rubax, a u.s manufacturer of athletic shoes, estiamtes the following linear trend model for shoe sales
Qt=a+bt+c1D1+c2D2+c3D3
where
Qt=sales of athletic shoes in the t-th quater
t=1,2,...,28[1998(I),1998(II),...,2004(IV)]
D1=1 if t is quater I (winter); 0 otherwisw
D2=1 if t is quater II (spring); o otherwise
D3=1 if t is quater III (summer); o otherwise
The regression analysis produces the following results:
Dependent Variable:QT R-Square F-Ratio P-Value on F
Observations: 28 0.9651 159.01
p-value on f 0.0001
variable parameter ESt. standarderror
intercept 184500 10310
T 2100 340
D1 3280 1510
D2 6250 2220
D3 7010 1580
T-ratio p-value
17.90 0.0001
6018 0.0001
2.17 0.0404
2.82 0.0098
4.44 0.0002
A. is there sufficient statiscal evidence of an upward trend in shoe sales?
B. do these data indicate a statistically significant seasonal pattern of sales for rubax shoes? if, so what is the seasonal pattern exhibited by the data?
C. using the estimated forecast equation, forecast sales of rubax shoes for 2005 (III) and 2006(II)
D.how might you improve this forecast equation?
I tried to line up the regression analysis i hope whom ever help me will see to make the regresssion line you correctly on their paper. thank you , signed the desperate Queen
On June 28, a Renee Williams posted this very question. Below is and was my response. Plz repost if you have questions.....
a) Look at the parameter and T-ratio for the T variable. The parameter is positive and the T-ratio is significant(as the P-value is .0001) so.....
b) Hummm. The appropriate test for multiple dummy variables, like your equation, is an F-test. Here, one would test whether the parameters for D1 D2 and D3, as a combination, are significantly different from zero. However, since the parameters for each D1,D2,D3 are each significant, then almost certainly, combined they would be different.
As for the seasonal pattern summer is the best quarter, autum is the worst.
c) Plug the appropriate values into the equation and solve....
d) There are a plethora of variables one could use to improve the forecast. How bout price, price of competitors, advertising expenses, advertising by competitors, population of young adults, number of retailers, number of shoe styles produced, etc.
A. Yes, there is sufficient statistical evidence of an upward trend in shoe sales. This can be determined by looking at the parameter and T-ratio for the T variable. The parameter is positive and the T-ratio is significant (p-value of 0.0001), indicating a statistically significant upward trend in shoe sales.
B. Yes, the data indicate a statistically significant seasonal pattern of sales for Rubax shoes. The parameters for D1, D2, and D3 are all significant, with p-values of 0.0404, 0.0098, and 0.0002 respectively. The seasonal pattern exhibited by the data is that sales are highest in Quarter III (summer) and lowest in Quarter II (autumn).
C. To forecast sales of Rubax shoes for 2005 (III) and 2006 (II), we can use the estimated forecast equation. Plug in the values for T and the appropriate D variables (D1, D2, and D3) for the corresponding quarters, and solve for Qt.
For 2005 (III), we have T = 28 and D1 = D2 = 0, D3 = 1.
Forecasted sales for 2005 (III) = 184500 + 2100(28) + 0(3280) + 0(6250) + 7010(1) = 244,910.
For 2006 (II), we have T = 29 and D1 = 1, D2 = D3 = 0.
Forecasted sales for 2006 (II) = 184500 + 2100(29) + 3280(1) + 0(6250) + 0(7010) = 248,140.
D. To improve the forecast equation, additional variables could be included such as price, price of competitors, advertising expenses, advertising by competitors, population of young adults, number of retailers, number of shoe styles produced, etc. By including these variables, the forecast equation could better capture the factors that influence shoe sales and lead to more accurate predictions.
A. To determine if there is sufficient statistical evidence of an upward trend in shoe sales, we need to look at the parameter and T-ratio for the T variable. In this case, the parameter is 2100 and the T-ratio is 17.90. The T-ratio provides a measure of how many standard errors the estimated parameter is away from zero. Since the T-ratio is significant (as the P-value is 0.0001), we can conclude that there is sufficient statistical evidence of an upward trend in shoe sales.
B. To determine if there is a statistically significant seasonal pattern of sales for Rubax shoes, we need to examine the parameters and T-ratios for the D1, D2, and D3 variables. Each of these variables represents a different quarter (winter, spring, summer). The parameters and T-ratios for these variables are as follows:
D1: Parameter = 3280, T-ratio = 2.17, P-value = 0.0404
D2: Parameter = 6250, T-ratio = 2.82, P-value = 0.0098
D3: Parameter = 7010, T-ratio = 4.44, P-value = 0.0002
To determine if there is a statistically significant seasonal pattern, we need to check if the parameters are significantly different from zero. In this case, all three parameters have T-ratios greater than 2 and corresponding P-values less than 0.05, indicating that there is a statistically significant seasonal pattern in shoe sales for Rubax.
The seasonal pattern exhibited by the data can be determined by comparing the parameter estimates for each quarter. The largest parameter is for D3 (summer) with a value of 7010, followed by D2 (spring) with a value of 6250, and D1 (winter) with a value of 3280. This indicates that the summer quarter has the highest sales, followed by spring and then winter.
C. To forecast sales of Rubax shoes for 2005 (III) and 2006(II), we can use the estimated forecast equation. The forecast equation is given by:
Qt = a + bt + c1D1 + c2D2 + c3D3
Plug in the values for 2005 (III) and 2006(II) into the equation and solve for the forecasted sales.
For 2005 (III):
t = 28 + 1 = 29
D1 = 0, D2 = 0, D3 = 1
Qt = a + b(29) + c1(0) + c2(0) + c3(1)
For 2006 (II):
t = 28 + 2 = 30
D1 = 0, D2 = 1, D3 = 0
Qt = a + b(30) + c1(0) + c2(1) + c3(0)
Substitute the values of a, b, c1, c2, c3 from the regression results into the equation and calculate the forecasted sales for each quarter.
D. To improve the forecast equation, we can consider adding additional variables that may have an impact on shoe sales. Some potential variables to consider are price, price of competitors, advertising expenses, advertising by competitors, population of young adults, number of retailers, and number of shoe styles produced. By including these variables in the regression analysis, we can assess their impact on shoe sales and potentially improve the accuracy of the forecast equation.