Using Multiple Regression Analysis in Real Estate Appraisal.abstract This paper illustrates, using a representative appraisal report, the pitfalls or mistakes that real estate appraisers can make when using multiple regression Multiple regression The estimated relationship between a dependent variable and more than one explanatory variable. analysis (MRA MRA Medical Record Administrator. MRA Magnetic resonance angiography, see MR angiography ) to estimate the value of real estate. The paper also presents techniques appraisers can use to avoid these pitfalls. Most of the pitfalls can be avoided by using a sufficient number of comparable sales. The study concludes with recommendations to appraisers regarding the application of MRA in the appraisal of real estate. This study addresses the pitfalls of using multiple regression analysis (MRA) in real estate appraisal Real estate appraisal An estimate of the value of property using various methods. . It also addresses techniques for avoiding these pitfalls. The study begins with a brief overview of the use of MRA in the appraisal of real estate. Then the study presents a representative application of MRA to the appraisal of an industrial property. Next, the study lists and discusses the pitfalls of using MRA in real estate appraisal with illustrations of these pitfalls taken from the representative appraisal report. The study continues with a discussion of the techniques for avoiding these mistakes and problems. Finally, the study concludes with recommendations to appraisers regarding the application of MRA in the appraisal of real estate. Overview of the Application of MRA in the Appraisal of Real Estate The idea that one can estimate the value of a highly heterogeneous product, such as real estate, by using statistical techniques to compare its value to its characteristics has been around for a long time. Court is generally given credit for the development of the notion that the price or value of a heterogeneous good (automobiles in Court's 1939 article) can be modeled as a function of the good's characteristics (make, model, year, and accessories). [1] Court calls his modeling of automobile prices a "hedonic he·don·ic adj. 1. Of, relating to, or marked by pleasure. 2. Of or relating to hedonism or hedonists. [Greek h " price index because his model is not strictly derived from rigorous theory. Although Webster's Ninth New Collegiate col·le·giate adj. 1. Of, relating to, or held to resemble a college. 2. Of, for, or typical of college students. 3. Of or relating to a collegiate church. Dictionary defines hedonic as "of, relating to relating to relate prep → concernant relating to relate prep → bezüglich +gen, mit Bezug auf +acc , or characterized char·ac·ter·ize tr.v. character·ized, character·iz·ing, character·iz·es 1. To describe the qualities or peculiarities of: characterized the warden as ruthless. 2. by pleasure," [2] the term hedonic is now used by urban economists and real estate analysts to describe any regression of the price of a parcel of real estate upon the characteristics of the real estate. The technique is called "hedonic" because the relationship between the price and the characteristics of the real estate is whatever fits the data. Thus, the technique is driven by data, not theory. Griliches popularized the technique by using it to develop an index of quality change in automobiles [3] Shenkel is one of the earliest appraisers to demonstrate that the technique can also be used to predict real estate values. [4] In 1978 the American Institute of Real Estate Appraisal gave multiple regression analysis even more visibility by including the technique in an appendix of the seventh edition of The Appraisal of Real Estate. The current (12th) edition of The Appraisal of Real Estate includes an example of how to use multiple regression analysis to estimate the value of real estate. More recently, Ramsland and Markham demonstrated and discussed how to apply MRA to the appraisal on an industrial property. [5] Also, MRA has become popular among tax assessors as a mass appraisal technique. MRA has become even easier to use as more and more personal computer software become available that can perform the necessary calculations quickly and accurately. Today, both of the two most popular spreadsheet packages (Lotus and Excel) contain builtin multiple regression functions. Access to these builtin multiple regression functions gives appraisers a powerful tool to use in the appraisal process. Unfortunately, powerful tools can produce powerful mistakes just as quickly as they can produce powerful results. As shown below, the complexities and data requirements of MRA make mistakes such as erroneous erroneous adj. 1) in error, wrong. 