II. Multivariate Regression: Demand Estimation Perform a (linear) multivariate r
ID: 3276042 • Letter: I
Question
II. Multivariate Regression: Demand Estimation Perform a (linear) multivariate regression analysis including consumption as dependent variable and the remaining three variables as independent variables.
1. Estimate the yearly per-capita demand for soft drinks as a (linear) function of 6-pack price, income per capita and mean temperature.
2. Give an economic interpretation of each of the estimated regression coefficients.
3. Which of the independent variables (if any) is statistically significant (at the 0.05 level) in explaining per capita demand for soft drink? (Hint: P-value)
4. What proportion of the total variation in demand for soft drink is explained by the regression model? (Hint: R Square)
5. What does the ANOVA (analysis of variance) suggest for the overall significance of the results (at the 0.05 level)? (Hint: Significance F statistic)
6. Based on the regression model, determine the best (point) estimate of yearly per-capital soft drink demand in a state where 6-pack price is $2.20, per capita income is $18 and mean temperature is 54F. Then, construct an approximate 95 percent prediction interval. 7. Determine the (point) price elasticity and (point) income elasticity, respectively, at the state mentioned in question 6. (Hint: use estimated demand equation.) Give an economic interpretation of each of the elasticities.
The following is the data
cans/capita/yr 6 pack price income/capita mean temp Alabama 200 2.19 13 66 Arizona 150 1.99 17 62 Arkansas 237 1.93 11 63 California 135 2.59 25 56 Colorado 121 2.29 19 52 Connecticut 118 2.49 27 50 Delaware 217 1.99 28 52 Florida 242 2.29 18 72 Georgia 295 1.89 14 64 Idaho 85 2.39 16 46 Illinois 114 2.35 24 52 Indiana 184 2.19 20 52 Iowa 104 2.21 16 50 Kansas 143 2.17 17 56 Kentucky 230 2.05 13 56 Louisiana 269 1.97 15 69 Maine 111 2.19 16 41 Maryland 217 2.11 21 54 Massachusetts 114 2.29 22 47 Michigan 108 2.25 21 47 Minnesota 108 2.31 18 41 Mississippi 248 1.98 10 65 Missouri 203 1.94 19 57 Montana 77 2.31 19 44 Nebraska 97 2.28 16 49 Nevada 166 2.19 24 48 New Hampshire 177 2.27 18 35 New Jersey 143 2.31 24 54 New Mexico 157 2.17 15 56 New York 111 2.43 25 48 North Carolina 330 1.89 13 59 North Dakota 63 2.33 14 39 Ohio 165 2.21 22 51 Oklahoma 184 2.19 16 82 Oregon 68 2.25 19 51 Pennsylvania 121 2.31 20 50 Rhode Island 138 2.23 20 50 South Carolina 237 1.93 12 65 South Dakota 95 2.34 13 45 Tennessee 236 2.19 13 60 Texas 222 2.08 17 69 Utah 100 2.37 16 50 Vermont 64 2.36 16 44 Virginia 270 2.04 16 58 Washington 77 2.19 20 49 West Virginia 144 2.11 15 55 Wisconsin 97 2.38 19 46 Wyoming 102 2.31 19 46Explanation / Answer
Solution:
The required output for the multiple regression model is given as below:
SUMMARY OUTPUT
Regression Statistics
Multiple R
0.835478305
R Square
0.698023997
Adjusted R Square
0.677434724
Standard Error
38.26108281
Observations
48
ANOVA
df
SS
MS
F
Significance F
Regression
3
148889.8565
49629.95217
33.9023141
1.64557E-11
Residual
44
64412.06016
1463.910458
Total
47
213301.9167
Coefficients
Standard Error
t Stat
P-value
Lower 95%
Upper 95%
Intercept
514.2669369
113.3315243
4.537721874
4.36383E-05
285.8622608
742.671613
6 pack price
-242.9707509
43.52628127
-5.582161944
1.38245E-06
-330.6922056
-155.2492962
income/capita
1.224163793
1.522612776
0.803988914
0.425725939
-1.844460582
4.292788167
mean temp
2.931228055
0.711458375
4.120027476
0.000164543
1.497377934
4.365078176
1. Estimate the yearly per-capita demand for soft drinks as a (linear) function of 6-pack price, income per capita and mean temperature.
Answer:
For estimation of the per capita demand for soft drinks, we have to use multiple linear regression model. For this regression model, the dependent variable or response variable is given as per capita demand while independent variables or explanatory variables are given as 6 pack price, income per capita and mean temperature. The required regression equation is given as below:
Cans/Capita/Year = 514.27 – 242.97*six pack price + 1.22*income per capita + 2.93*mean temperature
2. Give an economic interpretation of each of the estimated regression coefficients.
Answer:
The coefficient for the six pack price is negative which indicate the negative relationship with cans per capita per year. As the price of the six pack price is increased, the number of cans per capita per year would be decrease. The signs of the variables income per capita and mean temperature are positive which suggests that there is positive association between the cans per capita per year with the two variables income per capita and mean temperature.
3. Which of the independent variables (if any) is statistically significant (at the 0.05 level) in explaining per capita demand for soft drink? (Hint: P-value)
Answer:
The p-value for checking the significance of the coefficient of variable six pack price is given as 0.0000001382, which is very less than alpha value 0.05 and therefore coefficient for the variable six pack price is statistically significant. The p-value for the coefficient of variable income per capita is greater than alpha value 0.05, so coefficient of variable income per capita is not statistically significant. The p-value for mean temperature is less than alpha value, so this coefficient is statistically significant.
4. What proportion of the total variation in demand for soft drink is explained by the regression model? (Hint: R Square)
Answer:
The value of the coefficient of determination or R square is given as 0.698023997, which means about 69.80% of the total variation in demand for soft drink is explained by the regression model.
5. What does the ANOVA (analysis of variance) suggest for the overall significance of the results (at the 0.05 level)? (Hint: Significance F statistic)
Answer:
The p-value for the ANOVA for overall significance is given as 0.00 approximately which is less than alpha value, so we reject the null hypothesis that there is no any statistically significant relationship exists between the dependent variable cans per capita per year and independent variables six pack price, income per capita and mean temperature.
SUMMARY OUTPUT
Regression Statistics
Multiple R
0.835478305
R Square
0.698023997
Adjusted R Square
0.677434724
Standard Error
38.26108281
Observations
48
ANOVA
df
SS
MS
F
Significance F
Regression
3
148889.8565
49629.95217
33.9023141
1.64557E-11
Residual
44
64412.06016
1463.910458
Total
47
213301.9167
Coefficients
Standard Error
t Stat
P-value
Lower 95%
Upper 95%
Intercept
514.2669369
113.3315243
4.537721874
4.36383E-05
285.8622608
742.671613
6 pack price
-242.9707509
43.52628127
-5.582161944
1.38245E-06
-330.6922056
-155.2492962
income/capita
1.224163793
1.522612776
0.803988914
0.425725939
-1.844460582
4.292788167
mean temp
2.931228055
0.711458375
4.120027476
0.000164543
1.497377934
4.365078176