I think there are two main causes of the result discrepancy.
First, the GARCHFIT estimates an offset term (the intercept term in the mean equation) by default, while GARCH.ESTIMATE assumes the offset term is zero by default. The reason is that new GARCH is sometimes used as the error variance model of an ARIMA model. Therefore, GARCH itself does not include an offset term any more. So the translation from GARCHSET to GARCH is the following:
spec = garchset('VarianceModel','GARCH','P',1,'Q',1);
model = garch('ARCHLags',1,'GARCHLags',1,'Offset',NaN);
Then garchfit(spec, y) usually gives the same results as estimate(model,y).
Second, the Econometrics Toolbox relies on the Optimization Toolbox to maximize likelihood function of a time series model. However, the solvers in Optimization Toolbox have been improving over the years and may give better results (in term of higher likelihood values). In addition, the best-performed algorithm is chosen as the default solver. It is possible to have occasional discrepancy in the estimated coefficients, especially when the sample size is small and/or the data poorly fit the model. If we really want to replicate the old results, I think at least we may want to control the optimization option. For example, the default setting in GARCHSET looks like
Optimization = optimset(Optimization,'Algorithm','active-set');
While currently the default setting in GARCH.ESTIMATE looks like
options = optimoptions(options, 'Algorithm', 'sqp', 'TolCon' , 1e-7, ... 'Display' , 'off', 'Diagnostics', 'off');
Our apology for the inconvenience during the transition from the old GARCH to the new GARCH. I think the new GARCH suite is better than the old GARCH tools, so the old was retired in 2014.
Hang Qian
