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REYEM AFFAIR
REGRESSION CASEQUANTITATIVE METHODS II
TO
PROF. ARNAB BASU
ONOCTOBER 21, 2011
BY
INDIAN INSTITUTE OF MANAGEMENT, BANGALORE
Table of Contents
S.No Particulars Pages1. Executive Summary 3-42. Understanding of the Problem 43. Model Description 5-13
Model 1Prediction interval Vs Confidence IntervalStep wise Regression: A closer lookTest of Model: Analysis of Results
5-8678
Model 2Test of Model: Analysis of Results
9-1311-13
Other Models 134. Conclusions and Recommendations 145. Appendix
1. Variables Entered/Removed2. Model Summary3. ANOVA4. Coefficients5. Residual Statistics
15
Executive SummaryReyem Affiar has recently found the below described condominium in Mid-Cambridge that he wants to
purchase.
Street Address : 236 Ellery Street
Last Price : $169000
Area & Area Code : M/9
Bed : 2
Bath : 1
Rooms : 5
Interior : 1040
Condo : $175
Tax : $1121
RC : 1(Restrictions on monthly rent that owner may charge)
Even though Affiar is monetarily capable of paying the asking price of $169000, generally negotiations
from buyer’s agent keeps the selling price lower than the last asking price. Given the above information,
based on the data that Reyem Affiar has on condominiums sold in Cambridge the past five years, we
need to help Reyem Affiar to decide on a fair offer price.
Solution Approach
An estimate for selling price of the above condominium needs to be made. Hence selling price is clearly
the dependent variable ‘Y’ for the regression model. Clearly first date, close date and number of days
between the two (Days) cannot be part of the independent variable set since we do not have these
information for the 236 Ellery Steet Condominium yet (since the sale has not taken place yet). Further
the condominium of interest lies in area M (9), hence one could possibly analyze only the data on the
111 condominiums from the same area and ignore the rest. On the other hand, if we can set up
independent dummy variables for the area/area codes, these can be incorporated into our regression
model and then we will have a bigger sample of 456 data-points to make a better and more accurate
prediction for Affiar. This will be explained in detail in the model description. Stepwise regression in
SPSS has been adopted for variable selection. This method, being a combination of forward selection
and backward elimination techniques for variable selection, avoids the errors in regression model that
can be committed due to multi-collinearity.
Figure 11.45 from Pg 571
Understanding of the Problem
Selection of independent variables is the key to arriving at a good regression model. On first look at the
given data, one can clearly see that the possible independent variables that may be affecting the selling
price could be first price, last price, number of days between first and last date, location (Area), number
of bedrooms, number of bathrooms, number of rooms, interior space, condominium taxes, yearly
property tax and rent control. But we have assumed that the given asking price of $169000 for the
Ellery Street condominium is the last price since the transaction could possibly happen on the next day
(May 4, 1994). This means we don’t have information on the first price for the Ellery Street
condominium, hence we remove first price from our possible independent variable list. As stated before
in section 1.1, we cannot have number of days between first and last date as an independent variable
either since the sale of condominium has not happened and we don’t have information on the first date
the condominium was put on sale. Finally, we can intuitively see that there will be a positive correlation
between interior space and number of rooms, bathrooms and bedrooms. Since interior space can be
representative of all, to avoid the issue of multi-collinearity, interior space can very well act as a good
proxy in our regression model for number of rooms, bathrooms and bedrooms. We will also show this
through the output generated in the model description section. Further, one can also expect last price
and interior space to have positive coefficients while condominium taxes, property taxes and RC to have
negative coefficients. Effect of the other dummy variables for area/area codes need to be explored by
running the regression model.
We will start with a basic regression model, then will check the model for normality, linearity and in case
it does not pass the test we will transform the variables using Log, Square root or inverse.We will rerun
the regression model with transformations and try to find the outliers. If any outlier is found we will
remove that and then again run the regression model. Then we will check for Residuals normality and
homoscedasticty.If there is at least 2% increase in the R square value as compared to the baseline
regession then we will go with the regression model with transformed variable else we will go with
baseline model and mention the cautions for non normality etc.
Model DescriptionModel 1
Baseline regression model
The model(Appendix) can be described as follows (Exhibit A):
Sale Price = 0.333*Last Price + 35.947*Tax + 44.967*Interior + 105.108*Condo + 10992.327*RC +
12290.704*A2 + 29804.817*A5 – 27984.595*A12 – 12447.291*A16 - 15967.736
Where A2, A5, A12 and A16 are the dummy variables associated with areas Avon Hill, East Cambridge,
Porter Square and West Cambridge respectively. They will take values of 1 or 0 depending on whether
we are to predict the price of a condominium in that area. For 236 Ellery Street Apartment, we have
Sale Price = 0.333*169000 + 35.947*1121 + 44.967*1040 + 105.108*175 + 10992.327*1 –
15967.736 = 156757.758
95% prediction interval for the Selling price of 236 Ellery Street Condominium is given by:
= 156757.758 ±t[0.025,(456-10)](30268.701252 + 9.162 * 108)0.5
= 156757.758 ± 1.9653 *(30268.701252 + 9.162 * 108)0.5
= 156757.758 ± 84127.57
= {72630.188, 240885.328}
The standard error and MSE are taken from the regression output table (Appendix).
Now, a 95% Confidence Interval for the Selling Price (conditional mean) of 236 Ellery Street
Condominium would be given by:
= 156757.758 ±t[0.025,(456-10)](4021)
= 156757.758 ± 1.9653 *(4021)
= 156757.758 ±7902.471
= {148855.29, 164660.23}
The standard error of mean predicted value is taken from the Residual Statistics table (Appendix).
Exhibit 1: Regression Model Coefficients
Coefficientsa
Model
Unstandardized
Coefficients
Standardize
d
Coefficients
t Sig.
