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INFERENCE ON THE DIMENSION OF THE NONSTATIONARY SUBSPACE IN FUNCTIONAL TIME SERIES

Published online by Cambridge University Press:  28 March 2022

Morten Ørregaard Nielsen Open the ORCID record for Morten Ørregaard Nielsen [Opens in a new window] ,
Won-Ki Seo  and
Dakyung Seong
Morten Ørregaard Nielsen*
Affiliation:
Aarhus University
Won-Ki Seo
Affiliation:
University of Sydney
Dakyung Seong
Affiliation:
University of Sydney
*
Address correspondence to Morten Ørregaard Nielsen, Department of Economics and Business Economics, Aarhus University, Fuglesangs Alle 4, 8210 Aarhus V, Denmark; e-mail: mon@econ.au.dk
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Abstract

We propose a statistical procedure to determine the dimension of the nonstationary subspace of cointegrated functional time series taking values in the Hilbert space of square-integrable functions defined on a compact interval. The procedure is based on sequential application of a proposed test for the dimension of the nonstationary subspace. To avoid estimation of the long-run covariance operator, our test is based on a variance ratio-type statistic. We derive the asymptotic null distribution and prove consistency of the test. Monte Carlo simulations show good performance of our test and provide evidence that it outperforms the existing testing procedure. We apply our methodology to three empirical examples: age-specific U.S. employment rates, Australian temperature curves, and Ontario electricity demand.

Type
ARTICLES
Information
Econometric Theory , Volume 39 , Issue 3 , June 2023 , pp. 443 - 480
Copyright
© The Author(s), 2022. Published by Cambridge University Press

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Footnotes

We are grateful to the Editor, Co-Editor, two anonymous referees, Brendan Beare, Yoosoon Chang, Joon Park, Peter Phillips, Hanlin Shang, Yundong Tu, and seminar participants at Queen’s University, UC Davis, Australian National University, Boston University, Singapore (SMU/NUS), St. Petersburg State University, the 2019 Canadian Economics Association Conference, the 2019 Time Series and Forecasting Symposium at University of Sydney, the 2020 ANZESG conference, the 2020 Econometric Society World Congress, the 2020 (EC)2 Conference, and the CFE 2020 Conference for comments. An earlier version of this paper was circulated under the title “Variance ratio test for the number of stochastic trends in functional time series.” Nielsen thanks the Canada Research Chairs program and the Social Sciences and Humanities Research Council of Canada for financial support at Queen’s University. Seo thanks the Sir Edward Peacock Postdoctoral Fellowship at Queen’s University for financial support. Data and R code to replicate the empirical results in Table 5 are available on the authors’ websites.

