Predictability of Stock Returns: A Quantile Regression Approach

Transcription

1 Predictability of Stock Returns: A Quantile Regression Approach Tolga Cenesizoglu HEC Montreal Allan Timmermann UCSD April 13, 2007 Abstract Recent empirical studies suggest that there is only weak evidence of predictability of the mean or variance of stock returns at the monthly horizon. Little is known, however, about the extent to which other parts of the return distribution are predictable. We explore this issue in a quantile regression framework and looks into whether a range of common predictor variables proposed in the finance and macro literature are helpful in predicting specific quantiles of the stock return distribution. Out-of-sample forecasts suggest that tails of the return distribution can be predicted by means of time-varying state variables and that there are gains from combining forecasts from univariate quantile models. 1 Introduction Predictability of stock returns has been the subject of a large literature in financial economics. Most studies have focused on modeling the conditional mean of the ex post return distribution using full-sample estimates, see, e.g., Campbell (1987), Fama and French (1988, 1989), Ferson (1990), Ferson and Harvey (1991), Glosten, Jagannathan and Runkle (1993). Recent studies such as Goyal and Welch (2003, 2007) question whether time-variations in mean stock returns are predictable in an ex ante sense (i.e. out of sample) by means of state variables such as valuation ratios (e.g. the book-to-market ratio, earnings-price ratio or the dividend yield) or a host of other financial and macroeconomic indicators including default premia, short and long interest rates and inflation. 1 Largely overlooked in the recent debate on return predictability is the fact that investors generally are concerned not only with predictability of mean returns. Outside the classical mean variance framework, investors with more general preferences e.g. constant relative risk aversion or standard risk aversion 1 Pesaran and Timmermann (1995) and Bossaerts and Hillion (1999) also emphasize the need to use out-of-sample evidence to evaluate return predictability. Campbell and Thompson (2007) and Cochrane (2006) provide different perspectives on variations in the conditional mean of stock returns. 1

2 preferences as in Kimball (1993) need an estimate of the full return distribution to compute expected utility and derive their optimal portfolio holdings. While a large body of work has proposed models for the conditional mean and variance of stock returns, far less work has been undertaken towards modeling the full return distribution. Indeed, estimating this distribution is fraught with problems. For example, although a model for the full return distribution provides an estimate of the chances of extreme tail events, these tails are generally very difficult to estimate precisely. As a consequence, estimates of extreme tail events are effectively driven by parametric assumptions about the shape of the tail. An alternative to modeling the full distribution of asset returns is to extend the mean-variance analysis to consider the role that higher order moments such as the skew and kurtosis of returns play in determining investors portfolio choice. Indeed, accounting for higher order moments such as the skew and kurtosis can have important implications for the portfolio allocation of investors with power utility (Harvey, Liechty and Liechty (2004), Guidolin and Timmermann (2006)). It also appears to have implications for the cross-section of stock returns in the sense that exposure to the skew of the market portfolio or to downside risk more generally is priced in the cross-section (Harvey and Siddique (2000), Dittmar (2002) and Ang, Chen and Xi (2006)). Unfortunately, however, estimates of higher order moments of returns such as the skew and kurtosis of returns are known to be highly sensitive to outliers. In this paper we pursue a novel approach that focuses on predictability of quantiles located at different points of the return distribution. Given sufficiently many quantiles, we obtain a clear picture of how the return distribution depends on a set of state variables. Moreover, estimates of the quantiles are more robust with respect to the outliers known to affect stock returns and can thus be estimated with greater precision. In particular, we adopt quantile regressions that specify quantiles of the return distribution as a function of a set of (lagged) predictor variables proposed in the literature on return predictability. The conditional quantiles estimated in this manner encompass more conventional measures used to characterize the return distribution such as the mean, variance, skew and kurtosis. For example, quantiles can be used to capture location information through the median value, scale information through the inter-quartile range, skewness through differences between the mean and median value and kurtosis through tail quantiles such as the 5%, 10% and 90% or 95% quantiles. Our empirical analysis considers the predictability of monthly stock returns in the US over a long historical sample stretching back as far as Consistent with recent findings, few of the predictor variables considered in the literature summarized recently by Goyal and Welch (2006) have predictive power over the mean of the return distribution. However, this failure to predict the mean does not imply that these variables are useless in predicting other parts of the return distribution. By considering a wide range of quantiles, we document how various state variables forecast different points of the return distribution. We find that many 2

