Linear Regression with Many Controls of Limited Explanatory Power

We consider inference about a scalar coe¢ cient in a linear regression model. One previously considered approach to dealing with many controls imposes sparsity, that is it assumed known that nearly all control coe¢ cients are zero. We instead impose a bound on a weighted sum of squared control coe¢ cients, which is interpretable as a bound on the sample variation in the dependent variable induced by the controls. We develop a simple testing procedure that exploits this additional information in general heteroskedastic models. We also show that under asymptotics where the number of controls is a non-negligible fraction of the number of observations, and the bound is not too large, our suggested test comes close to being weighted average power maximizing in the Gaussian homoskedastic model. We compare our procedure to a sparsity-based approach in a Monte Carlo study and by revisiting the empirical relationship between crime and abortion. Keywords: high dimensional linear regression, limit of experiments, L2 bound, invariance to linear reparameterizations We thank participants at various workshops for useful comments and advice. Müller gratefully acknowledges …nancial support from the National Science Foundation through grant SES-1627660.