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I would like to forecast volatility with GARCH. I have taken portfolio consisting of 2 stock indices. One is equity, another is fixed income. I have tested the returns of both indices and equity indice is normal (95% confidence level) in terms of simple return, whereas another is not. But, fixed income returns are normal in terms of log returns whilst equity returns are not. What should I do in this situation?

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  • $\begingroup$ Perhaps you could come up with a more appropriate title. The current one does not seem to reflect your actual problem precisely. $\endgroup$ Commented Oct 14, 2015 at 13:26
  • $\begingroup$ I find it hard to believe that any return series are Gaussian. It's been shown over and over that the returns have fat tails. $\endgroup$
    – Aksakal
    Commented Oct 14, 2015 at 13:53

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If you intend to use a GARCH model, you should care about the distribution of standardized model residuals (i.e. residuals from the conditional mean model divided by fitted conditional standard deviations from the GARCH model) rather than the raw data -- because a GARCH model has distributional assumptions on standardized residuals rather than raw data.

If you fit a univariate GARCH model on a linear combination of the two series (the linear combination denoting portfolio returns) and the standardized residuals do not have the distribution that you have assumed, you have a problem. You then normally try other distributions as long as you find one that matches. But it is possible that you do not succeed in finding a matching distribution, which still leaves you another option: multivariate GARCH modelling. That is, you will model the two series separately.

There are many kinds of multivariate GARCH models, e.g. BEKK-GARCH, DCC-GARCH, GO-GARCH, etc; see Bauwens et al. "Multivariate GARCH models: a survey" (2006) or Silvennoinen & Terasvirta "Multivariate GARCH models" (2009). Again, your goal is to match the assumed distribution of standardized residuals with the resulting distribution that you get after having estimated the model. DCC is a flexible model allowing for different univariate distributions for the two series of standardized residuals. A DCC model works in two stages:

  1. Build univariate GARCH models for each individual series (in your case, two univariate GARCH models). The individual models can differ in their form (e.g. one can be simple GARCH, another can be EGARCH) and distributional assumptions (e.g. one can be normal while the other can be Student(4)), which is giving you quite a lot of flexibility.
  2. Put a GARCH-like structure on conditional correlations between the standardized residuals from the individual GARCH models. That makes it a DCC model.

Perhaps you could get away with the empirical distribution not matching the assumed distribution by invoking principles of quasi maximum likelihood estimation (QMLE) - see "Idea and intuition behind quasi maximum likelihood estimation (QMLE)".

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