Negative moment bounds for sample autocovariance matrices of stationary processes driven by conditional heteroscedastic errors and their applications

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Abstract
This paper addresses an important gap in time series analysis by establishing rigorous results on prediction and model selection for weakly stationary processes with conditionally heteroscedastic errors, using mean squared prediction error (MSPE) as the evaluation metric. A key contribution is the derivation of a negative moment bound for the minimum eigenvalue of the sample autocovariance matrix. This result enables an explicit asymptotic decomposition of the MSPE for least squares predictors into interpretable components reflecting model complexity, misspecification, and time-varying volatility. Leveraging this decomposition, we develop a model selection criterion that consistently identifies the MSPE-optimal subset autoregressive model, even in the presence of conditional heteroscedasticity and model misspecification. Simulation studies support the theoretical developments and demonstrate the practical effectiveness of the proposed method.

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