علیت گرنجر در ریسک و شناسایی سرایت ریسک شدید بین بازارهای مالی

Controlling and monitoring extreme downside market risk are important for financial risk management and portfolio/investment diversification. In this paper, we introduce a new concept of Granger causality in risk and propose a class of kernel-based tests to detect extreme downside risk spillover between financial markets, where risk is measured by the left tail of the distribution or equivalently by the Value at Risk (VaR). The proposed tests have a convenient asymptotic standard normal distribution under the null hypothesis of no Granger causality in risk. They check a large number of lags and thus can detect risk spillover that occurs with a time lag or that has weak spillover at each lag but carries over a very long distributional lag. Usually, tests using a large number of lags may have low power against alternatives of practical importance, due to the loss of a large number of degrees of freedom. Such power loss is fortunately alleviated for our tests because our kernel approach naturally discounts higher order lags, which is consistent with the stylized fact that today’s financial markets are often more influenced by the recent events than the remote past events. A simulation study shows that the proposed tests have reasonable size and power against a variety of empirically plausible alternatives in finite samples, including the spillover from the dynamics in mean, variance, skewness and kurtosis respectively. In particular, nonuniform weighting delivers better power than uniform weighting and a Granger-type regression procedure. The proposed tests are useful in investigating large comovements between financial markets such as financial contagions. An application to the Eurodollar and Japanese Yen highlights the merits of our approach.

Controlling and monitoring financial risk have recently received increasing attention from business practitioners, policy makers and academic researchers. For financial risk management and investment/portfolio diversification, it is important to understand the mechanism of how risk spillover occurs across different markets. When monitoring financial risk, the probability of a large adverse market movement is always of greater concern to practitioners (e.g., Bollerslev (2001)). When they occur, extreme market movements imply change hands of a huge amount of capital among market participants, unavoidably leading to bankruptcies due to various downside constraints. Market participants have been always aware of painful experiences when extreme adverse market movements occur, and their aversion to insolvency-type extreme risk is usually very high (e.g., Campbell and Cochrance (1999)). Large market movements have become commonplace nowadays. Examples include the 1994 Mexico Peso Crisis, the 1994 US bond debacle, the 1997–1998 Asian financial crisis, as well as the bankruptcies of the Long Term Capital Management, Enron, and Worldcom. Most of the existing literature uses volatility to measure risk and focuses on volatility spillover (e.g., Cheung and Ng, 1990 and Cheung and Ng, 1996), Engle et al. (1990), Engle and Susmel (1993), Granger et al. (1986), Hamao et al. (1990), King and Wadhwani (1990), King et al. (1994), Lin et al. (1994) and Hong (2001)). Volatility is an important instrument in finance and macroeconomics. However, it can only adequately represent small risks in practice (e.g., Gourieroux and Jasiak (2001, p. 427)). Volatility alone cannot satisfactorily capture risk in scenarios of occasionally occurring extreme market movements. For example, Longin (2000) and Bali (2000) point out that volatility measures based on asset return distributions cannot produce accurate estimates of market risks during volatile periods. Hong et al., 2004 and Hong et al., 2007 also find that the innovation distributions have heavier tails when the interest rate market and the foreign exchange market have higher volatilities. Moreover, volatility includes both gains and losses in a symmetric way, whereas financial risk is obviously associated with losses but not profits. Also, practical downside constraints often require asymmetric treatment between potential upside and downside risk. Therefore, a more sensible measure of risk should be associated with large losses, or large adverse market movements. In econometrics and statistics, left tail probabilities are closely related to the likelihoods of extreme downward market movements (e.g., Embrechts et al. (1997)). Although not a perfect measure of extreme market risk, Value at Risk (VaR), originally proposed by J.P. Morgan in 1994, has become a standard synthetic measure of extreme market risk (e.g., Duffie and Pan (1997) and Engle and Manganelli (2004)). It measures how much a portfolio can lose within a given time period, with a prespecified probability. It has become an essential part of financial regulations for setting risk capital requirements so as to ensure that financial institutions can survive after a catastrophic event (e.g., the Basel Committee on Banking Supervision, 1996 and Basel Committee on Banking Supervision, 2001).1 Intuitively, VaR measures the total risk in a portfolio of financial assets by summarizing many complex undesired outcomes in a single monetary number. It naturally represents a compromise between the needs of different users. The conceptual simplicity and compromise have made VaR the most popular measure of risk among practitioners in spite of its weakness.2 In this paper, we will develop econometric tools for investigating comovements of large changes between two time series. A leading motivating example is the spillover of extreme downside movements between financial markets when markets are integrated and suffer from the same global shock, or due to “market contagion”. Using VaR as a measure of extreme downside market risk, we first introduce a new concept of Granger causality in risk, where a large risk is said to have occurred at a prespecified level if actual loss exceeds VaR at the given level. As is well-known, Granger causality (Granger, 1969 and Granger, 1980) is not a relationship between “causes”and “effects”. Instead, it is defined in terms of incremental predictive ability. This concept is suitable for the purpose of predicting and monitoring risk spillover and can provide valuable information for investment decisions, risk capital allocation and external regulation. We then propose a class of econometric procedures to detect Granger causality in risk between financial markets. Utilizing the most concerned information, it checks whether the past history of the occurrences of large risks in one market has predictive ability for the future occurrences of large risks in another market. We emphasize that the scope of the applicability of the concept of Granger-causality in risk is not limited to financial markets and financial positions (e.g., investment portfolios). For example, it can also be used to investigate the spillover of international business cycles, where the understanding of the mechanism of how a large negative shock transmits across different economies is vital to international policy coordination to alleviate its adverse impact on the world economy. Our proposed procedure has a number of appealing features. First, it checks an increasing number of lags as the sample size TT grows. This ensures power against a wide range of alternatives of extreme downside risk spillover. Secondly, our frequency domain kernel-based approach naturally discounting higher order lags alleviates the loss of a large number of degrees of freedom and thus enhances good power of the test, which many chi-square tests with a large number of lags (e.g., Box and Pierre’s (1970) portmanteau test) suffer. Downward weighting for higher order lags is consistent with the stylized fact that today’s financial markets are often more influenced by the recent events than by the remote past events. Indeed, simulation shows that nonuniform weighting is more powerful than uniform weighting and a Granger-type regression-based procedure. Finally, our procedure is easy to implement, particularly since the VaR calculation has been available in the standard toolbox of risk managers’ desk. In Section 2, we describe the concept of Granger causality in risk and discuss its differences from the concepts of Granger causality in mean (Granger, 1969), Granger causality in variance (Granger et al., 1986) and general Granger causality (Granger, 1980). In Section 3, we use a cross-spectral approach to test one-way Granger causality in risk. The kernel method is used. Section 4 develops the asymptotic theory, and Section 5 considers extensions to bilateral Granger causalities in risk. In Section 6, a simulation study examines the finite sample performance of the proposed procedures. Section 7 presents an empirical application to the Eurodollar and the Japanese Yen. It is found that a large downward movement in the Eurodollar Granger-causes a large downward movement in the Japanese Yen, and the causality is stronger for larger movements. On the other hand, Granger causality in risk from the Japanese Yen to the Eurodollar is much weaker or nonexistent. Section 8 concludes the paper. All mathematical proofs are collected in the Appendix. Throughout, ΔΔ and Δ0Δ0 denote bounded constants; →d→d and →p→p convergences in distribution and in probability respectively; and ‖A‖‖A‖ the usual Euclidean norm of AA. Unless indicated, all limits are taken as the sample size T→∞T→∞. A GAUSS code for implementing the proposed procedures is available from the authors.

