Posted by Ed on Sunday, June 10, 2007 at 12:24pm.
I have put together my set of questions that I posted and answered them with yor help and direction, would you mind looking at the finished product for correctness. I still got lost on some of it I think. I could send them to you.
EY
Post them. We do not engage in Email with students for security sake.
Ok, understand that perhaps I can post?
Ok, I understand...Economyst, you helped me get in the right direction, can you see if I am on the right path?
1)The director of marketing at Vanguard Corporation believes the sales of the company’s Bright Side
Laundry detergent (S) are related to Vanguard’s own advertising expenditure (A), as well as the combined
advertising expenditures of its three biggest rival detergents (R). The marketing director collects 36 weekly observations on S, A, and R to estimate thefollowing multiple regression equation:
S = a + bA + cR
Where S, A, R are measured in dollars per week. Vanguard’s marketing director is comfortable
using parameter estimates that are statistically significant at the 10 percent level or better.
a)What sign does the marketing director expecta, b, and c to have?
ANSWER:
The director would expect his own advertising to have a positive effect and the competitor’s advertising to
have a negative effect. He should expect some level of brand loyalty, but his advertising should have a positive effect.
b)Interpret the coefficients a, b, and c?
ANSWER:
S = a + bA + cR
Here “a” will be the intercept parameter and b, along
with c, will be the slope parameter. Vanguard’s own advertising would be a positive effort and the competitor’s would be negative.
END
The regression output from the computer is as follows:
Dependent Variable: S
R-Square: 0.2247 F ratio: 4.781
P-Value on F: 0.0150
Observations: 36
Variable: Intercept
Parameter Estimate: 175086.0
Standard Error: 63821.0
T ratio: 2.74 P-Value: 0.0098
Variable: A
Parameter Estimate: 0.8550
Standard Error: 0.3250
T ratio: 2.63 P-Value: 0.0128
Variable: R
Parameter Estimate: - 0.284
Standard Error: 0.164
T ratio: - 1.73 P-Value: 0.0927
c)Does Vanguard’s advertising expenditure have a statistical significant effect on the sales of Bright Side detergent?
ANSWER:
Yes, at the 5% level, there is statistical significance at the 5% level.
Explain, using appropriatep-value…
ANSWER:
A 0.0128 p-value means the exact level of significance for a T-Ratio of 2.63 is 1 % and the level of confidence
is 99%. Stating b is statistically significant.
d)Does the advertising by its three largest rivals affect sales of Bright Side detergent in a statistical
significant way?
ANSWER:
P-Value and T-Ratio show that the competitor’s advertising has a negative effect.
Explain using the appropriate p-value…
ANSWER:
The high P-value indicates that the negative T-ratio has a high probability of competitor’s advertising effecting sales of Bright Side negatively.
e)What fraction of the total variation in sales of Bright Side remains unexplained?
ANSWER:
22%
What can the marketing director do to increase the explanatory power of the sales equation?
ANSWER:
He could look at the prices charged by the competitor and Vanguard and add this to the equation as well as log variables on advertising expenses.
What other explanatory variables might be added to this equation?
ASNWER:
Other variables might include family size, loads of laundry done during the summer vs. the winter.
f)What is the expected level of sales each week when Vanguard spends $40,000 per week and the combined advertising expenditures for the three rivals are $100,000 per week?
ASNWER:
S = a + b($40,000) + c($100,000)
S = 175086.0 + 0.85550($40,000)
+ - 0.284($100,000)
S = 175086.0 + $34,000 + - $28,400
S = $209,306 + (- $28,400)
S = $180,906
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.
ANSWER:
Logarithms must be taken of the equation to transform the log-linear model into a non-linearequation:
Q = aHbSc would be:
In Y = (In a) = b (In H) + c (In S)
We define the following:
Q’ = In Q
H’ = In H
S’ = In S
The linear equation is: Q = a’ + bH’ + cS’
The computer output for the multiple regression analysis is shown below:
Dependent Variable: LNQ
R-Square: 0.5452
F-Ratio:29.97
P-Value on F: 0.0001
Observations: 53
Variable: Intercept
Parameter Estimate: 0.9162
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 Estimate: 0.2250
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 thepercentage increase in daily sales?
ANSWER:
The parameter estimates of b and c are elastic. A 20% Increase in salespeople would result in a decrease in daily sales.
c)Test the overall model for statistical significance at the 5% level?
ASNWER:
1 – P-Value = level of confidence with the F-Stat
53 Observations – 3 Parameters = 50, the critical T-value is 2.0. The T-Ratios are over this so the significance level is strong a 5%.
d)What percent of the total variation in daily auto sales is explained by this equation?
ANSWER:
54%
What could you suggest to increase this
percentage?
ANSWER:
Perhaps pricing of the product could be used as a determinant of sales.
e)Test the intercept for statistical significance at the 5% level of significance.
ANSWER:
Values are significant at the 5% level.
If H and S both equal 0, are the sales expected to be 0? Explain why or why not…..
ANSWER:
If H and S are equal to 0 the explanatory variables which b is a coefficient, is not relative to the dependant variable.
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?
ANSWER:
The coefficient b is statistically significant at the 5% level. If hours of operation are decreased the number of sales will decrease.
Thanks for your help!
EY
Ok, EY, for the most part, you are on the right track. That said, i think it would be helpful if you better understood what linear regression analyses is trying to do and what the statistics mean.
For a parameter estimate, the T ratio is simply the parameter estimate divided by the standard error. A negative T ratio simply means the parameter estimate is negative. The test is: is the parameter estimate significantly different from zero. You could take the T ratio and find a probability in a cumulative normal distribution table. However, most stat packages do it for you; hence the P-Value. The P-Value gives the probability that the "true" parameter is zero (or worse, the opposite sign). The F statistic tests whether the explanatory power of the overall model is significantly different from zero.
Under d) in the first problem, you answered
"The high P-value indicates that the negative T-ratio has a high probability of competitor’s advertising effecting sales of Bright Side negatively."
No No No. The High P Value of .09 means there is a high (9%) probability the parameter estimate is zero or worse. It is not significant. Further, the sign has nothing to do with the significance.
The R-squared gives a measure of the total variation in y that is explained by the specified model. An R^2 of .22 means the 22% of the variation is explained, which means that 78% is unexplained. Check your answer e)
To increase the explanatory power, you suggested using logged variables. This may be true, but why? Taking logs of variables is one way of converting a non-linear relationship into a linear relationship. As I posted earlier, I suggested lagged variables. That is, advertising this week affects not only sales this week but sales next week and the week after.
Now equation 2) In my earlier post, I said that in a log model, parameter estimates were elasticities (not elastic). (See your basic economics text if you are confused about what an elasticity is and represents). So, the estimated elasticity of car sales (Q) to sales people (S) is .225. So if S goes up by 20%, we would expect Q to INCREASE by 20%*.225 = 4.5%.
Answer c) For the overall significance of a model, look at the P-value on the F-statistic. Since the P value is so very low, we can say the model is significant; it does explain at least some of the variation in the y variable.
Answer e). Your original specification is Q=aHS. So, if H and S are zero, the Q must be zero.
Answer f). Again, the parameter estimate is and elasticity. If H goes down by 10%, then Q decreases by 10%*.3517 = 3.517%.
Well, thanks, I think you pointed out I am still confused, this stuff is hard to understand. I meant log-linear variables, I thought that was where you were pointing me because I saw these in the book.
I will go back and look at these, it appears I only understood about half and got the other reveresed, like p-values.
Now on to theory of Consumer behavior with marginal utility, market demand, and income.
EY
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