In this article, we will follow Woolridge (2002) procedure to estimate a set of equations with nonlinear functional forms for panel data using the two-stage least squares estimator. It has to be mentioned that this topic is quite uncommon and not used a lot in applied econometrics, this is due that instrumenting the nonlinear terms might be somewhat complicated.
Assume a two-equation system of the form:
Where the y’s represents the endogenous variables, Z represents the exogenous variables taken as instruments and u are the residuals for each equation. Notice that y2 is in a quadratic form in the first equation but also present in linear terms on the second equation.
Woolridge calls this model as nonlinear in endogenous variable, yet the model still linear in the parameters γ making this a particular problem where we need to somehow instrument the quadratic term of y2.
Finding the instruments for the quadratic term is a particular challenge than already it is for linear terms in simple instrumental variable regression. He suggests the following:
“A general approach is to always use some squares and cross products of the exogenous variables appearing somewhere in the system. If something like exper2 appears in the system, additional terms such as exper3 and exper4 would be added to the instrument list.” (Wooldridge, 2002, p. 235).
Therefore, it worth the try to use nonlinear terms of the exogenous variables from Z, in the form of possible Z2 or even Z3. And use these instruments to deal with the endogeneity of the quadratic term y2. When we define our set of instruments, then any nonlinear equation can be estimated with two-stage least squares. And as always, we should check the overidentifying restrictions to make sure we manage to avoid inconsistent estimates.
The process with an example.
Let’s work with the Example of a nonlinear labor supply function. Which is a system of the form:
Some brief description of the model indicates that for the first equation, the hours (worked) are a nonlinear function of the wage, the level of education (educ), the age (age), the kids situation associated to the age, whether if they’re younger than 6 years old or between 6 and 18 (kidslt6 and kidsge6), and the wife’s income (nwifeinc).
On the second equation, the wage is a function of the education (educ), and a nonlinear function of the exogenous variable experience (exper and exper2).
We work on the natural assumptions that E(u|z)=0 therefore the instruments are exogenous. Z in this case contains all the other variables which are not endogenous (hours and wage are the endogenous variables).
We will instrument the quadratic term of the logarithm of the wage in the first equation, and for such instrumenting process we will add three new quadratic terms, which are:
And we include those in the first-stage regression.
With Stata we first load the dataset which can be found here.
https://drive.google.com/file/d/1m4bCzsWgU9sTi7jxe1lfMqM2T4-A3BGW/view?usp=sharing
Load up the data (double click the file with Stata open or use some path command to get it ready)
use MROZ.dta
Generate the squared term for the logarithm of the wage with:
gen lwage_sq=lwage *lwage
Then, get ready to use the following command with ivregress, however, we will explain it in detail.
ivregress 2sls hours educ age kidslt6 kidsge6 nwifeinc (lwage lwage_sq = educ c.educ#c.educ exper expersq age c.age#c.age kidsge6 kidslt6 nwifeinc c.nwifeinc#c.nwifeinc), first
Which has the following interpretation. According to the syntaxis of Stata’s program. First, make sure you specify the first equation with the associated exogenous variables, we do that with the part.
ivregress 2sls hours educ age kidslt6 kidsge6 nwifeinc
Now, let’s tell to Stata that we have two other endogenous regressors, which are the wage and the squared term of the wages. We open the bracket and put
(lwage lwage_sq =
This will tell to Stata that lwage and lwage_sq are endogenous, part of the first equation of hours, and after the equal, we specify ALL the exogenous variables including the instruments for the endogenous terms, this will lead to include the second part as:
(lwage lwage_sq = educ c.educ#c.educ exper expersq age c.age#c.age kidsge6 kidslt6 nwifeinc c.nwifeinc#c.nwifeinc)
Notice that this second part will have a c.var#c.var structure, this is Stata’s operator to indicate a multiplication for continuous variables, (and we induce the quadratic terms without generating the variables with another command like we did with the wage).
So notice we have c.educ#c.educ which is the square of the educ variable, and c.age#c.age which is the square of the age, and we also square the wife’s income with c.nwifeinc#c.nwifeinc. These are the instruments for the quadratic term.
The fact that we have two variables on the left (lwage and lwage_sq) indicates that the set of instruments will hold first for an equation for lwage and second for lwage_sq given the exact same instruments.
We include the option , first to see what were the regressions in the first stage.
ivregress 2sls hours educ age kidslt6 kidsge6 nwifeinc (lwage lwage_sq = educ c.educ#c.educ exper expersq age c.age#c.age kidsge6 kidslt6 nwifeinc c.nwifeinc#c.nwifeinc), first
The output of the above model for the first stage equations is:
And the output for the two stage equation is:
Which yields in the identical coefficients in Woolridge’s book (2002, p- 236) also with some slightly difference in the standard errors (yet these slight differences do not change the interpretation of the statistical significance of the estimators).
In this way, we instrumented both endogenous regressors lwage and lwage_sq. Which are a nonlinear relationship in the model.
As we can see, the quadratic term is not statistically significant to explain the hours worked.
At last, we need to make sure that overidentification restrictions are valid. So we use after the regression
estat overid
And within this result, we cannot reject the null that overidentifying restrictions are valid.
Bibliography
Wooldridge, J. M. (2002). Econometric Analysis of Cross Section and Panel Data. Cam-bridge, MA: MIT Press.
