I am studying the relatively classical model of selection to estimate the union wage premium, with two equations of salary and one two step equation of selection
$\ln(w_{1it}) = X^{'}_{it} \beta_1 + \epsilon_{1it}$
$\ln(w_{0it}) = X^{'}_{it} \beta_0 + \epsilon_{0it}$
$union^*_{it} = \gamma X^{'}_{it} + \epsilon_{2it} $
$union_{it} = \mathbb{I}(union^*_{it}>0)$
I have two different questions:
1) When I implement it as a cross section estimate in Stata (with the heckman command), I have very different results whether I estimate it by Heckman two step method or by MLE.
Is that normal? In which case what is the theoretical reasons?
2) As noted in my equations, I have panel data.
Does the Heckman procedure still apply to it? Or does the within estimation remove endogeneity and solve it all?
I would say it still applies, but I did not find relevant literature and my courses only cover cross section data.
Best Answer
If I understand correctly, you are "tricking" the Heckman selection model to estimate a endogenous switching regression model, also known as the Roy model and Tobit Type 5. This trick is explained in Lee, Lung-Fei (1978) "Unionism and Wage Rates: A Simultaneous Equations Model with Qualitative and Limited Dependent Variables", International Economic Review, Vol. 19(2), pp. 415-433. You're interested if worker characteristics are rewarded differently in the two regimes/sectors ($\beta_0 - \beta_1 \ne 0$) and the correlation parameter $\rho$ tells you about the effect of self-selected union membership on wages in the two sectors. You do, however, may need to adjust the standard errors if the Heckman technique is to be used, or you loose consistency.
Alternatively, since you have an exclusion restriction, you can get the causal effect of union membership on wages using instrumental variables:
treatreg/etregressfor cross sectional data, and all kinds of panel IV methods likextivreg.Some observations. First, there's is a user-written Stata command called
movestaydesigned to estimate the endogenous switching regression model with cross sectional data. It is a full information ML approach, which relies on the multivariate normality of the the error terms assumption, as does the Heckman MLE method. If this is satisfied, both will be consistent, though themovestaywill be somewhat more efficient than doing it in two parts.The Heckman Two Step limited information ML estimator relies only on univariate normality of the marginal distribution, so it is expected to be more robust since that is a lower hurdle to clear. But if you do have joint normality, the Two Step is still consistent, but no longer efficient, especially relative to
movestay. However, if you only have univariate normality, then the Two Step remains consistent while the FIML approaches are not.In short, FIML and LIML approaches will usually differ since they have different information to work with, as I show below with an example. I think this explains question (1).
Now for (2). As far as I know, there is no off-the-shelf panel version of
heckmanormovestay, though both allow you to cluster the standard errors on the panel id. That's not strictly correct, but may be good enough. There might also be a way to hack it usinggllamm, though I have never done this myself since it appears non-trivial. Some notes on that here and Statalist threads here.I am not really answering the second part of (2) since it is not clear to me how the fixed effects enter the model and how they are related to union membership. With panel methods I suggested above, you give up estimating different parameters in the two regimes/sectors. Depending on the details of your model, you may not even need to instrument if you can difference away the pesky effect. The details depend on your data and models.
Finally, you might consider changing your notation to add instrument(s) Z (something that alters union membership, but is not related to wages directly) and the fixed effects.
Here's some code showing the
movestayand Heckman MLE equivalence, along with the problem you have in (1). I am modeling wages with endogenous participation in public/union and private sectors. My instruments are marital status and number of job holders in the household. They are likely to be not very good.Here's the output:
The $\rho$s are the union effects. $\rho_0$ is positive and significant, so folks who choose to work in the public sector earn lower wages in that sector than a random individual from this sample. Those working in the private sector fare no better or worse than a random individual. The signs are a tad counterintuitive, but the authors of the Stata code parameterized $\rho$ as negative (see the conditional expectations part of the
movestaypaper). The likelihood-ratio test for joint independence of the three equations is reported in the last line of the output. The "frontslashed" parameters are ancillary. Some folks prefer to multiply $\rho$ and $\sigma$ to get $\lambda$, with standard errors estimated using the delta method.Now you can get the same estimates using
heckman(though the sign on $\rho$ and the two instruments flips since the command is parameterized a bit differently). You see the same negative effect:Using the two step just kills the results:
Now we replicate the second equation, with similar results: