Hostname: page-component-76fb5796d-r6qrq Total loading time: 0 Render date: 2024-04-25T09:17:48.984Z Has data issue: false hasContentIssue false
  • English
  • Français

SHRINKAGE ESTIMATION OF REGRESSION MODELS WITH MULTIPLE STRUCTURAL CHANGES

Published online by Cambridge University Press:  23 June 2015

Junhui Qian  and
Liangjun Su
Junhui Qian
Affiliation:
Shanghai Jiao Tong University
Liangjun Su*
Affiliation:
Singapore Management University
*
*Address correspondence to Liangjun Su, School of Economics, Singapore Management University, 90 Stamford Road, Singapore 178903; e-mail: ljsu@smu.edu.sg.
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper, we consider the problem of determining the number of structural changes in multiple linear regression models via group fused Lasso. We show that with probability tending to one, our method can correctly determine the unknown number of breaks, and the estimated break dates are sufficiently close to the true break dates. We obtain estimates of the regression coefficients via post Lasso and establish the asymptotic distributions of the estimates of both break ratios and regression coefficients. We also propose and validate a data-driven method to determine the tuning parameter. Monte Carlo simulations demonstrate that the proposed method works well in finite samples. We illustrate the use of our method with a predictive regression of the equity premium on fundamental information.

Type
ARTICLES
Information
Econometric Theory , Volume 32 , Issue 6 , December 2016 , pp. 1376 - 1433
Copyright
Copyright © Cambridge University Press 2015 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Andrews, D.W.K. (1993) Tests for parameter instability and structural change with unknown change point. Econometrica 61, 821–856.CrossRefGoogle Scholar
Andrews, D.W.K. (2003) End-of-sample instability tests. Econometrica 71, 1661–1694.CrossRefGoogle Scholar
Andrews, D.W.K. & Ploberger, W. (1994) Optimal tests when a nuisance parameter is present only under the alternative. Econometrica 62, 1383–1414.Google Scholar
Angelosante, D. & Giannakis, G.B. (2012) Group Lassoing change-points in piecewise-constant AR processes. EURASIP Journal on Advances in Signal Processing 1(70), 1–16.Google Scholar
Bai, J. (1995) Least absolute deviation estimation of a shift. Econometric Theory 11, 403–436.Google Scholar
Bai, J. (1997a) Estimation of a change point in multiple regression models. Review of Economics and Statistics 79, 551–563.Google Scholar
Bai, J. (1997b) Estimating multiple breaks one at a time. Econometric Theory 13, 315–352.CrossRefGoogle Scholar
Bai, J. (1998) Estimation of multiple-regime regressions with least absolute deviation. Journal of Statistical Planning and Inference 74, 103–134.Google Scholar
Bai, J. (2010) Common breaks in means and variances for panel data. Journal of Econometrics 157, 78–92.CrossRefGoogle Scholar
Bai, J., Lumsdaine, R.L., & Stock, J. (1998) Testing and dating common breaks in multivariate time series. Review of Economic Studies 65, 395–432.Google Scholar
Bai, J. & Perron, P. (1998) Estimating and testing liner models with multiple structural changes. Econometrica 66, 47–78.Google Scholar
Bai, J. & Perron, P. (2003a) Computation and analysis of multiple structural change models. Journal of Applied Econometrics 18, 1–22.Google Scholar
Bai, J. & Perron, P. (2003b) Critical values for multiple structural change tests. Econometrics Journal 6, 72–78.CrossRefGoogle Scholar
Bai, J. & Perron, P. (2006) Multiple structural change models: A simulation analysis. In Corbae, D., Durlauf, S.N., & Hansen, B.E. (eds.), Econometric Theory and Practice. Cambridge University Press.Google Scholar
Baltagi, B.H., Feng, Q., & Kao, C. (2014) Estimation of Heterogeneous Panels with Structural Breaks. Working paper, Syracuse University.Google Scholar
