Question

Use a bivariate linear regression trend model to estimate this trend equation: SALES = a + b(TIME) Is the sign of b what you would expect? Is b significantly different from zero? What is the coefficient of determination for this model? Explain your answers.

Date	Sales	Time
Mar-12	3,372.5	1
Jun-12	3,472.7	2
Sep-12	3,534.8	3
Dec-12	3,602.6	4
Mar-13	3,675.1	5
Jun-13	3,725.4	6
Sep-13	3,749.0	7
Dec-13	3,772.4	8
Mar-14	3,794.1	9
Jun-14	3,826.5	10
Sep-14	3,877.8	11
Dec-14	3,905.5	12
Mar-15	3,962.4	13
Jun-15	3,950.6	14
Sep-15	3,978.3	15
Dec-15	3,986.8	16
Mar-16	4,019.9	17
Jun-16	4,073.8	18
Sep-16	4,119.6	19
Dec-16	4,189.0	20

Answer 1

In the trend equation the dependent variable is sales while the independent variable is time. We are trying to estimate how sales will change over time. We carry out the regression in excel and get the following result.

SUMMARY OUTPUT
Regression Statistics
Multiple R	0.98
R Square	0.96
Adjusted R Square	0.96
Standard Error	43.16
Observations	20.00
ANOVA
	df	SS	MS	F	Significance F
Regression	1.00	883291.04	883291.04	474.19	0.00
Residual	18.00	33528.97	1862.72
Total	19.00	916820.01
	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%	Lower 95.0%	Upper 95.0%
Intercept	3446.76	20.05	171.92	0.00	3404.64	3488.89	3404.64	3488.89
Time	36.45	1.67	21.78	0.00	32.93	39.96	32.93	39.96

We see that b = 36.45. The sign of b is positive and this is expected since we see from the data that sales increase over time. This is shown in the graph below.

Sales 4,400.00 4,200.00 4,000.00 3,800.00 3,600.00 3,400.00 3,200.00 3,000.00 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Sales

We see the p value is 0 which indicates that b is significantly different from 0. P value gives the probability of accepting the null, here null being b= 0.

The coefficient of determination is given by R-square which is 0.96 for this model.