##
**Bernoulli vector autoregressive model.**
*(English)*
Zbl 1435.62322

Summary: In this paper, we propose a vector autoregressive (VAR) model of order one for multivariate binary time series. Multivariate binary time series data are used in many fields such as biology and environmental sciences. However, modeling the dynamics in multiple binary time series is not an easy task. Most existing methods model the joint transition probabilities from marginals pairwisely for which the resulting cross-dependency may not be flexible enough. Our proposed model, Bernoulli VAR (BerVAR) model, is constructed using latent multivariate Bernoulli random vectors. The BerVAR model represents the instantaneous dependency between components via latent processes, and the autoregressive structure represents a switch between the hidden vectors depending on the past. We derive the mean and matrix-valued autocovariance functions for the BerVAR model analytically and propose a quasi-likelihood approach to estimate the model parameters. We prove that our estimator is consistent under mild conditions. We perform a simulation study to show the finite sample properties of the proposed estimators and to compare the prediction power with existing methods for binary time series. Finally, we fit our model to time series of drought events from different regions in Mexico to study the temporal dependence, in a given region and across different regions. By using the BerVAR model, we found that the cross-region dependence of drought events is stronger if a rain event preceded it.

### MSC:

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

62M20 | Inference from stochastic processes and prediction |

62H12 | Estimation in multivariate analysis |

62P12 | Applications of statistics to environmental and related topics |

### Keywords:

cross-dependency; multivariate binary time series; multivariate Bernoulli; quasi-likelihood; vector autoregressive process### Software:

HIBITS
PDFBibTeX
XMLCite

\textit{C. Euán} and \textit{Y. Sun}, J. Multivariate Anal. 177, Article ID 104599, 19 p. (2020; Zbl 1435.62322)

Full Text:
DOI

### References:

[1] | Awale, M.; Ramanathan, T. V.; Kale, M., Coherent forecasting in integer-valued AR(1) models with geometric marginals, J. Data Sci., 15, 1, 95 (2017) |

[2] | Azzalini, A., Logistic regression and other discrete data models for serially correlated observations, J. Ital. Stat. Soc., 3, 2, 169-179 (1994) · Zbl 1446.62233 |

[3] | Berchtold, A.; Raftery, A., The mixture transition distribution model for high-order markov chains and non-gaussian time series, Statist. Sci., 17, 3, 328-356 (2002) · Zbl 1013.62088 |

[4] | Cessie, S. L.; Houwelingen, J. C.V., Logistic regression for correlated binary data, J. R. Stat. Soc. Ser. C. Appl. Stat., 43, 1, 95-108 (1994) · Zbl 0825.62509 |

[5] | Cox, D. R.; Reid, N., A note on pseudolikelihood constructed from marginal densities, Biometrika, 91, 3, 729-737 (2004) · Zbl 1162.62365 |

[6] | Czado, C.; Gneiting, T.; Held, L., Predictive model assessment for count data, Biometrics, 65, 4, 1254-1261 (2009) · Zbl 1180.62162 |

[7] | Dai, B.; Ding, S.; Wahba, G., Multivariate bernoulli distribution, Bernoulli, 19, 4, 1465-1483 (2013) · Zbl 1440.62227 |

[8] | (Davis, R. A.; Holan, S. H.; Lund, R.; Ravishanker, N., Handbook of Discrete-Valued Time Series. Handbook of Discrete-Valued Time Series, CRC Handbooks of Modern Statistical Methods, vol. 1 (2016), Chapman and Hall/CRC) · Zbl 1331.62003 |

[9] | Favre, A.-C.; Quessy, J.-F.; Toupin, M.-H., The new family of fisher copulas to model upper tail dependence and radial asymmetry: Properties and application to high-dimensional rainfall data, Environmetrics, 29, 3, Article e2494 pp. (2018) |

[10] | Fontana, R.; Semeraro, P., Characterization of multivariate Bernoulli distributions with given margins (2017), ArXiv e-prints |

[11] | Freeland, R. K.; McCabe, B., Forecasting discrete valued low count time series, Int. J. Forecast., 20, 3, 427-434 (2004) |

[12] | Gao, X.; Shahbaba, B.; Ombao, H., Modeling binary time series using gaussian processes with application to predicting sleep states, J. Classification, 35, 3, 549-579 (2018) · Zbl 1422.62219 |

[13] | Hyndman, R. J., Nonparametric additive regression models for binary time series, (Proceedings: ESAM99 (Econometric Society Australasian Meeting) (1999), University of Technology: University of Technology Sydney), 1-17 |

[14] | McKenzie, E., Some simple models for discrete variate time series, Water Resour. Bull., 21, 4, 645-650 (1985) |

[15] | Nicolau, J., A new model for multivariate markov chains, Scand. J. Stat., 41, 4, 1124-1135 (2014) · Zbl 1305.60061 |

[16] | Nicolau, J., A simple nonparametric method to estimate the expected time to cross a threshold, Statist. Probab. Lett., 123, 146-152 (2017) · Zbl 1463.62097 |

[17] | Raftery, A. E., A model for high-order Markov chains, J. R. Stat. Soc. Ser. B Stat. Methodol., 47, 3, 528-539 (1985) · Zbl 0593.62091 |

[18] | Sun, Y.; Genton, M. G., Functional boxplots, J. Comput. Graph. Statist., 20, 2, 316-334 (2011) |

[19] | Sun, Y.; Stein, M. L., A stochastic space-time model for intermittent precipitation occurrences, Ann. Appl. Stat., 9, 4, 2110-2132 (2015) · Zbl 1397.62489 |

[20] | Teugels, J. L., Some representations of the multivariate bernoulli and binomial distributions, J. Multivariate Anal., 32, 2, 256-268 (1990) · Zbl 0697.62042 |

[21] | Varin, C.; Czado, C., A mixed autoregressive probit model for ordinal longitudinal data, Biostatistics, 11, 1, 127-138 (2009) · Zbl 1437.62640 |

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.