Abstract
The specification and estimation of an ARMA(p,q) model for a given realization involves several intermingled steps. First one must determine the orders p and q. Given the orders one can then estimate the parameters ϕ j , \(\theta _{j}\) and \(\sigma ^{2}\). Finally, the model has to pass several robustness checks in order to be accepted as a valid model. These checks may involve tests of parameter constancy, forecasting performance or tests for the inclusion of additional exogenous variables. This is usually an iterative process in which several models are examined. It is rarely the case that one model imposes itself. All too often, one is confronted in the modeling process with several trade-offs, like simple versus complex models or data fit versus forecasting performance. Finding the right balance among the different dimensions therefore requires some judgement based on experience.
Keywords
- Likelihood Function
- Bayesian Information Criterion
- Maximum Likelihood Estimator
- Forecast Performance
- ARMA Model
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- 1.
Note that the application of the estimator introduced in Sect. 4.2 guarantees that \(\widehat{\Gamma }_{p}\) is always invertible.
- 2.
If the process does not have a mean of zero, we can demean the data in a preliminary step.
- 3.
In Sect. 2.4 we showed how the autocovariance function γ and as a consequence \(\Gamma _{T}\), respectively G T can be inferred from a given ARMA model, i.e from a given β.
- 4.
One such algorithm is the innovation algorithm. See Brockwell and Davis (1991, section 5) for details.
- 5.
- 6.
See Brockwell and Davis (1991) for details and a deeper appreciation.
- 7.
See, for example, the section on the unit root tests 7.3.
- 8.
An exact definition will be provided in Chap. 7 In this chapter we will analyze the consequences of non-stationarity and discuss tests for specific forms of non-stationarity.
- 9.
The confidence regions are determined by the delta-method (see Appendix E).
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Neusser, K. (2016). Estimation of ARMA Models. In: Time Series Econometrics. Springer Texts in Business and Economics. Springer, Cham. https://doi.org/10.1007/978-3-319-32862-1_5
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