A Bayesian analysis of moving average processes with time-varying parameters. (English) Zbl 1452.62683
Summary: A new Bayesian method is proposed for estimation and forecasting with Gaussian moving average (MA) processes with time-varying parameters. The focus is placed on MA models of order one, but a general result is given for an MA process of an arbitrary known order. A multiplicative model for the evolution of the squares of the parameters is introduced following Bayesian conjugacy through beta and truncated gamma distributions and a discount factor. Two new distributions are proposed providing the prior and posterior distributions of the parameters of the model and the one-step forecast distribution of the process. Several well-known distributional results are extended by replacing the gamma distribution with the truncated gamma distribution. The proposed methodology is illustrated with two examples consisting of simulated data and of aluminium spot prices of the London metal exchange.
MSC:
| 62M10 | Time series, auto-correlation, regression, etc. in statistics (GARCH) |
| 62F15 | Bayesian inference |
| 62P20 | Applications of statistics to economics |
| 62-08 | Computational methods for problems pertaining to statistics |
Keywords:
Bayesian models; forecasting; time series; moving average; time-varying parameters; locally stationary processes; London Metal ExchangeReferences:
| [1] | Anderson, P. L.; Meerschaert, M. M., Parameter estimation for periodically stationary time series, J. Time Ser. Anal., 26, 489-518 (2005) · Zbl 1090.62090 |
| [2] | Barnett, G.; Kohn, R.; Sheather, S., Robust Bayesian estimation of autoregressive-moving-average models, J. Time Ser. Anal., 18, 11-28 (1997) · Zbl 0936.62097 |
| [3] | Berger, J. O., Statistical Decision Theory and Bayesian Analysis (1985), Springer: Springer New York · Zbl 0572.62008 |
| [4] | Bibi, A.; Francq, C., Consistent and asymptotically normal estimators for cyclically time-dependent linear models, Ann. Inst. Statist. Math., 55, 41-68 (2003) · Zbl 1049.62094 |
| [5] | Box, G. E.P.; Jenkins, G. M.; Reinsel, G. C., Time Series Analysis, Forecasting and Control (1994), Prentice-Hall: Prentice-Hall Englewood Cliffs, NJ · Zbl 0858.62072 |
| [6] | Chapman, D. G., Estimating the parameters of a truncated gamma distribution, Ann. of Math. Statist., 27, 498-506 (1956) · Zbl 0072.36208 |
| [7] | Chatfield, C., The Analysis of Time Series (1996), Chapman & Hall: Chapman & Hall New York · Zbl 0870.62068 |
| [8] | Coffey, C. S.; Muller, K. E., Properties of doubly-truncated gamma variables, Comm. Statist. Theory Methods, 29, 851-857 (2000) · Zbl 0992.62012 |
| [9] | Dahlhaus, R., Fitting time series models to nonstationary processes, Ann. Statist., 25, 1-37 (1997) · Zbl 0871.62080 |
| [10] | Damien, P.; Walker, S. G., Sampling truncated normal, beta, and gamma densities, J. Comput. Graph. Statist., 10, 206-215 (2001) |
| [11] | DeJong, D. N.; Whiteman, C. H., Estimating moving average parameters: classical pileups and Bayesian posteriors, J. Bus. Econom. Statist., 11, 311-317 (1993) |
| [12] | Durbin, J.; Koopman, S. J., Time Series Analysis by State Space Methods (2001), Oxford University Press: Oxford University Press Oxford · Zbl 0995.62504 |
| [13] | Fildes, R., The evaluation of extrapolative forecasting methods, Int. J. Forecasting, 8, 69-80 (1992) |
| [14] | Foschi, D. A.; Belsley, D. A.; Kontoghiorghes, E. J., A comparative study of algorithms for solving seemingly unrelated regression models, Comput. Statist. Data Anal., 44, 3-35 (2003) · Zbl 1429.62289 |
| [15] | Francq, C.; Gautier, A., Large sample properties of parameter least squares estimates for time-varying ARMA models, J. Time Ser. Anal., 25, 765-783 (2004) · Zbl 1062.62172 |
| [16] | Francq, C.; Zakoan, J. M., Stationarity of multivariate Markov-switching ARMA models, J. Econometrics, 102, 339-364 (2001) · Zbl 0998.62076 |
| [17] | Harvey, A. C., Forecasting, Structural Time Series Models and the Kalman Filter (1989), Cambridge University Press: Cambridge University Press Cambridge |
