Considering the properties of a stationary time series, which equations are true?

For a stationary time series, certain statistical properties remain constant over time. These properties are crucial for understanding and analyzing time series data.

The first equation states that the expected value at time t plus 4 is the same regardless of the time index, which holds true for stationary processes. This means that the mean of the series does not depend on when it is measured in time, hence E(yt+4) equals E(ys+4).

The second equation regarding covariance indicates that the relationship between two points in time depends only on the time difference between them, not the actual time points. Since both yt and ys are part of a stationary process, Cov(yt, yt+3) equals Cov(ys+2, ys) confirms that the covariance is consistent across different time frames.

The third equation about variance states that the variance remains constant throughout the series. Thus, Var(yt) equals Var(ys) implies that regardless of which point in time you look at, the variability within the series remains unchanged.

All of these statements reflect fundamental properties of a stationary time series, establishing that they hold true for such processes. Hence, it can be concluded that all equations are indeed true for stationary processes.