Optimal method in multiple regression with structural changes. (English) Zbl 1388.62159
Summary: In this paper, we consider an estimation problem of the regression coefficients in multiple regression models with several unknown change-points. Under some realistic assumptions, we propose a class of estimators which includes as a special cases shrinkage estimators (SEs) as well as the unrestricted estimator (UE) and the restricted estimator (RE). We also derive a more general condition for the SEs to dominate the UE. To this end, we generalize some identities for the evaluation of the bias and risk functions of shrinkage-type estimators. As illustrative example, our method is applied to the “gross domestic product” data set of 10 countries whose USA, Canada, UK, France and Germany. The simulation results corroborate our theoretical findings.
MSC:
| 62H12 | Estimation in multivariate analysis |
| 62J07 | Ridge regression; shrinkage estimators (Lasso) |
| 62P20 | Applications of statistics to economics |
Keywords:
ADB; ADR; change-points; multiple regression; pre-test estimators; restricted estimator; shrinkage estimators; unrestricted estimatorReferences:
| [1] | Bai, J. and Perron, P. (2003). Computation and analysis of multiple structural change models. J. Appl. Econometr. 18 1-22. |
| [2] | Baranchick, A. (1964). Multiple regression and estimation of the mean of a multivariate normal distribution. Technical Report No. 51, Dept. Statistics, Stanford Univ. |
| [3] | Braun, J.V. and Muller, H.G. (1998). Statistical methods for DNA sequence segmentation. Statist. Sci. 13 142-162. · Zbl 0960.62121 · doi:10.1214/ss/1028905933 |
| [4] | Broemeling, L.D. and Tsurumi, H. (1987). Econometrics and Structural Change. Statistics : Textbooks and Monographs 74 . New York: Dekker, Inc. · Zbl 0624.62110 |
| [5] | Fu, Y.-X. and Curnow, R.N. (1990). Locating a changed segment in a sequence of Bernoulli variables. Biometrika 77 295-304. · Zbl 0714.62019 · doi:10.1093/biomet/77.2.295 |
| [6] | Fu, Y.-X. and Curnow, R.N. (1990). Maximum likelihood estimation of multiple change points. Biometrika 77 563-573. · Zbl 0724.62025 · doi:10.1093/biomet/77.3.563 |
| [7] | Hossain, S., Doksum, K.A. and Ahmed, S.E. (2009). Positive shrinkage, improved pretest and absolute penalty estimators in partially linear models. Linear Algebra Appl. 430 2749-2761. · Zbl 1160.62066 · doi:10.1016/j.laa.2008.12.015 |
| [8] | James, W. and Stein, C. (1961). Estimation with quadratic loss. In Proc. 4 th Berkeley Sympos. Math. Statist. and Prob. I 361-379. Berkeley, CA: Univ. California Press. · Zbl 1281.62026 |
| [9] | Judge, G.G. and Bock, M.E. (1978). The Statistical Implications of Pre-Test and Stein-Rule Estimators in Econometrics . Amsterdam: North-Holland. · Zbl 0395.62078 |
| [10] | Judge, G.G. and Mittelhammer, R.C. (2004). A semiparametric basis for combining estimation problems under quadratic loss. J. Amer. Statist. Assoc. 99 479-487. · Zbl 1117.62367 · doi:10.1198/016214504000000430 |
| [11] | Lombard, F. (1986). The change-point problem for angular data: A nonparametric approach. Technometrics 28 391-397. · Zbl 0616.62059 · doi:10.2307/1268988 |
| [12] | McLeish, D.L. (1977). On the invariance principle for nonstationary mixingales. Ann. Probab. 5 616-621. · Zbl 0367.60021 · doi:10.1214/aop/1176995772 |
| [13] | Nkurunziza, S. (2011). Shrinkage strategy in stratified random sample subject to measurement error. Statist. Probab. Lett. 81 317-325. · Zbl 1205.62035 · doi:10.1016/j.spl.2010.10.020 |
| [14] | Nkurunziza, S. (2012). The risk of pretest and shrinkage estimators. Statistics 46 305-312. · Zbl 1241.62095 · doi:10.1080/02331888.2010.508561 |
| [15] | Nkurunziza, S. and Ahmed, S.E. (2010). Shrinkage drift parameter estimation for multi-factor Ornstein-Uhlenbeck processes. Appl. Stoch. Models Bus. Ind. 26 103-124. · Zbl 1224.91173 · doi:10.1002/asmb.775 |
| [16] | Perron, P. and Qu, Z. (2006). Estimating restricted structural change models. J. Econometrics 134 373-399. · Zbl 1418.62273 · doi:10.1016/j.jeconom.2005.06.030 |
| [17] | Perron, P. and Yabu, T. (2009). Testing for shifts in trend with an integrated or stationary noise component. J. Bus. Econom. Statist. 27 369-396. · Zbl 1431.62410 |
| [18] | Saleh, A.K.Md.E. (2006). Theory of Preliminary Test and Stein-Type Estimation with Applications. Wiley Series in Probability and Statistics . Hoboken, NJ: Wiley. · Zbl 1094.62024 · doi:10.1002/0471773751 |
| [19] | Tan, Z. (2014). Improved minimax estimation of a multivariate normal mean under heteroscedasticity. Bernoulli . · Zbl 1311.62086 · doi:10.3150/13-BEJ580 |
| [20] | Zeileis, A., Kleiber, C., Krämer, W. and Hornik, K. (2003). Testing and dating of structural changes in practice. Comput. Statist. Data Anal. 44 109-123. · Zbl 1429.62307 |
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