Getting started with regression

Getting Started with Regression

Presented By: Tim Wilmath, MAI

Prepared For: Hillsborough County Property Appraiser’s Office

Predicted Values

$600,000$500,000$400,000$300,000$200,000$100,000

Sal

es P

rices

$700,000

$600,000

$500,000

$400,000

$300,000

$200,000

$100,000

History of Regression

James Galton created Regression Analysis in 1885 when he was attempting to predict a person’s height based on the height of his or her parent.


Galton found that children born to tall parents would be

shorter than their parents - and children born to short

parents would be taller than their parents. Both groups

of children regressed toward the mean height of all children.


In 1922, a PhD student by the name of Casper G. Haas suggested using regression for farm land valuation.


What is truly remarkable is that Mr. Haas was using a technique that required significant

amounts of calculations, calculations that today are done by sophisticated computer programs

in seconds. Mr. Haas did these calculations by hand. In looking at this excerpt, it is remarkable

how the nomenclature and the statistical output has varied so little in more than 85 years.

Uses of Regression

Predicting the Weather

Uses of Regression

Predicting Election Results

Uses of Regression

Predicting Sales Prices

What is Regression?

When Regression Analysis is used to predict sales prices or establish assessments it becomes an

Automated Sales Comparison Approach

Steps in Regression

1. Data Exploration and cleanup

2. Specifying the model

3. Calibrating the model

4. Interpreting the results

Data Exploration & CleanupIs there a pattern suggesting a

relationship between variables?

Because of the potential for extreme values to influence the mean,

modelers often remove or “trim” extreme values.

HEATED AREA

70006000500040003000200010000

SA

LES

PR

ICE

800000

700000

600000

500000

400000

300000

200000

100000

0

Note the outliers.

These will adversely

affect our final values

if we don’t deal with

them now

Model Specification

• Additive - Most common for residential properties

• Multiplicative- Often used for land valuation

• Hybrid - Most advanced

Specifying the model means picking the appropriate

equation and which variables that will be used.

We are going to use an Additive Model

in this presentation

Models can be:

Independent Variables: • Size• Age• Location• Condition• Lot size• Construction• Quality• Amenities

Regression Components

Dependent Variable:• Sales Price

Simple Regression includes one Dependent Variable (sales price) and only one Independent Variable - such as Square Footage.

Simple Regression

HEATED AREA

500040003000200010000

SA

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PR

ICE

500000

400000

300000

200000

100000

0

Using this model,

a 1,000 sf home would

be valued at $75,000

Simple Regression using only size as the independent variable will predict sales prices, however, it will treat all homes with the same size equally.

Simple Regression

1,000 square feet - $75,000?1,000 square feet - $75,000

Multiple RegressionWe know square footage is an important variable

but what other variables should we include and how do we decide?

Heated Area

QualityLot Size

Exterior Wall Type

Actual Age

Effective Age

Roof Type

Heat/Ac Type

Swimming Pool

GarageScreen Porch

View Location

Correlation Analysis

Correlations SALEPRICE BLDSIZE BEDROOMS DOCKSALEPRICE Pearson Correlation 1 0.855 0.557 0.142

Sig. (2-tailed) . 0 0 0N 1367 1367 1367 1367

BLDSIZE Pearson Correlation 0.855 1 0.659 0.062Sig. (2-tailed) 0 . 0 0.021N 1367 1367 1367 1367

BEDROOMS Pearson Correlation 0.557 0.659 1 0.037Sig. (2-tailed) 0 0 . 0.176N 1367 1367 1367 1367

DOCK Pearson Correlation 0.142 0.062 0.037 1Sig. (2-tailed) 0 0.021 0.176 .N 1367 1367 1367 1367

Notice the high

correlation between

sales price and size

Pearson’s Correlation tells you the degree of relationships between variables.

Very little

correlation between

sales price and dockCorrelation Analysis also helps identify “Collinearity”, which is a correlation between 2 independent variables. For example, the living area

of a home is highly correlated to the number of bedrooms. It would only be necessary to have one of these variables in the model.

