Generalised Autoregressive Conditional Heteroscedasticity Modelling of Drought Series in Northern Nigeria

Abstract

The various physical mechanisms governing the dynamics of drought series act on a seemingly wide range of temporal and spatial scales; almost all the mechanisms involved present some degree of nonlinearity. Against the back drop of these issues. This paper dealt with modelling the heteroscedasticity in the residuals of the Autoregressive Integrated Moving Average (ARIMA) model using a Generalised Autoregressive Conditional Heteroscedasticity (GARCH) model. Attempt was made to critically evaluate the subject generalised autoregressive conditional heteroscedasticity (GARCH) or volatility of drought series at SPI-3 and SPI-9 timescale resolution. It was also evident that the traditional seasonal Autoregressive Moving Average (ARMA) models are inadequate in describing ARCH effect in SPIs series process. For instance, the squared residuals (SR) and Standardised Squared Residual (SSR) are clearly correlated and appeared to be identically same, there is no distinctive different between SR and SSR in both SPI-3 and SPI-9 as the autocorrelation structures of both squared residual series still exhibit traces of strong seasonality with a lot of Spikes exceeding the confident bounds at 5% significant limits. As the P-values of the Engle’s test, as all the values are less than 0.05 significant level. The physical implication of this is that the variance of residual series is conditional on its past history; that is, the residual series exhibited ARCH effect. Therefore, the GARCH modelling approach was introduced i.e.  for ARIMA (1,1,3) x (1,1,1)₁₂-GARCH (5,3) and ARIMA (2,1,2)-GARCH (1,1) for the SPI-3 and SPI-9 respectively which captured the heteroscedasticity remaining in the residuals of the ARIMAs model. Considering this, the potential for a hybrid Autoregressive Moving Average (ARIMA) and Generalised Autoregressive Conditional Heteroscedasticity (GARCH)-type models should be further explored and probably embraced for modelling higher temporal accumulation of SPI-12, SPI-24 and SPI-48. in view of the relevance of statistical modelling in hydrology.