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

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

(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]

Endogeneity in Poisson regression edit

Wooldridge and Terza provide a methodology to both deal with and test for endogeneity within the exponential regression framework, which the following discussion follows closely.^[8] While the example focuses on a Poisson regression model, it is possible to generalize to other exponential regression models, although this may come at the cost of additional assumptions (e.g. for binary response or censored data models).

Assume the following exponential regression model, where ${\displaystyle a_{i}}$  is an unobserved term in the latent variable. We allow for correlation between ${\displaystyle a_{i}}$  and ${\displaystyle x_{i}}$  (implying ${\displaystyle x_{i}}$  is possibly endogenous), but allow for no such correlation between ${\displaystyle a_{i}}$  and ${\displaystyle z_{i}}$ .

${\displaystyle \operatorname {E} [y_{i}\mid x_{i},z_{i},a_{i}]=\exp(x_{i}b_{0}+z_{i}c_{0}+a_{i})}$ 

The variables ${\displaystyle z_{i}}$  serve as instrumental variables for the potentially endogenous ${\displaystyle x_{i}}$ . One can assume a linear relationship between these two variables or alternatively project the endogenous variable ${\displaystyle x_{i}}$  onto the instruments to get the following reduced form equation:

${\displaystyle x_{i}=z_{i}\Pi +v_{i}}$

(1)

The usual rank condition is needed to ensure identification. The endogeneity is then modeled in the following way, where ${\displaystyle \rho }$  determines the severity of endogeneity and ${\displaystyle v_{i}}$  is assumed to be independent of ${\displaystyle e_{i}}$ .

${\displaystyle a_{i}=v_{i}\rho +e_{i}}$ 

Imposing these assumptions, assuming the models are correctly specified, and normalizing ${\displaystyle \operatorname {E} [\exp(e_{i})]=1}$ , we can rewrite the conditional mean as follows:

${\displaystyle \operatorname {E} [y_{i}\mid x_{i},z_{i},v_{i}]=\exp(x_{i}b_{0}+z_{i}c_{0}+v_{i}\rho )}$

(2)

If ${\displaystyle v_{i}}$  were known at this point, it would be possible to estimate the relevant parameters by quasi-maximum likelihood estimation (QMLE). Following the two step procedure strategies, Wooldridge and Terza propose estimating equation (1) by ordinary least squares. The fitted residuals from this regression can then be plugged into the estimating equation (2) and QMLE methods will lead to consistent estimators of the parameters of interest. Significance tests on ${\displaystyle {\hat {\rho }}}$  can then be used to test for endogeneity within the model.

The original Heckit procedure makes distributional assumptions about the error terms, however, more flexible estimation approaches with weaker distributional assumptions have been established.^[9] Furthermore, Blundell and Powell show how the control function approach can be particularly helpful in models with nonadditive errors, such as discrete choice models.^[10] This latter approach, however, does implicitly make strong distributional and functional form assumptions.^[5]

• Two-stage least squares – Technique in statisticsPages displaying short descriptions of redirect targets
• Heckman correction – Statistical technique correcting sampling bias

• 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. arXiv:1602.01051.
• 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. S2CID 119604644.