'''Control functions''' are statistical methods to correct for [[endogeneity (econometrics)endogeneity]] problems by modelling the endogeneity in the [[errors and residualserror term]]. The approach thereby differs in important ways from other models that try to account for the same [[econometric]] problem. [[Instrumental variable]]s, for example, attempt to model the endogenous variable ''X'' as an often [[invertible]] model with respect to a relevant and [[exogenous]] instrument ''Z''. [[Panel data]] use special data properties to difference out unobserved heterogeneity that is assumed to be fixed over time.
Control functions were introduced by [[James HeckmanHeckman]] and [[Rafael RobbRobb]],<ref>{{cite book last=Heckman, first=J. J., and first2=R. last2=Robb (year=1985): chapter=Alternative Methods for Evaluating the Impact of Interventions. In title=Longitudinal AnalaysiAnalysis of Labor Market Data., ed. by editorfirst=J. editorlast=Heckman and editor2first=B. editor2last=Singer. publisher=CUP. isbn= }}</ref> although the principle can be traced back to earlier papers.<ref>{{cite journal last=Telser, first=L. G. (authorlink=Lester G. Telser year=1964): title=Iterative Estimation of a Set of Linear Regression Equations. journal=[[Journal of the American Statistical Association,]] volume=59, pp.issue=307 pages=845–862 doi=10.1080/01621459.1964.10480731 }}</ref> A particular reason why they are popular is because they work for noninvertible models (such as [[discrete choice model]]s) and allow for [[heterogeneous]] effects, where effects at the individual level can differ from effects at the aggregate.<ref>{{cite web last=Arellano, first=M. (authorlink=Manuel Arellano date=2008): title=Binary Models with Endogenous Explanatory Variables. work=Class notes: httpurl=https://www.cemfi.es/~arellano/binaryendogeneity.pdf ,}}</ref> Famous examples using the control function approach is the [[Heckit]] model and the [[Heckman correction]].
==Formal definition==
==Extensions==
The original Heckit procedure makes [[distributional assumption]]s about the error terms, however, more flexible estimation approaches with weaker distributional assumptions have been established.<ref>{{cite journal last=Matzkin, first=R. L. (year=2003): title=Nonparametric Estimation of Nonadditive Random Functions. journal=[[Econometrica,]] volume=71( issue=5), pppages=1339–1375 doi=10.1111/14680262.00452 1339–1375}}</ref> Furthermore, Blundell and Powell show how the control function approach can be particularly helpful in models with [[nonadditive error]]s, such as discrete choice models.<ref>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.</ref> This latter approach, however, does implicitly make strong distributional and [[functional form]] assumptions.<ref>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.</ref>
==References==
