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Control function (econometrics)
Controls functions are approaches to dealing with endogeneity that differ in important ways from other models that try to account for this 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 data use special data properties to difference out unobserved heterogeneity that is assumed to be fixed over time. Control functions take another approach and try to model the endogeneity in the error term. They were introduced by Heckman and Robb〔 Heckman, J. J., and R. Robb (1985): Alternative Methods for Evaluating the Impact of Interventions. In Longitudinal Analaysi of Labor Market Data., ed. by J. Heckman and B. Singer. CUP. 〕, although the principle can be traced back to earlier papers. 〔 Telser, L. G. (1964): Iterative Estimation of a Set of Linear Regression Equations. Journal of the American Statistical Association, 59, pp. 845-862 〕 A particular reason why they are popular is because they work for non-invertible models (such as discrete choice models) and allow for heterogeneous effects, where effects at the individual level can differ from effects at the aggregate. 〔Arellano, M. (2008): Binary Models with Endogenous Explanatory Variables. Class notes: http://www.cemfi.es/~arellano/binary-endogeneity.pdf ,〕 Famous examples using the control function approach is the Heckit model and the Heckman correction.
Assume we start from a standard endogenous variable set-up with additive errors, where ''X'' is an endogenous variable, ''Z'' is an exogenous variable that can serve as an instrument.
''Y = g(X) + U''
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