Abstract
Autoregressive moving-average models have become the predominant approach in the analysis of economic, especially macroeconomic time series. The success of these parametric models is due to a mature and by now well-understood statistical theory which has been the subject of this book. The main assumption behind this theory is its linear structure. Although convenient, the assumption of a constant linear structure turned out to be unrealistic in many empirical applications. The evolution of economies and the economic dynamics are often not fully captured by constant coefficient linear models. Many time series are subject to structural breaks which manifest themselves as a sudden change in the model coefficients by going from one period to another. The detection and dating of such structural breaks is the subject of Sect. 18.1. Alternatively, one may think of the model coefficients as varying over time. Such models have proven to be very flexible and able to generate a variety of non-linear features. We present in Sects. 18.2 and 18.3 two variants of such models. In the first one, the model parameters vary in a systematic way with time. They are, for example, following an autoregressive process. In the second one, the parameters switch between a finite number of states according to a hidden Markov chain. These states are often identified as regimes which have a particular economic meaning, for example as booms and recessions. Further parametric and nonparametric methods for modeling and analyzing nonlinear time series can be found in Fan and Yao (2003).
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Notes
- 1.
Generalization to higher order VAR models is straightforward. For changes in the covariance matrix \( \Sigma \) see Bai (2000). For the technical details the reader is referred to the relevant literature.
- 2.
As the asymptotic theory requires that t b ∕T does not go to zero, one has to assume that both the number of periods before and after the break go to infinity.
- 3.
A textbook version of the test can be found in Stock and Watson (2011).
- 4.
Assuming a trimming value of 0. 10 Andrews (2003, table I) reports critical values of 18.86 for p = 5 which corresponds to changes in the intercept only and 27.27 for p = 10 which corresponds to changes in intercept and time trend.
- 5.
See Perron (2006) for a discussion of multiple breaks.
- 6.
Potter (2000) discusses the primal problems of defining impulse responses in a nonlinear context.
- 7.
They allow for a correlation between V t and Z t .
- 8.
It is possible to consider other short-run type identification schemes (see Sect. 15.3) than the Cholesky factorization.
- 9.
The matrix exponential of a matrix A is defined as \( \exp (A) =\sum _{ i=0}^{\infty }\tfrac{1} {i!}A^{i} \) where A is any matrix. Its inverse \( \log (A) \) is defined only for \( \|A\| < 1 \) and is given by \( \log (A) =\sum _{ i=1}^{\infty }\tfrac{(-1)^{i-1}} {i} A^{i} \).
- 10.
A chain is called ergodic or irreducible if for every states i and j there is a strictly positive probability that the chain moves from state i to state j in finitely many steps. A chain is called aperiodic if it can return to any state i at irregular times. See, among others, Norris (1998) and Berman and Plemmons (1994) for an introduction to Markov chains and its terminology.
- 11.
References
Andrews DWK (1993) Tests for parameter instability and structural change with unknown change point. Econometrica 61:821–856
Andrews DWK (2003) Tests for parameter instability and structural change with unknown change point: A corrigendum. Econometrica 71:395–397
Andrews DWK, Ploberger W (1994) Optimal tests when a nuisance parameter is present only under the alternative. Econometrica 62:1383–1414
Aue A, Horváth L (2011) Structural breaks in time series. J Time Ser Anal 34:1–16
Bai J (2000) Vector autoregressive models with structural changes in regression coefficients and in variance-covariance matrices. Ann Econ Finance 1:303–339
Bai J, Lumsdaine RL, Stock JH (1998) Testing for and dating common breaks in multivariate time series. Rev Econ Stud 65:395–432
Baker A (2002) Matrix groups – an introduction to Lie Group theory. Springer, London
Berman A, Plemmons RJ (1994) Nonnegative matrices in the mathematical sciences. No. 9 in Classics in Applied Mathematics, Society of Industrial and Applied Mathematics, Philadelphia
