ARMA and GARCH models for silver, nickel and copper price returns

Abstract: This thesis compares Auto Regressive Moving Average (ARMA) and Generalized Auto Regressive Conditional Heteroscedacity (GARCH) models for three metal commodities. ARMA models have an unconditionally non-random and constant variance, which typically serves well in effectively representing homoscedastic data. The GARCH models feature variable variance that is non-random when conditioning on the past. Thus these models are often used to represent heteroscedastic data. It is documented that financial data, including metal commodities frequently exhibit heteroscedacity. This thesis investigates if this heteroscedacity in the observed historical data is shown in the quality of its ARMA and GARCH fits. The data used for comparison involve three time series of logarithmic price return for silver, nickel and copper. In the hypothesis it is assumed that GARCH is more efficient than ARMA. The efficient market hypothesis is also tested. The logarithmic price returns are stationary which is confirmed by statistical tests. Thereby, it is appropriate to fit ARMA and GARCH models. The ARMA and GARCH models with the lowest Akaike’s Information Criterion (AIC) are selected from each series. The models forecasted values and running standard deviations are cross-validated with the observed historical data using three measures. These measures are Mean Absolute Scaled Error (MASE), symmetric Mean Absolute Percentage Error (sMAPE) and correct pairs of sign which all provide different assessment of magnitude of error in estimation of the observed historical records. The correct pairs of sign are then tested against the efficient market hypothesis. The error in estimation for forecast values does not yield a difference between ARMA and GARCH models by MASE. For the running standard deviation, both measures MASE and sMAPE are applied. The GARCH model is then more efficient than ARMA. In this sense, the thesis confirms the increased efficiency of using GARCH models for metal commodities. According to correct pairs of sign measure, nickel has no arbitrage opportunities for logarithmic price return. This is expected according to the efficient market hypothesis. However, the test indicates it is possible to predict correct sign of logarithmic price return for copper and silver, which indicates that the efficient market hypothesis does not always apply.