If the PGI is uncorrelated with the covariates, then the estimator will inflate the naïve OLS estimate β^g and its standard error by the factor ρ. If, in addition, the covariates are uncorrelated with each other, then the estimator will also inflate δ^g and its standard error by the factor ρ. Because the regression coefficients and standard errors are inflated by the same factor, the t-statistics and p-values for the corrected estimates will be identical to those for the uncorrected estimates. Correlation between the PGI and the covariates and correlation among the covariates will lead to deviations from this “rule of thumb” adjustment (and can lead to the adjustment being different across regression coefficients and standard errors).
