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Econometric issues in the analysis of regressions with generated regressors. (English) Zbl 0547.62078
Efficiency and consistency properties of the estimators of a regression model are discussed here, when the regressor variables are generated either through another regression equation or by residuals from an estimated equation. A common use of the generated regressors is in the following type of econometric model: \[ y=\delta z^*+X\gamma +e,\quad z=z^*+\eta =W\alpha +\eta. \] In the rational expectation models in econometrics, \(z^*\) is taken to be an expected value of (for example) inflation, z is actual inflation and W some variables describing the formation of expectations.
For this type of model, a two-step estimator (\({\hat\delta }\),\({\hat\gamma }\)) is usually derived as follows: z is regressed against W to get an estimate of the expectation \(\hat z\) and then y is regressed against \(\hat z\) and X to give the two-step estimator. It is proved that \(T^{1/2}\left( \begin{matrix} {\hat\delta }-\delta\\ {\hat\gamma }- \gamma\end{matrix} \right)\) and \(T^{1/2}\left( \begin{matrix} {\tilde\delta }- \delta\\ {\tilde\gamma }-\delta\end{matrix} \right)\) do not generally have the same limiting distribution, unless X appears in W, or \(X^ tW=0\). Here \({\tilde\delta }\), \({\tilde\gamma }\) are the single-step maximum likelihood estimators. Some cases are discussed where the efficiency loss of two-step estimators may be quite small.
Reviewer: J.K.Sengupta

MSC:
62P20 Applications of statistics to economics
62J05 Linear regression; mixed models
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