4) The manager of Collins Import Autos
believes that the number of cars
sold in a day (Q) depends on two
factor: (1) the number of hours the
dealership is open (H) and (2) the
number of salespersons working that
day (S). After collecting the data
for two months (53 days), the
manager estimates the following log-
linear model:
b c
Q = aH S
a) Explain how to transform this log-
linear model into a linear form
that can be estimated using
multiple regression analysis.
The computer output for the multiple regression analysis is shown below:
Dependant Variable: LNQ
R-Square: 0.5452 F-Ratio: 29.97
P-Value on F: 0.0001
Observations: 53
Variable: Intercept
Parameter Est: o.9162
Standard Error: 0.2413
T-Ratio: 3.80
P-Value: 0.0004
Variable: LNH
Parameter Estimate: 0.3517
Standard Error: 0.1021
T-Ratio: 3.44
P-Value: 0.0012
Variable: LNS
Parameter Est: 0.2550
Standard Error: 0.0785
T-Ratio: 3.25
P-Value: 0.0021
b) How do you interpret coefficients
b and c?If the dealership increases
the number of salespersons by 20
percent, what will the percentage
increase in daily sales?
c) Test the overall model for
statistical significance at the 5%
level?
d) What percent of the total variation
in daily auto sales is explained by
this equation?
What could you suggest to increase
this percentage?
e) Test the intercept for statistical
significance at the 5% level
Significance. If H and S both equal
0, are the sales expected to be 0?
Explain why or why not…..
f) Test the estimated coefficient b
for statistical significance. If
the Dealership decreases its hours
of operation by 10%, what is the
expected impact on daily sales?
Thanks,
EY
a) transform natural log values to linear values using e^x.
b) Parameter estimates in a log-linear function are elasticities.
c) what does the F statistic tell you?
d) what does the R^2 statistic tell you.
e) what does the T-ratio statistics on the parameter estimates tell you?
f) re-examine answers b and e
a) To transform the log-linear model into a linear form for multiple regression analysis, you need to convert the natural log values back to linear values. This can be done using the exponential function, denoted as e^x. In this case, you would apply e^Q to the dependent variable, e^H to the hours variable, and e^S to the salespersons variable.
b) In a log-linear function, the parameter estimates (coefficients) represent elasticities. Elasticities measure the percentage change in the dependent variable for a 1% change in the independent variable. So, the coefficient b represents the percentage change in daily sales for a 1% change in the number of hours the dealership is open, and the coefficient c represents the percentage change in daily sales for a 1% change in the number of salespersons working.
If the dealership increases the number of salespersons by 20%, you can calculate the percentage increase in daily sales by multiplying the coefficient c by 20. For example, if c is 0.2550, the expected increase in daily sales would be 0.2550 * 20 = 5.1%.
c) The F statistic tests the overall statistical significance of the model. In this case, the F-Ratio is 29.97 and the p-value on F is 0.0001. A small p-value (less than the chosen significance level, such as 0.05) indicates that the model is statistically significant. Therefore, you can conclude that there is a significant relationship between the independent variables (number of hours and number of salespersons) and the dependent variable (number of cars sold).
d) The R-squared statistic (R^2) represents the proportion of the total variation in the dependent variable that is explained by the regression equation. In this case, the R^2 value is 0.5452, or 54.52%. This means that approximately 54.52% of the total variation in daily auto sales can be explained by the number of hours the dealership is open and the number of salespersons working.
To increase this percentage, you could consider adding additional independent variables that may have an impact on daily auto sales. For example, you might include variables such as advertising expenses, customer demographics, or economic factors like interest rates or unemployment rates.
e) The T-ratio statistics on the parameter estimates measure the statistical significance of each coefficient. They tell you whether each coefficient is significantly different from zero. If the T-ratio is larger than 2 or smaller than -2 (assuming a 5% significance level), the coefficient is considered statistically significant.
Regarding the intercept, if both H (number of hours) and S (number of salespersons) are equal to zero, the sales are not expected to be zero. This is because the intercept term captures the baseline level of sales even when there are no hours or salespersons, indicating some inherent level of demand or other factors not captured in the model.
f) To test the estimated coefficient b for statistical significance, you can examine its T-ratio and corresponding p-value. A small p-value (less than the chosen significance level, such as 0.05) indicates that the coefficient is statistically significant.
If the dealership decreases its hours of operation by 10%, you can calculate the expected impact on daily sales by multiplying the coefficient b by -10. For example, if b is 0.3517, the expected decrease in daily sales would be 0.3517 * -10 = -3.517%.