Instrumental Variable - Estimation

Estimation

Suppose the data are generated by a process of the form

where i indexes observations, is the dependent variable, is an independent variable, is an unobserved error term representing all causes of other than, and is an unobserved scalar parameter. The parameter is the causal effect on of a one unit change in, holding all other causes of constant. The econometric goal is to estimate . For simplicity's sake assume the draws of are uncorrelated and that they are drawn from distributions with the same variance, that is, that the errors are serially uncorrelated and homoskedastic.

Suppose also that a regression model of nominally the same form is proposed. Given a random sample of T observations from this process, the ordinary least squares estimator is

where x, y and denote column vectors of length T. When x and are uncorrelated, under certain regularity conditions the second term has an expected value conditional on x of zero and converges to zero in the limit, so the estimator is unbiased and consistent. When x and the other unmeasured, causal variables collapsed into the term are correlated, however, the OLS estimator is generally biased and inconsistent for β. In this case, it is valid to use the estimates to predict values of y given values of x, but the estimate does not recover the causal effect of x on y.

An instrumental variable z is one that is correlated with the independent variable but not with the error term. Using the method of moments, take expectations conditional on z to find

The second term on the right-hand side is zero by assumption. Solve for and write the resulting expression in terms of sample moments,

When z and are uncorrelated, the final term, under certain regularity conditions, approaches zero in the limit, providing a consistent estimator. Put another way, the causal effect of x on y can be consistently estimated from these data even though x is not randomly assigned through experimental methods.

The approach generalizes to a model with multiple explanatory variables. Suppose X is the T × K matrix of explanatory variables resulting from T observations on K variables. Let Z be a T × K matrix of instruments. Then it can be shown that the estimator

is consistent under a multivariate generalization of the conditions discussed above. If there are more instruments than there are covariates in the equation of interest so that Z is a T × M matrix with M > K, the generalized method of moments can be used and the resulting IV estimator is

where . The second expression collapses to the first when the number of instruments is equal to the number of covariates in the equation of interest.