Abstract
This paper attempts to provide a comprehensive review of nonlinear time series models, starting with the rationale for such models, their superiority over their linear counterparts, and issues surrounding their analysis especially in terms of the simultaneous examination of nonlinear and nonstationary properties of the data. The study provides a detailed typology of various univariate nonlinear time series models, the aspects that it helps capture in data and their estimation procedures. The paper then provides an exposition of the concept of nonlinear cointegration in a multivariate context and some of the issues therein. As an illustrative example, the study estimates a SETAR model for the Indian money multiplier and provides a brief analysis. We conclude with the relevance and applicability of these models in further understanding the dynamics in economic data.
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Notes
- 1.
Disclaimer: The views expressed here do not reflect the views of the Indian Institute of Technology Bombay, Mumbai. The author is extremely grateful to Prof. Gilles Dufrenot and Prof. Takashi Matsuki for accepting this paper as a chapter. This paper was part of my PhD thesis submitted and defended at the Indira Gandhi Institute of Development Research, Mumbai. Responsibility for any remaining shortcomings and errors rests solely with the author.
- 2.
Refer to Appendix 1 for definition of a limit cycle.
- 3.
It is not necessary that the presence of such a function indicates a nonlinear relationship with certainty. Such relations can also be analyzed under the linear time series modeling framework by transforming the relation into a linear one.
- 4.
Deterministic time trend implies the trend in the time series is a deterministic function of time; stochastic time trend implies that the trend is not predictable (Gujarati and Porter 2008, pp. 745).
- 5.
The auxiliary regression is (Teräsvirta et al. 1994,Teräsvirta 1994):
\(\hat{\varepsilon }_{t} = \hat{\varvec{z}}_{{1\varvec{t}}}^{{\prime }} \tilde{\beta }_{1} + \hat{z}_{2t} \left(\varvec{\pi}\right)\tilde{\beta }_{2} + u_{t} \left(\varvec{\pi}\right),t = 1, \ldots ,T\) where \(\tilde{\beta }_{1} = \left( {\tilde{\beta }_{11} , \ldots ,\tilde{\beta }_{1,p + 1} } \right)^{{\prime }}\) and \(u_{t} \left(\varvec{\pi}\right)\) is the error term.
- 6.
The auxiliary regression is then formulated as:
\(\hat{\upsilon }_{t} = \tilde{\beta }_{1}^{{\prime }} \hat{\varvec{z}}_{1t} + \tilde{\beta }_{2}^{{\prime }} \varvec{w}_{\varvec{t}} y_{t - d} + \tilde{\beta }_{3}^{{\prime }} \varvec{w}_{\varvec{t}} y_{t - d}^{2} + e_{t}^{{\prime }} ,t = 1, \ldots ,T\) where \(\tilde{\beta }_{1} = (\tilde{\beta }_{10} ,\tilde{\beta }_{1}^{{\prime }} )^{{\prime }}\), \(\tilde{\beta }_{10} = \theta_{10} - (c^{*} )^{2} \theta_{20} ,e_{t}^{{\prime }}\) is the error term and \(\beta_{2} = 2c^{*}\varvec{\theta}_{2} - \theta_{20} \varvec{e}_{d}\) and \(\beta_{3} = -\varvec{\theta}_{2}\).
- 7.
The power spectrum is defined as:
$$s\left( \omega \right) = \frac{1}{2\pi }\mathop \sum \limits_{k = - \infty }^{\infty } R_{k} e^{{ - ik_{\infty } }}$$where Rk are the k autocovariances of xt which is a zero mean linear stationary process.
- 8.
Mixing processes (Dufrénot and Mignon 2002): Mixing is a concept used to measure the degree of dependence in the memory of a time series. Strong mixing can be understood as short-range dependence. Mixing implies that as the time span between two events increases, the dependence between past and future events becomes negligible.
Refer to Footnote 38 for formal definition of memory in time series.
- 9.
