fundamentals of real estate lecture 12 spring, 2003 copyright © joseph a. petry

Fundamentals of Real Estate

Lecture 12

Spring, 2003

Copyright © Joseph A. Petry

www.cba.uiuc.edu/jpetry/Fin_264_sp03

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Multiple Regression Models:

Sales Comparison Approach—Ch 12

y = 0 + 1x1+ 2x2 + …+ kxk +

Dependent variable Independent variables

Coefficients Random error variable

3

Multiple Regression Models:

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.

Example: Multiple Regression ResultsX Variables Beta Estimate t-statistic

Constant 1,034.99 14.12Living Area (sq ft) 64.06 22.32Age -1,540.50 4.59Site size 35,000.92 3.23R2 = .89; F-Statistic = 69.74; standard error = 6786.5; Dep. Variable = Value


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Rules as indicated in text: If |t stat| > 2, then variable significant and should be keptIf 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)

Price = 1034.99 + 64.06 * 2600 - 1540.5 * 10 + 35,000.92 * .5Price = 169,686.4595% Confidence Interval = 169,686.45 +/- 2 *6786.5; 95% Confidence Interval = [156,113.45, 183,259.45]


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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.


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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. 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.

2. 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.

3. 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.

Project Description

1. Develop a model that has a sound basis. Theoretical and practical inputs into model formation

– Working group of experts for brainstorming session– Literature review on factors influencing variable of interest

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”.

– 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?

3. Draw the scatter diagram to determine whether a linear model (or other forms) appears to be appropriate.

4. Estimate the model coefficients and statistics using statistical computer software.

Regression Analysis—Step by Step

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.

Notice the iterative nature of the process.

8. 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!

– 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

• Example—Motel Profitability

Profitability

Competition Market awareness

Demand Generators

Physical

Rooms Nearest Officespace

Collegeenrollment

Income Disttown

Distance to downtown.

Medianhouseholdincome.

Distance tothe nearestLa Quinta inn.

Number of hotels/motelsrooms within 3 miles from the site.

Demographics

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 =Rooms NearestOfficeCollege

+ 5Income + 6Disttwn +

INN MARGIN ROOMS NEAREST OFFICE COLLEGE INCOME DISTTWN1 55.5 3203 0.1 549 8 37 12.12 33.8 2810 1.5 496 17.5 39 0.43 49 2890 1.9 254 20 39 12.24 31.9 3422 1 434 15.5 36 2.75 57.4 2687 3.4 678 15.5 32 7.96 49 3759 1.4 635 19 41 4

INN MARGIN ROOMS NEAREST OFFICE COLLEGE INCOME DISTTWN1 55.5 3203 0.1 549 8 37 12.12 33.8 2810 1.5 496 17.5 39 0.43 49 2890 1.9 254 20 39 12.24 31.9 3422 1 434 15.5 36 2.75 57.4 2687 3.4 678 15.5 32 7.96 49 3759 1.4 635 19 41 4

Rooms (vertical axis) vs. Margin (horizontal)

y = -27.179x + 4228.4

R2 = 0.2212

1500

2000

2500

3000

3500

4000

4500

20 30 40 50 60 70

Nearest (vertical axis) vs. Margin (horizontal)

y = -0.0183x + 2.8274

R2 = 0.0257

0

1

2

3

4

5

20 30 40 50 60 70

Step #3: Draw Scatter Diagrams

SUMMARY OUTPUT

Regression StatisticsMultiple R 0.724611R Square 0.525062Adjusted R Square0.49442Standard Error5.512084Observations 100

ANOVAdf SS MS F Significance F

Regression 6 3123.832 520.6387 17.13581 3.03E-13Residual 93 2825.626 30.38307Total 99 5949.458

CoefficientsStandard Error t Stat P-value Lower 95%Upper 95%Intercept 72.45461 7.893104 9.179483 1.11E-14 56.78049 88.12874ROOMS -0.00762 0.001255 -6.06871 2.77E-08 -0.01011 -0.00513NEAREST -1.64624 0.632837 -2.60136 0.010803 -2.90292 -0.38955OFFICE 0.019766 0.00341 5.795594 9.24E-08 0.012993 0.026538COLLEGE 0.211783 0.133428 1.587246 0.115851 -0.05318 0.476744INCOME -0.41312 0.139552 -2.96034 0.003899 -0.69025 -0.136DISTTWN 0.225258 0.178709 1.260475 0.210651 -0.12962 0.580138

Step #4: Estimate Model

This is the sample regression equation (sometimes called the prediction equation)

MARGIN = 72.455 - 0.008ROOMS -1.646NEAREST + 0.02OFFICE +0.212COLLEGE - 0.413INCOME + 0.225DISTTWN

Step #5: Assess the Model

1. R2 (Coefficient of Determination)1b). Adusted R2

1c). Standard error of the estimate

2. F-Test for overall validity of the model

3. T-test for slope– using b (estimate of the slope)– Partial F-test to verify elimination of some independent

variables


1a. Coefficient of determination– The definition is

– 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” . . .

