kasinkei

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

Regression Statistics
Multiple R	0.952640
R2	0.907523
Adjusted R2	0.898275
Standard Error	8.135514
Observations	12
1st order autocorrelation coefficient of the residuals: −0.075

ANOVA
	df	SS
Regression	1	6495.203
Residual	10	661.8659
Total	11	7157.069

	Coefficients	Standard Error
Intercept	10.0015	5.0071
Trend	6.7400	0.6803

The analyst then estimates the following model:

(natural logarithm of sales)t = α + β × (Trend)t + εt

Regression Statistics
Multiple R	0.952028
R2	0.906357
Adjusted R2	0.896992
Standard Error	0.166686
Observations	12
1st order autocorrelation coefficient of the residuals: −0.348

ANOVA
	df	SS
Regression	1	2.6892
Residual	10	0.2778
Total	11	2.9670

	Coefficients	Standard Error
Intercept	2.9803	0.1026
Trend	0.1371	0.0140

Seerveld compares the results based upon the output statistics and conducts two-tailed tests at a 5% 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) Yes, both are significant.

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

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

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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 5% level of significance is 2.228, so both slope coefficients are statistically significant. (Study Session 3, LOS 13.a)

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

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

B) The simple linear trend.

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

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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. (Study Session 3, LOS 13.b)

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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.

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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. (Study Session 3, LOS 13.b)

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The primary limitation of both models is that:

A) each uses only one explanatory variable.

B) the results are difficult to interpret.

C) regression is not appropriate for estimating the relationship.

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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 gross domestic product (GDP) have a significant time trend, and a regression is an appropriate tool for measuring that trend. (Study Session 3, LOS 13.b)

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

A) $97.6 million.

B) $113.0 million.

C) $123.0 million.

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The forecast is 10.0015 + (13 × 6.7400) = 97.62. (Study Session 3, LOS 13.a)

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

A) $117.0 million.

B) $121.2 million.

C) $109.4 million.

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The forecast is e2.9803 + (13 × 0.1371) = 117.01. (Study Session 3, LOS 13.a)

kasinkei

In the time series model: yt=b0 + b1 t + εt, t=1,2,…,T, the:

A) change in the dependent variable per time period is b1.

B) disturbance terms are autocorrelated.

C) disturbance term is mean-reverting.

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The slope is the change in the dependent variable per unit of time. The intercept is the estimate of the value of the dependent variable before the time series begins. The disturbance term should be independent and identically distributed. There is no reason to expect the disturbance term to be mean-reverting, and if the residuals are autocorrelated, the research should correct for that problem.

kasinkei

Modeling the trend in a time series of a variable that grows at a constant rate with continuous compounding is best done with:

A) a log-linear transformation of the time series.

B) a moving average model.

C) simple linear regression.

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The log-linear transformation of a series that grows at a constant rate with continuous compounding (exponential growth) will cause the transformed series to be linear.