3 of the variables considered in the literature are useful in predicting either the left or right tails or the shoulders of the return distribution. For example, consistent with the traditional finding of a negative short-term relation between inflation and stock returns, higher inflation leads to systematically smaller values of the quantiles of the return distribution but the effect is only significant in the right tail of the distribution. The strongest effect of higher inflation is therefore to lower the upside potential of the market. Similarly, a higher default return spread has no impact on the central or left side of the return distribution but has a large negative effect on the right tail, lowering the prospects of large positive returns. Conversely, the term spread has no effect on the right side of the return distribution but has a large positive effect on the left side, thus reducing the prospects of large negative future returns. Increases in variables such as the stock variance or the net equity expansion also have the effect of shifting the left side of the return distribution further downwards, thus increasing the probability of large negative returns without similarly affecting the center or the right tail of the return distribution. The outline of the paper is as follows. Section 2 presents the quantile modeling approach. Section 3 introduces the data set and presents empirical estimation results. Section 4 conducts an out-of-sample forecasting experiment and compares the proposed models to alternatives from the existing literature. Section 5 concludes. 2 Modeling Quantiles of Stock Returns At the heart of any expected utility maximizing investor s portfolio selection problem lies a model of the distribution of returns on one or multiple risky assets. For some special cases, e.g. mean-variance preferences, only the first and second moment of the return distribution are required to solve this problem. In general, however, more detailed information on the return distribution is needed to solve for the optimal portfolio weights. As a step towards understanding how the return distribution may depend on economic state variables, an alternative approach is to focus on quantiles located at separate points of the return distribution. Let α (0, 1) represent a particular quantile of interest. By varying α from values near zero (representing the left tail) through middle values near one-half (representing the central part of the distribution) to values near one (representing draws from the right tail of the return distribution) we can track variations in the return distribution. Moreover, by considering a large number of quantiles, this analysis offers a much richer picture of variations in the return distribution than the usual focus on the mean and variance and can be used to indicate evidence of skew or kurtosis. It can also be used to uncover periods with the potential for unusually large negative or positive returns or to form confidence intervals for the return distribution (see, e.g., Taylor and Bunn (1999)). 3

4 2.1 Model Specification To understand how we obtain estimates of the quantiles, note that the conventional practice of forecasting the mean stock return assumes a squared loss function L(e t+1 ) = e 2 t+1, where e t+1 = r t+1 ˆf t is the forecast error and ˆf t is the forecast of the future return, r t+1. Under this loss function, the optimal return forecast is the conditional mean. Similarly, under mean absolute error loss (L(e) = e ), the optimal forecast is the conditional median. Following Koenker and Bassett (1978), quantile estimation proceeds from the so-called tick loss function L α (e t+1 ) = (α 1{e t+1 < 0})e t+1, (1) where α (0, 1) is the quantile. Under this loss function, the optimal forecast is the conditional quantile. This follows from noting that the first order condition of (1) with respect to the forecast, ˆf t, is α + F ( ˆf t ) = 0, where F is the distribution function of returns. It follows immediately that ˆf t = F 1 (α), i.e. the α quantile of the return distribution, see, e.g., Koenker (2005). More specifically, we model the conditional α-quantile of stock returns as a linear function of a set of potential predictor variables. Following Engle and Manganelli (2004), our specification also includes last period s conditional quantile and the absolute value of last period s return. These variables can be used to capture autoregressive dynamics similar to the persistence found in the literature on volatility of returns (e.g. Christoffersen and Diebold (2000); see also Koenker and Ahzo (1996)). Hence, the conditional α-quantile of stock returns, q α (r t+1 F t, θ α ), is modeled as q α (r t+1 F t, θ α ) = β 0,α + x t β 1,α + β 2,α q α (r t F t 1, θ α ) + β 3,α r t (2) where θ α = (β 0,α, β 1,α, β 2,α, β 3,α ) is the vector of model parameters, while q α (r t F t 1, θ α ) is the lagged α quantile, r t is the absolute return and x t is any state variable known at time t. Note that a model that assumes constant (time-invariant) quantiles arises as a special case of (2) with β 1,α = β 2,α = β 3,α = 0, q α (r t+1 F t, θ α ) = β 0,α. (3) 2.2 Scale-Location Estimates of Quantiles It is useful to compare this approach to the common practice of assuming that stock returns belong to a class of conditional scale-location families with a timeinvariant distribution of the residuals. Letting r t+1 be the stock return in period 4

5 t + 1, measured in excess of the risk-free rate, this approach assumes r t+1 = µ t + σ t ε t+1, (4) where µ t and σ t are the conditional mean and volatility, respectively. The conditional mean and variance are assumed to be based on investors information at time t, F t, which may comprise a set of state variables, X t. Since it has been normalized, the return innovation ε t+1 has (conditional) mean zero and variance one. For given values of the conditional mean and variance and the cumulative density function of the return innovation, ε t+1, F ε, the α quantile of r t+1, denoted q α, is given by q α (r t+1 F t, θ) = µ t + σ t F 1 ε (α). (5) Application of this result requires that (i) returns are drawn from a conditional scale-location family so only the mean and the variance are time-varying; (ii) estimates of both µ t and σ t are available; (iii) estimates of the distribution of the return innovation F ε can be computed. The most common example of this approach is the GARCH(1,1) model which takes the form r t+1 = µ t + σ t ε t+1, (6) σt 2 = λ 0 + λ 1 ε 2 t + λ 2 σt 1 2 ε t+1 N(0, 1). For a given set of parameter estimates, (5) and (6) can be used to to compute quantile estimates. 2.3 Estimation To obtain the parameters of the conditional quantile specification (2), we adopt the tick-exponential quasi maximum likelihood estimation (QMLE) approach to estimating conditional quantile models proposed in Komunjer (2005). The parameter estimates solve where ˆθ α,t = arg max T 1 θ α T ln ϕ α t (r t, q α (r t F t 1, θ α )) (7) t=1 ϕ α t (y, η) = exp( 1 1 (η y)1{y η} + (η y)1{y > η}). (8) α 1 α Here 1{ } is the indicator function, ϕ α t is a probability density from the tickexponential family and η is the α quantile of ϕ α t. This approach builds on the regression quantile method introduced by Koenker and Bassett (1978). In this setting, we restrict the parameter on the lagged conditional linear quantile, 5

