Question

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.)

x3 160 80 130 110 187 187 10.3684 0.6348 10.4458 10.2211 10.1797 10.6584 10.1388 | 1.0000 10.3310 10.1685 10.3059 10.1063 | 0.4726 10.1913 0.1054 0.1167 0.4044 0.2724 0.1014 0.1561 10.1108 0.3007 0.2534 0.1070 0.2704 0.2145 10.1359 x1 17.1167 16.7333 17.1500 16.1333 16.4167 17.4833 16.4167 17.3333 16.3000 15.7167 | 17.2667 15.3667 16.5333 16.2333 16.2000 | 16.5500 16.6667 16.7667 14.9833 17.0667 16.3000 16.9500 17.6500 15.5833 15.167 16.1667 16.0000 600 690 502 420 582 485 763 830 697 608 402 602 819 714 290 38 52 155 96 267 86 118 292 244 89 112 158 123 610 575 622 551 536 376 574 572 418 821 184 171 247 87 763 51 717 578 122 74

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, x_{2 ,}= 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)

Answer 1

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, x_{2 ,}= 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)