Journal of Real Estate Practice and Education: Mass Appraisal: An Introduction to Multiple Regression Analysis for Real Estate Valuation

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Mass Appraisal: An Introduction to Multiple Regression Analysis for Real Estate Valuation

Benjamin, John D

Focus

This case study presents an introduction to the basics of real estate appraisal and multiple regression analysis; in particular, as used in real estate valuation for mass property tax assessment. While real estate researchers, appraisers and some tax assessors have used multiple regression analysis for many years, its use by a large number of assessors is relatively new. The purpose of this case is to expose students to standard appraisal approaches including the market comparison technique as well as the advantages and disadvantages of using multiple regression analysis. In their answers to the case, students are encouraged to explore and develop solutions, so as to understand how to use the market comparison approach and multiple regression analysis for real estate valuation.

Setting

The real estate tax assessment process is used to provide an introduction to multiple regression analysis. The tax assessor's office in a small west Texas county has always assessed properties through manual market comparison analysis. This manual process uses recently sold properties that are in close proximity to the subject property to make corresponding weighted adjustments. After going to a seminar on multiple regression analysis for mass appraisals, the county tax assessor employs a university professor to explain how multiple regression analysis works for real estate valuation and mass assessment, as well as what its relative benefits are over the existing manual system. He invites his staff, the county commissioners, and others to a one-night seminar that explains multiple regression analysis. This seminar presented by a university professor to Texas participants is used educate case readers about real estate appraisal and multiple regression analysis.

Exhibits

Multiple regression handout presented in Appendix.

Availability

This case is available through the ARES clearing house.

Teaching Notes

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

Introduction

This case study presents an overview of the basics of multiple regression analysis and illustrates its use in real estate valuation for mass property tax assessment. While real estate researchers, appraisers and some tax assessors have used this methodology for many 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 number of properties quickly and economically, which helps explain its (growing) popularity with tax assessors." The Appraisal of Real Estate (2001), published by The Appraisal Institute, 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 analysis and 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 tax assessors' conference in Washington, D.C. Being the assessor of a small west Texas county, 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 is intrigued by the use of multiple regression analysis to estimate real estate value for taxation purposes. His county has always assessed properties the old-fashioned way-through manual market comparison analysis of recently sold properties that are in close proximity to the subject property, a costly and time-consuming process. Furthermore, it requires hiring additional appraisers during the reassessment process and it is prone to human bias and error.

Changing from his county's assessment methodology to one using a multiple regression 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, would be eliminated. Second, rather than "guesstimating" adjustments in magnitude and direction, the multiple regression software output statistically estimates the adjustments 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, such as a fireplace or a swimming pool's contributory value, would not be necessary. Unnecessary as well would be determining whether or not a characteristic is a positive or a negative attribute. In multiple regression analysis, the direction of the adjustment is determined simply by the sign of the coefficient from the regression equation's output. Last, matched pairs analysis-an appraisal technique used in the traditional market comparison approach-becomes unnecessary. Although switching to a statistical analysis methodology would be costly and may require staff training, the ease of mass appraisals for property tax assessments would largely overcome these costs in the long run. Local elected officials, however, would also have to be persuaded of the benefits for these changes.

Believing that his county is ripe for statistical modernization, Austin decides to hold a seminar for his staff and the relevant county officials. His mission is to educate them on the benefits of utilizing multiple regression analysis. Not feeling qualified to teach 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 expertise in real estate valuation. Having attended Kat's community outreach classes in real estate, Austin is confident that she is the right person to get his associates up to speed on the use of multiple regression analysis.

Austin and Kat meet at her very small, nondescript university office. She explains that to teach the class effectively to novice students, she would need to present an overview of the appraisal process in general and the market comparison approach (MCA) as a specific method to appraising real estate. This overview would include the advantages and disadvantages of the MCA technique, as well as a demonstration of 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 a superior tool for assessing thousands of properties annually. Austin agrees with Kat's outline and suggests that she present a seminar. She agrees to do so in a couple of weeks, 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 meets with Austin's staff, several county commissioners, the Citizens for Financial Integrity Committee, and some interested appraisers and Realtors for the seminar. In order to provide the background necessary for appreciating the need for statistical valuation techniques, Kat presents an overview of the basics of real estate appraisal. She begins her presentation by explaining the appraisal process.

The first step in the appraisal process is to define the problem by identifying the property to be appraised, the property rights and the valuation date. One must also define the use and scope of the appraisal, as well as stating the appraisal's limiting conditions. In this case, the appraisals are to be used for real estate tax assessment purposes.

