Revisiting the Canadian Lynx Time Series Analysis Through TARMA Models
Keywords:Population dynamics, Predator-prey interaction, Canadian lynx time series, Nonlinear time series, TARMA processes, Asymmetric cycle
The class of threshold autoregressive models has been proven to be a powerful and appropriate tool to describe many dynamical phenomena in different fields. In this work, we deploy the threshold autoregressive moving-average framework to revisit the analysis of the benchmark Canadian lynx time series. This data set has attracted great attention among non-linear time series analysts due to its asymmetric cycle that makes the investigation very challenging. We compare some of the best threshold autoregressive models (TAR) proposed in literature with a selection of threshold
autoregressive moving-average models (TARMA). The models are compared under different prospectives: (i) goodness-of-fit through information criteria, (ii) their ability to reproduce characteristic cycles, (iv) their capability to capture multimodality and (iii) forecasting performance. We found TARMAmodels that perform better than TAR models with respect to all these aspects.
H. AN, S. CHEN (1997). A note on the ergodicity of non-linear autoregressive model. Statistics & Probability Letters, 34, no. 4, pp. 365–372.
F. BEC, M. BEN SALEM, M. CARRASCO (2004). Tests for unit-root versus threshold specification with an application to the purchasing power parity relationship. Journal of Business & Economic Statistics, 22, no. 4, pp. 382–395.
F. BEC, A. GUAY, E. GUERRE (2008). Adaptive consistent unit-root tests based on autoregressive threshold model. Journal of Econometrics, 142, no. 1, pp. 94–133.
P. J. BROCKWELL, R. J.WILLIAMS (1997). On the existence and application of continuous time threshold autoregressions of order two. Advances in Applied Probability, 29, no. 1, p. 205–227.
M.CANER, B.HANSEN (2001). Threshold autoregression with a unit root. Econometrica, 69, no. 6, pp. 1555–1596.
K.-S. CHAN (1990). Testing for threshold autoregression. The Annals of Statistics, 18, no. 4, pp. 1886–1894.
K.-S. CHAN (1993). Consistency and limiting distribution of the least squares estimator of a threshold autoregressive model. The Annals of Statistics, 21, no. 1, pp. 520–533.
K.-S. CHAN, S. GIANNERINI, G. GORACCI, H. TONG (2019). Testing for threshold regulation with measurement error. Tech. rep., University of Iowa and University of Bologna.
K.-S. CHAN, S. GIANNERINI, G. GORACCI, H. TONG (2020). Unit-root test within a threshold ARMA framework. arXiv:2002.09968 [stat.ME]. URL https://arxiv.org/abs/2002.09968.
K.-S. CHAN, G. GORACCI (2019). On the ergodicity of first-order threshold autoregressive moving-average processes. Journal of Time Series Analysis, 40, no. 2, pp. 256–264.
K.-S. CHAN, H. TONG (1985). On the use of the deterministic Lyapunov function for the ergodicity of stochastic difference equations. Advances in Applied Probability, 17, no. 3, pp. 666–678.
K.-S. CHAN, H. TONG (2001). Chaos: A Statistical Perspective. Springer Series in Statistics. Springer-Verlag, New York.
K.-S. CHAN, H. TONG (2010). A note on the invertibility of nonlinear ARMA models. Journal of Statistical Planning and Inference, 140, no. 12, pp. 3709–3714.
K. S. CHAN, R. S. TSAY (1998). Limiting properties of the least squares estimator of a continuous threshold autoregressive model. Biometrika, 85, no. 2, pp. 413–426.
C. CHEN, M. K. SO, F.-C. LIU (2011). A review of threshold time series models in finance. Statistics and its Interface, 4, no. 2, pp. 167–181.
R. CHEN, R. TSAY (1991). On the ergodicity of TAR(1) processes. The Annals of Applied Probability, 1, no. 4, pp. 613–634.
I. CHOI (2015). Almost All about Unit Roots: Foundations, Developments, and Applications. Themes in Modern Econometrics. Cambridge University Press, Cambridge.
D. CLINE (2009). Thoughts on the connection between threshold time series models and dynamical systems. In K.-S. CHAN (ed.), Exploration of a Nonlinear World. An Appreciation of Howell Tong’s Contributions to Statistics, World Scientific, Singapore, pp. 165–181.
