The appropriate model is estimated and a one to four step forecasting is undertaken to determine the appropriateness of the model.
We consider the price index of the Rio Tinto, the 5 day weekly stock price for the period 31st December 1999 to 31st December 2007 is used and the following chart summarizes the price index for the period.
From the above table it is evident that for the period 2000 to 2004 the price remained relatively stable deviating by small margins, however for the period 2005 to 2007 there was an increase in prices by larger margins. The following is an analysis of the Rio Tinto returns.
According to Woodridge (2006) dynamic heteroskedasticity can appear in regressions with no dynamic, in a regression if the Gauss Markov assumption holds then the estimators are BLUE (best linear unbiased estimator). However even when the homoskedasticity assumption that the error terms variance is constant across observations holds there could be still other forms of heteroskedasticity that may arise, heteroskedasticity can be tested using the white test or the Breusch pagan test. The following chart shows a case of homoskedasticity and heteroskedasticity:
From the above diagrams assuming that the 45 degree line is the fitted regression model, then the first diagram shows a case where as x increases the mean of y increases but the variance of y around its mean remains constant over time, for the second diagram a case where as x increases the mean of y increases and the variance of y around its mean does not remain constant and this shows heteroskedasticity.
There are a number of consequences of heteroskedasticity and they include the fact that:
Estimators are still linear functions of the independent variable
The estimators are not biased
Estimators no longer have minimum variance therefore are not efficient
The estimated variance of the estimators is biased because the formula to estimate them could over state or under state the true variance
The hypothesis test of the significance is unreliable given that the estimated variance is biased.
As a result Engel (1982) suggested the ARCH model that would consider a conditional error term variance that takes into consideration past error terms and this was the ARCH model. The ARCH and GARSH model are appropriate models that can be used in modeling financial data that exhibit volatility clustering, volatility clustering refers to a trend that shows that small increases or declines are followed by small increases or declines and that large increases or declines are followed by large increases or declines. From our price data chart it is evident that for the period 2000 to 2004 small increases and declines are followed by small increases or declines, however for the period 2005 and 2007 large increases are followed by large increases. This means that the ARCH and GARSH model are appropriate in estimating an appropriate model.
The following chart summarizes the returns mean, kurtosis and