6 β 2, to be between 0 and 1. Estimation of the conditional linear quantile model in (2) requires the choice of initial quantile, q α (r 1 F 0 ). In a setting without any covariates in the linear quantile specification, the natural choice is to set the initial quantile to the unconditional constant quantile of returns. However, in our setting with covariates, we recursively estimate the model several times using the conditional quantile in t = 2 from the previous estimation as the initial quantile where we use unconditional constant quantile as the initial quantile in the first estimation. Komunjer (2005) establishes conditions under which the QMLE estimates, ˆθα,T, are asymptotically normally distributed, T (ˆθ α,t d θ α ) N(0, Σ α ) where Σ α is the covariance matrix. A consistent estimator of the asymptotic covariance matrix can be obtained by numerical differentiation as discussed in Komunjer (2005). 3 Empirical Results In this section we present empirical results from applying the autoregressive quantile specification with time-varying predictor variables introduced in the previous section to data on US stock returns. 3.1 Data Our empirical analysis uses the data set comprising monthly stock returns along with a set of sixteen predictor variables analyzed in Goyal and Welch (2006). 2 The data sample varies across predictor variables with the longest sample spanning the period , while the shortest sample covers the period Stock returns are captured by the S&P500 index and includes dividends. A short T-bill rate is subtracted from stock returns so we are modeling excess returns. The predictor variables we consider along with the data samples are listed in Table 1. Most variables fall into three broad categories, namely (i) valuation ratios capturing some measure of fundamental to market value such as the dividend price ratio (d/p), the dividend yield (d/y), earnings- price ratio (e/p), 10-year earnings-price ratio (e10/p) or the book-to-market ratio (b/m); (ii) bond yield measures capturing level (the three-month T-bill rate (tbl) and yield (lty) on long term government bonds), slope (term spread (tms = lty tbl)) or default risk (default yield spread (dfy, the yield spread between BAA and AAA rated corporate bonds), and default return spread (df r, the difference between the yield on long-term corporate and government bonds)) effects; (iii) estimates of equity risk (cross-sectional equity premium (csp, the relative valuations of high- and low-beta stocks), long term return (ltr) and stock variance (svar, a volatility estimate based on daily squared returns)). Finally, two variables from corporate finance, namely the dividend payout ratio (d/e, the log of the dividend-earnings ratio), and net equity expansion (ntis, the ratio of 12-month 2 We are grateful to Amit Goyal for providing this data. 6

7 net issues by NYSE-listed stocks over the end-year market capitalization) and a macroeconomic variable (inflation (inf l, the rate of change in the consumer price index)) are considered. 3.2 Estimation Results As a precursor to our quantile analysis, we regress monthly stock returns on each of the individual predictor variables lagged one period. Table 2 presents OLS estimates for these univariate linear predictive regressions, in each case using the full set of available data. Only three of sixteen variables (inflation, cross-sectional premium and net equity expansion) have statistically significant predictive power for the mean of stock returns at the 10% critical level. At the 1% level, only one variable (net equity expansion) has a statistically significant coefficient. We conclude from these results that predictability of the mean of US stock returns is rather weak. This evidence is consistent with recent discussions of return predictability by Goyal and Welch (2003, 2006). However, we cannot conclude that these predictor variables fail to be useful for predicting other parts of the return distribution. For example, it could well be that a variable can predict events in the left tail (i.e. losses) although it fails to predict the center (mean) of the return distribution. To explore this possibility, we next performed a series of quantile regressions for the univariate specification (2). Besides the predictor variables listed in Table 1, these regressions also include a constant, the lagged quantile and the lagged absolute return. Our analysis considers quantiles in the range α {0.05, 0.10, 0.20,..., 0.90, 0.95}. Quantiles further out in the tails than 0.05 and 0.95 are not as precisely estimated and are hence omitted from our analysis. Our regressions are performed separately for each quantile. Table 3 reports estimates of the slope coefficients for each of the predictor variables along with the significance level based on the QMLE standard errors. Only the long term rate of return (ltr) appears to be highly significant for most of the quantiles. A more common finding is that a variable forecasts one but not both sides of the return distribution. For example, the term spread and earnings-price ratio have predictive power over the left tail of the return distribution (i.e., downmarkets), with higher values of these variables anticipating less negative left tails and thus reduced risk of loss. Net equity issues and stock variance also predict the left tail, although in this case larger values of these variables anticipate fatter (i.e. more negative) left tails and hence increased risk. Conversely, the default return spread, payout ratio and inflation appear mostly to predict events in the right tail of the return distribution. In all cases, increases in these variables lead to an anticipation of declines in the quantiles of the right tail. Common to all these variables is the asymmetry in their ability to predict stock returns: These variables work mostly either in downmarkets or in upmarkets, but not in both. The last two rows of Table 3 report the coefficients on the lagged quantile and absolute returns. The coefficients of the lagged quantile are highly significant for 7

