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I am working on ARMA+GARCH and wonder what the correct notation for a fitted model is. For example, my model is a jointly estimated AR(1)-GARCH(1,1) model with mean equation $$ r_t = 0.034 r_{t-1} + 0.0075 + a_t $$ and volatility equation $$ \sigma_t^2 = 0.0000794 + 0.122 a_{t-1}^2 + 0.855 \sigma_{t-1}^2. $$

Should I write \begin{aligned} r_t &= 0.034 r_{t-1} + 0.0075 + \hat{a}_t, \text{ where} \\ \hat{a}_t &= \hat{\sigma}_t \epsilon_t \text{ and} \\ \hat{\sigma}_t^2 &= 0.0000794 + 0.122 \hat{a}_{t-1}^2 + 0.855 \hat{\sigma}_{t-1}^2? \\ \end{aligned}

Or \begin{aligned} \hat{r_t} &= 0.034 r_{t-1} + 0.0075 \text{ and} \\ \hat{\sigma}_t^2 &= 0.0000794 + 0.122 \hat{a}_{t-1}^2 + 0.855 \hat{\sigma}_{t-1}^2? \\ \end{aligned}

Are these notations both correct? I think in the second notation the connection of both equations is not really clear.

Thanks for clarification.

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