You are given the following eight observations from a time series that follows a random walk model:

Time (t)	0	1	2	3	4	5	6	7
Observation ( [math]y_t[/math] )	3	5	7	8	12	15	21	22

You plan to fit this model to the first five observations and then evaluate it against the last three observations using one-step forecast residuals. The estimated mean of the white noise process is 2.25.

Let F be the mean error (ME) of the three predicted observations.

Let G be the mean square error (MSE) of the three predicted observations.

Calculate the absolute difference between F and G, | F − G | .

• 3.48
• 4.31
• 5.54
• 6.47
• 7.63

Copyright 2023. The Society of Actuaries, Schaumburg, Illinois. Reproduced with permission.

Key: D

The smoothed forecast at 100 is

0.2(100.2) + 0.8(95.1) = 96.1. 

This is also the forecast at 102. The smoothed values are at 99:

0.2(95.1) + 0.8(89.9) = 90.9. At 100: 0.2(96.1) + 0.8(90.9) = 91.9. 

The trend is

0.2(96.1 – 91.9)/.8 = 1.05. 

The intercept (value at 100) is

2(96.1) – 91.9 = 100.3. 

The forecast at time 102 is

100.3 + 2(1.05) = 102.4.

The difference is 6.3.

Copyright 2023. The Society of Actuaries, Schaumburg, Illinois. Reproduced with permission.