8 all quantiles. However, there is no systematic pattern in these coefficients across the various quantiles. In contrast, the coefficient on the lagged absolute return displays a nearly symmetric, U-shaped pattern with large negative values in the left tail and large positive values in the right tail. Large returns (measured in absolute value terms) in the current period thus tend to lead to translate into a greater spread in the predicted return distribution next period, with the largest effects discernible in the tails. Unlike for the case where interest is limited to predicting the mean of stock returns, it is more difficult to summarize the evidence on return predictability across several quantiles. Nevertheless, to address the question whether a particular predictor variable helps forecast (part of) the return distribution, the last column of Table 3 reports Bonferroni values for a test that a given predictor variable is (jointly) significant across all the quantiles under consideration. 3 The advantage of this test is that it is robust to arbitrary, but unknown, dependencies across individual test statistics. By this criterion, only four variables, namely the default yield, the dividend-price ratio, the dividend yield and the book-to-market ratio, fail to be significant at the 10% level whereas 10 of 16 of the variables are significant at the 5% level. This evidence stands in marked contrast to the earlier findings in Table 2 of weak (in-sample) predictability of the mean of stock returns. Furthermore, it indicates that many of the predictor variables proposed in the literature in fact contain valuable information for prediction parts of the return distribution. Finally, our finding that the coefficients on the predictor variables can differ quite significantly across different quantiles means that it is difficult to come up with a model that simultaneously covers all parts of the return distribution. To assist with the economic interpretation of the quantile estimates, Figure 1 plots the predicted 5%, 10%, 50%, 90% and 95% quantiles based on the linearautoregressive quantile specification in equation (2) where the covariate is the long term rate of return (ltr). Horizontal lines show the corresponding quantiles based on the model that assumes constant quantiles. In all cases, the quantiles are based on full-sample estimates of the model parameters. Several interesting findings emerge from this plot of the conditional quantiles. Increases in the spread between the top and bottom quantiles (or spikes in the individual tail quantiles) correspond to important historical events in the US economy such as the oil crisis of 1973, the change in the monetary policy regime around , the stock market crash of 1987 and the bear market of the early 2000s all periods reflecting uncertain market conditions. Consistent with the notion that return predictability is not as pronounced in the central part of the return distribution, the median value varies far less than the tails. Although some variation in the quantiles is due to volatility events, at other times the upper quantiles increased further than the lower quantiles decreased, indicating absence of symmetry in the return distribution. 4 3 This is computed as min(np min, 1), where n is the number of quantiles considered and p min is the smallest p value selected from the full set of quantiles. 4 Note also that there are very few crossings between the 90% and 95% quantile estimates or between the 5% and 10% quantile estimates. This is to be expected if our quantile model 8

9 To compare the linear autoregressive quantile forecasts with those implied by the GARCH(1,1) model (6), Figures 2-4 plot the 10%, 50% and 90% quantiles from the two models along with the corresponding quantiles from the constant distribution model. The quantile specification again uses the long term rate of return as the predictor variable. There clearly is a large common component in the estimates of the two tail quantiles, reflecting the presence of autoregressive terms in both specifications. However, there are also some significant differences. The GARCH forecasts of the tail quantiles are notably more extreme than the forecasts from the linear autoregressive quantile model sometimes by a considerable margin, e.g. in and in The same is true for the median quantile which fluctuates considerably more under the GARCH specification than under the linear quantile model. Since we have modeled such a wide range of quantiles, we can obtain a comprehensive picture of how the entire return distribution shifts when a predictor variable changes. Towards this end, Figure 5 shows the cumulative density function of returns computed under three sets of values for the long term return variable, ltr, namely a middle scenario that sets this variable at its sample mean and two scenarios representing values of the mean plus or minus two standard deviations of this variable. Increasing ltr has the effect of shifting the return distribution to the right. This means that large negative returns become less likely, while large positive returns become more likely. Conversely, if the long term return is reduced, the stock return distribution shifts to the left and large negative returns become more likely. 3.3 Relation to Moments of the Return Distribution Quantiles of stock returns capture features of the shape of the distribution of returns. Alternative measures such as the skew and kurtosis are conventionally estimated directly from a sample of T return observations r 1,..., r T ŜK = 1 T KR = 1 T T (r t ˆµ) 3 /ˆσ 3 t=1 T (r t ˆµ) 4 /ˆσ 4, (9) t=1 where ˆµ = 1 T T t=1 r t and ˆσ 2 = 1 T T t=1 (r t ˆµ) 2 are the the mean and variance estimates. The higher the order of the data used to compute the moments, the more these estimates will be influenced by outliers. This means that these moments can be very imprecisely estimated. This is even more of a concern when the moments are estimated conditionally in order to get a sense of timevariation in the higher order moments. is correctly specified since ˆq α1 < ˆq α2 for α 1 < α 2, even though we do not impose this in our estimation. 9