95% Confidence
Interval for B
Collinearity
Statistics
B Std. Error Beta
Lower
Bound
Upper
Bound
Toleranc
e VIF
9 (Constant
)
-
15967.7365913.780 -2.700 .007 -27590.071 -4345.402
LastPrice .333 .023 .403 14.763 .000 .289 .377 .335 2.988
Tax 35.947 3.136 .364 11.462 .000 29.783 42.110 .248 4.035
Interior 44.967 5.554 .173 8.097 .000 34.052 55.882 .549 1.821
Condo 105.108 21.268 .127 4.942 .000 63.311 146.906 .380 2.629
A12 -
27984.5958366.791 -.056 -3.345 .001 -44427.826 -11541.364 .902 1.108
A5 29804.817 6552.903 .084 4.548 .000 16926.416 42683.218 .738 1.354
RC 10992.327 3445.556 .059 3.190 .002 4220.785 17763.869 .726 1.378
A16 -
12447.2915480.634 -.037 -2.271 .024 -23218.366 -1676.216 .944 1.059
A2 12290.704 5486.742 .036 2.240 .026 1507.625 23073.784 .967 1.034
a. Dependent Variable: SalePrice
Step-wise regression: A closer look
Given the possible set of 23 independent variables (Last Price, Bed, Bath, Rooms, Interior, Condo, Tax,
RC, A1,A2, A3, A4, A5, A6, A7, A8, A10, A11, A12, A13, A14, A15, A16), the algorithm starts by finding the most
significant single-variable regression model. So Last Price with the highest F-value and hence a p-value <
pin enters the regression model (note pin = 0.05). Now the other 22 variables left out of the model are
checked via a partial F-test, and the most significant variable, Tax, is now added to the model.Now the
original variable Last Price is reevaluated to see if it meets the preset significance standard of p-value <
pout(note pin = 0.10). Since it meets this criterion, the variable is retained in the model. Now again the
other 21 variables outside the model are checked via a partial F-test, and the most significant variable,
now Interior, enters the model. All variables in the model, namely Last Price and Tax are now checked
again for staying significance. The procedure continues until there are no variables outside that should
be added to the model and no variables inside the model that should be out. On 9 th iteration, this
happens for Model 1 as shown in Appendix. To illustrate how the issue of multi-collinearity is inherently
taken of in this Step-wise regression technique, a regression analysis was done between rooms and
interior variables and it was found that these two were highly correlated (Appendix). Obviously, the
step-wise regression took the more significant variable “Interior” in the final regression model
eliminating the lesser significant highly correlated “Rooms” variable from the final regression model.
Let us check if the model’s regression assumptions are satisfied through Residual Analysis:
Complete stepwise multiple regression analysis: sample size
Since the number of cases is 456 and the number of independent variable is 9 the ratio is 50.66 which
passes the criteria of 50 is to 1.
Complete stepwise multiple regression analysis: assumption of normality
Sales price
As we can see from above that dependent does not pass the normality test
So Transform Salesprice to Log (salesprice) so that it follows normal distribution
Tax
It does not follow normal distribution as we can see below
After transformation to Sqrt(Tax) it follows normal distribution as shown below
Interior
The variable does not follow normal distribution as shown below
After transformation to Log (Interior) it follows normal distribution
Condo
It does not follow normal distribution as shown below
After transformation to Log(Condo)
Last Price
It also does not follow normal distribution
After transformation to 1/lastprice(Inverse) it follows normal distribution
Since all the other variables are ordinal we are not testing for normality
Test for Linearity
As we have transformed the independent variables test for linearity is not required.
TEST for OUTLIERS
After transformation for detecting the outliers we ran the regression(EXHIBIT B) again
with transformed variables and checked for outliers. Below was the result.
Casewise Diagnosticsa
Case
Number Std. Residual
LOG_SALEPRIC
E Predicted Value Residual
59 4.446 13.68 13.1891 .49288
217 3.660 11.03 10.6291 .40575
305 3.420 13.33 12.9502 .37916
306 3.162 13.35 12.9950 .35051
360 -8.181 11.73 12.6349 -.90689
408 -3.276 11.17 11.5336 -.36320
a. Dependent Variable: LOG_SALEPRICE
The above case number Std.Residual was outside + 3 and – 3 and hence were oitliers.
Deleted the above case numbers and rerun the regression again.
Complete stepwise multiple regression analysis: assumption of independence of errors
Also Durbin Watson value is 1.649 which is between 1.5 and 2.5
Step wise regression has taken care of multicollinearity which is tested at each stage with a Pin =
0.05 and Pout = 0.10. and it has eliminated Beds,Rooms,Bath which were collinear.
R square value = 0.949 and adjusted 0.948 which is more than 2 % higher than baseline
regression R square value of 0.889.Hence we will go with model with transformed variables
and outliers removed. Five outliers were removed based on the case diagnostic.
Transformation to Log has a base e that is natural log.
Modified Regression Model (Exhibit C)
The model (Appendix) can be described as follows (Exhibit C):
Coefficientsa
Model
Unstandardized
Coefficients
Standardize
d
Coefficients
t Sig.
95% Confidence
Interval for B
Collinearity
Statistics
B
Std.
