The Application of Hedonic Grid Regression to Commercial Real Estate Abstract This paper applies hedonic regression to estimate the grid adjustment factors for a national sample of commercial office properties. The paper demonstrates the viability of hedonic grid regression in commercial real estate. Several robustness tests are employed to test the reliability of the empirical results. The study finds that the hedonic approach yields slightly more accurate and stable prediction result than a basic matching model without hedonic adjustments. 1
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The Application of Hedonic Grid Regression to Commercial Real Estate
Abstract
This paper applies hedonic regression to estimate the grid adjustment factors for a national
sample of commercial office properties. The paper demonstrates the viability of hedonic
grid regression in commercial real estate. Several robustness tests are employed to test the
reliability of the empirical results. The study finds that the hedonic approach yields slightly
more accurate and stable prediction result than a basic matching model without hedonic
adjustments.
1
1 Introduction
In standard commercial real estate (CRE) appraisal real estate professionals generally
employ a carefully selected sample of comparable commercial properties to determine the
market value or likely rental rate for a subject property. The heterogeneity of commercial
property has not generally made it suitable for automated mass rent or value estimation
techniques.
By contrast, in the residential appraisal field statistical models are often used by tax
assessors and federal mortgage insurers in the mass appraisals of a large number of
properties within their taxing jurisdiction. Mass assessment does not lend itself to the hand
selection of comparables, hence the use of multiple regression techniques to adjust for
differences in property characteristics.
The objective of this research is to determine the effectiveness of estimating
commercial property rents using an automated matching procedure derived from grid
analysis and further augmented by hedonic regression techniques. Due to the increased
heterogeneity of commercial real estate relative to residential real estate, numerous
adjustments are made to create a viable model. In addition to customizing regression
coefficients for each market, stepwise models are implemented to determine ideal
comparable selection and additional techniques to determine the best comparables.
Furthermore, the majority of the literature uses sales to estimate value while this research
is among the few to use automated techniques to estimate fair market rent on a building
level. The models employ a large national cross-sectional sample of commercial office
transactions supplied by CoStar.i While the general principles would remain consistent,
some adaptions would be required to implement the model in industrial, retail or other
2
commercial property types. Additionally, the model tests whether the inclusion of green
variables materially impact the results, which they do not.
This study augments a current line of research evaluating the efficacy of various
appraisal methods. In commercial land appraisal, Shi (2015) proposes a new net rate
methodology comparing the residual land value of assessed property to a true sales
comparison approach. Gunterman et al (2015) investigate various methods to gauge the
impact of plottage and plattage on land values. Specifically in commercial real estate rent,
An et al (2015) use property level data to form a rental index, which would be a possible
application for this model. In the residential sector, Chen and Harding (2015) examine the
changes in attribute prices in hedonic and repeat sales methods. Cheng et al (2015) suggest
that the weighted repeated sales appraisal method is missing a market risk factor„ while
Gunterman, Liu and Novak (2015) propose modifications to the repeat sales procedure
incorporating nearest neighbor techniques.
In addition to filling a gap in the literature by examining CRE rental estimations with
hedonic grid regression, a number of potential academic and practical applications motivate
this research. On the academic side, much of the academic real estate literature relies on
various forms of hedonic regression modeling which estimate values for individual
characteristics bundled together to form a good or service, and are well-suited for real estate
applications. For example, the U.S. Consumer Price Index (CPI) uses hedonic models to
estimate the housing price component. At the same time, serious flaws can be introduced
when using hedonic regression in large scale studies at the national level.
Hedonic grid regression may offer alternative methods for analyzing data in CRE
for triangles, kriging, and cokriging. Artificial intelligence appraisal methods are compared
by Zurada et al.Zurada, Levitan, and Guan (2011) using a residential data set.
It appears that no research has been reported using a grid method with hedonic
adjustments in the commercial real estate field. However, one prior article, using residential
data, tested the grid method combined with hedonic regression to provide necessary
adjustment coefficients Kang and Reichert (1991).
Hedonic regression involves estimating the unique value for key property
characteristics such as property age. Once the comparable set is selected and the adjustment
coefficients estimated, the research still must decide the optimal weighting scheme for the
comparable set. Equally weighting each comparable may not be optimal because different
properties may be more representative of the subject property Colwell et al. (1983).
3 Model Specification The hedonic grid model described below closely follows the work Kang and Reichert
(1991), however the model is customized for commercial real estate applications, rather
than residential.
As a first step, a basic systematic matching approach is employed which focuses on the
most important property characteristics in the selection process. Table 2 provides
descriptive statistics on some key variables in the research sample such as building size,
age, and rent for 34 major US markets.