Based on a new concept of Granger causality in risk which focuses on the comovements between the tails of the two distributions, a class of kernel-based tests are proposed to test whether a large downside risk in one market will Granger-cause a large downside risk in another market. The proposed tests check a large number of lags but avoid suffering from severe loss of power due to the loss of a large number of degrees of freedom, thanks to the use of a downward weighting kernel function. This downward weighting is consistent with the stylized fact that today’s financial markets are more influenced by more recent events than by remote past events, thus enhancing the power of the proposed tests. This is expected to give good power against the alternatives with decaying cross-correlations as the lag order increases. Indeed, nonuniform weighting often delivers better power than uniform weighting, as is illustrated in a simulation study and an application to exchange rates. A Granger-type regression-based test (1969) is equivalent to the uniform-weighting-based test. Simulation studies show that the procedures have reasonable size and good power against a number of empirically plausible alternatives in finite samples, no matter whether risk spillover arises from spillover in mean, in variance, or in skewness and kurtosis. These procedures are therefore useful for investigating the comovements between large market changes such as financial contagions. We use the new tests to investigate intraday risk spillover causality between Euro/Dollar and Yen/Dollar. It is found that a past large downside movement of Euro Granger-causes a future large price fall of Japanese Yen, and the evidence is stronger for larger changes. On the other hand, only an extremely large past downside movement of Japanese Yen Granger-causes a future large price fall of Euro in risk.