mass appraisal.pdf

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Mass Appraisal: An Introduction toMultiple Regression Analysis for RealEstate Valuation

John D. Benjamin,* Randall S. Guttery** and C. F. Sirmans***

Focus

This case study presents an introduction to the basics of real estate appraisal andmultiple regression analysis; in particular, as used in real estate valuation for massproperty tax assessment. While real estate researchers, appraisers and some taxassessors have used multiple regression analysis for many years, its use by a largenumber of assessors is relatively new. The purpose of this case is to expose students

to standard appraisal approaches including the market comparison technique as wellas the advantages and disadvantages of using multiple regression analysis. In theiranswers to the case, students are encouraged to explore and develop solutions, so asto understand how to use the market comparison approach and multiple regressionanalysis for real estate valuation.

Setting

The real estate tax assessment process is used to provide an introduction to multipleregression analysis. The tax assessors office in a small west Texas county has always

assessed properties through manual market comparison analysis. This manual processuses recently sold properties that are in close proximity to the subject property tomake corresponding weighted adjustments. After going to a seminar on multipleregression analysis for mass appraisals, the county tax assessor employs a universityprofessor to explain how multiple regression analysis works for real estate valuationand mass assessment, as well as what its relative benefits are over the existing manualsystem. He invites his staff, the county commissioners, and others to a one-nightseminar that explains multiple regression analysis. This seminar presented by auniversity professor to Texas participants is used educate case readers about real estateappraisal and multiple regression analysis.

Exhibits

Multiple regression handout presented in Appendix.

Availability

This case is available through the ARES clearing house.

*American University, Washington, D.C. 20016 or [email protected].**University of North Texas, Denton, TX 76203 or [email protected].***University of Connecticut, Storrs, CT 06269 of [email protected].


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66 Journal of Real Estate Practice and Education

VOLUME 7, NUMBER 1, 2004

Teaching Notes

Teaching Notes are available and emphasize the objectives that the students areexpected to master. Generalized solutions for the case are included.

Introduction

This case study presents an overview of the basics of multiple regression analysis andillustrates its use in real estate valuation for mass property tax assessment. While realestate researchers, appraisers and some tax assessors have used this methodology formany years, its use by a large number of assessors is relatively new. Lusht (2001)suggests that multiple regression analysis . . . can be used to value a large numberof properties quickly and economically, which helps explain its (growing) popularitywith tax assessors. The Appraisal of Real Estate (2001), published by The AppraisalInstitute, offers an in-depth analysis of this methodology. Smith, Root, and Belloit(1995), Downing and Clark (1997), Allison (1998), Baldwin (1999), Betts and Ely(2001) and Ratterman (2001) investigate the worthiness of multiple regression analysisand its application to real estate valuation.

Background: Back to Texas with New Information

Mr. Austin Modano has recently returned from a business trip to the annual taxassessors conference in Washington, D.C. Being the assessor of a small west Texascounty, he is a bit in awe of the advances in software technology and statistical

techniques being used by his counterparts around the country. In particular, he isintrigued by the use of multiple regression analysis to estimate real estate value fortaxation purposes. His county has always assessed properties the old-fashionedwaythrough manual market comparison analysis of recently sold properties that arein close proximity to the subject property, a costly and time-consuming process.Furthermore, it requires hiring additional appraisers during the reassessment processand it is prone to human bias and error.

Changing from his countys assessment methodology to one using a multipleregression analysis for estimating value now seems preferable for several reasons.First, he realizes that in multiple regression analysis, data from all sales are utilized,rather than data from only three or four comparable properties that have sold recently.Appraiser bias with respect to choosing comparables or comps, therefore, wouldbe eliminated. Second, rather than guesstimating adjustments in magnitude anddirection, the multiple regression software output statistically estimates theadjustments through the values and signs of the regression coefficients. In other words,having to calculate the magnitude (i.e., the dollar value) of each characteristic, suchas a fireplace or a swimming pools contributory value, would not be necessary.Unnecessary as well would be determining whether or not a characteristic is a positiveor a negative attribute. In multiple regression analysis, the direction of the adjustmentis determined simply by the sign of the coefficient from the regression equationsoutput. Last, matched pairs analysisan appraisal technique used in the traditionalmarket comparison approachbecomes unnecessary. Although switching to a


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Mass Appraisal 67

statistical analysis methodology would be costly and may require staff training, theease of mass appraisals for property tax assessments would largely overcome thesecosts in the long run. Local elected officials, however, would also have to be persuadedof the benefits for these changes.

