Transcript Stone Arch Village

Fundamentals of Real Estate
Lecture 12
Spring, 2003
Copyright © Joseph A. Petry
www.cba.uiuc.edu/jpetry/Fin_264_sp03
Sales Comparison Approach—Ch 12
Multiple Regression Models:
Coefficients
Random error variable
y = b0 + b1x1+ b2x2 + …+ bkxk + e
Dependent variable
2
Independent variables
Sales Comparison Approach—Ch 12
Multiple Regression Models:
Example: Multiple Regression Results
X Variables
Beta Estimate t-statistic
Constant
1,034.99
14.12
Living Area (sq ft)
64.06
22.32
Age
-1,540.50
4.59
Site size
35,000.92
3.23
R2 = .89; F-Statistic = 69.74;
standard error = 6786.5; Dep. Variable = Value
You want to estimate the value of a house with 2600 sq ft, is 10
years old, and is on .5 acres. Value =
Create a 95% confidence interval around your estimate.
3
Sales Comparison Approach—Ch 12
Rules as indicated in text:
If |t stat| > 2, then variable significant and should be kept
If F stat > 3, model is significant and can be applied;
For 95% confidence interval, use predicted value +/- 2 * Se
(+/- 1 * Se gives 68% CI; +/- 3 * Se gives ~100% CI)
Example: Estimate home value which has 2,600sqft of livable space, is 10 years
old and is on .5 acres. Provide a 95% CI for this prediction.
Price = 1034.99 + 64.06 * LA - 1540.5 * AGE + 35,000.92 * Size
(t-stat) (14.12)
(22.32)
(4.59)
(3.23)
4
Price = 1034.99 + 64.06 * 2600 - 1540.5 * 10 + 35,000.92 * .5
Price = 169,686.45
95% Confidence Interval = 169,686.45 +/- 2 *6786.5;
95% Confidence Interval = [156,113.45, 183,259.45]
Sales Comparison Approach—Ch 12
Example #2: Estimate the value of a home which has 2,200sqft of
livable space, is 5 years old, is on 1.5 acres, has a 2 car
garage. Provide a 95% CI for this prediction.
5
Project Description
You and your team members are interested in investing in some
apartment buildings in Champaign-Urbana. Each team will
have narrowed down their choices to a few investment
opportunities, along with brief information about the current
owners. Your objective in the project is to:
1.
2.
3.
6
Use the market data that your team has already collected to
obtain solid estimates of the income potential of each property.
This should be done relying on a well-specified multiple
regression model.
Analyze each investment opportunity using the tools developed in
this class. To the extent you have expense data available for the
property, you can use it. Otherwise, you will have to depend on
reasonable estimates.
Establish the highest purchase price that you would be willing to
pay for each property. Develop a strategy of which property to
pursue, at what price and for how long. Develop a similar strategy
for the second property.
Regression Analysis—Step by Step
1.
Develop a model that has a sound basis.

Theoretical and practical inputs into model formation
–
–
2.
Gather data for the variables in the model.


Gather data for dependent and independent variables
If data cannot be found for the exact variable, use a “proxy”.
–
3.
4.
Working group of experts for brainstorming session
Literature review on factors influencing variable of interest
You believe sales of your product follows GDP growth, but you
want a model of monthly data, and GDP figures are quarterly.
What do you do?
Draw the scatter diagram to determine whether a linear
model (or other forms) appears to be appropriate.
Estimate the model coefficients and statistics using
statistical computer software.
5. Assess the model fit and usefulness using the
model statistics.


Use the three step process we developed with simple
linear regression.
Do the variables make sense? (significance, signs)
6. Diagnose violations of required conditions. Try to
remedy problems when identified.
7.
Assess the model fit and usefulness using the
model statistics.

