A Sad Tale of Homemade Forecasts

[Gregory]

I'm working with a tri-variate BEKK GARCH(1,1)-M model in which each of the three return series depend on their own current variance (similar to the one on page 431 of the User's Guide). It looks like this. The equation for H also accounts for asymmetries.

(1) Return equations

rx(t) = constantX + beta*H11(t) + ex(t)
ry(t) = constantY + beta*H22(t) + ey(t)
rz(t) = constantZ + beta*H33(t) + ez(t)

(2) Variance equation

H(t) = C'C + A'e(t-1)e'(t-1)A + B'H(t-1)B + D'u(t-1)u'(t-1)D

As described in some of my other recent posts, I checked the in-sample estimates by constructing functions (in Mathematica, but it could be done in RATS of course) that use the estimated coefficients to churn out the series of residuals and estimated Hs. No problem there. Got an exact match.

My big problem is with forecasting out-of-sample. I think I may be doing something wrong because the forecasts are producing weird results. If someone sees an obvious flaw in my approach, I'd very much appreciate knowing. Here's how I have been tackling it.

- I use a rolling or moving estimation to take in new observations and make a static one-period-ahead forecast.
- The in-sample has T observations {t=1, 2, 3, ..., T}.
- The first variance forecast, that is, for H(T+1), comes from using the estimated residuals, e(T)'s, and variance, H(T), from that original in-sample estimation.
- I then forecast expected return for each series, that is, E(r(T+1)), by using the forecasts for H11(T+1), H22(T+1), and H33(T+1) that I have just obtained.
- To forecast T+2, I re-estimate the model with the new observation, T+1, and drop the first observation t=1 {t=2,3,4, ..., T+1}, keeping the window a constant width.
- The process is repeated throughout my out-of-sample observations.

Well, the performance of my forecasts are in such stark contrast to the in-sample performance I obtain that I'm thinking that perhaps my approach to forecasting is wrong (wouldn't be the first time). Any thoughts on this would be most appreciated.

In the meantime, I'll probably run a simple GARCH(1,1), that is, a model where the returns do not depend on H, and see if I can replicate RATS' MVGARCHFORE.SRC procedure.