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Residual Empirical Processes and Weighted Sums for Time‐Varying Processes with Applications to Testing for Homoscedasticity
Author(s) -
Chandler Gabe,
Polonik Wolfgang
Publication year - 2017
Publication title -
journal of time series analysis
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.576
H-Index - 54
eISSN - 1467-9892
pISSN - 0143-9782
DOI - 10.1111/jtsa.12200
Subject(s) - homoscedasticity , mathematics , heteroscedasticity , residual , autoregressive model , context (archaeology) , parametric statistics , exponential function , variance (accounting) , method of mean weighted residuals , empirical distribution function , asymptotic distribution , statistics , econometrics , estimator , algorithm , mathematical analysis , paleontology , physics , accounting , nonlinear system , quantum mechanics , galerkin method , business , biology
In the context of heteroscedastic time‐varying autoregressive (AR)‐process we study the estimation of the error/innovation distributions. Our study reveals that the non‐parametric estimation of the AR parameter functions has a negligible asymptotic effect on the estimation of the empirical distribution of the residuals even though the AR parameter functions are estimated non‐parametrically. The derivation of these results involves the study of both function‐indexed sequential residual empirical processes and weighted sum processes. Exponential inequalities and weak convergence results are derived. As an application of our results we discuss testing for the constancy of the variance function, which in special cases corresponds to testing for stationarity.