Nonparametric estimates for conditional quantiles of time series
We consider the problem of estimating the conditional quantile of a time series <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$$\{ Y_t\}$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <mo stretchy="false">{</mo> <msub> <mi>Y</mi> <mi>t</mi> </msub> <mo stretchy="false">}</mo> </mrow> </math> </EquationSource> </InlineEquation> at time <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$$t$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mi>t</mi> </math> </EquationSource> </InlineEquation> given covariates <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$$\varvec{X}_{t}$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <msub> <mrow> <mi mathvariant="bold-italic">X</mi> </mrow> <mi>t</mi> </msub> </math> </EquationSource> </InlineEquation>, where <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$$\varvec{X}_{t}$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <msub> <mrow> <mi mathvariant="bold-italic">X</mi> </mrow> <mi>t</mi> </msub> </math> </EquationSource> </InlineEquation> can be either exogenous variables or lagged variables of <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$${ Y_t}$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <msub> <mi>Y</mi> <mi>t</mi> </msub> </math> </EquationSource> </InlineEquation> . The conditional quantile is estimated by inverting a kernel estimate of the conditional distribution function, and we prove its asymptotic normality and uniform strong consistency. The performance of the estimate for light and heavy-tailed distributions of the innovations is evaluated by a simulation study. Finally, the technique is applied to estimate VaR of stocks in DAX, and its performance is compared with the existing standard methods using backtesting. Copyright Springer-Verlag Berlin Heidelberg 2015
