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.
Optimal forecast for ARMA models
For a general ARMA process
 ( L) Z t  ( L)at
( L)
Zt 
at   ( L)at  at   1at 1   2 at 2  .....
 ( L)
Objective: given information up to time n, Z n , Z n 1 ,......Z1
want to forecast ‘l-step ahead’ Zˆ (l )
n
at t  n  l

l 1

Z l  n   j an l  j   j an l  j   j an l  j
j 0
j 0
j l

 

future a
ˆZn (l )   l *an  *l 1an1  *l 2an2  ......
past a
What is  * ?
Criterium: Minimize the mean square forecast error
ˆ (l )) 2
min
E
(
Z

Z
n l
n
*

l 1
E ( Z n l
 Zˆ n (l )) 2   2 a  j   a
j 0
2
2

 (
j 0
l j
 l  j )
* 2

E ( Z n l  Zˆ n (l )) 2
2
*
  a  2( l  j   l  j )( 1)  0 
*

j 0
 l j   l j
*
Zˆn (l )   l an  l 1an1  l 2an2  ........
Another interpretation of optimal forecast
Consider
l 1

j 0
j 0
Z l n   j an l  j   l  j an  j

E ( Z n l | Z n , Z n 1.....)   l  j an  j   l an   l 1an 1   l  2an 2  ....
j 0
Hence
Zˆ n (l )  E ( Z n l | Z n , Z n 1.....)
Given a quadratic loss function, the optimal forecast is a
conditional expectation, where the conditioning set is past
information
Properties of the forecast error
l 1

j 0
j 0
Zl n   j an l  j   l  j an j
Since Z n l  Zˆ n (l )  en (l )
l 1
forecast error  en (l )  Z n l  Zˆ n (l )   j an l  j MA(l-1)
j 0
1. The forecast Zˆ n (l ) and the forecast error en (l ) are uncorrelated
2.
E ( en (l ))  0
Unbiased
l 1
3.
Var( en (l ))   j  a
2
2
j 0
4.
en (l ) for l  1 are correlated
Properties of the forecast error (cont)
1-step ahead forecast errors, e n (1), e n 1(1).....e n  l (1) , are uncorrelated
en (1)  Zn1  Zˆn (1)  an1
In general, l-step ahead forecast errors (l>1) are correlated
en (l )  Z n l  Zˆ n (l )  an l   1an l 1  ...... l 1an 1
en  j (l )  Z n  j l  Zˆ n  j (l )  an  j l   1an  j l 1  ...... l 1an  j 1
cov( en (l ), en  j (l )) 
n-j
n l  j
2



