# Unit 33 Seasonal Forecasts

Consider a quarterly model:

$(1 - B^4) (X_t - \mu) = a_t$

If we rewrite the general AR forecast, since we still have a root at 1 (a fourth root), the xbar component still goes away.Then, we are left with

$\hat{X}_{t_0} (\ell) = \hat{X}_{t_0} (\ell - 4)$

This again makes a lot of sense. Each forecast is exactly the same as it was 4 times ago. A pretty simple and straightforward deal. We will after the test combine these models.

## 33.1 Airline models

In order to forecast into the future with a seasonal model, we want to not only include the existing seasonal trend, but also a 1-B wandering trend. Seasonal and ARIMA components. This way we get forecasts going off of themselves. This works because of

factor.wge(phi = c(rep(0, 11), 1))
##
## Coefficients of Original polynomial:
## 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 1.0000
##
## Factor                 Roots                Abs Recip    System Freq
## 1-1.0000B+1.0000B^2    0.5000+-0.8660i      1.0000       0.1667
## 1-1.0000B              1.0000               1.0000       0.0000
## 1-1.7321B+1.0000B^2    0.8660+-0.5000i      1.0000       0.0833
## 1+1.0000B+1.0000B^2   -0.5000+-0.8660i      1.0000       0.3333
## 1-0.0000B+1.0000B^2    0.0000+-1.0000i      1.0000       0.2500
## 1+1.7321B+1.0000B^2   -0.8660+-0.5000i      1.0000       0.4167
## 1+1.0000B             -1.0000               1.0000       0.5000
##
## 

We see that we have a 1-B component in there already, which means we can put it in there and it will get cancelled out without messing up out model!!!

The most general seasonal model we can have is:

$\phi(B)(1-B)(1-B^s)(X_t - \mu) = \theta(B)a_t$

The forecasts follow the example of the last s points, but it is not a replication. Models like this are known as airline models, specifically with s = 12. This is how you forecast the trend and forecast the seasonality.