Believing that his county is ripe for statistical modernization, Austin decides to holda seminar for his staff and the relevant county officials. His mission is to educatethem on the benefits of utilizing multiple regression analysis. Not feeling qualified toteach the seminar personally, he calls on a local real estate professor and friend, Dr.Katherine Kat Charbonneau, who teaches at a nearby university and has expertisein real estate valuation. Having attended Kats community outreach classes in realestate, Austin is confident that she is the right person to get his associates up to speedon the use of multiple regression analysis.

Austin and Kat meet at her very small, nondescript university office. She explainsthat to teach the class effectively to novice students, she would need to present anoverview of the appraisal process in general and the market comparison approach(MCA) as a specific method to appraising real estate. This overview would includethe advantages and disadvantages of the MCA technique, as well as a demonstrationof how its shortcomings could be overcome by using multiple regression analysis.Then she would define what regression analysis is, how it works and why it is asuperior tool for assessing thousands of properties annually. Austin agrees with Katsoutline and suggests that she present a seminar. She agrees to do so in a couple ofweeks, once final exams are graded and her semester at the university is completed.

Meeting with Seminar on the Appraisal Process

On a Wednesday night following the completion of her university semester, Kat meetswith Austins staff, several county commissioners, the Citizens for Financial IntegrityCommittee, and some interested appraisers and Realtors for the seminar. In order toprovide the background necessary for appreciating the need for statistical valuationtechniques, Kat presents an overview of the basics of real estate appraisal. She beginsher presentation by explaining the appraisal process.

The first step in the appraisal process is to define the problem by identifying theproperty to be appraised, the property rights and the valuation date. One must alsodefine the use and scope of the appraisal, as well as stating the appraisals limitingconditions. In this case, the appraisals are to be used for real estate tax assessmentpurposes.

The second step includes the preliminary analysis, data selection and data collection,both general and specific. General data is information related to environmental, social,economic and governmental trends in the local market area. These include, but arenot limited to, land use constraints, demographic changes, supply/demand factors andzoning changes. Specific data include such things as property location andimprovements. Data for these various attributes allow comparison of the subjectproperty to the other recent property sales.


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The third step in the appraisal process is highest and best use analysis. This appraisalprinciple requires the appraiser to consider the subject property as though its usegenerates the highest net return to the property over the holding period, given currentmarket conditions. To determine highest and best use, the use must be legally

permissible (e.g., adhere to zoning laws), physically probable (e.g., the size of theproperty must satisfy the use), financially feasible (i.e., benefits must exceed the costs)and maximally productive (i.e., the use chosen must satisfy the aforementioned threerequirement and maximize expected returns).

Land value estimation, the fourth step, assumes that the land is vacant and that theland is improved (ready to be built upon). Four methods available to the appraiser forland value estimation are: (1) the sales comparison method; (2) the value extractionmethod; (3) the land residual method; and (4) the ground rent capitalization method.Kat explains that she will not discuss the land valuation methods further, given that

they are primarily used for commercial real estate appraisals; instead, she will discussthe valuation of residential properties.

The fifth step is application of the three appraisal approaches: market comparison,income capitalization and cost. The market comparison approach suggests that theindicated value of the subject property equals the value-weighted cash sales prices ofsimilar properties that have sold recently and are in close proximity to the subjectproperty, plus/ minus adjustments for dissimilar characteristics. The incomecapitalization approach states that the indicated value of the subject property equalsthe present value of the expected future income stream generated from any income

producing real estate investment. The cost approach implies that the indicated valueof the subject property equals the value of the land as though it was vacant, plus thedepreciated value of the improvements permanently attached to the land. These threevaluation approaches are mostly important for commercial properties, and Katreiterates that she wants to focus on the residential valuation problem.

