A Course in Applied Econometrics 1 . Basic Issues and Quantities of Interest Lecture 6 : Nonlinear Panel Data Models � Let �� x it , y it � : t � 1,..., T � be a random draw from the cross section. Jeff Wooldridge IRP Lectures, UW Madison, August 2008 Typically interested in D � y it | x it , c i � (1) 1. Basic Issues and Quantities of Interest or some feature of this distribution, such as E � y it | x it , c i � , or a 2. Exogeneity Assumptions conditional median. 3. Conditional Independence � In the case of a mean, how do we summarize the partial effects? If x tj 4. Assumptions about the Unobserved Heterogeneity is continuous, then 5. Nonparametric Identification of Average Partial Effects � j � x t , c � � � m t � x t , c � , (2) � x tj 6. Dynamic Models 7. Applications to Specific Models or discrete changes. 1 2 � How do we account for unobserved c i ? Maybe we can insert � The difference between (3) and (4) can be nontrivial. In some leading meaningful values for c . For example, if � c � E � c i � , then we can cases, (4) is identified while (3) is not. � (4) is closely related to the notion of the average structural function compute the partial effect at the average (PEA) , (ASF) (Blundell and Powell (2003)). The ASF is defined as PEA j � x t � � � j � x t , � c � . (3) ASF � x t � � E c i � m t � x t , c i �� . (5) Of course, we need to estimate the function m t and � c . We might be � Passing the derivative through the expectation in (5) gives the APE. able to insert different quantiles, or a certain number of standard deviations from the mean. � Alternatively, we can average the partial effects across the distribution of c i : APE � x t � � E c i � � j � x t , c i �� . (4) 3 4
� How do APEs relate to parameters? Suppose � Altonji and Matzkin (2005) define the local average response (LAR) as opposed to the APE or PAE. The LAR at x t for a continuous variable m t � x t , c � � G � x t � � c � , (6) x tj is where, say, G ��� is strictly increasing and continuously differentiable. LAR j � x t � � � � m t � x t , c � d H t � c | x t � , Then (8) � x tj � j � x t , c � � � j g � x t � � c � , (7) where H t � c | x t � denotes the cdf of D � c i | x it � x t � . “Local” because it where g ��� is the derivative of G ��� . Then estimating � j means we can averages out the heterogeneity for the slice of the population described sign of the partial effect, and the relative effects of any two continuous by the vector x t . The APE is a “global average response.” variables. Even if G ��� is specified, the magnitude of effects cannot be � Definitions of partial effects do not depend on whether x t is estimated without making assumptions about the distribution of c i . correlated with c . Of course, whether and how we estimate them certainly does. 5 6 � The sequential exogeneity assumption is 2 . Exogeneity Assumptions � As in linear case, cannot get by with just specifying a model for D � y it | x i 1 ,..., x it , c i � � D � y it | x it , c i � . (11) D � y it | x it , c i � . Unfortunately, it is much more difficult to allow sequential exogeneity � The most useful definition of strict exogeneity for nonlinear panel in in nonlinear models. data models is � Neither (9) nor (10) allows for contemporaneous endogeneity of one D � y it | x i 1 ,..., x iT , c i � � D � y it | x it , c i � . (9) or more elements of x it , where, say, x itj is correlated with unobserved, Chamberlain (1984) labeled (9) strict exogeneity conditional on the time-varying unobservables that affect y it . (Later in control function unobserved effects c i . Conditional mean version: estimation.) E � y it | x i 1 ,..., x iT , c i � � E � y it | x it , c i � . (10) 7 8
