You are given:

• The random walk model
  [[math]]y_t = y_0 + c_1 + c_2 + \cdots c_t[[/math]]
where [math]c_t, t = 0,1,2,\cdots, T [/math] denote observations from a white noise process.
• The following nine observed values of [math]c_t[/math]:

  t	yt
  1	2
  2	5
  3	10
  4	13
  5	18
  6	20
  7	24
  8	25
  9	27
  10	30

• [math]y_0 = 0 [/math]
• The 9 step ahead forecast of [math]y_{19}[/math] , [math]\hat{y}_{19}[/math] , is estimated based on the observed value of [math]y_{10}[/math] .

Calculate the standard error of the 9 step-ahead forecast, [math]\hat{y}_{19}[/math] .

• 4/3
• 4
• 9
• 12
• 16

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

Key: B

[math]c= y_t − y_{t−1}[/math] and hence [math]c_1, c_2 ,\ldots c_{10} = 2,3,5,3,5, 2, 4,1, 2,3.[/math]

The mean of the [math]c[/math] values is 3, the variance is (1+0+4+0+4+1+1+4+1+0)/9 = 16/9. The standard deviation is 4/3. The standard error of the forecast is [math](4/3)\sqrt{9} = 4. [/math]

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