10 To deal with this problem, robust measures of skewness and kurtosis based on quantiles have been proposed in the literature. 5 Let Q i be the ith quartile of y, i.e. Q i = F 1 (i/4) (i = 1,.., 4). The coefficient of skewness proposed by Bowley (1920) takes the form SK = Q 3 + Q 1 2Q 2 Q 3 Q 1. As for kurtosis, the Crow and Siddiqui (1967) measure centered so as to be zero under the standard Gaussian distribution (Kim and White (2003)) takes the form KR = F 1 (1 α) F 1 (α) F 1 (1 β) F , (10) (β) Following Crow and Siddiqui (1967) and Kim and White (2003) we set α and β to and 0.25, respectively. Figures 7 and 8 plot the time series of skewness and kurtosis based on these measures using the conditional quantiles from the model that includes the long term rate of return as a predictor variable. The return distribution is negatively skewed most of the time although there were periods around the mid-eighties and mid-to-late nineties where the return distribution became right skewed. The strongest negative skew appeared after the oil shocks in the mid-seventies, around the early 80s (during the change in the monetary policy regime), after 1987 and during the bear market from Conversely, the kurtosis of the return distribution, plotted in Figure 8, is largely positive with peaks around the same periods where the return distribution had a negative skew, signalling greater risks during those points in time. 4 Predictability of the Quantiles of Stock Returns As discussed in the introduction, recent studies on predictability of stock returns have focused on whether the mean of stock returns is predictable out of sample, i.e. when the parameters of the forecasting model are estimated recursively to forecast one-step-ahead. The measure of return predictability used in this literature is the mean squared forecast error of a candidate forecasting model compared to the value generated by the prevailing mean, i.e. a model that assumes the mean stock return is constant through time. Goyal and Welch (2003, 2006) find that while stock returns can be predicted in-sample or ex-post using full sample information to estimate the parameters of a model relating stock returns to the dividend yield, they were not predictable in real time, if those parameters were based exclusively on historically available information. We are interested in discovering whether these findings carry over to other parts of the return distribution. To evaluate our quantile prediction models, we 5 For a survey, see Kim and White (2003). 10

11 estimate the parameters of the quantile prediction model using data from the start of the sample up to 1969:12. One-step-ahead forecasts are then generated for the period 1970:1. The following period we update the estimates by dropping the initial observation and adding data from 1970:01 and use the updated forecasting model to forecast returns in 1970:02. This rolling window forecasting procedure is repeated recursively up to the end of the sample, 2005:12, generating a set of out-of-sample forecasts for the period 1970: :12. We present results for three models used to forecast the quantiles of the return distribution, namely (i) the constant or prevailing quantile model which has no predictor variables (3); (ii) the GARCH(1,1) specification (5) - (6); (iii) the linear quantile specification (2). As a first measure of model fit, Table 4 reports out-of-sample coverage ratios, i.e. the percentage of times returns fall below the α quantile for α = {0.05, 0.1, 0.5, 0.9, 0.95}. For most linear quantile models the coverage ratios are close to their correct values, i.e. roughly 5% of the stock returns fall below most of the predicted quantiles for α = 0.05 etc. This also holds on average as witnessed by the equally weighted average line in Table 4. The GARCH model tends to overpredict the propensity of returns below the 90% and 95% quantiles or, equivalently, underpredict the probability of returns above these points in the return distribution. The coverage ratios reported in Table 4 provide a measure of average fit. A stricter test of a good quantile forecast is that the event that the actual return falls below the quantile should not itself be predictable by means of any other information. This is similar to the usual condition that forecast errors be unpredictable. The literature on predictability of the mean of stock returns suggests that few, if any, predictor variables can help reduce the error in predicting mean returns ex ante. A natural question to ask is therefore whether the predictor variables under consideration here add anything to the predictions from the constant quantile model which we shall refer to as the prevailing quantile model. To address this point, Table 5 reports the out-of-sample mean tick loss for the models under consideration. For each quantile the best overall model is marked in bold. Since different univariate linear specifications dominate for different quantiles, we also consider a simple equal-weighted quantile combination which is computed as ˆq α EW = (1/16) 16 i=1 ˆqi α. Univariate quantile specifications struggle in the left tails for α = 0.05 and α = 0.10 for which only five and six out of sixteen models, respectively, better the results produced by the simple prevailing quantile approach. Even worse performance is observed in the center of the return distribution where only two of sixteen univariate quantile models come out on top of the prevailing quantile model. Turning to the right tail of the return distribution, very different results are obtained. For α = 0.9 and α = 0.95 twelve and nine out of sixteen models, respectively, produce lower out-of-sample loss than the prevailing quantile method. Taking the simple equal-weighted average of quantile forecasts seems to work 11