SST

SSER 12

1b. The “Adjusted” Coefficient of Determination is defined as:

– 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”.

)]1/([

)]1/([1 Adjusted 2

nSST

knSSER

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:

– It is reported under “Regression Statistics”, as the “Standard Error” at the top of your output.

– Compare s to the mean value of y From the printout, Standard Error = 5.5121 Calculating the mean value of y we have

– Values of s will vary with each regression. While there are no set ranges for its value, it is a number that will often come in handy.

1knSSE

s

739.45y

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: 1 = 2 = … = k = 0

H1: At least one i is not equal to zero.

• If at least one i 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

– Rejection region

F>F,k,n-k-1

MSEMSR

F

MSR=SSR/k

MSE=SSE/(n-k-1)

SST = SSR + SSE. Large F results from a large SSR. Then, much of the variation in y is explained by the regression model. The null hypothesis shouldbe rejected; thus, the model is valid.



Excel provides the following ANOVA results


SSESSR

MSEMSR

MSR/MSE



F,k,n-k-1 = F0.05,6,100-6-1=2.17F = 17.14 > 2.17

Also, the p-value (Significance F) = 3.03382(10)-13

Clearly, = 0.05>3.03382(10)-13, and the null hypothesisis rejected.

Conclusion: There is sufficient evidence to reject the null hypothesis in favor of the alternative hypothesis. At least one of the i is not equal to zero. Thus, at least one independent variable is linearly related to y. This linear regression model is valid

Conclusion: There is sufficient evidence to reject the null hypothesis in favor of the alternative hypothesis. At least one of the i is not equal to zero. Thus, at least one independent variable is linearly related to y. This linear regression model is valid

CoefficientsStandard Error t Stat P-value Lower 95% Upper 95%Intercept 72.45461 7.893104 9.179483 1.11E-14 56.78048735 88.12874ROOMS -0.00762 0.001255 -6.06871 2.77E-08 -0.010110582 -0.00513NEAREST -1.64624 0.632837 -2.60136 0.010803 -2.902924523 -0.38955OFFICE 0.019766 0.00341 5.795594 9.24E-08 0.012993085 0.026538COLLEGE 0.211783 0.133428 1.587246 0.115851 -0.053178229 0.476744INCOME -0.41312 0.139552 -2.96034 0.003899 -0.690245235 -0.136DISTTWN 0.225258 0.178709 1.260475 0.210651 -0.12962198 0.580138

3a. Testing the coefficients– The hypothesis for each i


H0: i = 0H1: i = 0

Test statistic

d.f. = n - k -1ib

iis

bt

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 =Rooms NearestOfficeCollege + 5Income + 6Disttwn

– 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.

– 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).

5.72b

0076.b1

– 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???

– For each additional 1000 sq-ft of office space, the average increase in operating margin will be .02%.

– For additional thousand students MARGIN

increases by .21%.

– For additional $1000 increase in median

household income, MARGIN decreases by .41% ???

– For each additional mile to the downtown

center, MARGIN increases by .23% on the average???

65.1b2

02.b3

21.b4

41.b5

23.b6

– 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 StatisticsMultiple R 0.718990854R Square 0.516947848Adjusted R Square 0.491253584Standard Error 5.529320727Observations 100



Coefficients Standard Error t Stat P-value Lower 95% Upper 95%Intercept 75.13707499 7.624565098 9.854604692 3.74364E-16 59.99832996 90.27582ROOMS -0.007742161 0.001255304 -6.167558795 1.7296E-08 -0.010234595 -0.00525NEAREST -1.586922918 0.633058347 -2.506756172 0.013901469 -2.843874466 -0.329971OFFICE 0.019576011 0.00341778 5.727697298 1.21629E-07 0.012789931 0.026362COLLEGE 0.196384877 0.133283011 1.473442678 0.14397261 -0.068251531 0.461021INCOME -0.421411017 0.139833268 -3.013667795 0.003317336 -0.699053108 -0.143769

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”.

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.

