At the moment I am estimating GARCH models for financial return data in Stata. For one series, I checked the squared residuals for autocorrelation: there is. Also the ARCH-LM test affirms the finding, but with a very high lag order (see picture 1).
Now, when trying to fit a GARCH model to the data, I can hardly fit a statistically significant ARCH coefficient whereas the first GARCH term is already highly significant (see picture 2). Is it possible to have GARCH effects without ARCH effects present?
Also, I found an ARCH effect at lag $t = 3$ instead of $t = 1$. But how do I make that clear in the notation? GARCH(1,1) suggests lags $t = 1$ for both the autoregressive as well as moving average component. And GARCH(3,1) would suggest an ARCH model with three lags into the past instead of one at lag $t = 3$.