12 very well. Only four of the univariate models produce lower out-of-sample loss in the far left tail (α = 0.05), whereas between zero and two out of sixteen models better it for the other quantiles. This suggests that the simple average quantile forecast improves substantially on the individual univariate quantiles. Moreover, for all quantiles except the median, the equal-weighted quantile forecasts produce lower out-of-sample loss than both the forecasts based on the GARCH(1,1) and prevailing quantile methods. This again suggests that the simple equal-weighted quantile forecast offers a reasonable chance of successfully predicting stock returns, particularly in the tails of the return distribution. 4.1 Encompassing Tests and Combinations The previous analysis compared the out-of-sample performance of alternative approaches to forecasting the quantiles of stock returns but did not address whether these differences were statistically significant. To see whether any of the quantile prediction models can significantly improve upon the prevailing quantile forecasts, we adopt the approach proposed by Giacomini and Komunjer (2005). This approach tests whether one set of quantile forecasts encompasses another. Encompassing occurs when the second set of forecasts fail to add new information to the first set of quantile forecasts (or vice versa) in which case the first (second) quantile forecast is said to encompass the second (first). This is the natural perspective to adopt here since we are attempting to establish whether it is possible, out-of-sample, to improve upon a simple model that treats the return distribution as constant through time. To this end again we use the tick loss function and follow Giacomini and Komunjer in focusing on conditional expected loss rather than unconditional expected (or average) loss. Let ˆq α 1,t be the α quantile from a linear quantile prediction model while ˆq α 2,t is the competing quantile forecast produced by an alternative specification such as the constant( prevailing ) quantile model or the GARCH(1,1) model. We are interested in testing the null hypothesis that, ˆq α 1,t performs better in the sense of yielding a lower expected loss than any linear combination of ˆq α 1,t and ˆq α 2,t. This holds provided that, for all pairs of combination weights, γ α 1t and γ α 2t, E t [L α (r t+1 ˆq α 1,t)] E t [L α (r t+1 γ α 0,t γ α 1tˆq α 1,t γ α 2tˆq α 2,t)]. (11) This condition is satisfied if and only if the optimal weights (γ α 1t, γ α 2t ) = (1, 0), where (γ0t α, γ1t α, γ2t α ) = arg min γ α 0t,γα 1t,γα 2t E t [L α (r t+1 γ α 0,t γ α 1tˆq α 1,t γ α 2tˆq α 2,t)]. (12) Giacomini and Komunjer show that the vector of optimal forecast combination weights γt α = (γ0,t, α γ1t α, γ2t α ) satisfies the following first-order condition of the optimization in (12): E t [α 1{r t+1 γ α 0,t γ α 1tˆq α 1,t γ α 2tˆq α 2,t} < 0] = 0. (13) 12

13 Estimates of γ α 0,t, γ α 1t and γ α 2t can be obtained via the generalized method of moments (GMM) where the moment condition is the first-order condition in (13). Suppose ω t is a vector of instruments that are known at time t and define g(γ α t ; r t+1, w t ) = [α 1{r t+1 γ α t ˆq α t < 0}]ω t, where γ α t = (γ α 0,t, γ α 1,t, γ α 2,t) and ˆq α t = (1, ˆq α 1,t, ˆq α 2,t). Letting g n (γ α ) = n 1 T 1 t=t n g(γ α t ; r t+1, ω t ) be the sample moment condition based on a rolling window of n observations and Ŝn an estimator of the covariance matrix of moment conditions, γt α, can be obtained from the solution to the standard quadratic minimization problem { } min g n (γ α γt α t ) Ŝn 1 g n (γt α ). The asymptotic distribution of the GMM estimates of γt α require the moment conditions to be once differentiable. However, the indicator function in the moment conditions poses a problem in our setting. We follow Giacomini and Komunjer (2005) by replacing the moment condition with the following smooth approximation: g τ = [α (1 exp((r t+1 γ α t ˆq α t )/τ)]1{r t+1 γ α t ˆq α t < 0}w t. Here τ is the smoothing parameter which is set equal to GMM estimation of γ t and S is done recursively using the Newey-West optimal weighting matrix as discussed in Giacomini and Komunjer. Recursive GMM estimation of optimal forecast combination weights requires choices of instruments, initial combination weights and an initial weighting matrix. When analyzing the forecast combination of linear quantile models with the alternative forecast, we use a constant, the corresponding lagged covariate, the lagged return, the corresponding lagged linear quantile forecast and the lagged alternative forecast as instruments. 6 The initial weighting matrix is always set to the identity matrix whereas we do a global search for the best initial combination weights. We first generate 5000 random initial combination weights from a uniform distribution on [-2,2], sort them with respect to their out-of-sample tick exponential loss and choose those 500 initial values with the smallest out-of-sample tick exponential loss. We then estimate the optimal forecast combination weights via GMM for each of these 500 initial values and we report the combination weights with the smallest value of the objective function after the minimization. We consider two separate encompassing tests. The first test, labeled ENC1 α, addresses whether the linear quantile forecast (ˆq 1,t) α encompasses the alternative quantile forecast produced either by the constant quantile model or the 6 The lagged covariate is dropped from the instrument list when we analyze the forecast combination of the equal-weighted linear quantile forecast with the alternative forecast. 13

