Analysis of time series subject to changes in regime.

*(English)*Zbl 0723.62050The 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##### Software:

nlmdl
Full Text:
DOI

##### References:

[1] | Baum, Leonard E.; Petrie, Ted; Soules, George; Weiss, Norman: A maximization technique occuring in the statistical analysis of probabilistic functions of Markov chains. Annals of mathematical statistics 41, 164-171 (1970) · Zbl 0188.49603 |

[2] | Cecchetti, Stephen G., Pok-sang Lam, and Nelson Mark, forthcoming, Mean reversion in equilibrium asset prices, American Economic Review. |

[3] | Chiang, Chin Long: An introduction to stochastic processes and their applications. (1980) · Zbl 0427.60001 |

[4] | Cook, Timothy; Hahn, Thomas: The effect of changes in the federal funds rate target on market interest rates in the 1970’s. Journal of monetary economics 24, 331-351 (1989) |

[5] | Cosslett, Stephen R.; Lee, Lung-Fei: Serial correlation in discrete variable models. Journal of econometrics 27, 79-97 (1985) |

[6] | Degroot, Morris H.: Optimal statistical decisions. (1970) · Zbl 0225.62006 |

[7] | Dempster, A. P.; Laird, N. M.; Rubin, D. B.: Maximum likelihood from incomplete data via the EM algorithm. Journal of the royal statistical society B 39, 1-38 (1977) · Zbl 0364.62022 |

[8] | Engel, Charles and James D. Hamilton, forthcoming, Long swings in the exchange rate: Are they in the data and do markets know it?, American Economic Review. |

[9] | Engle, Robert F.: Autoregressive conditional heteroscedasticity with estimates of the variance of united kingdom inflation. Econometrica 50, 987-1007 (1982) · Zbl 0491.62099 |

[10] | Everitt, B. S.; Hand, D. J.: Finite mixture distributions. (1981) · Zbl 0466.62018 |

[11] | Fama, Eugene F.; French, Kenneth R.: Permanent and temporary components of stock prices. Journal of political economy 96, 246-273 (1988) |

[12] | Gallant, A. Ronald: Nonlinear statistical models. (1987) · Zbl 0611.62071 |

[13] | Goldfeld, Stephen M.; Quandt, Richard M.: A Markov model for switching regressions. Journal of econometrics 1, 3-16 (1973) · Zbl 0294.62087 |

[14] | Hamilton, James D.: Rational-expectations econometric analysis of changes in regime: an investigation of the term structure of interest rates. Journal of economic dynamics and control 12, 385-423 (1988) · Zbl 0661.62117 |

[15] | Hamilton, James D.: A pseudo-Bayesian approach to estimating parameters for mixtures of normal distributions. (1988) |

[16] | Hamilton, James D.: A new approach to the economic analysis of nonstationary time series and the business cycle. Econometrica 57, 357-384 (1989) · Zbl 0685.62092 |

[17] | Hassett, Kevin: Persistence and cyclicality in the aggregate labor market. (1988) |

[18] | Kiefer, Nicholas M.: Discrete parameter variation: efficient estimation of a switching regression model. Econometrica 46, 427-434 (1978) · Zbl 0408.62058 |

[19] | Kiefer, Nicholas M.: A note on switching regressions and logistic discrimination. Econometrica 48, 1065-1069 (1980) · Zbl 0448.62048 |

[20] | Lam, Pok-Sang: The generalized Hamilton model: estimation and comparison with other models of economic time series. (1988) |

[21] | Lee, Lung-Fei; Chesher, Andrew: Specification testing when score statistics are identically zero. Journal of econometrics 31, 121-149 (1986) · Zbl 0615.62146 |

[22] | Liporace, Louis A.: Maximum likelihood estimation for multivariate observations of Markov sources. IEEE transactions on information theory 28, 729-734 (1982) · Zbl 0499.62072 |

[23] | Jr., Robert E. Lucas: Asset prices in an exchange economy. Econometrica 66, 1429-1445 (1978) · Zbl 0398.90016 |

[24] | Perron, Pierre: The great crash, the oil price shock and the unit root hypothesis. Econometrica 57, 1361-1401 (1989) · Zbl 0683.62066 |

[25] | Poterba, James M.; Summers, Lawrence H.: Mean reversion in stock prices: evidence and implications. Journal of financial economics 22, 27-59 (1988) |

[26] | Quandt, Richard E.: The estimation of parameters of linear regression system obeying two separate regimes. Journal of the American statistical association 55, 873-880 (1958) · Zbl 0116.37304 |

[27] | Ruud, Paul A.: Extension of estimation methods using the EM algorithm. (1988) |

[28] | Theil, Henri: Principles of econometrics. (1971) · Zbl 0221.62002 |

[29] | Watson, Mark W.; Engle, Robert F.: Alternative algorithms for the estimation of dynamic factor, mimic, and varying coefficient regression models. Journal of econometrics 23, 385-400 (1983) · Zbl 0534.62083 |

[30] | Watson, Mark W.; Engle, Robert F.: Testing for regression coefficient stability with a stationary \(AR(1)\) alternative. Review of economics and statistics 67, 341-346 (1985) |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.