Truncated sum-of-squares estimation of fractional time series models with generalized power law trend. (English) Zbl 1493.62526

Summary: We consider truncated (or conditional) sum-of-squares estimation of a parametric fractional time series model with an additive deterministic structure. The latter consists of both a drift term and a generalized power law trend. The memory parameter of the stochastic component and the power parameter of the deterministic trend component are both considered unknown real numbers to be estimated and belonging to arbitrarily large compact sets. Thus, our model captures different forms of nonstationarity and noninvertibility as well as a very flexible deterministic specification. As in related settings, the proof of consistency (which is a prerequisite for proving asymptotic normality) is challenging due to non-uniform convergence of the objective function over a large admissible parameter space and due to the competition between stochastic and deterministic components. As expected, parameter estimates related to the deterministic component are shown to be consistent and asymptotically normal only for parts of the parameter space depending on the relative strength of the stochastic and deterministic components. In contrast, we establish consistency and asymptotic normality of parameter estimates related to the stochastic component for the entire parameter space. Furthermore, the asymptotic distribution of the latter estimates is unaffected by the presence of the deterministic component, even when this is not consistently estimable. We also include Monte Carlo simulations to illustrate our results.


62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
62F12 Asymptotic properties of parametric estimators
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[1] ABADIR, K. M., DISTASO, W. and GIRAITIS, L. (2007). Nonstationarity-extended local Whittle estimation. Journal of Econometrics 141 1353-1384. · Zbl 1418.62297
[2] ABRAMOWITZ, M. and STEGUN, I. A. (1970). Handbook of Mathematical Functions. National Bureau of Standards, Washington, D.C.
[3] BLOOMFIELD, P. (1973). An exponential model for the spectrum of a scalar time series. Biometrika 60 217-226. · Zbl 0261.62074
[4] BOX, G. E. P. and JENKINS, G. M. (1971). Time Series Analysis, Forecasting and Control. Holden Day, San Francisco.
[5] CAVALIERE, G., NIELSEN, M. Ø. and TAYLOR, A. M. R. (2015). Bootstrap score tests for fractional integration in heteroskedastic ARFIMA models, with an application to price dynamics in commodity spot and futures markets. Journal of Econometrics 187 557-579. · Zbl 1337.91138
[6] CAVALIERE, G., NIELSEN, M. Ø. and TAYLOR, A. M. R. (2017). Quasi-maximum likelihood estimation of bootstrap inference in fractional time series with heteroskedasticity of unknown form. Journal of Econometrics 198 165-188. · Zbl 1456.62182
[7] CAVALIERE, G., NIELSEN, M. Ø. and TAYLOR, A. M. R. (2022). Adaptive inference in heteroskedastic fractional time series models. Journal of Business and Economic Statistics 40 50-65.
[8] CHEN, W. W. and HURVICH, C. M. (2003). Estimating fractional cointegration in the presence of polynomial trends. Journal of Econometrics 117 95-121. · Zbl 1027.62066
[9] DAHLHAUS, R. (1989). Efficient parameter estimation for self-similar processes. Annals of Statistics 17 1749-1766. · Zbl 0703.62091
[10] FOX, R. and TAQQU, M. S. (1986). Large-sample properties of parameter estimates for strongly dependent stationary Gaussian time series. Annals of Statistics 14 517-532. · Zbl 0606.62096
[11] GIRAITIS, L. and SURGAILIS, D. (1990). A central limit theorem for quadratic forms in strongly dependent linear variables and its application to asymptotic normality of Whittle’s estimate. Probability Theory and Related Fields 86 87-104. · Zbl 0717.62015
[12] Hall, P. and Heyde, C. C. (1980). Martingale Limit Theory and Its Application. Academic Press, New York. · Zbl 0462.60045
[13] HANNAN, E. J. (1973). The asymptotic theory of linear time series models. Journal of Applied Probability 10 130-145. · Zbl 0261.62073
[14] HANSEN, B. E. (1996). Inference when a nuisance parameter is not identified under the null hypothesis. Econometrica 64 413-430. · Zbl 0862.62090
[15] HEYDE, C. C. and DAI, W. (1996). On the robustness to small trends of estimation based on the smoothed periodogram. Journal of Time Series Analysis 17 141-150. · Zbl 0845.62059
[16] HUALDE, J. and NIELSEN, M. Ø. (2020). Truncated sum of squares estimation of fractional time series models with deterministic trends. Econometric Theory 36 393-412. · Zbl 1447.62025
[17] HUALDE, J. and NIELSEN, M. Ø. (2022). Truncated sum-of-squares estimation of fractional time series models with generalized power law trend. CREATES Research Papers No. 2022-07, Aarhus University.
