The cross-quantilogram: Measuring quantile dependence and testing directional predictability between time series☆
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 does not improve the prediction about whether 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., for and the norm indicates the norm of a random vector , given by for . Let be the indicator function taking the value one when its argument is true, and zero otherwise. We use , and to denote the set of all real numbers, all integers and all positive integers, respectively. Let .
Section snippets
The cross-quantilogram
Let be a strictly stationary time series with and , where with for . We use to denote the conditional distribution function of the series given with density function , and the corresponding conditional quantile function is defined as for , for . Let 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 , , and and let and
The partial cross-quantilogram
We define the partial cross-quantilogram, which measures the relationship between two events and , while controlling for intermediate events between and as well as whether some state variables exceed a given quantile. Let be an vector for , where for and a vector (), and may include the quantile-hit processes based on some of the lagged predicted variables
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. where for .
DGP1: where is a 2×2 identity matrix. We let be mutually independent.
DGP2: where and
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 : 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
References (71)
- et al.
Bootstrap conditional distribution tests in the presence of dynamic misspecification
J. Econometrics
(2006) - et al.
Towards estimating extremal serial dependence via the bootstrapped extremogram
J. Econometrics
(2012) - et al.
Measures of serial extremal dependence and their estimation
Stochastic Process. Appl.
(2013) - et al.
Granger causality in risk and detection of extreme risk spillover between financial markets
J. Econometrics
(2009) - et al.
Spatial heteroskedasticity and autocorrelation consistent estimation of covariance matrix
J. Econometrics
(2011) - et al.
Empirical market microstructure: An analysis of the BRL/US$ exchange rate market
Emerg. Mark. Rev.
(2008) - et al.
Measuring and modeling variation in the risk-return trade-off
- et al.
The quantilogram: With an application to evaluating directional predictability
J. Econometrics
(2007) - et al.
Causality in temporal systems: Characterization and a survey
J. Econometrics
(1977) - et al.
Simple and powerful GMM over-identification tests with accurate size
J. Econometrics
(2012)
CoVaR, Tech. Rep
An introduction to functional central limit theorems for dependent stochastic processes
Internat. Statist. Rev.
Second order representations of the least absolute deviation regression estimator
Ann. Inst. Statist. Math.
Testing for parameter constancy in linear regressions: An empirical distribution function approach
Econometrica
Volatility, Correlation and Tails for Systemic Risk Management, Tech. Rep
Simple robust testing of hypotheses in nonlinear models
J. Amer. Statist. Assoc.
The Econometrics of Financial Markets
The relation between relative order imbalance and intraday futures returns: An application of the quantile regression model to Taiwan
Emerg. Mark. Financ. Trade
M tests with a new normalization matrix
Econometric Rev.
Some a posteriori probabilities in stock market action
Econometrica
The extremogram: a correlogram for extreme events
Bernoulli
La fonction de dépendance empirique et ses propriétés. Un test non paramétrique d’indépendance
Acad. Roy. Belg. Bull. Cl. Sci. (5)
Of copulas, quantiles, ranks, and spectra an L1 -approach to spectral analysis
Bernoulli
Generalized runs tests for heteroscedastic time series
Nonparametr. Stat.
Modelling Extremal Events for Insurance and Finance
CAViaR: conditional autoregressive value at risk by regression quantiles
J. Bus. Econom. Statist.
The behavior of stock market prices
J. Bus.
Weak convergence of empirical copula processes
Bernoulli
Testing linearity against threshold effects: uniform inference in quantile regression
Ann. Inst. Statist. Math.
A comprehensive look at the empirical performance of equity premium prediction
Rev. Financ. Stud.
Investigating causal relations by econometric models and cross-spectral methods
Econometrica
Cited by (0)
- ☆
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.