Does sequential augmenting of the simple linear heteroscedastic regression reduce variances of the Ordinary Least Squares?

Abstract

If uncorrelated random variables have a common expected value and decreasing variances then the variance of a sample mean is decreasing with the number of observations. Unfortunately, this natural and desirable Variance Reduction Property (VRP) by augmenting data is not automatically inherited by Ordinary Least Squares (OLS) estimators of parameters. In the paper we find conditions for the OLS to have the VRP. In the case of a straight line regression we show that the OLS estimators of intercept and slope have the VRP if the design points are increasing. This also holds true for alternating two-point experimental designs. The obtained results are useful in the cases where it is known that variances of the subsequent observations are non-increasing, but the ratios of the decrease are not available to use sub-optimal or optimal Weighted Least Squares estimators.