Hostname: page-component-cd9895bd7-q99xh Total loading time: 0 Render date: 2024-12-27T17:18:03.501Z Has data issue: false hasContentIssue false
  • English
  • Français

TESTING FOR STRICT STATIONARITY VIA THE DISCRETE FOURIER TRANSFORM

Published online by Cambridge University Press:  28 October 2022

Zhonghao Fu ,
Shang Gao ,
Liangjun Su  and
Xia Wang
Zhonghao Fu
Affiliation:
Fudan University and Shanghai Institute of International Finance and Economics
Shang Gao
Affiliation:
Fudan University
Liangjun Su
Affiliation:
Tsinghua University
Xia Wang*
Affiliation:
Renmin University of China
*
Address correspondence to Xia Wang, School of Economics, Renmin University of China, Beijing, China; e-mail: [email protected].
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

This paper proposes a model-free test for the strict stationarity of a potentially vector-valued time series using the discrete Fourier transform (DFT) approach. We show that the DFT of a residual process based on the empirical characteristic function weakly converges to a zero spectrum in the frequency domain for a strictly stationary time series and a nonzero spectrum otherwise. The proposed test is powerful against various types of nonstationarity including deterministic trends and smooth or abrupt structural changes. It does not require smoothed nonparametric estimation and, thus, can detect the Pitman sequence of local alternatives at the parametric rate $T^{-1/2}$, faster than all existing nonparametric tests. We also design a class of derivative tests based on the characteristic function to test the stationarity in various moments. Monte Carlo studies demonstrate that our test has reasonarble size and excellent power. Our empirical application of exchange rates strongly suggests that both nominal and real exchange rate returns are nonstationary, which the augmented Dickey–Fuller and Kwiatkowski–Phillips–Schmidt–Shin tests may overlook.

Type
ARTICLES
Information
Econometric Theory , Volume 40 , Issue 3 , June 2024 , pp. 511 - 557
Copyright
© The Author(s), 2022. Published by Cambridge University Press

Footnotes

Fu acknowledges financial support from the National Science Foundation of China (NSFC) under Nos. 71903032 and 72121002. Gao acknowledges financial support from the NSFC under No. 71973030. Su gratefully acknowledges the NSFC for financial support under No. 72133002. Wang acknowledges financial support from the NSFC under No. 71873151.

