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Nonstationary Binary Choice

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Abstract

This paper develops an asymptotic theory for time series binary choice models with nonstationary explanatory variables generated as integrated processes. Both logit and probit models are covered. The maximum likelihood (ML) estimator is consistent but a new phenomenon arises in its limit distribution theory. The estimator consists of a mixture of two components, one of which is parallel to and the other orthogonal to the direction of the true parameter vector, with the latter being the principal component. The ML estimator is shown to converge at a rate of n^{3/4} along its principal component but has the slower rate of n^{1/4} convergence in all other directions. This is the first instance known to the authors of multiple convergence rates in models where the regressors have the same (full rank) stochastic order and where the parameters appear in linear forms of these regressors. It is a consequence of the fact that the estimating equations involve nonlinear integrable transformations of linear forms of integrated processes as well as polynomials in these processes, and the asymptotic behavior of these elements are quite different. The limit distribution of the ML estimator is derived and is shown to be a mixture of two mixed normal distributions with mixing variates that are dependent upon Brownian local time as well as Brownian motion. It is further shown that the sample proportion of binary choices follows an arc sine law and therefore spends most of its time in the neighbourhood of zero or unity. The result has implications for policy decision making that involves binary choices and where the decisions depend on economic fundamentals that involve stochastic trends. Our limit theory shows that, in such conditions, policy is likely to manifest streams of little intervention or intensive intervention.

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File URL: http://cowles.econ.yale.edu/P/cd/d12a/d1223.pdf
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Bibliographic Info

Paper provided by Cowles Foundation for Research in Economics, Yale University in its series Cowles Foundation Discussion Papers with number 1223.

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Length: 35 pages
Date of creation: Jun 1999
Date of revision:
Publication status: Published in Econometrica (September 2000), 68(5): 1249-1280
Handle: RePEc:cwl:cwldpp:1223

Note: CFP 1003.
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Postal: Cowles Foundation, Yale University, Box 208281, New Haven, CT 06520-8281 USA

Related research

Keywords: Binary choice model; Brownian motion; Brownian local time; dual convergence rates; Integrated time series; maximum likelihood estimation;

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  1. Park, Joon Y. & Phillips, Peter C.B., 1989. "Statistical Inference in Regressions with Integrated Processes: Part 2," Econometric Theory, Cambridge University Press, vol. 5(01), pages 95-131, April.
  2. Peter C.B. Phillips & Joon Y. Park, 1998. "Asymptotics for Nonlinear Transformations of Integrated Time Series," Cowles Foundation Discussion Papers 1182, Cowles Foundation for Research in Economics, Yale University.
  3. Peter C.B. Phillips & Joon Y. Park, 1998. "Nonstationary Density Estimation and Kernel Autoregression," Cowles Foundation Discussion Papers 1181, Cowles Foundation for Research in Economics, Yale University.
  4. Peter C.B. Phillips & Victor Solo, 1989. "Asymptotics for Linear Processes," Cowles Foundation Discussion Papers 932, Cowles Foundation for Research in Economics, Yale University.
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