9. In an errors-in-variables simple regression model, the least squares estimate of the regression slope () is biased toward 0, an example of attenuation. Specifically,

9. In an errors-in-variables simple regression model, the least squares estimate of the regression slope (β) is biased toward 0, an example of attenuation. Specifically, if the true regression (through the origin) is Y = xβ + , but Y is regressed on X, with X = x + δ, then the least squares estimate (βˆ) has expectation: E[βˆ] ≈ ρβ, with ρ = σ2 x

σ2 x+σ2

δ

≤ 1.

If ρ is known or well-estimated, one can correct for attenuation and produce an unbiased estimate by using β/ρ ˆ to estimate β. However, this estimate may have poor MSE properties. To minimize MSE, consider an estimator of the form: βˆc = c(β/ρ ˆ ).

(a) Let σ2 = V ar[βˆ] and find the c that minimizes MSE.

(b) Discuss the solution’s implications on the variance/bias tradeoff and the role of shrinkage in estimation.