[18] HUALDE, J. and ROBINSON, P. M. (2011). Gaussian pseudo-maximum likelihood estimation of fractional time series models. Annals of Statistics 39 3152-3181. · Zbl 1246.62186
[19] IACONE, F. (2010). Local Whittle estimation of the memory parameter in presence of deterministic components. Journal of Time Series Analysis 31 37-49. · Zbl 1224.62057
[20] JOHANSEN, S. and NIELSEN, M. Ø. (2010). Likelihood inference for a nonstationary fractional autoregressive model. Journal of Econometrics 158 51-66. · Zbl 1431.62389
[21] Johansen, S. and Nielsen, M. Ø. (2012a). Likelihood inference for a fractionally cointegrated vector autoregressive model. Econometrica 80 2667-2732. · Zbl 1274.62598 · doi:10.3982/ECTA9299
[22] Johansen, S. and Nielsen, M. Ø. (2012b). A necessary moment condition for the fractional functional central limit theorem. Econometric Theory 28 671-679. · Zbl 1251.60028 · doi:10.1017/S0266466611000697
[23] JOHANSEN, S. and NIELSEN, M. Ø. (2016). The role of initial values in conditional sum-of-squares estimation of nonstationary fractional time series models. Econometric Theory 32 1095-1139. · Zbl 1441.62757
[24] JOHANSEN, S. and NIELSEN, M. Ø. (2019). Nonstationary cointegration in the fractionally cointegrated VAR model. Journal of Time Series Analysis 40 519-543. · Zbl 1421.62122
[25] LI, W. K. and MCLEOD, A. I. (1986). Fractional time series modelling. Biometrika 73 217-221.
[26] NIELSEN, M. Ø. (2015). Asymptotics for the conditional sum-of-squares estimator in multivariate fractional time series models. Journal of Time Series Analysis 36 154-188. · Zbl 1326.62191
[27] PHILLIPS, P. C. B. (2007). Regression with slowly varying regressors and nonlinear trends. Econometric Theory 23 557-614. · Zbl 1237.62085
[28] ROBINSON, P. M. (1994). Efficient tests of nonstationary hypotheses. Journal of the American Statistical Association 89 1420-1437. · Zbl 0813.62016
[29] ROBINSON, P. M. (2005). Efficiency improvements in inference on stationary and nonstationary fractional tim series. Annals of Statistics 33 1800-1842. · Zbl 1078.62096
[30] ROBINSON, P. M. (2012). Inference on power law trends. Bernoulli 18 644-677. · Zbl 1238.62106
[31] ROBINSON, P. M. and IACONE, F. (2005). Cointegration in fractional systems with deterministic trends. Journal of Econometrics 129 263-298. · Zbl 1335.62144
[32] ROBINSON, P. M. and MARINUCCI, D. (2000). The averaged periodogram for nonstationary vector time series. Statistical Inference for Stochastic Processes 3 149-160. · Zbl 0966.62061
[33] VELASCO, C. (1999a). Non-stationary log-periodogram regression. Journal of Econometrics 91 325-371. · Zbl 1041.62533
[34] VELASCO, C. (1999b). Gaussian semiparametric estimation of non-stationary time series. Journal of Time Series Analysis 20 87-127. · Zbl 0922.62093
[35] WHITE, H. and GRANGER, C. W. J. (2011). Consideration of trends in time series. Journal of Time Series Econometrics 3. Issue 1, article 2. · Zbl 1266.62071
[36] WU, C. (1981). Asymptotic theory of nonlinear least squares estimation. Annals of Statistics 9 501-513. · Zbl 0475.62050
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