 1963 to September 1986 and from October 1986 to December Regression results for the two subsample periods are reported in Panels A and B of Table 10, respectively. The dependent variable is the monthly CRSP value-weighted index excess return. We use two sets of dual predictors, which are constructed from within-month daily stock residual returns relative to the within-month Carhart model or the daily rolling-window six-factor APT model. Insert Table 10 Approximately Here In general, the two sets of dual predictors produce similar results in each subsample period. We thus focus our discussion on the dual predictor based on the within-month Carhart model. In the first regression of Panel A of Table 10, the dual predictor together with the realized market variance explains 2.17% of the variation in future market excess returns in the first subsample period. In Panel B of the table, the same regression explains 4.25% of the variation in future market excess returns in the second subsample period. In contrast, the existing predictors have much weaker performance in the recent subsample period than in the early subsample period. 45 When all predictors are included in the regression, the adjusted R 2 increases to 10.78% for the first subsample period, but decreases to 3.64% for the second subsample period. These results suggest that our dual predictor has more stable predictive power over the full sample period and has become increasingly important in the second subsample period. Nevertheless, the magnitude of the coefficient estimate of the signal predictor in the dualpredictor regression over the first half of the sample period is about two to three times as large as that for the second half of the sample period. 46 Hence, the risk premium per unit of undiversified idiosyncratic risk has dropped over time. Despite the declining risk premium, the total risk premium for one standard deviation change in undiversified idiosyncratic risk may still be comparable across the two subsample periods, because the volatilities of both the signal and noise predictors have tripled in the second half of the sample period. In addition to the CRSP index, it is also interesting to know if idiosyncratic risk matters to other major equity indexes. We perform similar analysis in Panels A through C of Table In unreported results, we find that all control variables (including the realized market variance) collectively explain 7.97% and 1.61% of the excess return variation in the first and second subsample periods, respectively. 46 The level of noise reduction in the dual-predictor regression should be similar for the two subsample periods because there is no substantial change in the correlations between the signal and noise predictors (0.80 in the first half vs in the second half). 43

46 for the NYSE/AMEX Composite Index, the Nasdaq Composite Index, and the S&P 500 Index, respectively. 47 This exercise allows us to show that our main results in Table 5 are not confined to small stocks in the market. For brevity, we limit our attention to the dual predictor based on the within-month Carhart model. 48 Insert Table 11 Approximately Here We highlight three features of the results in Table 11. First, we reject the null hypothesis that idiosyncratic risk does not matter in all three indexes, regardless of whether other popular predictors are controlled for. Second, as expected, the results are much stronger for the Nasdaq index because it contains more small and young firms. In particular, the dual predictor and the realized market variance jointly explain 5.04% of the variation in future excess returns on the Nasdaq Index. Third, the results for the excess returns on the NYSE/AMEX and S&P 500 indexes are slightly weaker than, but comparable to, those for the CRSP value-weighted index excess returns in Panel A of Table 5, consistent with the fact that large and mature firms are dominant in the two indexes. Nevertheless, the lower bound estimate for the risk premium κ is still close to 1. It is possible that the pricing effect remains strong for indexes that concentrate on large stocks, but the dual predictor is ineffectively constructed. Consequently, in Section 4.10 we re-examine the issue with an alternative dual predictor, which is expected to achieve better noise reduction Dual Predictors Based on Monthly Residual Returns The results we have presented so far exclusively use the aggregate