8.
Notice the iterative nature of the process.
If the model passes the assessment tests, use it to:



Predict the value of the dependent variables
Provide interval estimates for these predictions
Provide insight into the impact of each independent
variable on the dependent variable.
Remember: Statistics informs judgment, it does
not replace it. Use your common sense when
developing, finalizing and employing a model!
• Example—Motel Profitability
–
–
La Quinta Motor Inns is planning an expansion.
Management wishes to predict which sites are likely
to be profitable.
Step #1: Develop a model with a sound basis
–
Several predictors of profitability which can be
identified include:





Competition
Market awareness
Demand generators
Demographics
Physical quality
Profitability
Competition
Rooms
Number of
hotels/motels
rooms within
3 miles from
the site.
Market
awareness
Nearest
Distance to
the nearest
La Quinta inn.
Demand
Generators
Office
space
Demographics
College
enrollment
Income
Median
household
income.
Physical
Disttown
Distance to
downtown.
At this stage, you should also assign your “a priori” expectations of the sign of each
coefficient for each independent variable. We’ll use this information when we
“assess” the model.
Step #2: Gather Data
–
Data was collected from randomly selected 100
inns that belong to La Quinta, and ran for the
following suggested model:
Margin =b0 + b1Rooms + b2Nearest + b3Office +
+ b5Income + b6Disttwn +
INN
1
2
3
4
5
6
b4College
e
MARGIN ROOMS NEAREST OFFICE COLLEGE INCOME DISTTWN
55.5
3203
0.1
549
8
37
12.1
33.8
2810
1.5
496
17.5
39
0.4
49
2890
1.9
254
20
39
12.2
31.9
3422
1
434
15.5
36
2.7
57.4
2687
3.4
678
15.5
32
7.9
49
3759
1.4
635
19
41
4
Step #3: Draw Scatter Diagrams
Rooms (vertical axis) vs. Margin (horizontal)
4500
4000
y = -27.179x + 4228.4
R2 = 0.2212
3500
3000
2500
2000
1500
20
30
40
50
60
70
Nearest (vertical axis) vs. Margin (horizontal)
5
y = -0.0183x + 2.8274
R2 = 0.0257
4
3
2
1
0
20
30
40
50
60
70
Step #4: Estimate Model
This is the sample regression equation (sometimes called the prediction equation)
SUMMARY OUTPUT
Regression Statistics
Multiple R 0.724611
R Square 0.525062
Adjusted R Square
0.49442
Standard Error
5.512084
Observations
100
MARGIN = 72.455 - 0.008ROOMS -1.646NEAREST
+ 0.02OFFICE +0.212COLLEGE
- 0.413INCOME + 0.225DISTTWN
ANOVA
df
Regression
Residual
Total
SS
MS
F
Significance F
6 3123.832 520.6387 17.13581 3.03E-13
93 2825.626 30.38307
99 5949.458
Coefficients
Standard Error t Stat
Intercept
72.45461 7.893104 9.179483
ROOMS
-0.00762 0.001255 -6.06871
NEAREST -1.64624 0.632837 -2.60136
OFFICE
0.019766
0.00341 5.795594
COLLEGE 0.211783 0.133428 1.587246
INCOME
-0.41312 0.139552 -2.96034
DISTTWN 0.225258 0.178709 1.260475
P-value Lower 95% Upper 95%
1.11E-14 56.78049 88.12874
2.77E-08 -0.01011 -0.00513
0.010803 -2.90292 -0.38955
9.24E-08 0.012993 0.026538
0.115851 -0.05318 0.476744
0.003899 -0.69025
-0.136
0.210651 -0.12962 0.580138
Step #5: Assess the Model
1. R2 (Coefficient of Determination)
1b). Adusted R2
1c). Standard error of the estimate
2.
3.
F-Test for overall validity of the model
T-test for slope
–
–
using b (estimate of the slope)
Partial F-test to verify elimination of some independent
variables
Step #5: Assess the Model
1a. Coefficient of determination
–
The definition is
SSE
R  1
SST
2
–
–
–
From the printout, R2 = 0.5251
52.51% of the variation in the measure of profitability
is explained by the linear regression model formulated
above.
Notice that we are not using SSR/SST. This version of
the formula would still work for now, but it will not work
once we introduce “Adjusted R2” . . .
1b. The “Adjusted” Coefficient of Determination is defined as:
[ SSE /(n  k  1)]
AdjustedR  1 
[ SST /(n  1)]
2
–
–
–
–
As you add additional independent variables to your model, what
happens to SST, SSR, and SSE? What happens to R2? R2?
If all you cared about was a model with a high R2, you might be
tempted to increase the number of independent variables almost
irrespective of the amount of significant explanatory power each
added. Adj R2 penalizes you a small amount for each additional
independent variable you add. The new variable must significantly
contribute to explaining SST, before Adj R2 will go up.
From the printout, Adj R2 ( R2 )= 0.4944 or 49.44%
49.44% of the variation in the measure of profitability is explained by
the linear regression model formulated above after “adjusting for the
degrees of freedom”, or the “number of independent variables”.
1c.
Standard Error of the Estimate
–
–
Recall that the Standard Error is the standard deviation of the data
points around the regression line.
We modify the formula slightly from that when using simple
regression to account for the varying number of independent
variables (k) used in the model:
SSE
se 
n  k 1
–
–
It is reported under “Regression Statistics”, as the “Standard Error”
at the top of your output.
Compare se to the mean value of y


