ivprobit and firm FE?

[Mahtab Karimi]

Hello all,

I am estimating a probit model for a set of firms; I have a panel dataset of firms where they experience or do not experience a shock in each year. Technically, there is no estimator for a fixed effect model. So I do not include any firm FE in the model.

Now, I have an IV for an endogenous variable, and I am using ivprobit. Should I include a firm FE in my ivprobit? If I don't, does that cause problems for exclusion restriction? What if Z(the IV) can affect the probability of S==1 through the firm FE(unobserved time-invariant heterogeneity)?

This is the probit model:
P(S=1)=a+b_1A_(i,t)+X_it+year FE+Industry FE+e

This is the First stage:
A_(i,t)=a'+yZ_(i,t)+X_it+year FE+Industry FE+e'
Z is the IV for A.

Thank you,
Mahtab

[Maxence Morlet]

1. there is now sufficient statistic for probit model, you're right, but there is one in logit. You could however try the Chamberlain-Mundlak device in probit

2. linear probability model is also an option; the IPP does not apply there

3. if the exclusion restriction only holds conditional on firm FE, this may make reviewers question the overall exogeneity of your instrument...

[Mahtab Karimi]

Thank you. LPM is not a good choice because the number of ones is much smaller than the number of zeros...

[Maxence Morlet]

A good solution to that is the trimeed estimator of Horrace and Oaxaca (2006).

Also a bit of self-promotion regarding LPM: in the following paper, section 6.3.:
Gendered choices of labour market integration programmes: evidence from the United States | Emerald Insight

[Jeff Wooldridge]

A couple of comments. First, I agree with Maxence about the correlated random effects approach. This is what Papke and Wooldridge (2008, Journal of Econometrics) proposed. There is a more recent discussion and slightly different approach that I prefer with endogenous variables in Lin and Wooldridge (2019). Putting in lots of fixed effects is not advised. The conditional logit estimator is sensitive to serial correlation and, besides, you can only estimate coefficients, not magnitudes of average partial effects.

I do think it's often reasonably to assume instruments are uncorrelated with shocks but not with unobserved heterogeneity. The CRE approach is valid in this case with a time-varying instrument.

In recent work, Chen, Martin, and Wooldridge (2023), we show that having a large fractions of ones or zeros is neither necessary no sufficient for the LPM to be unreliable for estimating the average marginal (partial) effects. It's more about the distribution of X. Even with few ones, I would apply the linear fixed effects IV estimator and compare it with the CRE/control function estimator in Lin and Wooldridge.