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.
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 (2100) and the T-ratio (6.18) is significant as the P-value is 0.0001.
b) To determine if there is a statistically significant seasonal pattern in sales, one would typically conduct an F-test. However, in this case, the parameters for each seasonal variable (D1, D2, D3) are individually significant. Thus, it is highly likely that the combined effect of these variables would also be significant. As for the seasonal pattern exhibited by the data, the sales are highest in the summer quarter (D3) and lowest in the autumn quarter (not explicitly shown in the given data).
c) To forecast sales of Rubax shoes for 2005(III) and 2006(II), you would need to plug the appropriate values into the estimated forecast equation. The equation is Q1 = a + bt + c1D1 + c2D2 + c3D3. Since the data only goes up to 2004(IV), it is not explicitly stated what the values of D1, D2, and D3 would be for 2005(III) and 2006(II). Thus, without knowing the values of these variables, it is not possible to provide an accurate forecast using the estimated equation.
d) To improve the forecast equation, there are several variables that could be considered. Some possible variables to include would be price (of Rubax shoes), price of competitors, advertising expenses (by Rubax), advertising expenses by competitors, population of young adults, number of retailers, and number of shoe styles produced. By including additional relevant variables, the forecast equation will have a more comprehensive set of predictors, resulting in a potentially more accurate forecast.