The Box-Cox transformation is given as:
\(\begin{aligned} \varvec{y}_{\varvec{t}} \left(\varvec{\lambda}\right) & = \frac{{\varvec{y}_{\varvec{t}}^{\varvec{\lambda}} - 1}}{\varvec{\lambda}},\varvec{\lambda}\ne 0,\varvec{y}_{\varvec{t}} \ge 0 \\ & = \log \varvec{y}_{\varvec{t}} ,\varvec{\lambda}= 0,\varvec{y}_{\varvec{t}} > 0 \\ \end{aligned}\)
where t represents the inclusion of a time trend and λ denotes the set of parameters that enter in the nonlinear model.
- 10.
- 11.
Econometricians refer to conditional variance while dealing with the volatility of the time series and the time varying volatility is referred to as conditional heteroscedasticity (Harris and Sollis 2006).
- 12.
The conditional mean (and variance) of a time series are the mean (and variance) conditional on the information set available at time t (Harris and Sollis 2006).
- 13.
The AR(p) process can also be replace by series of exogenous variables which include lagged dependent values of the dependent variable as well.
- 14.
Ergodicity: It is an attribute of stochastic systems; generally, a system that tends in probability to a limiting form that is independent of the initial conditions.
- 15.
Markovian property implies that the current value of the state variable depends on its immediate past value.
- 16.
Quasi-maximum likelihood estimators refer to the maximum likelihood estimators obtained when normality is assumed but the true conditional distribution is non-normal (Harris and Solis 2006).
- 17.
Hamilton (1996) defines the conditional score statistic as the derivative of the conditional log-likelihood of the tth observation with respect to the parameter vector. This score can be calculated using the procedure for smoothed probabilities; thus, it does not require estimating additional parameters by maximum likelihood.
- 18.
Mixing processes (Dufrénot and Mignon 2002): Mixing is a concept used to measure the degree of dependence in the memory of a time series. Mixing implies that as the time span between two events increases, the dependence between past and future events becomes negligible.
- 19.
Aparicio and Escribano (1998), pp. 121 suggest the general characterization of mean reversion, long- and short- memory and integrated of order d where \(i_{x} \left( {\tau ,t} \right)\) is considered to be a non-negative measure of serial dependence which captures higher-order dependency structure in the series.
- 20.
Hansen and Seo (2002): The notation of Xt-1(β) implies that the variables are evaluated at generic values and not the true values of β. The variables evaluated at the true values are denoted by Xt−1. A similar argument holds for the ECT term.
- 21.
The sup-LM value is the maximal value for which the test is most favourably rejected. A supremum statistic is an aggregation possibility in case of an unknown threshold parameter (which would result in a non-standard distribution and the threshold parameter thus being unidentified under the null).
- 22.
Residual-based bootstrap: The time series under consideration, yt are nonstationary and cannot be resampled directly. Given the assumption that ut are iid (pp. 72) and unobservable, the least-squares residuals of the TVECM are resampled independently with replacement. This is called the residual-based bootstrap (Seo 2006).
- 23.
Transactions costs lead to large (more than proportional) changes in real exchange rates which is captured using an exponential functional form in the STAR model.
- 24.
Thus, resulting in a complete peak-trough cycle.
- 25.
Data source—Reserve Bank of India (2013): Handbook of Statistics on the Indian Economy, Reserve Bank of India, Mumbai.
- 26.
The results of the stationarity and nonlinearity tests are available on request. The augmented Dickey Fuller test for unit roots was also conducted for the sake of completeness.
- 27.
The results, code are available on request from the author.
References
Ahmad, Y., & Glosser, S. (2007). Searching for nonlinearities in real exchange rates, Manuscript. Whitewater, WI: University of Wisconsin.
Aparicio, F., & Escribano, A. (1998). Information-theoretic analysis of serial dependence and cointegration. Studies in Nonlinear Dynamics and Econometrics, 3(3), 119–140.
