Measuring Granger causality in quantiles.

We consider measures of Granger causality in quantiles, which detect and quantify both linear and nonlinear causal effects between random variables. The measures are based on nonparametric quantile regressions and defined as logarithmic functions of restricted and unrestricted expectations of quantile check loss functions. They can consistently be estimated by replacing the unknown expectations of check loss functions by their nonparametric kernel estimates. We derive a Bahadur-type representation for the nonparametric estimator of the measures. We establish the asymptotic distribution of this estimator, which can be used to build tests for the statistical significance of the measures. Thereafter, we show the validity of a smoothed local bootstrap that can be used in finite-sample settings to perform statistical tests. A Monte Carlo simulation study reveals that the bootstrap-based test has a good finite-sample size and power properties for a variety of data generating processes and different sample sizes. Finally, we provide an empirical application to illustrate the usefulness of measuring Granger causality in quantiles. We quantify the degree of predictability of the quantiles of equity risk premium using the variance risk premium, unemployment rate, inflation, and the effective federal funds rate. The empirical results show that the variance risk premium and effective federal funds rate have a strong predictive power for predicting the risk premium when compared to that of the predictive power of the other two macro variables. In particular, the variance risk premium is able to predict the center, lower and upper quantiles of the distribution of the risk premium; however, the effective federal funds rate predicts only the lower and upper quantiles. Nevertheless, unemployment and inflation rates have no effect on the risk premium