Control functions 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 data use special data properties to difference out unobserved heterogeneity that is assumed to be fixed over time.
Contents
Control functions were introduced by Heckman and Robb, although the principle can be traced back to earlier papers. 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. Famous examples using the control function approach is the Heckit model and the Heckman correction.
Formal definition
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 (1)X = π(Z) + V (2)
E[U ∨ Z,V] = E[U ∨ V] (3) E[V ∨ 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
E[Y ∨ X,V] = g(X) + E[U ∨ X,V] = g(X) + E[U ∨ Z,V] = g(X) + E[U ∨ 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.
In a 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
E[Y1 ∨ X,Z,D = 1] = μ1(X) + E[U ∨ D = 1] (6)as long as the potential outcomes Y0 and Y1 are independent of D conditional on X and Z.
Extensions
The original Heckit procedure makes distributional assumptions about the error terms, however, more flexible estimation approaches with weaker distributional assumptions have been established. Furthermore, Blundell and Powell show how the control function approach can be particularly helpful in models with nonadditive errors, such as discrete choice models. This latter approach, however, does implicitly make strong distributional and functional form assumptions.