Aparicio, F., Escribano, A., & GarcÃa, A. (2003). Range unit root (RUR) tests. Working paper 03–11 Statistics and Econometrics Series 26, Universidad Carlos III de Madrid.
Aparicio, F., Escribano, A., & Siplos, A. (2006). Range unit root (RUR) tests: robust against nonlinearities, error distributions, structural breaks and outliers. Journal of Time Series Analysis, 27, 545–576.
Ashley, R., Patterson, D. M., & Hinich, M. J. (1986). A diagnostic test for nonlinear serial dependence in time series fitting errors. Journal of Time Series Analysis, 7, 165–178.
Balke, N. S., & Fomby, T. B. (1997). Threshold cointegration. International Economic Review, 38(3), 627–645.
Basu, D. (2008). Essays on dynamic nonlinear time series models and on gender inequality. PhD Dissertation. The Ohio State University.
Bollerslev, T. (1986). Generalized autoregressive conditional heteroskedasticity. Journal of Econometrics, 31, 307–327.
Box, G., & Jenkins, GM. (1970). Time series analysis: forecasting and control. San Francisco: Wiley.
Brillinger, D. (1965). An Introduction to polyspectra. Annals of Mathematical Statistics, 36(5), 1351–1374.
Brock , W., Dechert, W., & Scheinkman, J. (1987). A test for independence based on the correlation dimension. Madison, Mimeo: Department of Economics, University of Wisconsin.
Bureau of Labor Statistics. (2020). Labour force statistics, current population survey. US Department of Labor, on the Internet at https://www.bls.gov/cps/data.htm.
Carnero, M., Peña, D., & Ruiz, E. (2004). Persistence and kurtosis in GARCH and stochastic volatility models. Journal of Financial Econometrics, 2(2), 319–342.
Chan, K., & Tong, H. (1986). A note on certain integral equations associated with nonlinear time series analysis. Probability Theory and Related Fields, 73, 153–158.
Chan, K. S. (1990), Testing for Threshold Autoregression, The Annals of Statistics, Vol. 18, No. 4, pp. 1886-1894.
Chen, R., & Tsay, R. S. (1991). On the ergodicity of TAR(1) processes. The Annals of Applied Probability, 1(4), 613–634.
Chitre, V. (1986). Quarterly predictions of reserve money multiplier and money stock in India. Arthavijnana, 28, 1–119.
Darbha, G. (2002). Testing for long-run stability—an application to money multiplier in India. Applied Economics Letters, 9, 33–37.
Davies, R. (1977). Hypothesis testing when a nuisance parameter is present only under the alternative. Biometrika, 74(1), 33–43 1987. https://doi.org/10.1093/biomet/74.1.33.
Gersovitz, M., & MacKinnon, J. M. (1978). Seasonality in regression: An application of smoothness priors, Journal of the American Statistical Association, 73(362), 264–73.
Dufrénot, G., & Mignon, V. (2002). Recent developments in nonlinear cointegration with applications to macroeconomics and finance. Dordrecht, Netherlands: Kluwer Academic Press.
Dua, P., & Kumawat, L. (2005). Modelling and forecasting seasonality in Indian macroeconomic time series, Centre for Development Economics, Working Paper 136.
Engle, R. (1982). Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom inflation. Econometrica, 50, 987–1007.
Engle, R. (1984). Wald, likelihood ratio, and lagrange multiplier tests in econometrics. In Z. Griliches, & M. Intriligator (Eds.), Chapter 13: Handbook of Econometrics, (Vol. 2, pp. 775–826). Amsterdam: Elsevier.
Escribano and Mira. (2002). Nonlinear error correction models. Journal of Time Series Analysis, 23(5), 509–522.
Franses, P. H., & McAleer, M. (1998). Testing for unit roots and nonlinear transformations. Journal of Time Series Analysis, 19(2), 147–164.
Goldfeld, S. M., & Quandt, R. (1973). A Markov model for switching regressions. Journal of Econometrics, 1(1), 3–16.
