رابطه علیت ـ گرنجر در کوناتایل بین بازارهای مالی: با استفاده از روش عضو رابط

This paper considers the Granger-causality in conditional quantile and examines the potential of improving conditional quantile forecasting by accounting for such a causal relationship between financial markets. We consider Granger-causality in distributions by testing whether the copula function of a pair of two financial markets is the independent copula. Among returns on stock markets in the US, Japan and U.K., we find significant Granger-causality in distribution. For a pair of the financial markets where the dependent (conditional) copula is found, we invert the conditional copula to obtain the conditional quantiles. Dependence between returns of two financial markets is modeled using a parametric copula. Different copula functions are compared to test for Granger-causality in distribution and in quantiles. We find significant Granger-causality in the different quantiles of the conditional distributions between foreign stock markets and the US stock market. Granger-causality from foreign stock markets to the US stock market is more significant from UK than from Japan, while causality from the US stock market to UK and Japan stock markets is almost equally significant.

A causal relationship in a system of economic or financial time series has been widely studied. Following a series of seminal papers by Granger, 1969, Granger, 1980 and Granger, 1988, Granger-causality (GC) test becomes a standard tool to detect causal relationship. Granger-causality in mean (GCM) is widely analyzed between macroeconomic variables, such as between money and income, consumption and output, etc. cf. Sims, 1972 and Sims, 1980, Stock and Watson (1989). In financial markets, a growing interest in volatility spill-over promotes the development of Granger-causality tests in volatility. cf. Granger, Robins, and Engle (1986), Lin, Engle, and Ito (1994), Cheung and Ng (1996), Comte and Lieberman (2000). Most tests of Granger-causality assume a bivariate Gaussian distribution and focus on Granger-causality in mean or variance. A Gaussian distribution cannot capture asymmetric dependence between financial markets. For instance, co-movements between different financial markets behave differently in a bull market and in a bear market. Ang and Chen (2002) assert that non-Gaussian dependence between economic variables or financial variables is prevalent. Associated with the non-elliptical distribution, causality may matter in higher moments or in the dependence structure in a joint density. Thus, it is more informative to test Granger-causality in distribution (GCD) to explore a causal relationship between two financial time series. We apply a copula-based approach to model the causality and dependence between a pair of two financial time series. Using copula density functions, we construct two tests for GCD. The first test is nonparametric, following Hong and Li (2005), to compare the copula density in quadratic distance with the independent copula density. The second test is parametric; noting that different parametric copula functions imply different dependence structures, we design a method to compare them in an entropy with the independent copula density. Both tests compare out-of-sample predictive ability of copula functions relative to the benchmark independent copula density. GCD implies Granger-causality in some quantiles. In financial risk management and portfolio management, it is useful to know which quantile leads to the GCD. In particular, Value-at-Risk (VaR) is a quantile in tail that is widely used in capital budgeting and risk control. We are interested in exploring the potential of improving quantile forecasting of a trailing variable Y using information of a preceding variable X. We define Granger-causality in quantile (GCQ), for which quantile forecasts are computed from inverting a conditional copula distribution, and we develop a test for GCQ. In our empirical application, these copula-based methods are applied to analyze the pair-wise GCD from the Japan stock market to the US stock market (Japan–US), from the UK stock market to the US stock market (UK–US), from the US stock market to the Japan stock market (US–Japan), and from the US stock market to the UK stock market (US–UK). We find significant GCD in these four data sets and all sample periods considered (seven different subsample periods), as the benchmark independent copula is clearly rejected in all data sets and subsamples. For GCQ, we compare predictive performance of various copula functions with the benchmark independent copula function over different quantiles of the conditional distribution of one market conditional on another market. It is found that GCQ is significant from US to foreign stock markets and from UK to the US stock market, but not from Japan to US. The result is robust over the seven subsamples. The rest of the paper is organized as follows. Section 2 introduces two tests of GCD based on copula density functions. Both tests are based on the distance measures and thus measure the strength of GCD. Section 3 defines GCQ and develops a method to test for GCQ. Section 4 reports empirical findings on GCD and GCQ. Section 5 concludes. Appendix A reviews some basic results on copula functions.

Instead of testing Granger-causality in conditional mean or conditional variance, we consider testing for GCD by checking for independence in the copula function. Among the returns in three major stock markets, we find a significant GCD. We also estimate several copula functions to model the GCD between financial markets, and invert the estimated conditional copula distribution function to obtain the conditional quantile functions, which enable us to examine the Granger-causality in various quantiles. We find that causality to the US stock market is more significant from UK than from Japan, while causality from the US stock market to UK and Japan stock markets is significant almost equally or slightly more to Japan than to UK. The literature on GCQ is young and thin. The first paper on GCQ is Lee and Yang (2012), who explore money-income Granger causality in the conditional quantile and find that GCQ is significant in tail quantiles whereas it is not significant near the middle quantiles of the conditional distribution. Lee and Yang (2012) use the quantile regression to compute the out-of-sample quantile forecasts. While not reported in the paper for space, we have also computed the quantile forecasts from the quantile regressions instead of inverting the conditional copula functions. The results showed that the inverting conditional copula distribution is the superior approach to the quantile regressions especially towards both tails. Even in the middle of the distribution, the quantile regression was often dominated by the copula approach. This result was robust with all subsamples and data sets. This indicates that using a copula based approach has great potential to improve the quantile forecasting especially in the tails (such as VaRs) and perhaps even in the center of the distribution. Another advantage of inverting the conditional copula functions to obtain quantile forecasts instead of using the regression quantiles is that the former can avoid entirely the quantile-crossing problem while the latter methods require some correction or adjustment as studied by Chernozhukov, Fernandez-Val, and Galichon (2009). Jeong, Härdle, and Song (2012) consider a nonparametric test for in-sample GCQ and examine the causal relations between the crude oil price, the USD/GBP exchange rate, and the gold price in the gold market. The contribution of the current paper is two-fold. First, we show how to test for GCD using the copula function and how to compare different parametric copula functions. These methods are based on quadratic distance or Hellinger entropy, and on the KLIC cross-entropy. The distance or entropy provides measures of the strength in GCD, as shown in Section 2. Second, we then show how to invert the predictive conditional copula functions to obtain conditional quantile forecasts and how to test for the out-of-sample GCQ, as shown in Section 3. Ashley, Granger, and Schmalensee (1980) advocate that a test for Granger-causality of X for Y be conducted for out-of-sample predictive content of the conditioning variable X for Y.