9
The characteristics used for matching are selected through a combination of stepwise
regression, professional judgment, and informal practitioner interviews. The results of
stepwise regression on lnrent (natural log of rent) for the entire database are shown in Table
1.iv Based on these regression results and practitioner judgment the final matching criteria
includes the following variables: 1) distinct geographic submarkets, 2) property size (square
footage), and 3) property class (A, B and C class buildings)
INSERT TABLE 1
The triple net variable (NNN)v was not included as a matching parameter because it
might unnecessarily limit the matching process. Furthermore, the hedonic-grid adjustment
technique itself accounts for a portion of the rent differentials between NNN and full service
gross (FSG) properties, and submarkets tend to have some degree of homogeneity regarding
lease structure.
While there may be some small variations the following procedure is commonly
employed by many practitioners in selecting comps. They begin with either an explicit or
perhaps implicit set of screening criteria to narrow the field of all possible comps.
Depending upon the number of comps this initial screening procedure identified the
appraiser may then reduce the number of comps by further analysis and ,where possible, by
quantifying key differences in property characteristics between the subject property and
each perspective comp. To arrive at an estimated or appraised value or rent the values and/or
rents associated with the final set of comps are then averaged to obtain the appraised value
or rent of the subject property.
To approximate this comp selection approach an hedonic regression model is estimated
using the variables found to be statistically significant in the stepwise regression model
(Table 1). Thus, these explanatory variables are regressed with the natural log of property
10
rent as the dependent variable. The regression coefficients for a given property
characteristic, say building size, is then multiplied by the difference between the subject
property size and the size of each potential comp in the market.
Differences are then calculated in a similar fashion for all of the property characteristics
in the hedonic model. These differences are then squared and summed to yield a
comparability index, using a sum of squares method from Colwell et al. (1983). Following
this procedure, all the comps in the relevant market have an assigned comparability index.
(Note: Later in this paper this index is referred to as ANET and is more formally depicted
in Equation 3). This index is then arrayed from smallest to highest with a smaller value
indicating greater comparability.
The number of comparables (N) associated with each subject property varies but the
minimum number is three. If less than three comparables exist the subject property is
excluded from the sample. The maximum number of comparables is set at ten. When more
than ten comparables are identified in the matching process the number is narrowed to ten
based upon the ten smallest comparability indices
Table 3 summarizes several important sample characteristics. For example it shows the
average number of comps selected for each major sub-market (Mean Comps). The Mean N
for T-Test column represents the average number of subject properties which experienced
a successful comparable selection per random draw. The Mean N% of draw column
represents the Mean N for T-Test as a percentage of the total properties drawn. For example,
a 90% value indicates that only 10% of the properties selected failed to draw three or more
comps. All Mean N less than 30 were highlighted in bold print to indicate potential small
sample bias.
Insert Table 3
11
3.1 Basic Matching Model The basic matching model simply compares properties but does not adjust for property level
differences. In the basic model, RSj is defined as the rent for the subject property j. First, a
set of comparables, RCj for RS
j based on the vector of control variables, Xi, is determined.
There are three to ten (N) comps for each each jth observation, such that RCj = [Rc
1j, Rc2j, ...,
RcNtj].
Second, the model estimates 𝑅𝑅�𝑗𝑗𝑠𝑠 based on the simple average rent for the selected
comps:
𝑅𝑅�𝑗𝑗𝑠𝑠 =∑ 𝑅𝑅𝑛𝑛𝑛𝑛
𝑐𝑐𝑁𝑁1
𝑁𝑁 (1)
Finally, to test whether the expected rent, 𝑅𝑅�𝑗𝑗𝑠𝑠, is different from the observed rent, a
paired t-test for dependent populations is performed.
𝑡𝑡 = 𝐷𝐷�𝜎𝜎𝐷𝐷√𝑁𝑁
(2)
Where: 𝐷𝐷� represents the mean of the differenced data (RSj −𝑅𝑅�𝑗𝑗𝑠𝑠), σD represents the
standard deviation of the differenced data (RSj −𝑅𝑅�𝑗𝑗𝑠𝑠) and N is the number of random draws
being tested.
The paired t-test is employed to test whether a statistically significant difference exists
between the observed Rj for a random sample of buildings and the estimated 𝑅𝑅𝚥𝚥� from its
comparable set of buildings.
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A failure to reject the null hypothesis of no difference indicates model success. That is,
if the model’s expected rent fails to show a sizable prediction error then the expected rent
is not statistically different from the observed rent; hence the model is effective in
generating accurate rental estimations.
