Discussion
Suppose, in a regression model, \(Y_{i} = \alpha + \beta X_{i} + e_{i}\); covariance between X and e is not zero. Then :
Answer: A
The OLS estimates are biased and inconsistent.
OLS requires the regressor to be uncorrelated with the error term. If \(\mathrm{Cov}(X, e) \neq 0\), the regressor is endogenous, and the estimator of the slope converges to
\[ \begin{aligned} \hat{\beta} &\to \beta \\ &+ \frac{\mathrm{Cov}(X, e)}{\mathrm{Var}(X)} \end{aligned} \]
which differs from \(\beta\). The error does not vanish as the sample grows, so the estimator is inconsistent as well as biased. Instrumental variables are the usual remedy.
Bias in small samples with consistency in large ones is the case of a lagged dependent variable with serially uncorrelated errors, not of a correlated regressor. The estimates can still be computed; they are simply wrong on average.
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