Switching regression model and Heckman procedure

[Marry Lee]

Hello!

I have a big confusion about the use of Switching regression model with endogenous switching and Heckman procedure.
Is a Switching regression model with endogenous switching always defined as:

• One selection equation
• Two outcome equations for each regime of the two possible regimes

This can be estimated using a Heckman procedure right?
So the Switching regression model with endogenous switching is the model and the Heckman procedure is the estimation method, right?

Because I found some papers describing the Heckman method (talking about Inverse Mills Ratios, etc) and saying this is the Switching regression model.

Can you help me please?

[Maxence Morlet]

In a Heckman model with endogenous switching, you have two equations: the outcome equation, generally estimated using OLS, and the selection equation, generally estimated using probit.

For instance, you will only observe someone's salary if they are employed (legally). Suppose you want to study the impact of health on wages (second step equation). However, people self-select into employment in an endogenous manner. Simply regressing wages on health using OLS will yield a substantial bias because of this endogenous selection.

What Heckman proposes is to add a selection equation (first step). In this example, this equation will try to predict labour market participation.

Kennedy (2006) says that you need at least one exclusion restriction, i.e. in this example a variable that will affect only affect wages through labour market participation. Classically, the number of children in a household has been used as an exclusion restriction.

We're unfortunately not out the woods yet, as this no longer suffices for causality. You still have to argue the exogeneity of your regressor of interest, in the above example health.

Hope this helps

[Jeff Wooldridge]

There are at least two uses of the Heckman approach to accounting for selection bias. Maxence is describing when it is used to account for unobservability of a variable. That can be applied essentially twice in the treatment effects case, where you observe some units in the treatment and some units in a control state. Then, yes, the Heckman approach can be applied separately to each of the groups separately. This still requires estimating a "treatment" equation (more generally, an equation that determines the regime). Then, one needs to properly compute the inverse Mills ratio terms in each case: for the control and for the treated. I discuss this in Chapter 21 of my MIT Press book and also in my 2015 Journal of Human Resources article on control functions. I can make the JHR Stata code available. Vella and Verbeek contain an earlier discussion in their 1999 Journal of Business and Economic Statistics article.

I often implement the method with a single pooled regression, including the so-called generalized residual in a flexible way.

[Marry Lee]

Thank you so much Jeff Wooldridge for your answer.
That's the use of Heckman that I am referring to.
I am interested in Lee (1978)'s method, where, he investigates the expected income differential on the fact of being a union member:
Because of self-selection,

• in the first step, he estimates a reduced form decision equation (be in a union or not)
• Predicts the inverse Mills ratios
• then introduce them in the income equations of each group/ regime
• then produces the predicted values of the income variables
• compute the differencial of the predicted income variables between the two regimes
• and introduce this differential in his structural decision equation (be in a union or not).

Part of this method includes Heckman technique (three first points).


On the other hand, Lokshin, M., & Sajaia, Z. (2004) say that "Models with endogenous switching can be fitted one equation at a time by either two-step least squares or maximum likelihood estimation. However, both of these estimation methods are inefficient and require potentially cumbersome adjustments to derive consistent standard errors. The movestay command, on the other hand, implements the full-information ML method (FIML) to simultaneously fit binary and continuous parts of the model in order to yield consistent standard errors"

Some papers even say: "The drawback of this two-step approach is that it generates residuals that are heteroskedastic and as a result cannot be used to obtain consistent standard errors without cumbersome adjustments (Lokshin and Sajaia 2004). The full information maximum likelihood method suggested by Lokshin and Sajaia (2004) overcomes the problem through a simultaneous estimation of the two equations"

1. First, I don't understand why Heckman technique will produce heteroskedacticity, is it because we are introducing the estimated version of inverse Mills ratios instead of their true values? wouldn't bootstrap correct for this?
2. why some papers applying the Lee (1978) method still use the same method from 1978 if it induces heteroskedastic residuals?

I would be very grateful if you can help me understand some of these issues and if you can give me the JHR Stata code.
Thank you!


References

Lee, L. F. (1978). Unionism and wage rates: A simultaneous equations model with qualitative and limited dependent variables. International economic review, 415-433.

Lokshin, M., & Sajaia, Z. (2004). Maximum likelihood estimation of endogenous switching regression models. The Stata Journal, 4(3), 282-289.

[Marry Lee]

Dear Jeff Wooldridge, this is a gentle reminder if you have forgotten to answer me.

[Jeff Wooldridge]

The heteroskedasticity introduced is not a big deal; it comes from conditioning on the regime. It's not due to estimating the inverse Mills ratio. But estimating the IMR does mean that something like bootstrapping is needed. It's true that the movestay command does joint MLE and so the standard errors don't need adjusting for the two-step procedure. But the bootstrap is not difficult to implement here. The claim by Lockshin and Sajaia is misleading: two-step estimation error is not the same as heteroskedasticity. If you send me an email I can send you my Stata code that bootstraps the two steps.

[Ahmed Tritah]

Dear Prof Wooldridge, thank you for your kind assistance; I will really much apprecciate if you could send me the (1) the stata code that bootstrap the two steps; and also the code you suggested earlier when saying:

"I discuss this in Chapter 21 of my MIT Press book and also in my 2015 Journal of Human Resources article on control functions. I can make the JHR Stata code available.". I think applying the generalized residual approach as suggested in a pooled regressions (i.e. the adopters and non adopters of a new technology in my case) is indeed more straightforward approach.
In an earlier post you said : "I often implement the method with a single pooled regression, including the so-called generalized residual in a flexible way".
I am not sure of what you mean by "flexible way"

In my case this means first estimating the generalised residuals :

probit y x
predict xb, xb
gen gen_residual = cond(foreign == 1,
normden(xb)/norm(xb), -normden(xb)/(1-norm(xb)))

Then regressing my outcome variable on controls, a dummy for (endogeneous) adoption and a polynomial of gen_residual (generalized residual). Am I right?
Again many thank for your help,
Best
Ahmed