2) not according to established law, particularly in a legal decision or court ruling. applications, erroneous conclusions, and unsupported interpretations of the results all too easy for the unsuspecting appraiser A person selected or appointed by a competent authority or an interested party to evaluate the financial worth of property. Appraisers are frequently appointed in probate and condemnation proceedings and are also used by banks and real estate concerns to determine the market to commit. A Representative Application of MRA in Real Estate Appraisal The appraisal of a manufacturing plant near a small, Midwestern city is selected as representative of the application of MRA in real estate appraisal. This particular appraisal is selected for three reasons: 1. The appraisal is performed by a welltrained appraiser. 2. The appraiser follows and cites Ramsland and Markham [6] in the report. 3. The final estimate of value is based entirely upon MRA. The subject property is a manufacturing plant site that consists of 24 buildings containing 2.2 million square feet of floor space on 178 acres of land. The site is connected by a rail siding A siding, in general rail terminology, refers to a section of track distinct from a through route such as a main line or branch line or spur. It may connect to through trackage or to other sidings at one or both ends. to a major rail line, and it has several accesses to a nearby state highway. The site is zoned heavy industrial, and it is assessed for 1997 property tax purposes at $16.58 million. The subject property is being appraised as part of the owner's efforts to reduce the assessed value of the property. The appraiser is statecertified, holds the MAI MAI Mail (File Name Extension) MAI Multilateral Agreement on Investment MAI Maius (Latin: May) MAI Ministerul Administratiei si Internelor (Romanian) , SREA SREA Saskatchewan Real Estate Association (Canada) SREA Supplier Request for Engineering Approval SREA Structural Roof Erectors Association (Portland, OR) SREA Smith Rea Energy Associates Ltd , and CRE CRE Commercial Real Estate CRE Corporate Real Estate CRE Commission for Racial Equality (Scotland) CRE CCD (Charge Coupled Device) and Readout Electronics CRE Camp Response Element designations, has a bachelor's degree from a major state university, and has considerable experience appraising industrial property. The appraiser applies the cost approach and market [7] approach (using MRA) but not the income capitalization capitalization n. 1) the act of counting anticipated earnings and expenses as capital assets (property, equipment, fixtures) for accounting purposes. 2) the amount of anticipated net earnings which hypothetically can be used for conversion into capital assets. approach to the subject property. The cost approach estimate is given no weight in the final estimate of value. Thus, the final estimate of value is based completely on the MRA performed by the appraiser. The results of the appraiser's MRA is shown in Table 1. Table 1 is an exact replica Earlier document exchange software from Farallon Communications, Inc. that converted a Windows or Mac document into a proprietary viewing format. The viewer could be distributed separately or embedded within the document itself, turning it into a singledocument viewer. of the page in the subject appraisal report that contains the results of the appraiser's MRA. The format of Table 1 (and the exhibit in the appraiser's repot Verb 1. repot  put in a new, usually larger, pot; "The plant had grown and had to be repotted" pot  plant in a pot; "He potted the palm" ) is nearly identical to the format Ramsland and Markham use in their article on applying multiple regression analysis. In deposition, the appraiser reports that Lotus software Lotus Software (called Lotus Development Corporation before its acquisition by IBM) is an American software company with its headquarters in Cambridge, Massachusetts. was used to perform the calculations. In an effort to support and bolster This article is about the pillow called a bolster. For other meanings of the word "bolster", see bolster (disambiguation). A bolster (etymology: Middle English, derived from Old English, and before that the Germanic word bulgstraz the estimate of value, the appraiser reports that the [R.sup.2] of the regression model (0.8503) "means that 85% of the differences between the independent variables and dependent variable can be explained by the data." The appraiser also states that the standard error of the regression model (1.4078) indicates that "a range in per square foot values of $1.41 per square foot above and below the predicted value of the subject can be supported." The final estimate of value based upon the regression model is $4.45 per square foot or about $9.735 million (41.3% below its assessed value) as of January 1, 1997. The Pitfalls In any application of MRA to real estate, the major pitfalls can be found in two areas: * The model specification * The robustness (sensitivity to changes in underlying assumptions, data, and procedures) of the results of the regression Although the terms model specification and robustness are well known in statistics, appraisers in general are not familiar with them. As a result, further discussion of these terms is warranted. Model specification is concerned primarily with: 1. The choice of dependent and independent variables In mathematics, an independent variable is any of the arguments, i.e. "inputs", to a function. These are contrasted with the dependent variable, which is the value, i.e. the "output", of the function. 