Measuring nonlinear Granger causality in mean

Transcription

1 Measuring nonlinear Granger causality in mean Xiaojun Song Universidad Carlos III de Madrid Abderrahim Taamouti Universidad Carlos III de Madrid May 20, 2013 Economics Department, Universidad Carlos III de Madrid. Address: Departamento de Economía, Universidad Carlos III de Madrid Calle Madrid, Getafe (Madrid) España. Economics Department, Universidad Carlos III de Madrid. Address: Departamento de Economía, Universidad Carlos III de Madrid Calle Madrid, Getafe (Madrid) España. TEL: ; FAX: ; 1

2 ABSTRACT We propose model-free measures for Granger causality-in-mean. Unlike the existing measures, ours are able to detect and quantify nonlinear Granger causality-in-mean between random variables. The new measures are based on nonparametric regressions and defined as a logarithmic function of restricted and unrestricted mean square forecast errors. They are easily and consistently estimated by replacing the unknown mean square forecast errors by their nonparametric kernel estimates. We establish the asymptotic normality of the nonparametric estimator of causality measures, which is used to build t-tests for the statistical significance of the measures. A desirable property of these t-tests is that they have nontrivial power against T -local alternatives, where T is the sample size. Monte Carlo simulations reveal that the t-tests behave very well and have quite good finite sample size and power properties for a variety of typical data generating processes and different sample sizes. Moreover, since testing that the value of the measure is equal to zero is equivalent to testing for non-causality in mean, we also consider additional simulation exercises where we compare the empirical size and power of our test with one of Nishiyama et al. (2011) who propose a nonparametric test for Granger causality-in-mean. The simulation results indicate that our test has comparable size, but much better power than Nishiyama et al. s (2011) test. We also establish the validity of smoothed local bootstrap that one can use in finite sample settings to test for Granger causality measures. Finally, the empirical importance of measuring nonlinear causality-in-mean is also illustrated in the present paper. We quantify the degree of nonlinear predictability of risk premium (expected stock excess returns) using what is called variance risk premium (the difference between implied and realized volatilities). Our empirical results indicate that the variance risk premium is a very good predictor of risk premium at horizons less than six months. Contrary to Bollerslev, Tauchen and Zhou (2009), we find that there is a very high degree of predictability at horizon one-month which can be attributed to a nonlinear causal effect. Keywords: Granger causality measures; nonlinear causality in mean; nonparametric estimation; time series; bootstrap; volatility index; realized volatility; variance risk premium; risk premium. Journal of Economic Literature classification: C12; C14; C15; C19; G1; G12; E3; E4.

3 1 Introduction The concept of causality introduced by Wiener (1956) and Granger (1969) constitutes a basic notion for studying dynamic relationships between time series. In studying Wiener-Granger causality, predictability is the central issue which is of great importance to the economists, policymakers and investors. Much research has been devoted to building and applying tests of non-causality in mean between random variables. However, once we have concluded that a causal relation is present, it is usually important to assess the strength of this relationship. Only few papers [see Geweke (1982, 1984) and Dufour and Taamouti (2010)] have been proposed to measure the causality-in-mean between the variables of interest. Furthermore, those papers assume a parametric linear model for the conditional mean, consequently the proposed measures ignore nonlinear causal relationships, which might lead to invalid causal analysis. Thus, we simply cannot use the existing causality measures to quantify the strength of nonlinear causality-in-mean. The present paper aims to propose a model-free measures that quantify nonlinear causality-in-mean. Wiener-Granger analysis distinguishes between three basic types of causality: from Y to X, from X to Y, and instantaneous causality. In practice, it is possible that all three causality relations coexist, hence the importance of finding means to quantify their degree. Unfortunately, causality tests fail to accomplish this task, because they only provide evidence on the presence of causality. A large effect may not be statistically significant (at a given level), and a statistically significant effect may not be large from an economic viewpoint (or more generally from the viewpoint of the subject at hand) or relevant for decision making. Hence, as emphasized by McCloskey and Ziliak (1996), it is crucial to distinguish between the numerical value of a parameter and its statistical significance. Thus, beyond accepting or rejecting non-causality hypotheses which state that certain variables do not help forecasting other variables we wish to assess the magnitude of the forecast improvement, where the latter is defined in terms of some loss function (mean-square forecast errors). Even if the hypothesis of no improvement (non-causality) cannot be rejected from looking at the available data (for example, because the sample size or the structure of the process does allow for high test power), sizeable improvements may remain consistent with the same data. Or, by contrast, a statistically significant improvement which may easily be produced by a large data set - may not be relevant from a practical viewpoint. In this paper we propose measures that are able to quantify nonlinear Granger causality-in-mean. Those measures are model-free, therefore they do not require the specification of the model linking the variables of interest. The new measures are based on nonparametric regressions and are defined as a logarithmic function of mean square forecast errors of the restricted and unrestricted nonparametric regression models. They are easily estimated by replacing the unknown restricted and unrestricted mean square forecast errors by their nonparametric kernel estimators. 1

