A Shortcut to LAD Estimator Asymptotics

Using generalized functions of random variables and generalized Taylor series expansions, we provide quick demonstrations of the asymptotic theory for the LAD estimator in a regression model setting. The approach is justified by the smoothing that is delivered in the limit by the asymptotics, whereby the generalized functions are forced to appear as linear functionals wherein they become real valued. Models with fixed and random regressors, and autoregressions with infinite variance errors are studied. Some new analytic results are obtained including an asymptotic expansion of the distribution of the LAD estimator.

[1]  H. L. Le Roy,et al.  Proceedings of the Fifth Berkeley Symposium on Mathematical Statistics and Probability; Vol. IV , 1969 .

[2]  D. Andrews.  Asymptotics for Semiparametric Econometric Models: I. Estimation , 1990 .

[3]  P. Hall,et al.  Martingale Limit Theory and its Application. , 1984 .

[4]  J. Neyman,et al.  Proceedings of the Fourth Berkeley Symposium on Mathematical Statistics and Probability , 1963 .

[5]  P. J. Huber.  The behavior of maximum likelihood estimates under nonstandard conditions , 1967 .

[6]  G. Walter,et al.  Probability Density Estimation Using Delta Sequences , 1979 .

[7]  Peter C. B. Phillips,et al.  Approximations to Some Finite Sample Distributions Associated with a First-Order Stochastic Difference Equation , 1977 .

[8]  R. Koenker,et al.  Asymptotic Theory of Least Absolute Error Regression , 1978 .

[9]  Jr. Gilbert W. Bassett.  A p -subset property of L 1 and regression quantile estimates , 1988 .

[10]  W. Newey,et al.  A Simple, Positive Semi-Definite, Heteroskedasticity and Autocorrelationconsistent Covariance Matrix , 1986 .

[11]  J. Neyman.  Second Berkeley Symposium on Mathematical Statistics and Probability , 1951 .

[12]  D. Pollard.  Convergence of stochastic processes , 1984 .

[13]  D. Pollard,et al.  Cube Root Asymptotics , 1990 .

[14]  H. Daniels.  The Asymptotic Efficiency of a Maximum Likelihood Estimator , 1961 .

[15]  D. Pollard.  New Ways to Prove Central Limit Theorems , 1985, Econometric Theory.

[16]  W. Steiger,et al.  Least Absolute Deviations: Theory, Applications and Algorithms , 1984 .

[17]  D. Pollard.  Asymptotics via Empirical Processes , 1989 .

[18]  P. Hall,et al.  Martingale Limit Theory and Its Application , 1980 .

[19]  J. D. Sargan,et al.  Econometric Estimators and the Edgeworth Approximation , 1976 .

[20]  Peter C. B. Phillips,et al.  Time Series Regression With a Unit Root and Infinite-Variance Errors , 1990, Econometric Theory.

[21]  P. Billingsley,et al.  Convergence of Probability Measures , 1969 .

[22]  K. Knight.  Limit theory for autoregressive‐parameter estimates in an infinite‐variance random walk , 1989 .

[23]  P. Phillips.  Small Sample Distribution Theory in Econometric Models of Simultaneous Equations , 1982 .

[24]  Prakasa Rao.  Estimation of a unimodal density , 1969 .

[25]  D. McFadden.  A Method of Simulated Moments for Estimation of Discrete Response Models Without Numerical Integration , 1989 .

[26]  D. Pollard,et al.  Simulation and the Asymptotics of Optimization Estimators , 1989 .

[27]  D. Pollard.  Asymptotics for Least Absolute Deviation Regression Estimators , 1991, Econometric Theory.

[28]  W. Newey,et al.  A Simple, Positive Semi-Definite, Heteroskedasticity and Autocorrelationconsistent Covariance Matrix , 1986 .