Estimation and inference in unstable nonlinear least squares models

Boldea, Otilia and Hall, Alastair R. (2010): Estimation and inference in unstable nonlinear least squares models.

Abstract

In this paper, we extend Bai and Perron's (1998, Econometrica, pp. 47-78) method for detecting multiple breaks to nonlinear models. To that end, we consider a nonlinear model that can be estimated via nonlinear least squares (NLS) and features a limited number of parameter shifts occurring at unknown dates. In our framework, the break-dates are estimated simultaneously with the parameters via minimization of the residual sum of squares. Using new uniform convergence results for partial sums, we derive the asymptotic distributions of both break-point and parameter estimates and propose several instability tests. We provide simulations that indicate good finite sample properties of our procedure. Additionally, we use our methods to test for misspecification of smooth-transition models in the context of an asymmetric US federal funds rate reaction function and conclude that there is strong evidence of sudden change as well as smooth behavior.

• Item Type: MPRA Paper
• Language: English
• Keywords: Multiple Change Points, Nonlinear Least Squares, Smooth Transition
• Subjects: C - Mathematical and Quantitative Methods > C3 - Multiple or Simultaneous Equation Models ; Multiple Variables > C32 - Time-Series Models ; Dynamic Quantile Regressions ; Dynamic Treatment Effect Models ; Diffusion Processes ; State Space Models; C - Mathematical and Quantitative Methods > C1 - Econometric and Statistical Methods and Methodology: General > C13 - Estimation: General; C - Mathematical and Quantitative Methods > C1 - Econometric and Statistical Methods and Methodology: General > C12 - Hypothesis Testing: General; C - Mathematical and Quantitative Methods > C2 - Single Equation Models ; Single Variables > C22 - Time-Series Models ; Dynamic Quantile Regressions ; Dynamic Treatment Effect Models ; Diffusion Processes

References

Anderson, G. J., and Mizon, G. E. (1983). `Parameter Constancy Tests: Old and New', Discussion Paper 8325, Economics Department, University of Southampton.

Andreou, E., and Ghysels, E. (2002). `Detecting Multiple Breaks in Financial Market Volatility Dynamics', Journal of Applied Econometrics, 17: 579-600.

Andrews, D. W. K. (1993). `Tests for Parameter Instability and Structural Change with Unknown Change Point', Econometrica, 61: 821-856.

Andrews, D. W. K. (2003). `End-of-Sample Instability Tests', Econometrica, 71: 1661-1694.

Andrews, D. W. K., and Fair, R. C. (1988). `Inference in Nonlinear Econometric Models with Structural Change', The Review of Economic Studies, 55: 615-639.

Bai, J. (1994). `Least Squares Estimation of a Shift in Linear Processes', Journal of Time Series Analysis, 15: 453-472.

Bai, J. (1995). `Least Absolute Deviation Estimation of a Shift', Econometric Theory, 11: 403-436.

Bai, J. (1997). `Estimation of a Change Point in Multiple Regression Models', Review of Economics and Statistics, 79: 551-563.

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

Bai, J., and Perron, P. (2003a). `Critical Values for Multiple Structural Change Tests', The Econometrics Journal, 6: 72-78.

Bai, J., and Perron, P. (2003b). `Multiple Structural Change Models: A Simulation Analysis', in Econometric Theory and Practice, pp. 212-237. Cambridge Univ. Press, Cambridge.

Banerjee, A., and Urga, G. (2005). `Modeling Structural Breaks, Long Memory and Stock Market Volatility: An Overview', Journal of Econometrics, 129: 1-34.

Bates, D. M., and Watts, D. (1988). Nonlinear Regression Analysis and Its Applications. John Wiley and Sons, New York.

Bec, F., Salem, M., and Collard, F. (2002). `Asymmetries in Monetary Policy Reaction Function: Evidence for the US, French and German Central Banks', Studies in Nonlinear Dynamics and Econometrics, 6: Art. 3.

Bhattacharya, P. K. (1994). `Some Aspects of Change-Point Analysis', in E. Carlstein, H.-G. Muller, and D. Siegmund (eds.), Change-Point Problems, vol. 23 of IMS Lecture Notes Monograph Series, pp. 28-56.

Caner, M. (2007). `Boundedly Pivotal Structural Change Tests in Continuous Updating GMM with Strong, Weak identification and Completely Unidentified Cases', Journal of Econometrics, 137: 28-67.

Carrasco, M., and Chen, X. (2002). `Mixing and Moment Properties of Various GARCH and Stochastic Volatility Models', Econometric Theory, 18: 17-39.

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

Csorgo, M., and Horvath, L. (1997). Limit Theorems for Change Point Analysis. Chichester-Wiley, Chichester.

Davidson, J. (2002). `Establishing Conditions for the Functional Central Limit Theorem in Nonlinear and Semiparametric Time Series Processes', Journal of Econometrics, 106: 179-190.

Davis, R. A., Lee, T. C. M., and Rodriguez-Yam, G. A. (2006). `Structural Break Estimation for Nonstationary Time Series Models', Journal of the American Statistical Association, 101: 223-239.

Dolado, J., Maria-Dolores, R., and Ruge-Murcia, M. (2004). `Nonlinear Monetary Policy Rules: Some New Evidence from US', Studies in Nonlinear Dynamics and Econometrics, 8: Art. 2.

Dufour, J.-M., and Ghysels, E. (1996). `Editor`s Introduction. Recent Developments in the Econometrics of Structural Change', Journal of Econometrics, 70: 1-8.

Eitrheim, ., and Terasvirta, T. (1996). `Testing the Adequacy of the Smooth Transition Autoregressive Models', Journal of Econometrics, 74: 59-75.

