Applying the Inverse PIT to Simulated GARCH-Copula Returns

Summary

The document asks how to convert simulated uniform values from a fitted multivariate copula back into return observations. Its setup fits an AR(1)-GARCH(1,1) model to each return series, applies the probability integral transform to the standardized residuals, fits a copula to the transformed data, and simulates new points from that copula.

To recover returns, apply each series’ fitted marginal inverse distribution to its simulated uniform value, then restore the conditional mean and volatility from the AR-GARCH model as appropriate for the forecast period. The document itself does not provide this procedure or specify how to handle conditional dynamics; it is a question rather than a worked explanation. The exact inverse also depends on the chosen residual distribution and the time step being simulated, so the model’s marginal and conditional components must be kept consistent.

Key ideas

• The setup transforms fitted AR-GARCH residuals to uniform values before fitting a copula.
• Copula simulations must be mapped back through each series’ fitted marginal quantile function.
• Recovering returns may also require restoring the AR conditional mean and GARCH conditional volatility.
• The document poses the inverse-transform question but does not supply an answer.

Tags

# Marginal Distribution using GARCH model: How to do inverse probability transform?


# Marginal Distribution using GARCH model: How to do inverse probability transform?












I have $n$ return series. I fitted AR(1)-GARCH(1,1) to each return series. Then used probability integral transform, PIT(residuals), to transform the residuals to have a uniform distribution. Then I fitted an $n$-dimensional copula to the data. I simulated 1000 points from the copula. Now, how can I transform these simulated points in [0,1], marginal $u$, back using the inverse probability transform (inverse of marginal fitted by AR-GARCH), $F^{-1}(u)$.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.