 UID
 223330
 帖子
 360
 主题
 131
 注册时间
 2011711
 最后登录
 2013818

4^{#}
发表于 2012327 13:34
 只看该作者
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 twotailed tests at a 5% level of significance. One concern is the possible problem of autocorrelation, and Seerveld makes an assessment based upon the firstorder 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 loglinear trend regression. 
 C)
 The simple trend regression is not, but the loglinear trend regression is. 

The respective tstatistics are 6.7400 / 0.6803 = 9.9074 and 0.1371 / 0.0140 = 9.7929. For 10 degrees of freedom, the critical tvalue for a twotailed test at a 5% level of significance is 2.228, so both slope coefficients are statistically significant. (Study Session 3, LOS 13.a)
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 loglinear trend have equal explanatory power. 
 B)
 The simple linear trend. 
 C)
 The cause cannot be determined using the given information. 

To actually determine the explanatory power for sales itself, fitted values for the loglinear 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)
With respect to the possible problems of autocorrelation and nonstationarity, using the loglinear 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. 

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 DurbinWatson test because there are fewer than 15 observations, which is the lower limit of the DW table. Looking at the firstorder autocorrelation coefficient, however, we see that it increased (in absolute value terms) for the loglinear equation. If anything, therefore, the problem became more severe. (Study Session 3, LOS 13.b)
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. 

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)
Using the simple linear trend model, the forecast of sales for Very Vegan for the first outofsample period is:
The forecast is 10.0015 + (13 × 6.7400) = 97.62. (Study Session 3, LOS 13.a)
Using the loglinear trend model, the forecast of sales for Very Vegan for the first outofsample period is:
The forecast is e2.9803 + (13 × 0.1371) = 117.01. (Study Session 3, LOS 13.a) 
