TS module 15 MA(2) forecasting practice problems
(The attached PDF file has better formatting.)
** Exercise 15.1: MA(2) parameters and forecasts
An MA(2) process has a mean
ì of 100 and the expected and actual values below in periods T-4 through T.
Period | Expected Value | Actual Value |
T-4 | 100 | 100 |
T-3 | 100 | 100 |
T-2 | 100 | 101 |
T-1 | 100.2 | 101 |
T | 99.8 | 99 |
What is the value of
è1?
What is the value of
è2?
What is the forecast for period T+1?
What is the forecast for period T+2?
Solution 15.1: Add residuals to the table and use periods T-1 and T to determine
è1 and è2.
Period | Expected Value | Actual Value | Residual |
T-4 | 100 | 100 | 0 |
T-3 | 100 | 100 | 0 |
T-2 | 100 | 101 | 1 |
T-1 | 100.2 | 101 | 0.8 |
T | 99.8 | 99 | -0.8 |
Part A:
From the expected value in Period T-1 we solve for
è1:
100 –
è1 × 1 – è2 × 0 = 100.2 è1 = –0.2
Part B:
From the expected value in Period T and the value of
è1 we solve for è2:
100 –
è1 × 0.8 – è2 × 1 = 99.8 = 100 – 0.2
-0.2 × -0.8 –
è2 × 1 = –0.2
0.16 –
è2 = –0.2
è
2 = +0.36
Part C:
The forecast for period T+1 = 100 – (-0.8) × -0.2 – 0.8 × 0.36 = 99.552
Part D:
The fitted value in Period T+1 is the best estimate, so the expected residual is zero. The forecast for Period T+2 is
100 –
è2 × –0.8 = 100 – 0.36 × (–0.8) = 100.288
** Exercise 15.2: MA(2) process
An MA(2) process has a mean
ì of 100 and the expected and actual values below in periods T-4 through T.
Period | Expected Value | Actual Value |
T-4 | 100 | 100 |
T-3 | 100 | 100 |
T-2 | 100 | 101 |
T-1 | 100.2 | 101 |
T | 99.8 | 99 |
What is the forecast for period T+1?
What is the forecast for period T+2?
Part A:
Add residuals to the table and use periods T-1 and T to determine
ö1 and ö2.
Period | Expected Value | Actual Value | Residual |
T-4 | 100 | 100 | 0 |
T-3 | 100 | 100 | 0 |
T-2 | 100 | 101 | 1 |
T-1 | 100.2 | 101 | 0.8 |
T | 99.8 | 99 | -0.8 |
Period T-1: 100 –
è1 × 1 – è2 × 0 = 100.2 è1 = –0.2
Period T: 100 –
è1 × 0.8 – è2 × 1 = 99.8 = 100 – 0.2
-0.2 × -0.8 –
è2 × 1 = –0.2
0.16 –
è2 = –0.2
è
2 = +0.36
The forecast for period T+1 = 100 + (-0.8) × -0.2 – 0.8 × 0.36 = 99.872
Part B:
For Period T+2, we assume the residual in Period T+1 is zero. The forecast for Period T+2 is
100 –
è2 × –0.8 = 100 – 0.36 × (–0.8) = 100.288