2. The functional form of the relationship between these variables 3. The statistical significance of the independent variables By examining these three aspects of a MRA, a judgement can be made regarding the trustworthiness trustworthiness Ethics A principle in which a person both deserves the trust of others and does not violate that trust of the model specification. Robustness of the results of the regression analysis In statistics, a mathematical method of modeling the relationships among three or more variables. It is used to predict the value of one variable given the values of the others. For example, a model might estimate sales based on age and gender. is concerned primarily with: 4. The multivariate The use of multiple variables in a forecasting model. normality normality, in chemistry: see concentration. of the data [9] 5. The sensitivity of the results to variations in the individual sales used in the analysis 6. Measurement errors in the data Examination of these three aspects of a MRA gives insight into the robustness of the results. Examination of these six critical aspects sheds light upon the extent that the results of any MRA support the resultant This article is about the resultant of polynomials. For the result of adding two or more vectors, see Parallelogram rule. For the technique in organ building, see Resultant (organ). In mathematics, the resultant of two monic polynomials estimate of value. Appraisers who ignore these six critical aspects of MRA risk falling into the pitfalls found therein. Each of these six critical aspects is examined further below with illustrations of the pitfalls taken from the representative appraisal report. Model Specification Analysis of the specification of any multiple regression model focuses upon three primary issues: 1. The choice of dependent and independent variables 2. The functional form of the relationship, between these variables 3. The statistical significance of the independent variables The functional relationship between the dependent and independent variables in a wellspecified model will be: * Supported by theory to the fullest extent possible * Free of builtin, spurious relationships In statistics, a spurious relationship (or, sometimes, spurious correlation) is a mathematical relationship in which two occurrences have no causal connection, yet it may be inferred that they do, due to a certain third, unseen factor (referred to as a "confounding factor" Moreover, the independent variables in a wellspecified model will be statistically significant at some acceptable level of confidence. (Statistical inference Inferential statistics or statistical induction comprises the use of statistics to make inferences concerning some unknown aspect of a population. It is distinguished from descriptive statistics. and confidence levels are discussed further below.) Pitfall pit·fall n. 1. An unapparent source of trouble or danger; a hidden hazard: "potential pitfalls stemming from their optimistic inflation assumptions" New York Times. Number One: The Choice of Dependent and Independent Variables Most appraisers are very good at selecting relevant and important dependent and independent variables to include in an MRA model. Appraisal textbooks and literature are full of rich examples of what units of comparison to use in appraising most any type of property. This same literature is also rich in examples of what property characteristics to include in the appraiser's analysis of comparable sales. Thus, appraisers usually do not fall into this particular pitfall. Indeed, in the representative appraisal report, the appraiser selects a defendable dependent variable and a defendable set of property characteristics for inclusion in the MRA model as independent variables. Pitfall Number Two: The Functional Form of the Model In the representative appraisal report, the appraiser's MRA model actually includes location, date of sale, effective age, landtobuilding area ratio, and the log of building area as the independent variables with price per square foot of building area as the dependent variable. Upon first glance, the model appears to contain a logical set of independent variables. Certainly, factors such as location, date of sale, building size, and effective age have an influence on value. The problem with