Application of a Log Likelihood Object In GARCH with T-distributed errors and EGARCH With Generalised Error Distribution Model of the Spot AUD/USD Exchange Rate Volatility

In this article, we have tested the volatility of the returns of the spot exchange rate of AUD/USD for changing conditional variances by using a log likelihood model. Generalized autoregressive conditional heteroskedastic models, (GARCH) with t-distributed errors, and exponential generalized autoregressive conditional heteroskedastic model, (EGARCH) with generalised error distribution take into account the non-linearity that arises in the financial time series. The aim is to compare and select the maximum value of the log likelihood estimation of the AUD/USD spot exchange rate. The log likelihood model take into account autoregressive, (AR), moving average, (MA), and monthly seasonal moving average, (SMA) factors that could better explain volatility clusters. We have selected the model with the best forecasting ability in terms of the lowest value of the Akaike information criterion, the Schwarz criterion, the Hannan – Quinn criterion. The best model will help the arbitrageurs to better craft their investment strategy in terms of holding, buying or selling portfolios of foreign currencies. The software that we have used is EViews 6. We have concluded that the best fit model is the GRACH model with a t-distribution, as it has the maximum log likelihood estimation of -500.354. The average log likelihood is -1.81. The Akaike information criterion, the Schwarz criterion and the Hannan – Quinn criterion have the lowest error estimates. Their values are 3.60, 3.56 and 3.59 respectively. In terms of gradients at the estimated parameters, we have found that there are outlier values and significant fluctuations at the various observations of the coefficients vectors, C(1), C(2), and C(3) of the gradients. Finally, the analytic derivatives were calculated based on the specified values. The real and minimum step sizes are identical for all coefficients vectors and very close to zero. We have used one-sided numeric derivative. The data that we have used are monthly returns starting from 01/01/1990 to 01/01/2013, which total to 276 observations. The total dataset includes 277 observations. The data was obtained from the Federal Reserve Statistical Release Department and the symbol of the series is H.10