TAILIEUCHUNG - Book Econometric Analysis of Cross Section and Panel Data By Wooldridge - Chapter 5

Instrumental Variables Estimation of Single-Equation Linear Models In this chapter we treat instrumental variables estimation, which is probably second only to ordinary least squares in terms of methods used in empirical economic research. The underlying population model is the same as in Chapter 4, but we explicitly allow the unobservable | Instrumental Variables Estimation of Single-Equation Linear Models In this chapter we treat instrumental variables estimation which is probably second only to ordinary least squares in terms of methods used in empirical economic research. The underlying population model is the same as in Chapter 4 but we explicitly allow the unobservable error to be correlated with the explanatory variables. Instrumental Variables and Two-Stage Least Squares Motivation for Instrumental Variables Estimation To motivate the need for the method of instrumental variables consider a linear population model y b0 x1 b2x2 bKxK u 5-1 E u 0 Cov xy u 0 j 1 2 . K - 1 5-2 but where xK might be correlated with u. In other words the explanatory variables x1 x2 . xK_1 are exogenous but xK is potentially endogenous in equation . The endogeneity can come from any of the sources we discussed in Chapter 4. To fix ideas it might help to think of u as containing an omitted variable that is uncorrelated with all explanatory variables except xK. So we may be interested in a conditional expectation as in equation but we do not observe q and q is correlated with xk . As we saw in Chapter 4 OLS estimation of equation generally results in inconsistent estimators of all the bj if Cov xK u 0 0. Further without more information we cannot consistently estimate any of the parameters in equation . The method of instrumental variables IV provides a general solution to the problem of an endogenous explanatory variable. To use the IV approach with xK endogenous we need an observable variable z1 not in equation that satisfies two conditions. First z1 must be uncorrelated with u Cov z1 u 0 5-3 In other words like x1 . xK_1 z1 is exogenous in equation . The second requirement involves the relationship between z1 and the endogenous variable xK. A precise statement requires the linear projection of xK onto all the exogenous variables xk 0 hx1 82x2 8k-1 xk-1 01Z1 tk 5-4 where by definition

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