the specification of this model is that it uses building area three times: 1. As the denominator denominator the bottom line of a fraction; the base population on which population rates such as birth and death rates are calculated. denominator of the dependent variable (price per square foot of building area) 2. In the denominator of one of the independent variables (landtobuilding area ratio) 3. In log form (log of building area) as another independent variable This multiple use of building area has the potential of distorting the true relationship between value and building area as well as introducing a spurious relationship between value and building area in this model. Pitfall Number Three: Statistical Significance The quintessential quin·tes·sen·tial adj. Of, relating to, or having the nature of a quintessence; being the most typical: "Liszt was the quintessential romantic" Musical Heritage Review. tests of model specification are the t and ftests of the statistical significance of the parameter (1) Any value passed to a program by the user or by another program in order to customize the program for a particular purpose. A parameter may be anything; for example, a file name, a coordinate, a range of values, a money amount or a code of some kind. estimates of the model. A 95% confidence level (maximum of a 5% error) is used in this study to distinguish between significant and insignificant results. [10] That is, the chance that a particular parameter estimate is equal to zero can be rejected with at least 95% confidence (no more than a 5% chance of incorrectly rejecting the hypothesis that the coefficient coefficient /co·ef·fi·cient/ (ko?ahfish´int) 1. an expression of the change or effect produced by variation in certain factors, or of the ratio between two different quantities. 2. is zero). In Table 1, all of the parameter estimates (called X coefficients in Table 1) in the regression model as well as the fvalue of the regression are statistically insignificant with 95% confidence. The fact that the X coefficients or parameter estimates are statistically insignificant means that the set of independent variables used by the appraiser contribute very little to the final estimate of value. In addition, the appraiser uses an adjustment factor for date of sale (6.3% per year) that cannot be supported by the regression model because the X coefficient for date variance (date of sale) is statistically insignificant (again, with 95% confidence). Indeed, the regression model implies that no adjustment be made for date of sale or time because the X coefficient for date of sale is statistically insignificant (no different than zero). In short, the X coefficients in the appraiser's MRA model are suspect. Moreover, Table 1 reveals that the standard errors are so large and t and fstatistics so low that anyone wellversed in MRA would not rely on these results for a supportable estimate of value. The appraiser has fallen deep into the model specification pit because the multiple regression results suffer from poor model specification. However, it is somewhat understandable why the appraiser succumbed to this particular pitfall. It is apparent that the appraiser followed the example MRA found in Ramsland and Markham. Unfortunately, Ramsland and Markham fall into this same pit, leading other appraisers to do likewise. Robustness of the Results of the Regression Another extremely important aspect of regression analysis is to examine the robustness of the results of the results of the regression. The use of MRA to predict value is an application of statistical inference. Statistical inference is a type of reasoning that proceeds from the particular to the general or, in other words Adv. 1. in other words  otherwise stated; "in other words, we are broke" put differently , from a few cases to the universal situation. That is, based upon the relationship between a dependent variable (such as value or price) and a set of independent variables (such as property characteristics), the appraiser infers an estimate of value for a subject property (the general situation) from the selling prices of a set of comparable properties (the particular situations). This inference (logic) inference  The logical process by which new facts are derived from known facts by the application of inference rules. See also symbolic inference, type inference. works best when the results are insensitive in·sen·si·tive adj. 1. Not physically sensitive; numb. 2. a. Lacking in sensitivity to the feelings or circumstances of others; unfeeling. b. to departures from the assumptions used in obtaining the results. The fundamental assumption underlying all MRA is that the data used to fir the regression model form a multivariate normal distribution