 i n i n  j
i  n 1
n
n-j+l n+l
jl
Forecast of an AR(1) process
Z t  Z t 1  at  Zˆ n (l )  ?
l 1
Z n 1  Z n  an 1
E ( Z n 1 |  n )  Z n
l2
Z n  2  Z n 1  an  2
E ( Z n2 |  n )   2 Z n
l
ˆ
for any l Z n (l )   Z n
The forecast decays geometrically as l increases
Forecast of an AR(p) process
Z t  1Z t 1  2 Z t 2  ........ p Z t  p  at 
Zˆ n (l )  E ( Z n l | Z n , Z n 1 ,.....)  ?
l  1 Z n 1  1Z n  2 Z n 1  ........ p Z n  p 1  an 1
Zˆ n (1)  E ( Z n 1 |  n )  1Z n  2 Z n 1  ........ p Z n  p 1
l  2 Z n  2  1Z n 1  2 Z n  ........ p Z n  p  2  an  2
Zˆ n (2)  E ( Z n  2 |  n )  1Zˆ n (1)  2 Z n  ........ p Z n  p 2
for any l
Zˆ n (l )  1Zˆ n (l  1)  2 Zˆ n (l  2)  ........ p Zˆ n (l  p )
You need to calculate the previous forecasts l-1,l-2,….
Forecast of a MA(1)
Z t  at  at 1
Zˆn (l )  E( Znl | I n )  ?
l 1
Z n 1  an 1  an
Zˆ n (1)  E ( Z n 1 )  an
l2
l 1
Z n  2  an  2  an 1
Zˆ n 2   0
Zˆ n (l )  0
Zn
an 
1  L
That is the mean of the process
Forecast of a MA(q)
Z t  (1  1L   2 L  ...... q L )at
2
q
( l   l 1L   l 2 L2  .... q Lql )an l  q
Zˆ n (l )  E ( Z n l | I n )  
lq
0
1
where an 
Z
q n
1  1L  .... q L
Forecast of an ARMA(1,1)
(1  L) Z t  (1  L)at
Z n l  Z n l 1  an l  an l 1
1  L
whe re an 
Zn
1  L
l  2 Zˆ n ( 2)  Zˆ n (1)   (Z n  an )
l  2 Zˆ n (l )  Zˆ n (l  1)   2 Zˆ n (l  2)  .... l 1Zˆ n (1)
l  1 Zˆ n (1)  Z n  an
Forecast of an ARMA(p,q)
 p ( L) Z t  q ( L)at
Z nl  1Z nl 1  ..... p Z n l  p  an l  1an l 1  .....   qanl q
Zˆ n (l )  1Zˆ n (l  1)  ....   p Zˆ n (l  p)  aˆn (l )  1aˆn (l  1)  ....   qaˆn (l  q)
where
Zˆ n ( j )  E ( Z n  j | Z n , Z n 1.....)
Zˆ n ( j )  Z n  j
aˆn ( j )  0
j 1
j0
j 1
aˆn ( j )  an  j  Z n  j  Zˆ n  j 1 (1)
j0
Example: ARMA(2,2)
Zt  1Zt 1  2 Zt 2  at  1at 1   2at 2
l  1 Z n1  1Z n  2 Z n1  an !  1an   2an1
Zˆ (1)  E ( Z | I )   Z   Z   aˆ   aˆ
n
n 1
n
1 n
where
Zˆ n ( 0)  Z n
Zˆ ( 1)  Z
n
ˆn 
a
ˆ n 1
a
2 n 1
n 1
 2 ( L)
Zn
2 ( L )
 Z
 Zˆ
n 1
1 n
n 2
(1)
2 n 1
Updating forecasts
Suppose you have information up to time n, such that
Zˆn (1), Zˆn (2),......Zˆn (l )
When new information comes, Z n 1
can we update the previous forecasts?
1.
2.
en (l )  Z n  l  Zˆ n ( l ) 
en 1 ( l  1) 
en 1 ( l  1) 
l 11

j 0
l 1

j
j
l 1

j 0
j
an  l  j
an 1 l 1 j 
l

j 0
j
an  l  j
an  l  j   l an  en ( l )   l an
j 0
3.
Z n  l  Zˆ n 1 ( l  1)  Z n  l  Zˆ n ( l )   l an
Zˆ ( l )  Zˆ ( l  1)   a
n
n 1
l
n
Zˆ n 1 ( l )  Zˆ n (l  1)   l an 1
Problems
P1: For each of the following models:
(i) (1 - 1L) Z t  a t
(ii ) (1  1L   2 L2 ) Z t  a t
(iii ) (1 - 1L)(1  L) Z t  a t
(a) Find the l-step ahead forecast of Zn+l
(b) Find the variance of the l-step ahead forecast error for l=1, 2,
and 3.
P2: Consider the IMA(1,1) model
(1  L)Z t  (1  L)a t
(a) Write down the forecast equation that generates the forecasts
(b) Find the 95% forecast limits produced by this model
(c) Express the forecast as a weighted average of previous
observations
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.
Appendix I: The Annihilation operator
We are looking for a compact lag operator expression to be used to
express the forecasts
(L)
 L s  1L1s  ...  s 1L1  s L0  s 1L1  s  2 L2  ...
Ls
The annihilation operator is
[
Then if
 (L)
]  s L0  s 1L1  s  2 L2  ..
Ls
Z t  (L)a t , E[ Z t  s | a t , a t 1 , ...]  [
(L)
] a t
Ls
Appendix II: Forecasting based on lagged Z´s
Let
 ( L) Z t  a t
Z t  ((L)) 1 a t   (L)a t
Then
(L)
1
E[ Z t  s | Z t , Z t 1 , ...]  [
]
Zt
(L)
Ls
Wiener-Kolmogorov
Prediction Formula