(On Problem 6 of the worksheet) Is
{
𝑋
𝑡
}
stationary?

Question

(On Problem 6 of the worksheet) Is 
{
𝑋
𝑡
}
 stationary?

BlackTom AI · Accepted Answer

We start by recalling the typical condition for stationarity in a simple AR(1) process: X_t = α X_{t-1} + ε_t, where ε_t is a white noise error term. The process is weakly stationary if and only if the absolute value of the autoregressive coefficient is strictly less than 1, i.e., |α| < 1. With that in mind, let’s evaluate each option one by one. Option 1: 'The process is stationary if α < 1.' This statement misses the important absolute value condition. If α = -1.5, for example, we would have |α| > 1 and the process would be explosive rather than stationary. The inequality α < 1 is necessary but not sufficient, and it overlooks the behavior when α is negative. Therefore, this option is incorrect. Option 2: 'The process is stationary provided α > 0.' This asserts stationarity only when α is positive and greater than zero. However, stationarity for AR(1) does not require α to be positive; negative values with small magnitude (e.g., α = -0.5) still yield a stationary process. This option is incorrect because it unnecessarily restricts α to positive values and ignores the magnitude condition. Option 3: 'The process is stationary.' This blanket statement claims stationarity in all cases, which is false. If α is, say, 1.2 and there is a tendency to grow without bound, the process is non-stationary. Without the |α| < 1 condition, we cannot guarantee stationarity in all scenarios, so this option is incorrect. Option 4: 'The process is stationary provided |α| < 1.' This aligns with the standard criterion for AR(1) processes: the series is weakly stationary exactly when the absolute value of the coefficient is less than one. This ensures the effects of shocks decay over time and the variance remains finite. This option is correct. Option 5: 'The process is non-stationary.' This contradicts the known criterion when |α| < 1. If that condition holds, the process is stationary, so labeling it always as non-stationary is incorrect. This option is incorrect. In summary, the only option that correctly captures the necessary and sufficient condition for stationarity in this context is the one stating |α| < 1, while all other options fail to properly characterize the stationarity condition or make overly broad claims.

STAT_V 443 202 2024W2 Activity 5: Autoregressive Processes

(On Problem 6 of the worksheet) Is { 𝑋 𝑡 } stationary?

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