4 Another main contribution of the present paper is that we establish the asymptotic normality of nonparametric estimator of measures of Granger causality-in-mean. This result is used to build t-test for the statistical significance of our causality measures. A desirable property of the t-test is that it has nontrivial power against T -local alternatives, where T is the sample size. Monte Carlo simulation study reveals that this t-test which is based on the asymptotic normality behaves very well and has desirable finite sample size and power properties for a variety of typical data generating processes and different sample sizes. This is very interesting because we do not need to use bootstrap, and thus the implementation of the test is very easy. Moreover, since testing that the value of the measure is equal to zero is equivalent to testing for non-causality in mean, we also consider additional simulation exercises where we compare the empirical size and power of our test with one of Nishiyama et al. (2011) who propose a nonparametric test for Granger causality-in-mean. To do so, we consider quite similar data generating processes to the ones in Nishiyama et al. (2011). The simulation results indicate that our test has comparable size to the one of Nishiyama et al. (2011), but with much higher power than Nishiyama et al. s (2011) test. Finally, we establish the validity of smoothed local bootstrap that one can use in finite sample settings to test for Granger causality measures. The empirical importance of measuring nonlinear causality-in-mean is also illustrated in the present paper. We quantify the degree of nonlinear predictability of risk premium (expected stock excess returns) using what is called variance risk premium [see Bollerslev, Tauchen and Zhou (2009)]. The latter is defined as the difference between the risk-neutral and objective expectations of realized variance, where the riskneutral expectation of variance is measured as the end-of-month Volatility Index-squared de-annualized and the realized variance is the sum of squared 5-minute log returns of the S&P 500 index over the month. Our empirical results indicate that the variance risk premium is a very good predictor of risk premium at horizons less than six months. Contrary to Bollerslev, Tauchen and Zhou (2009), we find that there is a very high degree of predictability at horizon one-month which can be attributed to a nonlinear causal effect. Finally, the topic of measuring the causality has attracted much less attention. Geweke (1982, 1984) introduced measures of causality-in-mean based on linear parametric autoregressive models. Dufour and Taamouti (2010) extended Geweke (1982, 1984) measures by proposing a short and long run causality-inmean measures in the context of parametric ARMA models. Gouriéroux, Monfort, and Renault (1987) have proposed causality measures based on the Kullback information criterion and provided a parametric estimation. Polasek (1994, 2002) showed how causality measures can be computed using the Akaike Information Criterion (AIC) and a Bayesian approach. Taamouti, Bouezmarni and El Ghouch (2012) have proposed a nonparametric estimator and a nonparametric test for Granger causality measures that quantify linear and nonlinear Granger causality in distribution. However, the main problem with their measures is that they cannot inform us about levels of distribution where the causality exist. The plan of the paper is as follows. Section 2 presents the general theoretical framework which un- 2

5 derlies the definition of causality-in-mean. In Section 3 and 4, we define the theoretical nonparametric measures of Granger causality-in-mean. In Section 5 we introduce a consistent nonparametric estimator of causality measures based on kernel estimation of mean-square forecast errors of restricted and unrestricted nonparametric regressions. We also establish the asymptotic distribution of the nonparametric estimator of causality-in-mean measures and discuss the asymptotic validity of a smoothed local bootstrap assisted test. In Section 6 we extend our results to the case where the random variables of interest are multivariate. In Section 7 we provide a simulation exercise to investigate the finite sample properties of the t-test for causality measures based on the asymptotic normality result. Section 8 is devoted to an empirical application and the conclusion relating to the results is given in Section 9. Proofs appear in the Appendix A. 2 Framework The notion of non-causality studied here is defined in terms of orthogonality conditions between subspaces of a Hilbert space of random variables with finite second moments. We denote L 2 L 2 (Ω, A, Q) a Hilbert space of real random variables with finite second moments, defined on a common probability space (Ω, A, Q). If E and F are two Hilbert subspaces of L 2, we denote E + F the smallest subspace of L 2 which contains both E and F, while E\F represents the smallest Hilbert subspace of L 2 which contains the difference E F = E F = {x : x E, x / F }.[If E F is empty, we set E\F = {0}.] Information is represented here by nondecreasing sequences of Hilbert subspaces of L 2. In particular, we consider a sequence I of reference information sets I(t), I = {I(t) : t Z, t > ω} with t < t I(t) I(t ) for all t > ω, (1) where I(t) is a Hilbert subspace of L 2, ω Z { } represents a starting point, and Z is the set of the integers. The starting point ω is typically equal to a finite initial date (such as ω = 1, 0 or 1) or to ; in the latter case I(t) is defined for all t Z. We also consider two stochastic processes X = {X t : t Z, t > ω}, Y = {Y t : t Z, t > ω} and a (possibly empty) Hilbert subspace H of L 2, whose elements represent information available at any time, such as time independent variables (e.g., the constant in a regression model) and deterministic processes (e.g., deterministic trends). We denote X(ω, t] the Hilbert space spanned by the components x i (τ), i = 1,..., m 1, of X(τ), ω < τ t, and similarly for Y (ω, t] : X(ω, t] and Y (ω, t] represent the information contained in the history of the variables X and Y respectively up to time t. Finally, the information sets obtained by adding X(ω, t] to I(t) and Y (ω, t] to I X (t) are defined as I X (t) = I(t) + X(ω, t], I XY (t) = I X (t) + Y (ω, t], (2) 3

6 For any information set B t [some Hilbert subspace of L 2 ], we denote P [X t+1 B t ] the best nonlinear forecast of X t+1 based on the information set B t, u[x t+1 B t ] = X t+1 P [X t+1 B t ] the corresponding prediction error, and σ 2 [X t+1 B t ] = E { u[x t+1 B t ] 2}. P [X t+1 B t ] is the orthogonal projection of X t+1 on the subspace B t. Following the definitions in Dufour and Taamouti (2010), characterization of non-causality can be expressed in terms of the variance of the forecast errors. Definition 1 Y does not cause X given I iff σ 2 [X t+1 I X (t)] = σ 2 [X t+1 I XY (t)], t > ω, where σ 2 [X t+1 ] = E { u[x t+1 ] 2}. This definition corresponds to causality from Y to X. It means that Y causes X if the past of Y improves the forecast of X t+1 based on the information in I(t) and X(ω, t]. We can in a similar way characterize the non-causality from X to Y as follows: Definition 2 X does not cause Y given I iff σ 2 [Y t+1 I Y (t)] = σ 2 [Y t+1 I XY (t)], t > ω, where σ 2 [Y t+1 ] = E { u[y t+1 ] 2}. 3 Causality measures The nonparametric causality measures that we propose here are defined using similar measure function as in Geweke (1982, 1984) and Dufour and Taamouti (2010). Important properties of these measure function include: (1) it is nonnegative, and (2) it cancels only when there is no causality. Specifically, the measure function of Granger causality that we use are defined as follows, where by convention ln(0/0) = 0 and ln(x/0) = + for x > 0. Definition 3 The function [ σ 2 ] [X t+1 I X (t)] C(Y X I) = ln σ 2 [X t+1 I XY (t)] is the mean-square causality measure from Y to X, given I. (3) 4

