# Computing one-step ahead forecast for AR and ARMA models

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?

• 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 $n=1$. Nov 3, 2017 at 23:18
• Maybe this could be useful: stats.stackexchange.com/questions/419313/…
– Bert
Jul 26, 2019 at 15:12

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.