Instrumental Variables

Example

Suppose that we are interested in estimation of the demand curve for a good, and gather data for the price P and the quantity purchased Q of the good. One might set up the following model for the demand curve:

and run a regression of log quantity on log price. However, although the above equation describes the functional relationship between the price P and the quantity demanded QD, the collected price and quantity data are equilibrium prices and equilibrium quantities, that is, the solutions to the simultaneous equations

where the first equation is the demand equation, and the second is for supply. In other words, although we are interested in the slope of the demand curves D1, D2, D3, and so on in Figure 1, the observed data are the equilibria E1, E2, E3, and so on, which are affected by the demand shocks uD and the supply shocks uS. Therefore, least squares regression using the observed data is unlikely to yield an unbiased estimator for the demand function.

Figure 1 Demand and Supply When Both Demand and Supply Shocks Are Present

An intuitive solution to this problem is to consider a third variable that shifts the supply curve but does not affect the demand. Then, the equilibrium prices and quantities trace out the demand curve as shown in Figure 2, and proper exploitation of the additional variable will lead to a consistent estimator. This variable, the instrumental variable, is correlated with the price P, but is uncorrelated with the demand shocks uD.

Figure 2 Demand and Supply With Only Supply Shocks

Note: A variable that affects only supply is an instrumental variable.

Instrumental Variable Estimation

Consider the linear regression model

where the variables in x1t are exogenous and x2t is the vector of endogenous regressors. By definition, the exogenous variables x1t are all uncorrelated with the error term, so these are all instruments. In addition, we may consider collecting additional variables z2t, which are also uncorrelated with the errors. Together, the instruments are zt = (x′1t, z′2t)′, and by assumption, these are uncorrelated with the errors ut; that is, Eztut = 0, or

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