–
From the printout, Standard Error = 5.5121
Calculating the mean value of y we have y  45 .739
Values of se will vary with each regression. While there are no set
ranges for its value, it is a number that will often come in handy.
2. The F-Test for Overall Validity of the Model
•
•
•
In conducting this test, we are posing the question:
Is there at least one independent variable linearly
related to the dependent variable?
To answer the question, we test the hypothesis:
H0: b1 = b2 = … = bk = 0
H1: At least one bi is not equal to zero.
If at least one bi is not equal to zero, the model is
valid.


To test these hypotheses we perform an
analysis of variance procedure.
The F test
–
Construct the F statistic
MSR=SSR/k
SST = SSR + SSE.
Large F results from a large SSR.
Then, much of the variation in y is
explained –by Rejection
the regression region
model.
The null hypothesis should
be rejected; thus, the model is valid.
MSR
F
MSE
MSE=SSE/(n-k-1)
F>Fa,k,n-k-1
• Example—Motel Profitability
Excel provides the following ANOVA results
MSR/MSE
ANOVA
df
Regression
Residual
Total
SSE
SSR
SS
MS
F
Significance F
6 3123.832 520.6387 17.13581 3.03382E-13
93 2825.626 30.38307
99 5949.458
MSE
MSR
ANOVA
df
Regression
Residual
Total
SS
MS
F
Significance F
6 3123.832 520.6387 17.13581 3.03382E-13
93 2825.626 30.38307
99 5949.458
Fa,k,n-k-1 = F0.05,6,100-6-1=2.17 Also, the p-value (Significance F) = 3.03382(10)-13
F = 17.14 > 2.17
Clearly, a = 0.05>3.03382(10)-13, and the null hypothesis
is rejected.
Conclusion: There is sufficient evidence to reject
the null hypothesis in favor of the alternative hypothesis.
At least one of the bi is not equal to zero. Thus, at least
one independent variable is linearly related to y.
This linear regression model is valid
3a.
Testing the coefficients
–
The hypothesis for each bi
•
H0: bi = 0
H1: bi = 0
Example—Motel Profitability
Coefficients
Standard Error t Stat
Intercept
72.45461 7.893104 9.179483
ROOMS
-0.00762 0.001255 -6.06871
NEAREST -1.64624 0.632837 -2.60136
OFFICE
0.019766 0.00341 5.795594
COLLEGE 0.211783 0.133428 1.587246
INCOME
-0.41312 0.139552 -2.96034
DISTTWN 0.225258 0.178709 1.260475
Test statistic
b b
t i i
sb i
P-value
1.11E-14
2.77E-08
0.010803
9.24E-08
0.115851
0.003899
0.210651
d.f. = n - k -1
Lower 95% Upper 95%
56.78048735 88.12874
-0.010110582 -0.00513
-2.902924523 -0.38955
0.012993085 0.026538
-0.053178229 0.476744
-0.690245235
-0.136
-0.12962198 0.580138
3b.
Do the Variables Make Sense?
–
–
–

When you establish which variables you want to use, you
should also establish your “a priori” assumptions regarding
the expected sign of the slope coefficients.
You do this prior to obtaining your actual model results so
the actual numbers do not influence your expectations.
By establishing these expectations, you are more able to
identify surprises in your results. These surprises may lead
you to additional insight into your model, or may lead you to
question your results. Either is useful. Retrieve your
expectations from an earlier slide, and place them here.
Example—Motel Profitability
Margin =b0 + b1Rooms + b2Nearest + b3Office + b4College + b5Income + b6Disttwn
–
b 0  72.5 This is the intercept, the value of y when
all the variables take the value zero. Since the
data range of all the independent variables do not
cover the value zero, do not interpret the intercept.
– b1  .0076
In this model, for each additional 1000
rooms within 3 mile of the La Quinta inn, the
operating margin decreases on the average by
7.6% (assuming the other variables are held
constant).
–
–
–
b 2  1.65
In this model, for each additional mile that
the nearest competitor is to La Quinta inn, the average
operating margin decreases by 1.65%. Sensible???
b 3  .02 For each additional 1000 sq-ft of office space,
the average increase in operating margin will be .02%.
b 4  .21
For additional thousand students MARGIN
increases by .21%.
–
b 5  .41 For additional $1000 increase in median
household income, MARGIN decreases by .41% ???
–
b 6  .23
For each additional mile to the downtown
center, MARGIN increases by .23% on the average???
–
Based on the t-tests, one should consider getting
rid of both “College” and “Disttwn”.


