A Test of Linearity for Functional Autoregressive Models

We propose a new test for linearity in time series. We consider an asymptotically stationary functional AR(p) model on ℜd of the form X n = f(Xn−1, ..., Xn−p) + ξn (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 ℜdp. The estimators are f^n, a recursive version of the non‐parametric kernel estimator of f, and Ân, a least squares estimator well suited to the linear case. The main result states that the test statistic has a χ2 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