ABy Admin
May 25'23

Exercise

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.

ABy Admin
May 26'23

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.

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