Byron Gangnes

Econ 427 lecture 14 slidesEcon 427 lecture 14 slides

Forecasting with MA Models

Byron Gangnes

Optimal forecastOptimal forecast• One which, given the available information, has the

smallest average loss. – This will normally be the conditional mean (the mean given that

we are a particular time period, t; i.e. given an “information set”):

E( yT +h |ΩT )

• The best linear forecast will then be the linear approximation to this, called the linear projection,

P( yT +h |ΩT )

Remember the info set will generally be current and past values of y and innovations (epsilons).

Byron Gangnes

Optimal forecast for MAOptimal forecast for MA• Like before (Chs. 5 and 6), we calculate the optimal point

forecast by writing down the process at period T+h and then “projecting it” on the information available at time T.– Book’s MA(2) example:

• Write out the process at time T+1:

yt=εt +θ1εt−1 +θ2εt−2

εt∼WN(0,σ 2 )

yT +1 =εT+1 +θ1εT +θ2εT−1

• Projecting this on the time T info set,

(remember that expectations of future innovs are zero)

Byron Gangnes

Optimal forecast for MAOptimal forecast for MA• For period T+2:

• Projecting this on the time T info set,

• And so forth… So for periods beyond T+2,

Why is that? an MA(q) process is not forecastable more than q

steps ahead. Why? Recall Autocorr function drops to 0 after q steps

yT +2 =εT+2 +θ1εT+1 +θ2εT

y

T +2,T =θ2εT

y

T +h,T =0, for h> 2

Byron Gangnes

Uncertainty around optimal forecastUncertainty around optimal forecast• 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,

Why are errors serially correlated? Why can’t we use this info to improve forecast?

• Same for all forecasts T+h, h>2

e

T +h,T =yT+h −yT+h,T

e

T +1,T = εT+1 +θ1εT +θ2εT−1( )− θ1εT +θ2εT−1( ) =εT+1     (WN)

e

T +2,T = εT+2 +θ1εT+1 +θ2εT( )− θ2εT( )                   =εT+2 +θ1εT+1     (MA(1))

e

T +3,T = εT+3 +θ1εT+3 +θ2εT+1( )− 0( )                       =εT+3 +θ1εT+3 +θ2εT+1     (MA(2))

Byron Gangnes

Uncertainty around optimal forecastUncertainty around optimal forecast• forecast error variance is the variance of eT+h,T

We can use these conditional variances to construct confidence intervals. What will they look like?

Notice that the error variance is less than the underlying variability of the series yt for h < q. Note that because of its MA(2) form, the variance of yt equals

σ12 =σ 2

σ 22 =σ 2(1+θ1

2 )

σ h2 =σ 2(1+θ1

2 +θ22 ), h> 2

e

T +1,T =εT+1

e

T +2,T =εT+2 +θ1εT+1    

var( yt) =σ 2(1+θ1

2 +θ22 )