Multinomial logit with binary endogenous explanatory variable

[Maria Ferreyra]

Hello,

I am estimating a multinomial logit model where one of the explanatory variables is binary ** and ** endogenous. To be more precise:

1. Y = f(X, W) + epsilon2 --> this gives rise to a multinomial logit model

1. W = g(X, Z) + epsilon1; W is 0 or 1 --> this gives rise to a binomial model (linear probability, probit, etc)

Note: correlation (epsilon2, epsilon1) is different from zero.

How can I handle this in Stata? I can think of the approaches listed below; please let me know if there is another:

1. IV estimation:
   1. Use a linear probability model to estimate (2)
   2. Obtain predicted values for W
   3. Estimate (1) using X and predicted W as right-hand side variables
   4. Bootstrap standard errors, or apply asymptotic variance correction formulas.
2. Control function approach:
   1. Use a linear probability model to estimate (2)
   2. Obtain residuals for W
   3. Estimate (1) using X, W, and the residuals for W as right-hand side variables
   4. Bootstrap the standard errors

Would these approaches work?

Also, are they well-suited to a binary endogenous variable?

Thank you very much,

Maria

[Junran Cao]

Maria:

What I suggest below does not answer your question in its entirety but hopefully it's of some help until someone more knowledgeable comes along.

According to Baum et al. (2012), it is inadvisable to use control function if your endogenous variable is binary (slide 23). The authors also illustrate the dangers of using LPM via an example in which the treatment effect - the coefficient for the regressor of interest - is constructed to be nonnegative (slides 14-18). You would need to decide if this is a relevant concern for your project.

The proposed solution is Lewbel's special regressor approach (Lewbel 2014). A special regressor is a regressor that is exogenous, additive, continuous and has a large support (i.e. domain) (pp.5-6, Lewbel 2014). The special regressor approach can be implemented in Stata via the user-written -sspecialreg-.

Using the example from -help sspecialreg-, suppose your i) outcome variable is 'migrate' ii) the special regressor is the age of household head -ageh- iii) the endogenous binary regressor is homeownership -homeowner- & iv) the instrument is welfare receipts -welfare-. You would do something akin to the following (-bs- refers to bootstrap SE)

ssc install sspecialreg

use http://fmwww.bc.edu/ec-p/data/wooldridge/sspecialreg_sample.dta, clear

clonevar migrate = D3
label define migrate 1 "migrated" 0 "otherwise"
label values migrate migrate

sspecialreg migrate ageh, exog(married child educat) endog(homeowner) iv(welfare) bs

Unfortunately here the outcome variable - your Y - is binary instead of having >2 categories.

However, if you consult Dong and Lewbel (2012, p.6) you can see the other referenced works including those modelling multinomial choice.


REFERENCES:

Baum, CF, Dong, Y., Lewbel, A., Yang, T., 2012. Binary Choice Models with Endogenous Regressors.
http://repec.org/san2012/baum.san2012.pdf.

Dong, Yingying, and Arthur Lewbel. 2012. ‘Simple Estimators for Binary Choice Models With Endogenous Regressors’, 30. https://www.economics.uci.edu/files/...12/dong-04.pdf

Lewbel, Arthur. 2014. ‘An Overview of the Special Regressor Method’. In The Oxford Handbook of Applied Nonparametric and Semiparametric Econometrics and Statistics, edited by Jeffrey S. Racine, Liangjun Su, and Aman Ullah. Oxford University Press. https://doi.org/10.1093/oxfordhb/9780199857944.013.002.

[Maria Ferreyra]

Dear Junran,

Thank you very much for your response, and I apologize for my late reply.

Your response led me to do some additional reading. My impression now is that one ** can ** use a control function for an endogenous binary variable, as outlined in Wooldridge (JHR, 2015).

Thanks, again, for your prompt response.

Best,

Maria

[Junran Cao]

Dear Maria

You are very welcome. And thank you for mentioning Wooldridge (JHR, 2015) - You are absolutely right. Section II.B deals with this question. Professor Wooldridge comments on this forum from time to time, hopefully he would come across your question and provide some comments.

Regards,
Junran

[Mehwish Ali]

Hi Maria

Just wondering what your final solution was as I am in a similar predicament.