Granger, C. W., & Teräsvirta, T. (1993). Modelling nonlinear economic relationships. Oxford: Oxford University Press.
Granger, C. W. J., & Hallman, J. (1991). Nonlinear transformations of integrated time series. Journal of Time Series Analysis, 12(3), 207–218.
Gujarati, D., & Porter, D. C. (2008). Basic econometrics (5th ed.). New York: McGraw-Hill.
Hamilton, J. D. (1989). A new approach to the economic analysis of nonstationary time series and the business cycle. Econometrica, 57(2), 357–384.
Hamilton, J. D. (1994). Time series analysis. Princeton, NJ: Princeton University Press.
Hamilton, J. D. (1996). Specification testing in Markov-switching time series models. Journal of Econometrics, 70(1), 127–157.
Hamilton, J. D. (2005). Regime-switching models. In S. Durlauf, & L. Blume (Eds.), New palgrave dictionary of economics (2nd ed) London: Palgrave McMillan.
Hansen, B. E. (1992). The likelihood ratio test under nonstandard conditions: testing the Markov switching model of GNP. Journal of Applied Econometrics, 7(S1), S61–S82.
Hansen, B. (1997). Inference in TAR models. Studies in Nonlinear Dynamics and Econometrics, 2(1), 1–14.
Hansen, B. (2011). Threshold autoregression in economics. Statistics and Its Interface, 4, 123–127.
Hansen, B., & Seo, B. (2002). Testing for two-regime threshold cointegration in vector error-correction models. Journal of Econometrics, 110(2), 293–318.
Harris, R., & Sollis, R. (2006). Applied time series modelling and forecasting. Singapore: Wiley.
Hinich, M. J. (1982). Testing for gaussianity and linearity of a stationarv time series. Journal of Time Series Analysis, 3, 169–176.
Hinich, M. J. (1996). Testing for dependence in the input to a linear time series model. Nonparametric Statistics, 6, 205–221.
Jha, R., & Rath, D. P. (2001). On the endogeneity of the money multiplier in India, ASARC Working Papers 2001-12, Australia South Asia Research Centre, The Australian National University, available at: https://taxpolicy.crawford.anu.edu.au/acde/asarc/pdf/papers/conference/CONF2001_06.pdf.
Kar, S. (2010). A periodic autoregressive model of Indian WPI inflation, Margin—The Journal of Applied Economic Research, 4(3), 279–292.
Keenan, D. M. (1985). A Tukey nonadditivity-type test for time series nonlinearity. Biometrika, 72(1), 39–44.
Kilian, L., & Taylor, M. (2003). Why is it so difficult to beat the random walk forecast of exchange rates? Journal of International Economics, 60(1), 85–107.
Kim, C. J., Piger, J., & Startz, R. (2008). Estimation of Markov regime-switching regression models with endogenous switching. Journal of Econometrics, 143(2), 263–273. https://doi.org/10.1016/j.jeconom.2007.10.002.
Kirchler, M., & Huber, J. (2007). Fat tails and volatility clustering in experimental asset markets. Journal of Economic Dynamics and Control, 31(6), 1844–1874.
Krishnan, R. (2007). Seasonal characteristics of Indian time series. Indian Economic Review, 42(2) (July-December), 191–210.
Kuan, Chung-Ming. (2002). Lecture on the Markov switching model. Academia Sinica, Taipei: Institute of Economics.
Lahtinen, M. (2006). The purchasing power parity puzzle: a sudden nonlinear perspective. Applied Financial Economics, 16(1), 119–125.
Mandelbrot, B. (1963). The variation of certain speculative prices. Journal of Business, XXXVI, 392–417.
Miller, J. (2006). A random coefficients autoregressive model with exogenously-driven stochastic unit roots. In 2006 International Symposium on Econometric Theory and Applications, Xiamen, China.
Mizrach, B., & Watkins, J. (1999). A Markov switching cookbook. In P. Rothman (Ed.), Nonlinear time series analysis of economic and financial data. New York: Springer.