Error Beta
Lower
Bound
Upper
Bound
Toleranc
e VIF
5 (Constant)12.059 .178
67.59
8
.00
011.708 12.409
INVERSE_LASTRIC
E
-
133487.06
9
3284.28
2-.808
-
40.64
4
.00
0
-
139941.77
9
-
127032.36
0
.2943.40
3
SQRT_TAX.004 .001 .108 5.711
.00
0.003 .006 .324
3.08
9
A5.119 .019 .074 6.363
.00
0.082 .155 .869
1.15
1
LOG_CONDO.034 .011 .042 3.032
.00
3.012 .057 .596
1.67
7
LOG_INTERIOR.062 .021 .052 2.904
.00
4.020 .104 .368
2.72
0
a. Dependent Variable: LOG_SALEPRICE
Log(Sale Price) = -133487.069 (1 / Lastprice) + .004 * SQRT(Tax) + .119A5 + .034 *Log(Condo) +
.062 * Log(Interior)+ 12.059
Where A5 are the dummy variables associated with East Cambridge, This will take values of 1 or 0
depending on whether we are to predict the price of a condominium in that area. For 236 Ellery Street
Apartment, we have
Log(Sale Price) = -133487.069 (1 / 169000) + .004 * SQRT(1121) + .119* 0 + .034 *Log(175) +
.062 * Log(1040)+ 12.059 = 164288.0015
95% prediction interval for the Selling price of 236 Ellery Street Condominium is given by:
= 164288.0015±t[0.025,(450-10)]( .091892 + .008)0.5
= 164288.0015± 1.9653 *( .091892 + .008)0.5
= 164288.0015± 0.5844
= {164287.4172, 164288.586}
The standard error and MSE are taken from the regression output table (Appendix).
Now, a 95% Confidence Interval for the Selling Price (conditional mean) of 236 Ellery Street
Condominium would be given by:
= 164288.0015 ±t[0.025,(450-10)]( .010)
= 164288.0015± 1.9653 *(.010)
= 164288.0015±.019653
= {164287.9818, 164288.0212}
The standard error of mean predicted value is taken from the Residual Statistics table (Appendix).
Prediction interval Vs Confidence Interval
We have calculated the prediction interval and confidence interval for E(Sale Price) for the Ellery street
condominium for the given input independent variables (Section 1). While the predicted value and the
estimate of the mean value of Y(Sale Price here) are equal, the prediction interval is wider than a
confidence interval for E(Y) using the same confidence level. There is more uncertainty about the
predicted value than there is about the average value of Y given the values of X i. Based on the
confidence interval, the recommendation for Affiar would be to not bid more than the upper limit value
of $164288.0212 since he can be confident to a level of 97.5% (100% – 5%/2) that the final selling price
(mean) of the condominium would be below this number. So 164288.0212 is the maximum that he
should bid on the condominium. If he were to be more conservative in his bid, then he can go by the
prediction interval. Since the upper limit of the prediction interval $164288 is lower than the asking
price of $169000, his bid should be 164288 in this case. The maximum he can afford to bid for the house
with a 95% confidence level would be $164288.
Residuals do follow normal distribution as shown below
Lastly homoscedasticity can be seen from the residual scatter plot where the residuals are scattered
around the mean 0 in a random fashion with no observable pattern or heteroscedasticity
Test of Model: Analysis of Results
Significance of model: From Appendix, ANOVA table shows that that F-value for model 2 is 1632 with a
significant p-value of 0. Since p-value < 0.05, we reject the null hypothesis (β1= ……..= β11 = 0) and hence
there is atleast one βi that is significant. We will look at the coefficients table to ensure the coefficients
are significantly different from zero. As we can see from the coefficients table for Model 1, the p-values
for coefficients are lesser than 0.05 (alpha value). Hence we reject the null-hypothesis for each βi(i.e. βi
= 0) and thus the coefficients are significant. Finally we look at the Adjusted-R2 (since this accounts for
the increase in R2 due to an increase in number of independent variables) values for goodness-of-fit test.
A high Adjusted R2 value of 0.948 in this case (Appendix) suggests that 94.8% of the variation in Sale
Price is explained by the regression model.
Model 2:
In Model 1, we have clearly accounted for the areas/area codes of condominiums by starting with the 15
dummy variables for our step-wise regression analysis. One could very well argue that condominiums
outside of Mid-Cambridge should not be considered for analysis. Hence step-wise regression was run
with only the 111 data points from Mid-Cambridge condominiums. The step-wise regression was
started with the input independent variables including Last Price, Bed, Bath, Rooms, Interior, Condo, Tax
and RC. But Last Price and RC were the only independent variables that seem to have a significant
impact on the Selling Price. The step-wise regression with a P in = 0.05 and Pout = 0.10 was carried out, as
we can see from Appendix, Last Price and RC were the only independent variables with a significant
impact (based on step-wise partial F-test) on Selling Price. The model can be summarized as below:
Selling Price = 0.96 * Last Price + 1935.903 * RC – 2181.178
For the Ellery Street condominium, we have:
Selling Price = 0.96 * 169000 + 1935.903 * 1 – 2181.178
= $161,994.725
Similar to model 1, 95% prediction interval for the Selling price of 236 Ellery Street Condominium is
given by :
= 161,994.725±t[0.025,(111-3)](4422.9452 + 1.956 * 107)0.5
= 161,994.725± 1.98217 *(4422.9452 + 1.956 * 107)0.5
= 161,994.725 ± 12398.064
= {149596.661, 174392.7892}
The standard error and MSE are taken from the regression output table (Appendix).
Now, a 95% Confidence Interval for the Selling Price (conditional mean) of 236 Ellery Street
Condominium would be given by:
= 161,994.725±t[0.025,(111-3)](698.994)
= 161,994.725 ± 1.98217 *(698.994)
= 161,994.725±1385.525
= {160609.2,163380.25}
The standard error of mean predicted value is taken from the Residual Statistics table (Appendix).
As explained for model 1, there is more uncertainty about the predicted value than there is about the
average value of Y given the values of X i. Based on the confidence interval, the recommendation for
Affiar would be to not bid more than the upper limit value of $163,380 since he can be confident to a
level of 97.5% (100% – 5%/2) that the final selling price (mean) of the condominium would be below this
number. So $163,380 is the maximum that he should bid on the condominium. If he were to be more
conservative in his bid, then he can go by the prediction interval. Since the upper limit of the prediction
interval $174,393 is greater than the asking price of $169000, his bid should be $169,000 in this case.
