Rubax, a US manufacturer of athletic shoes, estimates the following linear trend model for shoe sales.
Q1=a+bt+c1D1+c2D2+c3D3
where
Q1=sales of athletic shoes in the t-th quarter
t= 1,2,...,28{1998(I), 1998(II),...2004(IV)}
D1= 1 if t is quarter I (winter); 0 otherwise
D2= 1 if t is quarter II (spring); 0 otherwise
D3= 1 if t is quarter III(summer); 0 otherwise
The regression analysis produces the following results.
Dependent Variable: QT R-Square F-Ratio P-Value of F
Observations: 28 R-Square 0.9651
F-Ratio 159.01 P-Value = 0.0001
Variable Parameter Standard T-Ratio
Intercept 184500 10310 17.90
T 2100 340 6.18
D1 3280 1510 2.17
D2 6250 2220 2.82
D3 7010 1580 4.44
P-Value
0.0001
0.0001
0.0404
0.0098
0.0002
(a) is there sufficient statistical evidence of an upward trend in shoe sales?
(b) Do this 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 forecase equation?
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.
Again, see my post to your later postings
a) Yes, there is sufficient statistical evidence of an upward trend in shoe sales. This is indicated by the positive parameter for the T variable and the significant T-ratio (as the P-value is 0.0001).
b) The data indicates a statistically significant seasonal pattern of sales for Rubax shoes. This is determined by testing whether the parameters for D1, D2, and D3 (representing winter, spring, and summer quarters) are significantly different from zero. Since the parameters for each of these variables are significant, it suggests that there is a seasonal pattern. Specifically, the data shows that summer is the best quarter for sales, while autumn is the worst.
c) To forecast sales of Rubax shoes for 2005(III) and 2006(II), plug the appropriate values into the estimated forecast equation:
For 2005(III), D1 = 0, D2 = 0, D3 = 1
Sales = a + b(t) + c1(D1) + c2(D2) + c3(D3)
Sales = 184500 + 2100(28) + 6250(0) + 7010(1) = 238,770
For 2006(II), D1 = 0, D2 = 1, D3 = 0
Sales = a + b(t) + c1(D1) + c2(D2) + c3(D3)
Sales = 184500 + 2100(31) + 6250(1) + 7010(0) = 252,110
So, the forecasted sales for Rubax shoes for 2005(III) is 238,770 and for 2006(II) is 252,110.
d) To improve this forecast equation, you could consider incorporating additional variables such as price, price of competitors, advertising expenses, advertising by competitors, the population of young adults, the number of retailers, and the number of shoe styles produced. These variables may have an impact on sales and can provide more accurate forecasts when included in the equation.
a) To determine if there is sufficient statistical evidence of an upward trend in shoe sales, we need to examine the parameter and T-ratio for the T variable. In this case, the parameter is 2100 and the T-ratio is 6.18. The T-ratio indicates the significance of the parameter, and in this case, the T-ratio is significant as the P-value is 0.0001. Therefore, there is sufficient statistical evidence of an upward trend in shoe sales.
b) To determine if there is a statistically significant seasonal pattern in the sales of Rubax shoes, we need to perform an F-test on the parameters for D1, D2, and D3. Since each of the parameters for D1, D2, and D3 are individually significant, it is likely that the combination of them is also significant. Therefore, there is a statistically significant seasonal pattern in the sales of Rubax shoes.
As for the specific seasonal pattern exhibited by the data, the parameter estimates for D1, D2, and D3 are as follows: D1 = 3280, D2 = 6250, D3 = 7010. This indicates that sales tend to increase in the spring (D2), summer (D3), and winter (D1) quarters, with the largest increase in sales observed in the summer quarter.
c) To forecast sales of Rubax shoes for 2005(III) and 2006(II), we need to use the estimated forecast equation.
The forecast equation is given by: Q1 = a + bt + c1D1 + c2D2 + c3D3
For 2005(III), we set t = 28 and D1 = 0, D2 = 0, D3 = 1. Plugging in these values, we get:
Q1 = 184500 + 2100(28) + 3280(0) + 6250(0) + 7010(1) = 184500 + 58800 + 7010 = 251310
Therefore, the forecasted sales for 2005(III) are 251310.
For 2006(II), we set t = 29 and D1 = 0, D2 = 1, D3 = 0. Plugging in these values, we get:
Q1 = 184500 + 2100(29) + 3280(0) + 6250(1) + 7010(0) = 184500 + 60900 + 6250 = 253650
Therefore, the forecasted sales for 2006(II) are 253650.
d) To improve this forecast equation, we can consider incorporating additional variables that may impact shoe sales. Some potential variables to consider are price of the shoes, price of competitor's shoes, advertising expenses, advertising by competitors, population of young adults, number of retailers, and number of shoe styles produced. By including these variables and estimating their effects on shoe sales, we can create a more comprehensive and accurate forecast equation.