Unit Root Tests: The Role of the Univariate Models Implied by Multivariate Time Series
Abstract
:1. Introduction
2. The Model
3. A Simulation Experiment
- MC(a):
- MC(b):
- as in Gonzalo [18], we set , and and consider the following values for the remaining parameters: , and , for a total of 18 experiments. The root of the common autoregressive component (one root is always equal to 1) and the coefficients of the distinct MA components of the univariate models implied by the multivariate DGP are reported in Table 2.
4. Conclusions
Acknowledgments
Author Contributions
Conflicts of Interest
References
- W. Schwert. “Testing for unit roots: a montecarlo investigation.” J. Bus. Econ. Stat. 7 (1989): 147–159. [Google Scholar]
- P. Perron, and S. Ng. “Useful modifications to some unit root tests with dependent errors and their local asymptotic properties.” Rev. Econ. Stud. 63 (1996): 435–563. [Google Scholar] [CrossRef]
- W.R. Reed. Unit Root Tests, Size Distortion, and Cointegrated Data. Working Paper; Christchurch, New Zealand: University of Canterbury, 2014. [Google Scholar]
- W.R. Reed. Univariate Unit root Tests Perform Poorly When Data are Cointegrated. Working Paper; Christchurch, New Zealand: University of Canterbury, 2016. [Google Scholar]
- A. Zellner, and F. Palm. “Time series analysis and simultaneous equation econometric models.” J. Econom. 2 (1974): 17–54. [Google Scholar] [CrossRef]
- R.S. Tsay. Analysis of Financial Time Series, 3rd ed. New York, NY, USA: John Wiley & Sons, 2010. [Google Scholar]
- G. Cubadda, A. Hecq, and F. Palm. “Studying co-movements in large multivariate data prior to multivariate modelling.” J. Econom. 148 (2009): 25–35. [Google Scholar] [CrossRef] [Green Version]
- G. Cubadda, and U. Triacca. “An alternative solution to the autoregressive paradox in time series analysis.” Econ. Model. 28 (2011): 1451–1454. [Google Scholar] [CrossRef]
- A. Maravall, and A. Mathis. “Encompassing univariate models in multivariate time series.” J. Econom. 61 (1994): 197–233. [Google Scholar] [CrossRef]
- S.E. Said, and D.A. Dickey. “Testing for unit roots in autoregressive-moving average models of unknown order.” Biometrika 71 (1984): 599–607. [Google Scholar] [CrossRef]
- P.C.B. Phillips, and P. Perron. “Testing for a unit root in time series regression.” Biometrika 75 (1987): 335–346. [Google Scholar] [CrossRef]
- J.H. Stock. “A Class of Tests for Integration and Cointegration.” In Cointegration, Causality and Forecasting: A Festschrift for Clive W.J. Granger. Oxford, UK: Oxford University Press, 1999, pp. 135–167. [Google Scholar]
- G. Elliott, T.J. Rothenberg, and J.H. Stock. “Efficient tests for an autoregressive unit root.” Econometrica 64 (1996): 813–836. [Google Scholar] [CrossRef]
- S. Ng, and P. Perron. “Lag length selection and the construction of unit root tests with good size and power.” Econometrica 69 (2001): 1519–1554. [Google Scholar] [CrossRef]
- P. Perron, and Z. Qu. “A simple modification to improve the finite sample properties of ng and perron’s unit root tests.” Econ. Lett. 94 (2007): 12–19. [Google Scholar] [CrossRef]
- F.C. Palm, S. Smeekes, and J. Urbain. “Bootstrap unit root tests: Comparisons and extensions.” J. Time Ser. Anal. 29 (2008): 371–401. [Google Scholar] [CrossRef]
- Y. Chang, and J.Y. Park. “A sieve bootstrap for the test of a unit root.” J. Time Ser. Anal. 24 (2003): 379–400. [Google Scholar] [CrossRef]
- J. Gonzalo. “Five alternative methods of estimating long-run equilibrium relationships.” J. Econom. 60 (1994): 203–233. [Google Scholar] [CrossRef]
- Z. Psaradakis. “Bootstrap tests for an autoregresive unit root in the presence of weakly dependent errors.” J. Time Ser. Anal. 22 (2001): 577–594. [Google Scholar] [CrossRef]
- 1In general, considering a k-dimensional VAR process the univariate models will be at most ARMA, all univariate processes share the same AR component, and an MA component is present in each univariate model. See [5] for a general treatment of this issue.