The maximum he can afford to bid for the house with a 95% confidence level would be $174,393.
Coefficientsa
Model
Unstandardized Coefficients
Standardized
Coefficients
t Sig.B Std. Error Beta
1 (Constant) -544.824 1357.461 -.401 .689
LastPrice .958 .008 .996 123.128 .000
2 (Constant) -2181.178 1541.383 -1.415 .160
LastPrice .960 .008 .998 124.529 .000
RC 1935.903 909.479 .017 2.129 .036
a. Dependent Variable: SalePrice
Let us check if the model’s regression assumptions are satisfied through Residual Analysis:
From the normality histogram for residuals shown in the figure below, it is clear that the normality
assumption is satisfied since the residuals (standardized) seem to be normally distributed. The normal
P-P graph also confirms the same. Lastly homoscedasticity can be seen from the residual scatter plot
where the residuals are scattered around the mean 0 in a random fashion with no observable pattern or
heteroscedasticity. Finally the independence assumption between the independent variables is
inherently taken care of in the step-wise regression technique which checks for multi-collinearity after
each stage (as shown in Figure 1) with a P in = 0.05 and Pout = 0.10. Hence the algorithm automatically
kicks out of the model variables that are correlated to each other and keeps only the most significant
independent variables inside the model. The individual residual plots of residual error Vs each
independent variable is shown in Appendix.
The step-wise regression method adopted works the same way as it was explained for model-1. Here
only 2 iterations were required to arrive at the final model as shown in Appendix.
Test of Model: Analysis of Results
Significance of model: From Appendix, ANOVA table shows that that F-value for model 2 is 7828 with a
significant p-value of 0. Since p-value < 0.05, we reject the null hypothesis ( β1= β2= β3 = 0) and hence
there is at least one βi that is significant. We will look at the coefficients table to ensure the coefficients
are significantly different from zero. As we can see from the coefficients table for Model 2, the p-values
for coefficients are lesser than 0.05 (alpha value). Hence we reject the null-hypothesis for each βi(i.e. βi
= 0) and thus the coefficients are significant. Finally we look at the Adjusted-R2 (since this accounts for
the increase in R2 due to an increase in number of independent variables) values for goodness-of-fit test.
A high Adjusted R2 value of 0.993 in this case (Appendix) suggests that 99.3% of the variation in Sale
Price is explained by the regression model.
Other Models:
In addition to the above 2 best-fit models, a number of other regression models with different
combinations of input independent variables were tried. For instance, areas based on location (with the
help of the map provided) were grouped to form lesser number of dummy variables (e.g., grouping
Agassiz, Harvard Square and Radcliffe). Multiple such combinations were formed to see how area can
be best-fit into the model. ‘Rooms’ was tried as proxy for interior (due to their high correlation as seen
in Appendix). Best fit test for each model based on R2 values, significance of coefficients, residual plots
was conducted and the best 2 models have been presented in the case solution. Also in each model, the
given price for the Ellery street condominium has been assumed as the Last Price as stated before.
Conclusions and recommendationsTwo regression models were presented to fit the given data in order to predict the sale price for the 236
Ellery Street condominium. The summary of the offer price that Affiar should be making on the
condominium based on the two models is shown in the table below:
Mean Selling
Price ($)Prediction Interval ($) Confidence Interval ($)
Recommend
ed bid price
($)
Max.
Conservativ
e bid price
($)
Model
1
164288.0015 {164287.4172,
164288.586}
{164287.9818,
164288.0212}164,288.02
12
164288.58
6
Model
2
161,994.725 {149596.661,174392.789} {160609.2,163380.25} 163,380 174,393
Comparing the Adjusted R2 values of the two models, we see that Model 2 is able to explain 99.3% of
variation in Sale price against Model 1’s 94.9%. Hence one might be tempted to use Model 2. But on a
closer look at the independent variables in model 2, Last Price and RC are the only independent
variables used. In this case there is not a large difference between the recommended prices for Affiar
using model 1 or model 2, but in reality buyer can’t base his/her offer just by the seller’s stated Last
price. Obviously a number of other factors like interior space, tax, apartment maintenance fee, area,
etc., need to be considered. From the given data, model 1 has made a comprehensive attempt to form
the best possible regression fit by use of maximum data points. Hence the recommendation would be
to go by model 1, but in this specific case of the Ellery Street house, since the variation for the predicted
selling price from the two models is not much, it is left to Affiar to either make an initial offer of
$164,288 or $163,380.
AppendixEXHIBIT A (BASELINE REGRESSION)
Variables Entered/Removeda
Model
Variables
Entered
Variables
Removed Method
1
LastPrice .
Stepwise
(Criteria:
Probability-
of-F-to-
enter
<= .050,
Probability-
of-F-to-
remove
>= .100).
2
Tax .
Stepwise
(Criteria:
Probability-
of-F-to-
enter
<= .050,
Probability-
of-F-to-
remove
>= .100).
3
Interior .
Stepwise
(Criteria:
Probability-
of-F-to-
enter
<= .050,
Probability-
of-F-to-
remove
>= .100).
4
Condo .
Stepwise
(Criteria:
Probability-
of-F-to-
enter
<= .050,
Probability-
of-F-to-
remove
>= .100).
5
A12 .
Stepwise
(Criteria:
Probability-
of-F-to-
enter
<= .050,
Probability-
of-F-to-
remove
>= .100).
6
A5 .
Stepwise
(Criteria:
Probability-
of-F-to-
enter
<= .050,
Probability-
of-F-to-
remove
>= .100).
7
RC .
Stepwise
(Criteria:
Probability-
of-F-to-
enter
<= .050,
Probability-
of-F-to-
remove
>= .100).
