Estimating and Testing Threshold Regression Models with Multiple Threshold Variables

Chong, Terence Tai Leung and Yan, Isabel K. (2014): Estimating and Testing Threshold Regression Models with Multiple Threshold Variables.

Abstract

Conventional threshold models contain only one threshold variable. This paper provides the theoretical foundation for threshold models with multiple threshold variables. The new model is very different from a model with a single threshold variable as several novel problems arisefrom having an additional threshold variable. First, the model is not analogous to a change-point model. Second, the asymptotic joint distribution of the threshold estimators is difficult to obtain. Third, the estimation time increases exponentially with the number of threshold variables. This paper derives the consistency and the asymptotic joint distribution of the threshold estimators. A fast estimation algorithm to estimate the threshold values is proposed. We also develop tests for the number of threshold variables. The theoretical results are supported by simulation experiments. Our model is applied to the study of currency crises.

• Item Type: MPRA Paper
• Language: English
• Keywords: Threshold Model; Multiple Threshold Variables; Currency Crisis; Panel Data
• Subjects: C - Mathematical and Quantitative Methods > C1 - Econometric and Statistical Methods and Methodology: General > C12 - Hypothesis Testing: General; C - Mathematical and Quantitative Methods > C1 - Econometric and Statistical Methods and Methodology: General > C13 - Estimation: General; C - Mathematical and Quantitative Methods > C3 - Multiple or Simultaneous Equation Models ; Multiple Variables > C33 - Panel Data Models ; Spatio-temporal Models; F - International Economics > F3 - International Finance; F - International Economics > F3 - International Finance > F31 - Foreign Exchange; F - International Economics > F3 - International Finance > F37 - International Finance Forecasting and Simulation: Models and Applications

References

Astatkie, T., D. G. Watts, and W. E. Watt (1997). Nested Threshold Autoregressive (NeTAR) Models. International Journal of Forecasting 13, 105-116.

Bai, J. (1997). Estimating Multiple Breaks one at a time. Econometric Theory 13, 315-52.

Bai, J. and P. Perron (1998). Estimating and Testing Linear Models with Multiple Structural Changes. Econometrica 66(1), 47-78.

Bautista, C. (2000). Boom-bust Cycles and Crisis Periods in the Philippines, a Regime-switching Analysis. Working paper, the University of the Philippines.

Bhattacharya, P. K, and P. J. Brockwell (1976). The Minimum of an Additive Process with Applications to Signal Estimation and Storage Theory. Z. Wahrschein. Verw. Gebiete, 37, 51-75.

Billingsley, P. (1968). Convergence of Probability Measures. New York: John Wiley and Sons.

Calvo, G. and G. Mendoza (2000). Capital Markets Crises and Economic Collapse in Emerging Markets: An Informational-Frictions Approach. American Economic Review Papers and Proceedings 90(2), 59-64.

Chan, K. S. and H. Tong (1986). On Estimating Thresholds in Autoregressive Models. Journal of Time Series Analysis 7, 179-190.

Chan, K. S. and R. S. Tsay (1998). Limiting Properties of the Least Squares Estimator of a Continuous Threshold Autoregressive Model. Biometrika 85, 413-426.

Chang, R. and A. Velasco (1998a). Financial Fragility and the Exchange Rate Regime, NBER WP6469.

Chang, R. and A. Velasco (1998b). Financial Crises in Emerging Markets: A Canonical Model, Federal Reserve Bank of Atlanta, working paper.

Chen, H., T. T. L. Chong and J. Bai (2012). Theory and Applications of TAR Model with two Threshold Variables. Econometric Reviews 31(2), 142--170.

Chen, R. and S. Tsay (1993). Functional-coefficient Autoregressive Models. Journal of the American Statistical Association 88, 298-308.

Chong, T. T. L. (2001). Structural Change in AR(1) Models. Econometric Theory 17, 87-155.

Christiano, L. J. (1992). Searching for a Break in GNP. Journal of Business & Economic Statistics 10, 237-250.

Dornbusch, R., I. Goldgajn and R. O. Valdes (1999). Currency Crises and Collapses. Brookings Papers on Economic Activity, No.2, 219-95.

Durlauf, S. N. and P. A. Johnson (1995). Multiple Regimes and Cross-country Growth Behavior. Journal of Applied Econometrics 10, 365-384.

Edison, H. J. (2000). Do Indicators of Financial Crises Work? An Evaluation of an Early Warning System. Board of Governors of Federal Reserve System, International Finance Discussion Papers Number 675.

Edwards, S. (1989). Real Exchange Rates, Devaluations, and Adjustment: Exchange Rate Policy in Developing Countries, Cambridge, Massachusetts: MIT Press.

