1．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 = b₀ + b₁x_t-1 + ε_t.

C)   x_t = x_t-1 + ε_t.

D)   x_t = b₀ + b₁ x_t.

2．ich of the following statements regarding time series analysis is FALSE?

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

B)   A time series that has a unit root is not covariance stationary.

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

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

3．vid 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 forecast for next period is 2.2.

B)   The residuals of the forecasting model are autocorrelated.

C)   The mean reverting level is undefined.

D)   The process is not covariance stationary.

4．ven 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)   b₀ = 1,000.

B)   b₁ = 1.

C)   the long run mean is b₀ + b₁.

D)   E(e_t)=0.

1．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 = b₀ + b₁x_t-1 + ε_t.

C)   x_t = x_t-1 + ε_t.

D)   x_t = b₀ + b₁ x_t.

The correct answer was B)

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.

2．ich of the following statements regarding time series analysis is FALSE?

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

B)   A time series that has a unit root is not covariance stationary.

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

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

The correct answer was C)

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.

3．vid 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 forecast for next period is 2.2.

B)   The residuals of the forecasting model are autocorrelated.

C)   The mean reverting level is undefined.

D)   The process is not covariance stationary.

The correct answer was B)

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 values do not revert back to a long-run average and the variance changes with the value of the observation. However, the error term e_t = x_t - x_t-1 is not autocorrelated.

4．ven 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)   b₀ = 1,000.

B)   b₁ = 1.

C)   the long run mean is b₀ + b₁.

D)   E(e_t)=0.

The correct answer was C)

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). The intercept coefficient, b₀, can be a value other than zero. If b₀ ≠ 0, then it is a random walk with a “drift”, which means the value drifts by b₀ each period by that value on average.