Y = b0 + b1 X1 + b2 X2 + . . . + bK XK

Regression Equations

Y=mx+b

Running RegressionStatistical Software makes using Regression much easier,

performing the necessary calculations quickly and accurately.

Let’s Run

This!

Regression Results

Model Summary

Model R R SquareAdjusted R

SquareStd. Error of the

Estimate1 .855(a) .732 .731 25406.53266545

a Predictors: (Constant), BLDSIZE

Coefficients(a)

Unstandardized CoefficientsStandardizedCoefficients

Model B Std. Error Beta t Sig.(Constant) 6838.585 2195.717 3.115 .0021

BLDSIZE 75.068 1.231 .855 60.997 .000

a Dependent Variable: SALEPRIC

The closer the

Adj. R-Square is to “1”

the better

And - it gives us the coefficients (or adjustments)

$6,838

+ Bldsize x $75.07

= Property Value

Model 1

The adjusted R2 statistic measures the amount of total variation explained by the Regression Model. It ranges from 0.00 to 1.00 with 1.00

being the desired value. A high number, say 0.910 means that approximately 91% of the value can be explained by the model.

The Output tells us how good our model is working

Regression Results

Coefficients(a)


Model B Std. Error Beta t Sig.(Constant) 6838.585 2195.717 3.115 .0021

BLDSIZE 75.068 1.231 .855 60.997 .000


The output includes the coefficient and the “Constant”

The “Constant” represents the un-explained

value that is not included in the model.

Running RegressionLet’s add another variable to the model - Say Land Size

Let’s run

this model and

see if results

improve.

Regression Results

Model 2

We also have new coefficients (or adjustments)

Model Summary



Estimate1 .895(a) .801 .801 21864.78975921

a Predictors: (Constant), LANDSF, BLDSIZE

Our Adj. R2 went up from

.731 to .801!

Coefficients(a)


Model B Std. Error Beta t Sig.(Constant) 6119.232 1889.914 3.238 .001BLDSIZE 72.660 1.065 .828 68.237 .000

1

LANDSF .382 .017 .266 21.887 .000


$6,119

+ Bldsize x $72.66

+ Landsf x $0.382

= Property Value

Running RegressionLet’s add Age to the model

If Age is

significant

to value, the model

should improve.

Let’s run it.

Regression Results

Model 3

Notice the age coefficient is negative

Model Summary



Estimate1 .912(a) .832 .832 20114.04445033

a Predictors: (Constant), AGE, LANDSF, BLDSIZE


.801 to .832!

Coefficients(a)


Model B Std. Error Beta t Sig.(Constant) 22855.587 2036.809 11.221 .000BLDSIZE 67.276 1.037 .767 64.856 .000LANDSF .444 .017 .309 26.868 .000

1

AGE -630.763 39.991 -.189 -15.773 .000


$22,855

+ Bldsize x $67.28

+ Landsf x $0.44

+ Age x ($630.76)

= Property Value

Running RegressionLet’s add Building Quality to the model

We may have

a problem.

Let’s run it

and see.

Regression Results

Notice the constant is now negative - that’s not good!

Model Summary



Estimate1 .924(a) .854 .853 18784.15717760

a Predictors: (Constant), QUAL, LANDSF, AGE, BLDSIZE


.832 to .854 after

adding quality, but

Coefficients(a)


Model B Std. Error Beta t Sig.(Constant) -45723.503 5199.675 -8.794 .000BLDSIZE 59.808 1.103 .681 54.234 .000LANDSF .445 .015 .309 28.831 .000AGE -605.886 37.388 -.182 -16.205 .000

1

QUAL 26110.420 1842.475 .171 14.171 .000


What do we do with this

quality adjustment?

