×

Testing for boundary conditions in case of fractionally integrated processes. (English) Zbl 1457.62263

Summary: Bounded integrated time series are a recent development of the time series literature. In this paper, we work on testing the presence of unknown boundaries with particular attention to the class of fractionally integrated time series. We firstly show, via a preliminary Monte Carlo experiment, the effects of neglected boundaries conditions on the most commonly used estimators of the long memory parameter. Then, we develop a sieve bootstrap test to distinguish between unbounded and bounded fractionally integrated time series. We assess the finite sample performance of our test with a Monte Carlo experiment and apply it to the data set of the time series of the Danish Krone/Euro exchange rate.

MSC:

62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
62G10 Nonparametric hypothesis testing
62G09 Nonparametric statistical resampling methods
62P20 Applications of statistics to economics
60G22 Fractional processes, including fractional Brownian motion
65C05 Monte Carlo methods

Software:

R
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Andrews, D., Heteroskedasticity and autocorrelation consistent covariance matrix estimation, Econometrica, 59, 817-858 (1991) · Zbl 0732.62052 · doi:10.2307/2938229
[2] Bartlett, MS, Periodogram analysis and continuous spectra, Biometrika, 37, 1-16 (1950) · Zbl 0036.21701 · doi:10.1093/biomet/37.1-2.1
[3] Barkoulas, JT; Barilla, AG; Wells, W., Long-memory exchange rate dynamics in the euro era, Chaos Solitons Fractals, 86, 92-100 (2016) · Zbl 1415.91227 · doi:10.1016/j.chaos.2016.02.007
[4] Bickel, PJ; Bühlmann, P., A new mixing notion and functional central limit theorems for a sieve bootstrap in time series, Bernoulli, 5, 413-446 (1999) · Zbl 0954.62102 · doi:10.2307/3318711
[5] Blackman, RB; Tukey, JW, The measurement of power spectra from the point of view of communications engineering—part I, Bell Syst Tech J, 37, 185-282 (1958) · doi:10.1002/j.1538-7305.1958.tb03874.x
[6] Bühlmann, P., Sieve bootstrap for time series, Bernoulli, 3, 123-148 (1997) · Zbl 0874.62102 · doi:10.2307/3318584
[7] Cavaliere, G., Bounded integrated processes and unit root tests, Stat Methods Appl, 11, 41-69 (2002) · Zbl 1145.62366 · doi:10.1007/BF02511445
[8] Cavaliere, G., Limited time series with a unit root, Econ Theory, 21, 907-945 (2005) · Zbl 1081.62062
[9] Cavaliere, G., Testing mean reversion in target-zone exchange rates, Appl Econ, 37, 2335-2347 (2005) · doi:10.1080/00036840500359283
[10] Cavaliere, G.; Xu, F., Testing for unit roots in bounded time series, J Econ, 178, 259-272 (2014) · Zbl 1293.62169 · doi:10.1016/j.jeconom.2013.08.026
[11] Chang, Y.; Park, JY, A sieve bootstrap for the test of a unit root, J Time Ser Anal, 24, 379-400 (2003) · Zbl 1036.62070 · doi:10.1111/1467-9892.00312
[12] Dahlhaus, R., Efficient parameter estimation for self-similar processes, Ann Stat, 17, 1749-1766 (1989) · Zbl 0703.62091 · doi:10.1214/aos/1176347393
[13] Dickey, DA; Fuller, WA, Distribution of the estimators of an autoregressive time series with a unit root, J Am Stat Assoc, 74, 427-431 (1979) · Zbl 0413.62075
[14] Efron, B., Bootstrap methods: another look at the jackknife, Ann Stat, 7, 1-26 (1979) · Zbl 0406.62024 · doi:10.1214/aos/1176344552
[15] Fox, R.; Taqqu, MS, Large-sample properties of parameter estimates for strongly dependent stationary Gaussian time series, Ann Stat, 14, 517-532 (1986) · Zbl 0606.62096 · doi:10.1214/aos/1176349936
[16] Geweke, J.; Porter-Hudack, S., The estimation and application of long-memory time series models, J Time Ser Anal, 4, 221-237 (1983) · Zbl 0534.62062 · doi:10.1111/j.1467-9892.1983.tb00371.x