7 Since we consider here only mean-square measures, the term mean square causality measure will be abbreviated to causality measure. Clearly, C(Y X I) = 0 if Y (ω, t] I X (t), so C(Y X I) provides useful information mainly when Y (ω, t] I(t). C(Y X I) measures the causal effect from Y to X given I and the past of X. In terms of predictability, this can be viewed as the amount of information brought by the past of Y which can improve the forecast of X t+1. We now define an instantaneous causality measure between X and Y as follows. Definition 4 The function C(X Y I) = ln [ σ 2 [X t+1 I XY (t)] σ 2 ] [Y t+1 I XY (t)], det Σ[X t+1, Y t+1 I XY (t)] where Σ[X t+1, Y t+1 I XY (t)] = E { U[Z t+1 I XY (t)]u[z t+1 I XY (t)] } and Zt = ( X t, Y t ), is the mean-square instantaneous causality measure between Y and X. Observe that: det Σ[ ( X t+1, Y t+1 ) IXY (t)] = σ 2 [X t+1 I XY (t)] σ 2 [Y t+1 I XY (t)] ( cov[(x t+1, Y t+1 I XY (t)] ) 2, so that where [ C(X Y I) = ln ρ[x t+1, Y t+1 I XY (t)] = 1 1 ρ[x t+1, Y t+1 I XY (t)] 2 cov[x t+1, Y t+1 I XY (t)] σ[x t+1 I XY (t)]σ[y t+1 I XY (t)]. ], (4) is the conditional correlation coefficient between X t+1 and Y t+1 given the information set I XY (t). Thus, instantaneous causality increases with the absolute value of the conditional correlation coefficient. We also define a measure of dependence between X and Y. This will enable one to check whether the processes X and Y must be considered together or whether they can be treated separately. Definition 5 The function C(X, Y I) = C(X Y I) + C(Y X I) + C(X Y I) (5) is the intensity of the dependence between X and Y, given I. form: It is easy to see that the intensity of the dependence between X and Y can be written in the alternative C(X, Y I) = ln [ σ 2 [X t+1 I X (t)] σ 2 ] [Y t+1 I Y (t)], det Σ[X t+1, Y t+1 I XY (t)] where I Y (t) represents the Hilbert subspace spanned by the components of Y t and similarly for I X (t). 5

8 4 Causality measures for nonparametric regression models Let { (X t, Y t ) R R R 2, t = 0,..., T } be a sample of stationary stochastic process in R 2. For simplicity of exposition, here we consider the case of univariate Markov processes of order one. Later, see Section 6, we extend our results to the case where the variables of interest X and Y are multivariate Markov processes of order p, for p 1. We now focus on the following nonparametric regression Z t+1 = Φ (Z t )) + u t+1, (6) where Z t+1 = ( X t+1, Y t+1 ) and ut+1 = ( u X t+1, uy t+1). From the joint nonparametric regression (6) we can obtain the following marginal regression for X t+1 and Y t+1 : X t+1 = Φ 1 (Z t ) + u X t+1 and Y t+1 = Φ 2 (Z t ) + u Y t+1, where Φ 1 (Z t ) = E [X t+1 X t, Y t ] and Φ 2 (Z t ) = E [Y t+1 X t, Y t ]. In this case, we have σ 2 [X t+1 I XY (t)] = V ar [ u X t+1] = V ar [(Xt+1 Φ 1 (Z t ))]. Similarly, we have σ 2 [Y t+1 I XY (t)] = V ar [ u Y t+1] = V ar [(Yt+1 Φ 2 (Z t ))]. To quantify the degree of causality from Y to X and from X to Y we also need to consider the constrained regression models of the processes Y and X, respectively. The constrained models are: X t+1 = Φ 1 (X t ) + ū X t+1 and Y t+1 = Φ 2 (Y t ) + ū Y t+1, where Φ 1 (X t ) = E [X t+1 X t ] and Φ 2 (Y t ) = E [Y t+1 Y t ]. From the above notations and using the definitions in the previous sections, we have the following nonparametric regression based measures of Granger causality from X to Y and from Y to X. Proposition 1 Under assumption (6), C(Y X I ) = ln [ V ar [(Xt+1 E [X t+1 X t ])] ] V ar [(X t+1 E [X t+1 X t, Y t ])] Similarly, [ ] V ar [(Yt+1 E [Y t+1 Y t ])] C(X Y I) = ln. V ar [(Y t+1 E [Y t+1 Y t, X t ])] 6