total idiosyncratic variance measures constructed from daily stock or portfolio residual returns as dual predictors. We explore another way to sidestep the market microstructure noise issues by repeating our main exercise in Table 12, using the equal- and value-weighted aggregate total idiosyncratic variance measures constructed from monthly stock residual returns as dual predictors. The dependent variable is the monthly CRSP equal-weighted index excess return. Panels A and C of Table 12 display the results for the measures based on the monthly fixed-period Carhart model and the monthly fixed-period six-factor APT model, respectively. Panels D and E of the table report the results for the measures based on the monthly rolling-window Carhart model and the monthly rolling-window six-factor APT model, respectively. In Panel B of the table, we report the results for the measures based on the monthly fixed-period Carhart-EGARCH model. 47 We obtain similar results for the NYSE size decile value-weighted portfolios and the results are available from the authors. 48 The results are qualitatively the same if the dual predictor is instead constructed from within-month daily stock residual returns from the six-factor APT model. 44

47 Insert Table 12 Approximately Here A close look at Table 12 reveals that the results in all panels are qualitatively the same and that the coefficient estimate of the signal predictor in each panel is positive and statistically significant at the 10% level or better. Comparing the results for the measures based on the same estimation period but different factor models, we find that the coefficient estimate of the signal predictor based on the monthly six-factor APT model tends to be slightly less than that of the signal predictor based on the monthly Carhart model, regardless of the choice of fixed periods or rolling windows. This may be attributed to our conservative use of a multifactor model in removing additional common factors. The coefficient estimate of the signal predictor based on the monthly fixed-period Carhart-EGARCH model is about two to three times as much as that of the signal predictor based on the monthly fixed-period Carhart model or the monthly fixed-period six-factor APT model. This can be attributed to a much lower standard deviation of the signal predictor based on the monthly fixed-period Carhart-EGARCH model and a higher correlation of 0.88 between the signal and noise predictors based on the same model Dual Predictors Based on a Super-Value Weighting Scheme Our analysis so far relies on the standard equal- and value-weighting schemes to produce the dual predictor consisting of two highly correlated measures with different signal-to-noise ratios. As discussed in Sections 2.3 and 2.6, in the presence of the errors-in-variables problem, two highly correlated measures will create a noise cancellation mechanism, which is the fundamental idea behind the dual-predictor regression. Even though the standard errors of the coefficient estimates increase with multicollinearity in finite samples, our simulation results suggest that very good noise cancellation can still lead to high test power, even for measures with low signal-to-noise ratios. We thus use the value- and super-value-weighted aggregated total idiosyncratic variances as the signal and noise predictors in the dual-predictor regression. The two measures are constructed from within-month daily stock residual returns relative to the within-month Carhart model or the daily rolling-window six-factor APT model. The signal and noise predictors in each of the two sets of dual predictors are nearly perfectly correlated with each other (i.e., a correlation more than ) by construction. Results for this exercise are reported in Table 13. Insert Table 13 Approximately Here The dependent variables in Panels A and B of the table are the monthly excess returns on the CRSP value-weighted index and the largest NYSE size decile value-weighted portfolio, respectively. We find that both the signal and noise predictors have highly significant coefficient estimates with 45

