Can missing data cause endogeneity problem?

Question

I am a econometrics newbie. I would like to ask a question. In the standard Hausman Taylor model, we need to specify exogeneous/endogenous variables for estimation.

See: https://kb.iu.edu/d/bcfo

However, suppose we have missing data. For example , female data are more likely to be missing and only the most capable one are in the dataset. Lets assume some wild theory that females may take on motherhood duties and only the most capable females remain in the workforce and have higher wage. As a result, female in the dataset with missing data may be correlated with skills which is in the unobservable causing endogeneity problem.

Theoretically, female has no relationship with skills and are truly exogenous. But can a variable become endogenous due to data missing systematically ? And this needs to be corrected in the Hausman Taylor specification (e.g., make FEM an endogeneous variable) if the dataset suffers this problem ?

Thanks ! thanks !

Answer 1

Welcome.

So your question is: In the Hausman and Taylor (1981, HT hereafter) model without selectivity, FEM is usually specified as exogenous, but if selection involves, can we just specify FEM as endogenous?

To answer this question, we should understand the meaning of "endogenous" in the HT model. Let their model be written as $$y_{it} =x_{1,it} \beta_1 + x_{2,it}\beta_2 + z_{1,i}\gamma_1 + z_{2,i} \gamma_2 + \mu_i + e_{it},$$ where the "1" variables are exogenous and the "2" variables are endogenous. Importantly, all the regressors are independent of (or strictly exogenous to) the idiosyncratic error $e_{it}$ and they allow $x_{2,it}$ and $z_{2,i}$ correlated with $\mu_i$ only. That's the meaning of being "endogenous" in their model.

So if FEM is correlated with time-invariant skills $\mu_i$ (but not with $e_{it}$) due to selectivity, then, yes, FEM is endogenous in HT's sense. But specifying FEM as endogenous only does not solve the problem. Introducing selectivity comes at the cost that some of $x_{1,it}$ (specified as exogenous without sample selection) may become endogenous, so things become complicated. Note that $x_{1,it}$ is important in HT because $z_{2,i}$ is instrumented by $\bar{x}_{1,i}$. You need some time-varying exogenous variables unaffected by selection but cross-sectionally correlated with FEM. This would be hard. There is no magic.

Another issue is that selection may depend on the time-varying error $e_{it}$, that is, a woman works only in good years. Then some time varying regressors (the "$x$" variables) may be correlated with $e_{it}$, and you need genuine instruments (not in the equation) for them too. This would also be difficult.