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I have a series on which I fitted an ARIMA(4,0,4) model in R, and got the following estimations:

Coefficients:
      ar1     ar2      ar3      ar4     ma1     ma2     ma3     ma4  intercept
  -0.6498  0.0106  -0.7527  -0.8753  0.6727  0.0079  0.7486  0.8924     -1e-04
s.e.   0.0341  0.0274   0.0211   0.0530  0.0283  0.0275  0.0225  0.0497      1e-04

I then used the forecast library to get the next predicted value and got the following result

> forecast(ftfinal.arima, h=1)
 Point Forecast        Lo 80       Hi 80     Lo 95      Hi 95
3606   9.475018e-06 -0.007864678 0.007883628 -0.012033 0.01205195

This forecast result is different than the result I'm getting when I try to manually input the numbers into the ARIMA function, and I know that it's because there's something that I'm doing wrong but I don't really understand what it is.

let the ARIMA(4,0,4) function be:

$$X_t=c+\varepsilon_t+\sum^p_{i=1}\varphi_iX_{t-i}+\sum^q_{i=1}\theta_i\varepsilon_{t-1}.$$ where $p$ and $q$ both equal $4$.

and the most recent values of Xt are:

[3601]  1.502706e-03 -7.868107e-03  2.512803e-03  9.639389e-03  3.102150e-03

first of all for the AR part of the function, is the constant c the same as the "intercept" value that is output by the ARIMA model? secondly, is the epsilon sub t series calculated as Xt minus the expectation of the whole series?

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Epsilon sub t usually denotes the error in your model, thus it would not be included in any prediction, because if you knew the error, you would have 0 error.

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