1．ppose that the following time-series model is found to have a unit root: 

Sales_t = b₀ + b₁ Sales _t-1+ ε_t

What is the specification of the model if first differences are used?

A)   Sales_t = b₀ + b₁ Sales _t-1 + b₂ Sales _t-2 + ε_t. 

B)   (Sales_t - Sales_t-1)= b₀ + b₁ (Sales _t-1 - Sales _t-2) + ε_t.

C)   Sales_t = b₀ + b₁ ε_t-1. 

D)   Sales_t = b₁ Sales _t-1+ ε_t. 

The correct answer was B)

Estimation with first differences requires calculating the change in the variable from period to period.

2．time series that has a unit root can be transformed into a time series without a unit root through:

A)   first differencing.

B)   calculating moving average of the residuals.

C)   mean reversion.

D)   including a seasonality component.

The correct answer was A)

First differencing a series that has a unit root creates a time series that does not have a unit root.

3．rry Phillips, CFA, has estimated an AR(1) relationship (x_t = b₀ + b₁ × x_t-1 + e_t) and got the following result: x_t+1 = 0.5 + 1.0x_t + e_t. Phillips should:

A)   not first difference the data because b₀ = 0.5 < 1.

B)   first difference the data because b₁ = 1.

C)   not first difference the data because b₁ − b₀ = 1.0 − 0.5 = 0.5 < 1.

D)   first difference the data because b₀ = 0.5 < 1.

The correct answer was B)

The condition b₁ = 1 means that the series has a unit root and is not stationary. The correct way to transform the data in such an instance is to first difference the data.

4．rry Phillips, CFA, has the following time series observations from earliest to latest: (5, 6, 5, 7, 6, 6, 8, 8, 9, 11). Phillips transforms the series so that he will estimate an autoregressive process on the following data (1, -1, 2, -1, 0, 2, 0, 1, 2). The transformation Phillips employed is called:

A)   moving average.

B)   first differencing.

C)   seasonality.

D)   beta drift.

The correct answer was B)

Phillips obviously first differenced the data because the 1=6-5, -1=5-6, .... 1 = 9 - 9, 2 = 11 - 9.