Autoregressive conditional duration modeling and forecasting the earthquakes temporal data of the different regions of Pakistan

Pakistan is situated in an earthquake-prone area, and in the past, many destructive earthquakes have occurred including the most destructive 1935 Quetta, and 2005 Kashmir, earthquakes. Therefore, it is of vital importance to investigate the statistical properties of the earthquake temporal data of Pakistan. The present study brings novelty through the vast extension of the linear and nonlinear autoregressive conditional duration models in the field of seismology. A simple duration model capable of describing and forecasting the earthquake elapsed times data has been developed. Different specifications of the duration models were considered to completely extract the time-dependent structure in earthquake data. This study aims to identify the most suitable in-sample fitted and out-of-sample forecasting models for the earthquake elapsed times data of Pakistan. A variety of autoregressive conditional duration models were applied to the complete and updated earthquake catalog of Pakistan. The most suitable model was chosen through statistical model evaluation techniques. The method of maximum likelihood was used to estimate the model parameters. The adequacy of fitted models is assessed through residual analysis. The obtained results suggest that the Logarithmic Autoregressive Conditional Duration model of type 2 (LACD2) appeared as the most suitable in-sample fitted model for describing the earthquake temporal data of the different zones of Pakistan. Further, the simple autoregressive conditional duration (ACD) model outperforms the remaining considered models regarding 1-through-4 steps-ahead out-of-sample forecasting performance for both India–Eurasia collision (IEC) and Makran subduction (MS) zones. Autoregressive models with Burr and exponential distributions as assumed error distributions have appeared as the most suitable fitted models for IEC and MS zones, respectively. The residuals analysis results show that the most suitable fitted models are correctly identified. The results show that the earthquake short-term (h=1) forecasting with duration models is more accurate in comparison with earthquake long-term (h=1, 2, 3) forecasting. The forecasted elapsed times for IEC and MS zones are 0.60 and 2.94 years, respectively. The obtained results show that the autoregressive conditional models are a more useful tool for forecasting the earthquake elapsed times for short term in comparison with the long-term forecasting. Hence, autoregressive conditional models are capable of modeling and forecasting the earthquake temporal data of the different regions of Pakistan. Among the competing models, the best fit model can serve the purpose of data description, missing value estimation in earthquake catalogs and uncertainty quantification in the earthquake occurrence process. The most suitable forecasted model yields the future earthquake occurrence trend in the study region.