8
A16 .
Stepwise
(Criteria:
Probability-
of-F-to-
enter
<= .050,
Probability-
of-F-to-
remove
>= .100).
9
A2 .
Stepwise
(Criteria:
Probability-
of-F-to-
enter
<= .050,
Probability-
of-F-to-
remove
>= .100).
a. Dependent Variable: SalePrice
Model Summaryj
Model R R Square
Adjusted R
Square
Std. Error of
the Estimate
Change Statistics
R Square
Change F Change df1 df2
Sig. F
Change
9 .943i .889 .886 30268.70125 .001 5.018 1 446 .026
i. Predictors: (Constant), LastPrice, Tax, Interior, Condo, A12, A5, RC,
A16, A2
j. Dependent Variable: SalePrice
ANOVAj
Model Sum of Squares df Mean Square F Sig.
9 Regression 3.264E12 9 3.627E11 395.860 .000i
Residual 4.086E11 446 9.162E8
Total 3.673E12 455
i. Predictors: (Constant), LastPrice, Tax, Interior, Condo, A12, A5, RC, A16, A2
j. Dependent Variable: SalePrice
Correl
ations
Sale
Price
Last
Price
Inte
rior
Be
d
Ba
th
Ro
om
s
Co
ndo
Ta
x RC A1 A2 A3 A4 A5 A6 A7 A8
A1
0
A1
1
A1
2
A1
3
A1
4
A1
5
A1
6
Pearso
n
Correla
tion
Sale
Price
1.00
0.872
.65
2
.40
5
.53
4
.42
0
.71
3
.86
6
-.3
00
-.0
02
-.0
34
-.0
46
-.0
99
.40
3
-.0
20
.00
7
.09
8
-.0
49
-.0
12
-.0
95
-.0
23
-.0
61
-.1
12
.01
3
Last
Price.872
1.00
0
.57
4
.35
6
.51
0
.35
5
.64
3
.76
6
-.3
05
-.0
10
-.0
45
-.0
39
-.0
93
.37
6
-.0
15
-.0
01
.08
9
-.0
44
-.0
15
.03
3
-.0
33
-.0
61
-.0
97
.00
2
Interi
or.652 .574
1.0
00
.73
8
.56
5
.77
6
.28
9
.59
2
-.2
23
.03
8
-.0
82
.09
8
.01
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N Sale
Price456 456 456
45
6
45
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45
6
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6
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RC456 456 456
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6
45
6456 456
45
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A1456 456 456
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6456 456
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A2456 456 456
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6456 456
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A3456 456 456
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6456 456
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A4 456 456 456 45
6
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456 456 45
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6
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A5456 456 456
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6456 456
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A6456 456 456
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6456 456
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A7456 456 456
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A8456 456 456
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6456 456
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A10456 456 456
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6456 456
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A11456 456 456
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6456 456
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A12456 456 456
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A13456 456 456
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A14456 456 456
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A15456 456 456
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A16456 456 456
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6456 456
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6
Coefficientsa
Model
Unstandardized
Coefficients
Standardize
d
Coefficients
t Sig.
95% Confidence
Interval for B
Collinearity
Statistics
B Std. Error Beta
Lower
Bound
Upper
Bound
Toleranc
e VIF
9 (Constant
)
-
15967.7365913.780 -2.700 .007 -27590.071 -4345.402
LastPrice .333 .023 .403 14.763 .000 .289 .377 .335 2.988
Tax 35.947 3.136 .364 11.462 .000 29.783 42.110 .248 4.035
Interior 44.967 5.554 .173 8.097 .000 34.052 55.882 .549 1.821
Condo 105.108 21.268 .127 4.942 .000 63.311 146.906 .380 2.629
A12 -
27984.5958366.791 -.056 -3.345 .001 -44427.826 -11541.364 .902 1.108
A5 29804.817 6552.903 .084 4.548 .000 16926.416 42683.218 .738 1.354
RC 10992.327 3445.556 .059 3.190 .002 4220.785 17763.869 .726 1.378
A16 -
12447.2915480.634 -.037 -2.271 .024 -23218.366 -1676.216 .944 1.059
A2 12290.704 5486.742 .036 2.240 .026 1507.625 23073.784 .967 1.034
a. Dependent Variable: SalePrice
Excluded Variablesj
Model Beta In t Sig.
