Seminar 2023-10-06

Another Look at Assessing Goodness-of-Fit of Time Series Using Fitted Residuals

Richard Davis

Professor of Statistics

Department of Statistics

Columbia University

Date: Friday, October 6, 2023

Time: 11:00 A.M. – 12:00 P.M. Eastern Time

Location: HUB, Room 3 10423 Rivanna River Way

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 typically 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 parameter estimates. Thus, standard tests of serial independence, such as those based on the autocorrelation function (ACF) or distance correlation function (DCF) 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 case.

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 DCF.  Here the first a_n of the data points are used to estimate the parameters of the model and then using these parameter estimates, the last s_n of the data points are used to compute the estimated residuals.  Tests for serial independence are then based on these s_n residuals.  As long as the overlap between the a_n and s_n data splits is asymptotically ½, the ACF and DCF tests of serial independence tests often have the same limit distributions as though the underlying residuals are indeed iid.  This procedure ameliorates the need for adjustment to the construction of confidence bounds for both the ACF and DCF in goodness-of-fit testing.  (This is joint work with Leon Fernandes.)

Richard Davis is a Howard Levene Professor of Statistics at Columbia University, where he served as chair from 2013 to 2019. He was the President of the Institute of Mathematical Statistics (IMS) in 2016, as well as the Editor-in-Chief of Bernoulli (2010-2012). He is also a fellow of the American Statistical Association. His research interests lie primarily in the areas of applied probability, time series, and stochastic processes - much of which is strongly influenced by extreme value theory. 

Nicholas Rios

David Kepplinger