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How do I estimate the residual $\varepsilon_{t}$ of a Seasonal ARIMA model $\hat{Y}_t=\hat{\phi}{Y}_{t-1}+\hat{\Phi}{Y}_{t-12}+\varepsilon_{t}$?
If the MSE is 0.114, what does it mean?

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You can calculate $\varepsilon_{t}$ as follows. $$\varepsilon_{t} = {Y}_t - \hat{Y}_t $$

Mean squared Error can be calculated as

$$MSE = \frac{1} {n} \sum\limits_{t=1}^n ({Y}_t - \hat{Y}_t)^2$$

$$OR$$

$$MSE = \frac{1} {n} \sum\limits_{t=1}^n (\varepsilon_{t} )^2$$

where $\hat{Y}_t$ is the predicted value from your model, ${Y}_t$ is the actual values.

An MSE of $0$ would indicate that the model fits your data perfectly which happens very rarely in practice. You have a value of 0.114 which is close to $0$. I would say your model fits your data very well. I would consult any basic statistics book if you want to learn more about MSE.

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