- 3A general solution to this problem for an MA process of order q has been provided in Maravall and Mathis [9].
- 4For brevity, our discussion will refer to the model without trend, similar remarks apply when a trend component is included in the regression, simulation results in this case are available upon request from the authors.
DGP1 | β | ρ | ||||
, | 0.11 | 0 | −0.71 | 0.12 | 0.258 | |
DGP2 | ||||||
, | −0.2 | −0.92 | −0.69 | 0.821 | ||
DGP3 | ||||||
, | 3 | −0.2 | 0.14 | −0.52 | −0.195 | |
DGP4 | ||||||
, | −0.1 | −0.90 | −0.38 | 0.592 |
AR root | |||||||||
−0.5 | −0.972 | −0.978 | 0.911 | −0.902 | −0.942 | 0.029 | −0.885 | −0.924 | −0.635 |
0 | −0.975 | −0.975 | 0.887 | −0.930 | −0.930 | 0.025 | −0.910 | −0.910 | −0.583 |
0.5 | −0.978 | −0.972 | 0.911 | −0.942 | −0.902 | 0.029 | −0.924 | −0.885 | −0.635 |
AR root | |||||||||
−0.5 | −0.872 | −0.893 | 0.925 | −0.884 | −0.884 | 0.910 | −0.893 | −0.872 | 0.925 |
0 | −0.565 | −0.717 | 0.158 | −0.666 | −0.666 | 0.142 | −0.717 | −0.565 | 0.158 |
0.5 | −0.451 | −0.626 | −0.539 | −0.565 | −0.565 | −0.500 | −0.626 | −0.451 | −0.539 |
Test | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
OLS Detrending | ||||||||||||
0.029 | 0.058 | 0.037 | 0.030 | 0.044 | 0.027 | 0.018 | 0.054 | 0.049 | 0.048 | 0.065 | 0.023 | |
0.013 | 0.025 | 0.014 | 0.014 | 0.018 | 0.018 | 0.009 | 0.029 | 0.021 | 0.027 | 0.032 | 0.009 | |
0.043 | 0.070 | 0.047 | 0.044 | 0.055 | 0.046 | 0.028 | 0.066 | 0.061 | 0.068 | 0.099 | 0.051 | |
0.004 | 0.014 | 0.005 | 0.008 | 0.004 | 0.003 | 0.004 | 0.006 | 0.014 | 0.014 | 0.012 | 0.002 | |
0.020 | 0.037 | 0.027 | 0.024 | 0.035 | 0.023 | 0.013 | 0.039 | 0.035 | 0.041 | 0.041 | 0.013 | |
0.038 | 0.067 | 0.043 | 0.040 | 0.047 | 0.042 | 0.029 | 0.060 | 0.050 | 0.057 | 0.063 | 0.044 | |
GLS detrending | ||||||||||||
0.047 | 0.084 | 0.063 | 0.048 | 0.088 | 0.046 | 0.040 | 0.077 | 0.070 | 0.070 | 0.098 | 0.043 | |
0.041 | 0.071 | 0.052 | 0.041 | 0.077 | 0.039 | 0.034 | 0.07 | 0.059 | 0.062 | 0.083 | 0.036 | |
0.005 | 0.009 | 0.008 | 0.004 | 0.007 | 0.008 | 0.006 | 0.012 | 0.010 | 0.013 | 0.008 | 0.004 | |
0.051 | 0.083 | 0.062 | 0.051 | 0.090 | 0.052 | 0.038 | 0.084 | 0.073 | 0.071 | 0.102 | 0.044 | |
0.039 | 0.069 | 0.045 | 0.036 | 0.071 | 0.039 | 0.029 | 0.062 | 0.050 | 0.058 | 0.082 | 0.034 | |
