Dynamic unobserved effects model: Difference between revisions

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The “dynamic” here means the dependence of the dependent variable on its past history, this is usually used to model the “state dependence” in economics. For instance, a person who cannot find a job this year, it will be hard for her to find a job next year because the fact that she doesn’t have a job this year will be a very negative signal for the potential employers. The “unobserved effects” means that one or some of the explanatory variables are unobservable. For example, one’s preference affects quite a lot her consumption choice of the ice cream with a certain taste, but preference is unobservable. A typical dynamic unobserved effects model is represented <ref> Wooldridge, J. (2002): Econometric Analysis of Cross Section and Panel Data, MIT Press, Cambridge, Mass, pp 495.</ref> as:

The “dynamic” here means the dependence of the dependent variable on its past history, this is usually used to model the “state dependence” in economics. For instance, a person who cannot find a job this year, it will be hard for her to find a job next year because the fact that she doesn’t have a job this year will be a very negative signal for the potential employers. The “unobserved effects” means that one or some of the explanatory variables are unobservable. For example, one’s preference affects quite a lot her consumption choice of the ice cream with a certain taste, but preference is unobservable. A typical dynamic unobserved effects model is represented <ref> Wooldridge, J. (2002): Econometric Analysis of Cross Section and Panel Data, MIT Press, Cambridge, Mass, pp 495.</ref> as:

P(y_it = 1│y_i,t-1, … , y_i,0 , z_i , c_i ) = G (z_it δ + ρ y_i,t-1 + c_i)

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where c_i is a unobservable explanatory variable, z_it is explanatory variables which are exogenous conditional on the c_i, and G(∙) is a [[cumulative distribution function]].

In this type of model, economists have a special interest in ρ, which is used to characterize the state dependence. For example, ''y_i,t'' can be a woman’s choice whether work or not, ''z_it'' includes the ''i''-th individual’s age, education level, numbers of kids and so on. ''c_i'' can be some individual specific characteristic which cannot be observed by economists .<ref>James J. Heckman (1981): Studies in Labor Markets, University of Chicago Press, Chapter Heterogeneity and State Dependence</ref>. It is a reasonable conjecture that one’s labor choice in period ''t'' should depend on his or her choice in period ''t'' - 1 due to habit formation or other reasons. This is dependence is characterized by parameter ''ρ''.

There are several [[MLE]]-based approaches to estimate ''δ'' and ''ρ'' consistently. The simplest way is to treat ''y_i,0'' as non-stochastic and assume ''c_i'' is [[independent]]{{dndisambiguation needed|date=December 2015}} with ''z_i''. Then integrate ''P(y_i,t , y_i,t-1 , … , y_i,1 | y_i,0 , z_i , c_i)'' against the density of ''c_i'', we can obtain the conditional density P(y_i,t , y_i,t-1 , … , y_i,1 |y_i,0 , z_i). The objective function for the conditional [[MLE]] can be represented as: ''<math> \sum_{i=1}^N </math> log (P (y_i,t , y_i,t-1, … , y_i,1 | y_i,0 , z_i)).''

Treating ''y_i,0'' as non-stochastic implicitly assumes the independence of ''y_i,0'' on ''z_i''. But in most of the cases in reality, ''y_i,0'' depends on ''c_i'' and ''c_i'' also depends on ''z_i''. An improvement on the approach above is to assume a density of ''y_i,0'' conditional on (''c_i, z_i'') and conditional likelihood ''P(y_i,t) , y_i,t-1 , … , y_t,1,y_i,0 | c_i, z_i)'' can be obtained. Integrate this likelihood against the density of ''c_i'' conditional on ''z_i'' and we can obtain the conditional density ''P(y_i,t , y_i,t-1 , … , y_i,1 , y_i,0 | z_i)''. The objective function for the [[conditional MLE]] <ref> Greene, W. H. (2003), Econometric Analysis , Prentice Hall , Upper Saddle River, NJ .</ref> is ''<math> \sum_{i=1}^N </math> log (P (y_i,t , y_i,t-1, … , y_i,1 | y_i,0 , z_i)).''

Based on the estimates for (''δ, ρ'') and the corresponding variance, test about the coefficients can be implemented <ref> Whitney K. Newey, Daniel McFadden, Chapter 36 Large sample estimation and hypothesis testing, In: Robert F. Engle and Daniel L. McFadden, Editor(s), Handbook of Econometrics, Elsevier, 1994, Volume 4, Pages 2111-2245, ISSN 1573-4412, ISBN 9780444887665,</ref> and the average partial effect can be calculated. <ref>Chamberlain, G. (1980), “Analysis of Covariance with Qualitative Data,” Journal of Econometrics 18, 5-46</ref>

Treating ''y_i,0'' as non-stochastic implicitly assumes the independence of ''y_i,0'' on ''z_i''. But in most of the cases in reality, ''y_i,0'' depends on ''c_i'' and ''c_i'' also depends on ''z_i''. An improvement on the approach above is to assume a density of ''y_i,0'' conditional on (''c_i, z_i'') and conditional likelihood ''P(y_i,t) , y_i,t-1 , … , y_t,1,y_i,0 | c_i, z_i)'' can be obtained. Integrate this likelihood against the density of ''c_i'' conditional on ''z_i'' and we can obtain the conditional density ''P(y_i,t , y_i,t-1 , … , y_i,1 , y_i,0 | z_i)''. The objective function for the [[conditional MLE]] <ref> Greene, W. H. (2003), Econometric Analysis , Prentice Hall , Upper Saddle River, NJ .</ref> is ''<math> \sum_{i=1}^N </math> log (P (y_i,t , y_i,t-1, … , y_i,1 | y_i,0 , z_i)).''

Based on the estimates for (''δ, ρ'') and the corresponding variance, test about the coefficients can be implemented <ref> Whitney K. Newey, Daniel McFadden, Chapter 36 Large sample estimation and hypothesis testing, In: Robert F. Engle and Daniel L. McFadden, Editor(s), Handbook of Econometrics, Elsevier, 1994, Volume 4, Pages 2111-2245, ISSN 1573-4412, ISBN 9780444887665,</ref> and the average partial effect can be calculated. <ref>Chamberlain, G. (1980), “Analysis of Covariance with Qualitative Data,” Journal of Econometrics 18, 5-46</ref>

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