Eichengreen, B., Rose, A. and C. Wyplosz (1995). Exchange Market Mayhem: The Antecedents and Aftermath of Speculative Attacks. Economic Policy 21, 249-312.

Eichengreen, B., Rose, A., and C. Wyplosz (1996). Contagious Currency Crises, NBER WP5681.

Flood, R. and P. Garber (1984). Collapsing Exchange Rate Regimes: Some Linear Examples. Journal of International Economics 17, 1-13.

Frankel, J. A. and A. Rose (1996). Currency Crashes in Emerging Markets: Empirical Indicators, NBER WP5437.

Garcia, R. and P. Perron (1994). An Analysis of the Real Interest rate under Regime Shifts. Review of Economics and Statistics 78, 111-125.

Goldstein, M., G. Kaminsky and C. Reinhart (2000). Assessing Financial Vulnerability -- An Early Warning System for Emerging Markets, Institute for International Economics 51, 145-168.

Gonzalo, J. and J. Pitarakis (2002). Estimation and Model Selection Based Inference in Single and Multiple Threshold Models. Journal of Econometrics 110, 319-352.

Hali, J. E. (2000). Do Indicators of Financial Crises Work? An evaluation of the early warning system. Board of Governors of Federal Reserve System, International Finance Discussion Paper #675.

Hansen, B. E. (1999). Threshold Effect in Non-dynamic Panels: Estimation, Testing, and Inference. Journal of Econometrics 93, 345-368.

Hansen, B. E. (2000). Sample Splitting and Threshold Estimation. Econometrica 68, 575-603.

Henry, Ó., N. Olekaln and P. M. Summers (2001). Exchange Rate Instability: a Threshold Autoregressive Approach. Economic Record 77, 160-166.

Kaminsky, G. L. (1998). Currency and Banking Crises: The Early Warnings of Distress. Board of Governors of the Federal Reserve System, International Finance Discussion Papers Number 629.

Kaminsky, G. L., S. Lizondo and C. M. Reinhart (1997). Leading Indicators of Currency Crises. IMF Staff Papers 45, 1-48.

Kakkar, V. and I. Yan (2012). Real Exchange Rates and Productivity: Evidence From Asia. Journal of Money Credit and Banking 44(2-3), 301-322.

Krugman, P. (1979). A Model of Balance of Payments Crises. Journal of Money, Credit and Banking, 311-325.

Krugman, P. (1999). Balance Sheets, the Transfer Problem, and Financial Crisis. Paper prepared for the volume in honor of Robert Flood.

Kumhof, M. , S. Li and I. Yan (2007). Balance of Payments Crises under Inflation Targeting. Journal of International Economics 72(1), 242-264.

Mankiw, N. G., D. Romer and D. N. Weil (1992). A Contribution to the Empirics of Economic Growth. Quarterly Journal of Economics 107(2), 407-437.

McKinnon, I. Ronald. and P. Huw (1996). Credible Economic Liberalizations and Overborrowing. American Economic Review 87(2), 189-193.

Mishkin, F. S. (1990). What does the Term Structure Tell us about Future Inflation? Journal of Monetary Economics 25, 77-95.

Obstfeld, M. (1986). Rational and Self-Fulfilling Balance of Payment Crises. American Economic Review 76(1), 72-81.

Pollard, D. (1984), Convergence of Stochastic Processes, Springer-Verlag Inc., New York.

Potter, S. M. (1995). A Nonlinear Approach to US GNP. Journal of Applied Econometrics 2, 109-125.

Qiu, P., Asano, C. and X. Li (1991). Estimation of Jump Regression Function. Bulletin of Informatics and Cybernetics, 24, No. 3-4, 197-212.

Quintos, C., Fan, Z. and P. C. B. Phillips (2001). Structural Change Tests in Tail Behavior and the Asian Crisis. Review of Economic Studies 68, 633-663.

Roley, V. V. and S. M. Wheatley (1996). Shifts in the Interest-rate Response to Money Announcements: What can We say about when they occur? Journal of Business & Economics Statistics 14, 135-138.

Sachs, J., A. Tornell and A. Velasco (1996). Financial Crises in Emerging Markets: The Lessons from 1995. Brookings Papers on Economic Activity No.1, 147-215.

Tong, H. (1983). Threshold Models in Nonlinear Time Series Analysis: Lecture Notes in Statistics 21. Berlon: Springer.

Tsay, R. S. (1998). Testing and Modeling Multivariate Threshold Models. Journal of the American Statistical Association 93, 1188-1202.

Xia, Y. and W. K. Li (1999). On Single-index Coefficient Regression Model, Journal of the American Statistical Association 94, 1275-1285.