J. CRYER, K.-S. CHAN (2008). Time Series Analysis - With Applications in R. Springer, New York, 2nd ed.
J. G. DE GOOIJER (2017). Elements of Nonlinear Time Series Analysis and Forecasting. Springer International Publishing, Cham, Switzerland.
R. DE JONG, C.-H. WANG, BAE (2007). Correlation robust threshold unit root tests. Mimeo, Ohio State University, Michigan.
J. L. DOOB (1936). Review: Vito Volterra, Leçons sur la Théorie Mathématique de la Lutte pour la Vie. Bulletin of the American Mathematical Society, 42, no. 5, pp. 304–305.
C. ELTON, M.NICHOLSON (1942). The ten-year cycle in numbers of the Lynx in Canada. Journal of Animal Ecology, 11, no. 2, pp. 215–244.
W. ENDERS, C. GRANGER (1998). Unit-root tests and asymmetric adjustment with an example using the term structure of interest rates. Journal of Business & Economic Statistics, 16, no. 3, pp. 304–311.
J. FAN, Q. YAO (2003). Nonlinear Time Series. Springer-Verlag, New York.
P. GALEANO, D. PEÑA (2007). Improved model selection criteria for SETAR time series models. Journal of Statistical Planning and Inference, 137, pp. 2802–2814.
S. GIANNERINI, G. GORACCI (2020). Forecasting with threshold ARMA models. Tech. rep., University of Bologna.
S. GIANNERINI, E. MAASOUMI, E. BEE DAGUM (2015). Entropy testing for nonlinear serial dependence in time series. Biometrika, 102, pp. 661–675.
F. GIORDANO, M. NIGLIO, C. VITALE (2017). Unit root testing in presence of a double threshold process. Methodology and Computing in Applied Probability, 19, no. 2, pp. 539–556.
J. GOOIJER (1998). On threshold moving-average models. Journal of Time Series Analysis, 19, no. 1, pp. 1–18.
G. GORACCI (2020). An empirical study on the parsimony and descriptive power of TARMA models. Statistical Methods & Applications. URL https://doi.org/10.1007/s10260-020-00516-8.
G. GORACCI, S. GIANNERINI (2020). Testing for threshold effects in ARMA models. Tech. rep., University of Bologna.
M. GUO, J. PETRUCCELLI (1991). On the null recurrence and transience of a first-order SETAR model. Journal of Applied Probability, 28, no. 3, pp. 584–592.
N. HALDRUP, M. JANSSON (2006). Improving size and power in unit root testing. In K. PATTERSON, T. MILLS (eds.), Palgrave Handbook of Econometrics: Volume 1: Econometric Theory, Palgrave Macmillan UK, Basingstoke, pp. 252–277.
B. HANSEN (2011). Threshold autoregression in economics. Statistics and its Interface, 4, no. 2, pp. 123–127.
A. S. HERBERT (1959). Book reviews: Elements of Mathematical Biology by Alfred J. Lotka. Econometrica, 27, no. 3, pp. 493–495.
H.-L. HSU, C.-K. ING, H. TONG (2019). On model selection from a finite family of possibly misspecified time series models. The Annals of Statistics, 47, no. 2, pp. 1061–1087.
Y. KAJITANI, K. W. HIPEL, A. I. MCLEOD (2005). Forecasting nonlinear time series with feed-forward neural networks: A case study of Canadian lynx data. Journal of Forecasting, 24, no. 2, pp. 105–117.
G. KAPETANIOS (2001). Model selection in threshold models. Journal of Time Series Analysis, 22, no. 6, pp. 733–754.
G. KAPETANIOS, Y. SHIN (2006). Unit root tests in three-regime SETAR models. The Econometrics Journal, 9, no. 2, pp. 252–278.
S. KONISHI, G. KITAGAWA (1996). Generalised information criteria in model selection. Biometrika, 83, no. 4, pp. 875–890.
D. LI, W. LI, S. LING (2011). On the least squares estimation of threshold autoregressive moving-average models. Statistics and its Interface, 4, pp. 183–196.
D. LI, S. LING (2012). On the least squares estimation of multiple-regime threshold autoregressive models. Journal of Econometrics, 167, no. 1, pp. 240 – 253.
D. LI, S. LING, W. K. LI (2013). Asymptotic theory on the least squares estimation of threshold moving-average models. Econometric Theory, 29, no. 3, p. 482–516.