3.2 Matching Model with Hedonic Coefficient Adjustment / Grid Method As previously mentioned, the hedonic grid regression method closely follows Kang and
Reichert (1991).
First, individual submarket hedonic regressions are used to estimate the appropriate
adjustment coefficients for each property attribute. The values by attribute are used for
creating the net adjustments in equation 3.vi
Second, a net adjustment factor (ANETi), is calculated based on the closeness of fit
between the subject property and each comparable as follows:
𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝑖𝑖 = ∑|𝛽𝛽𝑖𝑖(𝑋𝑋𝑖𝑖𝑠𝑠 − 𝑋𝑋𝑖𝑖𝑐𝑐)| (3)
Where Xi represents the i th explanatory variable in the model, βi represents the
regression-derived market value (hedonic price) for each property characteristic, s and c
stand for subject and comparable properties, respectively, and j indicates a specific
comparable property. ANETi is used to rank the quality of each comp with properties
having the smallest value representing the best match. In the third step, each comp set’s
rent is adjusted based on the estimated weight of the attribute, using the positive or negative
adjustment as opposed to the absolute value above. In mathematical terms:
6.1 Holdback Results To test their robustness, the models are validated using the 20% holdout sample reserved
at the beginning of the analysis. Due to the smaller number of available properties, the
number of iterations for each model is reduced to 200. While the subject properties are
drawn exclusively from the holdback sample, comparable properties are drawn from the
whole market. The regression coefficients for hedonic adjustments are those generated from
the 80% estimation sample. Detailed results are omitted due to space constraints.
In general the results are remarkably similar to the results obtained using the full
estimation sample. The larger markets consistently fail to reject the null at the five percent
on-tail T-test level. The stability of the holdout results indicate that the model design is not
sample specific and the model should continue to perform well when applied to comparable
data sets.
6.2 Less Stringent Matching Parameters In the results presented, if a comp fails to meet all the matching criteria, it is not selected;
if the subject property fails to generate at least three comps it is excluded from the analysis.
As a robustness test a more liberal matching process is employed as follows. If the matching
process does not generate at least 3 comps for a given subject property, the building class
matching requirement is dropped. This allows the matching process to draws from a wider
pool of comps although the quality of the selected comps would likely be lower. Thus, there
is a potential trade-off when a greater number of comps leads to less representative comps
being selected. However, in the case of the hedonic model the expected rent would still be
adjusted for building class. The robustness test is performed in the same manner as before
for each of the four model scenarios, with 500 random draws of a five percent sample for
21
each market. The results are omitted due to space constraints but they are qualitatively very
similar to the previous results.
6.3 Control Tests To further test the robustness of the model the matching process is intentionally “stressed”
to see how reliable the results are when estimating rent for clearly dissimilar properties. To
create this dissimilar sample, properties are considered comparable when the building is at
least 10 size categories larger (out of 20 total). In general, larger buildings tend to command
higher rents. The results are shown in Table 8.
The stress test results show that the basic model consistently rejects the null of no
difference in rent. The hedonic model does adjust for the differences in the buildings, but
performs more poorly than the original model. The stress results indicate that hedonic
adjustments significantly compensate for poorly matched properties, but the most efficient
model is includes well matched properties. This finding also shows that the basic model
can be used to identify potential market discounts or premiums.
INSERT TABLE 8
6.4 Whole Market Testing The primary purposes of testing random selections of a market is to ensure results are not
sample specific, to permit a holdback sample and to mirror real life applications with small
portfolio evaluation. However, further insight as to the robustness of the overall model is
gained through evaluating an entire market. Table 9 shows results from Equation 2, with
N as the entire market. They show that the model performs even better than in random
swatches and that the hedonic model clearly outperforms the basic model.
INSERT TABLE 9
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7 Conclusion This paper explores alternatives approaches to basic regression models for predicting
commercial real estate rents. The first model is a basic matching model based on the grid
method commonly used by appraisers. An automated algorithm selects three to ten
comparable properties and uses a simple average of the selected comps rent to estimate the
expected rent for the subject property. A second more sophisticated model uses hedonic
regression to determine the appropriate property adjustment coefficients to estimate rent
based upon a matched set of comps.
The various models consistently failed to reject the null hypothesis of no difference
between expected and observed rents which indicates success in the modeling effort. That
is, the paired differences between the expected and observed rents proved to be small and
statistically insignificant. Not surprisingly, the strongest performance is observed in larger
markets with more comps from which to select. The Hedonic model which adjusts the
expected rent using the hedonic coefficients as the adjustment parameters proved to be
somewhat more reliable overall. On the other hand, the Basic model also performed quite
well. As a secondary finding, the results indicate limited appraisal enhancements with the
inclusion of green variables.