Model 4

Regression ResultsCoefficients(a)


Model B Std. Error Beta t Sig.(Constant) -45723.503 5199.675 -8.794 .000BLDSIZE 59.808 1.103 .681 54.234 .000LANDSF .445 .015 .309 28.831 .000AGE -605.886 37.388 -.182 -16.205 .000

1

QUAL 26110.420 1842.475 .171 14.171 .000


Quality1 - Fair2 - Average3 - Good4 - Excellent5 - Superior

= 1 x $26,110 = $26,110

= 2 x $26,110 = $52,220

= 3 x $26,110 = $78,330= 4 x $26,110 = $104,440

= 5 x $26,110 = $130,550

Resulting Adjustment This doesn’t make

sense because the

codes 1,2,3, etc.

were not meant

to be a rank

A Note about Data Types

There are 3 primary types of property Characteristics:

• Continuous: Based on a size or measurement. Examples: Square Footage or Lot Size

• Discrete: Specific pre-defined value.Examples: Roof Material, Building Quality

• Binary: Either the item is present or notExamples: corner location, Lakefront Location

TransformationsTo solve the problem we need to convert the “discrete” variable Quality into individual “binary” variableswhich allows Regression to distinguish each type:

Fair - Yes/No Average - Yes/No Good - Yes/No Excellent - Yes/No Superior - Yes/No

“Quality” BECOMES

Running RegressionNow that we have transformed the variable

Quality

we can put it back in the model

Notice we left

“Average” out

Regression Results

Model Summary



Estimate1 .933(a) .870 .869 17717.09739523

a Predictors: (Constant), SUPERIOR, EXCEL, AGE, FAIR, GOOD, LANDSF, BLDSIZE


.832 to .869.

Coefficients(a)


Model B Std. Error Beta t Sig.(Constant) 35633.753 1922.792 18.532 .000BLDSIZE 58.537 1.045 .667 56.031 .000LANDSF .419 .016 .291 26.342 .000AGE -625.742 35.363 -.188 -17.695 .000FAIR -25511.289 8693.178 -.031 -2.935 .003GOOD 21095.623 1838.228 .127 11.476 .000EXCEL 75844.967 12720.934 .059 5.962 .000

1

SUPERIOR 305671.839 18494.059 .169 16.528 .000


These Quality

adjustments

are all relative to

“Average”

Model 5

Running Regression

Notice we left

out the“Base”

Neighborhood

(the most typical)

Let’s transform Neighborhood into a binary and add it to the model

Regression Results

Model Summary



Estimate1 .936(a) .875 .874 17391.93018134

a Predictors: (Constant), NB211006, BLDSIZE, EXCEL, FAIR, SUPERIOR, NB211002,NB211001, NB211005, AGE, LANDSF, GOOD, NB211003


.869 to .874.

Coefficients(a)


Model B Std. Error Beta t Sig.(Constant) 40799.859 2299.668 17.742 .000BLDSIZE 56.000 1.143 .638 48.980 .000LANDSF .423 .016 .294 25.753 .000AGE -671.493 37.221 -.201 -18.041 .000FAIR -33476.331 8602.963 -.041 -3.891 .000GOOD 17371.495 2023.937 .105 8.583 .000EXCEL 72617.618 12567.147 .057 5.778 .000SUPERIOR 313444.055 18313.237 .173 17.116 .000NB211001 14199.881 2321.457 .070 6.117 .000NB211002 -3514.034 1657.862 -.025 -2.120 .034NB211003 -1483.623 1244.877 -.015 -1.192 .234NB211005 4044.357 2266.186 .021 1.785 .075

1

NB211006 1915.755 2601.773 .008 .736 .462


These Neighborhood

adjustments

are all relative to

our “Base”

Neighborhood

Model 6

Running RegressionMultiplicative Transformations combine two variables into one

Square Footage x Quality = SQFT1Reflects the fact that quality may contribute greater value in larger homes and less value in smaller homes. In other words, without combining these variables, all Good Quality homes get the same adjustment regardless of their size. Let’s add this new combined variable to the model.

Since we combined SF

and Quality, we remove

them as stand-alone

variables

Regression ResultsModel Summary



Estimate1 .938(a) .880 .879 17065.96846831

a Predictors: (Constant), SQFT5, SQFT4, AGE, NB211002, SQFT2, SQFT1, NB211006,NB211001, NB211005, LANDSF, NB211003, SQFT3


.874 to .879.

