Abstract
Based on the quantile regression, we extend Koenker and Xiao (2004) and Ling and McAleer (2004)’s works from finite-variance innovations to infinite-variance innovations. A robust t-ratio statistic to test for unit-root and a re-sampling method to approximate the critical values of the t-ratio statistic are proposed in this paper. It is shown that the limit distribution of the statistic is a functional of stable processes and a Brownian bridge. The finite sample studies show that the proposed t-ratio test always performs significantly better than the conventional unit-root tests based on least squares procedure, such as the Augmented Dick Fuller (ADF) and Philliphs-Perron (PP) test, in the sense of power and size when infinite-variance disturbances exist. Also, quantile Kolmogorov-Smirnov (QKS) statistic and quantile Cramer-von Mises (QCM) statistic are considered, but the finite sample studies show that they perform poor in power and size, respectively. An application to the Consumer Price Index for nine countries is also presented.
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References
E Bofinger. Estimation of a density function using order statistics, Australian Journal of Statistics, 1975, 17: 1–7.
J Breitung, C Gouriéroux. Rank tests fot unit roots, Econometrics, 1997, 81(1): 7–27.
N H Chan, L T Tran. On the first-order autoregressive process with infinite variance, Econometric Theory, 1989, 5(3): 354–362.
N H Chan, R M Zhang. Quantile inference for near-integrated autoregressive time series under infinite variance and strong dependence, Stochastic Processes and their Applications, 2009, 119(12): 4124–4148.
N H Chan, R M Zhang. Inference for nearly nonstationary processes under strong dependence with infinite variance, Statistica Sinica, 2010, 19: 925–947.
W W Chatemza, D Hristova, P Burridge. Is inflation stationary?, Applied Economics, 2005, 37(8): 901–903.
D D Cox, I Llatas. Maximum likelihood type estimation for nearly nonstationary autoregressive time series, The Annals of Statistics, 1991, 19(3): 1109–1129.
R A Davis, T Hsing. Point process and partial sum convergence for weakly dependent random variables with infinite variance, The Annals of Probability, 1995, 23(2): 879–917.
D A Dickey, W A Fuller. Distribution of the estimators for autoregressive time series with a unit root, Journal of the American Statistical Association, 1979, 74(366a): 427–431.
D A Dickey, W A Fuller. Likelihood ratio statistics for autoregressive time series with a unit root, Econometrica, 1981, 49(4): 1057–1072.
B Falk, C H Wang. Testing long-run PPP with infinite-variance returns, Journal of Applied Econometrics, 2003, 18(4): 471–484.
E F Fama. The Behavior of Stock-Market Prices, Journal of Business, 1965, 38(1): 34–105.
B Finkenstädt, H Rootzén. Extreme Values in Finance, Telecommunications, and the Environment, New York: Chapman and Hall/CRC, 2003.
A F Galvao. Unit root quantile autoregression testing using covariates, Journal of Econometrics, 2009, 152(2): 165–178.
I Georgiev, P M M Rodrigues, A M R Taylor. Unit Root Tests and Heavy-Tailed Innovations, Journal of Time Series Analysis, 2017, 38(5): 733–768.
V Goodman. Distribution Estimates for Functionals of the Two-Parameter Wiener Process, The Annals of Probability, 1976, 4(6): 977–982.
P Hall, S J Sheather. On the Distribution of a Studentized Quantile, Journal of the Royal Statistical Society, 1988, 50(3): 381–391.
M N Hasan, R W Koenker. Robust rank tests of the unit root hypothesis, Econometrica, 1997, 65(1): 133–161.
K Knight. Limit theory for autoregressive-patameter estiamtes in an infinite-variance random walk, The Canadian Journal of Statistics, 1989, 17: 261–278.
K Knight. Limit theory for M-estimates in an integrated infinite variance processes, Econometric Theory, 1991, 7(2): 200–212.
K Knight. Limiting distributions for L1regression estimators under general conditions, The Annals of Statistics, 1998, 26(2): 755–770.
R Koenker, Z J Xiao. Unit root quantile autoregression inference, Journal of the American Statistical Association, 2004, 99(464): 775–787.
D Kwiatkowski, P C B Phillips, P Schmidt, Y Shin. Testing for the null hypothesis of stationary against the alternative of a unit root: How sure are we that economic time series have a unit root, Journal of Econometrics, 1992, 54(1–3): 159–178.
H Q Li, S Y Park. Testing for a unit root in a nonlinear quantile autoregression framework, Econometric Reviews, 2018, 37(8): 867–892.
S Q Ling, M McAleer. Regression quantiles for unstable autoregressive models, Journal of Multivariate Analysis, 2004, 89(2): 304–328.
A Lucas. Unit root tests based on M estimators, Econometric Theory, 1995, 11(2): 331–346.
B B Mandelbrot. The variation of certain speculative prices, Journal of Business, 1963, 36(4): 394–419.
B B Mandelbrot. How long is the coast of Britain? Statistical self similarity and fractional dimension, Science, 1967, 156(3775): 636–638.
P C B Phillips, P Perron. Testing for a unit root in time series regression, Biometrika Trust, 1988, 75(2): 335–346.
P C B Phillips. Time series regression with a unit root and infinite-variance errors, Econometric Theory, 1990, 6(1): 44–62.
D Pollard. Asymptotics for least absolute deviation regression estimators, Econometric Theory, 1991, 7(2): 186–199.
S Rachev, S Mittnik. Stable Paretian Models in Finance, Quantitative Finance, 2000.
S T Rachev, S Mittnik, J R Kim. Time series with unit roots and infinite-variance disturbances, Applied Mathematics Letters, 1998, 11(5): 69–74.
S I Resnick. Point Processes, Regular Variation and Weak Convergence, Advances in Applied Probability, 1986, 18(1): 66–138.
D M M Samarakoon, K Knight. A note on unit root tests with infinite variance noise, Econometric Reviews, 2009, 28(4): 314–334.
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Liu, Qm., Liao, Gl. & Zhang, Rm. Quantile inference for nonstationary processes with infinite variance innovations. Appl. Math. J. Chin. Univ. 36, 443–461 (2021). https://doi.org/10.1007/s11766-021-4187-6
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DOI: https://doi.org/10.1007/s11766-021-4187-6