Professor Orley Ashenfelter of Princeton University is a pioneer in the field of wine economics. He claims that, contrary to old orthodoxy, the quality of wine can be explained mostly in terms of weather conditions. Wine romantics accuse him of undermining the whole wine-tasting culture. In an interesting co-authored paper that appeared in Chance magazine in 1995, he ran a multiple regression model where quality, measured by the average vintage price relative to 1961, is used as the response variable y. The explanatory variables were the average temperature x1 (in degrees Celsius), the amount of winter rain x2 (in millimeters), the amount of harvest rain x3 (in millimeters), and the years since vintage x4. A portion of the data is shown in the accompanying table.
yˆy^ = + x1
+ x2
+ x3
+ x4
a-1. Estimate a linear model, y =
β0 +
β1x1 +
β2x2 +
β3x3 +
β4x4 + ε.
(Negative values should be indicated by a minus sign. Round
your answers to 4 decimal places.)
a-2. What is the predicted price if x1 = 16, x2 = 600, x3 = 120, and x4 = 20? (Round the regression estimates to at least 4 decimal places and answer to 2 decimal places.)
b-1. Estimate the exponential model, ln(y) = β0 + β1x1 + β2x2 + β3x3 + β4x4 + ε. (Negative values should be indicated by a minus sign. Round your answers to 4 decimal places.)
ln( yˆy^ ) = + x1 + x2 + x3 + x4
b-2. What is the predicted price if x1 = 16, x2 ,= 600, x3 = 120, and x4 = 20? (Round the regression estimates to at least 4 decimal places and answer to 2 decimal places.)
c. Use R2 to select the appropriate model for prediction.
Linear model because it has a lower R2 value (0.7361 < 0.8274)
Linear model because it has a lower R2 value (0.7361 < 0.8727)
Exponential model because it has a higher R2 value (0.8274 > 0.7361)
Exponential model because it has a higher R2 value (0.8727 > 0.7361)
a1
SUMMARY OUTPUT | |||||
Regression Statistics | |||||
Multiple R | 0.8580 | ||||
R Square | 0.7361 | ||||
Adjusted R Square | 0.6881 | ||||
Standard Error | 0.1175 | ||||
Observations | 27 | ||||
ANOVA | |||||
df | SS | MS | F | Significance F | |
Regression | 4 | 0.8467 | 0.2117 | 15.3420 | 3.93E-06 |
Residual | 22 | 0.3035 | 0.0138 | ||
Total | 26 | 1.1502 | |||
Coefficients | Standard Error | t Stat | P-value | ||
Intercept | -3.1753 | 0.6921 | -4.5881 | 0.0001 | |
x1 | 0.1906 | 0.0390 | 4.8838 | 0.0001 | |
x2 | 0.0006 | 0.0002 | 2.8504 | 0.0093 | |
x3 | -0.0010 | 0.0003 | -3.1210 | 0.0050 | |
x4 | 0.0080 | 0.0029 | 2.7332 | 0.0121 |
y = - 3.1753+ 0.1906 x1 + 0.0006 x2 -0.0010 x3 0.0080 x4
x1 = 16, x2 = 600, x3 = 120, and x4 = 20 y = 0.2483
b1)
SUMMARY OUTPUT | ||||
Regression Statistics | ||||
Multiple R | 0.9096 | |||
R Square | 0.8274 | |||
Adjusted R Square | 0.7961 | |||
Standard Error | 0.2866 | |||
Observations | 27 | |||
ANOVA | ||||
df | SS | MS | F | |
Regression | 4 | 8.6633 | 2.1658 | 26.3738 |
Residual | 22 | 1.8066 | 0.0821 | |
Total | 26 | 10.4699 | ||
Coefficients | Standard Error | t Stat | P-value | |
Intercept | -12.1436 | 1.6885 | -7.1920 | 0.0000 |
x1 | 0.6163 | 0.0952 | 6.4736 | 0.0000 |
x2 | 0.0012 | 0.0005 | 2.4204 | 0.0242 |
x3 | -0.0039 | 0.0008 | -4.7789 | 0.0001 |
x4 | 0.0239 | 0.0072 | 3.3281 | 0.0031 |
ln( y^ ) = -12.1436 + 0.6163 x1 + 0.0012 x2 -0.0039 x3 + 0.0239 x4
the predicted price if x1 = 16, x2 ,= 600, x3 = 120, and x4 = 20 ln(y) = -1.5692
y = 0.2082
c) Exponential model because it has a higher R2 value (0.8274 > 0.7361)
Professor Orley Ashenfelter of Princeton University is a pioneer in the field of wine economics. He claims that, contrar...
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