Bivariate probit model + Endogeinity --> which model to use?

[Elise Sobrie]

Dear all

I am trying a bivariate probit model for my master thesis. However, I find kind of two streams on the internet: just bivariate model applied in papers without taking into account endogeinity and then everything to do with endogeinity but not really applied in bivariate probit.

I am confused with the link of the bivariate model with endogenous variable (and the instrumental variable therewith). As you use bivariate probit model when the two dependent variables are interrelated. So isn't it normal that the one dependent variable is endogenous to the other one? Or does the command 'biprobit' already take care of this? However, I already read in posts on the forum that some people include this endogenous variabele and instrument in the biprobit model. Or maybe I am just really confused and nothing is right what I say.

To understand it better, my two dependente variables are 'Use' and 'Request'.
Use is a dummy variable, equal to one if the respondent(firm) indicated that he/she currently uses a loan, zero otherwise.
Request is also a dummy variable, equal to one if the respondent applied for a loan last quarter, zero otherwise.
Respondents are only able to answer the question about 'Request' if they indicated 'no' to the question of 'Use' (So if they don't use a loan, they will be asked if they applied for it). Because of this, 'Request' is endogenous to 'Use'.

I firstly set up these two models:
y1 = a + b*X + y2 + e1 - y1= Use; a = intercept; b*X= independent variables; e1 = error term
y2 = c + b*X + d*Z + e2 - y2= Request; c = intercept; b*X = independent variables; Z =instrument variable; e2 = error term

But after reading some papers, I came to this model:
Yij = Aj + (summation sign k = 1 to n) Bjk*Xik + Eij - A = intercept (i= 1...405 - number of respondents) (j=1,2 - Use and Request) (k= 1..n - number of independent variable)
So this model doesn't take into account the endogeinity

STATA:
I already tried this command:
biprobit Use Request WLB i.Industry i.COUNTRY FirmAge Size - (WLB = dummy equal to 1 if the firm is led by a woman)
In this model, no sign of the endogeinity and instrument

Or do I need to use this:
biprobit (Use = WLB Request i.Industry i.COUNTRY FirmAge Size) (Request = WLB i.Industry i.COUNTRY FirmAge Size Instrument)
So this is my first two models (with endogeinity aspect and instrument)

In short, I am confused with the link of endogeinity and bivariate probit model

Thank you in advance!
Elise

[Stephen Jenkins]

This is a complicated area, but my take (shooting from the hip; sorry, I can't recall textbook references), is that bivariate probit models are not identified when the observed outcome variable A appears as explanatory variable in the equation for observed binary outcome B and also observed outcome variable B appears as explanatory variable in the equation for observed binary outcome A. But one can fit recursive models (e.g. B appears in equation for A, or vice versa). And as usual having additional predictors in the equation for A that are not in the equation for B (and vice versa) helps identification. It'd be good to have a reminder of appropriate literature references from other posters.

A classic article related to this topic is: JJ Heckman (1978) Dummy endogenous regressors in a simultaneous equation system. Econometrica, 46(4), 931-959.

[Marcos Almeida]

Welcome to the Stata Forum / Statalist.

I assume you are using Stata 15.1.

If so, I strongly recommend to the a look at the extended regression models (ERMs).

In the Stata Manual, perhaps the third command line from Quick Start (page 111) is what you are looking for.

Hopefully that helps.

[Stephen Jenkins]

As an update on #2, recursive bivariate probit models are discussed by Greene in his Econometric Theory text (in my 6th edition, see section 23.8.4).

More generally, look at section 5.7 "Models with mixed structures; some consistent and inconsistent models" in GS Maddala (1983), Limited-Dependent and Qualitative Variables in Econometrics, Cambridge UP.

As far as I can see, the ERM examples that Marcos cites in #3 are special cases of the bivariate probit probit, e.g. cases in which there is sample selection in the Heckman sense. The models differ from the model specifications I was mentioning, I think. For these more general models, the key issue is whether the model is "consistent" as Maddala labels it.

[Elise Sobrie]

Thank you both for your help!

Stephen Jenkins, thank you for the references, I will look into it! And if I get you right, endogeneity is not captured by bivariate probit models and if I want to capture it I would need to use recursive models (with additional predictors)?