Control functions (also known as twostage residual inclusion) are statistical methods to correct for endogeneity problems by modelling the endogeneity in the error term. The approach thereby differs in important ways from other models that try to account for the same econometric problem. Instrumental variables, for example, attempt to model the endogenous variable X as an often invertible model with respect to a relevant and exogenous instrument Z. Panel analysis uses special data properties to difference out unobserved heterogeneity that is assumed to be fixed over time.
Control functions were introduced by Heckman and Robb^{[1]} although the principle can be traced back to earlier papers.^{[2]} A particular reason why they are popular is because they work for noninvertible models (such as discrete choice models) and allow for heterogeneous effects, where effects at the individual level can differ from effects at the aggregate.^{[3]} A wellknown example of the control function approach is the Heckman correction.
Formal definition
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.

(1)

(2)

(3)

(4)
A popular instrumental variable approach is to use a twostep 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

(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 Y_{1} is the outcome variable of people for who the participation indicator D equals 1, the control function approach leads to the following model

(6)
as long as the potential outcomes Y_{0} and Y_{1} are independent of D conditional on X and Z.^{[5]}
Variance correction
Since the secondstage regression includes generated regressors, its variancecovariance matrix needs to be adjusted.^{[6]}^{[7]}
Extensions
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]}
See also
References
 ^ Heckman, James J.; Robb, Richard (1985). "Alternative methods for evaluating the impact of interventions". Journal of Econometrics. Elsevier BV. 30 (1–2): 239–267. doi:10.1016/03044076(85)901393. ISSN 03044076.
 ^ Telser, L. G. (1964). "Iterative Estimation of a Set of Linear Regression Equations". Journal of the American Statistical Association. 59 (307): 845–862. doi:10.1080/01621459.1964.10480731.
 ^ Arellano, M. (2008). "Binary Models with Endogenous Explanatory Variables" (PDF). Class notes.
 ^ Arellano, M. (2003): Endogeneity and Instruments in Nonparametric Models. Comments to papers by Darolles, Florens & Renault; and Blundell & Powell. Advances in Economics and Econometrics, Theory and Applications, Eight World Congress. Volume II, ed. by M. Dewatripont, L.P. Hansen, and S.J. Turnovsky. Cambridge University Press, Cambridge.
 ^ ^{a} ^{b} Heckman, J. J., and E. J. Vytlacil (2007): Econometric Evaluation of Social Programs, Part II: Using the Marginal Treatment Effect to Organize Alternative Econometric Estimators to Evaluate Social Programs, and to Forecast the Effects in New Environments. Handbook of Econometrics, Vol 6, ed. by J. J. Heckman and E. E. Leamer. North Holland.
 ^ Murphy, Kevin M.; Topel, Robert H. (1985). "Estimation and Inference in TwoStep Econometric Models". Journal of Business & Economic Statistics. 3 (4): 370–379. JSTOR 1391724.
 ^ Gauger, Jean (1989). "The Generated Regressor Correction: Impacts Upon Inferences in Hypothesis Testing". Journal of Macroeconomics. 11 (3): 383–395. doi:10.1016/01640704(89)900657.
 ^ Matzkin, R. L. (2003). "Nonparametric Estimation of Nonadditive Random Functions". Econometrica. 71 (5): 1339–1375. doi:10.1111/14680262.00452. hdl:10908/409.
 ^ Blundell, R., and J. L. Powell (2003): Endogeneity in Nonparametric and Semiparametric Regression Models. Advances in Economics and Econometrics, Theory and Applications, Eight World Congress. Volume II, ed. by M. Dewatripont, L.P. Hansen, and S.J. Turnovsky. Cambridge University Press, Cambridge.
Further reading
 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.