Time series Mod 15 AR(1) forecasts.wpd
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** Exercise 15.1: AR(1) forecasts
An AR(1) model with T observations has
ì = 2, ö = 0.5, yT = 1.6, and ó2t = 2,
What is the one period ahead forecast (for Period 51)?
What is the two periods ahead forecast (for Period 52)?
What is the variance of the one period ahead forecast?
What is the variance of the two periods ahead forecast?
Solution 15.1: We solve this problem two equivalent ways; use whichever way is clearer to you.
Method 1: An AR(1) process is Yt =
è0 + ö × Yt-1 + åt, where è0 = ì × (1 – ö).
è
0 = ì × (1 – ö) = 2 × (1 – 0.5) = 1.
Part A:
The one period ahead forecast is 1 + 0.5 × 1.6 = 1.800.
Part B:
The two periods ahead forecast is 1 + 0.5 × 1.8 = 1.900.
Part C:
The one period ahead forecast is a fixed amount
è0 + ö × YT plus the stochastic term åt. The fixed amount has a variance of zero, so the variance of the one period ahead forecast is ó2t = 2.
Part D:
The two periods ahead forecast is
è0 + ö × YT+1 + åT+2 = è0 + ö × (è0 + ö × YT + åT+1) + åT+2 =
[
è0 + ö × è0 + ö2 × YT] + [ö × åT+1 + åT+2]
The terms in the first set of brackets are fixed, with no variance. The variance of the stochastic terms in the second set of brackets is
ö2 × ó2t + ó2t = (1 + 0.52) × 2 = 2.50.
Jacob:
How do we get the relation
è0 = ì × (1 – ö)?
Rachel:
The mean
ì does not depend on t: ì = E(Yt) = E(Yt-1). Take the expectation of the AR(1) equation:
E(Yt) = E(
è0 + ö × Yt-1 + åt)
ì
= è0 + ö × ì + 0
ì
× (1 – ö) = è0.
Method 2: An AR(1) process is Yt –
ì = ö × (Yt-1 – ì) + åt Yt = ì + ö × (Yt-1 – ì) + åt
Part A:
The one period ahead forecast is 2 + 0.5 × (1.6 – 2) = 1.800.
Part B:
The two periods ahead forecast is 2 + 0.5 × (1.8 – 2) = 1.900.
Parts C and D are the same as for Method 1.
** Exercise 15.2: AR(1) forecasts: deriving the mean and the
ö parameter
We can ask the same exercise in reverse, deriving the mean and
ö from the first two forecasts.
An AR(1) process of T observations has yT = 1.600. The one period ahead forecast is 1.800, and the two periods ahead forecast is 1.900.
What is the
ö parameter of this AR(1) process?
What is the mean
ì of this AR(1) process?
Solution 15.2: The arithmetic is slightly simpler with the AR(1) process written as Yt =
è0 + ö × Yt-1 + åt, though we could use the version with ì instead of è0 as well.
We write two linear equations:
1.8 =
è0 + ö × 1.6 + 0
1.9 =
è0 + ö × 1.8 + 0
Part A:
Subtracting the first from the second give (1.9 – 1.8) =
ö × (1.8 – 1.6) ö = 0.5
Part B:
Using the first equation and the value of
ö gives 1.8 = è0 + 0.5 × 1.6 è0 = 1.8 – 0.8 = 1.
We derive
ì as è0 / (1 – ö) = 1 / (1 – 0.5) = 2.