Mohanty, D., Chakraborty, A. B., Das, A. & John, J. (2011). Inflation threshold in India: an empirical investigation. RBI Working Paper Series (DEPR): 18/2011, Department of Economic and Policy Research. http://www.rbi.org.in/scripts/PublicationsView.aspx?id=13838.
Montgomery, A., Zarnowitz, V., Tsay, R., & Tiao, G. (1998, June). Forecasting the US unemployment rate. Journal of the American Statistical Association, 93(442), 478–493.
Morley, J., & Piger, J. (2010). The asymmetric business cycle. http://research.economics.unsw.edu.au/jmorley/abc.pdf .
Nachane, D. (1992). The money multiplier in India: short-run and long-run aspects. Journal of Quantitative Economics, 8(1), 51–66.
Nachane, D., & Ray, D. (1997). Non-linear dynamics of the money multiplier—selected case studies. Indian Economic Journal, 45(1), 36–53.
Nachane, D. (2006). Econometrics: theoretical foundations and empirical perspective. New Delhi: Oxford University Press.
Nachane, D. (2011). Selected problems in the analysis of nonstationary and nonlinear time series. Journal of Quantitative Economics, 9(1), 1–17.
Nachane, D., & Clavel, J. (2008). Forecasting interest rates: a comparative assessment of some second generation nonlinear models. Journal of Applied Statistics, 35(5), 493–514.
Nicholls, D. & Quinn, B. (1980). Random coefficient autoregressive models: an introduction. Berlin: Springer.
Petruccelli, J. D., & Woolford, S. W. (1984). A threshold AR(1) model. Journal of Applied Probability, 21(2), 270–286.
Priestley, M. B. (1988). Nonlinear and nonstationary time series analysis. New York, NY: Academic Press.
Ramsey, J. (1969). Tests for specification errors in classical linear least-squares regression analysis. Journal of the Royal Statistical Society Series B (Methodological), 31(2), 350–371.
Rangarajan, C., & Singh, A.. (1984, June). Reserve money: Concepts and Implications for India. Reserve Bank of India Occasional Papers.
Reserve Bank of India. (2004). Report on currency and finance. Mumbai: Reserve Bank of India.
Reserve Bank of India. (2013). Handbook of statistics of the Indian economy. Mumbai: Reserve Bank of India.
Saaty, T., & Bram, J. (1960). Nonlinear mathematics. Mineola: Courier Dover Publications.
Sarno, L. (2003). Nonlinear exchange rate models: a selective overview. IMF Working Paper WP/03/111.
Seo, M. (2006). Bootstrap testing for the null of no cointegration in a threshold vector error correction model. Journal of Econometrics, 127(1), 129–150.
Seo, M. (2009). Estimation of nonlinear error-correction models. Discussion paper. London School of Economics, http://personal.lse.ac.uk/seo/PDF/SEO-NECM.pdf.
Stigler, M. (2011). Threshold cointegration: overview and implementation. Working paper. http://cran.r-project.org/web/packages/tsDyn/vignettes/ThCointOverview.pdf.
Subba Rao, T., & Gabr, M. M. (1984). An introduction to bispectral analysis and bilinear time series models, Vol. 24 of lecture notes in statistics. New York: Springer.
Subba Rao, T., & Gabr, M. M. (1980). A test for linearity of stationary time series. Journal of Time Series Analysis, 1(2), 145–158.
Teräsvirta, T. (1994). Specification, estimation, and evaluation of smooth transition autoregressive models. Journal of the American Statistical Association, 89(425), 208–218.
Teräsvirta, T., Tjøstheim, D., & Granger, C. W. J. (1994). Aspects of modelling nonlinear time series. In R. F. Engle & D. McFadden (Eds.), Handbook of Econometrics (Vol. 4). Amsterdam: Elsevier.
Ters, K., & Urban, J. (2018). Estimating unknown arbitrage costs: Evidence from a 3-regime threshold vector error correction model, BIS Working Papers No. 689, Monetary and Economic Department, Bank for International Settlements.
