Revisiting The Bell Curve Debate Regarding the Effects of Cognitive Ability on Wages

Hausman and Taylor (1981) offer a solution to this problem. They propose a procedure to
estimate a random effects model in which the unobserved individual-level characteristics are
correlated with some of the independent variables. Their method applies a Generalized Least
Squares transformation with instrumental variables to obtain consistent coefficient estimates.

The model is as follows:

yit = τt + Ziγ + Xitβ + ai + uit

where Zi is a vector of time-invariant explanatory variables, Xit is a vector of time-variant
variables, ai is unobserved individual-level characteristics, and uit is the idiosyncratic error term.

Split Xit =[X1it X2it] Zit =[Z1i Z2i]

where X1it and Z1i are exogenous in the sense that

E[X1it ai] = E[Z1i ai] = 0 and

E[X1it uit] = E[Z1i uit] = 0

X2it and Z2i are endogenous in the sense that

E[X2it ai] ≠ 0 and E[Z2i ai] ≠ 0 but

E[X2it uit] = E[Z2i uit] = 0

The key assumption is that certain variables among the variables X1it and Z1i are
uncorrelated with ai. The variables of X1it “can serve two functions because of their variations
across both individuals and time: (i) using deviations from individual means, they produce
unbiased estimates of the β ’s, and (ii) using the individual means, they provide valid instruments
for the columns of Zi that are correlated with ai” (Hausman and Taylor [1981], p.3). Essentially
the Hausman-Taylor model employs instrumental variables from within the regression model
itself, which is an advantage of panel data. However, the authors caution that these instrumental
variables have to be chosen carefully to ensure that X1it and ai are not correlated. They also point

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