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SEQUENTIAL MONITORING OF CHANGES IN DYNAMIC LINEAR MODELS, APPLIED TO THE U.S. HOUSING MARKET

Published online by Cambridge University Press:  23 March 2021

Lajos Horváth ,
Zhenya Liu  and
Shanglin Lu
Lajos Horváth
Affiliation:
University of Utah
Zhenya Liu*
Affiliation:
Renmin University of China and Aix-Marseille University
Shanglin Lu*
Affiliation:
Renmin University of China
*
Address correspondence to Zhenya Liu and Shanglin Lu, School of Finance, Renmin University of China, Beijing 100872, China; e-mail: [email protected], [email protected].
Address correspondence to Zhenya Liu and Shanglin Lu, School of Finance, Renmin University of China, Beijing 100872, China; e-mail: [email protected], [email protected].
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Abstract

We propose a sequential monitoring scheme to find structural breaks in dynamic linear models. The monitoring scheme is based on a detector and a suitably chosen boundary function. If the detector crosses the boundary function, a structural break is detected. We provide the asymptotics for the procedure under the null hypothesis of stability. The consistency of the procedure is also proved. We derive the asymptotic distribution of the stopping time under the change point alternative. Monte Carlo simulation is used to show the size and the power of our method under several conditions. As an example, we study the real estate markets in Boston and Los Angeles, and at the national U.S. level. We find structural breaks in the markets, and we segment the data into stationary segments. It is observed that the autoregressive parameter is increasing but stays below 1.

Type
ARTICLES
Information
Econometric Theory , Volume 38 , Issue 2 , April 2022 , pp. 209 - 272
Copyright
© The Author(s), 2021. Published by Cambridge University Press

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Footnotes

Part of the research was done while Shanglin Lu was visiting the University of Utah. We appreciate the support of the Department of Mathematics. We are grateful to three referees for their useful comments. The detailed remarks of Professor Peter C.B. Phillips, Editor in Chief, helped us to provide more general results, remove some unnecessary conditions, and improve the presentation of our results.