Bougerol P, Picard N (1992b) Strict stationarity of generalized autoregressive processes. Ann Probab 20:1714–1730
Brandt A (1986) The stochastic equation y n+1 = a n y n + b n with stationary coefficients. Adv Appl Probab 18:211–220
Burren D, Neusser K (2013) The role of sectoral shifts in the decline of real GDP volatility. Macroecon Dyn 17:477–500
Canova F (2007) Methods for applied macroeconomic research. Princeton University Press, Princeton
Cogley T, Sargent TJ (2001) Evolving post-world war II U.S. inflation dynamics. In: Bernanke BS, Rogoff K (eds) NBER macroeconomics annual, vol 16. MIT Press, Cambridge, pp 331–373
Cogley T, Sargent TJ (2005) Drift and volatilities: Monetary policies and outcomes in the post WWII U.S. Rev Econ Dyn 8:262–302
Colonius F, Kliemann W (2014) Dynamical Systems and Linear Algebra. Graduate Studies in Mathematics, Vol. 158. American Mathematical Society, Providence, Rhode Island
Diebold FX, Lee JH, Weinbach GC (1994) Regime switching with time-varying transition probabilities. In: Hargreaves CP (ed) Nonstationary time series analysis and cointegration. Oxford Universiy Press, Oxford, pp 283–302
Doan T, Litterman RB, Sims CA (1984) Forecasting and conditional projection using realistic prior distributions. Econ Rev 3:1–100
Fan J, Yao Q (2003) Nonlinear time series. Springer, New York
Filardo AJ (1994) Business-cycle phases and their transitional dynamics. J Bus Econ Anal 12:299–308
Filardo AJ, FGordon S (1998) Business cycle duration. J Econ 85:99–123
Frühwirt-Schnatter S (2006) Finite mixture and markov switching models. Springer Science + Business Media LLC, New York
Goldfeld SM, Quandt RE (1973) A Markov model for switching regressions. J Econ 1:3–15
Goldfeld SM, Quandt RE (1976) Studies in nonlinear estimation. Ballinger Publishing, Cambridge
Gouriroux C, Jasiak J, Sufana R (2009) The Wishart autoregressive process of multivariate stochastic volatility. J Econ 150:167–181
Granger CWJ, Teräsvirta T (1993) Modelling nonlinear economic relationships. Oxford University Press, Oxford
Hamilton JD (1994b) Time series analysis. Princeton University Press, Princeton
Hamilton JD (1996) Specification testing in Markov-switching time-series models. J Econ 70:127–157
Kiefer NM (1978) Discrete parameter variation: Efficient estimation of a switching regression model. Econometrica 46:427–434
Kim CJ (1994) Dynamic linear models with Markov-switching. J Econ 60:1–22
Kim CJ, Nelson CR (1999) State-space models with regime-switching: classical and Gibbs-sampling approaches with applications. MIT Press, Cambridge
Koop G, Korobilis D (2009) Bayesian multivariate time series methods for empirical macroeconomics. Found Trends Econ 3:267–358
Krolzig HM (1997) Markov–switching vector autoregressions. Modelling, statistical inference, and application to business cycle analysis. Springer, Berlin
Maddala GS (1986) Disequilibrium, self-selection, and switching models. In: Griliches Z, Intriligator MD (eds) Handbook of econometrics, vol III. North-Holland, Amsterdam
Negro Md, Primiceri GE (2015) Time varying structural vector autoregressions and monetary polic: A corrigendum. Rev Econ Stud, forthcoming
Neusser K (2016) A topological view on the identification of structural vector autoregressions. Economics Letters forthcoming
Norris JR (1998) Markov chains. Cambridge series in statistical and probabilistic mathematics. Cambridge University Press, Cambridge
Perron P (2006) Dealing with structural breaks. In: Hassani H, Mills TC, Patterson K (eds) Palgrave handbook in econometrics, vol 1. Econometric theory. Palgrave Macmillan, Hampshire, pp 278–352
Potter SM (2000) Nonlinear impulse response functions. J Econ Dyn Control 24:1425–1446
Primiceri GE (2005) Time varying structural vector autoregressions and monetary policy. Rev Econ Stud 72:821–852
Quandt RE (1960) Tests of the hypothesis that a linear regression system obeys two separate regimes. J Am Stat Assoc 55:324–330
Stock JH, Watson MW (2011) Introduction to econometrics, 3rd edn. Addison Wesley, Longman
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Neusser, K. (2016). Generalizations of Linear Time Series Models. In: Time Series Econometrics. Springer Texts in Business and Economics. Springer, Cham. https://doi.org/10.1007/978-3-319-32862-1_18
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