Hostname: page-component-745bb68f8f-b95js Total loading time: 0 Render date: 2025-01-12T04:25:38.159Z Has data issue: false hasContentIssue false
  • English
  • Français

TESTING FOR A SHIFT IN TREND AT AN UNKNOWN DATE: A FIXED-B ANALYSIS OF HETEROSKEDASTICITY AUTOCORRELATION ROBUST OLS-BASED TESTS

Published online by Cambridge University Press:  25 March 2011

Özgen Sayginsoy  and
Timothy J. Vogelsang
Özgen Sayginsoy
Affiliation:
Barclays
Timothy J. Vogelsang*
Affiliation:
Michigan State University
*
*Address correspondence to Tim Vogelsang, Department of Economics, Michigan State University, 110 Marshall-Adams Hall, East Lansing, MI 48824-1038, USA; e-mail: [email protected].
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper analyzes tests for a shift in the trend function of a time series at an unknown date based on ordinary least squares (OLS) estimates of the trend function. Inference about the trend parameters depends on the serial correlation structure of the data through the long-run variance (zero frequency spectral density) of the errors. Asymptotically pivotal tests can be obtained by the use of serial correlation robust standard errors that require an estimate of the long-run variance. The focus is on the class of nonparametric kernel estimators of the long-run variance. Tests based on these estimators present two problems for practitioners. The first is the choice of kernel and bandwidth. The second is the well-known overrejection problem caused by strong serial correlation (or a possible unit root) in the errors.We provide solutions to both problems by using the fixed-b asymptotic framework of Kiefer and Vogelsang (2005, Econometric Theory, 21, 1130–1164) in conjunction with the scaling factor approach of Vogelsang (1998, Econometrica 65, 123–148). Our results provide practitioners with a family of OLS-based trend function structural change tests that are size robust to the presence of strong serial correlation or a unit root. Specific recommendations are provided for the tuning parameters (kernel and bandwidth) in a way that maximizes asymptotic integrated power.

Type
ARTICLES
Information
Econometric Theory , Volume 27 , Issue 5: SPECIAL ISSUE ON BOOTSTRAP AND NUMERICAL METHODS IN TIME SERIES , October 2011 , pp. 992 - 1025
Copyright
Copyright © Cambridge University Press 2011