Coefficients(a)


Model B Std. Error Beta t Sig.(Constant) 43999.158 2299.663 19.133 .000LANDSF .418 .016 .291 25.996 .000AGE -660.473 36.505 -.198 -18.092 .000NB211001 10975.273 2335.844 .054 4.699 .000NB211002 -3611.418 1624.028 -.026 -2.224 .026NB211003 -1250.573 1221.119 -.013 -1.024 .306NB211005 6350.688 2243.206 .033 2.831 .005NB211006 1923.311 2554.324 .008 .753 .452SQFT1 21.119 8.533 .026 2.475 .013SQFT2 53.673 1.169 .723 45.916 .000SQFT3 63.139 1.074 .964 58.806 .000SQFT4 77.267 3.557 .210 21.720 .000

1

SQFT5 108.100 2.941 .356 36.759 .000


Notice the adjustments

went from fixed dollar

amounts to

“per square foot”

Model 7

Advanced TransformationsExponential transformations - Raise variable to a power

Land Size x .75 = LAND75

Reflects the principle of diminishing returns. The unit price of land tends to decrease as size increases. Without this transformation land would get the same adjustment, regardless of size. Raising land size to the power of .75 reflects the curve shown below.

SINGLE FAMILY LOT PRICES

$2.40$2.45$2.50$2.55$2.60$2.65$2.70$2.75$2.80$2.85

LOT SIZE

PRIC

E P

ER

SF

Running RegressionLet’s add our new transformed land variable to the model

Regression ResultsModel 8

Coefficients(a)


Model B Std. Error Beta t Sig.(Constant) 40782.649 2277.915 17.903 .000AGE -731.178 36.549 -.219 -20.005 .000NB211001 10061.900 2314.108 .050 4.348 .000NB211002 -3196.888 1609.968 -.023 -1.986 .047NB211003 -1646.847 1211.025 -.017 -1.360 .174NB211005 6714.691 2224.018 .035 3.019 .003NB211006 -5595.936 2625.622 -.024 -2.131 .033SQFT1 30.298 8.324 .038 3.640 .000SQFT2 51.834 1.167 .698 44.421 .000SQFT3 60.732 1.081 .927 56.177 .000SQFT4 71.516 3.559 .194 20.094 .000SQFT5 104.644 2.937 .345 35.625 .000

1

LAND75 12.233 .459 .314 26.668 .000


Model Summary



Estimate1 .939(a) .882 .881 16919.04533480

a Predictors: (Constant), LAND75, NB211005, NB211001, SQFT4, NB211002, SQFT5,SQFT1, AGE, SQFT2, NB211006, NB211003, SQFT3


.879 to .881.

Running RegressionLet’s add garages, pools, and baths just to round out our model.

Regression ResultsModel 9

Coefficients(a)


Model B Std. Error Beta t Sig.(Constant) 29680.695 2885.532 10.286 .000AGE -705.817 38.491 -.212 -18.337 .000NB211001 12374.064 2176.815 .061 5.684 .000NB211002 -1094.891 1527.977 -.008 -.717 .474NB211003 -938.838 1136.671 -.010 -.826 .409NB211005 12639.946 2139.489 .066 5.908 .000NB211006 852.109 2535.266 .004 .336 .737SQFT1 31.388 7.815 .039 4.016 .000SQFT2 44.166 1.365 .595 32.349 .000SQFT3 52.939 1.265 .808 41.857 .000SQFT4 60.447 3.561 .164 16.974 .000SQFT5 94.723 2.943 .312 32.186 .000LAND75 11.788 .433 .303 27.240 .000BATHS 7714.093 1338.204 .076 5.765 .000POOL 13359.275 1184.469 .105 11.279 .000

1

GARAGE 10.750 3.137 .038 3.427 .001


Model Summary(b)



Estimate1 .947(a) .897 .895 15854.87728402


.881 to .895.




1

SUPERIOR 305671.839 18494.059 .169 16.528 .000


The “Beta” value in column 4 indicates the partial correlation

of the variable. It is used in stepwise regression in deciding

which variable to add next.

