You decide to estimate the following quarterly sales forecasting model for new boat sales in your local county:
Qt = a + bt + cD
The equation is estimated using quarterly data on new boat sales in the county from the 3rd quarter of 2001 to the 4th quarter of 2007 (t = 1,…,26). The variable D is a dummy variable for the second quarter, which is the “season” for selling new boats: D = 1 in the second quarter, and 0 otherwise. The results of the estimation are:
DEPENDENT VARIABLE: QT R-SQUARE F-RATIO P-VALUE ON F
OBSERVATIONS: 26 0.5216 12.54 0.0002
PARAMETER STANDARD
VARIABLE ESTIMATE ERROR T-RATIO P-VALUE
INTERCEPT 241.3 65.36 3.69 0.0012
T 47.52 13.55 3.51 0.0019
D 34.66 10.33 3.36 0.0027
a. Do the statistical estimates indicate a trend in sales? If so, what is the trend in units per quarter?
b. Do the statistical estimates indicate a seasonal effect on 2nd quarter sales? If so, what is the seasonal effect in units per quarter?
c. What are the forecasted sales for the 1st quarter of 2008?
d. What are the forecasted sales for the 2nd quarter of 2008?
e. What are the forecasted sales for the 3rd quarter of 2008?
f. What are the forecasted sales for the 4th quarter of 2008?
a. The statistical estimates indicate a trend in sales as the coefficient for the variable T is positive (47.52) and statistically significant (p-value < 0.05). This implies that there is a positive trend in sales over time. The trend in units per quarter is an increase of 47.52 units.
b. The statistical estimates indicate a seasonal effect on 2nd quarter sales as the coefficient for the dummy variable D is positive (34.66) and statistically significant (p-value < 0.05). This implies that there is an increase in sales during the 2nd quarter compared to other quarters. The seasonal effect in units per quarter is an increase of 34.66 units during the 2nd quarter.
c. To forecast sales for the 1st quarter of 2008, we can use the estimate equation:
Qt = a + bt + cD
Plugging in the values for the 1st quarter of 2008:
Qt = 241.3 + (47.52 * 27) + (34.66 * 0)
Qt = 241.3 + 1285.04 + 0
Qt = 1526.34
The forecasted sales for the 1st quarter of 2008 is approximately 1526 units.
d. To forecast sales for the 2nd quarter of 2008, we can again use the estimate equation:
Qt = a + bt + cD
Plugging in the values for the 2nd quarter of 2008:
Qt = 241.3 + (47.52 * 28) + (34.66 * 1)
Qt = 241.3 + 1332.56 + 34.66
Qt = 1608.52
The forecasted sales for the 2nd quarter of 2008 is approximately 1609 units.
e. To forecast sales for the 3rd quarter of 2008, we use the estimate equation again:
Qt = a + bt + cD
Plugging in the values for the 3rd quarter of 2008:
Qt = 241.3 + (47.52 * 29) + (34.66 * 0)
Qt = 241.3 + 1377.08 + 0
Qt = 1618.38
The forecasted sales for the 3rd quarter of 2008 is approximately 1618 units.
f. To forecast sales for the 4th quarter of 2008, we use the estimate equation:
Qt = a + bt + cD
Plugging in the values for the 4th quarter of 2008:
Qt = 241.3 + (47.52 * 30) + (34.66 * 0)
Qt = 241.3 + 1422.6 + 0
Qt = 1663.9
The forecasted sales for the 4th quarter of 2008 is approximately 1664 units.
a. The statistical estimates indicate a trend in sales. The trend is represented by the coefficient of the variable 'T'. According to the estimation results, the coefficient of 'T' is 47.52. This means that for each additional quarter (t), the sales (Qt) increase by an average of 47.52 units.
b. The statistical estimates also indicate a seasonal effect on 2nd quarter sales. The seasonal effect is represented by the coefficient of the dummy variable 'D'. According to the estimation results, the coefficient of 'D' is 34.66. This means that during the 2nd quarter (when 'D' is equal to 1), the sales increase by an average of 34.66 units compared to other quarters (when 'D' is equal to 0).
c. To forecast the sales for the 1st quarter of 2008, we can use the estimated equation:
Qt = a + bt + cD.
Since 't' represents the time period, we need to calculate the 't' value for the 1st quarter of 2008. The given data ranges from the 3rd quarter of 2001 (t=1) to the 4th quarter of 2007 (t=26).
To calculate 't' for the 1st quarter of 2008:
t = (year - base year) * number of quarters per year + 1
= (2008 - 2001) * 4 + 1
= 7 * 4 + 1
= 29.
Now we can substitute the values into the equation:
Qt = a + bt + cD
= 241.3 + 47.52 * 29 + 34.66 * 0
= 241.3 + 1377.08 + 0
= 1618.38.
Therefore, the forecasted sales for the 1st quarter of 2008 are approximately 1618.38 units.
d. Similarly, to forecast the sales for the 2nd quarter of 2008:
t = (2008 - 2001) * 4 + 2
= 29 + 2
= 31.
Substituting the values into the equation:
Qt = a + bt + cD
= 241.3 + 47.52 * 31 + 34.66 * 1
= 241.3 + 1472.12 + 34.66
= 1747.08.
The forecasted sales for the 2nd quarter of 2008 are approximately 1747.08 units.
e. To forecast the sales for the 3rd quarter of 2008:
t = (2008 - 2001) * 4 + 3
= 29 + 3
= 32.
Substituting the values into the equation:
Qt = a + bt + cD
= 241.3 + 47.52 * 32 + 34.66 * 0
= 241.3 + 1520.64 + 0
= 1761.94.
The forecasted sales for the 3rd quarter of 2008 are approximately 1761.94 units.
f. To forecast the sales for the 4th quarter of 2008:
t = (2008 - 2001) * 4 + 4
= 29 + 4
= 33.
Substituting the values into the equation:
Qt = a + bt + cD
= 241.3 + 47.52 * 33 + 34.66 * 0
= 241.3 + 1569.16 + 0
= 1810.46.
The forecasted sales for the 4th quarter of 2008 are approximately 1810.46 units.