Unit 32 ARIMA forecasts

These are too hard to do by hand, we do not need to worry about it. We will use fore.aruma/fore.arima instead. However, it is interesting as an excercise to do an ARIMA(0,1,0) forecast

32.1 An example:

Consider $$(1-0.9B)C_t = a_t$$ We have that $\phi_1$ = 1, as is the case with these models. If we recall the general AR(p) forecast equation:

$$\hat{X}_{t_0} (\ell) = \phi_1 \hat{X}_{t_0}(\ell -1) + ... + \phi_p \hat{X}_{t_0}(\ell - p) + \bar{X} (1 - \phi_1 - ... - \phi_p))$$ We see that the right hand term with Xbar goes to zero, and the phi value is one. This gives us $$\hat{X}_{t_0} (\ell) = \hat{X}_{t_0} (\ell - 1)$$

Which says that we are just going to forecast the last value we obseved over and over and over. Pretty boring, but that is fine with me. Note that the probability limits of these forecasts increase in width with no bound. Pretty crap forecast huh.