INFERENCE IN PREDICTIVE QUANTILE REGRESSIONS

Transcription

1 INFERENCE IN PREDICIVE QUANILE REGRESSIONS Alex Maynard University of Guelph Katsumi Shimotsu Hitotsubashi University August 23, 2010 Yini Wang Queen s University Abstract his paper studies inference in predictive quantile regressions when the predictive regressor has a near-unit root. We derive nonstandard distributions for the quantile regression estimator and t-statistic in terms of functionals of diffusion processes. he critical values are found to depend on both the quantile of interest and the local-to-unity parameter, which is not consistently estimable. Based on these critical values, we propose a valid Bonferroni bounds test for quantile predictability with persistent regressors. We employ this new methodology to test the ability of many commonly employed and highly persistent regressors, such as the dividend yield, earnings price ratio, book to market ratio, term spread and -bill rate, to predict the median, shoulders, and tails of the stock return distribution. JEL Classification: C12, C22, G1 Keywords: local-to-unity, quantile regression, Bonferroni method, predictability, stock return Preliminary version. We thank Chuan Goh for useful discussion, and olga Cenesizoglu for providing the data. We take full responsibility for any remaining errors. Department of Economics, McKinnan Building, University of Guelph, Guelph, Ontario N1G 2W1, Canada. maynard@uoguelph.ca Department of Economics, Hitotsubashi University, 2-1 Naka, Kunitachi, okyo , Japan. shimotsu@econ.hit-u.ac.jp Department of Economics, Queen s University, Kingston, Ontario K7L 3N6, Canada. wangyini@econ.queensu.ca 1

2 1 Introduction In this paper, we develop asymptotic theory in the context of predictive quantile regressions with nearly nonstationary regressors. his has important empirical applications such as testing the predictability of the stock return distribution. here has been an extensive literature on predictive tests for mean returns, debating whether predictors such as dividend yield can forecast stock returns. Earlier work adopts standard t-tests and finds strong evidence of predictability. For example, Shiller (1984), Campbell and Shiller (1988a, 1988b), Fama and French (1988) and Hodrick (1992) show that stock returns are highly forecastable by dividend yields, dividend price and earning price ratios. However, as Shiller (1984) points out, some regressors in the predictive regressions can be stochastic, so that the conventional t-tests are invalid. In particular, the predictability of the return with these predictors may be overstated. Mankiw and Shapiro (1986) investigates small sample properties of tests for rational expectation models. hey find that ordinary tests of orthogonality reject too frequently when the predictive variables are nearly nonstationary. Other papers, such as Stambaugh (1986), Cavanagh et al. (1995) and Stambaugh (1999), also emphasize the poor small sample properties in predictive regressions with highly persistent regressors. Conventional predictive tests of stock returns are not valid for the following reasons. Firstly, the predictor variables, such as dividend yields, dividend price and earning price ratios, are strongly autocorrelated. Secondly, although pre-determined, typically the predictor is not strictly exogenous since its innovation is often highly correlated with the error term in the predictive regression. For example, with financial data such as dividend price ratios and returns, it is reasonable to expect strong correlation between the regressor s innovation and the regression disturbance. his tends to inflate the t-statistic resulting in over-rejection. In linear predictive regressions, it is well-known that when the regressor is not strictly exogenous and has a largest autoregressive root close to one, the limiting distribution of the t-statistic will be nonstandard. In this case, the t-statistic is too large, and tests using the standard normal critical values will over-reject the null hypothesis of nonpredictability. A great deal of attention has been devoted to overcoming such size distortions in linear predictive regressions resulting in a rich literature. Cavanagh et al. (1995) develop inference in predictive regression under a local-to-unity specification of the regressors, and present three types of confidence intervals to control the size of test. Similarly, Campbell and Yogo (2006) use a pretest to determine whether the conventional t-test leads to invalid inference and develop a bound based efficient test to corrects the size. In contrast, Campbell and Dufour (1995 and 1997) provide finite-sample nonparametric sign and sign-rank tests of conditional independence and random walk. Wright (2000) presents stationarity tests in a cointegrating system. Lanne (2002) also proposes a test for stationarity and finds no predictability for stock return in the empirical study. Jansson and Moreira (2006) derive conditionally optimal inference using Gaussian 1

3 asymptotic power envelopes. Maynard and Shimotsu (2009) develop a covariance based orthogonality test. Nelson and Kim (1993) and Goetzmann and Jorion (1993) conduct bootstrap simulations. Likewise, Wolf (2000) applies a subsampling method and finds no solid evidence for predictability with dividend yields. Stambaugh (1999) derives small-sample Bayesian posterior distributions for the regression parameters. Lewellen (2004) performs finite sample correction by calculating a joint significance level from a combination of the conditional and unconditional tests. Amihud and Hurvich (2004) and Amihud et al. (2004) employ an augmented regression method to obtain reduced-bias estimated predictive coefficients for single and multiple regressor models. In contrast to this large literature devoted to testing the predictability of the mean of stock returns, there are few works studying the predictability of the entire return distribution using similar lagged predictors. Recent empirical work by Cenesizoglu and immermann (2008) employs the quantile regression method introduced by Koenker and Bassett (1978), and finds a number of pre-determined predictors having asymmetric effect on various quantiles in the return distribution. Whereas least square methods are confined to estimating the conditional mean, quantile regression methods are able to give a more informative picture of the entire distribution. Literature in quantile regressions has been developed rapidly. here are substantial recent works in both theoretical and empirical areas. For example, in time series, there are quantile autoregression (Koenker and Xiao 2006), unit root quantile autoregression (Koenker and Xiao 2004), conditional quantile estimation for GARCH models (Xiao and Koenker 2009) and Copula-based quantile autoregression (Chen et al. 2009). Besides, extremal quantile regressions are empirically relevant in many economic areas such as applied microeconomics and finance. Chernozhukov and Du (2006) and Chernozhukov (2010) developed the theory of conditional extremal quantiles with applications to value-at-risk and birthweights. Nonparametric methods are also used in quantile regresssions, such as Frolich and Melly (2008), Huber (2010), Jun et al. (2009) and Koenker (2010). Among many other most recent papers, Koenker (2008) proposed censored quantile regression estimation; Chernozhukov et al. (2009) derived finite sample inference for quantile regression models under minimal assumptions; Chernozhukov and Belloni (2010) studied l 1 -penalized quantile regression in high-dimensional sparse models. It is sometimes very useful to analyze how different positions in the distribution, such as the median, shoulders and tails, respond to changes in a predictor. However, we are aware of no theoretical work to date, that establishes valid econometric inference methods in quantile predictive regression with persistent regressors. In an attempt to fill this gap, we develop inference in short-horizon predictive quantile regressions with nearly integrated regressors which follow an AR(p) process in general, with p 1. We first derive the limit distribution of the quantile regression coefficients by generalizing results of Xiao (2009) 1 to the local-to-unity setting of 1 Xiao (2009) derives inference in a quantile regression with cointegrated time series. 2