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. (1991) Heteroskedasticity and autocorrelation consistent covariance matrix estimation. Econometrica 59, 817–854.CrossRefGoogle Scholar
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. & Ploberger, W. (1994) Optimal tests when a nuisance parameter is present only under the alternative. Econometrica 62, 1383–1414.CrossRefGoogle Scholar
Bai, J. (1994) Least squares estimation of a shift in linear processes. Journal of Time Series Analysis 15, 453–470.CrossRefGoogle Scholar
Bai, J., Lumsdaine, R.L., & Stock, J.H. (1998) Testing for and dating breaks in integrated and cointegrated time series. Review of Economic Studies 65, 395–432.CrossRefGoogle Scholar
Bernard, A.B. & Durlauf, S.N. (1995) Convergence in international output. Journal of Applied Econometrics 10, 161–173.CrossRefGoogle Scholar
Bernard, A.B. & Durlauf, S.N. (1996) Interpreting tests for the convergence hypothesis. Journal of Econometrics 91, 161–173.CrossRefGoogle Scholar
Breitung, J. (2002) Nonparametric tests for unit roots and cointegration. Journal of Econometrics 108, 343–363.CrossRefGoogle Scholar
Bunzel, H. & Vogelsang, T.J. (2005) Powerful trend function tests that are robust to strong serial correlation with an application to the Prebish Singer hypothesis. Journal of Business & Economic Statistics 23, 381–394.CrossRefGoogle Scholar
Carlino, G.A. & Mills, L.O. (1993) Are U.S. regional incomes converging? Journal of Monetary Economics 32, 335–346.CrossRefGoogle Scholar
Chen, C.L. & Tiao, G.C. (1990) Random level-shift time series models, ARIMA approximations, and level-shift detection. Journal of Business & Economic Statistics 8, 83–97.CrossRefGoogle Scholar
Chu, C.S.J. & White, H. (1992) A direct test for changing trend. Journal of Business & Economic Statistics 10, 289–300.CrossRefGoogle Scholar
Frisch, R. & Waugh, F.V. (1933) Partial time regressions as compared with individual trends. Econometrica 1, 387–401.CrossRefGoogle Scholar
Gregory, A. & Hansen, B.E. (1996) Residual-based tests for cointegration in models with regime shifts. Journal of Econometrics 70, 99–126.CrossRefGoogle Scholar
Grenander, U. & Rosenblatt, M. (1957) Statistical Analysis of Stationary Time Series. Wiley.CrossRefGoogle Scholar
Harvey, D., Leybourne, S.J., & Taylor, A.M.R. (2009) Simple, robust and powerful tests of the breaking trend hypothesis. Econometric Theory 25, 995–1029.CrossRefGoogle Scholar
Kiefer, N.M. & Vogelsang, T.J. (2005) A new asymptotic theory for heteroskedasticity-autocorrelation robust tests. Econometric Theory 21, 1130–1164.CrossRefGoogle Scholar
Kramer, W., Ploberger, W., & Alt, R. (1988) Testing for structural change in dynamic models. Econometrica 56, 1355–1369.CrossRefGoogle Scholar
Kwiatkowski, D., Phillips, P., 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, 154–178.CrossRefGoogle Scholar
Loewy, M.B. & Papell, D.H. (1996) Are U.S. regional incomes converging? Some further evidence, Journal of Monetary Economics 38, 587–598.CrossRefGoogle Scholar
Park, J.Y. (1990) Testing for unit roots and cointegration by variable addition. In Fomby, T. & Rhodes, R. (eds.), Advances in Econometrics: Cointegration, Spurious Regressions and Unit Roots, pp. 107–134. Jai Press.Google Scholar
Park, J.Y. & Choi, B. (1988) A New Approach to Testing for a Unit Root. CAE Working paper 88–23, Cornell University.Google Scholar
Perron, P. (1989) The Great Crash, the oil price shock and the unit root hypothesis. Econometrica 57, 1361–1401.CrossRefGoogle Scholar
Perron, P. (1990) Testing for a unit root in a time series regression with a changing mean. Journal of Business & Economic Statistics 8, 153–162.CrossRefGoogle Scholar
Perron, P. & Yabu, T. (2009) Testing for shifts in trend with an integrated or stationary noise component. Journal of Business & Economic Statistics 27, 369–396.CrossRefGoogle Scholar
Perron, P. & Zhu, X. (2005) Structural breaks with stochastic and deterministic trends. Journal of Econometrics 129, 65–119.CrossRefGoogle Scholar
Phillips, P.C.B. (1987) Time series regression with unit roots. Econometrica 55, 277–302.CrossRefGoogle Scholar
Phillips, P.C.B. & Solo, V. (1992) Asymptotics for linear processes. Annals of Statistics 20, 971–1001.CrossRefGoogle Scholar
Roy, A., Falk, B., & Fuller, W. (2004) Testing for trend in the presence of autoregressive error. Journal of the American Statistical Association 99, 1082–1092.CrossRefGoogle Scholar
Sayginsoy, O. (2004) Essays on nonstandard testing of trend functions in time series models. Dissertation Abstracts International 65, 3938–4167A. Ph.D. dissertation, Cornell University.Google Scholar
Sayginsoy, O. (2006) Joint Inequality Tests of Trend Function Parameters. Working paper, Department of Economics, SUNY–University at Albany.Google Scholar
Sayginsoy, O. and Vogelsang, T.J. (2008) Testing for a Shift in Trend at an Unknown Date: A Fixed- b Analysis of Heteroskedasticity Autocorrelation Robust OLS Based Tests. Working paper, Department of Economics, Michigan State University.Google Scholar
Shin, Y. & Schmidt, P. (1992) The KPSS stationarity test as a unit root test. Economics Letters 38, 387–392.CrossRefGoogle Scholar
Tanaka, K. (1990) Testing for a moving average root. Econometric Theory 6, 433–444.CrossRefGoogle Scholar
Tomljanovic, M. & Vogelsang, T.J. (2002) Are U.S. regions converging? Using new econometric methods to examine old issues. Empirical Economics 27, 49–62.CrossRefGoogle Scholar
Vogelsang, T.J. (1997) Wald-type tests for detecting shifts in the trend function of a dynamic time series. Econometric Theory 13, 818–849.CrossRefGoogle Scholar
Vogelsang, T.J. (1998a) Testing for a shift in mean without having to estimate serial correlation parameters. Journal of Business & Economic Statistics 16, 73–80.CrossRefGoogle Scholar
Vogelsang, T.J. (1998b) Trend function hypothesis testing in the presence of serial correlation. Econometrica 65, 123–148.CrossRefGoogle Scholar