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Sample Splitting and Threshold Estimation

  • Bruce E. Hansen

Threshold models have a wide variety of applications in economics. Direct applications include models of separating and multiple equilibria. Other applications include empirical sample splitting when the sample split is based on a continuously-distributed variable such as firm size. In addition, threshold models may be used as a parsimonious strategy for non-parametric function estimation. For example, the threshold autoregressive model (TAR) is popular in the non-linear time series literature. Threshold models also emerge as special cases of more complex statistical frameworks, such as mixture models, switching models, Markov switching models, and smooth transition threshold models. It may be important to understand the statistical properties of threshold models as a preliminary step in the development of statistical tools to handle these more complicated structures. Despite the large number of potential applications, the statistical theory of threshold estimation is undeveloped. The previous literature has demonstrated that threshold estimates are super-consistent, but a distribution theory useful for testing and inference has yet to be provided. This paper develops a statistical theory for threshold estimation in the regression context. We allow for either cross-section or time series observations. Least squares estimation of the regression parameters is considered. An asymptotic distribution theory for the regression estimates (the threshold and the regression slopes) is developed. It is found that the distribution of the threshold estimate is non- standard. Methods to construct asymptotic confidence intervals are introduced, using both a t-statistic and LR-statistic approach. Monte Carlo simulations are presented to assess the accuracy of the asymptotic approximations. The empirical relevance of the theory is illustrated through an application to the multiple equilibria growth model of Durlauf and Johnson (1995).

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Article provided by Econometric Society in its journal Econometrica.

Volume (Year): 68 (2000)
Issue (Month): 3 (May)
Pages: 575-604

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Handle: RePEc:ecm:emetrp:v:68:y:2000:i:3:p:575-604
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  1. Hardle, W., 1992. "Applied Nonparametric Methods," Papers 9206, Tilburg - Center for Economic Research.
  2. Potter, Simon M, 1995. "A Nonlinear Approach to US GNP," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 10(2), pages 109-25, April-Jun.
  3. Davidson, James, 1994. "Stochastic Limit Theory: An Introduction for Econometricians," OUP Catalogue, Oxford University Press, number 9780198774037.
  4. Hansen, B.E., 1991. "Inference when a Nuisance Parameter is Not Identified Under the Null Hypothesis," RCER Working Papers 296, University of Rochester - Center for Economic Research (RCER).
  5. Hansen, Bruce E, 1992. "Tests for Parameter Instability in Regressions with I(1) Processes," Journal of Business & Economic Statistics, American Statistical Association, vol. 10(3), pages 321-35, July.
  6. Durlauf, S.M. & Johnson, P.A., 1995. "Multiple Regimes and Cross-Country Growth Behavior," Working papers 9419r, Wisconsin Madison - Social Systems.
  7. Oliver LINTON, . "Applied nonparametric methods," Statistic und Oekonometrie 9312, Humboldt Universitaet Berlin.
  8. repec:cup:etheor:v:7:y:1991:i:2:p:213-21 is not listed on IDEAS
  9. Hansen, Bruce E., 2000. "Testing for structural change in conditional models," Journal of Econometrics, Elsevier, vol. 97(1), pages 93-115, July.
  10. Hansen, Bruce E., 1991. "Strong Laws for Dependent Heterogeneous Processes," Econometric Theory, Cambridge University Press, vol. 7(02), pages 213-221, June.
  11. Chu, Chia-Shang James & White, Halbert, 1992. "A Direct Test for Changing Trend," Journal of Business & Economic Statistics, American Statistical Association, vol. 10(3), pages 289-99, July.
  12. Jushan Bai, 1997. "Estimation Of A Change Point In Multiple Regression Models," The Review of Economics and Statistics, MIT Press, vol. 79(4), pages 551-563, November.
  13. Pham, Tuan D. & Tran, Lanh T., 1985. "Some mixing properties of time series models," Stochastic Processes and their Applications, Elsevier, vol. 19(2), pages 297-303, April.
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