On the asymptotic behavior of a finite section of the optimal causal filter

We establish an $L_1$-bound between the coefficients of the optimal causal filter applied to the data-generating process and its finite sample approximation. Here, we assume that the data-generating process is a second-order stationary time series with either short or long memory autocovariances. To derive the $L_1$-bound, we first provide an exact expression for the coefficients of the causal filter and their approximations in terms of the absolute convergent series of the multistep ahead infinite and finite predictor coefficients, respectively. Then, we prove a so-called uniform Baxter's inequality to obtain a bound for the difference between the infinite and finite multistep ahead predictor coefficients in both short and long memory time series. The $L_1$-approximation error bound for the causal filter coefficients can be used to evaluate the performance of the linear predictions of time series through the mean squared error criterion.