Q1. Which of the following statements regarding unit roots in a time series is FALSE?

A)     Even if a time series has a unit root, the predictions from the estimated model are valid.

B)     A time series that is a random walk has a unit root.

C)   A time series with a unit root is not covariance stationary.

Q2. The standard error of the slope coefficient is 0.15, and the number of observations is 60. Given a level of significance of 5%, which of the following can we NOT conclude about this model?

A)    The model has a unit root.

B)    The slope on lagged sales is not significantly different from one.

C)    The model is covariance stationary.

Q3. An AR(1) autoregressive time series model:

A)   cannot be used to test for a unit root.

B)   can be used to test for a unit root, which exists if the slope coefficient is less than one.

C)   can be used to test for a unit root, which exists if the slope coefficient equals one.

答案和详解如下：

Q1. Which of the following statements regarding unit roots in a time series is FALSE?

A)     Even if a time series has a unit root, the predictions from the estimated model are valid.

B)     A time series that is a random walk has a unit root.

C)   A time series with a unit root is not covariance stationary.

Correct answer is  A)

Marvin Greene is interested in modeling the sales of the retail industry. He collected data on aggregate sales and found the following:

Sales_t = 0.345 + 1.0 Sales_t-1

Q2. The standard error of the slope coefficient is 0.15, and the number of observations is 60. Given a level of significance of 5%, which of the following can we NOT conclude about this model?

A)    The model has a unit root.

B)    The slope on lagged sales is not significantly different from one.

C)    The model is covariance stationary.

Correct answer is C)

The test of whether the slope is different from one indicates failure to reject the null H₀: b₁=1 (t-critical with df = 58 is approximately 2.000, t-calculated = (1.0 - 1.0)/0.15 = 0.0).  This is a 2-tailed test and we cannot reject the null since 0.0 is not greater than 2.000. This model is nonstationary because the 1.0 coefficient on Sales_t-1 is a unit root. Any time series that has a unit root is not covariance stationary which can be corrected through the first-differencing process.

Q3. An AR(1) autoregressive time series model:

A)   cannot be used to test for a unit root.

B)   can be used to test for a unit root, which exists if the slope coefficient is less than one.

C)   can be used to test for a unit root, which exists if the slope coefficient equals one.

Correct answer is C)

If you estimate the following model x_t = b₀ + b₁ × x_t-1 + e_t and get b₁ = 1, then the process has a unit root and is nonstationarity.