Econ 427 lecture 15 slides
Forecasting with AR Models
Byron Gangnes
Optimal forecast
• Remember, the best linear forecast is often the linear
projection,
P( yT  h | T )
Where the info set will generally be current and past
values of y and innovations (epsilons).
• For forecasting AR processes, we will proceed as we did for
MA:
• Write out the process at time T+1
• Projecting this on the time T info set
• We could rewrite the cov. Stationary AR in MA form
• But there is a simpler way—the chain rule of forecasting
Byron Gangnes
Optimal forecast for AR
• Consider the AR(1) process:
yt   yt 1   t
 t WN (0,  2 )
• To get optimal fcst for t=T+1, write out the process at time
T+1:
yT 1   yT   T 1
• Projecting this on the time T info set,
yT 1,T  P( yT 1 | T )   yT
(remember that expectations of future innovs are zero)
Byron Gangnes
Optimal forecast for AR
• For T+2:
yT  2   yT 1   T  2
• Projecting this on the time T info set,
yT  2,T  P( yT 2 | T )   yT 1
• But we already have an optimal fcst of yT+1. Substituting:
 
yT  2,T   yT 1,T    yT   2 yT
• Similarly, for a 3-step-ahead forecast, we would get:


yT 3,T   yT  2,T    yT 1   3 yT
• Generally:
yT  h,T   h yT
Byron Gangnes
More complicated AR forecasts
• What if we had a higher-order AR(p) time series?
– There would be p terms in each time period
• What if we had both MA and AR terms?
– We would combine the two methods—see pp. 178-79 in the book
Byron Gangnes
Uncertainty around optimal forecast
• Again, we would like to know how much uncertainty
there will be around point estimates of forecasts.
• To see that, let’s look at the forecast errors, eT  h,T  yT  h  yT  h,T

  

 
eT 1,T   yT   T 1   yT   T 1
eT  2,T   yT 1   T  2  yT  2,T
(WN )

  yT 1  yT  2,T   T  2


 
  yT 1   yT 1,T   T  2

  yT 1  yT 1,T
T 2
  T 1   T  2
Can you show that the error for a 3-step-ahead fcst is:
eT 3,T   2 T 1   T  2   T 3
MA(2)
Byron Gangnes
MA(1)
Uncertainty around optimal forecast
• In general:
eT  h,T   h1 T 1  ...   T  h1   T  h
MA(h  1)
• Note that the errors are serially correlated but don’t drop off
Byron Gangnes
Uncertainty around optimal forecast
• forecast error variance is the variance of eT+h,T
eT 1,T   T 1
12   2
eT  2,T  T 1  T  2
 22   2 (1  2 )
eT 3,T   2 T 1   T  2   T 3
 32   2 (1  2   4 )
Generally,
h1
 h2   2 (1  2   4  ...   2( h1) )   2  2i
i0
And we can use these conditional variances to construct
confidence intervals. What will they look like?
Byron Gangnes