References

REFERENCES

Andrews, D.W.K. (1987) Least squares regression with integrated or dynamic regression under weak error assumptions. Econometric Theory 3(1), 98–116.CrossRefGoogle Scholar
Aue, A., Hörmann, S., Horváth, L., & Hušková, M. (2014) Dependent functional linear models with applications to monitoring structural change. Statistica Sinica 24(3), 1043–1073.Google Scholar
Aue, A., Hörmann, S., Horváth, L., Hušková, M., & Steinebach, J.G. (2012) Sequential testing for the stability of portfolio betas. Econometric Theory 28(4), 804–837.CrossRefGoogle Scholar
Aue, A. & Horváth, L. (2004) Delay time in sequential detection of change. Statistics & Probability Letters 67(3), 221–231.CrossRefGoogle Scholar
Berkes, I., Gombay, E., Horváth, L., & Kokoszka, P. (2004) Sequential change-point detection in GARCH $\left(p,q\right)$ models. Econometric Theory 20(6), 1140–1167.CrossRefGoogle Scholar
Billingsley, P. (1968) Convergence of Probability Measures. Wiley.Google Scholar
Brown, R.L., Durbin, J., & Evans, J.M. (1975) Techniques for testing the constancy of regression relationships over time (with discussion). Journal of the Royal Statistical Society: Series B 37(2), 149–163.Google Scholar
Caron, E. (2019) Asymptotic distribution of least square estimators for linear models with dependent errors. Statistics 53(4), 885–902.CrossRefGoogle Scholar
Case, K.E. & Shiller, R.J. (1989) The efficiency of the market for single-family homes. The American Economic Review 79(1), 125–137.Google Scholar
Case, K.E. & Shiller, R.J. (2003) Is there a bubble in the housing market? Brookings Papers on Economic Activity 2, 299–362.CrossRefGoogle Scholar
Chu, C.S.J., Stinchcombe, M., & White, H. (1996) Monitoring structural change. Econometrica 64, 1045–1065.CrossRefGoogle Scholar
Davis, M.A. & Heathcote, J. (2007) The price and quantity of residential land in the United States. Journal of Monetary Economics 54(8), 2595–2620.CrossRefGoogle Scholar
Dümbgen, L. (1991) The asymptotic behavior of some nonparametric change-point estimators. Annals of Statistics 19(3), 1471–1495.CrossRefGoogle Scholar
Gallin, J. (2006) The long-run relationship between house prices and income: Evidence from local housing markets. Real Estate Economics 34(3), 417–438.CrossRefGoogle Scholar
Ghysels, E. & Lieberman, O. (1996) Dynamic regression and filtered data series: A Laplace approximation to the effects of filtering in small samples. Econometric Theory 12(3), 432–457.CrossRefGoogle Scholar
Guay, A. & Guerre, E. (2006) A data-driven specification test for dynamic regression model. Econometric Theory 22(4), 543–586.CrossRefGoogle Scholar
Gyourko, J., Mayer, C., & Sinai, T. (2013) Superstar cities. American Economic Journal: Economic Policy 5(4), 167–199.Google Scholar
Himmelberg, C., Mayer, C. & Sinai, T. (2005) Assessing high house prices: Bubbles, fundamentals and misperceptions. Journal of Economic Perspectives 19(4), 67–92.CrossRefGoogle Scholar
Hlávka, Z., Hušková, M., Kirch, C., & Meintanis, S. (2012) Monitoring changes in the error distribution of autoregressive models based on Fourier methods. Test 21(4), 605–634.CrossRefGoogle Scholar
Homm, U. & Breitung, J. (2012) Testing speculative bubbles in stock markets: A comparison of alternative methods. Journal of Financial Econometrics 10(1), 198–231.CrossRefGoogle Scholar
Hörmann, S. & Kokoszka, P. (2010) Weakly dependent functional data. Annals of Statistics 38(3), 1845–1884.CrossRefGoogle Scholar
Horváth, L., Hušková, M., Kokoszka, P., & Steinebach, J. (2004) Monitoring changes in linear models. Journal of Statistical Planning and Inference 126(1), 225–251.CrossRefGoogle Scholar
Horváth, L., Kokoszka, P., & Steinebach, J. (2007) On sequential detection of parameter changes in linear regression. Statistics & Probability Letters 77(9), 885–895.CrossRefGoogle Scholar
Horváth, L., Kokoszka, P., & Zhang, A. (2006) Monitoring constancy of variance in conditionally heteroscedastic time series. Econometric Theory 22(3), 373–402.CrossRefGoogle Scholar
Horváth, L., Liu, Z., Rice, G., & Wang, S. (2020) Sequential monitoring for changes from stationarity to mild non-stationarity. Journal of Econometrics 215(1), 209–238.CrossRefGoogle Scholar