In probability theory and statistics, a multivariate normal distribution, also sometimes called a multivariate Gaussian distribution . (See footnote Text that appears at the bottom of a page that adds explanation. It is often used to give credit to the source of information. When accumulated and printed at the end of a document, they are called "endnotes." 9 for further discussion of multivariat e normal distributions.) When the results of a MRA are insensitive to slight departures from the assumption of multivariate normality, the results are said to be robust. The appraiser does not report any examination of the robustness of the results. Therefore, it is necessary to replicate rep·li·cate v. 1. To duplicate, copy, reproduce, or repeat. 2. To reproduce or make an exact copy or copies of genetic material, a cell, or an organism. n. A repetition of an experiment or a procedure. the MRA results in order to investigate their robustness. This is done using version 6.07 of TS301 SAS (1) (SAS Institute Inc., Cary, NC, www.sas.com) A software company that specializes in data warehousing and decision support software based on the SAS System. Founded in 1976, SAS is one of the world's largest privately held software companies. See SAS System. computer software." The raw data used by the appraiser are fitted to a model identical to that used by the appraiser. The findings of this analysis are reported below. Pitfall Number Four: Multivariate Normality First and foremost, robustness requires that the distribution of the data be multivariate normal or at least closely multivariate normal. Tests for multivariate normality are not well developed, but tests for univariate normality are available. Data that is multivariate normal is always univariate normal, but not vice versa VICE VERSA. On the contrary; on opposite sides. . Thus, if a data set is not univariate normal, it cannot be multivariate normal. The ShapiroWilk test In statistics, the ShapiroWilk test tests the null hypothesis that a sample x_{1}, ..., x_{n} came from a normally distributed population. It was published in 1965 by Samuel Shapiro and Martin Wilk. for univariate normality is particularly useful in testing for multivariate normality. Unfortunately, the MRA features of the standard spreadsheet packages do not include this important test statistic statistic, n a value or number that describes a series of quantitative observations or measures; a value calculated from a sample. statistic a numerical value calculated from a number of observations in order to summarize them. . Therefore, it is calculated using the SAS software package cited above. For any data set to be multivariate normal, it must also be univariate normal one variable at a time. Thus, if a ShapiroWilk statistic is not statistically significant, then the variable is not distributed normally, and the set of independent variables cannot form a multivariate normal distribution. If all of the ShapiroWilk statistics are statistically significant, then the independent variables might form a multivariate normal distribution, but there is no guarantee of it and further testing is required to confirm the presence of multivariate normality. Testing the appraiser's independent variables using the ShapiroWilk statistic reveals that the appraiser's comparable sales data set does not form a multivariate normal distribution. Only one of the independent variables is distributed normally, namely the log of building size. So, the log of building size should improve the chances of the data set being multivariate normal, but it does not achieve this end. The lack of multivariate normality is not always a serious problem in MRA. If the sample size of the data set is large enough, the results often are asymptotically multivariate normal. That is, as more and more observations are added to the data set, the data becomes more and more multivariate normal. Unfortunately, the appraiser's set of comparable sales is far too small to ensure asymptotic multivariate normality. Hair, Anderson, Tathum, and Black suggest that a data set should have at least ten observations for each independent variable in the model, [12] while others call for 30 observations for each independent variable. Thus, the results of the MRA lack robustness due to the lack of multivariate normality and the small number of comparable properties the appraiser uses to conduct the MRA. Pitfall Number Five: The Use of IIIConditioned Comparable Sales Robustness of the results of a MRA also requires a data set that is wellconditioned. That is, the results of the regression analysis should not be sensitive to the deletion deletion /de·le·tion/ (dele´shun) in genetics, loss of genetic material from a chromosome. de·le·tion n. Loss, as from mutation, of one or more nucleotides from a chromosome. of one of the observations in the data set. One way in which a data set can be compromised is by something called illconditioned data. Further examination of the 10 comparable sales using