Abstract
A basic idea in time series analysis is to construct more complex processes from simple ones. In the previous chapter we showed how the averaging of a white noise process leads to a process with first order autocorrelation. In this chapter we generalize this idea and consider processes which are solutions of linear stochastic difference equations. These so-called ARMA processes constitute the most widely used class of models for stationary processes.
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Notes
- 1.
One technical advantage of using the double-infinite index set \(\mathbb{Z}\) is that the lag operators form a group.
- 2.
Phillips and Sul (2007) provide an application and an in depth discussion of the hypothesis of economic growth convergence.
- 3.
The reader is invited to verify this.
- 4.
In the case of multiple roots one has to modify the formula according to Eq. (B.2).
- 5.
The use of the partial derivative sign actually represents an abuse of notation. It is inspired by an alternative definition of the impulse responses: \(\psi _{j} = \frac{\partial \tilde{\mathbb{P}}_{t}X_{t+j}} {\partial x_{t}}\) where \(\tilde{\mathbb{P}}_{t}\) denotes the optimal (in the mean squared error sense) linear predictor of X t+j given a realization back to infinite remote past \(\{x_{t},x_{t-1},x_{t-2},\ldots \}\) (see Sect. 3.1.3). Thus, ψ j represents the sensitivity of the forecast of X t+j with respect to the observation x t . The equivalence of alternative definitions in the linear and especially nonlinear context is discussed in Potter (2000).
- 6.
Without the qualification strict, the miniphase property allows for roots of \(\Theta (z)\) on the unit circle. The terminology is, however, not uniform in the literature.
- 7.
In case of multiple roots the formula has to be adapted accordingly. See Eq. (B.2) in the Appendix.
References
Brockwell PJ, Davis RA (1991) Time series: theory and methods, 2nd edn. Springer, New York
Phillips PCB, Sul D (2007) Some empirics on economic growth under heterogeneous technology. J Macroecon 29:455–469
Potter SM (2000) Nonlinear impulse response functions. J Econ Dyn Control 24:1425–1446
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Neusser, K. (2016). Autoregressive Moving-Average Models. In: Time Series Econometrics. Springer Texts in Business and Economics. Springer, Cham. https://doi.org/10.1007/978-3-319-32862-1_2
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DOI: https://doi.org/10.1007/978-3-319-32862-1_2
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