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In this thesis we evaluate fixed effects linear panel data models where the sample within variation of the explanatory variable is significantly small. We notice that when a covariate is exhibiting high degrees of low longitudinal variation (LLV), this indicates a nearly singular design.
Consistently with the related literature, we demonstrate that this design affects
the rate of convergence of the FE estimator. The new rate is slower than n and depends critically on the severity of the LLV problem. By extending the subsampling methodology for the linear panel data context, we show how to estimate this unknown rate of convergence.
We subsequently begin to evaluate alternative ways of estimating the parameter of interest when the explanatory variable is characterized by LLV. In particular, we demonstrate how it is possible to obtain a shrinkage estimator whose Mean Square Error always dominates, under appropriate conditions, the Mean Square Error of the fixed effect estimator. We demonstrate the importance of this result with a specific empirical example.