Question

Bottled Water

Shown above is the demand for bottled water in thousands of Gallons for 110 consecutive weeks. From weeks 75 through 84, there was a severe flood in the area. Shown below are two regression results using this data. The “Week” variable is an index of weeks from 1 through 109. The “Intervention” variable is a dummy variable equaling one during the intervention and zero otherwise.

Regression #1

Audit Trail — ANOVA Table (Multiple Regression Selected)
Source of variation	SS			df			MS			SEE
Regression		33,763.50			1			33,763.50
Error		6,697.97			108			62.02			7.88
Total		40,461.47			109

Audit Trail — Coefficient Table (Multiple Regression Selected)
Series Description	Included in Model			Coefficient			Standard Error			T-test			P-value				Elasticity				Overall F-test
Week		Yes			0.55			0.02			23.33			0.00				0.25
Demand		Dependent			94.28			1.51			62.35			0.00								544.41

Audit Trail — Correlation Coefficient Table
Series Description	Included in Model			Week			Demand
Week		Yes			1.00			0.91
Demand		Dependent			0.91			1.00

Audit Trail - Statistics
Accuracy Measures	Value		Forecast Statistics	Value
AIC	766.17		Durbin Watson	0.60
BIC	768.87		Mean	124.90
Mean Absolute Percentage Error (MAPE)	4.00	%	Standard Deviation	19.27
R-Square	83.45	%	Max	167.08
Adjusted R-Square	83.29	%	Min	89.55
Mean Square Error	60.89		Range	77.54
Root Mean Square Error	7.80

Method Statistics	Value
Method Selected	Multiple Regression

Regression #2

Audit Trail — ANOVA Table (Multiple Regression Selected)
Source of variation	SS			df			MS			SEE
Regression		38,993.33			2			19,496.66
Error		1.468.14			107			13.72			3.70
Total		40,461.47			109

Audit Trail — Coefficient Table (Multiple Regression Selected)
Series Description	Included in Model			Coefficient			Standard Error			T-test			P-value				Elasticity			Overall F-test
Week		Yes			0.50			0.01			43.50			0.00				0.22
Demand		Dependent			95.00			0.71			133.39			0.00							1,420.94
Intervention		Yes			24.70			1.27			19.52			0.00				0.22

Audit Trail — Correlation Coefficient Table
Series Description	Included in Model			Week			Demand			Intervention
Week		Yes			1.00			0.91			0.24
Demand		Dependent			0.91			1.00			0.57
Intervention		Yes			0.24			0.57			1.00

Audit Trail - Statistics
Accuracy Measures	Value		Forecast Statistics	Value
AIC	599.21		Durbin Watson	1.84
BIC	601.91		Mean	124.90
Mean Absolute Percentage Error (MAPE)	2.47	%	Standard Deviation	19.27
R-Square	96.37	%	Max	167.08
Adjusted R-Square	96.30	%	Min	89.55
Mean Square Error	13.35		Range	77.54
Root Mean Square Error	3.65

Method Statistics	Value
Method Selected	Multiple Regression

Examine the Akaike Information Criterion for both Regression #1 and Regression #2 above.

Answer 1

Here, we are asked to examine the Akaike Information Criterion for Regression #1 and Regression #2.

AIC is a method based on an out-sample fit to estimate the likelihood function of a given model to predict the future values of the dependent variables in the model. For comparison of two or more models, a good model is the one that has the least AIC among the models.

[ Bayesian Information Criterion (BIC) is also used to compare different models.A good model is the one that has the least BIC among the models. But BIC penalizes the model complexity more than AIC, that is BIC keeps a check on the number of parameters in the model. A more complex model will be preferred by AIC but not by BIC. ]

Here, Regression #1 has AIC 766.17 and Regression #2 has AIC 599.21.

So, Regression #2 has minimum AIC among the both and is thus considered a better model among the two.

[ Regression #1 has BIC 768.87 and Regression #2 has AIC 601.91.

So, Regression #2 has minimum BIC among the both and is thus again considered a better model among the two, if we consider BIC as the compaison measue. ]