[Lieke Holt]

Dear Jeff,

Thank you for your reply. Please find below some tables with corresponding regression equations (I believe I included the full set of time averages everywhere) and some notes and questions along with them.
I am not sure about the preferred way to present results here, but I hope these screenshots will do.

Table 1 gives an example of FE logit versus RE CRE logit versus pooled CRE logit estimates (columns in that order).
All coefficients except the one corresponding to the 2016 dummy and the meanincome regressor are statistically insignificant. These estimates are based on the following code:

clogit lead_don income pwhrs i.year, group(id) robust
xtlogit lead_don income pwhrs i.year i.tobs meanincome meanhrs, re vce(cluster id)
logit lead_don income pwhrs i.year i.tobs meanincome meanhrs, vce(cluster id)

Table 1:

The FE logit estimates are based on a sample size more than half as small because of dropping the individuals with all 0 or all 1 outcomes.

Table 2 presents the equivalent for Poisson estimates for the whole sample, so not only focusing on the second stage. Despite the different properties of RE Poisson relative to linear RE models the estimates are in line with the FE ones. The p-value of pwhrs is nearly equal to 1 in the first two columns, so this imprecise estimation probably explains why the estimate is so different for the last column. But which one to trust?

xtpoisson lead_don income pwhrs i.year, fe vce(robust)
xtpoisson lead_don income pwhrs i.year i.tobs meanincome meanhrs, re vce(cluster id)
poisson lead_don income pwhrs i.year i.tobs meanincome meanhrs, vce(cluster id)

Table 2:

The differences blow up when including an if lead_don > 0 condition in my code, see Table 3.
Table 3:

As a consequence, my final two-stage results including the interaction term differ quite a bit between the pooled CRE and the other two estimation methods (columns 1-3):

egen mean_volbin = mean(volbin), by(id)
gen meanvolbinxhrs = mean_volbin*meanhrs
xtpoisson lead_don income pwhrs volbin 1.volbin#c.pwhrs i.year if lead_don>0, fe vce(robust)
xtpoisson lead_don income pwhrs volbin 1.volbin#c.pwhrs i.year i.tobs meanincome meanhrs meanvolbinxhrs if lead_don>0, re vce(cluster id)
poisson lead_don income pwhrs volbin 1.volbin#c.pwhrs i.year i.tobs meanincome meanhrs meanvolbinxhrs if lead_don>0, vce(cluster id)

I understand your comment about the necessity to include the interaction term, but volbin is a dummy for engaging in volunteering last-period and this is highly correlated with my outcome variable. Column 4 stands for CRE pooled estimates without the volbin regressor and column 5 for CRE RE estimates without. Since the interaction term here is of main importance for my research, I am very curious to hear which of the presented models would be most accurate.

Table 4:

Thank you so much in advance!

Best wishes,

Lieke

[Lieke Holt]

I'm sorry, I just found out that I inserted the wrong picture for Table 3. The one below (in this post) is the correct version instead. pwhrs' p-values are almost approaching 1 in all three models, so I guess that differences are not too important when a coefficient is this insignificant. Still, for my final model(s) that I presented in Table 4 the differences between the third and and other columns become of importance again.
Table 3:

[Lieke Holt]

Sorry, I have one more table correction because I forgot to include the volbin individual mean in Table 4 before. Adding this to the regression yields the following results.
Given the properties of the CRE pooled model I would definitely prefer to use this estimation strategy, by now I also see the necessity of including the volbin dummy.
However, I am concerned about the interpretation of the positive coefficient in the interaction term. It turned significant only when using the CRE pooled model and that is why I wanted to double check if this would be the way to go.

Table 4 (corrected):

[Jeff Wooldridge]

I don't see any serious conflicts across all results. First, you should read about how the magnitudes of the coefficients for logit using joint MLE, pooled MLE, and fixed effects are not comparable. It's possible to rescale the joint MLE to make them comparable to pooled logit. That's why the focus should be on the average partial effects. You can't get APEs for the fixed effects logit -- not convincing ones, anyway. The pooled logit estimates are always smaller than joint MLE, but this doesn't mean the effect is smaller.

Having said that, income and pwhrs have no discernable effect on the participation equation. Compute the APEs for the joint and pooled logit and I bet you find they're similar. But even if they're aren't, it is no surprise to see insignificant estimates bounce around. Even change signs. That's what the insignificance is telling you: there's no evidence that the underlying population coefs are different from zero. For the amount equation, I see nothing that surprises me.

One thing you might try: use log(income) as then its coefficient in the exponential part will be an elasticity. I'd use it in the participation model, too.

[Lieke Holt]

Thank you so much, your comments about the first stage logit clarify a lot!

Personally I was quite surprised by the amount equation outcome that the interaction term coefficient is (marginally) significant in the pooled CRE model estimation despite being very insignificant in the other two models' estimations (see corrected Table 4 in my last post). Is it fair to interpret the coefficient of that term (or at least its sign), arguing that the pooled CRE model is most accurate? Or should I conclude on an indeterminate effect given that the other two model estimations do not confirm this significant positive interaction term coefficient? I still do not quite get where the difference is arising from given that the pooled CRE estimates should be the most flexible of the two in terms of imposed assumptions.

The income elasticity entry is a good suggestion, thank you.

[Jeff Wooldridge]

The pooled CRE estimator is a nice compromise, as it does not require nearly as many assumptions as the joint MLE. It requires more assumptions than Poisson FE but maybe the FE approach is too costly in terms of efficiency in this case. I like the way you have all results in a table. The CRE approach imposes functional form on how the heterogeneity depends on the covariates but that's all.

[Lieke Holt]

Got it! Thanks again, your input has been super helpful.

[Ruzica Savcic]

Hello! Jeff Wooldridge, which result do you refer to when suggesting using Poisson over Tobit? Does it apply to, say, hours of work, as well? I get my data from a monthly panel, not from daily dairies, which is why I thought Tobit estimation would be a correct approach.

[Jeff Wooldridge]

For effects on the mean I’d mainly rely on FE Poisson. But the CRE pooled Tobit approach is also sound. You can compute percentage effects and hopefully they’re reasonably close to FE Poisson, which directly gives percentage effects.