In an earlier post we examined our estimated series of the dynamic trading risk factor returns for trending in the context of building an ARMA model. This was done via a Box-Jenkins estimation procedure and our methodology assumed the innovations where homoskedastic.

The obvious counterpoint, particularly in the context of financial data series, is whether the data is actually homoskedastic or whether it is more accurately modelled by an ARCH or GARCH type structure (and GARCH(1,1) should really be our null hypothesis for financial time series).

There are several tests for heteroskedasticity in the literature. A common method is to divide the data into two (or more) groups and examine the ratio of the subsample variances (this is the Goldfield-Quandt test for two groups, or the likelihood ratio test) or to perform an regression of the squared residuals onto lagged regressors and their cross products (White's test).

This discussion notwithstanding, in financial data the evidence of heteroskedasticity is often so compelling that little need of a formal test exists outside the desired to publish. We do know, whatever the underlying process, that for a homoskedastic process the residual sum of squares should accumumulate linearly with the sample size. The chart below is a nice exposition of this for our dynamic trading risk factor data. Plotted is the ratio of the cumulative residual sum of squares (CRSS) to the total residual sum of squares (TRSS) --- i.e. the variance x (T-1) where T is the sample size. Also plotted are the expected linear accumulation and a monte-carlo sample path for an IID N(0,1) process. The vertical bar represents the location of the maximum absolute difference between the CRSS and the expectation.
(Clearly this analysis is motivated by the Kolmogorov-Smirnov test, although I am not aware of the distribution of the test statistic d(T) = max |CRSS(t)/CRSS(T)-t/T|.)

With this graph cheering us on, for it does appear to indicate a notable departure from linear growth, I fitted the data to an AR(1)xGARCH(1,1) model with IID innovations drawn from the Generalized Error Distribution. The results, are shown below.

The regression prefers the inclusion of the GARCH terms, although from the 96 datapoints we have the significance is not overwhemling. The result is a slight downweighting of the AR(1) term in the process vs. the estimate for the homoskedastic model. Since the significance is low (greater than 1% of rejecting the null by chance), I will stick with the prior process specificiation for the time being.