Elliott, G., and Muller, U. (2007). `Condence Sets for the Date of a Single Break in Linear Time Series Regressions', Journal of Econometrics, 141: 1196-1218.

Fan, J., and Yao, Q. (2003). Nonlinear Time Series: Nonparametric and Parametric Methods. Springer Series in Statistics, New York.

Florio, A. (2006). `Asymmetric Interest Rate Smoothing: The Fed Approach', Economic Letters, 93: 190-195.

Gallant, A. R. (1987). Nonlinear Statistical Models, Wiley Series in Probability and Mathematical Statistics. John Wiley and Sons, New York.

Gallant, A. R., and White, H. (1988). A Unied Theory of Estimation and Inference for Nonlinear Dynamic Models. Basil Blackwell, Oxford, UK.

Ghysels, E., and Hall, A. R. (1990). `A Test for Structural Stability of Euler Conditions Parameters Estimated via the Generalized Method of Moments Estimator', International Economic Review, 31: 355-364.

Hall, A. R., Han, S., and Boldea, O. (2009). `Inference Regarding Multiple Structural Changes in Linear Models with Endogenous Regressors', Discussion Paper, Department of Economics, North Carolina State University.

Boldea, O., Hall, A. and Han, S.(2010). `Asymptotic Distribution Theory for Break Point Estimators in Models Estimated via 2SLS', Discussion Paper, Department of Economics, North Carolina State University.

Hall, A. R., and Sakkas, N. (2010). `Approximate p-values of Certain Tests Involving Hypotheses about Multiple Breaks', work in progress.

Hall, A. R., and Sen, A. (1999). `Structural Stability Testing in Models Estimated by Generalized Method of Moments', Journal of Business & Economic Statistics, 17: 335-348.

Hansen, B. E. (2000). `Testing for Structural Change in Conditional Models', Journal of Econometrics, 97: 93-115.

Kesriyeli, M., Osborn, D., and Sensier, M. (2006). `Nonlinearity and Structural Change in Interest Rate Reaction Functions for the US, UK and Germany', in C. Milas, D. van Dijk, and P. Rothman (eds.), Nonlinear Time Series Analysis of Business Cycles, pp. 283-310.

Kim, D., Osborn, D., and Sensier, M. (2005). `Nonlinearity in the Fed's Monetary Policy Rule', Journal of Applied Econometrics, 20: 621-639.

Kokoszka, P., and Leipus, R. (2000). `Change-point Estimation in ARCH models', Bernoulli, 6: 513-539.

Krishnaiah, P. R., and Miao, B. Q. (1988). `Review about Estimation of Change Points', in P. R. Krishnaiah and C. P. Rao (eds.), Handbook of Statistics, vol. 7, pp. 375-402. New York: Elsevier.

Lavielle, M., and Moulines, E. (2000). `Least-Squares Estimation of an Unknown Number of Shifts in a Time Series', Journal of Time Series Analysis, 21: 33-59.

Lucas, R. (1976). `Econometric Policy Evaluation: A Critique', in K. Brunner and A. Melzer (eds.), The Phillips Curve and Labor Markets, vol. 1 of Carnegie- Rochester Conference Series on Public Policy, pp. 19-46.

Luukkonen, R., Saikkonen, P., and Teräsvirta, T. (1988). `Testing Linearity Against Smooth Transition Autoregressive Models', Biometrika, 75: 491-499.

Mokkadem, A. (1985). `Le Modele Non Lineaire AR(1) General. Ergodicite et Ergodicite Geometrique', Comptes Rendus de l'Academie des Sciences. Serie I. Mathematique, 331: 889-892.

Perron, P., and Qu, Z. (2006). `Estimating Restricted Structural Change Models', Journal of Econometrics, 134: 373-399.

Qu, Z., and Perron, P. (2007). `Estimating and Testing Multiple Structural Changes in Multivariate Regressions', Econometrica, 75: 459-502.

Quandt, R. E. (1958). `The Estimation of the Parameters of a Linear Regression System Obeying Two Separate Regimes', Journal of the American Statistical Association, 53: 873-880.

Quandt, R. E. (1960). `Tests of the Hypothesis that a Linear Regression System Obeys Two Separate Regimes', Journal of the American Statistical Association, 55: 324-330.

Rissanen, J. (1989). Stochastic Complexity in Statistical Inquiry, vol. 15 of World Scientic Series in Computer Science.World Scientic Publishing, Teaneck, NJ.

Rosenblatt, M. (1971). Markov processes: Structure and Asymptotic Behavior. Springer, New York.

Schaling, E. (2004). `The Nonlinear Phillips Curve and In ation Forecast Targeting: Symmetric versus Asymmetric Monetary Policy Rules', Journal of Money, Credit and Banking, 36: 361-386.

Sowell, F. (1996). `Optimal Tests for Parameter Instability in the Generalized Method of Moments Framework', Econometrica, 64: 1085-1107.

Stock, J. H., and Watson, M. W. (2002). `Has the Business Cycle Changed and Why?', NBER Macroeconomics Annual, pp. 159-218.

van Dijk, D., Terasvirta, T., and Franses, P. (2002). `Smooth Transition Autoregressive Models: A Survey of Recent Developments', Econometric Reviews, 21: 1-47.

Yao, Y.-C. (1988). `Estimating the Number of Change-Points via Schwarz' Criterion', Statistics and Probability Letters, 6: 181-189.

Zacks, S. (1983). `Survey of Classical and Bayesian Approaches to the Change-Point Problem: Fixed Sample and Sequential Procedures of Testing and Estimation', in Recent Advances in Statistics, pp. 245-269. Academic Press, New York.