Choosing Degrees of Freedom for Multivariate t Return Simulations

Summary

The document asks how to estimate the degrees of freedom for a multivariate Student t distribution used to simulate asset returns. The questioner already has a sample mean vector and covariance matrix and notes that, when the degrees of freedom exceed two, the t distribution's scale matrix differs from its covariance matrix by a factor involving the degrees of freedom. The simulation setup therefore requires choosing a tail parameter as well as matching the covariance.

No estimation procedure or answer is included, so the document does not establish how to select that parameter in practice. The key modeling implication is that degrees of freedom control tail heaviness, while the scale adjustment is needed to make simulated returns have the intended covariance. Any choice should be supported by return data and checked against the intended horizon and dependence assumptions; the prompt itself provides no empirical comparison or simulation results.

Key ideas

• The multivariate t simulation requires a degrees-of-freedom parameter in addition to a mean vector and covariance estimate.
• Degrees of freedom govern tail heaviness and affect the relationship between scale and covariance.
• For finite covariance, the degrees of freedom must exceed two.
• The document asks for an estimation method but provides no answer or empirical guidance.

Tags

# Degree of freedom input for Monte Carlo simulation of asset returns with multivariate t distribution


# Degree of freedom input for Monte Carlo simulation of asset returns with multivariate t distribution












How do I calculate or estimate the degrees of freedom in order to perform a Monte Carlo simulation of asset returns with multivariate t distribution using R functions? I am able to calculate the mean vector of asset returns `mu`, as well as the covariance matrix `covmat` of asset returns. I also understand that `Sigma` is the scale matrix which is `covmat * df/(df-2)`

```
library(mvtnorm)
sim <- rep(mu, each = n) + rmvt(n, sigma=Sigma, df=df)
#df is degrees of freedom, which is a required input in the function rmvt and
required to calculate the scale matrix
```

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