Control function (econometrics)

Assume we start from a standard endogenous variable setup with additive errors, where X is an endogenous variable, and Z is an exogenous variable that can serve as an instrument.

${\displaystyle Y=g(X)+U}$

(1)

${\displaystyle X=\pi (Z)+V}$

(2)

${\displaystyle E[U\mid Z,V]=E[U\mid V]}$

(3)

${\displaystyle E[V\mid Z]=0}$

(4)

A popular instrumental variable approach is to use a two-step procedure and estimate equation (2) first and then use the estimates of this first step to estimate equation (1) in a second step. The control function, however, uses that this model implies

${\displaystyle E[Y\mid X,V]=g(X)+E[U\mid X,V]=g(X)+E[U\mid Z,V]=g(X)+E[U\mid V]=g(X)+h(V)}$

(5)

The function h(V) is effectively the control function that models the endogeneity and where this econometric approach lends its name from.[4]

In a Rubin causal model potential outcomes framework, where Y₁ is the outcome variable of people for who the participation indicator D equals 1, the control function approach leads to the following model

${\displaystyle E[Y_{1}\mid X,Z,D=1]=\mu _{1}(X)+E[U\mid D=1]}$

(6)

as long as the potential outcomes Y₀ and Y₁ are independent of D conditional on X and Z.[5]

Since the second-stage regression includes generated regressors, its variance-covariance matrix needs to be adjusted.[6][7]

The original Heckit procedure makes distributional assumptions about the error terms, however, more flexible estimation approaches with weaker distributional assumptions have been established.[8] Furthermore, Blundell and Powell show how the control function approach can be particularly helpful in models with nonadditive errors, such as discrete choice models.[9] This latter approach, however, does implicitly make strong distributional and functional form assumptions.[5]

• Two-stage least squares
• Heckman correction

• Guo, Zijian; Small, Dylan S. (2016). "Control Function Instrumental Variable Estimation of Nonlinear Causal Effect Models". Journal of Machine Learning Research. 17 (100): 1–35.
• Wooldridge, Jeffrey M. (2015). "Control Function Methods in Applied Econometrics". Journal of Human Resources. 50 (2): 420–445. doi:10.3368/jhr.50.2.420.