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Abstract

We consider least absolute error estimation in a nonlinear dynamic model with neither independent nor identically distributed errors. Under the null hypothesis and local alternatives, the estimator is shown to be consistent and asymptotically normal, with asymptotic covariance matrix depending upon the heights of the density functions of the errors at their median (zero). A consistent estimator of the asymptotic covariance matrix of the estimator is given, and the Wald, Lagrange multiplier and Likelihood ratio tests for linear restrictions on the parameters in the regression equation are discussed. The Wald and Lagrange multiplier tests are distributed as central x2 under the null and non-central x2 under local alternatives. The Likelihood ratio test on the other hand is not always equivalent to the other two tests and may not be asymptotically distributed as x2. A simple artificial regression form of the Lagrange multiplier test is available in omitted variables problems, making this test attractive in many testing situations. Since the form of the covariance matrix and tests depend on the presence or absence of heteroscedasticity, a Lagrange multiplier test based on the absolute residuals is analysed.

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