14 GARCH(1,1) model (ˆq α 2,t). The second test, labeled ENC α 2, addresses whether the alternative quantile forecast encompasses the linear quantile forecast. The test statistics considered are therefore ENC α 1 = n((ˆγ α 1, ˆγ α 2 ) (1, 0))ˆΩ 1 α ((ˆγ α 1, ˆγ α 2 ) (1, 0)) (14) ENC α 2 = n((ˆγ α 1, ˆγ α 2 ) (0, 1))ˆΩ 1 α ((ˆγ α 1, ˆγ α 2 ) (0, 1)). (15) Here ˆΩ α is a consistent estimate of the variance matrix of the second and third elements of γ α. Under the null that the linear quantile forecast encompasses the alternative forecast ENC1 α is asymptotically distributed with a χ 2 2 distribution. The same holds true for ENC2 α under the null that the alternative forecast encompasses the linear quantile forecast Empirical Findings Tables 6 reports empirical estimates of the combination weights when these methods are applied to our data. In the left tail (α = 0.05 and α = 0.10) and the center of the return distribution there are few cases with significant weights on the quantile predictions, whereas there are many instances where the weight on the prevailing quantile forecasts are significant (e.g., 9 of 16 cases for α = 0.05 and 16 of 16 cases for α = 0.10 at the 10% significance level). The same holds true for the median (α = 0.5) where 14 of 16 models give rise to weights on the prevailing quantile forecasts that are significant at the 10% level. Very different conclusions emerge for the right tail of the stock return distribution (α = 0.90 and α = 0.95) where 12 of 16 of the linear univariate quantile forecasts generate significant weights. Moreover, these weights are frequently quite large and always positive. Turning to the formal encompassing tests (reported in Table 7), these suggest that in general neither the linear quantile model nor the prevailing quantile model encompasses the other in a statistically significant sense for the left and central parts of the return distribution. In this sense, both models are useful to a forecaster. Conversely, in the right tail of the return distribution there are a few cases where the forecasts from the linear quantile model appear to encompass the prevailing quantile forecasts. For example, for α = 0.9 and α = 0.95, linear quantile forecasts based on either the payout ratio (d/e) or on inflation encompass those from the prevailing quantile model. In another five cases the linear quantile forecasts encompass those generated by the prevailing quantile model mostly for univariate models representing valuation ratios such as the dividend yield, the dividend-price or earnings-price ratio. This suggests that when the valuation ratios are very high, there is a good chance of a surge in market prices. Table 5 suggested that the equal-weighted combination of quantile forecasts performs quite well. We therefore compare the out-of-sample performance for the equal-weighted quantile forecast to that produced by forecasts based on the prevailing quantile or GARCH(1,1) models. Table 8 presents the outcome of this analysis. As revealed by their high combination weights, equal-weighted quantile 14

15 forecasts dominate prevailing quantile forecasts in the far tails, i.e. for α = 0.05 and α = Moreover, tests that the linear quantile forecasts encompass the prevailing quantile forecasts are not rejected for these tail quantiles, whereas the converse proposition that the prevailing quantile forecasts encompass the linear quantile forecasts is rejected very strongly. There is also some evidence that the equal-weighted quantile forecast dominates the prevailing quantile for α = 0.9. Compared to the GARCH forecasts, the equal-weighted quantile forecasts dominate in the left tail (α = 0.05 and α = 0.10) and in the far right tail (α = 0.95) where the former give rise to large positive weights in the combination. Furthermore, the equal-weighted quantile forecasts encompass the GARCH forecasts for two of the tail quantiles (α = 0.1 and α = 0.95), whereas the converse is rejected. The GARCH forecasts do, however, appear to be better than the equal-weighted quantile forecast at the center of the return distribution. 5 Conclusion We used quantile regression methods to explore the extent to which different parts of the distribution of monthly stock returns are predictable. Most previous work on return predictability at the monthly horizon has focused on modeling the conditional mean. Consistent with earlier studies we find little evidence to suggest that the center of the return distribution can be predicted. However, our findings also suggest that the tails of stock returns can be predicted by means of state variables proposed in the literature. We find evidence that an equal-weighted combination of quantile forecasts does very well compared to alternative approaches that either assume that the distribution of stock returns is time-invariant or that any variation takes the form of time-varying volatility. Interestingly, for the univariate quantile prediction models the evidence in support of predictability of stock returns is strongest in the right tail of the return distribution. While most previous work has focused on downside risk, the possibility of predicting periods with strong upside potential has not received nearly as much attention. However, our results suggest that the strongest predictability may in fact occur during such states of the economy. References [1] Ang, A., J. Chen and Y. Xing, 2006, Downside Risk. Forthcoming in Review of Financial Studies. [2] Bossaerts, P. and P. Hillion, 1999, Implementing Statistical Criteria to Select Return Forecasting Models: What do we Learn? Review of Financial Studies 12(2),

16 [3] Bowley, A.L., 1920, Elements of Statistics. New York: Charles Scribner s Sons. [4] Campbell, J.Y., 1987, Stock Returns and the Term Structure. Journal of Financial Economics 18(2), [5] Campbell, J.Y. and S.B. Thompson, 2007, Predicting the Equity Premium Out of Sample: Can Anything Beat the Historical Average? Working Paper, Harvard University. [6] Christoffersen, P.F., 1998, Evaluating Interval Forecasts. International Economic Review 39, [7] Christoffersen, P.F., and F. Diebold, 2000, How Relevant is Volatility Forecasting for Financial Risk Management?, Review of Economics and Statistics, 82, [8] Cochrane, J.H., 2006, The Dog That Did Not Bark: A Defense of Return Predictability. Working Paper, University of Chicago. [9] Crow, E.L. and M.M. Siddiqui, 1967, Robust Estimation of Location. Journal of the American Statistical Association 62, [10] Dittmar, R., 2002, Nonlinear Pricing Kernels, Kurtosis Preference, and Evidence from the Cross Section of Equity Returns, Journal of Finance, 57, [11] Engle, R. F. and S. Manganelli, 2004, CAViaR: Conditional Autoregressive Value at Risk by Regression Quantiles. Journal of Business and Economic Statistics 22(4), [12] Fama, E.F. and K.R. French, 1988, Dividend Yields and Expected Stock Returns. Journal of Financial Economics 22(1), [13] Fama, E.F. and K.R. French, 1989, Business Conditions and Expected Returns on Stocks and Bonds. Journal of Financial Economics 25(1), [14] Ferson, W., 1990, Are the Latent Variables in Time-Varying Expected Returns Compensation for Consumption Risk?, Journal of Finance, 45, [15] Ferson, W., and C., Harvey, 1993, The Risk and Predictability of International Equity Returns, Review of Financial Studies, 6, [16] Giacomini, R. and I. Komunjer, 2005, Evaluation and Combination of Conditional Quantile Forecasts. Journal of Business and Economic Statistics 23(4), [17] Goyal, A. and I. Welch, 2003, Predicting the Equity Premium with Dividend Ratios. Management Science 49(5),