9 Using Equation (4), a nonparametric regression based measure of the instantaneous causality between X and Y is given by the following proposition. Proposition 2 Under assumption (6), [ C(X Y I)=ln 1 1 ρ[(x t+1 E [X t+1 X t, Y t ]), (Y t+1 E [Y t+1 Y t, X t ]) ] 2 The nonparametric regression based measure of dependence between X and Y can be deduced from its decomposition given by Equation (5). In the next section, we propose nonparametric estimators for the above Granger causality measures and we provide their asymptotic distributions. We also establish the local power property of the test for non-causality and a bootstrap procedure to improve finite sample performance. The basic idea is to consider nonparametric estimation of the following restricted and unrestricted nonparametric residuals: X t+1 E [X t+1 X t ], Y t+1 E [Y t+1 Y t ], X t+1 E [X t+1 X t, Y t ], and Y t+1 E [Y t+1 Y t, X t ]. Thus, the above measures can easily and consistently estimated by replacing the unknown mean square forecast errors by their nonparametric kernel estimates. ]. 5 Estimation and inference Henceforth, without confusion, we will omit the conditioning set I in the Granger causality measure developed in previous sections. To show the consistency and asymptotic normality of estimator of proposed nonparametric Granger causality measures, we need to control the amount of dependence in the processes of interest. In what follows, we consider β-mixing dependent random variables. Let us recall the definition of a β-mixing process (see e.g., Doukhan (1994); Fan and Yao (2003), among others). For { Z t = (X t, Y t ) ; t 0 } a strictly stationary stochastic process and F s t a sigma algebra generated by (Z s,..., Z t ) for s t, the process Z is called β-mixing or absolutely regular, if β (l) = sup s N E sup A F + s+l ( ) P A F s P (A) 0, a.s. l. The β-mixing condition is required to show the asymptotic normality of nonparametric estimator of Granger causality measure [see Tenreiro (1997) and Fan and Li (1999)]. 5.1 Estimation In Section 4 we have shown that the Granger causality measures can be written in terms of variances of the restricted and unrestricted nonparametric residuals. Hereafter, we focus on the estimation of Granger causality measures from Y to X, C(Y X), defined in Proposition 1. Estimators of measures of Granger 7

10 causality from X to Y and of the instantaneous Granger causality between X and Y can be obtained in similar way. Recall Z t = (X t, Y t ). To estimate C(Y X), we need to find consistent estimates of the quantities σ 2 = V ar[ū X t ] (i.e. the restricted mean square forecast errors) and σ 2 = V ar[u X t ] (i.e. the unrestricted mean square forecast errors). To get ˆ σ 2 and ˆσ 2, we first need to estimate the restricted conditional regression function Φ 1 (X t 1 ) and the unrestricted conditional regression function Φ 1 (Z t 1 ) nonparametrically and then get the nonparametric residuals ˆū X t and û X t. The natural nonparametric estimators for regression functions are the Nadaraya-Watson kernel estimators [see Nadaraya (1964) and Watson (1964)]. To this end, we define ( ) W t (x, h) K x Xt 1 h = ( ) T s=1 K x Xs 1 h as the Nadaraya-Watson weights for estimation of the restricted conditional regression function Φ 1 ( ) = E(X t X t 1 = ), where h = h T R + is a sequence of smoothing parameters (i.e. bandwidth) and K is a univariate kernel function, usually a density function. We have and ˆ σ 2 := 1 T ˆ Φ 1 (x) = t=1 (ˆū X t ) 2 = 1 T t=1 W t (x, h)x t (X t ˆ Φ1 (X t 1 )) 2 as the Nadaraya-Watson estimator of Φ 1 ( ) and nonparametric residual-based estimator of σ 2, respectively. ( ) Similarly, with z = (x, y) and h = (h 1, h 2 ), we define W t (z, h) = K z Zt 1 h / ( ) T s=1 K z Zs 1 h to ( ) estimate the unrestricted conditional regression function Φ 1 ( ) = E(X t Z t 1 = ), where K z Zt 1 h = ( ) ( ) k x Xt 1 h 1 k y Yt 1 h 2 is a two-product kernel function and k is a univariate kernel function. Likewise, we have and ˆσ 2 := 1 T ˆΦ 1 (z) = t=1 W t (z, h)x t t=1 (û X t ) 2 = 1 T t=1 (X t ˆΦ 1 (Z t 1 )) 2 as the Nadaraya-Watson estimator of Φ 1 ( ) and nonparametric residual-based estimator of σ 2, respectively. t=1 Note that we have adopted three different bandwidths h, h 1 and h 2 to take into account of the possible data heterogeneity among X t and Y t in the restricted and unrestricted nonparametric estimation problems. However, as is well understood in the nonparametric estimation literature, the choices of kernel functions are not so important as that of bandwidths. Therefore, in general, we can use the same univariate kernel function k( ) in the above estimators. To further simplify our asymptotic analysis and to minimize the notation, we will adopt h 1 = h 2 = h in the unrestricted estimation part outlined above. However, different 8

11 bandwidths could be considered and the asymptotic theory will still be valid when h 1 h 2 with a bit more complexity. Thus, based on the previous proposed nonparametric estimators ˆ σ 2 and ˆσ 2 of the loss functions defined as the restricted and unrestricted mean square forecast errors, respectively, a natural estimator of Granger causality measure from Y to X, C(Y X), is given by ( ˆ σ 2 ) ( 1 T T t=1 Ĉ(Y X) := ln ˆσ 2 = ln (X ) t ˆ Φ1 (X t 1 )) 2 T t=1 (X t ˆΦ, (7) 1 (Z t 1 )) 2 where ˆ Φ1 ( ) and ˆΦ 1 ( ) are kernel estimators of the unknown regression functions. 1 T The most basic property that the above estimator should have is consistency. To show consistency, we need to impose some regularity conditions on the stochastic processes generating the data, and bandwidth parameters and possible kernel choice in the Nadaraya-Watson estimators. Assumption A.1 (Assumptions on the stochastic process) A.1.1 {(X t, Y t ) R R R 2, t 0} is a strictly stationary, ergodic and absolutely regular (β-mixing) process. A.1.2 The marginal density f X ( ) of X t and the joint density f Z ( ) of Z t = (X t, Y t ) are bounded away from zero and bounded above. Assumption A.2 (Assumptions on the bandwidth parameters and kernel function) A.2.1 The kernel functions K( ) and K( ) are two-product and univariate kernel functions, respectively and they are symmetric and bounded. That is, K(u 1, u 2 ) = k(u 1 )k(u 2 ) and K(u) = k(u), where k( ) satisfies k(u) du = 1, uk(u) du = 0 and u 2 k(u) du <. A.2.2 The bandwidth parameters h and h satisfy h 0, h 0, as T. Further, T h and T h 2, as T. Assumption A.1.1 is standard in asymptotic theory of nonparametric regression for dependent data and is satisfied by many processes such as ARMA and ARCH processes. It is possible to relax this assumption to strong mixing under which we expect to complicate the proof further. We need Assumptions A.2.1 and A.2.2 to show the asymptotic normality of our test statistics, where Assumption A.2.1 is needed to alleviate the bias terms of nonparametric variance estimators constructed from the (restricted and/or unrestricted) nonparametric residuals. Note that for first order Markov case in this section, the Gaussian kernel function (r = 2) suffices. Assumption A.2.2 is a common and minimal assumption in nonparametric regression literature. Observe that the assumption A.1.2, which requires that the densities f X ( ) and f Z ( ) have to be bounded away from zero, is for convenient purposes only. This assumption eases greatly the derivation of the asymptotic theory. It can be weakened by employing, for example, indicator function as a trimming function like in Robinson (1988) to trim out near zero densities in the nonparametric estimation procedure. We can also 9

