Solved – Bivariate probit versus 2SLS, contradictory results (sign)

2slsbivariateinstrumental-variablesstata

I am currently facing puzzle and I hope some of you will be able to provide me some insights. I have this model: y: binary variable, x1: binary variable (endogenous), z: binary instrument, x2: controls, i.year: dummies for each year, i.region: dummies for each region with in total about 4,000 observations

I first run my 2SLS and I obtained that the effect of x1(instrumented by z) on y is positive.

But my problem is when I turn to a bivariate probit, using the following stata code:

biprobit (y=x1 x2 i.year i.region) (x1= z x2 i.year i.region), cluster(region)
margins, dydx(x) predict(pmarg1) force

Now I obtain a negative effect, even though from what I've seen I shoud at least get the same sign in the biprobit and the 2SLS.

Does anyone has any idea for what reason I have this problem, or have some suggestion as to what I should do?

Best Answer

It's probably the controls.

If you omit the controls, the marginal effects from the bivariate probit regression should be quite similar to the 2SLS estimates. See page 203 of Mostly Harmless Econometrics for an example.

Related Solutions

Probit Two-Stage Least Squares (2SLS) – Overview and Application

What was proposed to you is sometimes referred to as a forbidden regression and in general you will not consistently estimate the relationship of interest. Forbidden regressions produce consistent estimates only under very restrictive assumptions which rarely hold in practice (see for instance Wooldridge (2010) "Econometric Analysis of Cross Section an Panel Data", p. 265-268).

The problem is that neither the conditional expectations operator nor the linear projection carry through nonlinear functions. For this reason only an OLS regression in the first stage is guaranteed to produce fitted values that are uncorrelated with the residuals. A proof for this can be found in Greene (2008) "Econometric Analysis" or, if you want a more detailed (but also more technical) proof, you can have a look at the notes by Jean-Louis Arcand on p. 47 to 52.

For the same reason as in the forbidden regression this seemingly obvious two-step procedure of mimicking 2SLS with probit will not produce consistent estimates. This is again because expectations and linear projections do not carry over through nonlinear functions. Wooldridge (2010) in section 15.7.3 on page 594 provides a detailed explanation for this. He also explains the proper procedure of estimating probit models with a binary endogenous variable. The correct approach is to use maximum likelihood but doing this by hand is not exactly trivial. Therefore it is preferable if you have access to some statistical software which has a ready-canned package for this. For example, the Stata command would be ivprobit (see the Stata manual for this command which also explains the maximum likelihood approach).

If you require references for the theory behind probit with instrumental variables see for instance:

Newey, W. (1987) "Efficient estimation of limited dependent variable models with endogenous explanatory variables", Journal of Econometrics, Vol. 36, pp. 231-250
Rivers, D. and Vuong, Q.H. (1988) "Limited information estimators and exogeneity tests for simultaneous probit models", Journal of Econometrics, Vol. 39, pp. 347-366

Finally, combining different estimation methods in the first and second stages is difficult unless there exists a theoretical foundation which justifies their use. This is not to say that it is not feasible though. For instance, Adams et al. (2009) use a three-step procedure where they have a probit "first stage" and an OLS second stage without falling for the forbidden regression problem. Their general approach is:

use probit to regress the endogenous variable on the instrument(s) and control variables
use the predicted values from the previous step in an OLS first stage together with the control (but without the instrumental) variables
do the second stage as usual

A similar procedure was employed by a user on the Statalist who wanted to use a Tobit first-stage and a Poisson second stage (see here). The same fix should be feasible for your estimation problem.

2SLS – How to Use 2SLS with Second Stage Probit?

Your case is less problematic than the other way round. The expectations and linear projections operators go through a linear first stage (e.g. OLS) but not not through non-linear ones like probit or logit. Therefore it's not a problem if you first regress your continous endogenous variable $X$ on your instrument(s) $Z$, $$X_i = a + Z'_i\pi + \eta_i$$ and then use the fitted values in a probit second stage to estimate $$\text{Pr}(Y_i=1|\widehat{X}_i) = \text{Pr}(\beta\widehat{X}_i + \epsilon_i > 0)$$

The standard errors won't be right because $\widehat{X}_i$ is not a random variable but an estimated quantity. You can correct this by bootstrapping both first and second stage together. In Stata this would be something like

// use a toy data set as example
webuse nlswork

// set up the program including 1st and 2nd stage
program my2sls
    reg grade age race tenure
    predict grade_hat, xb

    probit union grade_hat age race
    drop grade_hat
end

// obtain bootstrapped standard errors
bootstrap, reps(100): my2sls

In this example we want to estimate the effect of years of education on the probability of being in a labor union. Given that years of education are likely to be endogenous, we instrument it with years of tenure in the first stage. Of course, this doesn't make any sense from the point of interpretation but it illustrates the code.

Just make sure that you use the same exogenous control variables in both first and second stage. In the above example those are age, race whereas the (non-sensical) instrument tenure is only there in the first stage.

Best Answer

Related Solutions

Probit Two-Stage Least Squares (2SLS) – Overview and Application

2SLS – How to Use 2SLS with Second Stage Probit?

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