True/False: You are doing econometric analysis. If your dependent variable only takes on values of zero or one (for "no" and "yes"), and your independent variable has a continuous distribution, then regressing the dependent variable on the independent variable will unbiased coefficient and standard error estimates.

Answer:
False. There are many problems with such regressions. For example, the model will very likely suffer from severe heteroskedasticity, making standard error estimates biased.

True/False: Suppose you are analyzing panel data and are unsure whether the fixed effect and independent variables are correlated. Then, it is best to use fixed effect models.

Answer:
True. Random effects models impose the strong assumption that the fixed effect and independent variables are uncorrelated. These coefficients may be biased if the assumptions does not hold. Therefore, it is best to play safe and use fixed effects model. You potentially sacrifice some efficiency if your assumption is not correct, but your estimates are unbiased if all other assumptions are met.

True/False: Heteroskedasticity causes OLS coefficient estimates to be biased.

Answer:
False. OLS standard errors are biased, but not OLS coefficient estimates. These are still unbiased (although OLS is no longer BLUE, meaning it no longer is the best linear unbiased estimator. In other words, if errors are heteroskedastic, there exist other linear unbiased estimators that have lower variance than OLS. Consider generalized least squares)

True/False: With an omitted variable, OLS is no longer unbiased, but it is still consistent.

Answer:
False. OLS is inconsistent with an omitted (relevant) variable.

True/False: The two stage least squares estimator is unbiased.

Answer:
False. The two stage least squares estimator is consistent (meaning that as the sample size goes to infinity, the bias goes away). However, it is biased in finite ("small") samples.