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	[math]c_t[/math]
  11	2
  12	3
  13	5
  14	3
  15	4
  16	2
  17	4
  18	1
  19	2

• The average value of [math]c_1, c_2 , \ldots , c_{10}[/math] is 2.
• 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 forecast error, [math]y_{19} - \hat{y}_{19}[/math].

• 1
• 2
• 3
• 8
• 18

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

Key: D

[[math]] \begin{aligned} y_{10} &= y_0 + c_1 + \cdots + c_{10} = y_0 + 20 \\ y_{19} &= y_{10} + c_{11} + \cdots + c_{19} = y_0 + 20 + c_{11} + \cdots + c_{19} = y_0 + 20 + 26 = y_0 + 46 \\ \hat{y}_{19} &= y_{10} + \hat{c}_{11} + \cdots + \hat{c}_{19} = y_0 + 20 + 9(2) = y_0 + 38 \\ y_{19} − \hat{y}_{19} &= (y_0 + 46) − (y_0 + 38) = 8. \end{aligned} [[/math]]

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