Bayesian analysis of multivariate threshold autoregressive models with missing data

Abstract

In some fields, we are forced to work with missing data in multivariate time series, unfortunately the analysis in this context cannot be done as in the case of complete data. Bayesian analysis of multivariate thresholds autoregressive models(MTAR) with exogenous inputs and missing data is carried out. MCMC methods are used to obtain samples from the marginal posterior distributions, including threshold values and missing data. In order to identify autoregressive orders, we adapt the Bayesian variable selection method to the MTAR models. The number of regimes is estimated using marginal likelihood and product space strategies. The forecasting of the output vector is implemented finding its predictive distributions. Simulation experiments and real data examples are presented. Resumen: En algunos campos, nos vemos forzados a trabajar con datos faltantes en series de tiempo multivaridas, desafortunadamente el análisis en este contexto no puede ser hecho como en caso completo. El análisis de modelos multivaridos autoregresivos de umbrales(MTAR) con entradas exogenas y datos faltantes es llevado a cabo vía el enfoque Bayesiano. Los métodos MCMC son usados para obtener muestras de las distribuciones marginales aposteriori, incluyendo los valores de los umbrales y los datos faltantes. Con el objetivo de identificar losórdenes autoregresivos, el método Bayesiano de selección de varibales es adaptado para modelos MTAR. El número de regímenes es estimado usando la versimilitud marginal y las estrategias de espacio producto. El pronóstico para el vector de salida es implementado encontrando sus densidades predictivas. Experimentos de simulación y ejemplos con datos reales son presentados.

Bayesian Analysis of Multiplicative Seasonal Threshold Autoregressive Processes

“…So & Chen (2003) use the SSVS method to determine the autoregressive orders and the remaining parameters, when the number of regimes in the SETAR models is known. Calderón & Nieto (2017) use the KM and GVS methods in multivariate TARX models.…”

“…Additionally, we set φ ij = 25, η ij = 1.50 for all i, j, χ uj = 25, ξ uj = 1.50 for all u, j, ψ vj = 25, λ vj = 1.50 for all v, j, i = 0, 1, 2, u = 1, 2, v = 1, 2, 3, and j = 1, 2, 3, 4, 5. These values are the same as those chosen by Calderón & Nieto (2017). Furthermore, ν j and λ j , j = 1, 2, 3, 4, 5; a, b, H and d 0 are the same hyperparameters that were assumed for the TARX model used in the seasonality test.…”

“…The TAR models can be easily extended, including some exogenous time series and called TARX. The use of Bayesian methods applied to the TAR and SETAR models, can be found in Chen & Lee (1995), So & Chen (2003), Nieto (2005), Nieto (2008), Nieto, Zhang & Li (2013), Zhang & Nieto (2015) and Calderón & Nieto (2017). While to the TARX models, can be found in Chen (1998), So, Chen & Liu (2006), , Chen, Gerlach & Lin (2010), among other works.…”

Forecasting with Multivariate Threshold Autoregressive Models

“…Wu & Lee (2011) also developed a Bayesian methodology to analyze MTAR models with conditional heteroscedasticity without exogenous variables as covariates. Calderon & Nieto (2017) carried out the analysis of MTAR models with missing data in the output and exogenous vectors. Variable selection methodology was employed to identify the autoregressive orders; marginal likelihood and the Metropolised Carlin-and-Chib gave in (Dellaportas et al, 2002) were used to determinate the number of regimes.…”

“…Nevertheless, these forecasting procedures do not incorporate, neither, the forecasting of the exogenous variables nor the uncertainty of autoregressive orders. Therefore, in this paper, a methodology is proposed to compute the point forecasts of variables-of-interest that it is based on Calderon & Nieto's (2017) model. Additionally, the uncertainty of the point forecasts is obtained using the predictive distributions.…”