# Constructing estimated ARMA series

I have a time sequence $y_t$ and I want to estimte it via an ARMA(2,2) model, i. e.

$$y_t = \phi_1y_{t-1}+\phi_2y_{t-2}+\theta_1 \epsilon_{t-1}+\theta_1 \epsilon_{t-2}+\epsilon_t$$

I was able to estimate the coefficients using an MLE on a Kalman Filter of $y_t$. $$[\phi_1,\phi_2,\theta_1,\theta_2]=[.72,.23,.46,.35]$$

My question is now that I have the coefficients, how do I construct $\hat{y_t}$ ?

$y_{t-1}, y_{t-2}$ are lags but how do I get the residuals to construct the series? Do I regress $y_t$ on $y_{t-1}, y_{t-2}$ and the residuals give me $\epsilon_{t},\epsilon_{t-1},\epsilon_{t-2}$ which then I multiply by the estimated coefficients to construct the series? I know the software packages can do it for you but I simply wan to know the process of how to build out the estimated series after the coefficients have been estimated. Thank you.

• What exactly do you mean by $\hat{y}_t$? How do you want to use it? For example, in the Kalman filter setting, typically the "residuals" that we use for model diagnostics are the one-step-ahead forecast errors on $y_{t+1}$, which would mean that a natural meaning to give to $\hat{y}_t$ is the one-step-ahead forecast. Other meanings would be the expected value of $y_t$ given past and current observations, or all observations (Kalman smoother). – Chris Haug Mar 18 '17 at 17:55
• Thank you Chris, I see. The way I constructed the Kalman filter is I first guessed the coefficients, then generated 1-step forcasts using those coefficient and constructed a Likelihood function which then I maximized it. So to get the estimated y hat, I should run this process over again but then using the maximized coefficient values from the MLE which will generate the optimal 1-step ahead forcasts. Am i understanding this correctly? – jessica Mar 18 '17 at 18:01
• Jessica, you need to add @ before the user name, otherwise the user is not notified of your comment. Also, please do not post the same question twice (unless the other question is sufficiently different from this one). Cc @ChrisHaug – Richard Hardy Mar 20 '17 at 9:27
• Yes, that's correct, run the Kalman filter with your MLE parameters. A typical Kalman filter implementation will compute and output the state estimate both given only past information (i.e. the one-step-ahead state forecast) and given past and current information (the "filtered" state estimate). If you apply the observation matrix to either of these you will get some version of $\hat{y}_t$. Which one you want depends on what you want to use it for, as I said above. – Chris Haug Mar 20 '17 at 12:20