Regression Results

Coefficients(a)


Model B Std. Error Beta t Sig.(Constant) 29680.695 2885.532 10.286 .000AGE -705.817 38.491 -.212 -18.337 .000NB211001 12374.064 2176.815 .061 5.684 .000NB211002 -1094.891 1527.977 -.008 -.717 .474NB211003 -938.838 1136.671 -.010 -.826 .409NB211005 12639.946 2139.489 .066 5.908 .000NB211006 852.109 2535.266 .004 .336 .737SQFT1 31.388 7.815 .039 4.016 .000SQFT2 44.166 1.365 .595 32.349 .000SQFT3 52.939 1.265 .808 41.857 .000SQFT4 60.447 3.561 .164 16.974 .000SQFT5 94.723 2.943 .312 32.186 .000LAND75 11.788 .433 .303 27.240 .000BATHS 7714.093 1338.204 .076 5.765 .000POOL 13359.275 1184.469 .105 11.279 .000

1

GARAGE 10.750 3.137 .038 3.427 .001


Rule of Thumb:

“t” scores should

be 2.0 or greater

The significance of each variable to the model can be determined

by looking at the “t” values.

NB211002

NB211003

NB211006

are insignificant




1

SUPERIOR 305671.839 18494.059 .169 16.528 .000


The “t-statistic” is calculated by dividing the coefficient of

a variable by its standard error. For example: for the variable

BLDSIZE, the “t-statistic” is calculated as follows:

58.537 / 1.045 = 56.0

Regression ResultsModel Summary(b)



Estimate1 .947(a) .897 .895 15854.87728402

The “Standard Error of the Estimate” in the regression model tells us

how much a sale estimate will vary from its actual value.

This number alone is meaningless unless related to the average

sales price in the sale sample. Dividing the Standard Error by

the Average SalesPrice produces the Coefficient of Variation (COV)

$15,854 / $134,043 = 11.82% COV

Regression Options“Enter” is the default regression method in most statistical software programs. This method includes all variables “entered” by the modeler.

“Stepwise” multiple regression automatically eliminates

redundant or insignificant variables.Coefficients(a)

Model: 4


B Std. Error Beta t Sig.(Constant) 28624.283 2584.025 11.077 .000AGE -697.862 37.689 -.209 -18.516 .000NB211001 12794.553 2071.093 .063 6.178 .000NB211005 13302.885 1969.163 .069 6.756 .000SQFT1 31.406 7.797 .039 4.028 .000SQFT2 44.305 1.354 .597 32.723 .000SQFT3 53.134 1.249 .811 42.525 .000SQFT4 60.544 3.557 .164 17.023 .000SQFT5 94.884 2.924 .313 32.446 .000LAND75 11.891 .393 .305 30.243 .000BATHS 7732.836 1332.987 .076 5.801 .000POOL 13317.394 1179.165 .105 11.294 .000GARAGE 10.586 3.047 .037 3.474 .001


Notice that Stepwise

Regression

“kicked out” the

neighborhoods that had

low “t-scores"

Creating New Assessments

Once you have calibrated

your model, the Regression

software allows you to predict

the new values (or assessments)

using the coefficients

(or adjustments) you created.

Reviewing Ratio Statistics

Once the new assessments are created using our final model, we can

review the accuracy of our new values using traditional ratio statistics.

Ratio Statistics for ASSESS Unstandardized Predicted Value / SALEPRIC

Weighted Mean 1.000 Price Related Differential 1.008 Coefficient of Dispersion .079

Mean Centered 11.1% Coefficient of Variation

Median Centered 11.2%

Valuing the PopulationValuing the population requires transforming the same variables

you used in the model, then applying the coefficients to those variables.

This can be done internally within some CAMA systems, using

Microsoft Excel or other spreadsheet software, or within the

regression software.

Valuing the population is one of the most difficult aspects

of regression modeling because changes in the physical attributes of

any one parcel often requires re-running the entire model and

re-calculating values.

ConclusionPredicting assessments using Regression requires the appraiser to:

• Explore data to determine relationships and cleanup outliers

• Specify which model and variables will be used

• transform variables and run regression

• Review Results, modify or add variables

• Create predicted assessments and review ratio statistics

• Value Population using final coefficients

The End

Predicted Values

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Real Estate

Getting started with regression