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We establish an $L_1$-bound between the coefficients of the optimal causalfilter applied to the data-generating process and its finite sampleapproximation. Here, we assume that the data-generating process is asecond-order stationary time series with either short or long memoryautocovariances. To derive the $L_1$-bound, we first provide an exactexpression for the coefficients of the causal filter and their approximationsin terms of the absolute convergent series of the multistep ahead infinite andfinite predictor coefficients, respectively. Then, we prove a so-called uniformBaxter's inequality to obtain a bound for the difference between the infiniteand finite multistep ahead predictor coefficients in both short and long memorytime series. The $L_1$-approximation error bound for the causal filtercoefficients can be used to evaluate the performance of the linear predictionsof time series through the mean squared error criterion.

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