Research Method Seasonally Restacked Multi-series STSM Analogous to the multivariate structural time series model, this study proposes a seasonally restacked multi-series structural time series model (M-STSM), in which the vector “multi-series” is formed by restacking the quarterly components that are split from the original quarterly series. As discussed above, the SSOE formulate is more general than its MSOE counterpart in terms the parameter space and it generates better forecasts (de Silva, Hyndman, and Snyder 2010). Therefore, the SSOE scheme is adopted in this study. Following the notation introduced by Anderson and Moore (1979), the general form of the M-STSM can be formulated as: , where is a 4 × 1 vector of extracted quarterly components of the tourism demand series at time t. For instance, refers to a time series of tourism demand in the th quarter in the sample. i Equations (1) and (2) are known as the measurement equation and transition equation respectively. The structure matrix and transition matrix are typically fixed with dimensions H F of 4 × and × respectively, and is × 1 vector of states. The innovations is a 4 × 1 vector m m m m that follows a normal distribution Following the suggestion by de Silva, Hyndman, and N(0, Σ). Snyder (2010) the covariance matrix of innovations is assumed diagonal. is an × 4 impact K m Σ matrix, in which the non-zero off-diagonal elements capture the inter-quarter associations at the latent component level. Through the impact matrix an innovation is allowed to influence other K, three series in addition to its own. Based on the general form of the M-STSM, three special cases can be considered. The multi-series local trend model is specified as below:
Research Method Seasonally Restacked Multi-series STSM Analogous to the multivariate structural time series model, this study proposes a seasonally restacked multi-series structural time series model (M-STSM), in which the “multi-series” vector is formed by restacking the quarterly components that are split from the original quarterly series. As discussed above, the SSOE formulate is more general than its MSOE counterpart in terms the parameter space and it generates better forecasts (de Silva, Hyndman, and Snyder 2010). Therefore, the SSOE scheme is adopted in this study. Following the notation introduced by Anderson and Moore (1979), the general form of the M-STSM can be formulated as: , where is a 4 × 1 vector of extracted quarterly components of the tourism demand series at time t. For instance, refers to a time series of tourism demand in the i th quarter in the sample. Equations (1) and (2) are known as the measurement equation and transition equation respectively. The structure matrix H and transition matrix F are typically fixed with dimensions of 4 × m and m × m respectively, and is m × 1 vector of states. The innovations is a 4 × 1 vector that follows a normal distribution N(0, Σ). Following the suggestion by de Silva, Hyndman, and Snyder (2010) the covariance matrix Σ of innovations is assumed diagonal. K is an m × 4 impact matrix, in which the non-zero off-diagonal elements capture the inter-quarter associations at the latent component level. Through the impact matrix K, an innovation is allowed to influence other three series in addition to its own. Based on the general form of the M-STSM, three special cases can be considered. The multi-series local trend model is specified as below:
Forecasting%20Seasonal%20Tourism%20Demand%20Using%20a%20Multi-Series%20Structural%20Time%20Series%20Method.docx