The models were estimated using the CoStar commercial real estate database. Tests
using separate estimation and holdout samples performed equally well, suggesting that the
models are sufficiently flexible and adaptable for analyzing other data sets. Furthermore,
the matching models were “stressed” to see if observable differences could be captured and
potentially normalized. The Basic model clearly showed the difference in rent between
purposefully mismatched buildings, demonstrating the potential of the model to identify
23
statistically significant attribute differences. The Hedonic model effectively adjusted for
many of those differences.
Multiple research opportunities could stem from this paper. In addition to the further
refinement of the model, confirmation or rejection of a wide range of prior findings can be
tested using this method. Differences in size, stories, view premiums, or other building
attributes could be purposefully mismatched to identify market premiums. Another
potential research avenue is a detailed comparison of stand-alone hedonic regression to
matching in a constructed data set. In addition, models indicating whether a building
captures its “fair share” of market rent could be created. This foundation could potentially
be adapted to commercial real estate bond analysis.
Increased data availability for real estate researchers may represent the beginning of
new analysis techniques in real estate research. No longer confined to private, one-off data
sets, researchers can now begin to investigate the best ways to research. Alternative
research methods may open the door to a host of fresh findings, and new ideas in the field.
While the hedonic/grid approach will not necessarily replace the more traditional
approach to appraisal, this paper suggest the feasibility for practitioners and researchers to
at least consider it a possible option if they have sufficient data and expertise
24
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8 TABLES
Table 1: This table reports results of a stepwise regression with ln(Rent) as the dependent variable and key property characterisitcs as the explanatory variables. The variables are ordered in decreasing explanatory power. Lnsize is Ln(square feet), NNN is a dummy variable for NNN lease, A and B class are building designations with C class as the omitted, Percent Leased is the occupancy of the building, Stories is the number of stories, Lnage is Ln(years since construction), and FSG is a dummy variable for Full Service Gross lease with modified as omitted.
Step Variable Partial Cumulative C(p) F Value Pr > F
Table 2: This table shows basic descriptive statistics for the commercial properties included in this study. The sample is provided by CoStar from Q4 2011. The largest 34 MSA’s in the US constitute the final sample. Rent is rent per square foot, size is in square feet, and age is years since construction.
Mean Mean Market N Size Rent Age Market N Size Rent Age Atlanta 1,742 78,704 16 22 Long Island (NY) 867 67,622 25 34
Baltimore 785 51,504 19 25 Los Angeles 2,447 80,940 26 28 Boston 1,386 56,108 18 37 Minneapolis/St
Paul 1,009 84,177 13 30
Charlotte 651 72,528 17 20 New York City 777 218,797 52 68 Chicago 2,606 97,882 17 26 Northern New
Denver 1,177 75,138 17 25 Sacramento 876 46,799 19 21 Detroit 1,300 61,628 15 26 San Diego 983 51,303 22 24
East Bay/Oakland
676 58,254 21 30 Seattle/Puget Sound
1,187 65,783 19 25
Houston 1,323 113,418 17 22 South Bay/San Jose
617 52,784 24 29
Indianapolis 620 56,957 15 25 South Florida 1,945 59,343 19 22 Inland
Empire (CA) 801 31,493 17 19 St. Louis 881 59,465 16 26
Kansas City 804 59,003 16 29 Tampa/St Petersburg
769 53,190 16 22
Las Vegas 689 38,458 17 15 Washington DC 2,455 95,161 27 23
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Table 3: This table summarizes the mean information used in the market analysis. The Mean comps per property column represents the mean number of comparables used for each individual property. The “Mean N for T-Test column indicates the average number of subject properties with successful comp selection per random draw. The % of Draw Matched column indicates the percent of properties successfully paired with comps per random draw. Note that values for Mean N of less than 30 are highlighted in bold print as having potential small sample problems
983 San Diego 9.19 35.25 95% 9.20 35.24 37 881 St. Louis 8.85 30.03 91% 8.85 30.08 33 876 Sacramento 9.47 31.74 96% 9.46 31.81 33 867 Long Island (New York) 9.40 32.06 97% 9.38 32.02 33 804 Kansas City 8.76 26.97 90% 8.80 26.91 30 801 Inland Empire (California) 9.87 29.96 100% 9.85 29.95 30 785 Baltimore 7.78 24.24 81% 7.83 24.26 30 777 New York City 9.66 27.99 97% 9.63 27.93 29 769 Tampa/St Petersburg 9.14 27.46 95% 9.14 27.46 29 714 Portland 8.92 23.82 88% 8.92 23.85 27 689 Columbus 8.92 23.97 92% 8.89 23.81 26 689 Las Vegas 9.21 23.66 91% 9.18 23.75 26 676 East Bay/Oakland 8.96 24.79 95% 8.98 24.81 26 652 Cleveland 7.53 22.03 88% 7.51 22.04 25 651 Charlotte 8.07 21.31 85% 8.11 21.25 25 642 Cincinnati/Dayton 8.45 21.97 92% 8.42 21.91 24 620 Indianapolis 8.31 19.59 82% 8.33 19.55 24 617 South Bay/San Jose 7.73 20.05 87% 7.68 19.99 23