Thurner, S., Farmer, J., & Geanakoplos, J. (2012). Leverage causes fat tails and clustered volatility. Quantitative Finance, 12(5), 695–707.
Tong, H. (1983). Threshold models in nonlinear time series analysis. Berlin: Springer.
Tsay, R. (1986). Nonlinearity tests for time series. Biometrika, 73(2), 461–466.
Tsay, R. (1987). Conditional heteroscedastic time series models. Journal of the American Statistical Association, 82, 590–604.
Tsay, R. (2002). Analysis of financial time series (1st ed.). New York: Wiley.
Wang, D., & Ghosh, S. (2004). Bayesian analysis of random coefficient autoregressive models, Institute of Statistics Mimeo Series, NC State University Libraries, available at: https://repository.lib.ncsu.edu/bitstream/handle/1840.4/184/ISMS2566.pdf?sequence=1&isAllowed=y.
Zhang, W.-B. (1988). Limit cycles in van der ploeg’s model of economic growth and conflict over the distribution of Income. Journal of Economics, 48(2), 159–173
Zivot, E. (2008). Practical issues in the analysis of univariate GARCH models. In T. G. Andersen, R. A. Davis, J. P. Kreiss, & T. V. Mikosch (Eds.), Handbook of financial time series. Berlin: Springer. http://faculty.washington.edu/ezivot/research/practicalgarchfinal.pdf.
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Appendix 1
Appendix 1
Limit cycles
Threshold models have the distinction of being able to demonstrate limit cycle behavior (under suitable conditions), encountered in the study of nonlinear differential equations. If et is set to zero for t > t0 (t0 is a threshold beyond which noise et is ‘switched off’), then Eq. (24) may possess a solution \(\tilde{X}_{t}\), which has an asymptotic periodic form. This solution is closely related to the forecast function of the model.
Note: Though \(\tilde{X}_{t} = E[X_{{t_{0} + 1}} |X_{{t_{0} }} , {\text{X}}_{{t_{0} - 1}} , \ldots ]\) but \(\tilde{X}_{t} \ne E[X_{t} |X_{{t_{0} }} , {\text{X}}_{{t_{0} + 1}} , \ldots ] , {\text{t}} > t_{0} + 1\).
This assumption clears the fact that the solution would then closely help in describing the cyclical phenomena.
Thus, such models help describe cyclical phenomena better than the conventional models based on superposition of harmonic components on a linear stationary residual i.e. a nonlinear model may provide satisfactory description of the data in case of a series exhibiting cyclical form with asymmetrical cycles.
Definition of a limit cycle: (for discrete time system) (Tong 1983): Let \(\varvec{x}_{\varvec{t}} \in {\Re }^{k}\) denote a k-dimensional vector satisfying the recurrence relation
where f(.) is a [k] vector-valued function. Let \(\varvec{f}^{{({\text{j}})}}\) denote the jth iterate of f, i.e.
Then a [k] vector c1 is called a stable periodic point of period T w.r.t a domain \({\text{D}} \subset {\Re }^{k}\) if
T being the smallest integer for which such convergence holds. In this case, \(c_{1} ,f^{\left( 1 \right)} \left( {c_{1} } \right),f^{\left( 2 \right)} \left( {c_{1} } \right), \ldots ,f^{{\left( {T - 1} \right)}} \left( {c_{1} } \right)\) are all distinct stable limit points of period T. Then rewriting these distinct limit points in a recursive relation as
the set of vectors (c1, c2, …, cT) is called a stable limit cycle of period T (w.r.t. D).
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Chaubal, A. (2021). Typology of Nonlinear Time Series Models. In: Dufrénot, G., Matsuki, T. (eds) Recent Econometric Techniques for Macroeconomic and Financial Data. Dynamic Modeling and Econometrics in Economics and Finance, vol 27. Springer, Cham. https://doi.org/10.1007/978-3-030-54252-8_13
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