Hušková, M. & Kirch, C. (2012) Bootstrapping sequential change-point tests for linear regression. Metrika 75(5), 673–708.CrossRefGoogle Scholar
Ibragimov, I.A. (1959) Some limit theorems for strict-sense stationary stochastic processes (in Russian). Doklady Akademii Nauk SSSR 125, 711–714.Google Scholar
Ibragimov, I.A. (1962) Some limit theorems for stationary processes. Theory of Probability and Its Applications 7(4), 349–382.CrossRefGoogle Scholar
Kirch, C. (2007) Block permutation principles for the change analysis of dependent data. Journal of Statistical Planning and Inference 137(7), 2453–2474.CrossRefGoogle Scholar
Kirch, C. (2008) Bootstrapping sequential change-point tests. Sequential Analysis 27(3), 330–349.CrossRefGoogle Scholar
Knight, K. (1993) Estimating in dynamic regression model with infinite variance errors. Econometric Theory 9(4), 570–588.CrossRefGoogle Scholar
Krämer, W., Ploberger, W., & Alt, R. (1988) Testing for structural change in dynamic models. Econometrica 56, 1355—1370.CrossRefGoogle Scholar
Kwiatkowski, D., Phillips, P.C.B., Schmidt, P., & Shin, Y. (1992) Testing the null hypothesis of stationarity against the alternative of a unit root: How sure are we that economic time series have a unit root? Journal of Econometrics 54(1–3), 159–178.CrossRefGoogle Scholar
Leisch, F., Hornik, K., & Kuan, C-H. (2000) Monitoring structural changes with generalized fluctuation test. Econometric Theory 16(6), 835–854.CrossRefGoogle Scholar
Linton, O. (2019) Financial Econometrics: Models and Methods . Cambridge University Press.CrossRefGoogle Scholar
Phillips, P.C.B. & Shi, S. (2018) Financial bubble explosion and reverse regression. Econometric Theory 34(4), 705–753.CrossRefGoogle Scholar
Phillips, P.C.B., Shi, S., & Yu, J. (2014) Specification sensitivity in right-tailed unit root testing for explosive behaviour. Oxford Bulletin of Economics and Statistics 76(3), 315–333.CrossRefGoogle Scholar
Phillips, P.C.B., Shi, S., & Yu, J. (2015a) Testing for multiple bubbles: Historical episodes of exuberance and collapse in the S&P 500. International Economic Review 56(4), 1043–1078.CrossRefGoogle Scholar
Phillips, P.C.B., Shi, S., & Yu, J. (2015b) Testing for multiple bubbles: Limit theory of real-time detectors. International Economic Review 56(4), 1079–1134.CrossRefGoogle Scholar
Phillips, P.C.B. & Yu, J. (2011) Dating the timeline of financial bubbles during the subprime crisis. Quantitative Economics 2(3), 455–491.CrossRefGoogle Scholar
Piazzesi, M. & Schneider, M. (2009) Momentum traders in the housing market: Survey evidence and a search model. American Economic Review 99(2), 406–411.CrossRefGoogle Scholar
Picard, D. (1985) Testing and estimating change-points in time series. Advances in Applied Probability 17, 841–867.CrossRefGoogle Scholar
Saiz, A. (2010) The geographic determinants of housing supply. The Quarterly Journal of Economics 125(3), 1253–1296.CrossRefGoogle Scholar
Shiller, R.J. (2008) Historic turning points in real estate. Eastern Economic Journal 34(1), 1–13.CrossRefGoogle Scholar
Shiller, R.J. (2015) Irrational Exuberance: Revised and Expanded, 3rd Edition. Princeton University Press.CrossRefGoogle Scholar
Steland, A. (2007) Monitoring procedures to detect unit roots and stationarity. Econometric Theory 23(6), 1108–1135.CrossRefGoogle Scholar
Vogelsang, T.J. (1997) Wald-type tests for detecting breaks in the trend function of a dynamic time series. Econometric Theory 13(6), 818–848.CrossRefGoogle Scholar
Wu, W.B. (2007) M-estimates of linear models with dependent errors. Annals of Statistics 35(2), 495–521.CrossRefGoogle Scholar
Zeckhauser, R. & Thompson, M. (1970) Linear regression with non-normal error terms. Review of Economics and Statistics 52, 280–286.CrossRefGoogle Scholar
Zeileis, A., Leisch, F., Kleiber, C., & Hornik, K. (2005) Monitoring structural change in dynamic econometric models. Journal of Applied Econometrics 20(1), 99–121.CrossRefGoogle Scholar
Zheng, S., Sun, W., & Kahn, M.E. (2016) Investor confidence as a determinant of China’s urban housing market dynamics. Real Estate Economics 44(4), 814–845.CrossRefGoogle Scholar
Zhou, Z. & Shao, X. (2013) Inference for linear models with dependent errors. Journal of the Royal Statistical Society: Series B 75(2), 323–343.CrossRefGoogle Scholar