The cross-quantilogram: Measuring quantile dependence and testing directional predictability between time series

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Abstract

This paper proposes the cross-quantilogram to measure the quantile dependence between two time series. We apply it to test the hypothesis that one time series has no directional predictability to another time series. We establish the asymptotic distribution of the cross-quantilogram and the corresponding test statistic. The limiting distributions depend on nuisance parameters. To construct consistent confidence intervals we employ a stationary bootstrap procedure; we establish consistency of this bootstrap. Also, we consider a self-normalized approach, which yields an asymptotically pivotal statistic under the null hypothesis of no predictability. We provide simulation studies and two empirical applications. First, we use the cross-quantilogram to detect predictability from stock variance to excess stock return. Compared to existing tools used in the literature of stock return predictability, our method provides a more complete relationship between a predictor and stock return. Second, we investigate the systemic risk of individual financial institutions, such as JP Morgan Chase, Morgan Stanley and AIG.

Introduction

Linton and Whang (2007) introduced the quantilogram to measure predictability in different parts of the distribution of a stationary time series based on the correlogram of “quantile hits”. They applied it to test the hypothesis that a given time series has no directional predictability. More specifically, their null hypothesis was that the past information set of the stationary time series {yt} does not improve the prediction about whether yt will be above or below the unconditional quantile. The test is based on comparing the quantilogram to a pointwise confidence band. This contribution fits into a long literature of testing predictability using signs or rank statistics, including the papers of Cowles and Jones (1937),  Dufour et al. (1998), and Christoffersen and Diebold (2002). The quantilogram has several advantages compared to other test statistics for directional predictability. It is conceptually appealing and simple to interpret. Since the method is based on quantile hits it does not require moment conditions like the ordinary correlogram and statistics like the variance ratio that are derived from it, Mikosch and Starica (2000), and so it works well for heavy tailed series. Many financial time series have heavy tails, see, e.g.,  Mandelbrot (1963),  Fama (1965),  Rachev and Mittnik (2000), Embrechts et al. (1997),  Ibragimov et al. (2009), and Ibragimov (2009), and so this is an important consideration in practice. Additionally, this type of method allows researchers to consider very long lags in comparison with regression type methods, such as Engle and Manganelli (2004).

There have been a number of recent works either extending or applying this methodology. Davis and Mikosch (2009) have introduced the extremogram, which is essentially the quantilogram for extreme quantiles, and Davis et al. (2012) has provided the inference methods based on bootstrap and permutation for the extremogram. See also Davis et al. (2013). Li, 2008, Li, 2012 has introduced a Fourier domain version of the quantilogram while Hong (2000) has used a Fourier domain approach for test statistics based on distributions. Further development in the Fourier domain approach has been made by Hagemann (2013) and Dette et al. (2015). See also Li (2014) and Kley et al. (2016). The quantilogram has recently been applied to stock returns and exchange rates, Laurini et al. (2008) and Chang and Shie (2011).

Our paper addresses three outstanding issues with regard to the quantilogram. First, the construction of confidence intervals that are valid under general dependence structures. Linton and Whang (2007) derived the limiting distribution of the sample quantilogram under the null hypothesis that the quantilogram itself is zero, in fact under a special case of that where the process has a type of conditional heteroskedasticity structure. Even in that very special case, the limiting distribution depends on model specific quantities. They derived a bound on the asymptotic variance that allows one to test the null hypothesis of the absence of predictability (or rather the special case of this that they work with). Even when this model structure is appropriate, the bounds can be quite large especially when one looks into the tails of the distribution. The quantilogram is also useful in cases where the null hypothesis of no predictability is not thought to be true—one can be interested in measuring the degree of predictability of a series across different quantiles. We provide a more complete solution to the issue of inference for the quantilogram. Specifically, we derive the asymptotic distribution of the quantilogram under general weak dependence conditions, specifically strong mixing. The limiting distribution is quite complicated and depends on the long run variance of the quantile hits. To conduct inference we propose the stationary bootstrap method of Politis and Romano (1994) and prove that it provides asymptotically valid confidence intervals. We investigate the finite sample performance of this procedure and show that it works well. We also provide R code that carries out the computations efficiently.1 We also define a self-normalized version of the statistic for testing the null hypothesis that the quantilogram is zero, following Lobato (2001). This statistic has an asymptotically pivotal distribution, under the null hypothesis, whose critical values have been tabulated so that there is no need for long run variance estimation or even bootstrap.

Second, we develop our methodology inside a multivariate setting and explicitly consider the cross-quantilogram. Linton and Whang (2007) briefly mentioned such a multivariate version of the quantilogram but they provided neither theoretical results nor empirical results. In fact, the cross-correlogram is a vitally important measure of dependence between time series: Campbell et al. (1997), for example, use the cross autocorrelation function to describe lead lag relations between large stocks and small stocks. We apply the cross-quantilogram to the study of stock return predictability; our method provides a more complete picture of the predictability structure. We also apply the cross-quantilogram to the question of systemic risk. Our theoretical results described in the previous paragraph are all derived for the multivariate case.