Belloni, A., Chernozhukov, V., & Hansen, C. (2012) Sparse models and methods for optimal instruments with an application to eminent domain. Econometrica 80, 2369–2429.Google Scholar
Belloni, A., Chernozhukov, V., & Hansen, C. (2014) Inference on treatment effects after selection amongst high-dimensional controls. Review of Economic Studies 81, 608–650.Google Scholar
Bertsekas, D. (1995) Nonlinear Programming. Athena Scientific.Google Scholar
Bleakley, K. & Vert, J-P. (2011) The Group Fused Lasso for Multiple Change Point Detection. Working paper, INRIA Saclay, Orsay, France.Google Scholar
Caner, M. (2009) Lasso-type GMM estimator. Econometric Theory 25, 270–290.Google Scholar
Caner, M. & Fan, M. (2011) A Near Minimax Risk Bound: Adaptive Lasso with Heteroskedastic Data in Instrumental Variable Selection. Working paper, North Carolina State University.Google Scholar
Caner, M. & Knight, K. (2013) An alternative to unit root tests: Bridge estimators differentiate between nonstationary versus stationary models and select optimal lag. Journal of Statistical Planning and Inference 143, 691–715.Google Scholar
Chan, F., Mancini-Griffoli, T., & Pauwels, L.L. (2008) Stability Tests for Heterogenous Panel. Working paper, Curtin University of Technology.Google Scholar
Cheng, X., Liao, Z., & Schorfheide, F. (2014) Shrinkage Estimation of High-Dimensional Factor Models with Structural Instabilities. NBER Working Paper No. 19792.Google Scholar
De Watcher, S. & Tzavalis, E. (2005) Monte Carlo comparison of model and moment selection and classical inference approaches to break detection in panel data models. Economics Letters 99, 91–96.Google Scholar
De Watcher, S. & Tzavalis, E. (2012) Detection of structural breaks in linear dynamic panel data models. Computational Statistics and Data Analysis 56, 3020–3034.Google Scholar
Fan, J. & Peng, H. (2004) Nonconcave penalized likelihood with a diverging number of parameters. Annals of Statistics 32, 928–961.CrossRefGoogle Scholar
Friedman, J., Hastie, T., Höfling, H., & Tibshirani, R. (2007) Pathwise coordinate optimization. Annals of Applied Statistics 1, 302–332.Google Scholar
Grant, M., Boyd, S., & Ye, Y. (2009) CVX: Matlab Software for Disciplined Convex Programming. Mimeo.Google Scholar
Hall, P. & Heyde, C.C. (1980) Martingale Limit Theory and its Applications. Academic Press.Google Scholar
Harchaoui, Z. & Lévy-Leduc, C. (2010) Multiple change-point estimation with a total variation penalty. Journal of the American Statistical Association 105, 1481–1493.Google Scholar
Hsu, C-C. & Lin, C-C. (2012) Change-Point Estimation for Nonstationary Panel. Working paper, National Central University.Google Scholar
Kim, D. (2011) Estimating a common deterministic time trend break in large panels with cross sectional dependence. Journal of Econometrics 164, 310–330.Google Scholar
Kim, D. (2014) Common breaks in time trends for large panel data with a factor structure. The Econometrics Journal 17, 301–337.Google Scholar
Knight, K. & Fu, W. (2000) Asymptotics for Lasso-type estimators. Annals of Statistics 28, 1356–1378.Google Scholar
Kock, A.B. (2013) Oracle efficient variable selection in random and fixed effects panel data models. Econometric Theory 29, 115–152.Google Scholar
Kurozumi, E. (2012) Testing for Multiple Structural Changes with Non-Homogeneous Regressors. Working paper, Hitotsubashi University.Google Scholar
Kurozumi, E. & Arai, Y. (2006) Efficient estimation and inference in cointegrating regressions with structural breaks. Journal of Time Series Analysis 28, 545–575.Google Scholar
Lam, C. & Fan, J. (2008) Profile-kernel likelihood inference with diverging number of parameters. Annals of Statistics 36, 2232–2260.Google Scholar