Partial
Correlation
Collinearity Statistics
Tolerance VIF
Minimum
Tolerance
9 Bed -.010i -.408 .684 -.019 .414 2.416 .245
Bath -.002i -.070 .944 -.003 .436 2.295 .248
Rooms -.003i -.095 .924 -.005 .340 2.938 .246
A1 .001i .077 .939 .004 .971 1.030 .248
A3 -.025i -1.589 .113 -.075 .972 1.029 .248
A4 -.013i -.783 .434 -.037 .957 1.045 .247
A6 -.004i -.274 .784 -.013 .987 1.014 .248
A7 .010i .564 .573 .027 .818 1.223 .247
A8 -.019i -1.094 .275 -.052 .837 1.195 .248
A10 -.016i -.976 .329 -.046 .927 1.079 .246
A11 .003i .159 .873 .008 .983 1.017 .248
A13 .024i 1.411 .159 .067 .894 1.118 .245
A14 -.001i -.079 .937 -.004 .956 1.046 .248
A15 -.004i -.271 .786 -.013 .978 1.022 .246
a. Predictors in the Model: (Constant), LastPrice
b. Predictors in the Model: (Constant), LastPrice, Tax
c. Predictors in the Model: (Constant), LastPrice, Tax, Interior
d. Predictors in the Model: (Constant), LastPrice, Tax, Interior, Condo
e. Predictors in the Model: (Constant), LastPrice, Tax, Interior, Condo, A12
f. Predictors in the Model: (Constant), LastPrice, Tax, Interior, Condo, A12, A5
g. Predictors in the Model: (Constant), LastPrice, Tax, Interior, Condo, A12, A5, RC
h. Predictors in the Model: (Constant), LastPrice, Tax, Interior, Condo, A12, A5, RC, A16
i. Predictors in the Model: (Constant), LastPrice, Tax, Interior, Condo, A12, A5, RC, A16, A2
j. Dependent Variable: SalePrice
Residuals Statisticsa
Minimum Maximum Mean Std. Deviation N
Predicted Value 2.1894E4 7.3736E5 1.7108E5 84699.37571 456
Std. Predicted Value -1.761 6.686 .000 1.000 456
Standard Error of Predicted
Value1971.030 2.458E4 4.021E3 1982.252 456
Adjusted Predicted Value 1.6813E4 1.1794E6 1.7253E5 95574.81320 456
Residual -3.59573E5 1.37644E5 .00000 29967.84529 456
Std. Residual -11.879 4.547 .000 .990 456
Stud. Residual -20.352 4.861 -.017 1.268 456
Deleted Residual -1.05539E6 1.57295E5 -1.45182E3 55783.52632 456
Stud. Deleted Residual -76.135 4.990 -.139 3.664 456
Mahal. Distance .932 298.983 8.980 16.348 456
Cook's Distance .000 80.153 .179 3.753 456
Centered Leverage Value .002 .657 .020 .036 456
a. Dependent Variable: SalePrice
EXHIBIT B (REGRESSION WITH TRANSFORMED VARIABLE)
Variables Entered/Removeda
Model
Variables
Entered
Variables
Removed Method
1
INVERSE_LAST
RICE.
Stepwise
(Criteria:
Probability-
of-F-to-
enter
<= .050,
Probability-
of-F-to-
remove
>= .100).
2
SQRT_TAX .
Stepwise
(Criteria:
Probability-
of-F-to-
enter
<= .050,
Probability-
of-F-to-
remove
>= .100).
3
A5 .
Stepwise
(Criteria:
Probability-
of-F-to-
enter
<= .050,
Probability-
of-F-to-
remove
>= .100).
4
LOG_CONDO .
Stepwise
(Criteria:
Probability-
of-F-to-
enter
<= .050,
Probability-
of-F-to-
remove
>= .100).
5
LOG_INTERIOR .
Stepwise
(Criteria:
Probability-
of-F-to-
enter
<= .050,
Probability-
of-F-to-
remove
>= .100).
6
RC .
Stepwise
(Criteria:
Probability-
of-F-to-
enter
<= .050,
Probability-
of-F-to-
remove
>= .100).
7
A16 .
Stepwise
(Criteria:
Probability-
of-F-to-
enter
<= .050,
Probability-
of-F-to-
remove
>= .100).
a. Dependent Variable: LOG_SALEPRICE
Correlations
LOG_SALEP
RICE RC A2 A5 A12 A16
INVERSE_LAS
TRICE
SQRT_
TAX
LOG_INTE
RIOR
LOG_CO
NDO
Pearso
n
Correlat
ion
LOG_SALEPRI
CE1.000
-.27
8
-.00
3
.32
5
-.10
3
.04
8-.949 .816 .750 .502
RC-.278
1.0
00
.11
5
-.35
2
-.18
9
.10
2.244 -.378 -.248 -.332
A2-.003
.11
5
1.0
00
-.07
7
-.05
2
-.08
2-.016 -.102 -.088 -.066
A5.325
-.35
2
-.07
7
1.0
00
-.05
0
-.07
8-.240 .273 .150 .352
A12-.103
-.18
9
-.05
2
-.05
0
1.0
00
-.05
3.080 -.071 -.016 -.054
A16.048
.10
2
-.08
2
-.07
8
-.05
3
1.0
00-.067 .122 .078 .023
INVERSE_LAS
TRICE-.949
.24
4
-.01
6
-.24
0
.08
0
-.06
71.000 -.759 -.757 -.427
SQRT_TAX.816
-.37
8
-.10
2
.27
3
-.07
1
.12
2-.759 1.000 .655 .573
LOG_INTERIO
R
.750 -.24
8
-.08
8
.15
0
-.01
6
.07
8
-.757 .655 1.000 .204
LOG_CONDO.502
-.33
2
-.06
6
.35
2
-.05
4
.02
3-.427 .573 .204 1.000
Sig. (1-
tailed)
LOG_SALEPRI
CE.
.00
0
.47
1
.00
0
.01
4
.15
3.000 .000 .000 .000
RC.000 .
.00
7
.00
0
.00
0
.01
4.000 .000 .000 .000
A2.471
.00
7.
.05
1
.13
2
.04
0.366 .015 .030 .080
A5.000
.00
0
.05
1.
.14
4
.04
8.000 .000 .001 .000
A12.014
.00
0
.13
2
.14
4.
.12
8.044 .065 .369 .127
A16.153
.01
4
.04
0
.04
8
.12
8. .077 .005 .049 .314
INVERSE_LAS
TRICE.000
.00
0
.36
6
.00
0
.04
4
.07
7. .000 .000 .000
SQRT_TAX.000
.00
0
.01
5
.00
0
.06
5
.00
5.000 . .000 .000
LOG_INTERIO
R.000
.00
0
.03
0
.00
1
.36
9
.04
9.000 .000 . .000
LOG_CONDO.000
.00
0
.08
0
.00
0
.12
7
.31
4.000 .000 .000 .