| [18] | Hegde, L. M.; Dahiya, R. C., Estimation of the parameters of a truncated gamma distribution, Comm. Statist. Theory Methods, 18, 561-577 (1989) · Zbl 0696.62127 |
| [19] | Jawitz, J. W., Moments of truncated continuous univariate distributions, Adv. Water Resources, 27, 269-281 (2004) |
| [20] | Johnson, N. L.; Kotz, S.; Balakrishnan, N., Continuous Univariate Distributions, vol. 1 (1994), Wiley: Wiley New York · Zbl 0811.62001 |
| [21] | Kitagawa, G.; Gersch, W., Smoothness Priors Analysis of Time Series (1996), Springer: Springer New York · Zbl 0853.62069 |
| [22] | Koopman, S. J.; Ooms, M., Forecasting daily time series using periodic unobserved components time series models, Comput. Statist. Data Anal., 51, 885-903 (2006) · Zbl 1157.62505 |
| [23] | Leoneard, T.; Hsu, J. S.J., Bayesian Methods (1999), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0930.62023 |
| [24] | Marinucci, D.; Petrella, L., A Bayesian proposal for the analysis of stationary and nonstationary AR(1) time series, (Bernardo, J. M.; Berger, J. O.; Dawid, A. P.; Smith, A. F.M., Bayesian Statistics 6 (1999), Oxford University Press: Oxford University Press Oxford), 821-828 · Zbl 0976.62025 |
| [25] | Mercurio, D.; Spokoiny, V., Statistical inference for time-inhomogeneous volatility models, Ann. Statist., 32, 577-602 (2004) · Zbl 1091.62103 |
| [26] | Muirhead, R. J., Aspects of Multivariate Statistical Theory (1982), Wiley: Wiley New York · Zbl 0556.62028 |
| [27] | Nason, G. P.; von Sachs, R.; Kroisandt, G., Wavelet processes and adaptive estimation of the evolutionary wavelet spectrum, J. Roy. Statist. Soc. Ser. B, 62, 271-292 (2000) |
| [28] | Phillips, P. C.B., To criticize the critics: an objective Bayesian analysis of stochastic trend, J. Appl. Econometrics, 6, 333-364 (1991) |
| [29] | Prado, R.; Huerta, G., Time-varying autoregressions with model order uncertainty, J. Time Ser. Anal., 23, 599-618 (2002) · Zbl 1062.62201 |
| [30] | Prado, R.; West, M.; Krystal, A., Multi-channel EEG analyses via dynamic regression models with time-varying lag/lead structure, Appl. Statist., 50, 95-110 (2001) |
| [31] | Salvador, M.; Gargallo, P., Automatic monitoring and intervention in multivariate dynamic linear models, Comput. Statist. Data Anal., 47, 401-431 (2004) · Zbl 1429.62413 |
| [32] | Shaarawy, M.S., 1984. A Bayesian Analysis of Moving Average Processes. Unpublished Ph.D. Thesis, Oklahoma State University.; Shaarawy, M.S., 1984. A Bayesian Analysis of Moving Average Processes. Unpublished Ph.D. Thesis, Oklahoma State University. |
| [33] | Sims, C.; Uhlig, H., Understaing unit rooters: a helicopter tour, Econometrica, 59, 1591-1599 (1991) · Zbl 0739.62085 |
| [34] | Triantafyllopoulos, K., Multivariate control charts based on Bayesian state space models, Qual. Reliab. Eng. Int., 22, 693-707 (2006) |
| [35] | Tsay, R. S., Analysis of Financial Time Series (2002), Wiley: Wiley New York · Zbl 1037.91080 |
| [36] | Uhlig, H., On singular Wishart and singular multivariate beta distributions, Ann. Statist., 22, 395-405 (1994) · Zbl 0795.62052 |
| [37] | Watkins, C.; McAleer, M., Econometric modelling of non-ferrous metal prices, J. Econom. Surveys, 18, 651-701 (2004) |
| [38] | West, M.; Harrison, P. J., Bayesian Forecasting and Dynamic Models (1997), Springer: Springer New York · Zbl 0871.62026 |
| [39] | West, M.; Prado, R.; Krystal, A. D., Evaluation and comparison of EEG traces: latent structure in nonstationary time series, J. Amer. Statist. Assoc., 94, 375-387 (1999) |
| [40] | West, M.; Aguilar, O.; Prado, R.; Huerta, G., Bayesian inference on latent structure in time series (with discussion), (Bernardo, J. M.; Dawid, A. P.; Berger, J. C.; Smith, A. F.M., Bayesian Statistics 6 (1999), Oxford University Press: Oxford University Press Oxford), 3-26 · Zbl 0974.62066 |
| [41] | Zivot, E., A Bayesian analysis of the unit root hypothesis within an unobserved components model, Econometric Theory, 10, 552-578 (1994) |
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