4 Chan and Wei (1987), Chan (1988), Phillips (1987), Phillips (1988a, 1988b), and Nabeya and Sorensen (1994) 2. We then provide a test of quantile predictability based on an asymptotically valid Bonferroni bounds methodology in the spirit of Cavanagh et al. (1995). he remainder of the paper is organized as follows. In Section 2, the framework of the problem is established and the asymptotic theory for predictive quantile regression under a local-to-unity specification is developed. In Section 3, the Bonferroni method used to correct the size distortion is proposed. In Section 4, results from our simulation study are reported. In Section 5, the techniques are applied to test the predictability of the stock return distribution using various pre-determined predictive regressors. Section 6 concludes the paper. 2 heory 2.1 Assumptions Following Cavanagh et. al. (1995), we assume that the predictor x t is a finite order autoregressive process: x t = α 0 + v t, (1 α 1 L)b(L)v t = ε 1t (1) where b(l) = k i=0 b il i, b 0 = 1. Assume that the roots of b(l) are fixed and less than 1 in absolute value. Equation (1) can be rewritten as: x t = β 0 + β 1 x t 1 + ζ(l) x t 1 + ε 1t (2) where β 0 = (1 α 1 )b(1)α 0, β 1 = (α 1 1)b(1), and ζ(l) = k j=1 L 1 [b j (1 α 1 ) k i=j b i]l j. In this Augmented Dicky-Fuller representation, it is straightforward that x t follows an AR(k + 1) process with largest root α 1. Of particular interest in this process is the local-to-unity specification, α 1 = 1 + c, where c 0, and is the sample size. When c = 0, this generalizes to a unit root process. he mean reverting case c < 0 provides a useful large sample approximation for the case in which the largest root is very close to, but still less than one. A number of prior studies have used this framework to model predictors such as earnings and dividend price ratios which are highly persistent but a priori stationary on economic grounds. In this paper, we test whether a lagged regressor x t 1 can help to predict the τth quantile of the conditional distribution of y t. he data generating process of y t is the following: y t = γ 0 + γ 1 x t 1 + ε 2t = γ z t 1 + ε 2t, (3) where z t 1 = (1, x t 1 ) and γ = (γ 0, γ 1 ). Although this linear regression specification is more restrictive than required for the quantile estimation methods employed below, it provides a 2 hese works are all based on nearly nonstationary AR(1) models. 3

5 simple convenient framework in which to develop the limit theory. We also make the following assumptions on the innovations of (x t, y t ). Assumption 1 Denote F t 1 as information set up to time t. ε t = (ε 1t, ε 2t ) is a martingale difference sequence with E(ε t ε t F t 1 ) = Σ, with typical element σ ij. Assumption 2 he cumulative distribution function of ε 2t, F ( ), has a continuous density function f( ), which is positive on {ε 2t : 0 < F (ε 2t ) < 1}. Assumption 3 he conditional distribution function F t 1 ( ) = P r(ε 2t < F t 1 ) has derivative f t 1 ( ) a.s., and f t 1 (s n ) is uniformly integrable for any sequence s n F 1 (τ), and E[f ξ t 1 (F 1 (τ))] < for some ξ > 1. Xiao (2009) makes the same assumptions on the distribution functions of the error term. 2.2 Estimation method Least square methods have been widely used to estimate models for the conditional mean. However, in order to capture the predictability over the whole distribution, we generalize our predictive setting to the conditional quantiles. he median is the 50% quantile that divides the population into two equal portions. In a general case, the random variable at the τ th quantile of a distribution is larger than the τ proportion of the population but smaller than the (1 τ) proportion, with τ (0, 1). Consider the following quantile regression model, y t = γ 0 (τ) + γ 1 (τ)x t 1 + ε 2tτ = γ(τ) z t 1 + ε 2tτ (4) where γ 0 (τ) = γ 0 + F 1 (τ) and ε 2tτ = ε 2t F 1 (τ). Define Q ε2tτ (τ x t 1 ) as the τth quantile of ε 2tτ conditional on x t 1. Given Q ε2tτ (τ x t 1 ) = 0, we have the conditional quantile of y t, Q yt (τ x t 1 ) = γ 0 (τ) + γ 1 (τ)x t 1. he quantile regression coefficient estimates are given by ( ˆγ 0 (τ), ˆγ 1 (τ)) = arg min (γ 0,γ 1 ) R 2 ρ τ (y t γ 0 γ 1 x t 1 ). (5) ρ τ ( ) is the asymmetric absolute deviation loss function defined by ρ τ (u) = uψ τ (u), where ψ τ (u) = τ I(u < 0) (Koenker and Bassett 1978). For the median τ = 0.5, ρ τ (u) = 1/2 u is used for Laplaces median regression function. 4