–
The sign on “Disttwn” is also a bit unexpected as well—
though if you try hard you could justify it. These two
indications, reinforce one-another. Let’s get rid of it.
The “College” variable sign is what you would expect,
and it’s p-value, while not below 5%, is not that high.
Let’s keep this for now, and see what happens when we
eliminate “Disttwn”.
While Assumption Violations is officially a separate
step, it is usually best to be checking your
assumptions at this stage as well.

Recall how dramatically the model changed when we
had autocorrelation. Recall that Serious Multicollinearity
could also be leading me to get rid of some variables that
we might really want to keep.
SUMMARY OUTPUT
Regression Statistics
Multiple R
0.718990854
R Square
0.516947848
Adjusted R Square 0.491253584
Standard Error
5.529320727
Observations
100
Notice that when we get rid of “Disttwn”, both R2
AND Adj R2 went down, but the F stat went up. This
is where the “art” comes in. Despite the decline in
Adj R2, we will eliminate “Disttwn” on the basis of the
size of the p-value of the t-test, the sign being wrong
and the direction of the change in the F stat. You
could successfully argue to keep it as well based on
Adj R2. Notice the p-value on “College”.
ANOVA
df
Regression
Residual
Total
Intercept
ROOMS
NEAREST
OFFICE
COLLEGE
INCOME
5
94
99
SS
MS
3075.559456 615.1118912
2873.898444 30.5733877
5949.4579
Coefficients Standard Error
75.13707499
7.624565098
-0.007742161
0.001255304
-1.586922918
0.633058347
0.019576011
0.00341778
0.196384877
0.133283011
-0.421411017
0.139833268
t Stat
9.854604692
-6.167558795
-2.506756172
5.727697298
1.473442678
-3.013667795
F
Significance F
20.11919311
1.3555E-13
P-value
3.74364E-16
1.7296E-08
0.013901469
1.21629E-07
0.14397261
0.003317336
Lower 95% Upper 95%
59.99832996 90.27582
-0.010234595 -0.00525
-2.843874466 -0.329971
0.012789931 0.026362
-0.068251531 0.461021
-0.699053108 -0.143769
SUMMARY OUTPUT
Regression Statistics
0.711190008
Multiple R
0.505791227
R Square
Adjusted R Square 0.484982437
5.563295396
Standard Error
100
Observations
When we got rid of “Disttwn”, the p-value for College
actually increased, and now isn’t all that close to
5%. Consequently, we’ll get rid of it. Once we do, we
have a similar circumstance as last time, regarding
R2, adj R2 and the F stat. This could go either way
as well. In our case, we’ll keep “College” out, and do
a Partial F-test, and see what that suggests we do
about it.
ANOVA
df
Regression
Residual
Total
Intercept
ROOMS
NEAREST
OFFICE
INCOME
4
95
99
MS
SS
3009.183612 752.2959031
2940.274288 30.95025566
5949.4579
Coefficients Standard Error
7.429077126
77.93849422
0.00126034
-0.007862522
0.635316269
-1.653650492
0.003438713
0.019607492
0.13988637
-0.399387121
t Stat
10.49100621
-6.238412847
-2.602877611
5.701984661
-2.855082452
Significance F
F
7.22973E-14
24.30661354
P-value
1.48143E-17
1.22182E-08
0.010726099
1.33212E-07
0.005283347
Lower 95% Upper 95%
63.18992196 92.68707
-0.010364612 -0.00536
-2.914911849 -0.392389
0.012780787 0.026434
-0.677096479 -0.121678
3c. The Partial F-test.
–
–
How does one decide how many variables to
keep in your final model? Do you keep all the
variables, some of them?
While there is some “art” to this process as well,
we will use the following process.
1.
First, consider your individual t-test results.
•
•
•
2.
Which variables should you keep on this basis?
Are there any variables that officially should be
eliminated, but are close to having a small enough pvalue to be retained?
Are there any variables you believe strongly “must” be in
the model irrespective of the results of the t-test?
Once you have made your decisions, then conduct the
“Partial F-test” to verify your results.
H0: b1 = b2 = … = bi = 0
H1: At least one bi is not equal to zero.
Reject H0 if
(SSRf  SSRr ) / kd
 F( k ) d , ( nk 1) f
MSEf
Where:
bis refer only to those variables which were eliminated from the original
regression;
SSRf is from the full equation; SSRr is from the reduced equation;
MSEf is from the full equation; Kd is the number of variables eliminated.
The test statistic is determined by the difference in SSR (full model) vs. SSR
(reduced model). If there is a large difference, some of the variables you
eliminated have significant explanatory power. If this is the case, you will
reject H0, conclude some coefficients from the variables you eliminated are
non-zero, and use the “full model”.
This test will always be a one-sided upper tail test by its nature.
• Example—Motel Profitability

