You are given two models. Model L:

[[math]] y_t = \beta_0 + \beta_1t + \epsilon_t [[/math]]

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

[[math]] \begin{aligned} y_t &= y_0 + \mu_ct + \mu_t\\ c_t &= y_t - y_{t-1}\\ u_t &= \sum_{j=1}^t \epsilon_j \end{aligned} [[/math]]

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).

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

Key: D

I is true. See formula (7.8) in the Frees text.

II. is true. See formula (7.9) in the Frees text.

III is true. The only difference is the error terms, which are difficult to compare. See page 243 in the Frees text.

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