6 2.3 heoretical results Define δ = corr(ε 1t, ε 2t ). Let B = (B 1, B 2 ) denote a two-dimensional Brownian motion with covariance matrix Σ, where σ 11 = σ 22 = 1 and σ 12 = σ 21 = δ. We have 1 2 [ ] [ [ r] b 1 (L)ε 1t d ε 2t ωb1 (r) σ B 2 (r) ] = BM(0, Ω) (6) where Ω = ω2 ωσ δ with ω = ω 1 ωσ = σ b(1) δ σ 22 Let Z = (Z 1, Z 2 ) be a two-dimensional Brownian motion such that Z 1 = ωb 1 (r) and Z 2 = σ B 2 (r). Also, J c is a diffusion process defined by dj c (s) = cj c (s)ds + db 1 (s) with initial condition J c (0) = 0. Besides, we distinguish demeaned variables by superscript. For example, x t = x t ( 1) 1 t=2 x t 1 and J c (s) = J c (s) 1 0 J c(r)dr. In light of Phillips (1987), under α 1 = 1 + c, 1 2 x [ r] d ωj c 2 (x t 1 )2 d (ωj c ) 2 dr 1 x t 1 b 1 (L)ε 1t d ωj c dz 1 where [ ] denotes the greatest lesser integer function. Proposition 1 he asymptotic representation of the standard t-statistic used to test H 0 : β 1 = 0 in (2) is given by t β1 d [ c b(1) Since ψ τ (ε 2tτ ) = τ I(ε 2t < F 1 (τ)), we have Eψ τ (ε 2tτ ) = 0. ] 1 (J c ) 2 2 J c db 1 dr + [ (J c. (7) ) 2 dr] 1 2 Lemma 1 where Ω τ = [ 1 2 ] ω 11 ω 12. ω 21 ω 22 [ [ r] b 1 (L)ε 1t ψ τ (ε 2tτ ) ] d [ Z 1 (r) Z 2τ (r) ] = BM(0, Ω τ ) (8) It follows that 1 x t 1 ψ τ (ε 2tτ ) d ωj c dz 2τ. 5

7 Proposition 2 where D = [ [ 1 ( ˆγ 1 (τ) γ 1 (τ)) d f(f 1 (τ)) [ 1 D (ˆγ(τ) γ(τ)) d f(f 1 (τ)) ], and J c = (1, ωj c ). he standard error of ˆγ 1 (τ) is given by (ωj c ) 2 ] 1 [ J c J c ] 1 [ ωj c dz 2τ ] J c dz 2τ ] 1 f(f 1 (τ)) 2 [ (x t 1 )2 ] 1, where consistent estimator of f(f 1 (τ)). Hence, we have the following proposition. (9) (10) f(f 1 (τ)) is a Proposition 3 he asymptotic representation of the t-statistic to test H 0 : γ 1 (τ) = 0 3 for a given τ is 3 Methodology J c dz 2τ t γ1 (τ) d [ (J c. (11) ) 2 ] 1 2 In equation (2), if α 1 < 1 and is fixed, the unit root test will reject with probability of one asymptotically. Since x t is stationary in this case, the quantile regressions of equation (3) are well-behaved in large samples. It would be sensible to test H 0 : γ 1 (τ) = 0 using standard normal critical values. If α 1 = 1, which corresponds to c = 0, then x t has a unit root. When c < 0 and α 1 is large, x t has a near unit root. In the stock return predictability example, many predictors, such as the dividend price ratio, have large autoregressive roots that are very close to one. Hence, it is appropriate to model x t as a near unit root process. As Proposition 3 shows, in this case the limiting distribution of t γ1 (τ) is nonstandard and dependent on the nuisance parameter c. Moreover, the asymptotic distribution of t γ1 (τ) also depends on δ. If the two error terms in equation (2) and (3) are uncorrelated, t γ1 (τ) will have a standard normal asymptotic distribution. However, when the two error terms are correlated, the limiting distribution of the quantile regression coefficient is no longer standard normal, which causes size distortion of the predictability test. hus, standard normal critical values are not reliable for the test of interest, since they tend to over-reject the null. his is a problem in practice, because financial data such as prices and dividends do not satisfy strict exogeneity. he over-rejection is especially severe when the residual cross-correlation is large. herefore, this paper proposes a Bonferroni bounds method in the spirit of Cavanagh et al. (1995), which results in asymptotically valid test. he procedure has two steps. Because 3 Note that the null hypothesis can be generalized to γ 1(τ) = γ 1,0. 6

8 the local-to-unity parameter c is a nuisance parameter that cannot be consistently estimated, we first derive a 100(1 η 1 )% confidence interval for c, and denote it by CI c (η 1 ). his firststage confidence interval is computed by inverting a unit root test statistic as illustrated in Stock (1991). In the second stage, based on the previously derived asymptotic distribution, we compute the critical value for t γ1 (τ) of size η 2 at each point of c in the first-stage confidence interval. hen, the Bonferroni bounds [C l (η 1, η 2 ), C u (η 1, η 2 )] gives the asymptotically valid critical values for t γ1 (τ), where lower bound: C l (η 1, η 2 ) = min c CI c(η 1 ) C t γ1,c, η 2 2 upper bound: C u (η 1, η 2 ) = max c CI c(η 1 ) C t γ1,c,1 η 2 2, and C tγ1,c, η 2 2 is the 100 η 2 2 % quantile of the limiting distribution of t γ1 (τ) for a given δ. ests based on the Bonferroni bounds are conservative with size less than or equal to η, for η = η 1 + η 2. Although the Bonferroni interval is conservative, the size of the test can be adjusted by selecting appropriate (η 1, η 2 ). More specificly, one can set η 2 = η, and choose η 1 such that the asymptotic size equals to η. In our Monte Carlo experiments, we consider η = 10% as the size of the test, and choose η 1 under = 1000 such that the size is close to the benchmark case where c is known. 4 Simulation Study Our simulation study focuses on the demeaned case in which a regression contains a constant term (without a time trend term). he results are based on asymptotic results from 2000 Monte Carlo replications with sample size for each set of parameters (c, δ, τ), where c ranges from 0 to, δ ranges from 0 to 0.95, and τ ranges from 0.05 to In each replication, x t is generated according to equation (2). We set b j = 0 for j > 1. Under the local-to-unity specification, we have x t = (1 + c )x t 1 + ε 1t. y t is generated by equation (3) under the null H 0 : γ 1 = 0. We set γ 0 = 0 so that y t = ε 2t. ε 1t and ε 2t are both i.i.d. N(0, 1) with correlation coefficient δ. he level of the predictability test is 10%. As discussed in previous sections, conventional t-test from predictive quantile regression overreject the null, since with nearly integrated predictors the asymptotic distribution of t-statistics is not standard normal. As a preliminary experiment, we investigate the size distortion problem associated with the t-test based on standard normal critical values, ±1.65. As able 1 shows, when δ = 0, 0.25, the test is conservative for τ = 0.2 to 0.8 with large sample size such as = From able 2, even with small sample size as = 200, the rejection rates for τ = 0.2 to 0.8 are still close to 10%. hat is, when there is no or small correlation between ε 1t and ε 2t, there is no or very little over-rejection for inner quantiles. As the correlation coefficient increases, over-rejection becomes more severe. For δ = 0.50, 0.75, 0.95, the null is over-rejected 7

