Wald exogeneity test with -biprobit-

[Parul Gupta]

Dear Statalisters
Is there an in-built command/option with -biprobit- which shows the result for Wald's exogeneity test (or something similar). Is it straightforward enough to do "by hand" if no built-in command is available? My code is

biprobit (math $xlist tuition) (tuition= $zlist), nolog

Additionally, if there is not much difference between -ivprobit- and -biprobit- results, is it appropriate to use -ivprobit- even if the endogenous regressor is a dummy?

[Parul Gupta]

Could someone please advise on this?

[Jeff Wooldridge]

Parul: The Wald test is simply the test of rho = 0, which is reported in the output. Hopefully you get the same answer whether you use rho or the atan transformation of rho.

[Parul Gupta]

Professor Jeff Wooldridge,
I could never have imagined discussing a problem with my textbook's author! Thank you very much for your response. I was careless enough to have overlooked the last line of the output table which reports rho.

If you could kindly shed light on the advantage of using -biprobit- over -ivprobit- when the endogenous regressor is a dummy, I'll be extremely grateful. Specifically, I am trying to model a simultaneous equation framework where both the endogenous regressor and dependent variables are binary.

\[ Y= \alpha + \beta*X + \theta*Z_1 + \epsilon \]
\[ X= \delta + \phi*Y + \gamma*Z +\nu*Z_2+\eta \]

I am interested in the beta coefficient, hence am using an instrument (Z) for X. The instrument is continuous. I have run -ivprobit- as

 ivprobit Y  Z_1 (X=Z)

If I use -biprobit- instead with code:

 biprobit (Y X Z_1) (X= Z), nolog

will it be correct? Or should I include Z_2 in the second regression as well? Thank you for your time.
Regards
Parul

[Jeff Wooldridge]

Neither approach is suitable for your model when Y and X are both binary. In fact, that’s a tough model to estimate. There may not even exist a solution for X and Y. What are these variables? In many cases a simultaneous equation is not appropriate.

[Parul Gupta]

Some colleagues on the forum suggested -biprobit- may be used. I tried using -sspecialreg- as well but got some error.

I am estimating the impact of private tutoring on learning outcomes. Y is math proficiency (high or low), X is tuition participation (Yes or no). Z is average tuition participation among peers. Household, child and school variables are used as controls.

Should I use a continuous variable instead of a dummy as the regressor? I have data on monthly tuition expenditure as well.

[Jeff Wooldridge]

I don’t think tutoring should be a function of math proficiency. Presumably, from a causality perspective, we want to know if tutoring now improves future math performance. Of course tutoring is correlated with general math ability but that’s an omitted variable story. If you model math proficiency as a function of tutoring and tutoring as a function of z1 and z2 then you can use biprobit. That’s what I would do.

[Parul Gupta]

I would suspect students take tutoring to improve learning outcomes, hence I am modelling tutoring as a function of math proficiency (measured by test scores). Do you suggest that tutoring could be a function of past scores and not a function of contemporaneous math proficiency?

In the model you advised, would the coefficient on tutoring be identified, since there are omitted variables?

[Jeff Wooldridge]

Parul: I suggest you look at the literature on estimating causal effect with potential outcomes notation. Your problem fits. To say that the decision to participate in tutoring depends on current math proficiency is clearly true, but it's not the same thing as tutoring depending on the outcome of a test that hasn't yet been taken. That makes no sense. As I describe in both of my books, most applications of simultaneous equations models are ill advised. And I think yours is one of them. There is no "omitted variable" problem because your inclusion of Y in the X equation is inappropriate to begin with.

I the potential outcomes setting, Y(0) is the math test outcome without tutoring and Y(1) is the outcome with math tutoring. Now, from here, the tutoring assignment, X, can be correlated with Y(0) and Y(1). We then observe the outcome of the test, Y = (1 - X)*Y(0) + X*Y(1). This is precisely the setting of literally thousands of methodological and empirical papers. When you start from this, you are led to different possibilities. One is biprobit with Y as a function of X and then X following a reduced form probit. But you could also estimate separate probits for X = 0 and X = 1.

If you insist on the approach in post #4 then I can't help you.

JW

[Parul Gupta]

Thank you, professor, for pointing towards the potential outcomes approach.

[Nishant Kathuria]

Dear Statalisters,

I am writing to follow up on Proff. Jeff Wooldridge comment above "One is biprobit with Y as a function of X and then X following a reduced form probit." Does this mean we should use the model as: biprobit (Y= X1) (X1= X2 X3 Z) or (Y= X1 Z) (X1= X2 X3 Z)?

My model is same as the one Parul Gupta is handling- binary DV Y, binary endogenous regressor X1. Above that, my binary endogenous regressor X1 is an interaction between two other binary variables, say X4 and X5, where X1=X4*X5.

Any help in clarifying these concepts will be very much appreciated.

Thanks,
Nishant