Testing for Reverse Causation and Omitted Variable Bias in Regressions

Abstract

Textbook theory predicts that t-ratios decline towards zero in regressions when there is collinearity between two regressors. This paper shows that this often does not occur if the regression suffers from simultaneity or omitted variable bias. With more collinearity, t-ratios generally increase indefinitely or reach a plateau at a high level. Coefficients on the two regressors have opposite signs, and they increase indefinitely or reach a plateau. I use this phenomenon to develop a test for the presence of simultaneity or omitted variable bias, important and intractable problems in many disciplines. The test is simple: one selects a regressor and creates a second regressor that is highly correlated with the first. Simultaneity or omitted variable bias is indicated if t-ratios and coefficients undergo these trends with more collinearity. The test does not rely on lagged variables or instruments. Mathematical analysis of simultaneity and omitted variable bias show that many situations lead to unsolvable anomalies, which suggest the likelihood of unexpected results when there is collinearity. The test cannot be proved mathematically because of the anomalies, but it is substantiated with numerous simulations. I give several empirical examples of the test, including a test of causal assumptions in a Granger regression, a test of whether subjects are not randomly assigned in a randomized controlled experiment, and a test of whether instrumental variables in a two stage least square regression are endogenous.

Keywords: simultaneity, collinearity, multicollinearity, omitted variable bias, endogeneity, misspecification, simulations, Granger test, two stage least squares, random experiments, quadratic variables.

JEL Classification: C10, C15, C18, C29, K49