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Assessing Tail Risk Using Expectile Regressions with Partially Varying Coefficients

Abstract To characterize heteroskedasticity, nonlinearity, and asymmetry in tail risk, this study investigates a class of conditional (dynamic) expectile models with partially varying coefficients in which some coefficients are allowed to be constants, but others are allowed to be unknown functions of random variables. A three-stage estimation procedure is proposed to estimate both the parametric constant coefficients and nonparametric functional coefficients. Their asymptotic properties are investigated under a time series context, together with a new simple and easily implemented test for testing the goodness of fit of models and a bandwidth selector based on newly defined cross-validatory estimation for the expected forecasting expectile errors. The proposed methodology is data-analytic and of sufficient flexibility to analyze complex and multivariate nonlinear structures without suffering from the curse of dimensionality. Finally, the proposed model is illustrated by simulated data, and applied to analyzing the daily data of the S&P500 return series.