Abstract
In this study, we derive the limiting distribution of the least squares estimator (LSE) and the localized LSE for mildly explosive autoregressive models with locally stationary disturbance and verify that it is Cauchy as in the iid case. We also investigate the limiting distribution of two types of Dickey–Fuller unit root tests, designed for detecting a bubble period in economic time series data, and show that these tests are consistent. To evaluate the methods, we conduct a simulation study and carry out a data analysis using time series data on bitcoin prices.
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References
Anderson TW (1959) On asymptotic distributions of estimates of parameters of stochastic difference equations. Ann Math Stat 30:676–687
Aue A, Horváth L (2007) A limit theorem for mildly explosive autoregression with stable errors. Econ Theory 23:201–220
Chan NH, Wei CZ (1987) Asymptotic inference for nearly nonstationary AR(1) processes. Ann Stat 15:1050–1063
Cretarola A, Figa-Talamanca G (2019) Detecting bubbles in Bitcoin price dynamics via market exuberance. Online published in Ann Oper Res
Dahlhaus R (1996a) On the Kullback–Leibler information divergence of locally stationary processes. Stoch Process Appl 62:139–168
Dahlhaus R (1996b) Maximum likelihood estimation and model selection for locally stationary processes. J Nonparametr Stat 6:171–191
Dahlhaus R (1996c) Asymptotic statistical inference for nonstationary processes with evolutionary spectra. In: Robinson PM, Rosenblatt M (eds) Athens conference on applied probability and time series 2. Lecture Notes in Statistics, vol 115. Springer, New York, pp 145–159
Dahlhaus R (1997) Fitting time series models to nonstationary process. Ann Stat 25:1–37
Dahlhaus R (2012) Locally stationary processes. In: Subba Rao T, Subba Rao S, Rao CR (eds) Handbook of statistics: time series analysis: methods and applications, vol 30. North Holland, New York, pp 351–413
Hirukawa J (2017) Time series regression models with locally stationary disturbance. Stat Inference Stoch Process 20:329–346
Hirukawa J, Sadakata M (2012) Least squares estimators for unit root processes with locally stationary disturbance. Adv Decis Sci. https://doi.org/10.1155/2012/893497
Lee S, Wei CZ (1999) On residual empirical process of stochastic regression models with applications to time series. Ann Stat 27:237–261
Magdalinos T (2012) Mildly explosive autoregression under weak and strong dependence. J Econ 169:179–187
Magdalinos T, Phillips PCB (2008) Limit theory for cointegrated systems with moderately integrated and moderately explosive regressors. Econ Theory 25:482–526
Oh H, Lee S, Chan NH (2018) Mildly explosive autoregression with mixing innovations. J Korean Stat Soc 47:41–53
Phillips PCB (1987) Time series regression with a unit root. Econometrica 55:277–301
Phillips PCB, Magdalinos T (2007a) Limit theory for moderate deviations from a unit root. J Econ 136:115–130
Phillips PCB, Magdalinos T (2007b) Limit theory for moderate deviations from a unit root under weak dependence. In: Garry DAP, Tzavalis E (eds) The refinement of econometric estimation and test procedures: finite sample and asymptotic analysis. Cambridge University Press, Cambridge, pp 123–162
Phillips PCB, Wu Y, Yu J (2011) Explosive behavior in the 1990s NASDAQ: when did exuberance escalate asset values? Int Econ Rev 52:201–226
Priestley MB (1965) Evolutionary spectra and non-stationary processes. J R Stat Soc Ser B 27:204–237
Tanaka K (1996) Time series analysis: nonstationary and noninvertible distribution theory. Wiley, New York
Wang X, Yu J (2016) Double asymptotics for explosive continuous time models. J Econ 193:35–53
White JS (1958) The limiting distribution of the serial correlation coefficient in the explosive case. Ann Math Stat 29:1188–1197
Yu J, Phillips PCB (2009) Limit theory for dating the origination and collapse of mildly explosive periods in time series data. Unpublished manuscript, Sim Kee Boon Institute for Financial Economics, Singapore Management University
Acknowledgements
The authors thank the Editor, an AE, and two anonymous referees for their careful reading and valuable comments. This research is supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT and future Planning (No. 2018R1A2A2A05019433) and JSPS KAKENHI (Grant No. 16K00042 and 19K11857).
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Hirukawa, J., Lee, S. Asymptotic properties of mildly explosive processes with locally stationary disturbance. Metrika 84, 511–534 (2021). https://doi.org/10.1007/s00184-020-00782-2
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DOI: https://doi.org/10.1007/s00184-020-00782-2