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Lets say that I have a time series data $Y_{t}$. I'm trying to forecast using am moving average MA(1) model using Box Jenkins methodology. The following is the equation for an MA(1) obtained from a textbook.

$$ Y_{t} = \theta_{0} + e_{t} + \theta_{1} e_{t-1} $$

I assume we would use some type of nonlinear optimization to estimate coefficients $\theta_{0}$ and $\theta_{1}$ via maximum likelihood or conditional least squares. My question is about the very first error term $e$ when $t = 1$.

  1. Does the very first error term $e_{1}$ is also automatically determined by the optimization ?
  2. What would be the value for $e_{0}$ ?

Thanks so much

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(1) Yes, you will have an estimate of $e_1$ as soon as you have an estimate of $\theta_0, \theta_1$, and $e_0$.

(2) $e_0$ usually has to be estimated along with the other parameters if you do not use other considerations to pin it down. Consult, e.g., the documentation for R's ARIMA fitting function to see this.

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