Different Marginal Distributions in Multivariate GARCH Innovations
Summary
The document explains that multivariate GARCH innovations need not have identical marginal distributions across components. The innovation vector is expected to be independent and identically distributed over time, with zero mean and identity covariance, but that does not require each component to share the same univariate distribution. A model can therefore accommodate, for example, one return series with normal innovations and another with heavier tails.
It also clarifies that the stated return model is incomplete without specifying a conditional mean and variance dynamics characteristic of GARCH. Copula-GARCH is offered as one way to combine chosen marginal distributions with a dependence structure. The discussion is conceptual and does not provide estimation details or diagnostics; the required temporal independence and covariance assumptions still constrain the model.
Key ideas
- Identical distribution over time applies to the innovation vector and does not force identical component marginals.
- The innovations should have zero mean and identity covariance under the described formulation.
- A multivariate GARCH specification needs conditional mean and volatility dynamics.
- Copula-GARCH can pair different marginal distributions with a multivariate dependence model.
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# Can the white noise in multivariate GARCH have different distributions?
# Can the white noise in multivariate GARCH have different distributions?
I have two datasets of log returns, one is clearly normal while the other is t-distributed. I want to fit these with a mutlivariate GARCH model. A multivariate GARCH model is defined as $$\mathbf{r}_t=\mathbf{H}_t\boldsymbol{\epsilon}_t$$ Where $\mathbf{H}_t$ is the conditional covariance matrix and $\boldsymbol{\epsilon}_t$ is white noise. In the litterature most authors state that $\boldsymbol{\epsilon}_t$ is IID(0,1). I have not understood the identicality requirement. Is it possible to have different distributions for the elements in the vector $\boldsymbol{\epsilon}_t$? If no, why not and what would be a workaround in this case?
## Answer by Richard Hardy (score 1, accepted)
https://quant.stackexchange.com/a/78261
To answer your question briefly, yes, it possible to have different distributions for the elements in the vector $\mathbf{\epsilon}_t$. Now an elaboration:
Your model formulation is incomplete.
- The mean vector $\mathbf{\mu}_t$ of $\mathbf{r}_t$ may be nonzero and time-varying, turning $\mathbf{r}_t=\mathbf{H}_t\mathbf{\epsilon}_t$ into $\mathbf{r}_t=\mathbf{\mu}_t+\mathbf{H}_t\mathbf{\epsilon}_t$ and adding an equation for $\mathbf{\mu}_t$.
- The variance dynamics should be characteristic of GARCH. There should be an equation for $\mathbf{H}_t$ containing an autoregressive component (lag of $\mathbf{H}_t$) and a "moving-average" component (lag of $\mathbf{\epsilon}_t$), or something roughly equivalent to that. Otherwise, multivariate GARCH is not a relevant name for the model.
- The distribution of $\mathbf{\epsilon}_t$ must be i.i.d. with zero mean and identity variance matrix. The i.i.d. requirement is stricter than white noise that you refer to in the title of your question.
Referring to point 3., there is no requirement for the components of $\mathbf{\epsilon}_t$ to have identical univariate distributions. E.g. copula-GARCH models build on the multivariate distribution of $\mathbf{\epsilon}_t$ being specified using arbitrary marginal distributions and an arbitrary copula (but still obeying the requirements in point 3.).Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.