Dynamic unobserved effects model: Difference between revisions

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There are several [[Maximum likelihood|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 variable#Use in statistics|independent]] 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>