N LOG_SALEPRI
CE455 455 455 455 455 455 455 455 455 455
RC 455 455 455 455 455 455 455 455 455 455
A2 455 455 455 455 455 455 455 455 455 455
A5 455 455 455 455 455 455 455 455 455 455
A12 455 455 455 455 455 455 455 455 455 455
A16 455 455 455 455 455 455 455 455 455 455
INVERSE_LAS
TRICE455 455 455 455 455 455 455 455 455 455
SQRT_TAX 455 455 455 455 455 455 455 455 455 455
LOG_INTERIO
R
455 455 455 455 455 455 455 455 455 455
LOG_CONDO 455 455 455 455 455 455 455 455 455 455
Model Summaryh
Mode
l R
R
Square
Adjusted R
Square
Std. Error
of the
Estimate
Change Statistics
Durbin-
Watson
R Square
Change
F
Change df1 df2
Sig. F
Change
7 .965g .932 .931 .11086 .001 4.369 1 447 .037 1.615
g. Predictors: (Constant), INVERSE_LASTRICE, SQRT_TAX, A5, LOG_CONDO,
LOG_INTERIOR, RC, A16
h. Dependent Variable:
LOG_SALEPRICE
ANOVAh
Model Sum of Squares df
Mean
Square F Sig.
7 Regression75.497 7 10.785
877.61
8.000g
Residual 5.493 447 .012
Total 80.991 454
g. Predictors: (Constant), INVERSE_LASTRICE, SQRT_TAX, A5, LOG_CONDO, LOG_INTERIOR, RC, A16
Coefficientsa
Model
Unstandardized
Coefficients
Standardized
Coefficients
t Sig.
95% Confidence Interval
for B
B Std. Error Beta
Lower
Bound
Upper
Bound
7 (Constant) 11.619 .216 53.895 .000 11.195 12.043
INVERSE_LASTRIC
E
-
121950.2973836.638 -.726 -31.786 .000 -129490.386 -114410.209
SQRT_TAX .007 .001 .180 7.971 .000 .005 .009
ANOVAh
A5 .135 .023 .081 5.886 .000 .090 .181
LOG_CONDO .049 .014 .059 3.595 .000 .022 .076
LOG_INTERIOR .087 .026 .069 3.412 .001 .037 .138
RC .031 .012 .035 2.488 .013 .006 .055
A16 -.042 .020 -.026 -2.090 .037 -.081 -.002
a. Dependent Variable: LOG_SALEPRICE
Casewise Diagnosticsa
Case
Number
Std. Residual LOG_SALEPRICE Predicted Value Residual
59 4.446 13.68 13.1891 .49288
217 3.660 11.03 10.6291 .40575
305 3.420 13.33 12.9502 .37916
306 3.162 13.35 12.9950 .35051
360 -8.181 11.73 12.6349 -.90689
408 -3.276 11.17 11.5336 -.36320
a. Dependent Variable: LOG_SALEPRICE
Residuals Statisticsa
Minimum Maximum Mean Std. Deviation N
Predicted Value 10.5508 13.1891 11.9509 .40779 455
Std. Predicted Value -3.433 3.036 .000 1.000 455
ANOVAh
Standard Error of
Predicted Value.007 .033 .014 .005 455
Adjusted Predicted Value 10.5324 13.1544 11.9506 .40777 455
Residual -.90689 .49288 .00000 .11000 455
Std. Residual -8.181 4.446 .000 .992 455
Stud. Residual -8.391 4.600 .001 1.009 455
Deleted Residual -.95413 .52762 .00029 .11387 455
Stud. Deleted Residual -9.132 4.708 .000 1.028 455
Mahal. Distance .861 39.738 6.985 5.922 455
Cook's Distance .000 .458 .005 .025 455
Centered Leverage Value .002 .088 .015 .013 455
a. Dependent Variable: LOG_SALEPRICE
ANOVAh
EXHIBIT C (REGRESSION WITH TRANSFORMED VARIABLE & REMOVED OUTLIERS)
Variables Entered/Removeda
Model
Variables
Entered
Variables
Removed Method
1
INVERSE_LAST
RICE.
Stepwise
(Criteria:
Probability-
of-F-to-
enter
<= .050,
Probability-
of-F-to-
remove
>= .100).
2
SQRT_TAX .
Stepwise
(Criteria:
Probability-
of-F-to-
enter
<= .050,
Probability-
of-F-to-
remove
>= .100).
3
A5 .
Stepwise
(Criteria:
Probability-
of-F-to-
enter
<= .050,
Probability-
of-F-to-
remove
>= .100).
4
LOG_CONDO .
Stepwise
(Criteria:
Probability-
of-F-to-
enter
<= .050,
Probability-
of-F-to-
remove
>= .100).
5
LOG_INTERIOR .
Stepwise
(Criteria:
Probability-
of-F-to-
enter
<= .050,
Probability-
of-F-to-
remove
>= .100).
a. Dependent Variable: LOG_SALEPRICE
Correlations
LOG_SALEP
RICE RC A2 A5 A12 A16
INVERSE_LAS
TRICE
SQRT_
TAX
LOG_INTE
RIOR
LOG_CO
NDO
Pearso
n
Correlat
ion
LOG_SALEPRI
CE1.000
-.26
7
.00
0
.31
4
-.10
3
.05
5-.965 .800 .752 .472
RC-.267
1.0
00
.11
2
-.35
3
-.18
3
.10
0.229 -.365 -.238 -.319
A2.000
.11
2
1.0
00
-.07
7
-.05
1
-.08
3-.019 -.103 -.088 -.062
A5.314
-.35
3
-.07
7
1.0
00
-.04
8
-.07
8-.235 .266 .136 .346
A12 -.103 -.18
3
-.05
1
-.04
8
1.0
00
-.05
2
.115 -.075 -.017 -.052
A16.055
.10
0
-.08
3
-.07
8
-.05
2
1.0
00-.072 .139 .080 .030
INVERSE_LAS
TRICE-.965
.22
9
-.01
9
-.23
5
.11
5
-.07
21.000 -.761 -.757 -.416
SQRT_TAX.800
-.36
5
-.10
3
.26
6
-.07
5
.13
9-.761 1.000 .658 .543
LOG_INTERIO
R.752
-.23
8
-.08
8
.13
6
-.01
7
.08
0-.757 .658 1.000 .183
LOG_CONDO.472
-.31
9
-.06
2
.34
6
-.05
2
.03
0-.416 .543 .183 1.000
Sig. (1-
tailed)
LOG_SALEPRI
CE.
