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Unformatted text preview: Valid Inference for a Class of Models Where Standard Inference Performs Poorly; Including Nonlinear Regression, ARMA, GARCH, and Unobserved Components** Jun Ma Department of Economics, Finance and Legal Studies University of Alabama and Charles R. Nelson* Department of Economics University of Washington First draft June 18, 2006 This draft September 15, 2008 Copyright © 2006-2008 by Jun Ma and Charles R. Nelson JEL classification : C120, C220, C330 Keywords: ARMA, Unobserved Components, State Space, GARCH, Zero-Information- Limit-Condition. Abstract Standard inference works poorly in models of the form ε β γ + • = ) , ( x g y , because the standard error for ˆ β depends on ˆ γ . In this paper we show that this problem is usefully studied by working with the linearization of (.) g and the resulting reduced form regression. Bias and dispersion in ˆ β depend on correlation between the ‘regressors’ and on γ , as does the size of the t-test. A reduced form test however is exact when (.) g is linear and has nearly correct size in examples from non-linear regression, ARMA, GARCH, and Unobserved Components models. Further, its distribution does not depend on the identifying restriction ≠ γ . * Corresponding author: Box 353330, University of Washington, Seattle, WA 98195, USA. Tel: +1 206 543 5955; fax +1 206 685 7477. E-mail address: [email protected] . ** The authors gratefully acknowledge support from the Ford and Louisa Van Voorhis Endowment at the University of Washington. We thank the following for helpful comments: Michael Dueker, Walter Enders, Evan Koenig, Junsoo Lee, James Morley, Christian Murray, David Papell, Ruxandra Prodan, Bent Sørensen, Richard Startz, and Eric Zivot, but responsibility for all errors is ours. 2 1. Introduction This paper is concerned with inference in the class of models that have a representation of the form N i x g y i i i ,..., 1 ; ) , ( = + • = ε β γ . (1.1) The parameter of interest is β which is identified only if γ ≠ . Additional regressors and parameters would often be present in practice. We assume that errors i ε are i.i.d. N (0, 2 σ ) so that Maximum Likelihood estimates of ˆ γ and ˆ β are obtained by non-linear least squares, given data y and x . In addition to non-linear regression models, this class includes the workhorse ARMA model, where data x are lagged observations. By extension, the GARCH model and Unobserved Components State Space models for trend and cycle decomposition fall into this class as well. What these models have in common is that standard inference based on asymptotic theory often works poorly in finite samples, essentially because the estimated standard error for ˆ β depends on ˆ γ . Further, the distribution of ˆ β will generally be displaced away from the true value. Nelson and Startz (2007) – hereafter NS - show that the estimated standard error for ˆ β is generally too small. Although these two effects might seem to imply that the t...
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This note was uploaded on 12/26/2011 for the course ECON 245a taught by Professor Staff during the Fall '08 term at UCSB.
- Fall '08