Consider the wage equation used in Example 10.5. Suppose we have a variable designed to measure ABILITY. This variable is an index created using 10 different tests of cognitive ability. Using data on 2,178 white males in 1980 , the ability variable has a sample mean of 0.04 and a standard deviation of 0.96 .

a. The estimated relationship between years of education and the ability measure is \(\widehat{E D U C}=\) \(12.30+0.977\) ABILITY with a \(t\)-value of 25.81. Is this result consistent with the usual "omitted variables bias" explanation of the endogeneity of education? Explain.

b. Using these data and the model in Example 10.5, the estimated coefficient on EDUC is 0.0609 with standard error 0.005 . Adding ABILITY to the equation reduces the estimated coefficient on \(E D U C\) to 0.054 with standard error 0.006 . Is this the effect that you anticipate? Explain.

c. Assuming that ABILITY and EXPER are exogenous, along with instrumental variables MOTHEREDUC and FATHEREDUC, what is the specification of the first-stage equation? That is, what variables are on the right-hand side?

d. Estimating the first-stage equation in (c), we find that the \(t\)-values on MOTHEREDUC and FATHEREDUC are 2.55 and 4.72, respectively. The \(F\)-test of their joint significance is 33.82. Are these instruments adequately strong for their use in IV/2SLS? Explain.

e. Let \(\hat{v}\) denote the OLS residuals from part (d). If we estimate the model in Example 10.5, and include the variables ABILITY and \(\hat{v}\), the \(t\)-statistic for \(\hat{v}\) is -0.94 . What does this result tell us about the endogeneity of \(E D U C\) after controlling for ability?