Q1. To qualify as a covariance stationary process, which of the following does not have to be true?

A)   Covariance(x_t, x_t-1) = Covariance(x_t, x_t-2).

B)   E[x_t] = E[x_t+1].

C)   Covariance(x_t, x_t-2) = Covariance(x_t, x_t+2).

Q2. Which of the following is NOT a requirement for a series to be covariance stationary? The:

A)     expected value of the time series is constant over time.

B)     covariance of the time series with itself (lead or lag) must be constant.

C)     time series must have a positive trend.

Q3. Which of the following statements regarding covariance stationarity is TRUE?

A)     A time series that is covariance stationary may have residuals whose mean changes over time.

B)     The estimation results of a time series that is not covariance stationary are meaningless.

C)     A time series may be both covariance stationary and have heteroskedastic residuals.

答案和详解如下：

Q1. To qualify as a covariance stationary process, which of the following does not have to be true?

A)   Covariance(x_t, x_t-1) = Covariance(x_t, x_t-2).

B)   E[x_t] = E[x_t+1].

C)   Covariance(x_t, x_t-2) = Covariance(x_t, x_t+2).

Correct answer is A)

If a series is covariance stationary then the unconditional mean is constant across periods. The unconditional mean or expected value is the same from period to period: E[x_t] = E[x_t+1]. The covariance between any two observations equal distance apart will be equal, e.g., the t and t-2 observations with the t and t+2 observations. The one relationship that does not have to be true is the covariance between the t and t-1 observations equaling that of the t and t-2 observations.

Q2. Which of the following is NOT a requirement for a series to be covariance stationary? The:

A)     expected value of the time series is constant over time.

B)     covariance of the time series with itself (lead or lag) must be constant.

C)     time series must have a positive trend.

Correct answer is C)

A time series can be covariance stationary and have either a positive or a negative trend.

Q3. Which of the following statements regarding covariance stationarity is TRUE?

A)     A time series that is covariance stationary may have residuals whose mean changes over time.

B)     The estimation results of a time series that is not covariance stationary are meaningless.

C)     A time series may be both covariance stationary and have heteroskedastic residuals.

Correct answer is B)

Covariance stationarity requires that the expected value and the variance of the time series be constant over time.