(Instrumental variable method) Consider the linear regression model y = a + x; +ui One of the assumptions we have made is that x, are uncorrelated with the errors u. If x, are correlated with u,, we have to look for a variable that is uncorrelated with u, (but correlated with x). Let us call this variable zi. zi is called an instrumental variable. Note that, as explained in Section 3.3, the assumptions E(u) = 0 and cov(x, u) = 0 are replaced by =0 and However, since x and u are correlated, we cannot use the second condition. But since z and u are uncorrelated, we use the condition cov(z, u) =0. This leads to the normal equations =0 and = 0 The estimates of a and obtained using these two equations are called the instrumental variable estimates. From a sample of 100 observations, the following data are obtained: = 350 = 150 = 400 = 100 200 = . = 400 = 100 * = 100 = 50 Calculate the instrumental variable estimates of a and B. Let the instrumental variable estimator of be denoted by * and the least squares estimator of B be denoted by B. Show that B = Sy/Sx with Sy, and S, defined in a similar manner to Sxx, Sxy, and Sy,. Show that var(B)=2.. Szz (S) -() Hence, var() var(B). To test whether x and u are correlated or not, the following test has been suggested (Hausman's test, discussed in Chapter 12): B-B (B)-V(B) ~ N(0, 1) Apply this test to test whether cov(x, x) = 0 with the data provided.