Session: Technical Session III

Date & Time: Saturday, May 23 — 9:30 AM–11:15 AM

Talk Title

Sample Splitting and Assessing Goodness-of-fit of Time Series

Abstract

A fundamental and often final step in time series modeling is to assess the quality of fit of a proposed model to the data. Since the underlying distribution of the innovations that generate a model is often not prescribed, goodness-of-fit tests typically take the form of testing the fitted residuals for serial independence. However, these fitted residuals are inherently dependent since they are based on the same parameter estimates and thus standard tests of serial independence, such as those based on the autocorrelation function (ACF) or distance correlation function (ADCF) of the fitted residuals need to be adjusted. The sample splitting procedure in Pfister et al. (2018) is one such fix for the case of models for independent data, but fails to work in the dependent setting. In this paper sample splitting is leveraged in the time series setting to perform tests of serial dependence of fitted residuals using the ACF and ADCF. Here the first half (n/2) of the data points are used to estimate the parameters of the model and then using these parameter estimates, all of the data points are used to compute the estimated residuals. Tests for serial independence are then based on these n residuals. In particular, the ACF and ADCF tests of serial independence tests often (always for the ACF!) have the same limit distributions as though the underlying residuals are indeed iid. In particular, for the ACF of the residuals confidence bounds of plus/minus 1.96/sqrt(n) are legit for all lags. Moreover, one does not need to resort to the Ljung/Box test, which is not always asymptotically correct, for testing uncorrelatedness at multiple lags. Rather one can use a routine test based on sums of squares of the sample ACF which will have the requisite chi-squared limit. The sample splitting procedure ameliorates the need for adjustment in the construction of confidence bounds for both the ACF and ADCF in goodness-of-fit testing. Extensions of these tests to Kolmogorov-Smirnov style tests based on the empirical distribution of the fitted residuals will also be discussed. (This is joint work with Leon Fernandes and Harry Xi.)