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A Shortcut to LAD Estimator Asymptotics

Published online by Cambridge University Press:  11 February 2009

P.C.B. Phillips
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Abstract

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.

Type
Research Article
Information
Econometric Theory , Volume 7 , Issue 4 , December 1991 , pp. 450 - 463
Copyright
Copyright © Cambridge University Press 1991

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References

REFERENCES

1.Amemiya, T. Advanced Econometrics Cambridge: Harvard University Press, 1985.Google Scholar
2.Andrews, D.W.K. Asymptotics for semiparametric econometric models: I. Estimation, Cowles Foundation Discussion Paper No. 908, 1989.Google Scholar
3.Bassett, G. & Koenker, R.. Asymptotic theory for least absolute error regression. Journal of the American Statistical Association 73 (1978): 618–622.10.1080/01621459.1978.10480065CrossRefGoogle Scholar
4.Bassett, G.W. A p-subset property of L 1 and regression quantile estimates. Computational Statistics and Data Analysis 6 (1988): 297–304.10.1016/0167-9473(88)90008-4CrossRefGoogle Scholar
5.Billingsley, P. Convergence of Probability Measures. New York: Wiley, 1968.Google Scholar
6.Bloomfield, P. & Steiger, W.L.. Least Absolute Deviations: Theory, Applications and Algorithms Boston: Birkhauser, 1983.Google Scholar
7.Cramer, H. Mathematical Methods of Statistics Princeton: Princeton University Press, 1946.Google Scholar
8.Daniels, H.E. The asymptotic efficiency of a maximum likelihood estimator. Proceedings of the Fourth Berkeley Symposium on Mathematical Statistics and Probability, Vol. 1. Berkeley: University of California Press, 1961.Google Scholar
9.Gelfand, I.M. & Shilov, G.E.. Generalized Functions, Vol. 1. New York: Academic Press, 1964.Google Scholar
10.Hall, P. & Heyde, C.C.. Martingale Limit Theory and its Application New York: Academic Press, 1980.Google Scholar
11.Huber, P. J. The behaviour of maximum likelihood estimates under nonstandard conditions. Fifth Berkeley Symposium on Mathematical Statistics and Probability Berkeley: University of California Press, 1967.Google Scholar
12.Kim, J. & Pollard, D.. Cube root asymptotics. Annals of Statistics 18 (1990): 191–219.10.1214/aos/1176347498CrossRefGoogle Scholar
13.Knight, K. Limit theory for autoregressive parameters in an infinite variance random walk. Canadian Journal of Statistics(1989): 261–278.10.2307/3315522CrossRefGoogle Scholar
14.McFadden, D. A method of simulated moments for estimation of discrete response models without numerical integration. Econometrica 57 (1989): 995–1026.10.2307/1913621CrossRefGoogle Scholar
15.Newey, W.K. & West, K.D.. A simple positive definite heteroskedasticity and autocorrelation consistent covariance matrix. Econometrica 55 (1987): 703–708.10.2307/1913610CrossRefGoogle Scholar
16.Pakes, A. & Pollard, D.. Simulation and the asymptotics of optimization estimators. Econometrica 57 (1989): 1027–1058.10.2307/1913622CrossRefGoogle Scholar
17.Phillips, P.C.B. Approximations to some finite sample distributions associated with a first order stochastic difference equation. Econometrica 45 (1977): 463–486.10.2307/1911222CrossRefGoogle Scholar
18.Phillips, P.C.B. Small sample distribution theory in econometric models of simultaneous equations, Cowles Foundation Discussion Paper No. 617, 1982.Google Scholar
19.Phillips, P.C.B. Time series regression with a unit root and infinite variance errors. Econometric Theory 6 (1990): 44–62.10.1017/S0266466600004904CrossRefGoogle Scholar
20.Pollard, D. Convergence of Stochastic Processes. New York: Springer-Verlag, 1984.10.1007/978-1-4612-5254-2CrossRefGoogle Scholar
21.Pollard, D. New ways to prove central limit theorems. Econometric Theory 1 (1985): 295–314.10.1017/S0266466600011233CrossRefGoogle Scholar
22.Pollard, D. Asymptotics via empirical processes. Statistical Science 4 (1989): 341–366.Google Scholar
23.Pollard, D. Asymptotics for least absolute deviation estimators. Econometric Theory (1990)(forthcoming).Google Scholar
24. Prakasa, Rao, B.L.S. Estimation of a unimodal density. Sankhya Series A 31 (1969): 23–36.Google Scholar
25.Sargan, J.D. Econometric estimators and the Edgeworth Approximation. Econometrica 44 (1976): 421–428.10.2307/1913972CrossRefGoogle Scholar
26.Silverman, B.W. Density Estimation London: Chapman Hall, 1986.Google Scholar
27.White, H. Asymptotics Theory for Econometricians New York: Academic Press, 1984.Google Scholar
28.Walter, G. & Blum, J.. Probability density estimation using delta sequences. Annals of Statistics 7 (1979): 328–340.10.1214/aos/1176344617CrossRefGoogle Scholar