Case 1_QM

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


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}




= 164288.0015 ±t[0.025,(450-10)]( .010)

= 164288.0015± 1.9653 *(.010)

= 164288.0015±.019653

= {164287.9818, 164288.0212}


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}




= 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}


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


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).


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


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

9

.15

1

.05

4

-.1

78

.06

1

.14

8

.03

2

-.0

22

-.1

15

.05

3

-.0

84

.04

7

Bed.405 .356

.73

8

1.0

00

.37

7

.84

9

.06

0

.39

3

-.0

99

.06

1

-.1

04

.07

8

.11

7

-.0

02

.05

5

-.2

38

-.0

02

.09

2

.01

4

-.0

28

.01

8

.05

9

-.0

66

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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


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


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


EXHIBIT B (REGRESSION WITH TRANSFORMED VARIABLE)


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).


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


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




Predicted Value 10.5508 13.1891 11.9509 .40779 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



ANOVAh

EXHIBIT C (REGRESSION WITH TRANSFORMED VARIABLE & REMOVED OUTLIERS)


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).


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




Predicted Value 10.4796 12.9755 11.9452 .39215 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



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

Case 1_QM

Documents

Transcript of Case 1_QM