9 for all quantiles, even with = Particularly, when the correlation is as large as 0.95, rejection rates can be over 25%. Similar over-rejection problem is illustrated for linear predictive regression in Cavanagh et al. (1995) and Maynard and Shimotsu (2009): with = 500, 200, 100, conventional t-tests can lead to substantial size distortion, especially for large correlation coefficients. o control the size distortion, we employ asymptotically valid critical values according to the asymptotic representation stated in Section 2 for a number of parameter sets. Since the analytic critical values cannot be obtained directly from the distribution, we calculate the simulated critical values under a large sample size, and store them in computerized lookup tables. he simulation is based on Monte Carlo replications with = Again, x t is simulated under the local-to-unity specification α 1 = 1 + c, and y t is simulated under the null of no predictability. Since the limiting distribution of γ 1 (τ) depends on c, δ and τ, 42 percentiles from the empirical distribution of t γ1 (τ) are calculated on a grid of 281 values of c, from -100 to 9.5, with the grid most dense on [ 15, 5], for all δ and τ. o investigate the accuracy of the simulated critical values, we calculate the rejection frequencies under a set of known c with = 1000, 200 using the critical values from the lookup table. As shown in able 3, when = 1000, the rejection frequencies are close to 10% by construction, and we take this case as the benchmark. With = 200, able 4 shows that the over-rejection problem is lessened. Although for the outer quantiles moderate size distortion still exists with small sample size, the rejection rates are significantly reduced compared to those from the standard t-tests. hus, for example, if we consider the predictability of the tails in the return distribution, it is useful to perform size correction. For the inner quantiles, when δ is small, the rejection rates are sometimes slightly larger than those from the conventional t-tests; when δ is large, the rejection rates are smaller than those from the conventional t-tests. Hence, size correction is not beneficial when there is none or small correlation between the error terms, but it is necessary when the correlation is large. Since c is a nuisance parameter, which is generally unknown in practice, we next follow the two-step procedure described in the previous section. First, we run the OLS regression according to the ADF representation as in equation (2) to obtain ˆt β1. hen we construct the 100(1 η 1 )% confidence interval for c by inverting the t-statistic (Stock 1991). Next, we run the quantile regression (4) and obtain ˆt γ1 (τ) for each τ. hen we compute the second stage critical values, that is, the 100(1 η 2 )% Bonferroni bounds for ˆt γ1 (τ) over the confidence region of c from stage one. he rejection frequencies are calculated according to the following rejection rule: if ˆt γ1 (τ) lies outside its Bonferroni interval, we reject the null; otherwise, we fail to reject. he size of the test is adjusted by choosing η 1 and η 2 according to the previous section so that the test will not be too conservative. Given η 2 = 10%, η 1 is selected with = 1000 such that the rejection rates are close to the benchmark case. For δ = 0.25, 0.50, 0.75, 0.95, this leads us to select η 1 = 0.7, 0.5, 0.4, 0.3, respectively. able 5 to 6 summarize the results from per- 8

10 forming the size-adjusted Bonferroni method with large and small sample sizes. More precisely, under the level of η 2 = 10%, we plug in η 1 calculated above for each of value of δ, so that the size from Bonferroni correction is not too conservative. With large sample size =1000, the rejection rates of the null, γ 1 (τ) = 0, are between to , which are close to those from the benchmark case due to size adjustment. With smaller sample size, such as = 200, there is negligible size distortion for the inner quantiles, and moderate over-rejection for the outer quantiles, which is a finite sample problem 4. Compared to the experiments with known c, when performing the Bonferroni procedure, the rejection rates in able 6 are generally no larger than those from able 4. herefore, although using the Bonferroni bounds may not fully correct the size distortion for small samples, it is asymptotically valid and can effectively solve the over-rejection problem with large samples. Note that when δ = 0, there is no need for size adjustment. Instead, simulated i.i.d. critical values are used in the tests. Similarly to producing the lookup table, we compute the simulated i.i.d. critical values based on Monte Carlo replications with = 1000, and c =, and store them in a lookup table. Consequently, as the first panel in able 5 illustrates, the rejection frequencies range from to with = Again, for smaller sample sizes, the finite sample problem is more significant. he Bonferroni technique is also evaluated by simulations assuming the residual correlation is unknown, with the data generated under true δ = 0.5 and 0.9. δ is estimated by ˆδ = corr(ˆε 1t, ˆε 2t ), where ˆε 1t is the residual from regressing x t on a constant and its own lag, and ˆε 2t is the residual from regressing y t on a constant and x t 1. he Bonferroni critical values are calculated by interpolation with respect to ˆδ. he level of the predictability test, η 2, is 5%. For δ = 0.25, 0.50, 0.75, 0.95, η 1 = 0.40, 0.30, 0.25, 0.20, respectively. he results are reported in able 7 and 8. For = 1000, the rejection frequencies are close to 5%, the level of the test. he test is conservative for c of smaller absolute values. For = 200, the rejection frequencies are close to 5% for the central part of the distribution. here is some moderate size distortion for the outer quantiles. he over-rejection problem is more severe for the extreme quantiles. 5 Empirical Study his paper applies the Bonferroni technique to test the predictability at different points in the stock return distribution. he univariate quantile regressions of the return on 16 pre-determined predictors are analyzed by Cenesizoglu and immermann (2008) using the data from Goyal and Welch (2008). his paper will use the same data set that comprises monthly observations over the period from February 1871 to December 2005, with the shortest series from May 1937 to December As in Cenesizoglu and immermann (2008), the stock return is measured by the S&P500 index including dividends. he 16 predictor variables, including dividend price 4 When c = 0, the normal approximation is poor in extreme quantiles. See, for example, Chernozhukov (2005) for the case in which x t is I(0). 9

