Nonstationary Binary Choice
AbstractThis 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 are 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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Bibliographic InfoPaper provided by Institute of Economic Research, Seoul National University in its series Working Paper Series with number no5.
Date of creation: Mar 1999
Date of revision:
Binary choice model; Brownian motion; Brownian local time; Dual convergence rates; Integrated time series; Maximum likelihood estimation;
Other versions of this item:
- C22 - Mathematical and Quantitative Methods - - Single Equation Models; Single Variables - - - Time-Series Models; Dynamic Quantile Regressions; Dynamic Treatment Effect Models
- C25 - Mathematical and Quantitative Methods - - Single Equation Models; Single Variables - - - Discrete Regression and Qualitative Choice Models; Discrete Regressors; Proportions
This paper has been announced in the following NEP Reports:
- NEP-ALL-1999-07-28 (All new papers)
- NEP-DCM-1999-09-02 (Discrete Choice Models)
- NEP-ETS-1999-07-28 (Econometric Time Series)
Please report citation or reference errors to , or , if you are the registered author of the cited work, log in to your RePEc Author Service profile, click on "citations" and make appropriate adjustments.:
- 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.
- Park, Joon Y. & Phillips, Peter C.B., 1999. "Asymptotics For Nonlinear Transformations Of Integrated Time Series," Econometric Theory, Cambridge University Press, vol. 15(03), pages 269-298, June.
- Peter C.B. Phillips & Joon Y. Park, 1986.
"Statistical Inference in Regressions with Integrated Processes: Part 2,"
Cowles Foundation Discussion Papers
819R, Cowles Foundation for Research in Economics, Yale University, revised Feb 1987.
- 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.
- 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.
- 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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