The sixth and final step in the appraisal process is to reconcile the values of eachapproach and to determine a final value estimate. In each appraisal approach, theindicated value is value-weighted. Respective weights for each are then multiplied bytheir indicated values and summed to determine a final value estimate. For example,if an owner-occupied residential dwelling were appraised for $150,000 using themarket comparison approach and $155,000 using the cost approach, the appraiser mayplace a 70% weight on the market comparison approach value from subjectiveexperience on the job, but only a 30% weight on the cost value,1 for a final valueestimate of $151,500 [($150,000 * 70%) ($155,000 * 30%)].2 In the past, the countyhas been using a similar assessor-assigned weighted methodology to determineresidential valuation and, thus, the tax assessment value for each property. Theseminars participants realize that a statistical approach might offer an unbiasedimprovement over the existing subjective weighting method. A computer-basedapproach would also offer the potential for much quicker and less costly results.


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Mass Appraisal 69

Kats Presentation of the Market Comparison Approach

Kat then narrows her discussion by detailing the market comparison approach (MCA)for single-family residential properties. She reminds those attending the meeting thatthe indicated value of any subject property equals the value-weighted cash sales pricesof similar properties that have sold recently and are in close proximity to the subjectproperty, plus/ minus adjustments for dissimilar characteristics. She then explains thatthis approach is based in large part on the Principle of Substitution, which posits that. . . the value of a property tends to be set by the price that would be paid to acquirea substitute property of similar utility and desirability within a reasonable period oftime. Therefore, the reliability of the MCA is diminished if substitute properties arenot available in the market.3

Kat proceeds with a discussion of the various steps of the MCA. The first step is togather comparable sales data. This includes sales data for all comparable properties

(also known as comps) that have sold recently and are in close proximity to thesubject property. These data could be compiled from public records, the MultipleListing Service (MLS) database, lenders, builders, contractors and possibly appraisers.Many attendees at the meeting nod in agreement with Kats comments. She continuessaying that data would need to be cleaned for inaccuracies in the description ofthe propertys attributes. This would be labor intensive at first.

The second step is to choose the comps from Step 1 that are most similar to thesubject property. Some statisticians argue that this step minimizes the credibility ofthe MCA because the appraiser discards otherwise valuable data from omitted

comparable properties and reduces the sample size to as little as three observations.In appraising an owner-occupied residential dwelling, the appraiser most typically isrequired to retain only three or four recent sales (i.e., within six to nine months) thatare in close proximity to the subject property (i.e., within a three- to five-mile radius,if possible). But what if Step 1 produced 75 legitimate comps? Step 2 eliminatesinformation that otherwise could have been provided by the other 72 sales. Moreover,one likely will not convince a statistician that a sample size of three is statisticallysignificant, so inferences are weak at best and useless at worst. Certainly, the countywith its recent growth has sufficient sales comparables to merit a statistical analysis.

Step 3 requires the appraiser to adjust the comps sale prices for dissimilarcharacteristics, relative to the subject property. The five most common adjustmentsare:

1. Physical Characteristics: Valuation differences based on dissimilarphysical characteristics, such as square feet of living area, the number ofbedrooms and bathrooms, lot size, overall quality, age, the number ofdays the property was exposed to the market and other factors such asproperty condition.

2. Location: Valuation differences based solely on the desirability ofdifferent locations.

3. Market Conditions: Changes in the overall economy that may affectvalue.


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4. Financing Concessions: Special below-market seller or third-partyfinancing.

5. Conditions of Sale: Special sales concessions offered by the seller, suchas seller-paid closing costs or discount points, non-arms-length deals,

divorce or lawsuit settlements, condemnation sales, tax sales andforeclosure sales.

The appraiser then must quantify any adjustments by both magnitude and direction.The magnitude of the adjustment represents how much a characteristic contributes tooverall value. For example, an in-ground swimming pool may cost $35,000 to install,but if it contributes only $10,000 to overall value, then there would be only a $10,000adjustment to the comparables sales price.

The direction of the adjustment represents whether a comparables sales price shouldbe adjusted downward or upward by the dollar magnitude. If the comp has the

preferred characteristic over the subject property, then make a downward adjustment;if the subject has the preferred characteristic over the comparable property, then makean upward adjustment. The theory underlying this strategy is to transform thecomparable property to be like the subject property. For example, if the comparableproperty has the aforementioned pool but the subject property has no pool, then bytheoretically removing the pool, appraisal theory suggests that the comp would havesold for $10,000 less. If, on the other hand, the subject has a $2000 patio while thecomp has no patio, then theoretically transforming the comp so that it also has a patiowould result in the comp having sold for $2000 more.