[17] Granger, CWJ, Some thoughts on the development of cointegration, J Econ, 158, 3-6 (2010) · Zbl 1431.62617 · doi:10.1016/j.jeconom.2010.03.002
[18] Hurst, H., Long-term storage capacity of reservoirs, Trans Am Soc Civil Eng, 116, 770-799 (1951)
[19] Hurvich, C.; Ray, B., Estimation of the memory parameter for nonstationary or noninvertible fractionally integrated processes, J Time Ser Anal, 16, 17-41 (1995) · Zbl 0813.62081 · doi:10.1111/j.1467-9892.1995.tb00221.x
[20] Kapetanios G, Psaradakis Z (2006) Sieve bootstrap for strongly dependent stationary processes. Working Papers 552, Queen Mary University of London. School of Economics and Finance
[21] Kreiss JP (1992) Bootstrap procedures for AR \(( \infty )\)-processes. In: Lecture Notes in Economics and Mathematical Systems, vol 376: 107-113 (Proc. Bootstrapping and Related Techniques, Trier)
[22] Kunsch, HR, The jacknife and the bootstrap for general stationary observations, Ann Stat, 17, 1217-1241 (1989) · Zbl 0684.62035 · doi:10.1214/aos/1176347265
[23] Lahiri, SN, Resampling Methods for Dependent Data (2003), New York: Springer, New York · Zbl 1028.62002
[24] Lo, A., Long-term memory in stock market prices, Econometrica, 59, 1279-1313 (1991) · Zbl 0781.90023 · doi:10.2307/2938368
[25] Mandelbrot, B., Statistical methodology for nonperiodic cycles: from the covariance to r/s analysis, Ann Econ Soc Meas, 1, 259-290 (1972)
[26] Mandelbrot, B., Limit theorems of the self-normalized range for weakly and strongly dependent processes, Z Wahr verw Geb, 31, 271-285 (1975) · Zbl 0288.60033 · doi:10.1007/BF00532867
[27] Newey, WK; West, KD, A simple positive semi-definite heteroskedasticity and autocorrelation consistent covariance matrix, Econometrica, 55, 703-708 (1987) · Zbl 0658.62139 · doi:10.2307/1913610
[28] Palm, FC; Smeekes, S.; Urbain, JP, Bootstrap unit-root tests: comparison and extensions, J Time Ser Anal, 29, 371-400 (2008) · Zbl 1164.62051 · doi:10.1111/j.1467-9892.2007.00565.x
[29] Paparoditis, E., Bootstrapping autoregressive and moving average parameter estimates of infinite order vector autoregressive processes, J Multivar Anal, 57, 277-296 (1996) · Zbl 0863.62078 · doi:10.1006/jmva.1996.0034
[30] Parzen, E., An approach to time series analysis, Ann Math Stat, 32, 951-989 (1961) · Zbl 0107.13801 · doi:10.1214/aoms/1177704840
[31] Perron, P.; Ng, S., Useful modifications to some unit root tests with dependent errors and their local asymptotic properties, Rev Econ Stud, 63, 435-463 (1996) · Zbl 0872.62085 · doi:10.2307/2297890
[32] Phillips, PCB; Perron, P., Testing for unit root in time series regression, Biometrika, 75, 335-346 (1988) · Zbl 0644.62094 · doi:10.1093/biomet/75.2.335
[33] Poskitt, DS, Properties of the sieve bootstrap for fractionally integrated and non-invertible processes, J Time Ser Anal, 29, 224-250 (2008) · Zbl 1164.62053 · doi:10.1111/j.1467-9892.2007.00554.x
[34] Priestley, MB, Basic considerations in the estimation of spectra, Technometrics, 4, 551-564 (1962) · Zbl 0213.43802 · doi:10.1080/00401706.1962.10490039
[35] Psaradakis, Z., Bootstrap tests for an autoregressive unit root in the presence of weakly dependent errors, J Time Ser Anal, 22, 577-594 (2001) · Zbl 0979.62068 · doi:10.1111/1467-9892.00242
[36] R Core Team (2015) R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing, Vienna, Austria
[37] Robinson, PM, Log-periodogram regression of time series with long range dependence, Ann Stat, 23, 1048-1072 (1995) · Zbl 0838.62085 · doi:10.1214/aos/1176324636
[38] Robinson, PM, Gaussian semiparametric estimation of long range dependence, Ann Stat, 23, 1630-1661 (1995) · Zbl 0843.62092 · doi:10.1214/aos/1176324317
[39] Trokic, M., Regulated fractionally integrated processes, J Time Ser Anal, 34, 591-601 (2013) · Zbl 1282.62205 · doi:10.1111/jtsa.12036
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.