appropriate statistical techniques [13] reveals that the appraiser's data are illconditioned. That is, the parameter estimates and subsequent estimate of value of the subject property are significantly affected by some of the comparable properties, specifically comparable sales number 1, 9, and 10. The statistical tests for illconditioned data amount to the search for influential observations of comparable sales. One statistic that captures this effect is called the Dffits statistic. Unfortunately, standard spreadsheet packages do not calculate Difits statistics. Therefore, the Dffits values associated with each comparable sale are calculated using the previously cited SAS computer software. The Dffits values for comparable sales 1, 9, and 10 are 4.124, 4.735, and 10.375, respectively. In cases involving small sample size, Dffit values greater than 1.0 indicate influential sales, which in a regression analysis tend to distort the parameter estimates and estimates of value obtained from the model. Thus, the robustness of the results of the regression analysis is further weakened weak·en tr. & intr.v. weak·ened, weak·en·ing, weak·ens To make or become weak or weaker. weaken·er n. by the appraiser's use of illconditioned data. Pitfall Number Six: Measurement Errors Measurement errors in the data used in a regression analysis can also destroy the robustness of the results. It is important for any source of measurement error in the data to be carefully evaluated by the appraiser. Several of the property characteristics selected by the appraiser are worthy of this sort of evaluation. For example, the appraiser offers no objective data or information to support his measurement of two critically important property characteristics: location and age. Measurement errors are almost certainly present in these two property characteristics, due to the subjective nature of these measurements. The appraiser measures location using an ordinal scale ordinal scale (or´dn of 1.0 to 4.0, with 4.0 being the best location. Five of the comparable sales have a location score of 4.0, and the comparable sale in the poorest location has a score of 1.5. The appraiser gives the subject property a location score of 1.0, which lies outside the range of the location scores of the comparable properties. Thus, it is not possible for the appraiser to extract a defensible de·fen·si·ble adj. Capable of being defended, protected, or justified: defensible arguments. de·fen adjustment factor for location from the comparable properties. Moreover, half of the comparable properties have location scores four times that of the subject property. The appraiser should have found comparable properties with location scores both above and below the subject property's location score to properly adjust for location, and the appraiser should have used a more objective measure of location, such as proximity to nearby cities and major transit routes A sea route which crosses open waters normally joining two coastal routes. . In order to infer reasonably the adjustment factor for any property characteristic, it is necessary for the characteristics of the comparable sales to straddle In the stock and commodity markets, a strategy in options contracts consisting of an equal number of put options and call options on the same underlying share, index, or commodity future. the characteristic of the subject property. If this straddling strad·dle v. strad·dled, strad·dling, strad·dles v.tr. 1. a. To stand or sit with a leg on each side of; bestride: straddle a horse. b. is not achieved, the appraiser is extrapolating the adjustment factor beyond the range supported by the comparable sales data. [14] Thus, the location variable may contain some serious measurement errors. Effective age is another variable that may contain measurement errors. Chronological age chron·o·log·i·cal age n. Abbr. CA The number of years a person has lived, used especially in psychometrics as a standard against which certain variables, such as behavior and intelligence, are measured. may be a better measure, but there are some problems with using chronological age as an indicator of effective age. One way of dealing with this problem is to include chronological age squared as well as chronological age in the set of independent variables. [15] But the extent of these measurement errors is impossible to evaluate without onsite inspections of the comparable and subject properties to gather more objective data regarding the location and effective age of each property. The appraiser should have included a discussion of objective data that supports the subjective measurements in the report. But because this discussion is not included in the report, the full effect of these measurement errors cannot be assessed. The regression analysis uses data that is not multivariate normal, and the data contain influential observations