We're going to go a head and fit a single factor model to the Barclay Hedge Fund Index data. To do this, I'm going to exclude all the funds that are clearly composite (Fund of Funds, Multi-Strategy Funds, and the the full Barclay Hedge Fund Index). Let's follow the structure suggested by the Fama-French equity factor models and fit the following model to the returns.

r is the individual index return; R is the single global factor return; and, we assume the innovations are i.i.d. (I'm going to start of with least-squares, which is equivalent to assuming they are i.i.d. normal -- but that assumption is likely to prove false). The alpha, beta, and the entire factor return series R are to be estimated by our procedure. This is a slightly more complicated problem since the model is a little bit bi-linear in free parameters. (I say "a little bit bi-linear" because the factor at any time is common for all individuals, and the alpha and beta for any individual are common for all time, so providing the data is not too pathological, we should be ok.)

The approach is to start with an initial condition of the alpha and beta set to their estimates from the prior regression onto the VIX-GARCH variance spread, and the factor set to the returns of the overall aggregate index, and follow a simplex optimization with the factor fixed. I then free the factor and complete the simplex optimization. Finally, I used the simplex estimates as a starting point for a BFGS Steepest Descent optimization. This latter step is useful to confirm the minimum identified by the simplex method and also to build a Hessian matrix to allow error estimation. In all there are 1440 data points and 126 free variables (1314 d.o.f.), so we should not expect too much from the data.

The regression did actually converge. I won't present the full output, because there are so many free variables. Some interesting charts follow. The first is the series of the estimated factor returns.

The upper chart shows the time series of the estimated factor returns and the lower chart is a histogram of those returns. The histogram has been fitted to the Generalized Error Distribution. This is a p.d.f. with an adjustable kurtosis that I find quite suitable for modelling financial data.

Secondly we have two histograms of the estimated values of the alphas and betas from the regressions.

These parameters estimate the idiosyncratic drift of an investment style and it's exposure to the general factor (the risk premium that accrues to traders). With such a small dataset, we can't really compare the alphas and betas to the null hypothesis (that alpha=0 and beta=1 i.e. that style makes no difference and all hedge fund managers just expose themselves to the general risk factor) in a manner that gives us statistically meaningful statement. However, the data do seem consistent with the hypothesis that the style index returns are mostly due to the common factor with a little bit of positive alpha.

Finally let's examine the relationship between the factor series we just estimated and the VIX-GARCH premium. In this case I will use the simple premium (literally just VIX-GARCH), since this will make the regression coefficient dimensionless.

Here we do find a statistically significant covariance, at a significance level of 0.003.