References

REFERENCES

Bierens, H. (1982) Consistent model specification tests. Journal of Econometrics 20, 105–134.CrossRefGoogle Scholar
Bierens, H. (1990) A consistent conditional moment test of functional form. Econometrica 58, 1443–1458.CrossRefGoogle Scholar
Billingsley, P. (1999) Convergence of Probability Measures . Wiley.CrossRefGoogle Scholar
Busetti, F. & Harvey, A. (2010) Tests of strict stationarity based on quantile indicators. Journal of Time Series Analysis 31, 435–450.CrossRefGoogle Scholar
Busetti, F. & Harvey, A. (2011) When is a copula constant? A test for changing relationships. Journal of Financial Econometrics 9, 106–131.CrossRefGoogle Scholar
Busetti, F. & Taylor, A.M.R. (2003) Variance shifts, structural breaks, and stationarity tests. Journal of Business & Economic Statistics 21, 510–531.CrossRefGoogle Scholar
Busetti, F. & Taylor, A.M.R. (2004) Tests of stationarity against a change in persistence. Journal of Econometrics 123, 33–66.CrossRefGoogle Scholar
Cai, Z. (2007) Trending time-varying coefficient time series models with serially correlated errors. Journal of Econometrics 136, 163–188.CrossRefGoogle Scholar
Carrasco, M. & Chen, X. (2002) Mixing and moment properties of various GARCH and stochastic volatility models. Econometric Theory 18, 17–39.CrossRefGoogle Scholar
Cavaliere, G. & Taylor, A.M.R. (2005) Stationarity tests under time-varying second moments. Econometric Theory 21, 1112–1129.CrossRefGoogle Scholar
Cavaliere, G. & Taylor, A.M.R. (2009) A note on testing covariance stationarity. Econometric Reviews 28, 364–371.CrossRefGoogle Scholar
Chen, B. & Hong, Y. (2010) Characteristic function-based testing for multifactor continuous-time Markov models via nonparametric regression. Econometric Theory 26, 1115–1179.CrossRefGoogle Scholar
Chen, X. & Fan, Y. (2006) Estimation of copula-based semiparametric time series models. Journal of Econometrics 130, 307–335.CrossRefGoogle Scholar
Davidson, J. (1994) Stochastic Limit Theory . Oxford University Press.CrossRefGoogle Scholar
Davis, R.A., Matsui, M., Mikosch, T., & Wan, P. (2018) Applications of distance correlation to time series. Bernoulli 24, 3087–3116.CrossRefGoogle Scholar
Dickey, D.A. & Fuller, W.A. (1979) Distribution of the estimators for autoregressive time series with a unit root. Journal of the American Statistical Association 74, 427–431.Google Scholar
Dickey, D.A. & Fuller, W.A. (1981) Likelihood ratio statistics for autoregressive time series with a unit root. Econometrica 49, 1057–1072.CrossRefGoogle Scholar
Diebold, F.X., Gunther, T.A., & Tay, A.S. (1998) Evaluating density forecasts with applications to financial risk management. International Economic Review 39, 863–883.CrossRefGoogle Scholar
Diebold, F.X., Hahn, J., & Tay, A.S. (1999) Multivariate density forecast evaluation and calibration in financial risk management: High-frequency returns on foreign exchange. The Review of Economics and Statistics 81, 661–673.CrossRefGoogle Scholar
Doukhan, P. (1994) Mixing: Properties and Examples . Springer.CrossRefGoogle Scholar
Doukhan, P., Lang, G., Leucht, A., & Neumann, M.H. (2015) Dependent wild bootstrap for the empirical process. Journal of Time Series Analysis 36, 290–314.CrossRefGoogle Scholar
Dwivedi, Y. & Subba Rao, S. (2011) A test for second-order stationarity of a time series based on the discrete Fourier transform. Journal of Time Series Analysis 32, 68–91.CrossRefGoogle Scholar
Fan, J. & Yao, Q. (2003) Nonlinear Time Series: Nonparametric and Parametric Methods . Springer.CrossRefGoogle Scholar
Francq, C. & Zakoïan, J.-M. (2012) Strict stationarity testing and estimation of explosive and stationary generalized autoregressive conditional heteroscedasticity models. Econometrica 80, 821–861.Google Scholar
Fu, Z. & Hong, Y. (2019) A model-free consistent test for structural change in regression possibly with endogeneity. Journal of Econometrics 211, 206–242.CrossRefGoogle Scholar
Fu, Z., Hong, Y. and Wang, X. (2022a) On multiple structural breaks in distribution: An empirical characteristic function approach. Econometric Theory, published online 25 March 2022. doi:10.1017/S026646662200010X.CrossRefGoogle Scholar
Fu, Z., Hong, Y., & Wang, X. (2022b) Testing for structural changes in large dimensional factor models via discrete Fourier transform. Journal of Econometrics, published online 6 August 2022. doi:10.1016/j.jeconom.2022.06.005.CrossRefGoogle Scholar
Giné, E. & Zinn, J. (1990) Bootstrapping general empirical measures. The Annals of Probability 18, 851–869.CrossRefGoogle Scholar
González-Rivera, G. & Sun, Y. (2017) Density forecast evaluation in unstable environments. International Journal of Forecasting 33, 416–432.CrossRefGoogle Scholar
Guo, S., Li, D., & Li, M. (2019) Strict stationarity testing and GLAD estimation of double autoregressive models. Journal of Econometrics 211, 319–337.CrossRefGoogle Scholar
Hegwood, N.D. & Papell, D.H. (1998) Quasi purchasing power parity. International Journal of Finance & Economics 3, 279–289.3.0.CO;2-K>CrossRefGoogle Scholar
Hobijn, B., Franses, P.H., & Ooms, M. (2004) Generalizations of the KPSS-test for stationarity. Statistica Neerlandica 58, 483–502.CrossRefGoogle Scholar
Hong, Y., Wang, X., & Wang, S. (2017) Testing strict stationarity with applications to macroeconomic time series. International Economic Review 58, 1227–1277.CrossRefGoogle Scholar
Hong, Y. & White, H. (1995) Consistent specification testing via nonparametric series regression. Econometrica 63, 1133–1159.CrossRefGoogle Scholar
Inoue, A. (2001) Testing for distributional change in time series. Econometric Theory 17, 156–187.CrossRefGoogle Scholar
Jentsch, C. & Subba Rao, S. (2015) A test for second order stationarity of a multivariate time series. Journal of Econometrics 185, 124–161.CrossRefGoogle Scholar