Step 4 of the MCA is to determine the adjusted market price (AMP) for eachcomparable property. The AMP is simply a comps sales price, plus/minus alladjustments for dissimilar characteristics that are quantified in Step 3. Theoretically,the AMP represents the transformed value of the comp, as though it were now thesubject property. Step 5 is to value weight the comps AMPs. The comparable propertythat is considered to be the most similar to the subject property receives the highestweight and vice versa. These subjective weights are a function of the number ofadjustments for dissimilar characteristics and the magnitude of each adjustment (i.e.,the greater the similarity, the greater the weighting expressed in percentage terms).The sixth and final step is to determine the subjects indicated value, which is

calculated by summing all the weighted AMPs from Step 5.

Kat now explains to the group that the market comparison approach based on saleprices has several limitations. The past does not necessarily represent the future, butthe MCA analysis is based on past trends (i.e., historical data of recent sales), ratherthan current data or forecasts. In addition, it relies on sales data that may not exist insufficient quantities, particularly in less populated areas. Furthermore, even if thereare several recent sales, these properties may be so dissimilar to the subject propertythat the MCA is rendered useless. One or two of the meeting attendees nod becausethey know part of this west Texas county is still rural with limited residential sales.

Another drawback is that the appraisal becomes obsolete fairly quickly. Suppose anowner-occupied residential dwelling were to be appraised. If sold recently is defined


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Mass Appraisal 71

as no more than six to nine months, then in the best case scenario, all comparableproperties used for the MCA analysis would be outdated within only two to threequarters. A short time window exists for selecting comparables in order to make anappraisal.

Most importantly, the value-weighting process used in the MCA, as applied to theadjusted market prices, can be very subjective. Who is to say that Comp #1 shouldreceive a 40% weight, rather than a 25% weight? Multiple regression analysis,therefore, may likely overcome these deficiencies, particularly for tax assessors whomust assess thousands of properties annually.

Multiple regression analysis improves over the MCA approach by using many recentsales versus just a few. All sales are adjusted for statistically significant factors suchas living area. This statistical analysis decreases the likelihood of human error and

the problems of small samples.

At this point Kat encourages the group to take a coffee break prior to her beginningher presentation of her numerical illustration of multiple regression analysis.

The Nuts and Bolts of Multiple Regression Analysis: An Example

After the break, Kat begins her explanation of multiple regression analysis. She setsthe scene for the assembled group by telling them their task as a tax assessor is to

estimate the value of the subject property using the regression analysis outputprovided. You determine that the significant explanatory variables include square feetof living area, the number of days the property was on the market, square feet ofgarage area, whether there exists a fireplace and the age of the property. The subjectproperty has 1990 square feet of living area, was on the market for 76 days, has a450 square foot garage and a fireplace, and is 8 years old.

The regression equation to be estimated is:

SP LA DOM GARAGE FP AGE u , (1)i 0 1 i 2 i 3 i 4 i 5 i i

where SP is the response variable for the ith observation, 0, 1, . .. , 5 are theparameters that are estimated, and LA, DOM, GARAGE, FP and AGE (all importantproperty characteristics) are the independent or regressor variables. The error term, u,is the unknown error which represents the impact of all possible factors other thanthe explanatory variables on the response variable, SALES PRICE(SP). SALES PRICEis the variable that you as an assessor are trying to accurately estimate.

Using data supplied by Austin, Kat passes out a handout on the multiple regressioncomputer results (She also distributes an additional handout with more specificinformation regarding the mechanics of multiple regression analysis and this handoutis contained in the Appendix).


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Constant 235.32Standard error 115.22

R2 0.742No. of observations 185

Variable Coefficient Std. Error t-Statistic

Sq. Ft. of Living Area (LA) 64.46 24.02 2.68

Days on Market (DOM) 8.19 2.20 3.72

Sq. Ft. of Garage Area (GARAGE) 16.10 5.28 3.05

Fireplace (FP) 1245.12 1599.66 0.77

Age of Structure (AGE) 2555.02 742.11 3.44

The 185 observations are from recent residential sales within three miles of the subjectproperty. The comparables have sales prices within plus or minus $25,000 of thesubject property. The t-Statistic for each explanatory variable (i.e., the coefficientdivided by the standard error) is reported in the table above. All t-Statistics are greaterthan 2.57 , other than for FP, suggesting these regressors are significant at the 1%level in explaining SP. FP is insignificant, so it adds no statistically significantexplanatory power to SP. The R-squared statistic, also known as the coefficient ofdetermination, measures the correlation between the dependent and independentvariables. An R-squared statistic of .742 suggests that approximately 75% of the totalvariation in sales price is explained by the five independent variables (LA, DOM,GARAGE, FP and AGE). In other words, these are the variables upon which thecomparison of value hinges. In academic terms, it is known as the linear influence ofthe independent right-hand-side variables. One of the attendees laughs at Katsacademic jargon. She smiles and continues.