0.035 | 0.061 | 0.052 | 0.043 | 0.066 | 0.038 | 0.035 | 0.068 | 0.053 | 0.058 | 0.069 | 0.028 | |
0.030 | 0.059 | 0.037 | 0.033 | 0.058 | 0.035 | 0.027 | 0.053 | 0.040 | 0.052 | 0.070 | 0.024 | |
0.037 | 0.067 | 0.042 | 0.035 | 0.071 | 0.037 | 0.029 | 0.061 | 0.049 | 0.059 | 0.082 | 0.032 | |
0.048 | 0.078 | 0.058 | 0.047 | 0.081 | 0.046 | 0.038 | 0.069 | 0.065 | 0.069 | 0.090 | 0.040 | |
Bootstrap Tests | ||||||||||||
0.045 | 0.058 | 0.060 | 0.043 | 0.064 | 0.036 | 0.040 | 0.051 | 0.048 | 0.053 | 0.062 | 0.043 | |
0.044 | 0.062 | 0.062 | 0.056 | 0.070 | 0.040 | 0.042 | 0.061 | 0.059 | 0.057 | 0.064 | 0.044 | |
Johansen’s Trace Test | ||||||||||||
0.722 | 0.834 | 0.717 | 0.059 | 0.076 | 0.069 |
Test | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
OLS detrending | ||||||||||||
0.033 | 0.098 | 0.082 | 0.080 | 0.107 | 0.037 | 0.049 | 0.146 | 0.108 | 0.119 | 0.164 | 0.081 | |
0.014 | 0.057 | 0.045 | 0.040 | 0.062 | 0.020 | 0.026 | 0.090 | 0.060 | 0.071 | 0.087 | 0.038 | |
0.037 | 0.102 | 0.082 | 0.092 | 0.115 | 0.049 | 0.064 | 0.196 | 0.149 | 0.150 | 0.205 | 0.087 | |
0.003 | 0.030 | 0.017 | 0.023 | 0.029 | 0.009 | 0.010 | 0.048 | 0.029 | 0.038 | 0.056 | 0.016 | |
0.021 | 0.077 | 0.062 | 0.069 | 0.089 | 0.029 | 0.038 | 0.109 | 0.073 | 0.085 | 0.112 | 0.056 | |
0.034 | 0.083 | 0.072 | 0.082 | 0.094 | 0.049 | 0.056 | 0.101 | 0.085 | 0.088 | 0.117 | 0.068 | |
GLS detrending | ||||||||||||
0.064 | 0.156 | 0.137 | 0.122 | 0.184 | 0.069 | 0.065 | 0.155 | 0.126 | 0.156 | 0.184 | 0.100 | |
0.053 | 0.134 | 0.118 | 0.102 | 0.154 | 0.056 | 0.055 | 0.115 | 0.100 | 0.122 | 0.145 | 0.078 | |
0.004 | 0.030 | 0.019 | 0.018 | 0.028 | 0.007 | 0.013 | 0.038 | 0.031 | 0.033 | 0.043 | 0.016 | |
0.061 | 0.166 | 0.146 | 0.127 | 0.184 | 0.078 | 0.075 | 0.175 | 0.144 | 0.168 | 0.226 | 0.106 | |
0.046 | 0.123 | 0.110 | 0.096 | 0.142 | 0.050 | 0.055 | 0.108 | 0.094 | 0.116 | 0.133 | 0.080 | |
0.053 | 0.115 | 0.103 | 0.100 | 0.141 | 0.048 | 0.059 | 0.107 | 0.089 | 0.118 | 0.131 | 0.070 | |
0.042 | 0.107 | 0.100 | 0.085 | 0.121 | 0.042 | 0.047 | 0.094 | 0.082 | 0.104 | 0.109 | 0.069 | |
0.047 | 0.118 | 0.111 | 0.095 | 0.143 | 0.049 | 0.055 | 0.107 | 0.093 | 0.115 | 0.132 | 0.076 | |
0.049 | 0.149 | 0.126 | 0.115 | 0.167 | 0.066 | 0.066 | 0.119 | 0.113 | 0.129 | 0.148 | 0.091 | |