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 are not limited to, land use constraints, demographic changes, supply/demand factors and zoning changes. Specific data include such things as property location and improvements. Data for these various attributes allow comparison of the subject property to the other recent property sales.

The third step in the appraisal process is highest and best use analysis. This appraisal principle requires the appraiser to consider the subject property as though its use generates the highest net return to the property over the holding period, given current market 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 the property 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 three requirement and maximize expected returns).

Land value estimation, the fourth step, assumes that the land is vacant and that the land is improved (ready to be built upon). Four methods available to the appraiser for land value estimation are: (1) the sales comparison method; (2) the value extraction method; (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 discuss the 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 the indicated value of the subject property equals the value-weighted cash sales prices of similar properties that have sold recently and are in close proximity to the subject property, plus/minus adjustments for dissimilar characteristics. The income capitalization approach states that the indicated value of the subject property equals the present value of the expected future income stream generated from any income producing real estate investment. The cost approach implies that the indicated value of the subject property equals the value of the land as though it was vacant, plus the depreciated value of the improvements permanently attached to the land. These three valuation approaches are mostly important for commercial properties, and Kat reiterates 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 each approach and to determine a final value estimate. In each appraisal approach, the indicated value is value-weighted. Respective weights for each are then multiplied by their indicated values and summed to determine a final value estimate. For example, if an owner-occupied residential dwelling were appraised for $150,000 using the market comparison approach and $155,000 using the cost approach, the appraiser may place a 70% weight on the market comparison approach value from subjective experience on the job, but only a 30% weight on the cost value,1 for a final value estimate of $151,500 [($150,000 * 70%) + ($155,000 * 30%)].2 In the past, the county has been using a similar assessor-assigned weighted methodology to determine residential valuation and, thus, the tax assessment value for each property. The seminar's participants realize that a statistical approach might offer an unbiased improvement over the existing subjective weighting method. A computer-based approach would also offer the potential for much quicker and less costly results.

Kat's 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 that the indicated value of any subject property equals the value-weighted cash sales prices of similar properties that have sold recently and are in close proximity to the subject property, plus/minus adjustments for dissimilar characteristics. She then explains that this 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 acquire a substitute property of similar utility and desirability within a reasonable period of time. Therefore, the reliability of the MCA is diminished if substitute properties are not available in the market."3

Kat proceeds with a discussion of the various steps of the MCA. The first step is to gather 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 the subject property. These data could be compiled from public records, the Multiple Listing Service (MLS) database, lenders, builders, contractors and possibly appraisers. Many attendees at the meeting nod in agreement with Kat's comments. She continues saying that data would need to be "cleaned" for inaccuracies in the description of the property's attributes. This would be labor intensive at first.

The second step is to choose the comps from Step 1 that are most similar to the subject property. Some statisticians argue that this step minimizes the credibility of the 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 is required to retain only three or four recent sales (i.e., within six to nine months) that are 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 eliminates information 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 statistically significant, so inferences are weak at best and useless at worst. Certainly, the county with its recent growth has sufficient sales comparables to merit a statistical analysis.

Step 3 requires the appraiser to adjust the comps' sale prices for dissimilar characteristics, relative to the subject property. The five most common adjustments are:

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

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

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

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

5. Conditions of Sale: Special sales concessions offered by the seller, such as seller-paid closing costs or discount points, non-arms-length deals, divorce or lawsuit settlements, condemnation sales, tax sales and foreclosure sales.

The appraiser then must quantify any adjustments by both magnitude and direction. The magnitude of the adjustment represents how much a characteristic contributes to overall 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 $10,000 adjustment to the comparable's sales price.

The direction of the adjustment represents whether a comparable's sales price should be 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 make an upward adjustment. The theory underlying this strategy is to transform the comparable property to be like the subject property. For example, if the comparable property has the aforementioned pool but the subject property has no pool, then by theoretically removing the pool, appraisal theory suggests that the comp would have sold for $10,000 less. If, on the other hand, the subject has a $2000 patio while the comp has no patio, then theoretically transforming the comp so that it also has a patio would result in the comp having sold for $2000 more.