48 the predicted signs. Holding both the dependent variable and the estimation period unchanged, we show that the results for the new dual predictor in Panel A of the table are statistically and economically much stronger than those documented for other dual predictors in Panels B and D of Table 4, Panel A of Table 5, and Panel A of Table 6. Since Case IV of Section 2.4 is invoked, β 1,d + aβ 2,d is an improved lower bound for κ. Because the two measures in each dual predictor are very similar, we assume 0.98 a 0.99, 49 which leads to a relative signal-to-noise ratio in the range of 0.98σ 1 /σ 2 aθ 2 /θ σ 1 /σ 2. If we further assume σ(x i ) σ i, where i = 1, 2, the value range for the relative signal-to-noise ratio can be explicitly calculated. For example, the value range is aθ 2 /θ for the measures based on the within-month Carhart model. We report the results for the F tests of β 1,d +aβ 2,a = 0 for a = 0.98, , 0.985, , and In almost all cases, we reject the null hypothesis that the lower bound is zero at the 10% significance level or better. Because the value-weighted aggregate total idiosyncratic variance measure (i.e., the signal predictor in this table) has a much lower standard deviation than the equal-weighted measures (e.g., the signal predictor in Table 5) in other tables. The lower bound estimate ˆβ 1,d +a ˆβ 2,d should be adjusted for the difference in standard deviations to be comparable to the lower bound estimates in other tables. When the measures are based on the within-month Carhart model (the daily rolling-window six-factor APT model), the adjusted lower bound estimates from the complete model are between 0.59 and 3.53 (between 1.37 and 3.77) and between 0.62 and 3.07 (between 1.16 and 3.07) for excess returns on the CRSP value-weighted index and the largest NYSE size decile value-weighted portfolio, respectively. 51 Alternatively, if we are willing to assume 0.04 θ Max 0.10, we can use R 2 d /(1 R2 d )/θ Max as a lower bound for the underlying parameter (see the discussion in Section 2.4). The adjusted R 2 s look impressive across all the regressions. For example, in the upper part of Panel A of Table 13 the regression that employs the new dual predictor (based on the within-month Carhart model) and the realized market variance has an adjusted R 2 of 4.16%, compared to 2.74% for a similar regression in Panel B of Table 4. When all other predictors are controlled for, the adjusted R 2 increases to 7.21%, compared to 4.54% for a similar regression in Panel A of Table 5. Similarly, the adjusted R 2 of the complete model in the upper part of Panel B of the table is still as high as 5.88%, even for the index composed of the largest stocks. Using the marginal adjusted R 2 attributed to 49 Because ˆσ(x 2) 1, this value range is much more conservative than â = 2 10 ˆσ(x 1 ˆσ(x 2) ) 9 ˆσ(x , which is used in our ) simulation analysis. 50 We have also tested the null hypothesis of β 1,d + β 2,d = 0, and reject the null at the 10% significance level or better in all regressions. The estimate of β 1,d + β 2,d, however, is negative and cannot be a meaningful lower bound. 51 The adjusted lower bound estimate is ˆσ(x 1) ˆσ(x ( ˆβ 1 ) 1,d + a ˆβ 2,d ), where x 1 is the signal predictor in Table 13 and x 1 is the signal predictor in Table 5. 46

49 the new dual predictors in the complete model, we estimate the lower bound for κ to be between 1.98 and 4.95 (using the measures based on the Carhart model) or between 1.73 and 4.31 (using the measures based on the APT model) for the CRSP value-weighted index excess returns and between 1.74 and 4.34 (using the measures based on the Carhart model) or between 1.41 and 3.54 (using the measures based on the APT model) for the largest NYSE size decile value-weighted portfolio excess returns. Due to better noise reduction, the new estimate of 1.98 to 4.95 for the CRSP value-weighted index excess returns is much higher than the estimate of 1.26 from the complete model in Panel A of Table Other Robustness Checks It is well known that the coefficient estimates in the OLS regressions are sensitive to outliers, and thus our tests may be affected by highly skewed disturbance in finite samples. We further assess the robustness of our main results by adopting two approaches. 52 First, we repeat our analysis in Table 5 by removing the five market crashes or the five most volatile months over the sample period. 