as well as measurement errors. Thus, the lack of robust results implies that the MRA should not be used as a basis for estimating the value of the subject property. How to Avoid the Pitfalls The representative appraisal report illustrates exceptionally well the pitfalls that await AWAIT, crim. law. Seems to signify what is now understood by lying in wait, or waylaying. the unsuspecting appraiser who attempts to use MRA as a tool for estimating the value of real estate. It is understandable why the author of the representative appraisal report fell into so many of these pitfalls; until recently, the appraisal literature did not warn appraisers about them. Appraisers can avoid these pitfalls by doing two things. First and foremost, appraisers using MRA must use more comparable sales than they are accustomed to using. A good rule of thumb is to us at least ten, preferably more, comparable sales per independent variable included in the MRA model However, appraisers are often required to work with very little data. When this is the case, it is a gross misrepresentation misrepresentation In law, any false or misleading expression of fact, usually with the intent to deceive or defraud. It most commonly occurs in insurance and realestate contracts. False advertising may also constitute misrepresentation. and standards violation to employ a datahungry statistical technique like MRA present it as being reliable and accurate. Secondly, appraisers who employ MRA should conduct the same sort of statistical tests discussed it this paper. Of course, most appraisers do not have the training nor the computer software to perform the statistical tests necessary to avoid the pitfalls. Fortunately, Levine, Berenson, and Stephan contains both the information and spreadsheet software fox performing these tests. [16] This particular text should be easily understandable by most appraisers, because it is commonly used in undergraduate business statistics courses throughout the country. Hans R. Isakson, PhD, is a corecipient of the 1979 Arthur A. May Award and is a professor of economics in the department of finance at the University of Northern Iowa The University of Northern Iowa, in Cedar Falls, Iowa, was founded in 1876, as the Iowa State Normal School. It has colleges of Business Administration, Education, Humanities and Fine Arts, Natural Sciences, and Social and Behavioral Sciences, and a graduate school. , Cedar cedar, common name for a number of trees, mostly coniferous evergreens. The true cedars belong to the small genus Cedrus of the family Pinaceae (pine family). Fails, Iowa. Excerpt ex·cerpt n. A passage or segment taken from a longer work, such as a literary or musical composition, a document, or a film. tr.v. ex·cerpt·ed, ex·cerpt·ing, ex·cerpts 1. from Papers and Proceedings, published by Valuation 2000 in July 2000. (1.) A.T. court, "Hedonic Price Indexes with Automotive Examples," The Dynamics of Automobile Demand (New York New York, state, United States New York, Middle Atlantic state of the United States. It is bordered by Vermont, Massachusetts, Connecticut, and the Atlantic Ocean (E), New Jersey and Pennsylvania (S), Lakes Erie and Ontario and the Canadian province of : General Motors, 1939). (2.) Webster's Ninth New Collegiate Dictionary (Springfield, Massachusetts Springfield is a city in Massachusetts, United States. It is the county seat of Hampden County.^{GR6} In the 2000 census, the city population was 154,082. : MerriamWebster, Inc., 1990): 350. (3.) Z. Griliches, "On an Index of Quality Change," Journal of American Statistical Society (56: 1961): 535548. (4.) W.M. Shenkel, Modern Real Estate Appraisal (New York: McGrawHill, 1978). (5.) Maxwell O. Ramsland and Daniel E. Markham, "MarketSupported Adjustments Using Multiple Regression Analysis," The Appraisal Journal (April 1998): 181191. (6.) Ibid. (7.) The market approach to value is also known as the sales comparison approach The sales comparison approach (SCA) is one of the three major groupings of valuation methods, called the three approaches to value, commonly used in real estate appraisal. to value. (8.) Ibid. (9.) In order to make reliable inferences (i.e., predict the dependent variable) from the data being analyzed an·a·lyze tr.v. an·a·lyzed, an·a·lyz·ing, an·a·lyz·es 1. To examine methodically by separating into parts and studying their interrelations. 2. Chemistry To make a chemical analysis of. 3. , the data must be distributed multivariate normal. See Henri Theli, Principles of Econometrics econometrics, technique of economic analysis that expresses economic theory in terms of mathematical relationships and then tests it empirically through statistical research. (New York: John Wiley John Wiley may refer to:
