We construct limiting and small sample distributions of maximum likelihood estimators (mle) from the property that they satisfy the first order condition (foc). The foc relates the mle of the analyzed model to the mle of an encompassing model and shows that the mle of the analyzed model is a realization from the limiting or small sample distribution of the mle of the encompassing model given that the foc holds. We can thus use the unique conditional (limiting or small sample) density of the mle of the encompassing model given that the foc holds to construct the limiting or small sample density/distribution of the mle of the analyzed model. To proof the validity of this approach and thus of the concept of an unique conditional density, we use it to construct the small sample and limiting distribution of the limited information mle and show that they are identical to resp. the sampling density and the expression discussed elsewhere in the literature. We analyze the small sample density further and relate it to existing expressions and show its limiting behavior in case of weak and strong instruments.
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Paper provided by Erasmus University of Rotterdam - Econometric Institute in its series Papers with number
9844/a.
Find related papers by JEL classification: C52 - Mathematical and Quantitative Methods - - Econometric Modeling - - - Model Evaluation and Testing C51 - Mathematical and Quantitative Methods - - Econometric Modeling - - - Model Construction and Estimation