Residual Bootstrap Design for GARCH Return Models

# RUGARCH (output) and Residual Resampling using GARCH(1,1)


# RUGARCH (output) and Residual Resampling using GARCH(1,1)












I try to replicate the methodology proposed by Freedman and Peters (1984a, 1984b) which was applied in the famous paper by Brock, Lakonishok and LeBaron (1992) to generate many artificial log return series to test for robustness of different technical trading rules.

### Estimation-based bootstrap

The estimation-based bootstrap of Freedman and Peters works as follows:

- A model is fit to the original return series to obtain estimated parameters and residuals

- Residuals are standardized using estimated standard deviations for the error process (or, as stated by Marshall et al. (2008), by the conditional standard deviation)

- The standardized residuals are resampled (with replacemenet) to form a new scrambled series which is then used with the estimated parameters to form a new respresentative series for the given model.

Note: The standardized residuals are not restricted to a particular distribution, such as Gaussian, by this procedure. (Brock et al. (1992))

Question 1: Are "estimated standard deviations for the error process" the same as the "conditional standard deviation"

### Mean Equation specification in Rugarch

I fitted several regressions: GARCH(1,1), EGARCH, GARCH-M, CGARCH to my 4511 return observations using Rugarch in R.

In order to be able to follow the procedure mentioned above, I have to be sure how the mean equations are specified in R.

Question 2: Which of these two mean equation specifications does rugarch use when a mean (c) is included?

- (1) $r_{t}$ = c + $σ_{t}$[Θ]ɛ​

- (2) $r_{t}$ = c + ɛ​, where ɛ​ = $σ_{t}$z

and z ~ i.i.d. N(0,$σ^2$) or Student-t

In the rugarch manual "Introduction to the Rugarch" I was not able to find a clear response to my question.

### Outputs Rugarch

This is the output to my GARCH(1,1) which uses the following conditional variance equation

$σ_{t}[\hat{Θ}]$ = c + $\alpha$$ɛ​^2_{t-1}$ + $\beta$$σ^2_{t-1}$

Now, following the estimation-based bootstrap procedure I should first calculate the standardized Residuals. I think its right to assume, that "residuals" are the estimated residuals when looking at the output. However,

Question 3: Which part of the output is the "conditional standard deviation" vector i should use in order to generate standardized residuals?

Or more general:

Question 4: Whas does each output mean? The coefficients are clear. But I am somehow confused when looking at "cvar", "var", "sigma"

After having calculated the standardized residuals, they are resampled (with replacement) 500 times. Afterwards, each newly created scramled standardized residual vector $e^*$ is used with the estimated parameters to form a new artificial return series using the mean equation.

Under the assumption that rugarch specifies the mean equation as described in equation (1), a single new artificial return series would be generated using

$r_{t}^*$ = $\hat{c}$ + $σ_{t}[\hat{Θ}]$$e^*$​

Question 5: Then, If I assume rugarch to specifie the mean equation as state in equation (1), how can I calculate $σ_{t}[\hat{Θ}]$? Is there an output which gives me directly $σ_{t}[\hat{Θ}]$?

Thank you very much in advance for help!