Third, we explicitly allow the cross-quantilogram to be based on conditional (or regression) quantiles (Koenker and Bassett, 1978). Using conditional quantiles rather than unconditional quantiles, we measure directional dependence between two time-series after parsimoniously controlling for the information at the time of prediction.2 Moreover, we derive the asymptotic distribution of the cross-quantilogram that are valid uniformly over a range of quantiles.

The remainder of the paper is as follows: Section  2 introduces the cross-quantilogram and Section  3 discusses its asymptotic properties. For consistent confidence intervals and hypothesis tests, we define the bootstrap procedure and introduce the self normalized test statistic. Section  4 considers the partial cross-quantilogram and gives a full treatment of its behavior in large samples. In Section  5 we report results of some Monte Carlo simulations to evaluate the finite sample properties of our procedures. In Section  6 we give two applications: we investigate stock return predictability and system risk using our methodology. Appendix contains all the proofs.

We use the following notation: The norm denotes the Euclidean norm, i.e.,  z=(j=1dzj2)1/2 for z=(z1,,zd)Rd and the norm p indicates the Lp norm of a d×1 random vector z, given by zp=(j=1dE|zj|p)1/p for p>0. Let 1[] be the indicator function taking the value one when its argument is true, and zero otherwise. We use R, Z and N to denote the set of all real numbers, all integers and all positive integers, respectively. Let Z+=N{0}.

Section snippets

The cross-quantilogram

Let {(yt,xt):tZ} be a strictly stationary time series with yt=(y1t,y2t)R2 and xt=(x1t,x2t)Rd1×Rd2, where xit=[xit(1),,xit(di)]Rdi with diN for i=1,2. We use Fyi|xi(|xit) to denote the conditional distribution function of the series yit given xit with density function fyi|xi(|xit), and the corresponding conditional quantile function is defined as qi,t(τi)=inf{v:Fyi|xi(v|xit)τi} for τi(0,1), for i=1,2. Let T be the range of quantiles we are interested in evaluating the directional

Asymptotic properties

We next present the asymptotic properties of the sample cross-quantilogram and related test statistics. Since these quantities contain non-smooth functions, we employ techniques widely used in the literature on quantile regression, see Koenker and Bassett (1978) and Pollard (1991) among others.

Define yt,k=(y1t,y2,tk), xt,k=(x1t,x2,tk), qt,k(τ)=[q1,t(τ1),q2,tk(τ2)] and qˆt,k(τ)=[qˆ1,t(τ1),qˆ2,tk(τ2)] and let {yt,kqt,k(τ)}={y1tq1(τ1|x1t),y2,tkq2(τ2|x2tk)} and Fy|x(k)(|xt,k)=P(yt,k|x

The partial cross-quantilogram

We define the partial cross-quantilogram, which measures the relationship between two events {y1tq1,t(τ1)} and {y2,tkq2,tk(τ2)}, while controlling for intermediate events between t and tk as well as whether some state variables exceed a given quantile. Let zt[ψτ3(y3tq3,t(τ3)),,ψτl(yltql,t(τl))] be an (l2)×1 vector for l3, where qi,t(τi)=xitβi(τi) for τi and a di×1 vector xit (i=3,,l), and zt may include the quantile-hit processes based on some of the lagged predicted variables {y1,

Monte Carlo simulation

We investigate the finite sample performance of our test statistics. We adopt the following simple VAR model with covariates and consider two data generating processes for the error terms. y1t=0.1+0.3y1,t1+0.2y2,t1+0.3z1t+u1ty2t=0.1+0.2y2,t1+0.3z2t+u2t, where zitiidχ2(3)/3 for i=1,2.

DGP1: (u1t,u2t)iidN(0,I2) where I2 is a 2×2 identity matrix. We let (u1t,u2t,z1t,z2t) be mutually independent.

DGP2: (u1tu2t)=(σ1t001)(ε1tε2t) where (ε1t,ε2t)iidN(0,I2) and σ1t2=0.1+0.2u1,t12+0.2σ1,t12+u2,t

Stock return predictability

We apply the cross-quantilogram to detect directional predictability from an economic state variable to stock returns. The issue of stock return predictability has been very important and extensively investigated in the literature; see Lettau and Ludvigson (2010) for an extensive review. A large literature has considered predictability of the mean of stock return. The typical mean return forecast examines whether the mean of an economic state variable is helpful in predicting the mean of stock

Conclusion

We have established the limiting properties of the cross-quantilogram in the case of a finite number of lags. Hong (1996) established the properties of the Box–Pierce statistic in the case that p=pn: after a location and scale adjustment the statistic is asymptotically normal, see also Hong et al. (2009) for a related work. No doubt our results can be extended to accommodate this case, although in practice the desirability of such a test is questionable, and the chi-squared type limit in our

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    We thank a Co-Editor, Jianqing Fan, an Associate Editor and three anonymous referees for constructive comments. Han’s work was supported by the National Research Foundation of Korea (NRF-2013S1A5A8021502). Linton’s work was supported by Cambridge INET and the ERC (NAMSEF). Oka’s work was supported by Singapore Academic Research Fund (FY2013-FRC2-003). Whang’s work was supported by the National Research Foundation of Korea (NRF-2011-342-B00004) and Seoul National University.

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