Skip to main content


  • Fabrizio Iacone (a1), Stephen J. Leybourne (a2) and A.M. Robert Taylor (a3)

We develop a test, based on the Lagrange multiplier [LM] testing principle, for the value of the long memory parameter of a univariate time series that is composed of a fractionally integrated shock around a potentially broken deterministic trend. Our proposed test is constructed from data which are de-trended allowing for a trend break whose (unknown) location is estimated by a standard residual sum of squares estimator applied either to the levels or first differences of the data, depending on the value specified for the long memory parameter under the null hypothesis. We demonstrate that the resulting LM-type statistic has a standard limiting null chi-squared distribution with one degree of freedom, and attains the same asymptotic local power function as an infeasible LM test based on the true shocks. Our proposed test therefore attains the same asymptotic local optimality properties as an oracle LM test in both the trend break and no trend break environments. Moreover, this asymptotic local power function does not alter between the break and no break cases and so there is no loss in asymptotic local power from allowing for a trend break at an unknown point in the sample, even in the case where no break is present. We also report the results from a Monte Carlo study into the finite-sample behaviour of our proposed test.

Corresponding author
*Address correspondence to Robert Taylor, Essex Business School, University of Essex, Colchester, CO4 3SQ, UK; e-mail:
Hide All

We are grateful to the Editor, Peter Phillips, three anonymous referees and the Co-Editor, Anna Mikusheva, for their very helpful and constructive comments. Taylor gratefully acknowledges financial support provided by the Economic and Social Research Council of the United Kingdom under research grant ES/M01147X/1.

Hide All
Angeloni, I., Aucremanne, L., Ehrmann, M., Gali, J., Levin, A., & Smets, F. (2006) New evidence on inflation persistence and price stickiness in the Euro area: Implications for macro modeling. Journal of the European Economic Association 4, 562574.
Baillie, R.T., Chung, C.-F., & Tieslau, M.A. (1996) Analysing inflation by the fractionally integrated ARFIMA–GARCH model. Journal of Applied Econometrics 11, 2340.
Boivin, J. & Giannoni, M.P. (2006) Has monetary policy become more effective? Review of Economics and Statistics 88, 445462.
Busetti, F. & Harvey, A.C. (2001) Testing for the presence of a random walk in series with structural breaks. Journal of Time Series Analysis 22, 127150.
Busetti, F. & Harvey, A.C. (2003) Further comments on stationarity tests in series with structural breaks at unknown points. Journal of Time Series Analysis 24, 137140.
Chang, S.Y. & Perron, P. (2016) Inference on a structural break in trend with fractionally integrated errors. Journal of Time Series Analysis 37, 555574.
Clarida, R., Gali, J., & Gertler, M. (2000) Monetary policy rules and macroeconomic stability: Evidence and some theory. The Quarterly Journal of Economics 115, 147180.
Dickey, D.A. & Fuller, W.A. (1979) Distribution of the estimators for autoregressive time series with a unit root. Journal of the American Statistical Association 74, 427431.
Diebold, F.X. & Inoue, A. (2001) Long memory and structural change. Journal of Econometrics 105, 131159.
Fuhrer, J.C. (2011) Inflation persistence. Handbook of Monetary Economics 3, 423486.
Gourieroux, C. & Jasiak, J. (2001) Memory and infrequent breaks. Economics Letters 70, 2941.
Granger, C.W.J. & Hyung, N. (2004) Occasional structural breaks and long memory with an application to the S&P 500 absolute stock returns. Journal of Empirical Finance 11, 399421.
Hamilton, J.D. (1994) Time Series Analysis. Princeton University Press.
Harvey, A.C. (1993) Time Series Models. Harvester Wheatsheaf.
Hassler, U. & Wolters, J. (1995) Long memory in inflation rates: International evidence. Journal of Business and Economic Statistics 13, 3745.
Iacone, F. (2010) Local Whittle estimation of the memory parameter in presence of deterministic components. Journal of Time Series Analysis 31, 3749.
Iacone, F., Leybourne, S.J., & Taylor, A.M.R. (2013a) On the behavior of fixed-b trend break tests under fractional integration. Econometric Theory 29, 393418.
Iacone, F., Leybourne, S.J., & Taylor, A.M.R. (2013b) Testing for a break in trend when the order of integration is unknown. Journal of Econometrics 176, 3045.
Johansen, S. & Nielsen, M.Ø. (2010) Likelihood inference for a nonstationary fractional autoregressive model. Journal of Econometrics 158, 5166.
Johansen, S. & Nielsen, M.Ø. (2012) A necessary moment condition for the fractional functional central limit theorem. Econometric Theory 28, 671679.
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, 159178.
Lavielle, M. & Moulines, E. (2000) Least squares estimation of an unknown number of shifts in a time series. Journal of Time Series Analysis 21, 3359.
Marinucci, D. & Robinson, P.M. (2000) Weak convergence of multivariate fractional processes. Stochastic Processes and their Applications 86, 103120.
McCloskey, A. & Perron, P. (2013) Memory parameter estimation in the presence of level shifts and deterministic trend. Econometric Theory 29, 11961237.
Mikosch, T. & Starica, K. (2004) Nonlinearity in financial time series, the long range dependence and the I-GARCH effect. Review of Economics and Statistics 86, 378390.
Nielsen, M.Ø. (2004) Efficient likelihood inference in nonstationary univariate models. Econometric Theory 20, 116146.
Perron, P. (1989) The great crash, the oil price shock, and the unit root hypothesis. Econometrica 57, 13611401.
Perron, P. & Rodríguez, G. (2003) GLS detrending, efficient unit root tests and structural change. Journal of Econometrics 115, 127.
Qu, Z. (2011) A test against spurious long memory. Journal of Business and Economic Statistics 29, 423438.
Robinson, P.M. (1994) Efficient tests of nonstationary hypotheses. Journal of the American Statistical Association 89, 14201437.
Schwarz, G. (1978) Estimating the dimension of a model. Annals of Statistics 6, 461464.
Shimotsu, K. (2010) Exact local Whittle estimation of fractional integration with unknown mean and time trend. Econometric Theory 26, 501540.
Shimotsu, K. & Phillips, P.C.B. (2005) Exact local Whittle estimation of fractional integration. Annals of Statistics 33, 18901933.
Tanaka, K. (1999) The nonstationary fractional unit root. Econometric Theory 15, 549582.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Econometric Theory
  • ISSN: 0266-4666
  • EISSN: 1469-4360
  • URL: /core/journals/econometric-theory
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
Type Description Title
Supplementary materials

Iacone et al. supplementary material
Iacone et al. supplementary material 1

 PDF (399 KB)
399 KB


Altmetric attention score

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed