Question

(a) Derive the bias, e[b * 1] - β1, in terms of sample covariances. (b) Derive the asymptotic bias, p lim[b * 1 - β1], in terms of population covariances?

Answer 1

The bias, e[b * 1] - β1, in terms of sample covariances and asymptotic bias, p lim[b * 1 - β1], in terms of population covariances is:

(a) e[b * 1] - β1 = Cov(b, 1) / Var(1)

(b) p lim[b * 1 - β1] = ρ * σ_b / σ_1

Step-by-step explanation:

In part (a), the bias, e[b * 1] - β1, is derived in terms of sample covariances. The formula indicates that the bias is equal to the covariance between the estimated coefficient b and the predictor variable 1, divided by the variance of 1. This relationship reflects the impact of the covariance structure of the sample on the bias in estimating the coefficient.

Moving to part (b), the asymptotic bias is expressed in terms of population covariances. The formula involves the population correlation coefficient (ρ), the standard deviation of the estimated coefficient (σ_b), and the standard deviation of the predictor variable 1 (σ_1).

This representation emphasizes the role of population-level covariances and standard deviations in the asymptotic behavior of the bias. As the sample size approaches infinity, this formula provides insights into the convergence of the bias and its dependence on the underlying population covariance structure.