Recovering AR-GARCH Returns from Simulated Copula Uniforms
Summary
The document describes how to turn simulated uniform observations from a fitted copula into return innovations compatible with previously fitted AR-GARCH models. The proposed step is to apply the inverse cumulative distribution function of the conditional innovation distribution used in each marginal fit. For a Student-t specification, the response points to the fitted degrees-of-freedom parameter and a quantile transformation with standardized location and scale.
Once transformed into residuals, the innovations can be used with the fitted conditional mean and volatility dynamics to construct return paths. This depends on knowing the marginal distribution assumed during estimation; a copula alone specifies dependence, not the marginal shapes. The discussion is a brief answer rather than a complete simulation recipe: it does not detail recursion through the AR-GARCH equations or address uncertainty in estimated parameters. It cautions against assuming normal marginals without justification.
Key ideas
- A copula specifies dependence, so simulated uniform values need marginal distributions to become return innovations.
- Apply the inverse distribution function corresponding to the marginal model fitted for each series.
- For Student-t innovations, use the fitted shape parameter in the quantile transformation.
- Reconstruct returns by combining transformed innovations with the fitted conditional mean and volatility process.
Tags
Full text
# Fitting Copula and Simulation
# Fitting Copula and Simulation
I would greatly appreciate any insights into the problem described below, regarding using the data obtained from applying the functions of the `rugarch` package into those from the `copula` package.
- I fitted AR(1)-GARCH(1,1) to two return series u,v of length 500 each. using `rugarchfit`in R.
- I converted the residuals to uniform using `pit(residuals(fit,standardize=TRUE))`
- Then, I plugged these residuals (uniform using PIT) to a copula and got the parameters.
- I simulated 100 points (bivariate) from the fitted copula.
Now, I want to convert these 100 points which are uniformly distributed back to the originally distributed series. How can I do this? How can I convert them back to residual form and then applying the fitted AR-GARCH to get the original series form?
## Answer by dms_quant (score 2)
https://quant.stackexchange.com/a/25785
You need to estimate or assume a marginal distribution of the (u,v). Lets say you assume normality (don't do this), you would be able to perform a rosenblatt-transformation, to perform the task you describe.
https://en.wikipedia.org/wiki/Inverse_transform_sampling
This could be a useful resource.
## Answer by Hanjo Odendaal (score 1)
https://quant.stackexchange.com/a/26013
You need to know what your original conditional distribution was when you fitted the AR-GARCH(1,1). Assuming that you chose a `student-t` distribution, the reverse transformation after `step 4` in `R` would look as follows:
#### step 1: Fit Garch
```
fit <- rugarchfit
```
#### step 4: Simulate points
```
sim <- 'simulated 100 points'
```
#### step 5: Convert 100 uniformly distributed points back to the originally distributed series
```
shape <- coef(fit)['shape']
transformed_residuals <- qdist("std", mu=0, sigma=1, sim, shape = shape)
```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.