11 ratio, dividend yield and earnings price ratio, are listed in able 9. o examine the predictor variables, first we run OLS regression of each predictor on its own lags as in equation (2). As shown in able 9, the lag length is selected by the Bayes information criterion (BIC) with a maximum of four lags. In addition, the confidence interval for the local-to-unity parameter c of each predictor is calculated based on Stock (1991). From able 9, the largest autoregressive root α 1 is close to but smaller than one in many cases. Some predictors such as dividend price ratio, dividend yield and -bill rate have upper bounds of the 95% interval slightly larger than one. Default return spread, long term rate of return, stock variance, and inflation are close to stationary. Besides, the correlation coefficient δ is estimated by ˆδ = corr(ˆε 1t, ˆε 2t ), where ˆε 1t is the residual from the ADF regression of equation (2), and ˆε 2t is the residual from regressing the stock return on the predictor as in equation (3). As able 9 shows, ˆδ is negative for most predictors. Moreover, ˆε 1t is strongly correlated with ˆε 2t for dividend price ratio, earnings price ratio, smoothed earnings price ratio and book to market ratio. In contrast, for default return spread, long term rate of return and inflation, the cross-correlations are positive and small. o test the predictability of different parts in the return distribution, we then run quantile regression of the return on each predictor for eleven quantiles. he quantile coefficients are estimated for each quantile. he magnitude of the coefficient estimates are the same as those from Cenesizoglu and immermann (2008), where they have discussed the asymmetric effects of the predictive variables on different positions in the return distribution. For our predictive tests, the relevant Bonferroni critical values are interpolated from the lookup tables with respect to ˆδ. he test results from using these Bonferroni bounds are reported in able 10. At the 5% significance level, we find evidence of predictability at various points in the stock return distribution for most predictor variables. Some variables have significant effects on the outer quantiles of the return. For example, we find evidence of predictability of default yield spread in the tails and shoulders of the return distribution, but no evidence for the inner quantiles such as τ = 0.4, 0.5, 0.6, 0.7. For smoothed earnings price ratio and book to market ratio, there is only evidence of predictability in the two tails of the return. For long term rate of return, we reject the null only at τ = For default return spread, the null is only rejected at τ = 0.05, and for net equity expansion, in the left tail where τ = 0.05, 0.1, 0.2. Other variables have more effects on the central part of the distribution. For the long term yield we reject the null in the center and right shoulder. For -bill rate we reject when τ 0.4, and for inflation we reject when τ 0.3. Among all these predictors, inflation can predict most quantiles of the return distribution. -bill rate, long term yield, default yield spread and stock variance also have considerable predictive content compared with the other variables. On the other hand, dividend price ratio, dividend yield and earnings price ratio fail to predict any quantile in the return distribution. In many cases, our critical values give the same inferences as the standard normal critical values do. However, at the 5% significance level, there are discrepancies for smoothed 10

12 earnings price ratio, book to market ratio, cross sectional premium, stock variance, dividend payout ratio, long term yield and long term rate of return. Compared to our technique, the conventional tests over-reject for smoothed earnings price ratio, book to market ratio and stock variance, at some points in the return distribution, due to the large residual cross-correlations. On the contrary, our tests over-reject for long term yield and long term rate of return, where ˆδ is small. his corresponds to our simulation results that the Bonferroni procedure with our simulated critical values is beneficial when there is large residual cross-correlation. Furthermore, our empirical study has some similarities to that conducted by Cenesizoglu and immermann (2008). In their paper, the Bonferroni p-values are used as a summary measure in a joint test across all quantiles considered under the null that a given predictor does not predict any of the quantiles in the return distribution. According to their findings, eight of the predictors considered are significant at the 5% level, while the others are insignificant. In particular, the p-values are one for dividend price ratio and dividend yield, which indicates no predictability of any points in the return distribution using either one of the two variables. Unlike their approach, the Bonferroni methodology we proposed is used for a different purpose. Instead of testing the significance of each predictor across all quantiles jointly, our paper adopts a Bonferroni method in the local-to-unity context and tests whether the quantile regression coefficient is zero for each individual τ. he test results confirm nonpredictability of the return distribution using dividend price ratio and dividend yield, which is consistent with the results from Cenesizoglu and immermann (2008). However, as mentioned before, our empirical results also show that earnings price ratio has no predictive content of the return distribution. 6 Conclusion his paper develops inference in predictive quantile regressions with a nearly nonstationary regressor. he predictive setting of interest is a quantile regression of a dependent variable y t on a lagged regressor x t 1 which is possibly strongly autocorrelated and not strictly exogenous. We derive the limiting distributions of the quantile regression coefficient and its corresponding t- statistic under the local-to-unity specification that α = 1+ c. Both of these distributions depend on the nuisance parameter c and the residual cross-correlation coefficient δ. According to the asymptotic representation of the test statistic, we create computerized lookup tables consisting of simulated critical values for a number of (c, δ) pairs. We also present an asymptotically valid Bonferroni bounds methodology to test whether x t 1 has predictive content of the τth quantile of the conditional distribution of y t using the critical values from our look-up tables. simulation results from 2000 Monte Carlo replications show that it is worthwhile to adopt the Bonferroni procedure to correct the size distortion especially when there is considerable residual cross-correlation. Furthermore, despite the substantial literature on the predictability of the mean of stock return, few research has studied the predictability of the entire return distribution. he 11

