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**Analysis of time series subject to changes in regime.**
*(English)*
Zbl 0723.62050

The paper builds on an approach based on the author’s paper, Econometrica 57, No.2, 357-384 (1989; Zbl 0685.62092), for analyzing discrete shifts of time series. The parameters of a vector autoregression are regarded as subjected to occasional discrete shifts. The probability law governing these shifts is also stated explicitly and presumed to exhibit dynamic behaviour of its own. The task is then to determine when the shifts occurred and to estimate parameters characterizing the different regimes and the probability law for the transition between regimes.

The expressions in this paper permit analytic derivatives of the sample log-likelihood function to be calculated quite trivially from the smoothed inferences about the unobserved regime. Adding more parameters requires no changes in the routine for calculating smoothed probabilities, and thus has essentially no effect on the computation time required to calculate the gradient. In fact, the methods can maximize the likelihood function for a large vector system in less time than required for scalar systems, because the time required for iteration is basically independent of the size of the system and the number of iterations required can be lower.

The paper further shows how this class of models can be estimated by using the EM principle, presenting a summary of the particular algebraic results necessary to apply the named principle in the present context. Finally, there are some comments on use of Bayesian priors, alternative approaches to modeling nonstationary processes and hypothesis testing.

The expressions in this paper permit analytic derivatives of the sample log-likelihood function to be calculated quite trivially from the smoothed inferences about the unobserved regime. Adding more parameters requires no changes in the routine for calculating smoothed probabilities, and thus has essentially no effect on the computation time required to calculate the gradient. In fact, the methods can maximize the likelihood function for a large vector system in less time than required for scalar systems, because the time required for iteration is basically independent of the size of the system and the number of iterations required can be lower.

The paper further shows how this class of models can be estimated by using the EM principle, presenting a summary of the particular algebraic results necessary to apply the named principle in the present context. Finally, there are some comments on use of Bayesian priors, alternative approaches to modeling nonstationary processes and hypothesis testing.

Reviewer: J.C.Abril (Tucuman)

### MSC:

62M10 | Time series, auto-correlation, regression, etc. in statistics (GARCH) |

### Keywords:

maximum likelihood estimates; discrete-valued Markov process; discrete shifts of time series; vector autoregression; analytic derivatives of the sample log-likelihood function; smoothed inferences; smoothed probabilities; EM principle; Bayesian priors; nonstationary processes; hypothesis testing### Citations:

Zbl 0685.62092### Software:

nlmdl
Full Text:
DOI

### References:

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