1. The Instrumental Variables Estimator The instrumental variables (IV) estimator is an alternative to Ordinary Least Squares (OLS) which generates consistent estimates when the RHS variables are correlated with the error term. Although the IV estimator will produce consistent estimates they will typically be less efficient (have higher variance) than the OLS estimates. In the case of the simultaneous equations model, the equation must be identified before we can make use of the IV estimator.

2. Instrumental Variables We define the instrumental variable estimator to be: We wish to estimate: Suppose we find a variable Z such that:

3. Consistency of the IV estimator We have Therefore: Since Z i and u i are uncorrelated by assumption.

4. Variance of the IV estimator whereis the sample correlation between X and Z. This shows that the IV estimator must always have a higher variance than the OLS estimator. The lower is the correlation between X and Z then the higher will be the variance of the IV estimator.

5. The trade-off between bias and variance It is not always obvious that we want an unbiased estimator at all costs. An alternative criterion is to look for the estimator with the lowest mean square error.

6. The diagram shows the PDF for an unbiased estimator (the red line) and the PDF for a biased estimator (the blue line). The estimate obtained from the biased estimator is likely to have a lower mean square error.

7. Example: Estimating the consumption function The simplest Keynesian model has a pair of equations for consumption and income: These are the structural equations of the model. The reduced form equation for income can be derived as:

8. The reduced form equation shows that the change in income is correlated with the error term from the consumption function. It follows that OLS estimation of the consumption function will be inconsistent. An alternative is to use the change in investment as an instrument and use the IV estimator. The change in investment is correlated with the change in income but there is no reason to think it should be correlated with the error from the consumption function.

9. The Two Stage Least Squares Estimator If we have more potential instruments then there are endogenous variables then we can use the two stage least squares TSLS estimator: Regress the endogenous variable on all the instruments and compute the fitted values. Use these fitted values as instruments in a second stage regression. Stage 1: Stage 2:

10. The Hausman Specification Test 1. Regress the potentially endogenous variable X on the instrument Z and calculate the residuals. 2. Regress Y on X and the residuals created at stage 1. 3. Test the significance of the residuals in the regression estimated at stage 2. Hausman, J. A. (1978) ‘Specification Tests in Econometrics’, Econometrica, Vol 46, pp. 1251-1271.