Residual autocorrelation distribution in the validation data set (Q2703251)

The author considers a stationary invertible ARMAX model NEWLINE\[NEWLINEa(z)y_t=b(z)u_{t-1}+c(z)\varepsilon_t,NEWLINE\]NEWLINE where \(y_t\) and \(u_t\) are observable, \(\varepsilon_t\) are non-observable i.i.d. with variance \(\sigma^2\) and \(\varepsilon_t\) can be correlated with \(u_{t-h}\). The coefficients of the polynomials \(a\), \(b\) and \(c\) are unknown and form the parameter vector \(\vartheta\). It is supposed that they are estimated by \(T_1\) observations and the estimator \(\hat\vartheta\) is asymptotically normal. Then the model is validated by \(T_2=\tau T_1\) observations, and residuals \(e_t\) are defined by NEWLINE\[NEWLINE\hat a(z)y_t=\hat b(z)u_{t-1}+\hat c e_t,\quad g_e(h)=(1/\sqrt{T_2})\sum_{t=h+1}^{T_2}e_te_{y-h}.NEWLINE\]NEWLINE It is shown that \( g_e(h)=g_\varepsilon(h)+\sqrt{\tau}\rho(|h|)\Delta+o(1), \) where \(\Delta\) is the limit normal distribution of \(\sqrt{T_1}(\vartheta-\hat\vartheta_{T_1})\), and \(\rho(h)\) is some nonrandom vector valued function defined by \(a\), \(b\) and \(c\). Simulation results are presented.