(10.) See Appraisal Institute The Appraisal Institute (Institute), headquartered in Chicago, Illinois, is an international association of professional real estate appraisers.^{[1]} It was founded in January 1991 when the American Institute of Real Estate Appraisers (AIREA) and the , The Appraisal of Real Estate, 11th ed. (Chicago: Appraisal Institute, 1996): 731736 for a discussion of statistical inference and confidence levels. (11.) SAS Institute SAS Institute Inc., headquartered in Cary, North Carolina, USA, has been a major producer of software since it was founded in 1976 by Anthony Barr, James Goodnight, John Sall and Jane Helwig. , Inc., World Headquarters, SAS campus Drive, Cary, North Carolina Cary is the second largest municipality in Wake County, North Carolina and the third largest municipality in The Triangle (North Carolina) behind Raleigh and Durham. It is the seventh largest municipality in North Carolina. (1989). (12.) Hair, Anderson, Tathum, and Black, multivariate Data Analysis with Readings (New York: McMillan, 1992). (13.) See Belsley, Kuh, and Welsch for discussion of these techniques. (14.) See Neter, et al,, 8485 for a discussion of making inferences outside the range of the data set. (15.) Marvin L. Wolverton, "Empirical Analysis of the Breakdown Method of Estimating Physical Depreciation," The Appraisal Journal (April 1998): 163171. (16.) David M. Levine, Mark L. Berenson, and David Stephan, Statistic for Managers Using Microsoft Excel (tool) Microsoft Excel  A spreadsheet program from Microsoft, part of their Microsoft Office suite of productivity tools for Microsoft Windows and Macintosh. Excel is probably the most widely used spreadsheet in the world. Latest version: Excel 97, as of 19970114. (New Jersey: Prentice Hall Prentice Hall is a leading educational publisher. It is an imprint of Pearson Education, Inc., based in Upper Saddle River, New Jersey, USA. Prentice Hall publishes print and digital content for the 612 and higher education market. History In 1913, law professor Dr. , 1989). References Appraisal Institute. The Appraisal of Real Estate, 7th and 11th editions. Chicago: Appraisal Institute, 1978 and 1996. Belsley, D.A., Kuh, E. and Welsch, R.E. Regression Diagnostics. New York: John Wiley & Sons, 1980. Dilmore, Gene. "Of Regression Analysis, Business Valuation, Lotus 123, HewlettPackard, and William of Ockham." Business Valuation Review (June 1995): 7582. Table 1 MRA Results Taken from Representative Appraisal Base Date: 1/15/1992 Date of No. Location Sale When Built Sale Price Size 1 Ypsilanti Twp., MI 3/15/1997 1942, 78 $10,600,000 2,166,600 2 Silvis, IL 1/31/1997 1966 $2,625,000 751,658 3 Davenport, IA 9/14/1995 1967, 78 $10,500,000 2,422,650 4 Spirit Lake, IA 7/14/1995 1968, 72 $1,850,000 224,573 5 Columbus, OH 3/7/1995 1971 $20,000,000 3,917,800 6 Framingham, MA 12/21/1994 1947, 87 $8,000,000 2,866,526 7 Springfield, MO 6/22/1994 1967, 81 $10,000,000 1,698,161 8 Romulus, MI 7/15/1993 1942, 78 $6,670,000 1,046,260 9 Mentor, OH 5/17/1993 1969 $5,825,000 1,108,828 10 Underwood, IA 1/15/1992 1973 $4,517,275 405,160 Actual Price/ Predicted Precent Date No. Location Sq. Ft. Price/Sq. Ft. Variance Variance 1 Ypsilanti Twp., MI $4.89 $3.81 22.18% 62 2 Silvis, IL $3.49 $4.56 30.17% 60 3 Davenport, IA $4.33 $4.65 7.30% 43 4 Spirit Lake, IA $8.24 $7.94 3.59% 41 5 Columbus, OH $5.10 $4.97 2.70% 37 6 Framingham, MA $2.79 $2.90 4.00% 35 7 Springfield, MO $5.89 $5.43 7.79% 29 8 Romulus, MI $6.38 $5.49 13.95% 18 9 Mentor, OH $5.25 $7.25 38.03% 16 10 Underwood, IA $11.15 $10.52 5.66% 0 Eff. Land/Bldg. Log of No. Location Age Location Ratio Bldg. Size 1 Ypsilanti Twp., MI 30 4.0 2.0105234 6.3357787 2 Silvis, IL 30 2.0 3.5425952 5.8760203 3 Davenport, IA 28 3.0 3.6318432 6.3842907 4 Spirit Lake, IA 25 1.5 4.6532753 5.3513575 5 Columbus, OH 26 4.0 1.7118025 6.5930423 6 Framingham, MA 35 4.0 2.2794142 6.4573559 7 Springfield, MO 28 3.0 3.1225249 6.2299789 8 Romulus, MI 33 4.0 4.7712567 6.0196396 9 Mentor, OH 28 4.0 7.5689827 6.0448642 10 Underwood, IA 24 2.0 17.190246 5.6076266 Regression Output Constant Std Err of Y Est [R.sup.2] No. of Observations Degrees of Freedom X Coefficient Std Error tstat fstat Subject property 1/1/1996 $10,379,562 2,185,171 $.475 29.704 1.4087 0.8503 10 4 0.03 0.285 0.37122 0.1245317 2.716181 0.0445 0.1811 0.98715 0.2079735 2.3080811 0.68 1.574 0.37606 0.5987864 1.176814 4.54 Subject property 47 25 1.0 3.5469215 6.3394854 Note: This Exhibit is presented in the same format as that found in Ramsland and Markham.[8] The column labeled Actual Price/Sq. Ft. is the dependent variable. The last four columns from the left contain the independent vaiables. The output from the MRA is to the right. The line at the bottom of the spreadsheet contains data for the subject property, including the estimated value ($10,379,562) from the MRA. Contact the author for a copy of this spreadsheet containing the spreadsheet functions used that generate the Regression Output. 

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