Control Function Methods Slides

Control Function Methods Slides - Whats New in Econometrics...

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What’s New in Econometrics? Lecture 6 Control Functions and Related Methods Jeff Wooldridge NBER Summer Institute, 2007 1. Linear-in-Parameters Models: IV versus Control Functions 2. Correlated Random Coefficient Models 3. Some Common Nonlinear Models and Limitations of the CF Approach 4. Semiparametric and Nonparametric Approaches 5. Methods for Panel Data 1

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1 . Linear - in - Parameters Models : IV versus Control Functions Most models that are linear in parameters are estimated using standard IV methods – two stage least squares (2SLS) or generalized method of moments (GMM). An alternative, the control function (CF) approach, relies on the same kinds of identification conditions. Let y 1 be the response variable, y 2 the endogenous explanatory variable (EEV), and z the 1 L vector of exogenous variables (with z 1 1 : y 1 z 1 1 1 y 2 u 1 , (1) where z 1 is a 1 L 1 strict subvector of z . First consider the exogeneity assumption 2
E z u 1 0 . (2) Reduced form for y 2 : y 2 z 2 v 2 ,E z v 2 0 (3) where 2 is L 1. Write the linear projection of u 1 on v 2 , in error form, as u 1 1 v 2 e 1 , (4) where 1 E v 2 u 1 /E v 2 2 is the population regression coefficient. By construction, E v 2 e 1 0 and E z e 1 0 . Plug (4) into (1): y 1 z 1 1 1 y 2 1 v 2 e 1 , (5) where we now view v 2 as an explanatory variable in the equation. By controlling for v 2 , the error e 1 is uncorrelated with y 2 as well as with v 2 and z . Two-step procedure: (i) Regress y 2 on z and 3

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obtain the reduced form residuals, v ̂ 2 ; (ii) Regress y 1 on z 1 , y 2 , and v ̂ 2 . (6) The implicit error in (6) is e i 1 1 z i ̂ 2 2 , which depends on the sampling error in ̂ 2 unless 1 0. OLS estimators from (6) will be consistent for 1 , 1 , and 1 . Simple test for null of exogeneity is (heteroskedasticity-robust) t statistic on v ̂ 2 . The OLS estimates from (6) are control function estimates. The OLS estimates of 1 and 1 from (6) are identical to the 2SLS estimates starting from (1). Now extend the model: y 1 z 1 1 1 y 2 1 y 2 2 u 1 E u 1 | z 0. (7) (8) 4
z 2 be a scaler not also in z 1 . Under the (8) – which is stronger than (2), and is essential for nonlinear models – we can use, say, z 2 2 as an instrument for y 2 2 . So the IVs would be z 1 , z 2 , z 2 2 for z 1 , y 2 , y 2 2 . What does CF approach entail? We require an assumption about E u 1 | z , y 2 ,say E u 1 | z , y 2 E u 1 | v 2 1 v 2 , (9) where the first equality would hold if u 1 , v 2 is independent of z – a nontrivial restriction on the reduced form error in (3), not to mention the structural error u 1 . Linearity of E u 1 | v 2 is a substantive restriction. Now,

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Control Function Methods Slides - Whats New in Econometrics...

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