The point is that the influence on sales price of each explanatory variable, both indirection and magnitude, has been estimated by the model and, thus, not subject tohuman error. Every additional square foot of living area (LA) results in a $64.46increase in sales price. As expected, LA is positive because more LA is perceived asa positive effect on SP, all else held equal. Second, each additional day a property ison the market (DOM) results in an $8.19 decrease in sales price. DOM is negativebecause the longer a property is on the market, the greater the probability that theproperty is undesirable at its asking price. With respect to the size of the garage, everyadditional square foot of garage area results in a $16.10 increase in sales price.GARAGE is positive because more garage area is perceived as a positive effect onSP. If the property has a fireplace (FP), then the sales price would increase by$1245.12 because it is perceived to add value. From a statistical perspective, however,the variable insignificantly affects value, so the researcher may choose to rerun theregression equation with FP omitted. Finally, each additional year of age will causea $2555.02 decrease in sales price because older houses are less desirable than newones. This decrease is due to physical depreciation, functional obsolescence andexternal depreciation.


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Mass Appraisal 73

From the regression results above, the estimated assessed value of the subject propertywould equal $115,938. This value is calculated by multiplying each coefficientestimated from the equation with the subject propertys respective characteristic,summing these products and adding the intercept term, 0, which was estimated to be

$235.32. That is:

Assessed Value $235.32 ($64.46 1990 LA) ($8.19 76 DOM)

($16.10 450 GARAGE) ($1245.12 1 FP)

($2555.02 8 years) $115,938

Summary and Thoughts for Further Discussion

The group applauds Kats suggested solution to their mass appraisal needs. Kat again

says that, through its coefficient estimates, the multiple regression analysis makespossible factor weightings using a large number of comparable sales so that any oneproperty can be assigned accurate assessment value.

Several participants raise questions. One wants to know if this methodology reallycosts less in the long run, given the need to update data from recent sales and toinstall the multiple regression software with appropriate personnel training. Katresponds that there are benefits such as lower long-term costs, less human bias anderror when making adjustments for property differences, and more easily updatedassessment figures. Austin also notes that the county is occurring significant expenses

now when updating data for the old manual weighting system. He comments that acase-by-case system of human weighting will be replaced by a multiple regressionequation that would update itself over time. Adding recent sales data would allow theequation to update itself within seconds by way of the multiple regression softwareprogram.

Another participant questions the political ramifications of implementing this newsystem. What is the additional cost to the county and would the voters accept thisnew technology? Kat comments that few people know about how assessments areactually performedunlike the visible problems associated with hanging chads inpublic elections. This outlay for the updated assessment technology would be viewedas a beneficial investment that could easily be covered in the existing assessmentoffice budget. Austin agrees.

One county commissioner inquires if after the new multiple regression analysissoftware is up and running could the county actually reduce the number of employeesin the assessment office. Austin replies humorously that after the system isimplemented then the personnel needs for the office could be re-assessed.

Questions

1. Should the assessors office continue to value single-family residentialproperties using the manual method or should multiple regression


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analysis be utilized? What are the benefits and costs of changingmethodologies?

2. Kat describes the first step of the MCA. How do sold recently andin close proximity differ by property type? Give specific examples.

3. The MCAs third step is to quantify the magnitude of the adjustment.Explain at least two ways that appraisers estimate this magnitude.4. Highland Shores National Bank has employed you to review an appraisal

that was performed on a house in the Woodlawn subdivision. Sales forthe previous nine months in the area and the appropriate characteristicsare given in the table below. Using multiple regression analysis, evaluatethe previous appraisal of $174,600. Use your regression output to defendand explain your reasoning. Is the appraisal supported by yourregression? Explain.

