1. Forecasting II (forecasting with ARMA models)
“There are two kind of forecasters: those who don´t know and those who don´t know they don´t know”
John Kenneth Galbraith (1993)
Gloria González-Rivera
University of California, Riverside
and
Jesús Gonzalo U. Carlos III de Madrid
Spring 2002
Copyright(© MTS-2002GG): You are free to use and modify these slides for educational purposes, but please if you improve this material send us your new version.

2. Optimal forecast for ARMA models
For a general ARMA process
Objective: given information up to time n,
want to forecast ‘l-step ahead’

3. Criterium:
Minimize the mean square forecast error

4. Another interpretation of optimal forecast
Consider
Given a quadratic loss function, the optimal forecast is a
conditional expectation, where the conditioning set is past
information

5. Sources of forecast error
When the forecast is using :

6. Properties of the forecast error
MA(l-1)
The forecast and the forecast error are uncorrelated
Unbiased

7. Properties of the forecast error (cont)
1-step ahead forecast errors, , are uncorrelated
In general, l-step ahead forecast errors (l>1) are correlated
n-j n
n-j+l
n+l

8. Forecast of an AR(1) process
The forecast decays geometrically as l increases

9. Forecast of an AR(p) process
You need to calculate the previous forecasts l-1,l-2,….

10. Forecast of a MA(1)
That is the mean of the process

11. Forecast of a MA(q)

12. Forecast of an ARMA(1,1)

13. Forecast of an ARMA(p,q)
where

14. Example: ARMA(2,2)

15. Updating forecasts
Suppose you have information up to time n, such that
When new information comes,
can we update the previous forecasts?

16. Problems
P1: For each of the following models:
Find the l-step ahead forecast of Zn+l
Find the variance of the l-step ahead forecast error for l=1, 2, and 3.
P2: Consider the IMA(1,1) model
Write down the forecast equation that generates the forecasts
Find the 95% forecast limits produced by this model
Express the forecast as a weighted average of previous observations

17. Problems (cont)
P3: With the help of the annihilation operator (defined in the appendix) write down an expression for the forecast of an AR(1) model, in terms of Z.
P4: Do P3 for an MA(1) model.

18. Appendix I: The Annihilation operator
We are looking for a compact lag operator expression to be used to
express the forecasts
The annihilation operator is
Then if

19. Appendix II: Forecasting based on lagged Z´s
Let
Then
Wiener-Kolmogorov
Prediction Formula