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In this paper, we obtain generic bounds on the variances of estimation andprediction errors in time series analysis via an information-theoreticapproach. It is seen in general that the error bounds are determined by theconditional entropy of the data point to be estimated or predicted given theside information or past observations. Additionally, we discover that in orderto achieve the prediction error bounds asymptotically, the necessary andsufficient condition is that the "innovation" is asymptotically white Gaussian.When restricted to Gaussian processes and 1-step prediction, our bounds areshown to reduce to the Kolmogorov-Szeg\"o formula and Wiener-Masani formulaknown from linear prediction theory.

Generic Variance Bounds on Estimation and Prediction Errors in Time Series Analysis: An Entropy Perspective