A test for nonlinearity of time series with infinite variance (Q5942929)

For a time series \(X_1\), \(X_2\), \(\dots\) a linearity hypothesis \(H_0:\) \(X_t=\sum_{j=0}^\infty c_j Z_{t-j}\) is considered, where \(c_j\) are real-valued coefficients, and \(Z_j\) are i.i.d. heavy-tailed with infinite variance. Even in this case, the sample autocovariance function (ACF) of \(X_i\), namely, \[ \hat\rho_n(h)=\{\sum_{t=1}^{n-h}X_t X_{t+h}\} \{\sum_{t=1}^n X_t^2\}^{-1} \] converges (as \(n\to\infty\)) to an analogue of ACF for \(X_j\): \[ \rho(h)=\{\sum_{j=0}^\infty c_j c_{j+h}\} \{\sum_{j=0}^\infty c_j^2\}^{-1}. \] For heavy tailed nonlinear series (i.e. if \(H_0\) doesn't hold) the empirical ACF does not converge to any fixed number but in many cases it converges to some nondegenerate random variable. The authors propose a test in which estimates of the empirical ACF stability are used as test statistics for \(H_0\) and investigate it's properties.