Missing Data Lecture

Missing Data Lecture - Imbens/Wooldridge, Lecture Notes 12,...

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Imbens/Wooldridge, Lecture Notes 12, Summer ’07 What s New in Econometrics ? NBER , Summer 2007 Lecture 12 , Wednesday , August 1st , 11 - 12 . 30 pm Missing Data These notes discuss various aspects of missing data in both pure cross section and panel data settings. We begin by reviewing assumptions under which missing data can be ignored without biasing estimation or inference. Naturally, these assumptions are tied to “exogenous” sampling. We then consider three popular solutions to missing data: inverse probability weighting, imputation, and Heckman-type selection corrections. The first to methods maintain “missing at random” or “ignorability” or “selection on observables” assumptions. Heckman corrections, whether applied to cross section data or panel data, linear models or (certain) nonlinear models, allow for “selection on unobservables.” Unfortunately, their scope of application is limited. An important finding is that all methods can cause more harm than good if selection is on conditioning variables that are unobserved along with response variables. 1 . When Can Missing Data be Ignored ? It is easy to obtain conditions under which we can ignore the fact that certain variables for some observations, or all variables for some observations, have missing values. Start with a linear model with possibly endogenous explanatory variables: y i x i u i , (1.1) where x i is 1 K and the instruments z i are 1 L , L K . We model missing data with a selection indicator, drawn with each i . The binary variable s i is defined as s i 1 if we can use observation i , s i 0 if we cannot (or do not) use observation i .Inthe L K caseweuseIVon the selected sample, which we can write as ̂ IV N 1 i 1 N s i z i x i 1 N 1 i 1 N s i z i y i N 1 i 1 N s i z i x i 1 N 1 i 1 N s i z i u i (1.2) (1.3) For consistency, we essentially need rank E s i z i x i K (1.4) and E s i z i u i 0, (1.5) 1
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Imbens/Wooldridge, Lecture Notes 12, Summer ’07 which holds if E z i u i | s i 0,whichinturnisimpliedby E u i | z i , s i 0. (1.6) Sufficient for (1.6) is E u i | z i 0, s i h z i (1.7) for some function h  . Note that the zero covariance assumption, E z i u i 0, is not sufficient for consistency when s i h z i . A special case is when E y i | x i x i and selection s i is a function of x i . Provided the selected sample has sufficient variation in x , can consistently estimate by OLS on the selected sample. We can use similar conditions for nonlinear models. What is sufficient for consistency on
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Missing Data Lecture - Imbens/Wooldridge, Lecture Notes 12,...

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