A Test of Linearity for Functional Autoregressive Models

Abstract

We propose a new test for linearity in time series. We consider an asymptotically stationary functional AR( p ) model on ℜ d of the form X n = f ( X n −1 , ..., X n − p ) + ξ n ( n ∈ N). The testing procedure is based on a suitably normalized sum of quadratic deviations between two different estimates of the function f evaluated at q distinct points of ℜ dp . The estimators are f^ n , a recursive version of the non‐parametric kernel estimator of f , and  n , a least squares estimator well suited to the linear case. The main result states that the test statistic has a χ 2 limit distribution under the null hypothesis. A similar result is derived under the alternative hypothesis for the test statistic corrupted by a non‐linear term. Our simulations indicate that our asymptotic results hold for moderate sample sizes when the testing procedure is used carefully