A random walk is expressed as

[[math]] y_t = y_{t-1} + c_t, \, t = 1,2, \ldots [[/math]]

where

[[math]] \operatorname{E}(c_t) = \mu_c, \, \operatorname{Var}(c_t) = \sigma_c^2, \, t=1,2,\ldots [[/math]]

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).

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

Key: C

I is true because the mean [math]\operatorname{E}(y_t) = y_0 + t\mu_c[/math] depends on [math]t[/math].

II is false because the variance [math]\operatorname{Var}(y_t) = t\sigma_c^2 = 0 [/math] does not depend on [math]t[/math].

III is true because the variance depends on [math]t.[/math]

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