Economyst, please help
Rubax__ a U.S. manufacturer of athletic shoes, estimates the following linear trend model for shoe sales:
Qt= a + bt + c1D1 + c2D2 + c3D3
Where
Qt= sales of athletic shoes in the tth quarter
t= 1, 2,…., 28[2001(I), 2001(II), …., 2007(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 0.9651
F-Ratio 159.01
P-Value on F 0.0001
Intercept: Parameter Estimate 184500, Standard Error 10310, T-Ratio 17.90, P-Value 0.0001
T: Parameter Estimate 2100, Standard Error 340, T-Ratio 6.18, PValue 0.0001
D1: Parameter Estimate 3280, Standard Error 1510, T-Ratio 2.17, P-Value 0.0404
D2: Parameter Estimate 6250, Standard Error 2220, T-Ratio 2.82, P-Value 0.0098
D3: Parameter Estimate 7010, Standard Error 1580, T-Ratio 4.44, P-Value 0.0002
a. Is there sufficient statistical 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 2008(II) and 2009(II).
d. How might you improve this forecast question?
a. To determine if there is sufficient statistical evidence of an upward trend in shoe sales, we need to analyze the t-ratio and the associated p-value for the T variable.
From the results, the T variable has a parameter estimate of 2100, a standard error of 340, a t-ratio of 6.18, and a p-value of 0.0001.
Since the p-value is less than the typical significance level of 0.05, we can conclude that there is sufficient statistical evidence to support the presence of an upward trend in shoe sales. The positive parameter estimate for the T variable also indicates that as time increases (t), the sales of athletic shoes also increase.
b. To determine if there is a statistically significant seasonal pattern in the sales of Rubax shoes, we need to analyze the t-ratios and the associated p-values for the D1, D2, and D3 variables.
From the results, we can see that all three seasonal dummy variables (D1, D2, D3) have t-ratios and p-values.
D1 has a t-ratio of 2.17 and a p-value of 0.0404, D2 has a t-ratio of 2.82 and a p-value of 0.0098, and D3 has a t-ratio of 4.44 and a p-value of 0.0002.
Since the p-values for all three variables are less than 0.05, we can conclude that there is a statistically significant seasonal pattern in the sales of Rubax shoes.
The seasonal pattern can be observed by comparing the parameter estimates for the D1, D2, and D3 variables.
D1 has a parameter estimate of 3280, D2 has a parameter estimate of 6250, and D3 has a parameter estimate of 7010. This indicates that sales are higher in quarter II (spring) and quarter III (summer) compared to quarter I (winter).
c. To forecast the sales of Rubax shoes for 2008(II) and 2009(II), we can use the estimated forecast equation:
Qt = a + bt + c1D1 + c2D2 + c3D3
Plug in the values for t and the corresponding seasonal dummy variables:
For 2008(II):
t = 28 (since it is the 28th quarter)
D1 = 0
D2 = 1
D3 = 0
Replace the values into the forecast equation:
Q(28) = 184500 + 2100(28) + 3280(0) + 6250(1) + 7010(0)
Calculate the forecasted sales for 2008(II).
Similarly, you can repeat the process for 2009(II).
d. To improve this forecast, you can consider the following approaches:
1. Collect more recent data: The data provided only goes up to 2007(IV). Collecting more recent data will help capture any changes or shifts in sales patterns that may have occurred since then.
2. Include more variables: The current model only considers time and seasonal patterns. Including other relevant variables such as advertising expenditure, competitor sales, or economic indicators may improve the accuracy of the forecast.
3. Evaluate alternative models: Consider exploring different regression models such as polynomial regression or exponential smoothing to capture non-linear patterns or trends in the data.
4. Validate and update the model: Regularly validate the accuracy of the forecasting model using actual sales data. If necessary, update the model parameters to better reflect the changes in sales patterns.