A Multicollinearity and Measurement Error Statistical Blind Spot: Correcting for Excessive False Positives in Regression and PLS

Appendix A Deriving Equations for M+ME Biases and for t-statistic Overestimations Determining the Impact of VIF Bias on the t-statistic Consider first the situation where we have no measurement error. Using standard equations for the estimated standard error of β2 from any regression textbook, an unbiased and consistent estimate of the standard error is s = Σεi / (N – K) (A1-1) and the estimate of the variance of β2 is Var (β2) = [s / Σx2i] ∗ [1 / (1-ρ12Underlying)] (A1-2) When the error terms are normally distributed, the following has a t distribution: t-stat = (β2 β2) / {Var(β2)} ~ t N-K (A1-3) Substituting in the equations for Var ( β2) and s from above (A1-1 and A1-2), we can see the impact of correlated predictor variables on the t-stat. t-stat = (β2 β2) / { [s / Σx2i ] ∗ [ 1 /(1ρ12 Underlying)] } t-stat = (β2 β2) / { [{Σεi / (N – K)}] / Σx2i ]) ∗ [ 1 /(1ρ12 Underlying)] }