MIT14_382S17_lec8_22.pdf - Victor Chernozhukov and Ivan Fernandez-Val 14.382 Econometrics Spring 2017 Massachusetts Institute of Technology MIT

MIT14_382S17_lec8_22.pdf - Victor Chernozhukov and Ivan...

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Victor Chernozhukov and Ivan Fernandez-Val. 14.382 Econometrics . Spring 2017. Massachusetts Institute of Technology: MIT OpenCourseWare, . License: Creative Commons BY-NC-SA . 14.382 L8. LINEAR PANEL DATA MODELS UNDER STRICT AND WEAK EXOGENEITY VICTOR CHERNOZHUKOV AND IV ´ ANDEZ-VAL AN FERN ´ Abstract. We discuss basic examples of linear panel data models and their estimation via the “fixed effects”, differencing, and correlated random effects approaches. 1. A Structural Linear Panel Model 1.1. The Setting. Here we consider the linear structural equations model (SEM) Y it it α + W f it γ + E it , (1.1) = a i + D f it β + E it =: a i + X f E it ( X it , a i ) , where i = 1 , ..., n and t = 1 , ..., T . Here Y it is the outcome for an observational unit i at “time” t , D it is a vector of variables of interest or treatments, whose predictive effect α we would like to estimate, W it is a vector of covariates or controls including a constant, X it simply stacks together D it and W it , and E it is an error term normalized to have zero mean for each unit. We shall assume that the vectors Z i := { ( Y it , X f it ) f } T t =1 , that collect all the variables for the observational unit i , are i.i.d. across i . We note that this assumption does allow for arbitrary dependence of data for unit i across t , subject to other conditions specified below. In our analysis the temporal dimension T will be small and the cross-sectional dimension n will be large. Accordingly, we shall derive formal asymptotic results under the “large n , fixed T asymptotics, were n → ∞ and T is fixed. This type of scenario is often called the “short panel”. The orthogonality condition stated will be strengthened below to various assumptions, which permit application of common estimation methods for performing inference on the target parameter α . The random variable a i is the unobserved individual effect . It can be correlated to X it , and so we can not omit it without introducing omitted variable bias that leads to 1
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´ ´ 2 VICTOR CHERNOZHUKOV AND IVAN FERNANDEZ-VAL inconsistent estimates of the parameter of interest α . We can give context to this point by thinking of the case where a i is the unobserved individual’s innate ability, Y it is wage, D it is education, and W it are other characteristics of a person i at time t . Clearly omission of a i from the model would lead to an omitted variable bias and inconsistent estimation of the target parameter α for the usual reasons that we discussed in L2. Figure 1 illustrates the omitted variable bias problem in the linear panel model. An important point to make here is the following. Suppose D it is randomly assigned conditional on a i and W it , then α estimates a causal parameter the average treatment effect. This is merely one of many suffi- cient conditions for causal interpretability of α . An example of another condition is the assumption of parallel trends underlying the difference-in-difference approach, as described below.
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  • Summer '19
  • Econometrics, Estimation theory, Panel data, eit, Generalized method of moments, vit ⊥ Xit

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