12 use a general weighting function, like densities, to circumvent the problem of random denominator often encountered in the nonparametric estimation and testing literature. We now state the consistency of the estimator Ĉ(Y X) defined in (7). Proposition 3 Under Assumptions A.1-A.2, the estimator Ĉ(Y X) converges in probability to the true Granger causality measure C(Y X). The proof of Proposition 3 can be found in Appendix A. By inspecting the proof, we find that the consistency result can be strengthen to almost sure convergence without difficulty. In the next section we establish the asymptotic normality of the nonparametric estimator Ĉ(Y X) under the no Granger causality hypothesis. This will enable us to construct tests and provide confidence intervals for our Granger causality measures. 5.2 Inference The measures proposed in the previous sections can be used to test for the Granger non-causality in the mean between random variables. If there is no causality, we immediately have σ 2 = σ 2. Obviously, the null hypothesis of interest is given by H 0 : C(Y X) = 0, (8) which corresponds to Granger non-causality in mean from Y to X. property for our test statistic. We have the following asymptotic Theorem 1 Under Assumptions A.1.1-A.1.5 and under the null hypothesis of Granger non-causality from Y to X in (8), we have T Ĉ(Y X) d N (0, Ω), where ( (ū X Ω = κ + κ 2E t ) 2 (u X t ) 2 ) σ 2 σ 2. with κ = E(ū X t ) 4 / σ 4 and κ = E(u X t ) 4 /σ 4 are the kurtosis coefficients of ū X t and u X t, respectively. It is worthy to notice that, if we assume ū X t and u X t are jointly normally distributed and the correlation coefficient is zero between ū X t and u X t so that ū X t u X t, we have κ = κ = 3, and the asymptotic variance is simply Ω = 4. In this special case, it is no longer necessary to estimate Ω. In fact, we study this case extensively in our Monte Carlo section. However, even for the general case, consistent estimator of Ω is readily obtained since kurtosis estimators are widely available in the literature. In this paper, we propose to use ˆΩ = ˆ κ + ˆκ 2 1 n t=1 (ˆū X t ) 2 ˆ σ 2 (û X t ) 2 ˆσ 2, 10

13 with ˆ κ = 1 T n t=1 (ˆū X t ˆū X ) 4 ˆ σ 2, ˆκ = 1 T n t=1 (ûx t û X ) 4 ˆσ 2 the estimators of κ and κ, respectively, where ˆū X = T t=1 ˆū X t /T and û X = T t=1 ûx t /T denote the sample mean of the nonparametric residuals ˆū X t and û X t obtained from the restricted and unrestricted nonparametric regressions, respectively. The following proposition establishes the consistency property of the test statistic defined in Theorem 1. Proposition 4 If Assumptions A.1 and A.2 hold, then the test defined in Theorem 1 is consistent for any loss functions σ 2 and σ 2 such that: σ 2 σ 2 > 0, where σ 2 and σ 2 are the mean square forecast errors of ū X t and u X t from the restricted and unrestricted nonparametric regression functions defined in Section 4, respectively. 5.3 Local power analysis In this section, we discuss the power of our test under the T -local alternatives: H 1T : C(Y X) = 1 T µ, (9) where µ is a finite positive constant. The following proposition states that our test has non-trivial local power against the alternatives converging to the null at a parametric rate. Proposition 5 Under Assumptions A.1 and A.2, under the local alternatives in (9), we have T Ĉ(Y X) d N (µ, Ω), where Ω is defined in Theorem 1. We do not provide the proof of Proposition 5 because it is obvious from the proof of Proposition 4. We can immediately conclude that the limiting distribution of the nonparametric estimator Ĉ(Y X) is non-trivially shifted whenever µ > 0, and therefore the proposed test is able to detect local alternatives converging to the null at a parametric rate. 5.4 Smoothed local bootstrap It is important to notice that in general the result in Theorem 1 is valid only asymptotically and the asymptotic normal distribution might not work very well in finite samples. It is true that our simulation study indicates that the asymptotic normal distribution seems to approximate quite well the finite sample distribution in terms of empirical size and power. This is not surprising since we found, see sections