Comp

Sales Price

($)

Square Feet of

Living Area

SquareFeet of Net

Area

Month House

Sold

Age of House

in Years

Fireplace

(Y/N)

1 146,250 2,202 698 10 3 Y

2 137,675 2,343 1,058 07 10 Y

3 170,950 2,332 1,269 12 4 Y

4 147,375 2,478 960 09 12 Y

5 156,750 2,336 1,056 12 10 Y

6 153,000 2,336 1,056 12 10 Y

7 141,200 2,371 914 04 11 Y

8 142,550 2,137 860 11 4 N

9 136,625 2,375 903 08 11 Y

10 148,700 2,354 1,032 06 9 Y

11 140,500 2,260 979 07 8 Y

12 152,975 2,274 1,057 12 10 N

13 143,200 2,206 765 11 2 Y

14 154,675 2,394 1,220 04 9 Y

15 153,300 2,260 883 06 2 Y

16 162,875 2,747 928 03 10 Y

17 150,925 2,601 1,082 08 14 N

18 159,475 2,580 1,270 10 9 Y

19 161,325 2,388 803 05 2 Y

20 165,750 2,440 1,064 03 6 Y

21 150,650 2,430 1,178 08 8 Y

22 146,175 2,547 1,101 04 15 Y

23 140,325 2,563 1,032 03 12 Y

24 143,550 2,612 921 03 13 Y

25 180,450 2,545 991 12 5 Y

26 160,450 2,671 1,309 05 13 Y

Subject ofAppraisal

? 2,450 1,125 12 8 Y


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Appendix

Multiple Regression Analysis

Below is a detailed discussion multiple regression analysis (MRA). While most

academicians (and some students) are well-versed on this topic, it is included as areview for those who have not used MRA recently and is provided by Kat to theparticipants in the meeting. Recognizing that Austins group knows very little aboutthe subject, she begins her discussion of MRA with a basic overview. She explainsthat it is a way to show how a response variable such as Y will vary with a set ofindependent (sometimes know as explanatory) variables such as X1, X2, . . . , Xn.

Kats Handout: An Overview of Multiple Regression Analysis

When we estimate the regression equation through a computer statistical package or

program, we are modeling our response variable Y as a function of the independentvariables or Xs. The variable Y will be determined by two components: a systematiccomponent captured by the multiple regression equation and a random componentthat is unknown. The second or random component is the part of the model that doesnot explain or capture variable Ys response. The random component is usuallyrepresented by the error term, u.

Suppose that a response variable Y can be predicted by a linear combination (orequation) of some independent variables X1, X2, . . . , Xn. Using MRA contained instatistical or spreadsheet software, tax assessors or real estate appraisers can estimate

the coefficients or parameters in the equation. These coefficients or parametersquantify how much a particular characteristic (independent variable or X) influencesthe propertys sales price. Thus, a multiple regression equation for two Xs (twoindependent or explanatory variables) can be described as:

Y X X u , (1a)i 0 1 1i 2 2i i

where Yi represents the ith value of the dependent variable, and X1i represents the ithobservation of the first X independent variable or X1, and X2i represents the ithobservation of the second X independent variable or X2. Two subscripts are used foreach variable: the first subscript represents the variable number and the second onerepresents the observation number. This equation is a linear model.

The method of ordinary least squares or OLS is used to estimate the 0, 1 and 2parameters of the equation. This method is a statistical technique that finds the bestlinear unbiased estimates (BLUE) under classical statistical assumptions. This processestimates 0, 1 and 2 by minimizing the sum of squares of the errors between thevalues of Y predicted by the equation and the actual values of Y.4 The actual or truevalues of 0, 1 and 2 are unknown, but the multiple regression model will try toestimate them. The value 0 is a constant term in the model. The random part of theequation is captured by u

i, and it represents the impact of all other factors besides the

independent variables (i.e., omitted explanatory variables) on the response variable Y.


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Regression analysis might be used to find out how well a houses selling price orvalue (for tax assessment, appraisal, or other reasons) can be predicted if severalvariables that help explain sales price (such as the square footage of the house andthe number of bathrooms) are known.5 The process begins by collecting recent sales

prices for homes in a particular neighborhood or census tract as well as their squarefootage, the number of bathrooms, age, and so forth. The intercept 0 can be estimatedand the variables 1, 2, ... n of a sales price equation can be used:

SALES PRICE SQUARE FOOTAGE BATHROOMS ... u ,i 0 1 i 2 i i (2a)

where SALES PRICE is the response variable; 0, 1, ... , n are the parameters thatare estimated; and SQUARE FOOTAGE (X1i), BATHROOMS (X2i), etc. are theindependent or regressor variables. The error term or u is the unknown error whichrepresents the impact of all possible factors other than the explanatory variables on

the SALES PRICE.

