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

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

A random walk is expressed as

where

Determine which statements is/are true with respect to a random walk model.

• If [math]µ_c \neq 0[/math], then the random walk is nonstationary in the mean.
• If [math] \sigma_c^2 = 0[/math], then the random walk is nonstationary in the variance.
• If [math]\sigma_c^2 \gt 0[/math], then the random walk is nonstationary in the variance.

• None
• I and II only
• I and III only
• II and III only
• The correct answer is not given by (A), (B), (C), or (D).

Determine which of the following indicates that a nonstationary time series can be represented as a random walk

• A control chart of the series detects a linear trend in time and increasing variability.
• The differenced series follows a white noise model.
• The standard deviation of the original series is greater than the standard deviation of the differenced series.

• I only
• II only
• III only
• I, II and III
• The correct answer is not given by (A), (B), (C), or (D).

You are given two models. Model L:

where [math]\{\epsilon_t\}[/math] is a white noise process, for [math]t=0,1,2,\ldots [/math]. Model M:

where [math]\{\epsilon_t\}[/math] is a white noise process, for [math]t=0,1,2,\ldots [/math].

Determine which of the following statements is/are true.

• Model L is a linear trend in time model where the error component is not a random walk.
• Model M is a random walk model where the error component of the model is also a random walk.
• The comparison between Model L and Model M is not clear when the parameter [math]\mu_c = 0.[/math]

• I only
• II only
• III only
• I, II and III
• The correct answer is not given by (A), (B), (C), or (D).

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

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