Nonlinear Panel Data Models Lecture

Nonlinear Panel Data Models Lecture - Imbens/Wooldridge...

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Imbens/Wooldridge, Lecture Notes 4, Summer ’07 What s New in Econometrics ? NBER , Summer 2007 Lecture 4 , Monday , July 30th , 3 . 15 - 4 . 15 pm Nonlinear Panel Data Models These notes summarize some recent, and perhaps not-so-recent, advances in the estimation of nonlinear panel data models. Research in the last 10 to 15 years has branched off in two directions. In one, the focus has been on parameter estimation, possibly only up to a common scale factor, in semiparametric models with unobserved effects (that can be arbitrarily correlated with covariates.) Another branch has focused on estimating partial effects when restrictions are made on the distribution of heterogeneity conditional on the history of the covariates. These notes attempt to lay out the pros and cons of each approach, paying particular attention to the tradeoff in assumptions and the quantities that can be estimated. 1 . Basic Issues and Quantities of Interest Most microeconomic panel data sets are best characterized as having few time periods and (relatively) many cross section observations. Therefore, most of the discussion in these notes assumes T is fixed in the asymptotic analysis while N is increasing. We assume random sample in the cross section,  x it , y it : t 1,. .., T .Take y it to be a scalar for simplicity. If we are not concerned about traditional (contemporaneous) endogeneity, then we are typically interested in D y it | x it , c i (1.1) or some feature of this distribution, such as E y it | x it , c i , or a conditional median. In the case of a mean, how do we summarize the partial effects? Let m t x t , c be the mean function. If x tj is continuous, then j x t , c m t x t , c x tj , (1.2) or look at discrete changes. How do we account for unobserved c i ?Ifwewanttoestimate magnitudes of effects, we need to know enough about the distribution of c i so that we can insert meaningful values for c . For example, if c E c i , then we can compute the partial effect at the average (PEA) , j x t , c . (1.3) Of course, we need to estimate the function m t and the mean of c i . If we know more about the distribution of c i , we can insert different quantiles, for example, or a certain number of standard deviations from the mean. Alternatively, we can average the partial effects across the distribution of c i : 1
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Imbens/Wooldridge, Lecture Notes 4, Summer ’07 APE x t E c i j x t , c i  . (1.4) The difference between (1.3) and (1.4) can be nontrivial for nonlinear mean functions. The definition in (1.4) dates back at least to Chamberlain (1982), and is closely related to the notion of the average structural function (ASF) (Blundell and Powell (2003)). The ASF is defined as ASF x t E c i m t x t , c i  . (1.5) Assuming the derivative passes through the expectation results in (1.5), the average partial effect. Of course, computing discrete changes gives the same result always. APEs are directly across models, and APEs in general nonlinear models are comparable to the estimated coefficients in a standard linear model.
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Nonlinear Panel Data Models Lecture - Imbens/Wooldridge...

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