1．e table below shows the autocorrelations of the lagged residuals for the first differences of the natural logarithm of quarterly motorcycle sales that were fit to the AR(1) model: (ln sales_t – ln sales_t−1) = b₀ + b₁(ln sales_t−1 – ln sales_t−2) + ε_t. The critical t-statistic at 5 percent significance is 2.0, which means that there is significant autocorrelation for the lag-4 residual, indicating the presence of seasonality. Assuming the time series is covariance stationary, which of the following models is most likely to correct for this apparent seasonality?

A)   (ln sales_t - ln sales_t-4) = b₀ + b₁(ln sales_t-1 - ln sales_t-2) + _t.

B)   (ln sales_t - ln sales_{t - 1}) = b₀ + b₁(ln sales_t-1 - ln sales_t-2) + b₂(ln sales_t-4 - ln sales_t-5)+ €_t.

C)   (ln sales_t-1 - ln sales_t-4) = b₀ + b₁(ln sales_t-1) - b₂(ln sales_t-4) + _t.

D)   ln sales_t = b₀ + b₁(ln sales_t-1) - b₂(ln sales_t-4) + €_t.

2．ich of the following is a seasonally adjusted model?

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

B)   Sales_t = b₀ + b₁ ε_t-1.

C)   Sales_t = b₁ Sales _t-1+ ε_t.

D)   (Sales_t - Sales _t-1)= b₀ + b₁ (Sales _t-1 - Sales _t-2) + b₂ (Sales _t-4 - Sales _t-5) + ε_t.

3．hich of the following statements regarding seasonality is FALSE?

A)   Not correcting for seasonality when, in fact, seasonality exists in the time series results in a violation of an assumption of linear regression.

B)   The presence of seasonality makes it impossible to forecast using a time-series model.

C)   If seasonality is quarterly, the appropriate term to add to the model is the previous year's quarter's time-series value.

D)   A time series that is first differenced can be adjusted for seasonality by incorporating the first-differenced value for the previous year's corresponding period.

4．e table below shows the autocorrelations of the lagged residuals for quarterly theater ticket sales that were estimated using the AR(1) model: ln(sales_t) = b₀ + b₁(ln sales_t−1) + e_t. Assuming the critical t-statistic at 5 percent significance is 2.0, which of the following is the most likely conclusion about the appropriateness of the model? The time series:

A)   contains ARCH (1) errors.

B)   would be more appropriately described with an MA(4) model.

C)   contains seasonality.

D)   contains no significant serial correlation in the error terms.

5．rry Phillips, CFA, is analyzing quarterly data. He has estimated an AR(1) relationship (x_t = b₀ + b₁ × x_t-1 + e_t) and wants to test for seasonality. To do this he would want to see if which of the following statistics is significantly different from zero?

A)   Correlation(e_t, e_t-2).

B)   Correlation(e_t, e_t-5).

C)   Correlation(e_t, e_t-4).

D)   Correlation(e_t, e_t-1).

1．e table below shows the autocorrelations of the lagged residuals for the first differences of the natural logarithm of quarterly motorcycle sales that were fit to the AR(1) model: (ln sales_t – ln sales_t−1) = b₀ + b₁(ln sales_t−1 – ln sales_t−2) + ε_t. The critical t-statistic at 5 percent significance is 2.0, which means that there is significant autocorrelation for the lag-4 residual, indicating the presence of seasonality. Assuming the time series is covariance stationary, which of the following models is most likely to correct for this apparent seasonality?

A)   (ln sales_t - ln sales_t-4) = b₀ + b₁(ln sales_t-1 - ln sales_t-2) + _t.

B)   (ln sales_t - ln sales_{t - 1}) = b₀ + b₁(ln sales_t-1 - ln sales_t-2) + b₂(ln sales_t-4 - ln sales_t-5)+ €_t.

C)   (ln sales_t-1 - ln sales_t-4) = b₀ + b₁(ln sales_t-1) - b₂(ln sales_t-4) + _t.

D)   ln sales_t = b₀ + b₁(ln sales_t-1) - b₂(ln sales_t-4) + €_t.

The correct answer was B)

Seasonality is taken into account in an autoregressive model by adding a seasonal lag variable that corresponds to the seasonality. In the case of a first-differenced quarterly time series, the seasonal lag variable is the first difference for the fourth time period. Recognizing that the model is fit to the first differences of the natural logarithm of the time series, the seasonal adjustment variable is (ln sales_t-4 – ln sales_t-5).

2．ich of the following is a seasonally adjusted model?

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

B)   Sales_t = b₀ + b₁ ε_t-1.

C)   Sales_t = b₁ Sales _t-1+ ε_t.

D)   (Sales_t - Sales _t-1)= b₀ + b₁ (Sales _t-1 - Sales _t-2) + b₂ (Sales _t-4 - Sales _t-5) + ε_t.

The correct answer was D)

This model is a seasonal AR with first differencing.

3．hich of the following statements regarding seasonality is FALSE?

A)   Not correcting for seasonality when, in fact, seasonality exists in the time series results in a violation of an assumption of linear regression.

B)   The presence of seasonality makes it impossible to forecast using a time-series model.

C)   If seasonality is quarterly, the appropriate term to add to the model is the previous year's quarter's time-series value.

D)   A time series that is first differenced can be adjusted for seasonality by incorporating the first-differenced value for the previous year's corresponding period.

The correct answer was B)

Forecasting is no different in the case of seasonal component in the time-series model than any other forecasting.

4．e table below shows the autocorrelations of the lagged residuals for quarterly theater ticket sales that were estimated using the AR(1) model: ln(sales_t) = b₀ + b₁(ln sales_t−1) + e_t. Assuming the critical t-statistic at 5 percent significance is 2.0, which of the following is the most likely conclusion about the appropriateness of the model? The time series:

A)   contains ARCH (1) errors.

B)   would be more appropriately described with an MA(4) model.

C)   contains seasonality.

D)   contains no significant serial correlation in the error terms.

The correct answer was C)

The time series contains seasonality as indicated by the strong and significant autocorrelation of the lag-4 residual.

5．rry Phillips, CFA, is analyzing quarterly data. He has estimated an AR(1) relationship (x_t = b₀ + b₁ × x_t-1 + e_t) and wants to test for seasonality. To do this he would want to see if which of the following statistics is significantly different from zero?

A)   Correlation(e_t, e_t-2).

B)   Correlation(e_t, e_t-5).

C)   Correlation(e_t, e_t-4).

D)   Correlation(e_t, e_t-1).

The correct answer was C)

Although seasonality can make the other correlations significant, the focus should be on correlation(e_t, e_t-4) because the 4^th lag is the value that corresponds to the same season as the predicted variable in the analysis of quarterly data.