Which equations are true for a stationary time series?

Which equations are true for a stationary time series?

• A. E(yt+4)=E(ys+5
• B. Cov(yt,yt+3)=Cov(ys+3,ys)
• C. Var(yt)=Var(ys+2)
• D. All of the above

Answer: (D)

In the context of stationary time series, all the statements provided reflect the properties inherent to such series. For a stationary time series, the expected value (mean) at any time point does not change over time. This means that the expected value for any observation at time \( t+4 \) should be the same as that for any observation at a comparable position in the series, such as at \( s+5 \). Hence, the equation holds true as both sides pertain to the process's mean, which is constant for a stationary series. Covariance, which measures how two time series move together, also exhibits stationary properties. The covariance between observations at different times remains constant over time in a stationary series. Therefore, \( \text{Cov}(y_t, y_{t+3}) \) equals \( \text{Cov}(y_{s+3}, y_{s}) \) because they represent the same relative distance in terms of lag, reflecting that the relationship between values does not depend on when they are observed. Variance, which measures the dispersion of the time series values, is also constant. Thus, the variance of any observation at time \( t \) is equal to the variance of any observation at time \( s

In the context of stationary time series, all the statements provided reflect the properties inherent to such series.

For a stationary time series, the expected value (mean) at any time point does not change over time. This means that the expected value for any observation at time ( t+4 ) should be the same as that for any observation at a comparable position in the series, such as at ( s+5 ). Hence, the equation holds true as both sides pertain to the process's mean, which is constant for a stationary series.

Covariance, which measures how two time series move together, also exhibits stationary properties. The covariance between observations at different times remains constant over time in a stationary series. Therefore, ( \text{Cov}(y_t, y_{t+3}) ) equals ( \text{Cov}(y_{s+3}, y_{s}) ) because they represent the same relative distance in terms of lag, reflecting that the relationship between values does not depend on when they are observed.

Variance, which measures the dispersion of the time series values, is also constant. Thus, the variance of any observation at time ( t ) is equal to the variance of any observation at time ( s