For my variable $l_t$ I want to use an ARMA(p,q)-GARCH(r,s). So the mean equation follows an ARMA(p,q) and the conditional volatility is modelled by a GARCH(r,s). Is my notation correct?
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The GARCH filter does not depend on information at time $t$. You need to start your counter for $a_{t-i}$ at 1. Please see my answer [here](http://stats.stackexchange.com/a/41514/8141). – tchakravarty Feb 14 '15 at 09:14