Step 4 of the MCA is to determine the adjusted market price (AMP) for each comparable property. The AMP is simply a comp's sales price, plus/minus all adjustments for dissimilar characteristics that are quantified in Step 3. Theoretically, the AMP represents the transformed value of the comp, as though it were now the subject property. Step 5 is to value weight the camps' AMPs. The comparable property that is considered to be the most similar to the subject property receives the highest weight and vice versa. These subjective weights are a function of the number of adjustments 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 subject's 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 sale prices has several limitations. The past does not necessarily represent the future, but the MCA analysis is based on past trends (i.e., historical data of recent sales), rather than current data or forecasts. In addition, it relies on sales data that may not exist in sufficient quantities, particularly in less populated areas. Furthermore, even if there are several recent sales, these properties may be so dissimilar to the subject property that the MCA is rendered useless. One or two of the meeting attendees nod because they 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 an owner-occupied residential dwelling were to be appraised. If "sold recently" is defined as no more than six to nine months, then in the best case scenario, all comparable properties used for the MCA analysis would be outdated within only two to three quarters. A short time window exists for selecting comparables in order to make an appraisal.

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

Multiple regression analysis improves over the MCA approach by using many recent sales versus just a few. All sales are adjusted for statistically significant factors such as 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 beginning her 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 sets the 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 output provided. You determine that the significant explanatory variables include square feet of living area, the number of days the property was on the market, square feet of garage area, whether there exists a fireplace and the age of the property. The subject property has 1990 square feet of living area, was on the market for 76 days, has a 450 square foot garage and a fireplace, and is 8 years old.

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

The 185 observations are from recent residential sales within three miles of the subject property. The comparables have sales prices within plus or minus $25,000 of the subject property. The t-Statistic for each explanatory variable (i.e., the coefficient divided by the standard error) is reported in the table above. All t-Statistics are greater than |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 significant explanatory power to SP. The R-squared statistic, also known as the coefficient of determination, measures the correlation between the dependent and independent variables. An R-squared statistic of .742 suggests that approximately 75% of the total variation 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 the comparison of value hinges. In academic terms, it is known as the linear influence of the independent right-hand-side variables. One of the attendees laughs at Kat's academic jargon. She smiles and continues.

The point is that the influence on sales price of each explanatory variable, both in direction and magnitude, has been estimated by the model and, thus, not subject to human error. Every additional square foot of living area (LA) results in a $64.46 increase in sales price. As expected, LA is positive because more LA is perceived as a positive effect on SP, all else held equal. second, each additional day a property is on the market (DOM) results in an $8.19 decrease in sales price. DOM is negative because the longer a property is on the market, the greater the probability that the property is undesirable at its asking price. With respect to the size of the garage, every additional 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 on SP. 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 the regression equation with FP omitted. Finally, each additional year of age will cause a $2555.02 decrease in sales price because older houses are less desirable than new ones. This decrease is due to physical depreciation, functional obsolescence and external depreciation.

Summary and Thoughts for Further Discussion

The group applauds Kat's suggested solution to their mass appraisal needs. Kat again says that, through its coefficient estimates, the multiple regression analysis makes possible factor weightings using a large number of comparable sales so that any one property can be assigned accurate assessment value.

Several participants raise questions. One wants to know if this methodology really costs less in the long run, given the need to update data from recent sales and to install the multiple regression software with appropriate personnel training. Kat responds that there are benefits such as lower long-term costs, less human bias and error when making adjustments for property differences, and more easily updated assessment figures. Austin also notes that the county is occurring significant expenses now when updating data for the old manual weighting system. He comments that a case-by-case system of human weighting will be replaced by a multiple regression equation that would update itself over time. Adding recent sales data would allow the equation to update itself within seconds by way of the multiple regression software program.

Another participant questions the political ramifications of implementing this new system. What is the additional cost to the county and would the voters accept this new technology? Kat comments that few people know about how assessments are actually performed-unlike the visible problems associated with "hanging chads" in public elections. This outlay for the updated assessment technology would be viewed as a beneficial investment that could easily be covered in the existing assessment office budget. Austin agrees.

One county commissioner inquires if after the new multiple regression analysis software is up and running could the county actually reduce the number of employees in the assessment office. Austin replies humorously that after the system is implemented then the personnel needs for the office could be "re-assessed."

Questions

1. Should the assessor's office continue to value single-family residential properties using the manual method or should multiple regression analysis be utilized? What are the benefits and costs of changing methodologies?

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

3. The MCA's 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 for the previous nine months in the area and the appropriate characteristics are given in the table below. Using multiple regression analysis, evaluate the previous appraisal of $174,600. Use your regression output to defend and explain your reasoning. Is the appraisal supported by your regression? Explain.

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

Endnotes

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

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

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

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

5. Researchers have determined that several variables help explain a house's sales price. They include, 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, the location, 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), Belts and Ely (2001), Lusht (2001), Ratterman (2001) and The Appraisal of Real Estate (2001).