You are given a stationary AR(1) model,

[[math]] y_t = \beta_0 + \beta_1 y_{t-1} + \epsilon_t, \, t = 2, \ldots, T [[/math]]

Determine which or the following is always true.

• [math]\beta_0 \neq 0 [/math]
• [math]\beta_0 = 1[/math]
• [math]\beta_1 = 0[/math]
• [math]\beta_1 = 1[/math]
• [math]|\beta_1| \lt 1[/math]

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

Key: E

From the definition of a stationary AR(1) process on Page 254 of Frees, the parameter [math]\beta_0[/math] can be any fixed constant, ruling out answers (A) and (B). To be stationary it is necessary that [math]−1 \lt \beta_1 \lt 1 [/math] , which makes answer (E) correct.

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