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2023 | OriginalPaper | Chapter

2. ARMA(p,q) Processes

Author : John D. Levendis

Published in: Time Series Econometrics

Publisher: Springer International Publishing

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Abstract

An ARIMA model is made up of two components: an Autoregressive (AR) model and a Moving average (MA) model. Both rely on previous data to help predict future outcomes. AR and MA models are the building blocks of all our future work in this text. They are foundational, so we proceed slowly. First, we introduce the concept of stationarity and see what restrictions on the two models are required for stationarity. Then we turn to estimating the models, extracting the associated Impulse Response Functions, and using the models for forecating.

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Footnotes
1
The list of influential economists who have worked in some capacity at the Cowles Commission is astounding. All in all, 13 Nobel Prize-winning economists have worked at the commission including Maurice Allais, Kenneth Arrow, Gerard Debreu, Ragnar Frisch, Trygve Haavelmo, Lenoid Hurwicz, Lawrence Klein, Tjalling Koopmans, Harry Markowitz, Franco Modigliani, Edmund Phelps, Joseph Stiglitz, and James Tobin. Not all were involved in the econometric side of the Cowles Commission’s work.
 
2
For a brief discussion of the Cowles approach, see Fair (1992). Epstein (2014) provides much more historical detail. Diebold (1998) provides some historical context, as well as a discussion of the more current macroeconomic forecasting models that have replaced the Cowles approach.
 
3
This is reminiscent of adding epicycles to models of the geocentric universe. The basic model wasn’t fitting the data right, so they kept adding tweaks on top of tweaks to the model, until the model was no longer elegant.
 
4
I am indebted to Keele & Kelly (2005) who showed the algebra behind OLS’ bias when used with lagged dependent variables.
 
5
For AR(p) models, the requirements for stationarity are a little more stringent than they are for AR(1) processes. Necessary conditions include that the \(\beta \)s each be less than one in magnitude, they must not sum to anything greater than plus or minus one, and they cannot be more than one unit apart. We will explore the stationarity restrictions at greater length in Chap. 4.
 
6
Notice that it is critical that \(\beta _1\) not be equal to one, as you’d be dividing by zero and the expectation would not be defined. This is a familiar result: stationarity requires that we not have a unit root. We will explore the consequences of such “unit roots” in Chap. 5.
 
7
We do this because the earlier data have not yet converged to their long-run level. By keeping only the later observations, we ensure that the earlier data do not contaminate our analysis. It is probably overkill to drop so many of our initial observations, but we’re playing with lots of fake data anyway.
 
Literature
Ando, A., Modigliani, F., & Rasche, R. (1972). Appendix to part 1: Equations and definitions of variables for the FRB-MIT-Penn econometric model, November 1969. In Econometric Models of Cyclical Behavior (Vols. 1 and 2’, NBER, pp. 543–598).
Cooper, R. L. (1972). The predictive performance of quarterly econometric models of the united states. In Econometric models of cyclical behavior (Vols 1 and 2’, NBER, pp. 813–947).
Diebold, F. X. (1998), The past, present, and future of macroeconomic forecasting. Journal of Economic Perspectives, 12(2), 175–192.CrossRef
Epstein, R. J. (2014). A history of econometrics. Elsevier.
Fair, R. C. (1992), The Cowles Commission approach, real business cycle theories, and New-Keynesian economics. In The business cycle: Theories and evidence (pp. 133–157). Springer.
Keele, L., & Kelly, N. J. (2005). Dynamic models for dynamic theories: The ins and outs of lagged dependent variables. Political Analysis, 14(2), 186–205.CrossRef
Lucas, R. E. (1976). Econometric policy evaluation: A critique. In Carnegie-Rochester conference series on public policy (Vol. 1, pp. 19–46). Elsevier.
Naylor, T. H., Seaks, T. G., & Wichern, D. W. (1972). Box-Jenkins methods: An alternative to econometric models. International Statistical Review/Revue Internationale de Statistique, 40(2), 123–137.
Nelson, C. R. (1972). The prediction performance of the FRB-MIT-Penn model of the US economy. American Economic Review, 62(5), 902–917.
Metadata
Title
ARMA(p,q) Processes
Author
John D. Levendis
Copyright Year
2023
Book
Time Series Econometrics

Print ISBN: 978-3-031-37309-1

Electronic ISBN: 978-3-031-37310-7

Copyright Year: 2023

DOI
https://doi.org/10.1007/978-3-031-37310-7_2

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