Then a multiple regression analysis is used to test hypotheses about the relationshipbetween a dependent variable, Y, and two or more independent variables, Xs. Multipleregression can also be used to make predictions about the Y variable. That is why thestatistical technique is useful to determine a propertys likely sales price or worth.

For the case of independent or explanatory variables, the model can be written as:

Y X X n . . . X u , (3a)1 0 1 1i 2 2i n ni i

where Xni

, represents the ith observation on the independent variable Xn.

Several assumptions are made when performing MRA. First, all relevant independentvariables are included, and the functional form of the model is correct. Typically, thefunctional form of the model is linear. Second, the expected value of each error term,u, is zero (that is, they sum to zero, with negative error terms being offset by positiveerror terms) and it represents the impacts of all possible factors other than theexplanatory variables. Third, the variance of the error terms is constant, and they areuncorrelated (that is, if they were correlated or connected, then it may indicate thatan important independent variable is missing). Finally, an assumption is made that theerror terms are normally distributed (meaning that they are random and would fit anormal distribution curve).

Another assumption required for the multiple regression linear model is that there isno exact linear relationship between the Xs (the independent or explanatory variables).If two or more explanatory variables are perfectly linearly correlated, it will beimpossible to calculate parameters of the equation. If two or more explanatoryvariables are highly but not perfectly linearly correlated (e.g., including both bedroomsand bathrooms), then the parameter estimates can be calculated, but the effect of eachof the highly linearly correlated variables on the explanatory variable cannot beisolated. Thus, the less connected the variables are, the more representative the model.


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The error term (also known as the disturbance or stochastic term) measures thedeviation of each observed Y value from the true (but unobserved) estimated value orregression line. The error terms arise because: (1) numerous explanatory variableswith only slight, and irregular effects on Y are omitted from the exact linear

relationship; (2) there are possible errors of measurement in Y; and (3) random humanbehavior is present.6

Endnotes

1. If fewer sales were available, an appraiser may place less weight on the MCA becausereliability is diminished.

2. Because this is not an income producing property, the income capitalization approach is notindicated.

3. The Appraisal of Real Estate, 12th edition, 2001.

4. Graphically, these error terms are quantified as the vertical distance between a plottedobservation and the true regression line.

5. Researchers have determined that several variables help explain a houses sales price. Theyinclude, but are not limited to, square feet of living area, square feet of net area under roof,the number of bedrooms, the number of bathrooms, the age of the property, the lot size, thelocation, a time trend variable to proxy economic conditions and a swimming pool.

6. For more information on multiple regression analysis, see Smith, Root and Belloit (1995),Downing and Clark (1997), Allison (1998), Appraising Residential Properties (1999),Baldwin (1999), Betts and Ely (2001), Lusht (2001), Ratterman (2001) and The Appraisalof Real Estate (2001).

Suggested Readings

Allison, P. D., Multiple Regression: A Primer, The Pine Forge Press Series in Research Methodsand Statistics, 1998.

Appraising Residential Properties, 3rd Edition, The Appraisal Institute, 1999.

Baldwin, P. N., Statistics: Know-How Made Easy, LmIT Publishing Co., 1999.

Betts, R. M. and S. J. Ely, Basic Real Estate Appraisal, 5th Edition, Prentice-Hall, 2001.

Downing, D. and J. Clark, Statistics: The Easy Way, Barrons Educational Services, Inc., 1997.

Lusht, K. M., Real Estate Valuation, KLM Publishing, 2001.

Ratterman, M., Residential Sales Comparison Approach: Deriving, Documenting, andDefending Your Value Opinion, The Appraisal Institute, 2001.

Smith, H. C., L. C. Root and J. D. Belloit, Real Estate Appraisal, 3rd Edition, GorsuchScarisbrick Publishers, 1995.

The Appraisal of Real Estate, 12th Edition, The Appraisal Institute, 2001.

The authors acknowledge the helpful comments and suggestions of Bill Hardin and an

anonymous reviewer.


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