Leeb, H. & Pötscher, B.M. (2005) Model selection and inference: Facts and fiction. Econometric Theory 21, 21–59.Google Scholar
Leeb, H. & Pötscher, B.M. (2008) Sparse estimators and the oracle property, or the return of the Hodges estimator. Journal of Econometrics 142, 201–211.Google Scholar
Liao, Z. (2013) Adaptive GMM shrinkage estimation with consistent moment selection. Econometric Theory 29, 857–904.Google Scholar
Liao, Z. & Phillips, P.C.B. (2014) Automated estimation of vector error correction models. Econometric Theory. Forthcoming.Google Scholar
Liao, W. & Wang, P. (2012) Structural Breaks in Panel Data Models: A Common Distribution Approach. Working paper, HKUST.Google Scholar
Liu, Q. & Watbled, F. (2009) Exponential inequalities for martingales and asymptotic properties of the free energy of directed polymers in a random experiment. Stochastic Processes and Their Applications 119, 3101–3132.Google Scholar
Lu, X. & Su, L. (2013) Shrinkage Estimation of Dynamic Panel Data Models with Interactive Fixed Effects. Working paper, Singapore Management University.Google Scholar
Lu, X. & Su, L. (2015) Jackknife model averaging for quantile regressions. Journal of Econometrics 188, 40–58.Google Scholar
Merlevède, F., Peligrad, M., & Rio, E. (2009) Bernstein inequality and moderate deviations under strong mixing conditions. IMS collections. High Dimensional Probability 5, 273–292.Google Scholar
Merlevède, F., Peligrad, M., & Rio, E. (2011) A Bernstein type inequality and moderate deviations for weakly dependent sequences. Probability Theory and Related Fields 151, 435–474.Google Scholar
Ohlsson, H., Ljung, L., & Boyd, S. (2010) Segmentation of ARX-models using sum-of-norms regularization. Automatica 46, 1107–1111.Google Scholar
Perron, P. (2006) Dealing with structural breaks. In Mills, T.C. & Patterson, K. (eds.), Palgrave Handbook of Econometrics, Econometric Theory, vol. 1, pp. 278–352. Palgrave Macmillan.Google Scholar
Pötscher, B.M. & Leeb, H. (2009) On the distribution of penalized maximum likelihood estimators: The LASSO, SCAD, and thresholding. Journal of Multivariate Analysis 100, 2065–2082.Google Scholar
Pötscher, B.M. & Schneider, U. (2009) On the distribution of the adaptive LASSO estimator. Journal of Statistical Planning and Inference 139, 2775–2790.Google Scholar
Qian, J. & Su, L. (2014) Shrinkage Estimation of Regression Models with Multiple Structural Changes. Working paper, Singapore Management University.Google Scholar
Qu, Z. & Perron, P. (2007) Estimating and testing structural changes in multiple regressions. Econometrica 75, 459–502.Google Scholar
Rinaldo, A. (2009) Properties and refinement of the fused Lasso. Annals of Statistics 37, 2922–2952.Google Scholar
Su, L. & White, H. (2010) Testing structural change in partially linear models. Econometric Theory 26, 1761–1806.Google Scholar
Su, L., Xu, P., & Ju, H. (2013) Pricing for Goodwill: A Threshold Quantile Regression Approach. Working paper, Singapore Management University.Google Scholar
Tibshirani, R.J. (1996) Regression shrinkage and selection via the Lasso. Journal of the Royal Statistical Society, Series B 58, 267–288.Google Scholar
Tibshirani, R., Saunders, M., Rosset, S., Zhu, J., & Knight, K. (2005) Sparsity and smoothness via the fused Lasso. Journal of the Royal Statistical Society, Series B 67, 91–108.Google Scholar
Welch, I. & Goyal, A. (2008) A comprehensive look at the empirical performance of equity premium prediction. Review of Financial Studies 21, 1455–1508.Google Scholar
White, H. (2001) Asymptotic Theory for Econometricians, 2nd ed. Emerald.Google Scholar
Yuan, M. & Lin, Y. (2006) Model selection and estimation in regression with grouped variables. Journal of the Royal Statistical Society, Series B 68, 49–67.Google Scholar