An asymptotic behavior of a finite-section of the optimal causal filter

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