Multicolinearity and ridge regression: results on type I errors, power and heteroscedasticity

Let $ \beta _1, \ldots , \beta _p $ β1,…,βp be the slope parameters in a linear regression model and consider the goal of testing $ H_0: \beta _j=0 $ H0:βj=0 ( $ j=1, \ldots , p $ j=1,…,p). A well-known concern is that multicolinearity can inflate the standard error of the least squares estimate of $ \beta _j $ βj, which in turn can result in relatively low power. The paper examines heteroscedastic methods for dealing with this issue via a ridge regression estimator. A method is found that might substantially increase the probability of identifying a single slope that differs from zero. But due to the bias of the ridge estimator, it cannot reject $ H_0: \beta _j=0 $ H0:βj=0 for more than one value of j. Simulations indicate that the increase in power is a function of the correlations among the dependent variables as well as the nature of the distributions generating the data.