17 [18] Goyal, A. and I. Welch, 2006, A Comprehensive Look at the Empirical Performance of Equity Premium Prediction. Forthcoming in Review of Financial Studies. [19] Guidolin, M. and A. Timmermann, 2006, International Asset Allocation under Regime Switching, Skew and Kurtosis Preferences. Mimeo, Manchester University and UCSD. [20] Harvey, C., J., Liechty, M., Liechty, and P., Müller, 2004, Portfolio Selection with Higher Moments, mimeo, Duke University. [21] Harvey, C., and A., Siddique, 2000, Conditional Skewness in Asset Pricing Tests, Journal of Finance, 55, [22] Kim, T-H. and H. White, 2003, On More Robust Estimation of Skewness and Kurtosis: Simulation and Application to the S&P500 Index. Manuscript UCSD. [23] Kimball, M., 1993, Standard Risk Aversion, Econometrica, 61, [24] Koenker, R., 2005, Quantile Regression. Econometric Society Monographs. Cambridge University Press: New York. [25] Koenker, R. and G. Bassett, 1978, Regression Quantiles. Econometrica 46, [26] Koenker, R. and Q. Ahzo, 1996, Conditional Quantile Estimation and Inference for ARCH Models. Econometric Theory 12, [27] Komunjer, I., 2005, Quasi-Maximum Likelihood Estimation for Conditional Quantiles. Journal of Econometrics 128, [28] Pesaran, M.H. and A. Timmermann, 1995, Predictability of Stock Returns: Robustness and Economic Significance. Journal of Finance 50(4), [29] Taylor, J.W. and D.W. Bunn, 1999, A Quantile Regression Approach to Generating Prediction Intervals. Management Science 45,

18 Table 1: Predictor Variables Variable Description Sample d/e Dividend Payout Ratio 02/ /2005 svar Stock Variance 02/ /2005 dfr Default Return Spread 01/ /2005 lty Long Term Yield 01/ /2005 ltr Long Term Rate of Returns 01/ /2005 infl Inflation 02/ /2005 tms Term Spread 02/ /2005 tbl T-bill 02/ /2005 dfy Default Yield Spread 01/ /2005 d/p Dividend Price Ratio 02/ /2005 d/y Dividend Yield 02/ /2005 e/p Earnings Price Ratio 02/ /2005 b/m Book to Market Ratio 03/ /2005 e10/p Smoothed Earnings Price Ratio 12/ /2005 csp Cross Sectional Premium 05/ /2002 ntis Net Equity Expansion 12/ /

19 Table 2: OLS estimates of regression coefficients for univariate prediction models Variable Estimate Dividend Payout Ratio Stock Variance Default Return Spread Long Term Yield Long Term Rate of Returns Inflation * Term Spread T-bill Default Yield Spread Dividend Price Ratio Dividend Yield Earnings Price Ratio Book to Market Ratio Smoothed Earnings Price Ratio Cross Sectional Premium ** Net Equity Expansion *** Note: These coefficient estimates are based on full-sample estimates using the data samples listed in Table 1. * indicates significance at the 10% level ** indicates significance at the 5% level *** indicates significance at the 1% level 19

20 Table 3: Coefficient estimates for quantile prediction models (a) Quantiles 0.05, 0.1, 0.2, 0.3, 0.4, 0.5 Quantile d/e ** svar ** ** *** *** *** *** dfr * lty *** * ltr *** *** *** *** ** infl *** *** tms *** *** ** *** tbl ** ** *** ** dfy ** * d/p ** d/y ** e/p *** ** *** b/m e10/p ** csp ** *** ntis ** ** *** *** Lag q *** *** *** *** *** *** LAR *** *** *** *** (b) Quantiles 0.6, 0.7, 0.8, 0.9, 0.95 Quantile Bonferroni d/e *** ** *** svar *** dfr * ** ** *** *** lty * *** ltr * ** ** *** infl *** ** ** ** tms ** tbl ** *** dfy d/p d/y e/p b/m * e10/p ** *** csp * ** ntis Lag q *** *** *** *** *** LAR *** *** *** *** *** Note: For each quantile α = {0.05, 0.10,..., 0.90, 0.95} the table reports the estimated slope coefficient obtained from quasi-maximum likelihood estimation using the full sample listed in Table 1. The final column lists Bonferroni p-values for a joint test across all quantiles that the slope coefficients in the linear quantile model all are equal to zero. Lag q is the lagged quantile and LAR is the lagged absolute return. * indicates significance at the 10% level ** indicates significance at the 5% level *** indicates significance at the 1% level 20