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An empirical likelihood goodness-of-fit test for time series. (English) Zbl 1063.62064
Summary: Standard goodness-of-fit tests for a parametric regression model against a series of nonparametric alternatives are based on residuals arising from a fitted model. When a parametric regression model is compared with a nonparametric model, goodness-of-fit testing can be naturally approached by evaluating the likelihood of the parametric model within a nonparametric framework. We employ the empirical likelihood for an \(\alpha\)-mixing process to formulate a test statistic that measures the goodness of fit of a parametric regression model. The technique is based on a comparison with kernel smoothing estimators.
The empirical likelihood formulation of the test has two attractive features. One is its automatic consideration of the variation that is associated with the nonparametric fit due to empirical likelihood’s ability to Studentize internally. The other is that the asymptotic distribution of the test statistic is free of unknown parameters, avoiding plug-in estimation. We apply the test to a discretized diffusion model which has recently been considered in financial market analysis.

MSC:
62G10 Nonparametric hypothesis testing
62F03 Parametric hypothesis testing
62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
62G30 Order statistics; empirical distribution functions
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