SUMMARY OUTPUT




Coefficients Standard Error t Stat P-value Lower 95% Upper 95%Intercept 77.93849422 7.429077126 10.49100621 1.48143E-17 63.18992196 92.68707ROOMS -0.007862522 0.00126034 -6.238412847 1.22182E-08 -0.010364612 -0.00536NEAREST -1.653650492 0.635316269 -2.602877611 0.010726099 -2.914911849 -0.392389OFFICE 0.019607492 0.003438713 5.701984661 1.33212E-07 0.012780787 0.026434INCOME -0.399387121 0.13988637 -2.855082452 0.005283347 -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. • 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 p-value to be retained?

• Are there any variables you believe strongly “must” be in the model irrespective of the results of the t-test?

2. Once you have made your decisions, then conduct the “Partial F-test” to verify your results.

fd knkf

drfF

MSE

kSSRSSR)1()(0 ,

/)(if HReject

fd knk

f

drfF

MSE

kSSRSSR)1()(0 ,

/)(if HReject

H0: 1 = 2 = … = i = 0

H1: At least one i is not equal to zero.

Where:

is 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.

df SS MS F Significance FRegression 4 3009.184 752.2959 24.30661 7.23E-14Residual 95 2940.274 30.95026Total 99 5949.458

The ANOVA results for the reduced model are:


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.

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.

The test statistic for the Partial F-test:

[(3123.83-3009.184)/2]/30.95=57.323/30.95=1.852F,k,n-k-1 = F0.05,2,100-6-1=3.095; F = 1.852 < 3.1; therefore, DNR H0

• 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.


We now have our final model. You should be able to do the assessment on your own at this stage.

SUMMARY OUTPUT




Coefficients Standard Error t Stat P-value Lower 95% Upper 95%Intercept 77.93849422 7.429077126 10.49100621 1.48143E-17 63.18992196 92.68707ROOMS -0.007862522 0.00126034 -6.238412847 1.22182E-08 -0.010364612 -0.00536NEAREST -1.653650492 0.635316269 -2.602877611 0.010726099 -2.914911849 -0.392389OFFICE 0.019607492 0.003438713 5.701984661 1.33212E-07 0.012780787 0.026434INCOME -0.399387121 0.13988637 -2.855082452 0.005283347 -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 Bloomington Champaign

Rooms 2672 2,500 2,300

Competitor Distance 1.3 1.2 .5

Office Space (‘000s) 952 604 1,430

Students (‘000s) 42 21 45

Income (‘000s) 35 37 33.5

Dist to Downtown 3.4 4.5 1.4

Predicted Margin

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 3 29029.71625 9676.57208 5.974883 0.001315371Residual 56 90694.33308 1619.54166Total 59 119724.0493

Coefficients Standard Error t Stat P-value Lower 95% Upper 95%Intercept 51.391216 23.51650385 2.18532554 0.033064 4.282029664 98.5004Lot size 0.6999045 0.558855319 1.25238937 0.215633 -0.419616528 1.819425Trees 0.6788131 0.229306132 2.96029204 0.0045 0.219458042 1.138168Distance -0.3783608 0.195236549 -1.9379609 0.057676 -0.769466342 0.012745

SUMMARY OUTPUT



Regression 1 18122.63 18122.63 10.34545 0.002124116Residual 58 101601.4 1751.749Total 59 119724

CoefficientsStandard Error t Stat P-value Lower 95% Upper 95%Intercept 56.26122 10.90103 5.161092 3.13E-06 34.44044988 78.08198Trees 0.727649 0.226229 3.216434 0.002124 0.274803888 1.180494



Regression 2 26489.5 13244.75 8.097328 0.000803103Residual 57 93234.55 1635.694Total 59 119724

CoefficientsStandard Error t Stat P-value Lower 95% Upper 95%Intercept 75.52475 13.54641 5.575259 7.06E-07 48.39851987 102.651Trees 0.767051 0.219299 3.497737 0.000917 0.327911764 1.206191Distance -0.432673 0.191306 -2.261677 0.027549 -0.815756848 -0.049589

Price Trees Distance Lot sizePrice 1Trees 0.389063 1Distance -0.232614 0.079442 1Lot size 0.303525 0.285682 -0.189499 1

Histogram of Standardized Residuals

0

5

10

15

-1.7 -1.1 -0.6 0.0 0.5 1.1 1.6 More

Fre

qu

ency

Standard Residuals Vs. Predicted

-2

-1

0

1

2

3

40 60 80 100 120 140

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. What output should have been provided, but wasn’t?

2. What output was provided, that probably should not have been?

3. What output might be provided if the data was different, but wasn’t necessary to provide in this case?

4. Are any of the assumptions violated or other danger signals present?

============================================

1. Which model should you most likely use to make predictions?

2. 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.

3. 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?

fundamentals of real estate lecture 12 spring, 2003 copyright © joseph a. petry

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