References

REFERENCES

Ahn, S.K. & Reinsel, G.C. (1990) Estimation for partially nonstationary multivariate autoregressive models. Journal of the American Statistical Association 85, 813–823.10.1080/01621459.1990.10474945CrossRefGoogle Scholar
Andrews, D.W.K. (1991) Heteroskedasticity and autocorrelation consistent covariance matrix estimation. Econometrica 59, 817–858.CrossRefGoogle Scholar
Aue, A., Rice, G., & Sönmez, O. (2017). Detecting and dating structural breaks in functional data without dimension reduction. Journal of the Royal Statistical Society B 80, 509–529.CrossRefGoogle Scholar
Aue, A. & van Delft, A. (2020). Testing for stationarity of functional time series in the frequency domain. Annals of Statistics 48, 2505–2547.10.1214/19-AOS1895CrossRefGoogle Scholar
Beare, B.K., Seo, J., & Seo, W.-K. (2017). Cointegrated linear processes in Hilbert space. Journal of Time Series Analysis 38, 1010–1027.10.1111/jtsa.12251CrossRefGoogle Scholar
Beare, B.K. & Seo, W.-K. (2020) Representation of I(1) and I(2) autoregressive Hilbertian processes. Econometric Theory 36, 773–802.CrossRefGoogle Scholar
Berkes, I., Horváth, L., & Rice, G. (2013) Weak invariance principles for sums of dependent random functions. Stochastic Processes and their Applications 123, 385–403.10.1016/j.spa.2012.10.003CrossRefGoogle Scholar
Bewley, R. & Yang, M. (1995) Tests for cointegration based on canonical correlation analysis. Journal of the American Statistical Association 90, 990–996.10.1080/01621459.1995.10476600CrossRefGoogle Scholar
Bosq, D. (2000) Linear Processes in Function Spaces . Springer.10.1007/978-1-4612-1154-9CrossRefGoogle Scholar
Breitung, J. (2002) Nonparametric tests for unit roots and cointegration. Journal of Econometrics 108, 343–363.CrossRefGoogle Scholar
Chang, Y., Kaufmann, R.K., Kim, C.S., Miller, I.J., Park, J.Y., & Park, S. (2020). Evaluating trends in time series of distributions: a spatial fingerprint of human effects on climate. Journal of Econometrics 214, 274–294.10.1016/j.jeconom.2019.05.014CrossRefGoogle Scholar
Chang, Y., Kim, C.S., & Park, J.Y. (2016) Nonstationarity in time series of state densities. Journal of Econometrics 192, 152–167.10.1016/j.jeconom.2015.06.025CrossRefGoogle Scholar
Flood, S., King, M., Rodgers, R., Ruggles, S., & Warren, J.R. (2018) Integrated Public Use Microdata Series, Current Population Survey: Version 6.0 [dataset] . IPUMS.Google Scholar
Franchi, M. & Paruolo, P. (2020) Cointegration in functional autoregressive processes. Econometric Theory 36, 803–839.CrossRefGoogle Scholar
Harris, D. (1997) Principal components analysis of cointegrated time series. Econometric Theory 13, 529–557.CrossRefGoogle Scholar
Hörmann, S. & Kokoszka, P. (2010) Weakly dependent functional data. Annals of Statistics 38, 1845–1884.CrossRefGoogle Scholar
Horváth, L. & Kokoszka, P. (2012) Inference for Functional Data with Applications . Springer.CrossRefGoogle Scholar
Horváth, L., Kokoszka, P., & Rice, G. (2014) Testing stationarity of functional time series. Journal of Econometrics 179, 66–82.CrossRefGoogle Scholar
Johansen, S. (1995) Likelihood-Based Inference in Cointegrated Vector Autoregressive Models . Oxford University Press.CrossRefGoogle Scholar
Kokoszka, P. & Young, G. (2016) KPSS test for functional time series. Statistics 50, 957–973.CrossRefGoogle Scholar
Kwiatkowski, D., Phillips, P.C.B., Schmidt, P., & Shin, Y. (1992) Testing the null hypothesis of stationarity against the alternative of a unit root. Journal of Econometrics 54, 159–178.CrossRefGoogle Scholar
Mas, A. (2002) Weak convergence for the covariance operators of a Hilbertian linear process. Stochastic Processes and their Applications 99, 117–135.CrossRefGoogle Scholar
Müller, U.K. (2007) A theory of robust long-run variance estimation. Journal of Econometrics 141, 1331–1352.CrossRefGoogle Scholar
Müller, U.K. (2008) The impossibility of consistent discrimination between I(0) and I(1) processes. Econometric Theory 24, 616–630.CrossRefGoogle Scholar
Nelson, C.R. & Siegel, A.F. (1987) Parsimonious modeling of yield curves. Journal of Business 60, 473–489.CrossRefGoogle Scholar
Nielsen, M.Ø. (2009) A powerful test of the autoregressive unit root hypothesis based on a tuning parameter free statistic. Econometric Theory 25, 1515–1544.CrossRefGoogle Scholar
Nielsen, M.Ø. (2010) Nonparametric cointegration analysis of fractional systems with unknown integration orders. Journal of Econometrics 155, 170–187.CrossRefGoogle Scholar
Nyblom, J. & Harvey, A. (2000) Tests of common stochastic trends. Econometric Theory 16, 176–199.CrossRefGoogle Scholar
Pantula, S.G. (1989) Testing for unit roots in time series data. Econometric Theory 5, 256–271.CrossRefGoogle Scholar
Pedroni, P.L., Vogelsang, T.J., Wagner, M., & Westerlund, J. (2015) Nonparametric rank tests for non-stationary panels. Journal of Econometrics 185, 378–391.CrossRefGoogle Scholar
Petersen, A. & Müller, H.-G. (2016) Functional data analysis for density functions by transformation to a Hilbert space. Annals of Statistics 44, 183–218.CrossRefGoogle Scholar
Phillips, P.C.B. & Ouliaris, S. (1988) Testing for cointegration using principal components methods. Journal of Economic Dynamics and Control 12, 205–230.CrossRefGoogle Scholar
Phillips, P.C.B. & Solo, V. (1992) Asymptotics for linear processes. Annals of Statistics 20, 971–1001.CrossRefGoogle Scholar
Seo, W.-K. & Beare, B.K. (2019) Cointegrated linear processes in Bayes Hilbert space. Statistics & Probability Letters 147, 90–95.CrossRefGoogle Scholar
Stock, J.H. & Watson, M.W. (1988). Testing for common trends. Journal of the American Statistical Association 83, 1097–1107.CrossRefGoogle Scholar
Tanaka, K. (1990) Testing for a moving average unit root. Econometric Theory 6, 433–444.CrossRefGoogle Scholar
Taylor, A.M.R. (2005) Variance ratio tests of the seasonal unit root hypothesis. Journal of Econometrics 124, 33–54.CrossRefGoogle Scholar
Yao, J., Kammoun, A., & Najim, J. (2012) Eigenvalue estimation of parameterized covariance matrices of large dimensional data. IEEE Transactions on Signal Processing 60, 5893–5905.Google Scholar
Zhang, R., Robinson, P., & Yao, Q. (2019) Identifying cointegration by eigenanalysis. Journal of the American Statistical Association 114, 916–927.CrossRefGoogle Scholar
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