14 and 5.3, that the test of non-causality has parametric convergence rate and nontrivial power against T - local alternatives. However, particularly for high dimensional random variables, the asymptotic test is subject to size distortion because of possible finite sample bias in the nonparametric estimation due to curse of dimensionality. Furthermore, it is challenging to choose optimal bandwidths that maximize the test s performance. One way to improve the size and power performance for our proposed test is to use the smoothed local bootstrap developed in Paparoditis and Politis (2000). One major advantage for the smoothed local bootstrap procedure is that it can preserve the unknown dependence structure in the data, thus it can mimic the finite sample distribution of our test statistic. In the sequel, X f X means that the random variable X is generated from the density function f X. Let L 1 ( ), L 2 ( ) L 3 ( ) be three univariate kernels that satisfy Assumption A.2.1 and h be a smoothing parameter satisfying Assumption A.3 below. The smoothed local bootstrap in our context consists of the following four steps: (1) We draw a bootstrap sample {(X t, Y t )} T t=1. We first draw X t 1 as follows Xt 1 1 ( ) Xs 1 x T h L h ; s=1 then conditional on Xt 1, we draw X t and Yt 1 independently from the following nonparametric estimators T ( Xs 1 Xt 1 ( Xs y ( Xs 1 Xt 1 ) and X t 1 h Y t 1 1 h s=1 s=1 L h ) L ( Xs 1 X ) t 1 L L h h ( Ys 1 z h ) / ) / s=1 s=1 L h ( Xs 1 X ) t 1 L ; (2) Based on the bootstrap sample, we compute the bootstrapped version of test statistic Γ = T Ĉ (Y X)/ ˆΩ ; (3) Repeat the steps (1)-(2) B times so that we get Γ j, for j = 1,..., B; (4) We compute the bootstrapped p-value using p = B 1 B j=1 1(Γ j > Γ), where Γ = T Ĉ(Y X)/ ˆΩ is the test statistic based on the original sample, and for a given significance level α, we reject the null hypothesis if p < α. Notice that in the above bootstrap procedure, we have taken the same bandwidth h in the nonparametric kernel estimator of conditional density of X t given X t 1 (resp. Y t 1 given X t 1 ). However, using different bandwidth parameters will not invalidate the local bootstrap. Now we need to impose an additional assumption concerning the bandwidth parameter h in order to validate the smoothed local bootstrap discussed above. Assumption A.3 (Assumption on the bootstrap bandwidth parameter) The bootstrap bandwidth parameter h satisfies h 0 and T h 1+2r /(ln T ) γ C, for some γ > 0 and 0 < C <, as T. 12 h

15 Theorem 2 Under Assumptions A.1, A.2, and A.3, under the null hypothesis of Granger non-causality from Y to X in (8), we have T Ĉ (Y X) d N (0, Ω), with Ω is defined in Theorem 1. 6 Extension: High dimensional random variables In this section, we extend our analysis for high dimensional variables. However, due to the prevalent course of dimensionality, we shall focus our attention on d 8. As it has been well established in the nonparametric variance estimation literature, the accuracy T 1/2 of variance estimation can be achieved only under some restrictions on smoothness properties of the regression function and on the dimensionality of the model. For a higher dimensional model such as d > 8, the optimal accuracy of variance estimation is only T 4/d which is worse than T 1/2, so that for dimensionality larger than 8 we can not guarantee the T -convergence of the test statistics for testing Granger non-causality in mean. See Spokoiny (2002) for more details about variance estimation for high-dimensional regression models. Again, we focus on the estimation and inference for Granger causality measures from Y to X, C(Y X). Estimation and inference for measures of Granger causality from X to Y and of the instantaneous Granger causality between X and Y can be obtained in similar way. We consider the following set of standard assumptions. They are only mild modification of Assumptions A.1-A.2. Assumption A.1 (Assumptions on the stochastic process) A.1.1 {Z t = (X t, Y t ) R d 1 R d 2 R d, t 0} is a strictly stationary, ergodic and absolutely regular (β-mixing) process, where d = d 1 + d 2 8. A.1.2 The marginal density f X ( ) of X t and joint density f Z ( ) of Z t are bounded away from zero and bounded above. We also assume that both f X ( ) and f Z ( ) are r + 1-times continuously differentiable on their supports X and Z, respectively. Assumption A.2 (Assumptions on the bandwidth parameters and kernel function) A.2.1 The kernel functions K( ) and K( ) are d-product and d 1 -product kernel functions, respectively and they are symmetric and bounded. That is, K(u) = k(u j ) and K(u) = d 1 Π k(u j=1 j), where k( ) satisfies j=1 k(u) du = 1 and u i k(u) du = 0 for 1 i r 1 and u r k(u) du < with r 2. A.2.2 The bandwidth parameters h and h satisfy h 0, h 0, as T. Further, T h d 1, T h d 1+d 2, as T. d Π Proposition 6 Under Assumptions A.1 and A.2, the estimator the true Granger causality measure C(Y X). Ĉ(Y X) converges in probability to 13

16 Theorem 3 Under Assumptions A.1 and A.2 and under the null hypothesis of Granger non-causality from Y to X in (8), we have T Ĉ(Y X) d N (0, Ω) with Ω defined in a similar way as in Theorem 1. The proofs of Proposition 6 and Theorem 3 are similar to those of Proposition 3 and Theorem 1, respectively, hence we omit them. However, it is important to notice that, for high dimensional random variables, because of the prevalence of course of dimensionality in nonparametric estimation, the bias term resulting from the MSEs of kernel estimators for conditional regression functions is much larger than the low dimensional case, so we suggest to use our bootstrap assisted test to alleviate the bias term in the test statistic for small and mild large sample sizes. 7 Monte Carlo simulations: size and power In this section, we conduct an extensive Monte Carlo study to investigate the finite sample performance of our proposed test statistic. Throughout this section, we have two univariate time series processes, X t and Y t. Our primary interest is to know whether lags of Y t accounts for the variations of X t in the mean and whether lags of X t accounts for the variations of Y t in the mean. Since they are of the same nature, we focus only on the former case. Our null hypothesis is therefore H 0 : C(Y X) = 0. Let η t and ε t be two independent sequences of independently and identically distributed (i.i.d.) standard normal random variables. To examine the size of the test under the null hypothesis, we consider the following four data generating processes (DGPs): DGP S1: X t = 0.5X t 1 + η t and Y t = 0.5Y t 1 + ε t. DGP S2: X t = X t η t and Y t = 0.5Y t 1 + ε t. DGP S3: X t = 0.5X t 1 exp{ 0.5Xt 1 2 } + η t and Y t = 0.5Y t 1 + ε t. DGP S4: X t = sin(x t 1 ) + η t and Y t = 0.5Y t 1 + ε t. To examine the empirical power of our test, we consider the following DGPs: DGP P1: X t = 0.5X t Y t 1 + η t and Y t = 0.5Y t 1 + ε t. DGP P2: X t = 0.5X t Y t sin( 2Y t 1 ) + η t and Y t = 0.5Y t 1 + ε t. DGP P3: X t = 0.5X t Y 2 t 1 + η t and Y t = 0.5Y t 1 + ε t. DGP P4: X t = 0.5X t 1 Y t 1 + η t and Y t = 0.5Y t 1 + ε t. DGP P5: X t = 0.5X t Y t Yt 1 2 η t and Y t = 0.5Y t 1 + ε t. DGP P6: X t = sin(2(x t 1 + Y t 1 )) + η t and Y t = 0.5Y t 1 + ε t. 14