The ANOVA results for the reduced model are:
df
Regression
Residual
Total
SS
MS
F
Significance F
4 3009.184 752.2959 24.30661
7.23E-14
95 2940.274 30.95026
99 5949.458
The test statistic for the Partial F-test:
[(3123.83-3009.184)/2]/30.95=57.323/30.95=1.852
Fa,k,n-k-1 = F0.05,2,100-6-1=3.095; F = 1.852 < 3.1; therefore, DNR H0
Conclusion: There is insufficient evidence to reject the null hypothesis in favor of
the alternative hypothesis. The independent variables eliminated from the
regression do not appear to be different from 0, and hence have no explanatory
power.
The reduced model appears to be the most appropriate model in this case.
• Example
Assume you have conducted two regressions using the same
data. The first regression on the “full model” had 9 independent
variables, and a sample size of 200. You then run a “reduced
model” after eliminating 4 of the independent variables that
appeared insignificant on the basis of t-tests.
Data for Full Model
Data for Reduced Model
SSR = 95,532
SSR = 7,978
MSE = 654.
MSE = 13,431
Conduct a partial F-test. F4,190= 2.41918485.
Conduct the same test, this time assuming the SSR from your
reduced model was 92,300.
Step #6: Diagnose Violations of Required Conditions
–
–
–
We already did this in concert with Step #5, and that is
the way you really should do it. You cannot effectively
assess the model, without having considered whether
the assumptions have been violated.
We separate them into steps only because both are
so critical to constructing a useful regression model.
Having to combine these critical steps is another
manner in which the “art” of regression analysis
becomes obvious.
Step #7: Assess the Model
We now have our final model. You should be able to
do the assessment on your own at this stage.
SUMMARY OUTPUT
Regression Statistics
Multiple R
0.711190008
R Square
0.505791227
Adjusted R Square 0.484982437
Standard Error
5.563295396
Observations
100
ANOVA
df
Regression
Residual
Total
Intercept
ROOMS
NEAREST
OFFICE
INCOME
4
95
99
SS
MS
3009.183612 752.2959031
2940.274288 30.95025566
5949.4579
Coefficients Standard Error
77.93849422
7.429077126
-0.007862522
0.00126034
-1.653650492
0.635316269
0.019607492
0.003438713
-0.399387121
0.13988637
t Stat
10.49100621
-6.238412847
-2.602877611
5.701984661
-2.855082452
F
Significance F
24.30661354
7.22973E-14
P-value
1.48143E-17
1.22182E-08
0.010726099
1.33212E-07
0.005283347
Lower 95% Upper 95%
63.18992196 92.68707
-0.010364612 -0.00536
-2.914911849 -0.392389
0.012780787 0.026434
-0.677096479 -0.121678
Step #8: Use the Model
Example—Motel Profitability

–
Use the model to predict the profit margin of three possible
locations.
Characteristics
Ann Arbor
Rooms
2672
Competitor Distance 1.3
Office Space (‘000s) 952
Students (‘000s)
42
Income (‘000s)
35
Dist to Downtown
3.4
Predicted Margin
Bloomington
2,500
1.2
604
21
37
4.5
Champaign
2,300
.5
1,430
45
33.5
1.4
What are your expectations for profit margins in each
location?
Where should we recommend that to locate the next motel?
What seem to be the deciding factors in this case?
Reviewing Steps 1-8 of the modeling process.