.00
0
.49
6
.00
0
.01
4
.12
2.000 .000 .000 .000
RC.000 .
.00
9
.00
0
.00
0
.01
7.000 .000 .000 .000
A2.496
.00
9.
.05
3
.13
9
.03
9.343 .015 .032 .095
A5.000
.00
0
.05
3.
.15
5
.05
0.000 .000 .002 .000
A12.014
.00
0
.13
9
.15
5.
.13
5.008 .056 .358 .135
A16.122
.01
7
.03
9
.05
0
.13
5. .064 .002 .045 .265
INVERSE_LAS
TRICE.000
.00
0
.34
3
.00
0
.00
8
.06
4. .000 .000 .000
SQRT_TAX.000
.00
0
.01
5
.00
0
.05
6
.00
2.000 . .000 .000
LOG_INTERIO
R.000
.00
0
.03
2
.00
2
.35
8
.04
5.000 .000 . .000
LOG_CONDO.000
.00
0
.09
5
.00
0
.13
5
.26
5.000 .000 .000 .
N LOG_SALEPRI
CE449 449 449 449 449 449 449 449 449 449
RC 449 449 449 449 449 449 449 449 449 449
A2 449 449 449 449 449 449 449 449 449 449
A5 449 449 449 449 449 449 449 449 449 449
A12 449 449 449 449 449 449 449 449 449 449
A16 449 449 449 449 449 449 449 449 449 449
INVERSE_LAS
TRICE449 449 449 449 449 449 449 449 449 449
SQRT_TAX 449 449 449 449 449 449 449 449 449 449
LOG_INTERIO
R449 449 449 449 449 449 449 449 449 449
LOG_CONDO 449 449 449 449 449 449 449 449 449 449
Model Summaryf
Mode
l R
R
Square
Adjusted R
Square
Std. Error
of the
Estimate
Change Statistics
Durbin-
Watson
R Square
Change
F
Change df1 df2
Sig. F
Change
5 .974e .949 .948 .09189 .001 8.435 1 443 .004 1.649
e. Predictors: (Constant), INVERSE_LASTRICE, SQRT_TAX, A5, LOG_CONDO,
LOG_INTERIOR
f. Dependent Variable:
LOG_SALEPRICE
ANOVAf
Model Sum of Squares df Mean Square F Sig.
5 Regression 68.896 5 13.779 1.632E3 .000e
Residual 3.741 443 .008
Total 72.636 448
e. Predictors: (Constant), INVERSE_LASTRICE, SQRT_TAX, A5, LOG_CONDO,
LOG_INTERIOR
f. Dependent Variable: LOG_SALEPRICE
Coefficientsa
Model
Unstandardized
Coefficients
Standardize
d
Coefficients
t Sig.
95% Confidence
Interval for B
Collinearity
Statistics
B
Std.
Error Beta
Lower
Bound
Upper
Bound
Toleranc
e VIF
5 (Constant)12.059 .178
67.59
8
.00
011.708 12.409
INVERSE_LASTRIC
E
-
133487.06
9
3284.28
2-.808
-
40.64
4
.00
0
-
139941.77
9
-
127032.36
0
.2943.40
3
SQRT_TAX.004 .001 .108 5.711
.00
0.003 .006 .324
3.08
9
A5.119 .019 .074 6.363
.00
0.082 .155 .869
1.15
1
LOG_CONDO.034 .011 .042 3.032
.00
3.012 .057 .596
1.67
7
LOG_INTERIOR.062 .021 .052 2.904
.00
4.020 .104 .368
2.72
0
a. Dependent Variable: LOG_SALEPRICE
Residuals Statisticsa
Minimum Maximum Mean Std. Deviation N
Predicted Value 10.4796 12.9755 11.9452 .39215 449
Std. Predicted Value -3.737 2.627 .000 1.000 449
Standard Error of Predicted
Value.005 .028 .010 .004 449
Adjusted Predicted Value 10.4557 12.9635 11.9449 .39233 449
Residual -.26221 .37351 .00000 .09138 449
Std. Residual -2.853 4.065 .000 .994 449
Stud. Residual -2.922 4.137 .001 1.007 449
Deleted Residual -.27492 .38696 .00027 .09376 449
Stud. Deleted Residual -2.947 4.215 .003 1.012 449
Mahal. Distance .299 41.115 4.989 5.166 449
Cook's Distance .000 .182 .004 .015 449
Centered Leverage Value .001 .092 .011 .012 449
a. Dependent Variable: LOG_SALEPRICE
Interior Vs Rooms – Regression results showing correlation
SUMMARY OUTPUT
Regression Statistics
Multiple R 0.775952808R Square 0.60210276Adjusted R Square 0.601226335Standard Error 217.7411745Observations 456
ANOVAdf SS MS F Significance F
Regression 1 32571418.053257141
8686.998
1 6.7719E-93Residual 454 21524693.45 47411.22Total 455 54096111.51
Coefficients Standard Error t Stat P-value Lower 95% Upper 95%Intercept -76.7538578 42.08789622 -1.82366 0.068861 -159.4651166 5.957400971Rooms 235.8872688 8.999672999 26.21065 6.77E-93 218.2010847 253.5734529
0 20 40 60 80 100 1200
4000
Normal Probability Plot
Sample Percentile
Inte
rior
1 2 3 4 5 6 7 8 9 10-1000
0
1000
2000Rooms Residual Plot
Rooms
Resid
uals