17 Table 1: Empirical size of the test statistic for nonparametric measure of Granger causality in mean h = n 1/(2+δ), h = n 1/5 DGP S1 DGP S2 DGP S3 DGP S4 n = 100 δ = δ = δ = n = 200 δ = δ = δ = n = 300 δ = δ = δ = DGPs S1 and P1 are linear. All other DGPs are highly nonlinear. Note that all of them are stationary and ergodic processes. Three sample sizes, n = 100, 200, 300 are considered to check the finite sample performance of our test. For each DGP, we first generate n observations and then discard the first 200 observations to minimize the effect arising from the initial values. The number of Monte Carlo experiments is We consider a nominal size of 5% and results from other nominal sizes are similar. We use critical value from standard normal distribution, which is for 5% nominal size. For the implementation of the test, we use standard normal density as the univariate kernel function. The most important choice here is the bandwidth. In this simulation, we propose to adopt two different bandwidths, univariate bandwidth h = a n 1/(2+δ) for the restricted model and bivariate bandwidth h = b n 1/5 for the unrestricted model. We set a = b = 1. We have experimented various δ s. The choice δ = 0.6 or 0.8 seem to produce reasonable results. However, how to optimally choose the two bandwidths h and h in our testing framework to maximize the test s performance is not yet known and needs more attention in the future work. We conjecture that by using cross-validation method to choose the bandwidth, we can further improve the performance. We report the empirical sizes associated with the DGPs S1-S4 in Table 1. In Table 2 we report the empirical powers against the DGPs P1-P5. Generally speaking, our test performs well in terms of size and power in small sample size as small as 100. For δ = 1.0, it is oversized except DGP S3. But the overall performance of δ = 0.6 and δ = 0.8 is satisfactory. It is important to note that we are using the asymptotic critical value. One message from the simulation results for the size in Table 1 is that the asymptotic 15

18 Table 2: Empirical power of the test statistic for nonparametric measure of Granger causality in mean h = n 1/(2+δ), h = n 1/5 DGP P1 DGP P2 DGP P3 DGP P4 DGP P5 DGP P6 n = 100 δ = δ = δ = n = 200 δ = δ = δ = n = 300 δ = δ = δ = normal distribution represents the finite sample distribution in an acceptable way if we select the suitable bandwidths and we expect to have even better results once bootstrap procedures are employed. It is known that some appropriate bootstrap methods (e.g. local smooth bootstrap, moving block bootstrap, etc.) can handle unknown dependence in the data, thus the bootstrap gives an opportunity to eliminate or mitigate the finite sample bias that may affect the test statistic in different ways in finite sample. By observing the proof of our test, the finite sample bias arises when the mean squared errors (MSE) of kernel estimator is large. Bootstrap will capture this bias. Also the test is not so sensitive after using bootstrap. This is left for further studies. Finally, our test also has high power against many kinds of alternatives. We expect to improve the power performance too if using bootstrap. 8 Empirical application: Nonlinear predictability of risk premium Many empirical studies have investigated whether stock excess returns can be predictable. The econometric methodology used in this context is an ordinary least squares regression of stock returns onto the past of some financial variables. Fama and French (1988) argue that using the lagged dividend-price ratio as a predictor variable has a significant effect on stock returns. Campbell and Shiller (1988) find that the lagged dividendprice ratio together with the lagged dividend growth rate have a significant predictive power on stock returns. Since the publication of Fama and French (1988) and Campbell and Shiller (1988), the question of whether stock returns are predictable or not has attracted much more attention from economists; for review see 16

19 Lewellen (2004). The finding of Campbell and Shiller (1988) and Fama and French (1988) was confirmed by subsequent studies and considered to be a new stylized fact by Cochrane (1999) and Campbell (1999). This section aims to join the recent vast literature and study the predictive power of what is called variance risk premium for the expected stock excess returns. The variance risk premium is defined as the difference between the risk-neutral and objective expectations of realized variance, where the risk-neutral expectation of variance is measured as the end-of-month Volatility Index-squared de-annualized ( V IX2 12 ) and the realized variance is the sum of squared 5-minute log returns of the S&P 500 index over the month. Recently, many papers have shown the importance of using variance risk premium for predicting expected stock bond returns and exchange rates; see Bollerslev, Tauchen and Zhou (2009), Wang, Zhou, and Zhou (2013), Bollerslev, Marrone, Xu, and Zhou (2013), and Della Corte, Ramadorai, and Sarno (2013). Bollerslev, Tauchen and Zhou (2009) find that variance risk premium is able to explain a non-trivial fraction of the time series variation in post 1990 aggregate stock market returns, with high (low) premia predicting high (low) future returns. However, most existing works focus on linear predictability. In this section we examine the nonlinear predictability of expected stock excess returns (risk premium) using variance risk premium. The nonparametric Granger causality measures proposed in the previous sections do not impose any restriction on the model linking the dependent variable (stock excess return) to the independent variable (variance risk premium). 8.1 Variance Risk Premium Hereafter we define the variance risk premium that we use as a predictor of risk premium. To do this, we first need to define the model-free realized variance and implied variance. Let us first set some notations. We denote by p t the logarithmic price of the risky asset (at time t) and by r t+1 = p t+1 p t the continuously compounded return from time t to t + 1. We implicitly assume that the price process could belong to the class of continuous-time jump diffusion processes, dp t = µ t dt + σ t dw t + κ t dq t, 0 t T, (10) where µ t is a continuous and locally bounded variation process, σ t is the stochastic volatility process, W t denotes a standard Brownian motion, dq t is a counting process such that dq t = 1 represents a jump at time t (and dq t = 0 no jump) with jump intensity λ t. The parameter κ t refers to the size of the corresponding jumps. Further, we normalize the time-interval to unity and we divide it into h periods. Each period has length = 1/h. Let the discretely sampled -period returns be denoted by r (t, ) = p t p t. The realized variance over the discrete t to t+1 time interval is defined as the summation of the h high-frequency intradaily squared returns: h RV t,t+1 r(t+j, 2 ). j=1 17

