n the empirical exercises on earning and height in Chapters 4 and 5, you estimated a relatively large and statistically significant effect of a worker’s height on his or her earnings. One explanation for this result is omitted variable bias: Height is correlated with an omitted factor that affects earnings. For example, Case and Paxson (2008) suggest that cognitive ability (or intelligence) is the omitted factor. The mechanism they describe is straightforward: Poor nutrition and other harmful environmental factors in utero and in early childhood have, on average, deleterious effects on both cognitive and physical development. Cognitive ability affects earnings later in life and thus is an omitted variable in the regression. Suppose that the mechanism described above is correct. Explain ho
In the empirical exercises on earning and height in Chapters 4 and 5, you estimated a relatively large and statistically significant effect of a worker’s height on his or her earnings. One explanation for this result is omitted variable bias: Height is
Suppose that the mechanism described above is correct. Explain how this leads to omitted variable bias in the OLS regression of Earnings on Height. Does the bias lead the estimated slope to be too large or too small? [Hint: Review Equation (6.1)]
![Omitted Variable Bias in Regression
with a Single Regressor
6.1 Omitted Variable Bias
Omitted variable bias is the bias in the OLS estimator that arises when the regres-
sor, X, is correlated with an omitted variable. For omitted variable bias to occur,
conditions must be true:
two
1. X is correlated with the omitted variable.
2. The omitted variable is a determinant of the dependent variable, Y.
=
185
Because ui
and X; are correlated, the conditional mean of u; given X; is nonzero.
This correlation therefore violates the first least squares assumption, and the con-
sequence is serious: The OLS estimator is biased. This bias does not vanish even
in very large samples, and the OLS estimator is inconsistent.
KEY CONCEPT
6.1
A Formula for Omitted Variable Bias
The discussion of the previous section about omitted variable bias can be sum-
marized mathematically by a formula for this bias. Let the correlation between X;
and u; be corr(X¡, u¡) Pxu. Suppose that the second and third least squares
assumptions hold, but the first does not because pxu is nonzero. Then the OLS
estimator has the limit (derived in Appendix 6.1)
Ou
B₁ B₁ + PXux
(6.1)
That is, as the sample size increases, B₁ is close to B₁ + pxu(ou/ox) with increas-
ingly high probability.
The formula in Equation (6.1) summarizes several of the ideas discussed
above about omitted variable bias:
1. Omitted variable bias is a problem whether the sample size is large or small.
Because ₁ does not converge in probability to the true value B₁, B₁ is biased
and inconsistent; that is, ₁ is not a consistent estimator of B₁ when there is
omitted variable bias. The term pxu(ou/ox) in Equation (6.1) is the bias in
that persists even in large samples.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F30613526-87b9-48aa-9e9a-cda2327d4281%2F7603bae9-dd15-4f88-a3e5-46d7a2c6f6ec%2Fuq4tsawj_processed.jpeg&w=3840&q=75)
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