Modeling Quantile Dependence

Thesis advisor: Zhijie XiaoIn recent years, quantile regression has achieved increasing prominence as a quantitative method of choice in applied econometric research. The methodology focuses on how the quantile of the dependent variable is influenced by the regressors, thus providing the researcher with much information about variations in the relationship between the covariates. In this dissertation, I consider two quantile regression models where the information set may contain quantiles of the regressors. Such frameworks thus capture the dependence between quantiles - the quantile of the dependent variable and the quantile of the regressors - which I call models of quantile dependence. These models are very useful from the applied researcher's perspective as they are able to further uncover complex dependence behavior and can be easily implemented using statistical packages meant for standard quantile regressions. The first chapter considers an application of the quantile dependence model in empirical finance. One of the most important parameter of interest in risk management is the correlation coefficient between stock returns. Knowing how correlation behaves is especially important in bear markets as correlations become unstable and increase quickly so that the benefits of diversification are diminished especially when they are needed most. In this chapter, I argue that it remains a challenge to estimate variations in correlations. In the literature, either a regime-switching model is used, which can only estimate correlation in a finite number of states, or a model based on extreme-value theory is used, which can only estimate correlation between the tails of the returns series. Interpreting the quantile of the stock return as having information about the state of the financial market, this chapter proposes to model the correlation between quantiles of stock returns. For instance, the correlation between the 10th percentiles of stock returns, say the U.S. and the U.K. returns, reflects correlation when the U.S. and U.K. are in the bearish state. One can also model the correlation between the 60th percentile of one series and the 40th percentile of another, which is not possible using existing tools in the literature. For this purpose, I propose a nonlinear quantile regression where the regressor is a conditional quantile itself, so that the left-hand-side variable is a quantile of one stock return and the regressor is a quantile of the other return. The conditional quantile regressor is an unknown object, hence feasible estimation entails replacing it with the fitted counterpart, which then gives rise to problems in inference. In particular, inference in the presence of generated quantile regressors will be invalid when conventional standard errors are used. However, validity is restored when a correction term is introduced into the regression model. In the empirical section, I investigate the dependence between the quantile of U.S. MSCI returns and the quantile of MSCI returns to eight other countries including Canada and major equity markets in Europe and Asia. Using regression models based on the Gaussian and Student-t copula, I construct correlation surfaces that reflect how the correlations between quantiles of these market returns behave. Generally, the correlations tend to rise gradually when the markets are increasingly bearish, as reflected by the fact that the returns are jointly declining. In addition, correlations tend to rise when markets are increasingly bullish, although the magnitude is smaller than the increase associated with bear markets. The second chapter considers an application of the quantile dependence model in empirical macroeconomics examining the money-output relationship. One area in this line of research focuses on the asymmetric effects of monetary policy on output growth. In particular, letting the negative residuals estimated from a money equation represent contractionary monetary policy shocks and the positive residuals represent expansionary shocks, it has been widely established that output growth declines more following a contractionary shock than it increases following an expansionary shock of the same magnitude. However, correctly identifying episodes of contraction and expansion in this manner presupposes that the true monetary innovation has a zero population mean, which is not verifiable. Therefore, I propose interpreting the quantiles of the monetary shocks as having information about the monetary policy stance. For instance, the 10th percentile shock reflects a restrictive stance relative to the 90th percentile shock, and the ranking of shocks is preserved regardless of shifts in the shock's distribution. This idea motivates modeling output growth as a function of the quantiles of monetary shocks. In addition, I consider modeling the quantile of output growth, which will enable policymakers to ascertain whether certain monetary policy objectives, as indexed by quantiles of monetary shocks, will be more effective in particular economic states, as indexed by quantiles of output growth. Therefore, this calls for a unified framework that models the relationship between the quantile of output growth and the quantile of monetary shocks. This framework employs a power series method to estimate quantile dependence. Monte Carlo experiments demonstrate that regressions based on cubic or quartic expansions are able to estimate the quantile dependence relationships well with reasonable bias properties and root-mean-squared errors. Hence, using the cubic and quartic regression models with M1 or M2 money supply growth as monetary instruments, I show that the right tail of the output growth distribution is generally more sensitive to M1 money supply shocks, while both tails of output growth distribution are more sensitive than the center is to M2 money supply shocks, implying that monetary policy is more effective in periods of very low and very high growth rates. In addition, when non-neutral, the influence of monetary policy on output growth is stronger when it is restrictive than expansive, which is consistent with previous findings on the asymmetric effects of monetary policy on output.Thesis (PhD) — Boston College, 2009.Submitted to: Boston College. Graduate School of Arts and Sciences.Discipline: Economics