# Essay on Garch Model Fitting for Ibm

1742 Words Feb 20th, 2015 7 Pages
GARCH Model Fitting For IBM

Introduction:
Nowadays, many econometric studies have documented that financial return time series tend to be highly heteroskedastic, with varying variability or volatility and thick-tailed marginal distribution. Volatility is the conditional variance, which is unobservable and tends to be clustered together. Clustering is referred to as the phenomenon that large price changes of either signs tend to be followed by large changes, and similarly small changes follow small changes. Sudden bursts of volatility in financial returns exhibit strong dependence on the return time series, and a period of tranquility alternates with a period of volatility bursts. A popular ARCH, which is called Autoregressive
Therefore, with the optimal parameters above, we can fit the data to linear GARCH (1, 1) model, yielding: xt=0.000105 + yt σt2 = 0.000003+0.213663yt-12+0.655531σt-12
Moreover, since α1=0.213663, β1=0.655531 α1+β1 = 0.869194 <1, while α1>0, β1<1
We can ensure the weak stationary and finite second moment assumption for yt.
If, 1-2α12-α1+β12= 0.1532 > 0,
Kurtosis = Eyt4Eyt22 = 3*1-0.86919421-0.8691942-2*0.2136632 = 4.7880 > 3
We can conclude that the tail distribution of Garch (1, 1) process is heavier than that of a normal distribution.
Model Diagnosis: *

Plot Returns, the autocorrelation of log returns and the autocorrelation of log square returns
From the plot of ACF Log Return, we can find that for any k>j, there is no correlation between yt's, because as we know that: Covyi, yk=Eyiyk-EyiEyk=Eyiyk=0
Moreover, from the plot of ACF of Log Square Return, there is highly correlation among yt2's, because the lags in the plot of ACF Log Return have a more massive expression, and more lags are reach the dotted line. This indicates the correlation between the lags. However, in the ACF of Log Square Return IBM, there is much less lags reaching the dotted line. The ones, which are high correlated, have much longer lags. In addition the two ACF plots are quite good with a few lags out of the dotted line.

* Fit a GARCH model to the returns * GARCH (1,1)

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