ARMA Model Interpretation from R

Question

I tried my data into an $ARMA$ model which is turned out to be $ARMA(2,3)$. I want to extract the model from the parameters I got in the pic (without any transformation). Is it right if I write the model like this?

$Y_t = 0.6218 + 0.126Y_{t-1} + 0.3964Y_{t-2} + e_t + 0.1488e_{t-1} + 0.9569e_{t-2} - 0.1057e_{t-3}$

Furthermore, how do I get $e_t$ (error of prediction on period-t) since I haven't got the prediction value yet.. I mean, $Y_t$ is something I want to predict, right?

Answer 1

You will need to flip the signs on the MA coefficients. Here is the help page for arima():

Different definitions of ARMA models have different signs for the AR and/or MA coefficients. The definition used here has

 X[t] = a[1]X[t-1] + ... + a[p]X[t-p] + e[t] + b[1]e[t-1] + ... + b[q]e[t-q]

and so the MA coefficients differ in sign from those of S-PLUS.

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In forecasting, you will plug the last few residuals into $e_{t-3}$, $e_{t-2}$ and $e_{t-1}$. Get these using residuals(). For $e_t$, as you write correctly, we don't know this yet. But we assume normally distributed errors. So in forecasting, just plug in the expectation, which is zero.

Of course, if you are not interested in point forecasting, but in predictive densities or prediction-intervals, it makes sense to simulate and draw $e_t~\sim N(0,\widehat{\sigma}^2)$, with $\widehat{\sigma}^2=0.003064$ as per your output.