53 We have also tried to drop the 60 most influential observations sequentially in the regression analysis. In all these cases, we obtain qualitatively similar and statistically strong results. Second, because quantile regressions (see Koenker and Hallock, 2001) allow us to establish a linear relation conditional on the median or some other quantile, we use them to describe a more complete and robust linear relationship between the excess returns and the dual predictor. We find that the results for our primary dual predictor (based on the within-month Carhart model) are remarkably similar to those for the dual-predictor regressions in Panel A of Table 5, irrespective of whether the conditional quantile is the 25th percentile, the median, or the 75th percentile. Even though the adjusted R 2 of the dual-predictor regression provides only a lower bound for the population explanatory power of the priced component, the marginal adjusted R 2 s of 1-4% for most regressions are quite impressive. Nevertheless, there still might be a statistical concern as to whether the predictability is by chance. Foster, Smith and Whaley (1997) have shown that the classical goodness-of-fit can be misleading for statistical inference when researchers collectively publish only significant predictors out of a large pool of variables. As a result, they suggest a new cutoff value for R 2 based on the distribution of the maximal R 2 to eliminate the variable-selection problem. This issue is unlikely to be a concern in our study because we report results for so many 52 Due to space limitation, these results are removed from the final version of the paper, but are available upon request. 53 The five market crashes are defined as the five lowest monthly returns on the CRSP value-weighted index from July 1963 to December 2009, and they occurred in November 1973 (oil shocks), March 1980 (silver price crash), October 1987 (black Monday), August 1998 (Russian financial crisis), and October 2008 (financial crisis triggered by the subprime mortgage crisis). The five most volatile months are June 1970, November 1987, August 2002, November 2008, and December

50 different versions of dual predictors. To safeguard our findings, we have used the bootstrap method to compute the critical values for the adjusted R 2 s of the last regression in Panel B of Table 4, the first regression in Panel A of Table 5, and the first regression in Panel A of Table 13 under the assumption of no return predictability. The adjusted R 2 s of our dual-predictor regressions are significant at the 5% or 10% level. 5. Concluding Comments According to Merton (1987), when investors hold relatively diversified portfolios, it should be the case that only a small portion of total idiosyncratic risk is not diversified away and is thus priced in stock returns. However, the undiversified portion of idiosyncratic risk can still have a material economic effect on equity risk premiums, even though it may be relatively small for most stocks. Because the undiversified portion of idiosyncratic risk is unobservable, total idiosyncratic risk is commonly used as a proxy for its undiversified portion in empirical studies. Such a practice may, however, have introduced substantial noise into the existing empirical tests, thereby reducing their statistical power of detecting an economically significant effect of idiosyncratic risk on stock returns. Consequently, the extant evidence tends to support the conclusion that idiosyncratic risk has no pricing role. We propose a simple regression-based method that can, under realistic conditions, substantially increase the power of standard tests to detect the pricing role of idiosyncratic risk, while sacrificing little in the sizes of these tests. The methodology is developed in the context of the time-series analysis of whether aggregate undiversified idiosyncratic risk can predict future market excess returns. Specifically, we employ two different but highly correlated noisy measures of aggregate undiversified idiosyncratic risk in the same regression. The less noisy measure is treated as the signal predictor and the other one as the noise predictor. Highly correlated variables in the same regression induce multicollinearity, which decreases test power in the absence of the errors-invariables problem. Nevertheless, the high correlation between the two noisy predictors is mostly attributed to that between the two dominant noise components. In the presence of the errors-invariables problem, we show that multicollinearity can generally increase test power through noise cancellation. As a result, the dual-predictor regression improves test power over the conventional univariate-predictor regression. Our new approach can be applied to tests in other contexts, where large measurement noises deteriorate test power. In general, large stocks or stocks with high aggregate institutional ownership should have relatively small undiversified idiosyncratic risk components, as suggested by Merton (1987). We construct numerous aggregate total idiosyncratic variance measures with different weighting schemes, 48