G. LI,W. LI (2008). Testing for threshold moving average with conditional heteroscedasticity. Statistica Sinica, 18, pp. 647–665.
G. LI, W. LI (2011). Testing a linear time series model against its threshold extension. Biometrika, 98, no. 1, pp. 243–250.
K. S. LIM (1987). A comparative study of various univariate time series models for Canadian lynx data. Journal of Time Series Analysis, 8, no. 2, pp. 161–176.
T. C. LIN, M. POURAHMADI (1998). Nonparametric and non-linear models and data mining in time series: A case-study on the Canadian lynx data. Journal of the Royal Statistical Society, Series C, 47, no. 2, pp. 187–201.
S. LING, H. TONG (2005). Testing for a linear MA model against threshold MA models. The Annals of Statistics, 33, no. 6, pp. 2529–2552.
S. LING, H. TONG, D. LI (2007). Ergodicity and invertibility of threshold moving-average models. Bernoulli, 13, no. 1, pp. 161–168.
W. LIU, J.L., C. LI (1997). On a threshold autoregression with conditional heteroscedastic variances. Journal of Statistical Planning and inference, 62, no. 2, pp. 279–300.
J. PARK, M. SHINTANI (2016). Testing for a unit root against transitional autoregressive models. International Economic Review, 57, no. 2, pp. 635–664.
K. PATTERSON (2010). A Primer for Unit Root Testing. Palgrave Texts in Econometrics. Palgrave Macmillan UK, Basingstoke.
K. PATTERSON (2011). Unit Root Tests in Time Series Volume 1: Key Concepts and Problems. Palgrave Texts in Econometrics. Palgrave Macmillan UK, Basingstoke.
K. PATTERSON (2012). Unit Root Tests in Time Series Volume 2: Extensions and Developments. Palgrave Texts in Econometrics. Palgrave Macmillan UK, Basingstoke.
L. QIAN (1998). On maximum likelihood estimators for a threshold autoregression. Journal of Statistical Planning and Inference, 75, no. 1, pp. 21 – 46.
R. RUDNICKI (2003). Long-time behaviour of a stochastic prey–predator model. Stochastic Processes and their Applications, 108, no. 1, pp. 93 – 107.
M. SEO (2008). Unit root test in a threshold autoregression: Asymptotic theory and residual based block bootstrap. Econometric Theory, 24, no. 6, pp. 1699–1716.
N. C. STENSETH, W. FALCK, K.-S. CHAN, O. BJORNSTAD, M. O’DONOGHUE, H.TONG, R. BOONSTRA, S. BOUTIN, C.KREBS, N.YOCCOZ (1998). From patterns to processes: Phase and density dependencies in the Canadian lynx cycle. Proceedings of the National Academy of Sciences, 95, pp. 15430–15435.
M. STONE (1977). An asymptotic equivalence of choice of model by cross-validation and Akaike’s criterion. Journal of the Royal Statistical Society, Series B, 39, no. 1, pp. 44–47.
H. TONG (1978). On a threshold model. In C. CHEN (ed.), Pattern Recognition and Signal Processing, Sijthoff & Noordhoff, Amsterdam, NATO ASI Series E: Applied Sc. (29), pp. 575–586.
H. TONG (1983). Threshold models in non-linear time series analysis. Lecture Notes in Statistics. Springer-Verlag, New York.
H. TONG (1990). Non-linear Time Series: A Dynamical System Approach. Clarendon Press, Oxford.
H. TONG (2007). Birth of the threshold time series model. Statistica Sinica, 17, no. 1, pp. 8–14.
H. TONG (2011). Threshold models in time series analysis - 30 years on. Statistics and its Interface, 4, no. 2, pp. 107–118.
H. TONG (2017). Threshold models in time series analysis - some reflections. Journal of Econometrics, 189, no. 2, pp. 485 – 491.
H. TONG, K. LIM (1980). Threshold autoregression, limit cycles and cyclical data. Journal of the Royal Statistical Society, Series B, 42, no. 3, pp. 245–292.
Z. WANG, Y. XIE, J. LU, Y. LI (2019). Stability and bifurcation of a delayed generalized fractional-order prey–predator model with interspecific competition. Applied Mathematics and Computation, 347, pp. 360 – 369.
Y.WU, W. Q. ZHU (2008). Stochastic analysis of a pulse-type prey-predator model. Physical Review E, 77, p. 041911.
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