# Efficient online (rolling window) estimation of a GARCH model

I have a time series $$x_t$$ of length $$n$$. I would like to model it using rolling window approach with window length (width) $$w$$:

• window $$1$$: $$x_1,\dots,x_w$$,
• window $$2$$: $$x_2,\dots,x_{w+1}$$,
• $$\dots$$,
• window $$n-w+1$$: $$x_{n-w+1},\dots,x_n$$.

In each window, I would like to estimate a GARCH model. I could just do it using brute force. However, this is quite expensive computationally.

I wonder if I could borrow information from neighbouring windows and make the estimation more computationally efficient. Is there an algorithm available that is doing that?

(E.g. if I was estimating a regression model, I could use the ideas suggested in the thread "Efficient online linear regression").

• I have glimpsed at the source code of the functionugarchroll in the rugarch package in R and it seems it just uses brute force. But I may be mistaken. – Richard Hardy Dec 10 '19 at 16:52
• Hi: if you model garch using a state space formulation, then you have the updating equations ( the KF equations ) at your disposal which make computations convenient. I think there's a paper by harvey and ruiz on how to do that but it might be for arch. I forget exactly. If you google for "garch state space", I bet something will turn up. good luck. – mlofton Dec 10 '19 at 17:22
• @mlofton, thank you! I suspected the idea of Kalman filter might be worthwhile but was not aware of any existing attempts. I will look up the reference and see if it provides anything interesting. – Richard Hardy Dec 10 '19 at 17:23
• Hi Richard: I didn't realize that it was you asking the question. Note that the one drawback is that you're not going to get the results that a rolling ugarch estimation procedure would give. Duncan and Horne is the paper for doing regresssions by using the previous value's estimate. The problem is that GARCH is different enough from regression that I don't know how hard it would be to do the Duncan and Horne thing for GARCH. If you could do that then I think you could match the results of a rolling sum procedure. Let me find the link to that paper and I'll put it in another comment. – mlofton Dec 11 '19 at 3:38
• This is it but it doesn't look that easy to get. I had ( or have ?) a hardcopy somewhere so I might have the pdf somewhere also. If you have trouble, let me know and I can look to see if I have the pdf somewhere. tandfonline.com/doi/abs/10.1080/01621459.1972.10481299 – mlofton Dec 11 '19 at 3:46