I want to predict future returns over a 20 days horizon using an ARMA-GARCH model fitted to my data. The goal is to estimate different risk measures like VaR or CVar. In particular say I use AR(1) GARCH(1,1). The smaple I use for estimation has 500 observation of daily logreturns. That's what I usually do:
- Estimate AR, ARCH and GARCH coefficients
- Calculate standardized residual by dividing residuals by estimated conditional variances
- The standardized residuals constitute my INVARIANTS that is the i.i.d. series from which I extract bootstrap samples to generate scenarios.
The bootstrap samples are extracted by simulating a uniform between 1 and the sample size (500 in this case) and then taking the value corresponding to that position in the vector of the standardized residuals.
The problem is that I have only 500 standardized residuals and I think 500 is the maximum size of bootstrap samples I can extract.
My colleague instead extracts 100000 observations out of the original sample of $N=500$ observations.
I feel that this is somehow incorrect conceptually. Simulating only one step forward would produce exactly the same scenario as the initial one, but with repeted values that add no information.
My colleague claims that if he wants to project over a longer period, e.g. 20 days horizon, the 100000 extractions from the original sample of N=500 obs. would produce many different scenarios at the final horizon, providing a CDF that is smooth. Actually thit is true because, although the values are simply repeated in the first step, after that they can sum up in many different ways.
That being said I don't feel this is right. I proposed an alternative that is:
- From the standardized residuals create a smoothed empirical CDF, say kernel
- Exctract uniforms between 0 and 1 and feed it to the empirical smoothed CDF, i.e. inverse transform.
This way I feel more confortable to say that I can generate a bootstrab sample of size greater than the original one, but still I am not sure.
I am studying bootstrap theory on a book from Efron
Efron, Tibshirani - An Introduction to the Bootstrap - Springer US (1993)
but there are many concepts that I don't understand yet.
My question are:
- Would you give your opinion on the problem I just showed below?
- Would you suggest any valid matherial for studying bootstrap other than the book I mentioned?
- I think that this application of bootstrap is somehow different to the one explained in Efron's bookm, that is evaluating confidence intervals for estimated paramenters. What do you think about it?
Any comment would be much appreciated
I apologize for the length of the post but I tried to be as much concise as I could. Thank you