51 in which the weights are based on market capitalizations and/or the levels of aggregate institutional ownership. Different weighting schemes allow us to create highly correlated noisy predictors with different signal-to-noise ratios. Using these measures as the dual predictor, we strongly reject the null hypothesis that idiosyncratic risk does not matter in all major equity indexes, which include a few indexes that represent the largest stocks in the market. Interestingly, we find that the predictive power of market risk often improves in the presence of the priced component of aggregate total idiosyncratic risk, consistent with Merton s (1987) theory. In particular, we show that the dual predictor and the realized market variance can jointly explain more than 4% of the variation in future monthly CRSP value-weighted index excess returns. We estimate that more than 4% of the total market risk premium can be attributed to every one percent of aggregate total idiosyncratic risk that is not diversified away. Extensive robustness checks suggest that the pricing effect documented in our study cannot be attributed to common risk factors in stock returns or state variables. Since the noise reduction in our approach is imperfect in general, the estimated economic effect of aggregate undiversified idiosyncratic risk should largely account for only a small fraction of the true economic effect. Despite the remaining downward bias, the predictive power of the dual predictor is already quite strong when compared to that of other well-known predictors. In addition, the nonzero true coefficient of the signal predictor from our tests provides a basis for such unbiased estimation method as the higher-order moments approach of Erickson and Whited (2002) and Erickson, Jiang and Whited (2013) to be applied to the time-series analysis. We leave the unbiased estimation exercise for future research. Our new time-series evidence suggests that measurement issues can provide a rationale that helps to reconcile the debate over the mixed cross-sectional evidence. The spirit of our method could be useful for resolving the debate. Finally, our new time-series evidence also suggests that the dual predictor should help to estimate the time-varying conditional expected return more accurately. A closely related research question is whether the dual predictor can improve investors dynamic asset allocation decisions. It is interesting to see if reliable trading strategies based on the dual predictor can be designed and implemented. This issue can also be an interesting future research topic. 49

52 Appendix The simulations in Section 2.6 are implemented under the assumption that the signal component, v t, and the two noise components, s 1,t and s 2,t, follow first-order autoregressive processes with autocorrelation coefficients φ v, φ 1, and φ 2, respectively. For notational convenience, we define a vector m t = (r m,t v t s 1,t s 2,t ) to represent the market excess return, the signal component, and the two noise components. m t jointly evolves over time according to: m t = φm t 1 + ξ t, (22) where ξ t = (ε t ξ v,t ξ 1,t ξ 2,t ) is a vector of independently and identically distributed innovations with zero means at time t. The autocorrelation coefficients matrix φ can be written as follows: φ = 0 κ φ v φ φ 2. (23) The variance-covariance matrix of ξ t can be written as follows: σε Σ = E(ξ t ξ 0 σv(1 2 φ 2 v) 0 0 t) = σv (1 φ 2 θ1 2 1 ) ρσv 2 θ 1 θ 2 (1 φ 1 φ 2 ) ρσv σv θ 1 θ 2 (1 φ 1 φ 2 ) 2 (1 φ 2 θ2 2 2 ), (24) where θ 1 = σ v /σ 1 and θ 2 = σ v /σ 2. Under the null hypothesis of κ = 0, the standard test statistics do not depend on σε 2 and σv, 2 and we assume σε 2 = σv 2 = 1 if ε t follows a normal distribution or σv 2 = 1 if ε t follows a Student s t distribution with four degrees of freedom (i.e., σε 2 = 2). Under all alternative hypotheses of κ 0, we assume, without loss of generality, κ = 1. If ε t follows a normal distribution, we further assume, without loss of generality, σv 2 = 1 so that σε 2 = 1 1, where R 2 is the population explanatory R 2 power of the true predictive model. If ε t follows a Student s t distribution with four degrees of freedom (i.e., σε 2 = 2), we have σv 2 = 2R2. 