Robust spatial Durbin modelling on tuberculosis data using the MM-estimator method

In this study, we use daily gold log-returns to analyse the quality of forecasting expected shortfalls (ES) using volatility and models based on the extreme value theory (EVT). ES forecasts were calculated for conditional APARCH models formed on the entire distribution of returns, as well as for EVT models. The results of ES forecasts for each model were verified using the backtesting procedure proposed by Acerbi and Szekely. The results show that EVT models provide more accurate one-day ahead ES forecasts compared to the other models. Moreover, the asymmetric theoretical distributions for innovations of EVT models allow the improvement of the accuracy of ES forecasting.The spatial Durbin model (SDM) is a spatial regression model which shows the existence of spatial dependency on the response variable and predictor variables. However, SDM modelling may sometimes involve problems associated with e.g. the existence of spatial outliers. One way to overcome outliers in the SDM model is to use robust regression in the form of the robust spatial Durbin model (RSDM). This study aims to estimate the parameters of RSDM based on data on tuberculosis (TB) cases recorded in 2020 in the South Sulawesi Province in Indonesia and to identify the factors that affect the number of TB cases in the region. The MM-Estimator robust regression estimation method was used. It is a combination of a method involving a high breakdown value for the S-estimator and a high efficiency of the M-estimator. The results of the analysis show that RSDM can overcome outliers in spatial regression models. This is reflected in the value of the mean square error (MSE) of the RSDM, which is 6,461.734, i.e. smaller than the value of the SDM model, and the adjusted R^2 value of 99.52%, which is greater than that of the SDM model. The factors that influence the number of TB cases in the South Sulawesi Province are population density, the percentage of households leading a healthy lifestyle, the percentage of residents with Bacillus Calmette-Guérin (BCG) immunisation, and the percentage of those suffering from malnutrition.