Bootstrap Tests | ||||||||||||
0.049 | 0.130 | 0.105 | 0.091 | 0.125 | 0.052 | 0.067 | 0.096 | 0.077 | 0.079 | 0.099 | 0.061 | |
0.052 | 0.141 | 0.110 | 0.107 | 0.136 | 0.055 | 0.071 | 0.099 | 0.083 | 0.090 | 0.109 | 0.072 | |
Johansen’s Trace Test | ||||||||||||
0.710 | 0.829 | 0.742 | 0.056 | 0.062 | 0.084 |
Test | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
OLS Detrending | ||||||||||||
0.097 | 0.177 | 0.136 | 0.134 | 0.164 | 0.110 | 0.195 | 0.266 | 0.257 | 0.245 | 0.303 | 0.172 | |
0.061 | 0.108 | 0.086 | 0.086 | 0.098 | 0.072 | 0.095 | 0.140 | 0.129 | 0.134 | 0.153 | 0.088 | |
0.083 | 0.159 | 0.121 | 0.116 | 0.153 | 0.103 | 0.254 | 0.423 | 0.364 | 0.347 | 0.432 | 0.247 | |
0.028 | 0.053 | 0.040 | 0.034 | 0.052 | 0.037 | 0.052 | 0.101 | 0.096 | 0.090 | 0.106 | 0.050 | |
0.079 | 0.139 | 0.116 | 0.115 | 0.134 | 0.100 | 0.120 | 0.162 | 0.152 | 0.160 | 0.176 | 0.11 0 | |
0.072 | 0.141 | 0.108 | 0.100 | 0.124 | 0.089 | 0.125 | 0.158 | 0.148 | 0.149 | 0.161 | 0.112 | |
GLS detrending | ||||||||||||
0.197 | 0.292 | 0.233 | 0.235 | 0.241 | 0.195 | 0.191 | 0.262 | 0.219 | 0.207 | 0.279 | 0.189 | |
0.174 | 0.252 | 0.211 | 0.209 | 0.210 | 0.164 | 0.138 | 0.145 | 0.149 | 0.117 | 0.163 | 0.116 | |
0.035 | 0.047 | 0.037 | 0.030 | 0.035 | 0.037 | 0.044 | 0.076 | 0.069 | 0.060 | 0.086 | 0.04 | |
0.209 | 0.297 | 0.246 | 0.235 | 0.256 | 0.214 | 0.240 | 0.381 | 0.307 | 0.279 | 0.369 | 0.238 | |
0.157 | 0.234 | 0.192 | 0.197 | 0.194 | 0.164 | 0.134 | 0.130 | 0.138 | 0.116 | 0.155 | 0.106 | |
0.157 | 0.237 | 0.187 | 0.200 | 0.205 | 0.150 | 0.118 | 0.130 | 0.138 | 0.109 | 0.150 | 0.109 | |
0.137 | 0.200 | 0.171 | 0.172 | 0.170 | 0.138 | 0.119 | 0.112 | 0.122 | 0.099 | 0.132 | 0.089 | |
0.154 | 0.229 | 0.188 | 0.197 | 0.192 | 0.158 | 0.133 | 0.132 | 0.138 | 0.112 | 0.154 | 0.103 | |
0.188 | 0.271 | 0.223 | 0.221 | 0.235 | 0.199 | 0.149 | 0.168 | 0.156 | 0.126 | 0.176 | 0.122 | |
Bootstrap Tests | ||||||||||||
0.194 | 0.223 | 0.218 | 0.193 | 0.238 | 0.169 | 0.118 | 0.136 | 0.139 | 0.124 | 0.148 | 0.113 | |
0.200 | 0.239 | 0.226 | 0.206 | 0.251 | 0.188 | 0.129 | 0.149 | 0.135 | 0.130 | 0.153 | 0.121 | |
Johansen’s Trace Test | ||||||||||||
0.730 | 0.849 | 0.711 | 0.070 | 0.052 | 0.059 |
DGP1 | DGP2 | DGP3 | DGP4 | |||||