13 hus, we apply the asymptotic theory and Bonferroni methodology to test the predictability at different points in the return distribution using the 16 pre-determined predictor variables that have been investigated by some previous studies. From the test results at 5% significance level, except for dividend price ratio, dividend yield and earnings price ratio, all these variables have some predictive component of certain quantiles of the return distribution. 12

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16 Jun, Sung Jae, Joris Pinkse and Haiqing Xu (2009). Preprint. ighter bounds in triangular systems. Knight, Keith (1989). Limit theory for autoregressive-parameter estimates in an infinite-variance random walk. he Canadian Journal of Statistics 17, Koenker, R. (2008). Censored Quantile Regression Redux. Journal of Statistical Software 27(6). Koenker, R. (2010). Additive models for quantile regression: model selection and confidence bandaids. Preprint. Koenker, R and G. Bassettt (1978). Regression quantiles. Econometrica 46, Koenker, Roger and and Zhijie Xiao (2004). Unit root quantile autoregression inference. Journal of the American Statistical Association 99, Koenker, Roger and and Zhijie Xiao (2006). Quantile autoregression. Journal of the American Statistical Association 101, Lanne, Markku (2002). esting the predictability of stock returns. he Review of Economics and Statistics 84(3), Lewellen, Jonathan (2004). Predicting returns with financial ratios. Journal of Financial Economics 74(2), Mankiw, N.G. and M. Shapiro (1986). Do we reject too often? Small sample properties of tests of rational expectations models. Economics Letters 20, Maynard, Alex and Katsumi Shimotsu (2009). Covariance-based orthogonality tests for regressors with unknown persistence. Econometric heory 25(1), Nabeya, Seiji and Bent E. Srensen (1994). Asymptotic distributions of the least-squares estimators and test statistics in the near unit root model with non-zero initial value and local drift and trend. Econometric heory 10, Nelson, C. and M. Kim (1993). Predictable stock returns: he role of small sample bias. Journal of Finance 48,

17 Phillips, P.C.B. (1987). oward a unified asymptotic theory for autoregression. Biometrika 74, Shiller, Robert (1984). Stock prices and social dynamics. Brookings papers on Economic Activity 2, Stambaugh, R.F. (1986). Bias in regressions with lagged stochastic regressors. Center for Research in Security Prices Working Paper 156, University of Chicago. Stambaugh, Robert F. (1999). Predictive regressions. Journal of Financial Economics 54, Stock, J.H. (1991). Confidence intervals for the largest autoregressive root in U.S. economic time series. Journal of Monetary Economics 28 (3), Wolf, M. (2000). Stock returns and dividend yields revisited: A new way to look at an old problem. Journal of Business and Economic Statistics 18(1), Wright, Jonathan (2000). Confidence sets for cointegrating coefficients based on stationarity tests. Journal of Business and Economic Statistics 18(2), Xiao, Zhijie (2009). Quantile cointegrating regression. Journal of Econometrics 150, Xiao, Zhijie and Roger Koenker (2009). Conditional quantile estimation for GARCH models. Boston College Working Papers in Economics Number

18 A Proofs A.1 Proof of Proposition 1 Consider equation (2): x t = β 0 + β 1 x t 1 + ζ(l) x t 1 + ε 1t = β 0 + β 1 x t 1 + ζ 1 x t 1 + ζ 2 x t ζ k x t k + ɛ 1t where ζ(l) = k i=1 ζ il i 1. Use the Frisch-Waugh-Lovell (FWL) heorem: M ι x t = β 1 M ι x t 1 + ζ 1 M ι x t 1 + ζ 2 M ι x t ζ k M ι x t k + M ι ɛ 1t i.e. x t = β 1 x t 1 + ζ 1 x t 1 + ζ 2 x t ζ k x t k + ɛ 1t = X t B + ɛ 1t where X t = [x t 1, x t 1, x t 2,, x t k ] and B = [β 1, ζ 1, ζ 2,, ζ k ]. ɛ 1t = ɛ 1t 1 ɛ 1t. he OLS estimator is: ˆB = [ X t X t ] 1 [ X t x t ] = [ X t X t ] 1 [ X t (X t B + ɛ 1t )] = B + [ X t X t ] 1 [ X t ɛ 1t ]. Let Υ be a (k + 1) (k + 1) matrix such as Υ = , 17

19 Υ( ˆB B) = [Υ 1 X t X t Υ 1 ] 1 [Υ 1 X t ɛ 1t ] 2 (x t 1 )2 3 2 x t 1 x t x t 1 x t x t 1 x t k 3 2 x t 1 x t 1 1 ( x t 1 )2 1 x t 1 x t 2 1 x t 1 x t k = 3 2 x t 2 x t 1 1 x t 2 x t 1 1 ( x t 2 )2 1 x t 2 x t k x t k x t 1 1 x t k x t 1 1 x t k x t 2 1 ( x t k )2 1 x t 1 ɛ 1t 1 2 x t 1 ɛ 1t 1 2 x t 2 ɛ 1t x t k ɛ 1t 1 From Phillips (1987), under α 1 = 1 + c, 1 2 x [ r] d ωj c (r) 2 (x t 1 )2 d (ωj c ) 2 dr 1 x t 1 b 1 (L)ɛ 1t d ωj c dz 1. Recall that {x t } follows the real DGP such that: (1 α 1 L)b(L)(x t α 0 ) = ɛ 1t i.e. (1 α 1 L)x t = b 1 (L)ɛ 1t i.e. x t = β 1 x t 1 + b 1 (L)ɛ 1t where β 1 = α