Joe, H. (1997) Multivariate Models and Multivariate Dependence Concepts . CRC Press.Google Scholar
Kapetanios, G. (2009) Testing for strict stationarity in financial variables. Journal of Banking and Finance 33, 2346–2362.CrossRefGoogle Scholar
Koedijk, K.G., Schafgans, M.M.A., & de Vries, C.G. (1990) The tail index of exchange rate returns. Journal of International Economics 29, 93–108.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
Leucht, A. & Neumann, M.H. (2013) Dependent wild bootstrap for degenerate U- and V-statistics. Journal of Multivariate Analysis 117, 257–280.CrossRefGoogle Scholar
Leybourne, S.J. & McCabe, B.P.M. (1994) A consistent test for a unit root. Journal of Business & Economic Statistics 12, 157–166.CrossRefGoogle Scholar
Li, Q. & Racine, J.S. (2006) Nonparametric Econometrics: Theory and Practice . Princeton University Press.Google Scholar
Lin, C.-H. & Kao, T.-C. (2008) Multiple structural changes in the tail behavior: Evidence from stock index futures returns. Nonlinear Analysis: Real World Applications 9, 1702–1713.Google Scholar
Lothian, J.R. & Taylor, M.P. (1996) Real exchange rate behavior: The recent float from the perspective of the past two centuries. Journal of Political Economy 104, 488–509.CrossRefGoogle Scholar
Manner, H., Stark, F., & Wied, D. (2019) Testing for structural breaks in factor copula models. Journal of Econometrics 208, 324–345.CrossRefGoogle Scholar
Mark, N.C. (1990) Real and nominal exchange rates in the long run: An empirical investigation. Journal of International Economics 28, 115–136.CrossRefGoogle Scholar
Meese, R. & Rogoff, K. (1988) Was it real? The exchange rate-interest differential relation over the modern floating-rate period. The Journal of Finance 43, 933–948.Google Scholar
Papell, D.H. (1997) Searching for stationarity: Purchasing power parity under the current float. Journal of International Economics 43, 313–332.CrossRefGoogle Scholar
Phillips, P.C.B. & Perron, P. (1988) Testing for a unit root in time series regression. Biometrika 75, 335–346.CrossRefGoogle Scholar
Politis, D.N. & White, H. (2004) Automatic block-length selection for the dependent bootstrap. Econometric Reviews 23, 53–70.CrossRefGoogle Scholar
Quintos, C., Fan, Z., & Phillips, P.C.B. (2001) Structural change tests in tail behavior and the Asian crisis. The Review of Economic Studies 68, 633–663.CrossRefGoogle Scholar
Ramsay, J.O. (1991) Kernel smoothing approaches to nonparametric item characteristic curve estimation. Psychometrika 56, 611–630.CrossRefGoogle Scholar
Rho, Y. & Shao, X. (2019) Bootstrap-assisted unit root testing with piecewise locally stationary errors. Econometric Theory 35, 142–166.CrossRefGoogle Scholar
Robinson, P.M. (1989) Nonparametric estimation of time-varying parameters. In Hackl, P. (ed.), Statistical Analysis and Forecasting of Economic Structural Change , pp. 253–264. Springer.CrossRefGoogle Scholar
Robinson, P.M. (1991) Time-varying nonlinear regression. In Hackl, P. & Westlund, A. H. (eds.), Economic Structural Change , pp. 179–190. Springer.CrossRefGoogle Scholar
Rossi, B. (2013) Exchange rate predictability. Journal of Economic Literature 51, 1063–1119.CrossRefGoogle Scholar
Rossi, B. & Sekhposyan, T. (2013) Conditional predictive density evaluation in the presence of instabilities. Journal of Econometrics 177, 199–212.CrossRefGoogle Scholar
Shao, X. (2010) The dependent wild bootstrap. Journal of the American Statistical Association 105, 218–235.CrossRefGoogle Scholar
Sollis, R., Leybourne, S., & Newbold, P. (2002) Tests for symmetric and asymmetric nonlinear mean reversion in real exchange rates. Journal of Money, Credit and Banking 34, 686–700.CrossRefGoogle Scholar
Stinchcombe, M.B. & White, H. (1998) Consistent specification testing with nuisance parameters present only under the alternative. Econometric Theory 14, 295–325.CrossRefGoogle Scholar
Su, L. & White, H. (2007) A consistent characteristic function-based test for conditional independence. Journal of Econometrics 141, 807–834.CrossRefGoogle Scholar
Su, L. & White, H. (2010) Testing structural change in partially linear models. Econometric Theory 26, 1761–1806.CrossRefGoogle Scholar
Su, L. & Zheng, X. (2017) A martingale-difference-divergence-based test for specification. Economics Letters 156, 162–167.CrossRefGoogle Scholar
Székely, G.J., Rizzo, M.L., & Bakirov, N.K. (2007) Measuring and testing dependence by correlation of distances. Annals of Statistics 35, 2769–2794.CrossRefGoogle Scholar
Wang, X. & Hong, Y. (2018) Characteristic function based testing for conditional independence: A nonparametric regression approach. Econometric Theory 34, 815–849.CrossRefGoogle Scholar
White, H. (2001) Asymptotic Theory for Econometricians , Revised Edition. Academic Press.Google Scholar
Wu, Y. (1996) Are real exchange rates nonstationary? Evidence from a panel-data test. Journal of Money, Credit and Banking 28, 54–63.CrossRefGoogle Scholar
Xiao, Z. (2001) Testing the null hypothesis of stationarity against an autoregressive unit root alternative. Journal of Time Series Analysis 22, 87–105.CrossRefGoogle Scholar
Xiao, Z. & Lima, L.R. (2007) Testing covariance stationarity. Econometric Reviews 26, 643–667.CrossRefGoogle Scholar
Yang, S.C. (2007) Maximal moment inequality for partial sums of strong mixing sequences and application. Acta Mathematica Sinica, English Series 23, 1013–1024.CrossRefGoogle Scholar
Yokoyama, R. (1980) Moment bounds for stationary mixing sequences. Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete 52, 45–57.CrossRefGoogle Scholar
Zhou, Z. (2012) Measuring nonlinear dependence in time-series, a distance correlation approach. Journal of Time Series Analysis 33, 438–457.CrossRefGoogle Scholar
Supplementary material: PDF

Fu et al. supplementary material

Fu et al. supplementary material

Download Fu et al. supplementary material(PDF)
PDF 211.1 KB