Harmonically weighted processes. (English) Zbl 1444.62105
The article focuses mainly on two problems related to harmonically weighted processes: the persistence and long memory properties of harmonically weighted processes (HWP) that differ from features under fractional integration (FI) and the discrimination between HW and FI. One starts by formally introducing a harmonically weighted filter h(L) that defines a harmonically weighted process. Different assumptions and HWP properties as well as large sample properties of the sample mean of HWP are discussed. A functional central limit theorem is obtained. Results on the discrimination between HWP and FI are presented and computer experiments by means of Monte Carlo simulation are shown. A practical example on USA Inflation together with some Concluding remarks end the article. An Appendix to this paper contains the proofs of results.
Reviewer: Claudia Simionescu-Badea (Wien)
MSC:
| 62M10 | Time series, auto-correlation, regression, etc. in statistics (GARCH) |
| 62R10 | Functional data analysis |
| 60F17 | Functional limit theorems; invariance principles |
| 60G10 | Stationary stochastic processes |
| 62P20 | Applications of statistics to economics |
References:
| [1] | AbadirKM, DistasoW, GiraitisL, KoulHL. 2014. Asymptotic normality for weighted sums of linear processes. Econometric Theory30: 252-284. · Zbl 1316.60030 |
| [2] | AndrewsDWK. 1991. Heteroskedasticity and autocorrelation consistent covariance matrix estimation. Econometrica59: 817-858. · Zbl 0732.62052 |
| [3] | BaillieRT, ChungC‐F, TieslauMA. 1996. Analysing inflation by the fractionally integrated ARFIMA‐GARCH model. Journal of Applied Econometrics11: 23-40. |
| [4] | BeranJ, FengY, GhoshS, KulikR. 2013. Long‐Memory Processes: Probabilistic Properties and Statistical Methods: Springer. · Zbl 1282.62187 |
| [5] | Berenguer‐RicoV, GonzaloJ. 2014. Summability of stochastic processes: A generalization of integration for non‐linear processes. Journal of Econometrics178: 331-341. · Zbl 1293.62176 |
| [6] | BillingsleyP. 1968. Convergence of Probability Measures: Wiley. · Zbl 0172.21201 |
| [7] | BlagouchineI. V.2016. Two series expansions for the logarithm of the gamma function involving Stirling numbers and containing only rational coefficients for certain arguments related to π^−1. Journal of Mathematical Analysis and Applications442: 404-434. · Zbl 1339.33003 |
| [8] | BorweinD., BorweinJ. M.1995. On an intriguing integral and some series related to ζ(4). Proceedings of the American Mathematical Society123: 1191-1198. · Zbl 0840.11036 |
| [9] | BreitungJ, HasslerU. 2002. Inference on the cointegration rank in fractionally integrated processes. Journal of Econometrics110: 167-185. · Zbl 1038.62075 |
| [10] | BrillingerDR. 1975. Time Series: Data Analysis and Theory: Holt, Rinehart and Winston. · Zbl 0321.62004 |
| [11] | DelgadoMA, RobinsonPM. 1994. New methods for the analysis of long‐memory time‐series: Application to Spanish inflation. Journal of Forecasting13, 2: 97-107. |
| [12] | DemetrescuM, HasslerU, KuzinV. 2011. Pitfalls of post‐model‐selection testing: Experimental quantification. Empirical Economics40, 2: 359-372. |
| [13] | DemetrescuM, KuzinV, HasslerU. 2008. Long memory testing in the time domain. Econometric Theory24, 01: 176-215. · Zbl 1280.62024 |
| [14] | FullerWA. 1996. Introduction to Statistical Time Series, 2nd ed.: Wiley. · Zbl 0851.62057 |
| [15] | GewekeJ, Porter‐HudakS. 1983. The estimation and application of long memory time series models. Journal of Time Series Analysis4: 221-238. · Zbl 0534.62062 |
| [16] | GiraitisL, KoulHL, SurgailisD. 2012. Large Sample Inference for Long Memory Processes: Imperial College Press. · Zbl 1279.62016 |
| [17] | GrangerCWJ. 1980. Long memory relationships and the aggregation of dynamic models. Journal of Econometrics14: 227-238. · Zbl 0466.62108 |
| [18] | GrangerCWJ, JoyeuxR. 1980. An introduction to long‐memory time series models and fractional differencing. Journal of Time Series Analysis1: 15-29. · Zbl 0503.62079 |
| [19] | HaldrupN, Vera ValdésJE. 2017. Long memory, fractional integration, and cross‐sectional aggregation. Journal of Econometrics199: 1-11. · Zbl 1452.62642 |
| [20] | HasslerU. 2019. Time Series Analysis with Long Memory in View: Wiley. · Zbl 1407.62019 |
| [21] | HasslerU, WoltersJ. 1995. Long memory in inflation rates: International evidence. Journal of Business & Economic Statistics13: 37-45. |
| [22] | HurvichCM, DeoR, BrodskyJ. 1998. The mean squared error of Geweke and Porter‐Hudak’s estimator of the memory parameter of a long‐memory time series. Journal of Time Series Analysis19: 19-46. · Zbl 0920.62108 |
| [23] | KnoppK. 1951. Theory and Application of Infinite Series, 2nd ed.: Blackie & Son. · Zbl 0042.29203 |
| [24] | MarinucciD, RobinsonPM. 1999. Alternative forms of fractional Brownian motion. Journal of Statistical Planning and Inference80: 111-122. · Zbl 0934.60071 |
| [25] | MureşanM.2009. A Concrete Approach to Classical Analysis: Springer. · Zbl 1163.26001 |
| [26] | PalmaW. 2007. Long‐Memory Time Series: Theory and Methods: Wiley. · Zbl 1183.62153 |
| [27] | PhillipsPCB, SoloV. 1992. Asymptotics for linear processes. The Annals of Statistics20: 971-1001. · Zbl 0759.60021 |
| [28] | RobinsonPM. 1991. Testing for strong serial correlation and dynamic conditional heteroskedasticity in multiple regression. Journal of Econometrics47, 1: 67-84. · Zbl 0734.62070 |
| [29] | RobinsonPM. 1994. Efficient tests of nonstationary hypotheses. Journal of the American Statistical Association89: 1420-1437. · Zbl 0813.62016 |
| [30] | RobinsonPM. 1995. Log‐periodogram regression of time series with long range dependence. Annals of Statistics23: 1048-1072. · Zbl 0838.62085 |
| [31] | RobinsonPM. 2014. The estimation of misspecified long memory models. Journal of Econometrics178: 225-230. · Zbl 1293.62202 |
| [32] | ShimotsuK. 2010. Exact local Whittle estimation of fractional integration with unknown mean and trend. Econometric Theory26: 501-540. · Zbl 1185.62163 |
| [33] | ShimotsuK, PhillipsPCB. 2005. Exact local Whittle estimation of fractional integration. The Annals of Statistics33: 1890-1933. · Zbl 1081.62069 |
| [34] | TanakaK. 1999. The nonstationary fractional unit root. Econometric Theory15: 549-582. · Zbl 0985.62073 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