20 Under α 1 = 1 + c, x t = c x t 1 + b 1 (L)ɛ 1t. hen we have 1 x t i x t j = 1 ( c x t i 1 + b 1 (L)ɛ 1t i )( c x t j 1 + b 1 (L)ɛ 1t j ) = 1 ( c2 2 x t i 1x t j 1 + c x t i 1 b 1 (L)ɛ 1t j + c x t j 1 b 1 (L)ɛ 1t i +b 1 (L)ɛ 1t i b 1 (L)ɛ 1t j ) = 1 c 2 2 x t i 1 x t j 1 this term is O p ( 1 ) + 1 c 1 (1 αl) 1 b 1 (L)ɛ 1t i 1 b 1 (L)ɛ 1t j this term is O p ( 1 ) + 1 c 1 (1 αl) 1 b 1 (L)ɛ 1t j 1 b 1 (L)ɛ 1t i this term is O p ( 1 ) + 1 b 1 (L)ɛ 1t i b 1 (L)ɛ 1t j this term d γ i j = O p ( 1 ) + 1 b 1 (L)ɛ 1t i b 1 (L)ɛ 1t j p γ i j, where γ j = E(b 1 (L)ɛ 1t b 1 (L)ɛ 1t j ) for j = 1,, k, and 1 x t j = 1 ( c x t j 1 + b 1 (L)ɛ 1t j ) = 1 c 1 x t j b 1 (L)ɛ 1t j p 0, since 1 x t j 1 converges, 1 c 1 x t j 1 = O p( 1 ), and 1 b 1 (L)ɛ 1t j p 0. Also, 3 2 x t 1 x t j = 3 2 x t 1 ( c x t j 1 + b 1 (L)ɛ 1t j ) = 1 2 c 2 x t 1 x t j x t 1 b 1 (L)ɛ 1t j = O p ( 1 2 ) (1 αl) 1 b 1 (L)ɛ 1t 1 b 1 (L)ɛ 1t j = O p ( 1 2 ) + Op ( 1 2 ) p 0. In addition, for j = 1,, k, 1 2 x t j ɛ 1t = 1 c 2 ( x t j 1 + b 1 (L)ɛ 1t j )ɛ 1t = 1 2 c 1 x t j 1 ɛ 1t b 1 (L)ɛ 1t j ɛ 1t = O p ( 1 2 ) b 1 (L)ɛ 1t j ɛ 1t. Since 1 x t j 1 ɛ 1t converges, we have the following limiting distribution as in Hamilton 19

21 (1994) 5 : 1 2 t=2 x t 1 ɛ 1t 1 2 t=2 x t 2 ɛ 1t. d h 1 N(0, σ 11 V) 1 2 t=2 x t k ɛ 1t where V = γ 0 γ 1 γ k 1 γ 1 γ 0 γ k γ k 1 γ k 2 γ 0 hus, Υ( ˆB B) d [ = [ ω 2 (J c ) 2 dr 0 ] 1 [ σ ωj c db 1 0 V h 1 σ J c db 1 /ω ] (J c ) 2 dr. V 1 h 1 ] In particular, o test H 0 : β = 0, we consider ( ˆβ 1 β 1 ) d σ J c db 1 ω (J c ) 2 dr. ( ˆβ 0) d c + σ J c db 1 ω (J c ) 2 dr. Let ˆɛ 1t = x t X t ˆB be the residual from the FWL regression, then 5 Hamilton, James D. (1994). ime Series Analysis (Princeton University Press). 20

22 t β1 = = d = = ˆβ 1 0 [ 1 t=2 (ˆɛ 1t )2 ( 1 t=2 (x t 1 )2 ) 1 ] 1 2 ( ˆβ 1 0) (( 1 t=2 ɛ2 1t + o p(1))( 2 t=2 (x t 1 )2 ) 1 ) 1/2 c + b(1) J c db 1 / (J c ) 2 dr [σ 11 (ω 2 (J c ) 2 dr) 1 ] 1 2 c [σ 11 (ω 2 (J c ) 2 dr) 1 ] 1 2 c b(1) [ + σ1/2 11 (J c ) 2 dr] J c db 1 [ (J c ) 2 dr] 1 2 J c db 1 /(ω (J c ) 2 dr) [σ 11 (ω 2 (J c ) 2 dr) 1 ] 1 2 since ω = σ /b(1). A.2 Proof of Proposition 2 his proof follows the procedure in Xiao (2009). Denote û = D (ˆγ(τ) γ(τ)). ρ τ (y t ˆγ(τ) z t 1 ) = ρ τ (γ(τ) z t 1 + ε 2tτ ˆγ(τ) z t 1 ) = ρ τ (ε 2tτ (ˆγ(τ) γ(τ) )z t 1 ) = ρ τ (ε 2tτ (D 1 û) z t 1 ) hus, the minimization problem can be rewritten as min u G (u), where the objective function G (u) = [ρ τ (ε 2tτ (D 1 u) z t 1 ) ρ τ (ε 2tτ )]. From Xiao (2009) for any u 0, hen we have G (u) = ρ τ (u v) ρ τ (u) = vψ τ (u) + (u v)i(0 > u > v) I(0 < u < v). + = + (D 1 u) z t 1 ψ τ (ε 2tτ ) (ε 2tτ (D 1 (D 1 u) z t 1 ψ τ (ε 2tτ ) ((D 1 u) z t 1 )[I(0 > ε 2tτ > (D 1 u) z t 1 ε 2tτ )[I(0 < ε 2tτ < (D 1 u) z t 1 ) I(0 < ε 2tτ < (D 1 u) z t 1 )] u) z t 1 ) I(0 > ε 2tτ > (D 1 u) z t 1 )] 21

