Q1. Given an AR(1) process represented by x_t+1 = b₀ + b₁×x_t + e_t, the process would not be a random walk if:

A)   the long run mean is b₀ + b₁.

B)   b₁ = 1.

C)   E(e_t)=0.

Q2. David Brice, CFA, has tried to use an AR(1) model to predict a given exchange rate. Brice has concluded the exchange rate follows a random walk without a drift. The current value of the exchange rate is 2.2. Under these conditions, which of the following would be least likely?

A)   The residuals of the forecasting model are autocorrelated.

B)   The forecast for next period is 2.2.

C)   The process is not covariance stationary.

Q3. Which of the following statements regarding time series analysis is FALSE?

A)   We cannot use an AR(1) model on a time series that consists of a random walk.

B)   An autoregressive model with two lags is equivalent to a moving-average model with two lags.

C)   If a time series is a random walk, first differencing will result in covariance stationarity.

Q4. A time series x that is a random walk with a drift is best described as:

A)     x_t = b₀ + b₁ x_{t − 1}.

B)     x_t = x_{t − 1} + ε_t.

C)     x_t = b₀ + b₁x_{t − 1} + ε_t.

答案和详解如下：

Q1. Given an AR(1) process represented by x_t+1 = b₀ + b₁×x_t + e_t, the process would not be a random walk if:

A)   the long run mean is b₀ + b₁.

B)   b₁ = 1.

C)   E(e_t)=0.

Correct answer is A)

For a random walk, the long-run mean is undefined. The slope coefficient is one, b₁=1, and that is what makes the long-run mean undefined: mean = b₀/(1-b₁).

Q2. David Brice, CFA, has tried to use an AR(1) model to predict a given exchange rate. Brice has concluded the exchange rate follows a random walk without a drift. The current value of the exchange rate is 2.2. Under these conditions, which of the following would be least likely?

A)   The residuals of the forecasting model are autocorrelated.

B)   The forecast for next period is 2.2.

C)   The process is not covariance stationary.

Correct answer is A)

The one-period forecast of a random walk model without drift is E(x_t+1) = E(x_{t + et} ) = x_t + 0, so the forecast is simply x_t = 2.2. For a random walk process, the variance changes with the value of the observation. However, the error term e_t = x_t - x_t-1 is not autocorrelated.

Q3. Which of the following statements regarding time series analysis is FALSE?

A)   We cannot use an AR(1) model on a time series that consists of a random walk.

B)   An autoregressive model with two lags is equivalent to a moving-average model with two lags.

C)   If a time series is a random walk, first differencing will result in covariance stationarity.

Correct answer is B)         

An autoregression model regresses a dependent variable against one or more lagged values of itself whereas a moving average is an average of successive observations in a time series. A moving average model can have lagged terms but these are lagged values of the residual.

Q4. A time series x that is a random walk with a drift is best described as:

A)     x_t = b₀ + b₁ x_{t − 1}.

B)     x_t = x_{t − 1} + ε_t.

C)     x_t = b₀ + b₁x_{t − 1} + ε_t.

Correct answer is C)

The best estimate of random walk for period t is the value of the series at (t − 1). If the random walk has a drift component, this drift is added to the previous period’s value of the time series to produce the forecast.

thx