Interval Forecasts for MA and AR Models:II
Recall –
1. If yt is a covariance stationary process
whose innovations, εt, are normally
distributed, then
yˆT  h ,T  2 h
is an approximate 95-percent forecast
interval for yT+h, where
yˆT h.T  E( yT h yT , yT 1 ,...)
and
σh = s.d.(eT+h,T),
eT+h,T = yˆT  h ,T  yT  h
2. If yt = εt, i.e., if yt is a normally
distributed white noise process, the
95-percent forecast interval for yT+h is
[-2σ , 2σ]
where σ2 = var(εt) = var(yt) = E(yt2)
Although σ is unknown, we can replace
it with the consistent estimator:
1 T 2
̂ 
yt

T 1
3.If yt is a normally distributed MA(q)
process, i.e.,
yt = εt + θ1εt-1 + … + θqεt-q
For h < q: The 95% FI for yT+h is
yˆT  h,T  2 (1  12  ...   h21 ) 2
where
yˆ T  h ,T   h T   h1 T 1  ...   q  T q  h
For h > q: The 95% FI for yT+h is
0  2 (1  12  ...   q2 ) 2
To make these operational, we replace the
θ’s with their estimated values and replace
σ2 with the average of the squared
residuals from the fitted MA model.
Note –
There are two simple ways to obtain
̂ from the MA(q) estimation output:
1. Divide the SSR by T and take the
square root.
2. Use the standard error of the
Regression. This will actually be
equal to the square root of SSR/(T-q),
but for large T and relatively small q,
this will be, for practical purposes, the
same as SSR/T.
Now we turn to the construction of forecast
intervals for the AR(p) model. The general
formula for the 95-percent interval,
yˆT  h ,T  2 h
still applies and we have already worked out
the derivation of yˆT  h ,T for the AR(p) model.
Our current focus will be on the
computation of σh, the standard deviation of
the h-step ahead forecast.
Consider the AR(1) model:
yt = φyt-1 + εt
We know from previous work that in this
case
yT+1 = φyT + εT+1
yT+1,T = φyT
and so,
eT+1,T = εT+1
σ1 = σ
and
φyT + 2σ
is an approximate 95-percent forecast
interval for yT+1.
Similarly,
yT+2 = φyT+1 + εT+2
= φ2yT + εT+2 + φεT+1
yT+2,T = φ2yT
and so,
eT+2,T = εT+2 + φεT+1
σ2 = {E[(εT+2 + φεT+1)2]}1/2
= (1+φ2)1/2σ
Then
φ2yT + 2(1+φ2)1/2σ
is an approximate 95-percent forecast
interval for yT+2.
More generally, for the AR(1) process:
φhyT + 2(1+φ2+…+φh)1/2σ
is an approximate 95-percent forecast
interval for yT+h, h = 1,2,3,…
Next, consider the AR(2) process:
yt = φ1yt-1 + φ2yt-2 + εt
h = 1:
yT+1 = φ1yT + φ2yT-1 + εT+1
yT+1,T = φ1yT + φ2yT-1
eT+1,T = yT+1 - yT+1,T = εT+1
σ1 = σ
h = 2:
yT+2 = φ1yT+1 + φ2yT+ εT+2
yT+2,T = φ1yT+1,T + φ2yT
eT+2,T = yT+2 - yT+2,T = φ1(yT+1 - yT+1,T)+ εT+2
= φ1εT+1 + εT+2
σ2 = {E[(φ1εT+1 + εT+2)2]}1/2 = (1+ φ12)1/2σ
h = 3:
yT+3 = φ1yT+2 + φ2yT+1+ εT+3
yT+3,T = φ1yT+2,T + φ2yT+1,T
eT+3,T = yT+3 - yT+3,T
= φ1(eT+2,T)+ φ2(eT+1,T)+εT+3
= (φ12+ φ2) εT+1 + φ1εT+2 + εT+3
σ3 = [1+ φ12 +(φ12+ φ2)2]1/2σ
Notes –
 This procedure can be followed for
h=4,5,… to find σ4,σ5,…though the
formulas get messier.
 The formulas for σ1 and σ2 are the same
for any AR(p), p > 1. The formulas for
σ1,σ2,σ3 are the same for any AR(p),
p > 2. The formulas for σ1,σ2,σ3,σ4 are
the same for any AR(p), p > 3...
To make these operational, we need to
supply estimates of the φ’s and σ. These
come from the estimated autoregression.
The φ-hats are the point estimates of the AR
coefficients. σ-hat can be estimated by either
of the following
1.the standard error of the regression
2.
ˆ 
1
SSR
T