23 For the first term, since 1 x t 1 ψ τ (ε 2tτ ) d ωjc dz 2τ, D 1 [ z t 1 ψ τ (ε 2tτ ) = 1 2 ψ τ (ε 2tτ ) 1 x t 1ψ τ (ε 2tτ ) ] d [ ] dz2τ. ωjc dz 2τ For the second term, we start with W (u) = (u D 1 z t 1 ε 2tτ )I(0 < ε 2tτ < u D 1 Denote ξ t (u) = (u D 1 z t 1 ε 2tτ )I(0 < ε 2tτ < u D 1 z t 1). Define W m (u) = where ξ tm (u) = (u D 1 z t 1 ε 2tτ )I(0 < ε 2tτ < u D 1 z t 1). ξ tm(u), z t 1)I(u D 1 z t 1 m) is the truncation of u D 1 z t 1 at some finite number m > 0. Denote ξ tm (u) = E[ξ tm (u) F t 2 ] and W m (u) = ξ tm (u), where F t 2 is information set up to time t 1, and z t 1 F t 2. hus, we have W m (u) = = = = = = = = 1 2 E[(u D 1 E[(u D 1 [u D 1 z t 1 ε 2tτ )I(0 < ε 2tτ < u D 1 z t 1)I(u D 1 z t 1 m) F t 2 ] z t 1 + F 1 (τ) ε 2t )I(F 1 (τ) < ε 2t < u D 1 z t 1 + F 1 (τ))i(u D 1 z t 1 m) F t 2 ] z t 1+F 1 (τ)]i(u D 1 z t 1 m) F 1 (τ) r [u D 1 z t 1+F 1 (τ)]i(u D 1 z t 1 m) s F 1 (τ) [u D 1 z t 1+F 1 (τ)]i(u D 1 z t 1 m) F 1 (τ) [u D 1 z t 1+F 1 (τ)]i(u D 1 z t 1 m) F 1 (τ) [ [s F f t 1 (F 1 1 (τ)] 2 (τ)) 2 = 1 2 f t 1 (F 1 (τ))[u D 1 f t 1 (F 1 (τ))u [ D 1 [ [ [u D 1 z t 1+F 1 (τ)]i(u D 1 z t 1 m) F 1 f t 1 (r)dr]ds [s F 1 (τ)][ F t 1(s) F t 1 (F 1 (τ)) s F 1 (τ) [s F 1 (τ)]f t 1 (F 1 (τ))ds + op(1) [u D 1 z t 1+F 1 (τ)]i(u D 1 z t 1 m) F 1 (τ) z t 1] 2 I(u D 1 z t 1 m) + op(1) z t 1z t 1D 1 By Assumption 3 and stationarity of {f t 1 (F 1 (τ))}, for some ɛ > 0 ] + op(1) ds]f t 1 (r)dr ]ds ]ui(u D 1 z t 1 m) + op(1) sup 0 r 1 1 [ r] 1 ɛ [f t 1 (F 1 (τ)) f(f 1 (τ))] d 0. 22

24 herefore, we have W m (u) d 1 2 f(f 1 (τ))u [ := η m B z B z]i(0 < u B z m)u where B z = (1, ωj c ). Since (u D 1 z t 1)I(0 u D 1 z t 1 m) p 0 uniformly in t, E[ξ tm (u) 2 F t 2 ] max{(u D 1 p 0. z t 1)I(0 u D 1 z t 1 m)} ξtm (u) hus, [ξ tm(u) ξ tm (u)] p 0, where ξ tm (u) ξ tm (u) is a MDS. By the Asymptotic Equivalence lemma, the limiting distribution of ξ tm(u) is the same as that of ξ 1 tm (u), i.e. W m (u) d η m. As m, η m d 2 f(f 1 (τ))u [ B z B z]ui(u B z > 0) = η. We have lim m lim sup n P r[ W (u) W m (u) ɛ] = 0, since P r[ W (u) W m (u) 0] = P r[ (u D 1 P r[ {u D 1 z t 1 > m}] t = P r[max{u D 1 t z t 1} > m] lim P r[ sup u B z (r) > m] = 0. m 0 r 1 Also, W d η, i.e. z t 1 ε 2tτ )I(0 < ε 2tτ < u D 1 z t 1)I(u D 1 z t 1 > m)] (u D 1 z t 1 ε 2tτ )I(0 < ε 2tτ < u D 1 z 1 t 1) d 2 f(f 1 (τ))u [ B z B z]ui(u B z > 0). Similarly, (u D 1 z t 1 ε 2tτ )I(0 > ε 2tτ > u D 1 z 1 t 1) d 2 f(f 1 (τ))u [ B z B z]ui(u B z < 0). hus, the second term of the objective function converges to 1 2 f(f 1 (τ))u [ B z B z]u in distribution. he limiting distribution of the objective function is as follows, G (u) d u [ ] [ dz2τ + 1 ωjc dz 2τ 2 f(f 1 (τ))u ] 1 ωjc ωjc (ωjc ) 2 u := G(u). 23

25 Consider the first order condition of the minimization problem of G(u), G(u) u = the solution of which is the following u = [ ] [ dz2τ + f(f 1 (τ)) ωjc dz 2τ [ 1 f(f 1 (τ)) ] 1 ωjc ωjc (ωjc ) 2 u = 0, ] 1 [ ] 1 ωjc dz2τ ωjc (ωjc ) 2. ωjc dz 2τ G (u) is minimized at û = D (ˆγ(τ) γ(τ)), and û d u (Knight 1989, Xiao 2009), i.e. [ 1 D (ˆγ(τ) γ(τ)) d f(f 1 (τ)) ] 1 [ ] 1 ωjc dz2τ ωjc (ωjc ) 2. ωjc dz 2τ B ables 24

26 able 1: Standard t-tests: Finite-sample size ( = 1000) c τ = δ = δ = δ = δ = δ = Notes: he asymptotic results are based on 2000 Monte Carlo replications with sample size. Simulated data is generated according to: x t = (1 + c )x t 1 + ε 1t and y t = ε 2t, where ε 1t and ε 2t are i.i.d. N(0, 1) with correlation coefficient δ. he level of the test is 10%. 25