---|---|---|---|---|---|---|---|---|
Test | ||||||||
OLS Detrending | ||||||||
0.927 | 0.086 | 0.283 | 0.016 | 0.140 | 0.739 | 0.672 | 0.041 | |
0.084 | 0.024 | 0.112 | 0.009 | 0.017 | 0.020 | 0.188 | 0.011 | |
0.994 | 0.161 | 0.481 | 0.032 | 0.272 | 0.933 | 0.899 | 0.109 | |
0.070 | 0.010 | 0.075 | 0.003 | 0.010 | 0.012 | 0.154 | 0.003 | |
0.086 | 0.033 | 0.123 | 0.013 | 0.022 | 0.022 | 0.208 | 0.023 | |
0.136 | 0.056 | 0.134 | 0.03 | 0.048 | 0.072 | 0.228 | 0.037 | |
GLS detrending | ||||||||
0.338 | 0.078 | 0.270 | 0.036 | 0.062 | 0.273 | 0.216 | 0.033 | |
0.012 | 0.050 | 0.106 | 0.034 | 0.029 | 0.006 | 0.058 | 0.031 | |
0.033 | 0.006 | 0.053 | 0.004 | 0.003 | 0.006 | 0.065 | 0.003 | |
0.559 | 0.094 | 0.393 | 0.034 | 0.096 | 0.490 | 0.343 | 0.029 | |
0.012 | 0.047 | 0.101 | 0.032 | 0.028 | 0.006 | 0.057 | 0.028 | |
0.010 | 0.046 | 0.093 | 0.031 | 0.032 | 0.005 | 0.052 | 0.031 | |
0.012 | 0.035 | 0.085 | 0.029 | 0.027 | 0.006 | 0.054 | 0.024 | |
0.013 | 0.046 | 0.100 | 0.032 | 0.029 | 0.006 | 0.057 | 0.028 | |
0.062 | 0.059 | 0.133 | 0.028 | 0.051 | 0.052 | 0.093 | 0.039 | |
Bootstrap Tests | ||||||||
0.136 | 0.052 | 0.104 | 0.039 | 0.061 | 0.085 | 0.189 | 0.044 | |
0.122 | 0.051 | 0.113 | 0.034 | 0.056 | 0.078 | 0.188 | 0.041 | |
Johansen’s Trace Test | ||||||||
0.044 | 0.064 | 0.041 | 0.037 |
© 2016 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons by Attribution (CC-BY) license ( http://creativecommons.org/licenses/by/4.0/).
Share and Cite
Cappuccio, N.; Lubian, D. Unit Root Tests: The Role of the Univariate Models Implied by Multivariate Time Series. Econometrics 2016, 4, 21. https://doi.org/10.3390/econometrics4020021
Cappuccio N, Lubian D. Unit Root Tests: The Role of the Univariate Models Implied by Multivariate Time Series. Econometrics. 2016; 4(2):21. https://doi.org/10.3390/econometrics4020021
Chicago/Turabian StyleCappuccio, Nunzio, and Diego Lubian. 2016. "Unit Root Tests: The Role of the Univariate Models Implied by Multivariate Time Series" Econometrics 4, no. 2: 21. https://doi.org/10.3390/econometrics4020021
APA StyleCappuccio, N., & Lubian, D. (2016). Unit Root Tests: The Role of the Univariate Models Implied by Multivariate Time Series. Econometrics, 4(2), 21. https://doi.org/10.3390/econometrics4020021