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.

This appears to be repost of a set of questions that was presented months ago along with my answers. I stand by all of my answers. Did you have a follow-up question?

Yes can you elaborate on question A & C please Economyst?

. Is there sufficient statistical evidence of an upward trend in shoe sales?�

a) To determine if there is sufficient statistical evidence of an upward trend in shoe sales, we need to analyze the parameter and T-ratio for the T variable. In this case, the parameter is positive (2100) and the T-ratio is significant, indicated by the very low P-value (0.0001). This means that there is strong evidence of an upward trend in shoe sales for Rubax.

b) To determine if there is a statistically significant seasonal pattern of sales, we need to analyze the parameters and P-values for the dummy variables (D1, D2, D3). Since the parameters for each dummy variable are significant (P-values less than 0.05), it is likely that the combination of these variables is also significant. This indicates a statistically significant seasonal pattern in sales for Rubax shoes. From the parameter values, we can see that D3 (quarter III or summer) has the highest parameter value, followed by D2 (quarter II or spring), and D1 (quarter I or winter). This suggests that Rubax experiences higher shoe sales in summer, followed by spring and winter, with autumn having the lowest sales.

c) To forecast sales of Rubax shoes for 2005(III) and 2006(II), we need to use the estimated forecast equation: Q1 = a + bt + c1D1 + c2D2 + c3D3. Plug in the appropriate values for each variable:
- For 2005(III), t would be 28+1 = 29 (as it is the 29th quarter).
- For 2006(II), t would be 28+2 = 30 (as it is the 30th quarter).

Substitute the values into the equation and solve for the forecasted sales.

d) To improve the forecast equation, we can consider adding additional variables that may influence shoe sales. Some potential variables to consider are price (both Rubax's and competitors'), advertising expenses (both Rubax's and competitors'), population of young adults (target market), number of retailers, and number of shoe styles produced. By including these variables, we can better capture the various factors that may impact shoe sales and improve the accuracy of the forecast.