On Estimators of the Disturbance Variance in Econometric Models: Some Several Small-Sample Results on Bias and the Existence of Moments

We Present Several Small-Sample Results on the Distribution of Residuals and Estimators of the Disturbance Variance in Econometric Models. We Consider General Linear and Nonlinear Models with Stochastic Regressors and Possibly Nonlinear Restrictions on the Parameters. These Include Autoregressive Models and Stuctural Equations. for Models Estimated by Linear Or Nonlinear Least Squares, We Give Simple Bounds for the Expected Value (Or the Bias) of Standard Estimators of the Disturbance Variance. the Bounds Are Valide for Any Correlation Structure Between the Disturbances. We Give Simple Conditions That Ensure the Existence of Finite Moments for Residuals and Variance Estimators Up to Any Given Order. When the Disturbances Have a Normal Distribution, We Find That All the Moments Exist and Show That the Sum of Squared Residuals Is Bounded by a Chi-Square Random Variable Or by a Linear Combination of Independent Chi-Square Variables. We Also Present Analogous Results for a Number of Alternative Methods of Estimation: Generalized Least Squares (When Regression Coefficients and Parameters of the Covariance Matrix Are Estimated Jointly), Lp Estimation (Including Minimum Absolute Deviations) and Maximum Likelihood. in the Latter Case, We Give an Information Inequality Related to the Estimation of the Entropy of a Distribution. All the Proofs Are Simple.