I'm having trouble computing the one-step ahead forecast for the following time-series models.

For the following models, ${Z_t}$ is a whitenoise process with ${Z_t}$ ~ $WN(0, \sigma^2)$.

I am given the following set of $n=5$ data points: $$-0.93, -0.89, -0.63, -0.38, 0.76$$

and I am trying to forecast the 6th data point.

The models are:

$$AR(1): X_t = 0.9X_{t-1} + Z_t$$ $$ARMA(1,1): X_t = 0.3X_{t-1} + Z_t - Z_{t-1}$$ $$ARMA(2,1): X_t = 0.1X_{t-1} - 0.5X_{t-2} + Z_t + Z_{t-1}$$

I think I can compute the forecast for the $AR(1)$ process: the best linear predictor is $P_nX_{n+1} = a_n^{'}X_n = \phi X_n$, so the 6th data point $\hat{X}_6 = 0.9*0.76$

For the $ARMA(1,1)$ model, using the recursions of the innovations algorithm, I get $\hat{X}_{n+1} = \phi X_n + \theta_{n1}(X_n - \hat{X}_n)$, but do I have to recursively find the estimates for each data point? Also, how do I find out what the prediction for $\hat{X}_1$ is?

  • $\begingroup$ You should set $\hat{X}_1=0$ and run the recursions of the innovations algorithm. In order to get $\hat{X}_{n+1}$ you need $\hat{X}_n$, so you will have to start the recursions from $n=1$. $\endgroup$
    – javlacalle
    Commented Nov 3, 2017 at 23:18
  • $\begingroup$ Maybe this could be useful: stats.stackexchange.com/questions/419313/… $\endgroup$
    – Bert
    Commented Jul 26, 2019 at 15:12

1 Answer 1


Yes, you do need to work recursively, plugging in previous values. First you plug in $X_t$, and when these are not available any more, you plug in the point predictions.

To get the innovations, you are right that you need the first fitted value. There is no "right" way to get this. Some people (and some software) estimate the initial fits via maximum likelihood, others use heuristics and use, e.g., the first actual observation, or a heuristically smoothed value.

If this is self-study, do whatever your instructor suggested. If you are trying to recreate something your software did, you will need to find out how it fits these initial values. This is often not very well documented.


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