� In a parametric context, the CI assumption therefore reduces our task 3 . Conditional Independence � In linear models, serial dependence of idiosyncratic shocks is easily to specifying a model for D � y it | x it , c i � , and then determining how to dealt with, either by robust inference or GLS extensions of FE and FD. treat the unobserved heterogeneity, c i . � In random effects and correlated random effects frameworks, CI plays With strictly exogenous covariates, never results in biased estimation, even if it is ignored or improperly model. The situation is different with a critical role in being able to estimate the “structural” parameters and nonlinear models estimated by MLE. the parameters in distribution the of c i (and therefore, PAEs). In a � The conditional independence assumption is broad class of models, CI plays no role in estimating APEs. T D � y i 1 ,..., y iT | x i , c i � � � D � y it | x it , c i � (12) t � 1 (where we also impose strict exogeneity). 9 10 4 . Assumptions about the Unobserved Heterogeneity Correlated Random Effects Random Effects A CRE framework allows dependence between c i and x i , but restricted in some way. In a parametric setting, we specify a distribution for D � c i | x i 1 ,..., x iT � � D � c i � . (13) D � c i | x i 1 ,..., x iT � , as in Chamberlain (1980,1982), and much work Under (13), the APEs are nonparametrically identified from since. Can allow D � c i | x i 1 ,..., x iT � to depend in a “nonexchangeable” r t � x t � � E � y it | x it � x t � . (14) manner. (Chamberlain’s CRE probit and Tobit models.) Distributional � In some leading cases (RE probit and RE Tobit), if we want PEs for assumptions that lead to simple estimation – homoskedastic normal different values of c , we must assume more: strict exogeneity, with a linear conditional mean — are restrictive. � Possible to drop parametric assumptions with conditional independence, and (13) with a parametric distribution for D � c i � . D � c i | x i � � D � c i | x � i � , (15) without restricting D � c i | x � i � . 11 12
� As T gets larger, can allow c i to be correlated with features of the Fixed Effects covariates other than just the time average. Altonji and Matzkin (2005) The label “fixed effects” is used in different ways by different researchers. One view: c i , i � 1,..., N are parameters to be estimated. � i in equation (15) to be replaced by other functions of allow for x � x it : t � 1,..., T � , such as sample variances and covariance. Usually leads to an “incidental parameters problem” (which attentuates with large T � . Non-exchangeable functions, such as unit-specific trends, can be used, � A second meaning of “fixed effects” is that D � c i | x i � is unrestricted too. Generally, assume and we look for objective functions that do not depend on c i but still D � c i | x i � � D � c i | w i � . (16) identify the population parameters. Leads to “conditional maximum Practically, we need to specify w i and then establish that there is likelihood” if we can find a “sufficient statistic” such that enough variation in � x it : t � 1,..., T � separate from w i . D � y i 1 ,..., y it | x i , c i , s i � � D � y i 1 ,..., y it | x i , s i � . (17) � The CI assumption is usually maintained. 13 14 � Panel data example due to Hahn (2001): x it is a binary indicator and 5 . Nonparametric Identification of Average Partial Effects � Identification of PAEs can fail even under a strong set of parametric P � y it � 1| x i , c i � � � � � x it � c i � , t � 1,2. (20) assumptions. In the probit model � is not known to be identified in this model, even under conditional P � y � 1| x , c � � � � x � � c � , (18) independence and the random effects assumption D � c i | x i � � D � c i � . But the PE for a continuous variable x j is � j � � x � � c � . The PAE at the APE is � � E � � � � � c i �� � E � � � c i �� and is identified by a � c � E � c � � 0 is � j � � x � � . Suppose c | x ~Normal � 0, � c 2 � . Then difference of means for the treated and untreated groups, for either time period. P � y � 1| x � � � � x � / � 1 � � c 2 � 1/2 � , (19) � As shown in Wooldridge (2005a), identification of the APE holds if 2 � 1/2 is identified; � so only the scaled parameter vector � c � � / � 1 � � c we replace � with an unknown function G and allow and � j � � x � � are not identified. D � c i | x i � � D � c i | x � i � . � The APE is identified from P � y � 1| x � , and is given by � cj � � x � c � . 15 16
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