Let ${Y_t}$ be a stationary process with mean zero and let $a$ and $b$ be constants. Prove $X_t$ is a stationary

Question

Let ${Y_t}$ be a stationary process with mean zero and let $a$ and $b$ be constants. Show that $X_t = Y_t - Y_{t-1} - Y_{t-12} + Y_{t-13}$ is stationary.

I encountered this issue when working on a problem with the following solution:

Is there any theorem I can quickly reference for this? Checking the mean, variance, and covariance of $X_t$ is somewhat time-consuming (because I need to work quickly during the exam).

I really appreciate your help.

Answer 1

Hi: You don't need to calculate those things because

1. the $s_t$ have period 12 so the 12 lag difference on the second and third terms, namely $s_{t}-s_{t-12}$ and $s_{t-1} - s_{t-13}$, make them stationary. ( not sure how you got rid of them on the next line ? ).

2. The question says that $Y_t$ is stationary.

So, all three terms that are left are stationary ( of course $b \times 0 = 0$ ) and the sum of the three terms is stationary.