Chamberlain's approach to unobserved effects models

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In linear panel analysis, it can be desirable to estimate the magnitude of the fixed effects, as they provide measures of the unobserved components. For instance, in wage equation regressions, fixed effects capture unobservables that are constant over time, such as motivation. Chamberlain's approach to unobserved effects models is a way of estimating the linear unobserved effects, under fixed effect (rather than random effects) assumptions, in the following unobserved effects model

:$y\_\{it\}\; =\; x\_\{it\}\; b\; +\; c\_i\; +\; u\_\{it\}$

where c_i is the unobserved effect and x_it contains only time-varying explanatory variables.Wooldridge, J. (2002): Econometric Analysis of Cross Section and Panel Data, MIT Press, Cambridge, Mass. Rather than differencing out the unobserved effect c_i, Chamberlain proposed to replace it with the linear projection of it onto the explanatory variables in all time periods. Specifically, this leads to the following equation

:$c\_i\; =\; d\; +\; x\_\{i1\}\; \backslash lambda\_1\; +\; x\_\{i2\}\; \backslash lambda\_2\; +\; \backslash dots\; +\; x\_\{iT\}\; \backslash lambda\_T\; +\; e\_i$

where the conditional distribution of c_i given x_it is unspecified, as is standard in fixed effects models. Combining these equations then gives rise to the following model.Chamberlain, G. (1982): Multivariate Regression Models for Panel Data. Journal of Econometrics (18), pp. 5-46Chamberlain, G. (1984): Panel Data. Handbook of Econometrics, Volume 2, ed. Z. Griliches and M. D. Intriligator. Amsterdam: North Holland, pp. 1247-1318

:$y\_\{it\}\; =\; d\; +\; x\_\{i1\}\; \backslash lambda\_1\; +\; \backslash dots\; +\; x\_\{it\}\; (b\; +\; \backslash lambda\_t)\; +\; \backslash dots\; +\; x\_\{iT\}\; \backslash lambda\_T\; +\; e\_i\; +\; u\_\{it\}$

An important advantage of this approach is the computational requirement. Chamberlain uses minimum distance estimation, but a generalized method of moments approach would be another valid way of estimating this model. The latter approach also gives rise to a larger number of instruments than moment conditions, which leads to useful overidentifying restrictions that can be used to test the strict exogeneity restrictions imposed by many static Fixed Effects models.

Similar approaches have been proposed to model the unobserved effect. For instance, Mundlak follows a very similar approach, but rather projects the unobserved effect c_i onto the average of all x_it across all T time periods, more specifically Mundlak, Y. (1978): On the Pooling of Time Series and Cross Section Data. Econometrica (46), pp. 69-85

:$c\_i\; =\; d\; +\; \backslash overline\{x\}\_i\; \backslash lambda\; +\; e\_i$

It can be shown that the Chamberlain method is a generalization of Mundlak's model. The Chamberlain method has been popular in empirical work, ranging from studies trying to estimate the causal returns to union membership,Card, D. (1996): The effect of unions on the structure of wages: a longitudinal analysis. Econometrica (64), pp. 957-979 to studies investigating growth convergence,Islam, N. (1995): Growth Empirics: A Panel Data Approach. The Quarterly Journal of Economics (110), pp. 1127-1170 and estimating product characteristics in demand estimation.{{Cite journal|last=Nevo|first=Aviv|date=March 2001|title=Measuring Market Power in the Ready-to-Eat Cereal Industry |journal=Econometrica|volume=69|issue=2|pages=307–342|doi=10.1111/1468-0262.00194|s2cid=9708178 |issn=0012-9682}}