20 The realized variance satisfies lim RV t,t+1 = V ar t,t+1, (11) 0 where V ar t,t+1 is the variance of stock excess return between time t and t + 1. Equation (11) indicates that the realized variance is a consistent estimator of the true variance of stock excess return; see Andersen and Bollerslev (1998), Andersen et al. (2001b, 2010), Barndorff-Nielsen and Shephard (2002a), Barndorff- Nielsen and Shephard (2002b), and Comte and Renault (1998). The realized variance RV t,t+1 is known to be an accurate ex-post measure of the unobserved true variance of stock excess return, compared to the other traditional sample variances based on daily frequency returns such as ARCH and GARCH volatilities. However, it is also known that RV t,t+1 is very sensitive to the market microstructure noise that affects the financial intraday prices data. Fortunately, many approaches have been proposed to deal with microstructure noise problem; for review see Zhang et al. (2005), Christensen et al. (2010), Barndor -Nielsen et al. (2008b), and Podolskij and Vetter (2009) among others. We now define the model-free implied variance. Let C t (T, K) denotes the price of a European call option with time to maturity T and strike price K, and B(t, T ) denotes the price of a time t zero-coupon bond maturing at time T. Carr and Madan (1998), Demeterfi, Derman, Kamal, and Zou (1999) and Britten-Jones and Neuberger (2000), have shown that implied variance between time t and t + 1, IV t,t+1, can be replicated by a portfolio of European calls as follows: IV t,t+1 E Q t (V ar t,t+1) = 2 0 [ 1 K 2 C t (t + 1, ] K B(t, t + 1) ) C t(t, K) dk, (12) where E Q t denotes the conditional expectation with respect to risk-neutral probability[see also Bakshi and Madan (2000)]. Observe that Equation (12) depends on an increasing number of calls with strikes spanning zero to infinity. In practice the model-free implied variance IV t,t+1 must be constructed on the basis of a finite number of strikes. Several recent works have argued that even with relatively few different option strikes this tends to provide a fairly accurate approximation to the true risk-neutral expectation of the future market variance; for review see Jiang and Tian (2005), Carr and Wu (2008), and Bollerslev, Gibson, and Zhou (2008). We will now use the above model-free realized and implied variances to define variance risk premium. The latter is given by the difference between the ex-ante risk neutral expectation of the future stock return variance and the expectation of the stock return variance between time t and t + 1: V RP t E Q t (V ar t,t+1) E P t (V ar t,t+1 ), (13) where E P t denotes the conditional expectation with respect to physical probability. The V RP t in (13) is unobservable, since the quantities E Q t (V ar t,t+1) and E P t (V ar t,t+1 ) are unobservable. Estimating V RP t 18

21 depends on the estimation of risk neutral and physical expectations: V RP t ÊQ t (V ar t,t+1) ÊP t (V ar t,t+1 ). In practice, the risk-neutral expectation ÊQ t (V ar t,t+1) and the true variance V ar t,t+1 are commonly replaced by the squared-volatility Index (VIX) and the realized variance RV t,t+1, respectively. The VIX is provided by the Chicago Board Options Exchange (CBOE) in the US, and is calculated using the near term S&P 500 options markets. It is based on the highly liquid S&P500 index options along with the model-free approach. Furthermore, in the literature there is no unique approach for constructing the physical expectation Êt P (.). Bollerslev, Tauchen, and Zhou (2009) and Zhou (2010) have estimated a reduced-form multifrequency autoregression with potentially multiple lags for ÊP t (V ar t,t+1 ). Following Bollerslev, Tauchen, and Zhou (2009) and Zhou (2010), we use time-t realized variance RV t,t 1, which ensures that the variance risk premium proxy for predicting various risk premia is in the time t information set and would be a correct choice if the realized variance process were unit-root. As discussed in Zhou (2010), one could also use a moving average estimate of ÊP t (V ar t,t+1 ), say with a twelve lag, such that no parameters need to be estimated and that the predictor variable is within the current information set. 8.2 Data description We consider monthly aggregate S&P 500 composite index over the period January 1996 to September Our empirical analysis is based on the logarithmic return on the S&P 500 in excess of the 3-month T-bill rate. The excess returns are annualized. We also consider the monthly realized variance, implied variance, and variance risk premium that can be downloaded from Hao Zhou s website. Figure 1 plots the monthly time series of implied variance, realized variance, and variance risk premium. Both realized and implied variance measures are somewhat higher during the period and from 2007 to the end of the sample. The latter period corresponds to the recent financial crisis. The more distinct spikes in the two measures generally coincide. Further, the difference between implied and realized variances is almost always positive, and was also higher during and periods. In the next section we examine the nonlinear predictive power at different horizons of variance risk premium for risk premium. 8.3 Causality measures In Table 3 we report the results for measuring the Granger causality from variance risk premium to risk premium at horizons that go from one month to 9 months. These Granger causality measures quantify the degree of predictive power of variance risk premium for risk premium at different horizons. From table 3, 19