Abstract: We establish negative moment bounds for the minimum eigenvalue of the normalized Fisher information matrix in a stochastic regression model with a deterministic time trend. This result enables us to develop an asymptotic expression for the mean squared prediction error (MSPE) of the least squares predictor of the aforementioned model. Our asymptotic expression not only helps better understand how the MSPE is affected by the deterministic and random components, but also inspires an intriguing proof of the formula for the sum of the elements in the inverse of the Cauchy/Hilbert matrix from a prediction perspective.

Key words and phrases: Cauchy matrix, Hilbert matrix, mean squared prediction error, minimum eigenvalue, negative moment bound, stochastic regression model.