1 R 2 Evaluating the power of a test in the univariate- or dual-predictor regression under a particular alternative hypothesis requires a specification of the parameter set {R 2, θ 1, aθ 2, ρ, φ v, φ 1, φ 2 }. For the t test of β 1,d = 0 (see Eq. (8a)), we study both a special pure noise case (i.e., a = 0 or aθ 2 /θ 1 = 0) and a general mixed noise case (e.g., a = 1 or aθ 2 /θ 1 0). For the F test of β 1,d + aβ 2,d = 0 in the case of ρ = 0.99 or 0.999, we compare a case with known values of a (i.e., 0 < a 1) to a case with a conservative estimate of a (e.g., â = ˆσ(x 2) ˆσ(x 1 ) under the assumptions of 0 < θ and 0 < aθ 2/θ ; see Section 2.4). The size and power of a test are the rejection rates of the null hypothesis of κ = 0 at a nominal 5% significance level. In Figures 1 through 4, we generate 10, 000 independent samples of ε t, v t, s 1,t, and s t,2 by assuming ξ t N(0, Σ) for each set of parameter values under the alternative hypotheses. In Table 1, we generate 10, 000 independent samples of ε t, v t, s 1,t, and s t,2 for each set of parameter values under the alternative hypotheses by assuming that ε t follows a Student s t distribution with four degrees of freedom and that the innovations to the signal and two noise components follow a multivariate normal distribution. Each sample contains N observations with initial values v 0, s 1,0, and s 2,0 drawn randomly from the respective marginal distributions of ξ v,t, ξ 1,t, and ξ 2,t. 50

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56 Power Power ρ Sample Size ρ Sample Size (a) Univariate-Predictor Regression (b) Dual-Predictor Regression Figure 1: Test Power of the t Test in the Univariate- and Dual-Predictor Regressions For the alternative hypothesis of κ 0, we set κ = 1, R 2 = 0.10, θ 1 = 0.20, aθ 2/θ 1 = 0.20, ρ {0.75, 0.80, 0.85, 0.90, 0.95, 0.975, 0.99, 0.999}, and φ v = φ 1 = φ 2 = We use Monte Carlo experiments to simulate the rejection rates of the t test of β 1,u = 0 in the univariate-predictor regression and those of the t test of β 1,d = 0 in the dual-predictor regression in Panels (a) and (b), respectively. The rejection rates are based on 10,000 independent samples. The sample size increases from 100 to 600 by an increment of 100. Average Coefficient Average Coefficient Sample Size ρ Sample Size ρ (a) Univariate-Predictor Regression (b) Dual-Predictor Regression Average Coefficients of the Signal Predictor in the Univariate- and Dual-Predictor Regres- Figure 2: sions For the alternative hypothesis of κ 0, we set κ = 1, R 2 = 0.10, θ 1 = 0.20, aθ 2/θ 1 = 0.20, ρ {0.75, 0.80, 0.85, 0.90, 0.95, 0.975, 0.99, 0.999}, and φ v = φ 1 = φ 2 = We use Monte Carlo experiments to compute the average coefficient estimates of the signal predictor in the univariate- and dual-predictor regressions in Panels (a) and (b), respectively. The average coefficient estimates are based on 10,000 independent samples. The sample size increases from 100 to 600 by an increment of

57 Average Adj. R-squared (%) Sample Size ρ Average Adj. R-squared (%) Sample Size ρ (a) Univariate-Predictor Regression (b) Dual-Predictor Regression Figure 3: Average Adjusted R 2 s of the Univariate- and Dual-Predictor Regressions For the alternative hypothesis of κ 0, we set κ = 1, R 2 = 0.10, θ 1 = 0.20, aθ 2/θ 1 = 0.20, ρ {0.75, 0.80, 0.85, 0.90, 0.95, 0.975, 0.99, 0.999}, and φ v = φ 1 = φ 2 = We use Monte Carlo experiments to compute the average adjusted R 2 s of the univariate- and dual-predictor regressions in Panels (a) and (b), respectively. The average adjusted R 2 s are based on 10,000 independent samples. The sample size increases from 100 to 600 by an increment of 100. Power Power aθ2/θ1 Sample Size aθ2/θ1 Sample Size (a) ρ = 0.85 (b) ρ = 0.95 Figure 4: Test Power of the t Test in the Dual-Predictor Regression For the alternative hypothesis of κ 0, we set κ = 1, R 2 = 0.10, θ 1 = 0.20, aθ 2/θ 1 {0, 0.10, 0.20, 0.30, 0.40, 0.50, 0.60}, ρ {0.85, 0.95}, and φ v = φ 1 = φ 2 = ρ = 0.85 and 0.95 in Panels (a) and (b), respectively. We use Monte Carlo experiments to simulate the rejection rates of the t test of β 1,d = 0 in the dual-predictor regression. The rejection rates are based on 10,000 independent samples. The sample size increases from 100 to 600 by an increment of