1. Yolanda Seerveld is an analyst studying the growth of sales of a new restaurant chain called Very Vegan. The increase in the public’s awareness of healthful eating habits has had a very positive effect on Very Vegan’s business. Seerveld has gathered quarterly data for the restaurant’s sales for the past three years. Over the twelve periods, sales grew from $17.2 million in the first quarter to $106.3 million in the last quarter. Because Very Vegan has experienced growth of more than 500% over the three years, the Seerveld suspects an exponential growth model may be more appropriate than a simple linear trend model. However, she begins by estimating the simple linear trend model:

(sales)_t = α + β × (Trend)_t + ε_t

Where the Trend is 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12.

The analyst then estimates the following model:

(natural logarithm of sales)_t = α + β × (Trend)_t + ε_t

Seerveld compares the results based upon the output statistics and conducts two-tailed tests at a 5 percent level of significance. One concern is the possible problem of autocorrelation, and Seerveld makes an assessment based upon the first-order autocorrelation coefficient of the residuals that is listed in each set of output. Another concern is the stationarity of the data. Finally, the analyst composes a forecast based on each equation for the quarter following the end of the sample.

Are either of the slope coefficients statistically significant?

A)   The simple trend regression is, but not the log-linear trend regression.

B)   No, neither is significant.

C)   The simple trend regression is not, but the log-linear trend regression is.

D)   Yes, both are significant.

2.Based upon the output, which equation explains the cause for variation of Very Vegan’s sales over the sample period?

A)   The simple linear trend.

B)   The log-linear trend.

C)   The cause cannot be determined using the given information.

D)   Both the simple linear trend and the log-linear trend have equal explanatory power.

3.With respect to the possible problems of autocorrelation and nonstationarity, using the log-linear transformation appears to have:

A)   improved the results for autocorrelation but not nonstationarity.

B)   improved the results for nonstationarity but not autocorrelation.

C)   not improved the results for either possible problems.

D)   improved the results for both possible problems.

4.The primary limitation of both models is that:

A)   the results are difficult to interpret.

B)   each uses only one explanatory variable.

C)   regression is not appropriate for estimating the relationship.

D)   the methods used rarely yield significant results.

5.Using the simple linear trend model, the forecast of sales for Very Vegan for the first out-of-sample period is:

A)   $113.0 million.

B)   $97.6 million.

C)   $116.3 million.

D)   $123.0 million.

1. Yolanda Seerveld is an analyst studying the growth of sales of a new restaurant chain called Very Vegan. The increase in the public’s awareness of healthful eating habits has had a very positive effect on Very Vegan’s business. Seerveld has gathered quarterly data for the restaurant’s sales for the past three years. Over the twelve periods, sales grew from $17.2 million in the first quarter to $106.3 million in the last quarter. Because Very Vegan has experienced growth of more than 500% over the three years, the Seerveld suspects an exponential growth model may be more appropriate than a simple linear trend model. However, she begins by estimating the simple linear trend model:

(sales)_t = α + β × (Trend)_t + ε_t

Where the Trend is 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12.

The analyst then estimates the following model:

(natural logarithm of sales)_t = α + β × (Trend)_t + ε_t

Seerveld compares the results based upon the output statistics and conducts two-tailed tests at a 5 percent level of significance. One concern is the possible problem of autocorrelation, and Seerveld makes an assessment based upon the first-order autocorrelation coefficient of the residuals that is listed in each set of output. Another concern is the stationarity of the data. Finally, the analyst composes a forecast based on each equation for the quarter following the end of the sample.

Are either of the slope coefficients statistically significant?

A)   The simple trend regression is, but not the log-linear trend regression.

B)   No, neither is significant.

C)   The simple trend regression is not, but the log-linear trend regression is.

D)   Yes, both are significant.

The correct answer was D)

The respective t-statistics are 6.7400 / 0.6803 = 9.9074 and 0.1371 / 0.0140 = 9.7929. For 10 degrees of freedom, the critical t-value for a two-tailed test at a five percent level of significance is 2.228, so both slope coefficients are statistically significant.

2.Based upon the output, which equation explains the cause for variation of Very Vegan’s sales over the sample period?

A)   The simple linear trend.

B)   The log-linear trend.

C)   The cause cannot be determined using the given information.

D)   Both the simple linear trend and the log-linear trend have equal explanatory power.

The correct answer was C)

To actually determine the explanatory power for sales itself, fitted values for the log-linear trend would have to be determined and then compared to the original data. The given information does not allow for such a comparison.

3.With respect to the possible problems of autocorrelation and nonstationarity, using the log-linear transformation appears to have:

A)   improved the results for autocorrelation but not nonstationarity.

B)   improved the results for nonstationarity but not autocorrelation.

C)   not improved the results for either possible problems.

D)   improved the results for both possible problems.

The correct answer was C)

The fact that there is a significant trend for both equations indicates that the data is not stationary in either case. As for autocorrelation, the analyst really cannot test it using the Durbin-Watson test because there are fewer than 15 observations, which is the lower limit of the DW table. Looking at the first-order autocorrelation coefficient, however, we see that it increased (in absolute value terms) for the log-linear equation. If anything, therefore, the problem became more severe.

4.The primary limitation of both models is that:

A)   the results are difficult to interpret.

B)   each uses only one explanatory variable.

C)   regression is not appropriate for estimating the relationship.

D)   the methods used rarely yield significant results.

The correct answer was B)

The main problem with a trend model is that it uses only one variable so the underlying dynamics are really not adequately addressed. A strength of the models is that the results are easy to interpret. The levels of many economic variables such as the sales of a firm, prices, and GDP have a significant time trend, and a regression is an appropriate tool for measuring that trend.

5.Using the simple linear trend model, the forecast of sales for Very Vegan for the first out-of-sample period is:

A)   $113.0 million.

B)   $97.6 million.

C)   $116.3 million.

D)   $123.0 million.

The correct answer was B)

The forecast is 10.0015 + (13 × 6.7400) = 97.62.