Example—Vacation Homes
–
–
–
–
A developer who specializes in summer cottage
properties is looking at a lakeside tract of land for
possible development.
She wants to estimate the selling price for the
individual lots.
She knows from experience that sale price
depends upon lot size, number of mature trees,
and distance to the lake.
Establish your “a priori” expectations of the signs
of the coefficients:
–
–
–
Lot size (data in hundreds; 20 entered to represent 2,000
sq ft)
Number of mature trees
Distance to the lake (data in tens; 20 entered represents
200 ft)
SUMMARY OUTPUT
Regression Statistics
Multiple R
0.4924143
R Square
0.2424719
Adjusted R Square 0.20189
Standard Error
40.243529
Observations
60
ANOVA
df
Regression
Residual
Total
Intercept
Lot size
Trees
Distance
3
56
59
SS
MS
F
Significance F
29029.71625 9676.57208 5.974883 0.001315371
90694.33308 1619.54166
119724.0493
Coefficients Standard Error
t Stat
P-value
Lower 95% Upper 95%
51.391216
23.51650385 2.18532554 0.033064 4.282029664 98.5004
0.6999045
0.558855319 1.25238937 0.215633 -0.419616528 1.819425
0.6788131
0.229306132 2.96029204
0.0045 0.219458042 1.138168
-0.3783608
0.195236549 -1.9379609 0.057676 -0.769466342 0.012745
SUMMARY OUTPUT
Regression Statistics
Multiple R
0.389063
R Square
0.15137
Adjusted R Square
0.136738
Standard Error
41.8539
Observations
60
ANOVA
df
Regression
Residual
Total
Intercept
Trees
SS
MS
F
Significance F
1 18122.63 18122.63 10.34545 0.002124116
58 101601.4 1751.749
59
119724
Coefficients
Standard Error t Stat
P-value
56.26122 10.90103 5.161092 3.13E-06
0.727649 0.226229 3.216434 0.002124
Lower 95% Upper 95%
34.44044988 78.08198
0.274803888 1.180494
Regression Statistics
Multiple R
0.470377
R Square
0.221255
Adjusted R Square
0.19393
Standard Error
40.44371
Observations
60
ANOVA
df
Regression
Residual
Total
Intercept
Trees
Distance
SS
MS
F
Significance F
2 26489.5 13244.75 8.097328 0.000803103
57 93234.55 1635.694
59
119724
Coefficients
Standard Error t Stat
P-value
Lower 95% Upper 95%
75.52475 13.54641 5.575259 7.06E-07 48.39851987 102.651
0.767051 0.219299 3.497737 0.000917 0.327911764 1.206191
-0.432673 0.191306 -2.261677 0.027549 -0.815756848 -0.049589
Price
Price
Trees
Distance
Lot size
Frequency
15
Trees
1
0.389063
1
-0.232614 0.079442
1
0.303525 0.285682 -0.189499
Histogram of Standardized Residuals
10
5
0
-1.7
-1.1
-0.6
0.0
0.5
1.1
1.6
More
Standard Residuals Vs. Predicted
3
2
1
0
-1
-2
40
Distance
60
80
100
120
140
Lot size
1
What is the standard error of the estimate?
Interpret its value.
2.
What is the coefficient of determination? What does
this statistic tell you?
3.
What is the coefficient of determination, adjusted
for degrees of freedom? Why does this value differ
from the coefficient of determination? What does
this tell you about the model?
==========================================
1.
Test the overall validity of the model. What does the
p-value of the test statistic tell you?
2.
Interpret each of the coefficients. How do the signs
compare to your “a priori” assumptions?
3.
Test to determine whether each of the independent
variables is linearly related to the price of the lot.
1.
1.
2.
3.
4.
What output should have been provided, but wasn’t?
What output was provided, that probably should not have
been?
What output might be provided if the data was different, but
wasn’t necessary to provide in this case?
Are any of the assumptions violated or other danger signals
present?
============================================
1.
2.
3.
Which model should you most likely use to make predictions?
Which would you rather own, a lot with 20 trees, 250 feet from
the water, 2,500 square feet in size; or a lot with 16 trees, on
the water, with 1,800 square feet in size.
Which of the two lots should you buy, if you are interested in
resale value as your principal purchasing criteria, if you could
buy the lots for $77,000 and $87,000 respectively? Why?