Avoiding Invalid Instruments and Coping with Weak Instruments

A rchimedes said, “Give me the place to stand, and a lever long enough, and I will move the Earth” (Hirsch, Kett, and Trefil, 2002, p. 476). Economists have their own powerful lever: the instrumental variable estimator. The instrumental variable estimator can avoid the bias that ordinary least squares suffers when an explanatory variable in a regression is correlated with the regression’s disturbance term. But, like Archimedes’ lever, instrumental variable estimation requires both a valid instrument on which to stand and an instrument that isn’t too short (or “too weak”). This paper briefly reviews instrumental variable estimation, discusses classic strategies for avoiding invalid instruments (instruments themselves correlated with the regression’s disturbances), and describes recently developed strategies for coping with weak instruments (instruments only weakly correlated with the offending explanator). As an example of biased ordinary least squares, consider whether incarcerating more criminals reduces crime. To estimate the effect of increased incarceration on crime, an economist might specify a regression with the crime rate as the dependent variable and the incarceration rate as an explanatory variable. In this regression, the naı̈ve ordinary least squares regression could misleadingly indicate that high rates of incarceration are causing high rates of crime if the actual pattern is that more crime leads to more incarceration. Ordinary least squares provides a biased estimate of the effect of incarceration rates on crime rates in this case because the incarceration rate is correlated with the regression’s disturbance term. As another example, consider estimating consumption’s elasticity of intertemporal substitution (which measures the responsiveness of consumption patterns to changes in intertemporal prices). To estimate this elasticity, economists typically specify a linear relationship between the rate of growth in consumption and the