Abstract
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.
Similar content being viewed by others
Notes
For clarity’s sake, we stress once more that we refer here to the overall long memory of the process \(X_t\), denoted by \(d=d'+1\), where \(d'\) is the long memory parameter of \(\varDelta X_t=Z_t\)
We are well aware of the existence of many other estimation methods, but we believe that it is of interest to the reader the performance of the methods that are effectively the best-known and, also, implemented in the most common packages (e.g. R, Matlab,...).
All the codes (throughout the paper) are written in R language (R Core Team 2015) and are available upon request by the authors.
We concentrate only on the \(\hat{r}_{\mu }\) statistic proposed by Cavaliere (2002) and do not focus on the other versions (presented in the same article) accounting for a deterministic trend in the data, that is a case into which we are not interested.
Palm et al. (2008) also showed that for some data generating processes (not our case), residuals from a first order autoregression can lead to an even better performance of the sieve bootstrap in terms of asymptotic validity, compared to first difference.
Results with the censoring algorithm are substantially equivalent and are available upon request by the authors.
The euro is at the core of ERM 2, and the currencies of participating EU member states have central rates against the euro, but not against each other. The obligation to intervene–that is, to buy or sell currency to support the exchange rate–if a participating currency reaches a fluctuation limit depends only on the central bank of the relevant member state and the ECB. The other participating member states have no obligation to intervene. ERM 2 includes a provision on unlimited intervention credit between the ECB and the participating central banks in connection with intervention at the fluctuation limits. One of the convergence criteria for joining the euro area is to observe the normal fluctuation band within ERM 2 without severe tensions for at least two years. In the same period, the member state in question may not devalue its currency against the euro.
The ERM 1, antecedent to the ERM 2, is one of the arrangements implied by the European Monetary System (EMS), in place between 1979 and May 1998.
References
Andrews D (1991) Heteroskedasticity and autocorrelation consistent covariance matrix estimation. Econometrica 59:817–858
Bartlett MS (1950) Periodogram analysis and continuous spectra. Biometrika 37:1–16
Barkoulas JT, Barilla AG, Wells W (2016) Long-memory exchange rate dynamics in the euro era. Chaos Solitons Fractals 86:92–100
Bickel PJ, Bühlmann P (1999) A new mixing notion and functional central limit theorems for a sieve bootstrap in time series. Bernoulli 5:413–446
Blackman RB, Tukey JW (1958) The measurement of power spectra from the point of view of communications engineering—part I. Bell Syst Tech J 37:185–282
Bühlmann P (1997) Sieve bootstrap for time series. Bernoulli 3:123–148
Cavaliere G (2002) Bounded integrated processes and unit root tests. Stat Methods Appl 11:41–69
Cavaliere G (2005) Limited time series with a unit root. Econ Theory 21:907–945
Cavaliere G (2005) Testing mean reversion in target-zone exchange rates. Appl Econ 37:2335–2347
Cavaliere G, Xu F (2014) Testing for unit roots in bounded time series. J Econ 178:259–272
Chang Y, Park JY (2003) A sieve bootstrap for the test of a unit root. J Time Ser Anal 24:379–400
Dahlhaus R (1989) Efficient parameter estimation for self-similar processes. Ann Stat 17:1749–1766
Dickey DA, Fuller WA (1979) Distribution of the estimators of an autoregressive time series with a unit root. J Am Stat Assoc 74:427–431
Efron B (1979) Bootstrap methods: another look at the jackknife. Ann Stat 7:1–26
Fox R, Taqqu MS (1986) Large-sample properties of parameter estimates for strongly dependent stationary Gaussian time series. Ann Stat 14:517–532
Geweke J, Porter-Hudack S (1983) The estimation and application of long-memory time series models. J Time Ser Anal 4:221–237
Granger CWJ (2010) Some thoughts on the development of cointegration. J Econ 158:3–6
Hurst H (1951) Long-term storage capacity of reservoirs. Trans Am Soc Civil Eng 116:770–799
Hurvich C, Ray B (1995) Estimation of the memory parameter for nonstationary or noninvertible fractionally integrated processes. J Time Ser Anal 16:17–41
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
Kreiss JP (1992) Bootstrap procedures for AR(\(\infty \))-processes. In: Lecture Notes in Economics and Mathematical Systems, vol 376: 107–113 (Proc. Bootstrap** and Related Techniques, Trier)
Kunsch HR (1989) The jacknife and the bootstrap for general stationary observations. Ann Stat 17:1217–1241
Lahiri SN (2003) Resampling Methods for Dependent Data. Springer, New York
Lo A (1991) Long-term memory in stock market prices. Econometrica 59:1279–1313
Mandelbrot B (1972) Statistical methodology for nonperiodic cycles: from the covariance to r/s analysis. Ann Econ Soc Meas 1:259–290
Mandelbrot B (1975) Limit theorems of the self-normalized range for weakly and strongly dependent processes. Z Wahr verw Geb 31:271–285
Newey WK, West KD (1987) A simple positive semi-definite heteroskedasticity and autocorrelation consistent covariance matrix. Econometrica 55:703–708
Palm FC, Smeekes S, Urbain JP (2008) Bootstrap unit-root tests: comparison and extensions. J Time Ser Anal 29:371–400
Paparoditis E (1996) Bootstrap** autoregressive and moving average parameter estimates of infinite order vector autoregressive processes. J Multivar Anal 57:277–296
Parzen E (1961) An approach to time series analysis. Ann Math Stat 32:951–989
Perron P, Ng S (1996) Useful modifications to some unit root tests with dependent errors and their local asymptotic properties. Rev Econ Stud 63:435–463
Phillips PCB, Perron P (1988) Testing for unit root in time series regression. Biometrika 75:335–346
Poskitt DS (2008) Properties of the sieve bootstrap for fractionally integrated and non-invertible processes. J Time Ser Anal 29:224–250
Priestley MB (1962) Basic considerations in the estimation of spectra. Technometrics 4:551–564
Psaradakis Z (2001) Bootstrap tests for an autoregressive unit root in the presence of weakly dependent errors. J Time Ser Anal 22:577–594
R Core Team (2015) R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing, Vienna, Austria
Robinson PM (1995a) Log-periodogram regression of time series with long range dependence. Ann Stat 23:1048–1072
Robinson PM (1995b) Gaussian semiparametric estimation of long range dependence. Ann Stat 23:1630–1661
Trokic M (2013) Regulated fractionally integrated processes. J Time Ser Anal 34:591–601
Acknowledgements
We are very thankful to the Editor and two anonymous Referees for the helpful and constructive comments on a previous version of this paper.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Gerolimetto, M., Magrini, S. Testing for boundary conditions in case of fractionally integrated processes. Stat Methods Appl 29, 357–371 (2020). https://doi.org/10.1007/s10260-019-00474-w
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10260-019-00474-w