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Table 4: This table shows the distribution of T-test statistics from 500 random draws of five percent of each market for the basic model. The basic matching model simply compares properties but does not adjust for property level differences. The panels show results of significance tests for the lower and upper bounds of a two tailed T-test for the 500 random samples. A functioning model would have less than 500∝/2 of the draws reject at its significance level. For example, less than 2.5% of the random tests should reject at the upper or lower bounds at 5% significance level. Where results are higher than expected, numbers are bolded. Panel A shows results where property matching was narrowed using all coefficients, with and without the inclusion of green variables for comparable selection. Panel B shows results where property matching was narrowed using only significant coefficients, with and without green variables for comparable selection. No adjustments are made to the comparables in the Basic RHAT model. Table 4. A Basic Model- All coefficients This panel shows the distribution of T-statistics from 500 random draws of 5% of each market for Basic RHAT, or estimated rent without attribute adjustments. This panel uses all coefficients from the hedonic regressions for comparable selection to a maximum of 10. No adjustments are made to the comparables for rent comparison purposes.
This panel shows the distribution of T-statistics from 500 random draws of 5% of each market for Basic RHAT, or estimated rent without attribute adjustments. This panel uses significant coefficients from the hedonic regressions for comparable selection to a maximum of 10. No adjustments are made to the comparables for rent comparison purposes.
Table 5: This table shows the distribution of T-test statistics from 500 random draws of five percent of each market for the hedonic model. Or estimated rent using attribute adjustments on the comparable properties. The panel show significance tests for the lower and upper bounds of a two tailed T-Test for the 500 random samples. A functioning model would have less than 500∝/2 of the draws reject at its significance level. For example, less than 2.5% of the random tests should reject at the upper or lower bounds at 5% significance level. Where results are higher than expected, numbers are bolded. Panel A shows results using only significant coefficients, with and without the inclusion of green variables for comparable selection and adjustment. Panel B shows results using all coefficients, with and without green variables for comparable selection and adjustment.
Table 5.A Hedonic Model- All Coefficients
This panel shows the distribution of T-statistics from 500 random draws of 5% of each market for Hedonic RHAT, or estimated rent with attribute adjustments. Attribute adjustments are made using all coefficients in this panel.
Table 5.B Hedonic Model-Significant Coefficients This panel shows the distribution of T-statistics from 500 random draws of 5% each market for Hedonic RHAT, or estimated rent with attribute adjustments. Statistically significant coefficients are used to make adjustments in this panel.
Table 6: This table summarizes the results of the rejection of the null rates for all markets from the market by market 500 random draws displayed in Tables 4.A through 5.B. Rejection of the null indicates that expected rent is different than observed rent, and the model failed. An observed rejection rate of less than 500α/2, indicates model success. For example, less than 2.5% of the random tests should reject at the upper or lower bounds at a 5% significance level for a well behaved model. Where results are higher than expected, numbers are bolded. Summary results are shown for each model iteration.
All Coef. 0.003 0.014 0.031 0.017 0.009 0.002 All Coef. No Green 0.003 0.016 0.033 0.017 0.008 0.001
Sig. Coef 0.001 0.008 0.017 0.031 0.016 0.004 Sig. coef. No Green 0.001 0.007 0.015 0.033 0.017 0.003
Hedonic RHAT All Coef. 0.001 0.007 0.017 0.028 0.014 0.003
All Coef. No Green 0.001 0.008 0.017 0.030 0.015 0.003 Sig. Coef 0.001 0.007 0.016 0.034 0.018 0.004
Sig. coef. No Green 0.001 0.006 0.015 0.036 0.019 0.004
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Table 8. This table shows the distribution of T-test statistics from 200 random draws of five percent of each market in the holdout sample. Comparable properties are purposely mismatched or “stress” tested so that they are a minimum of 10 ( of 20 total) categories larger in size. The table show significance for the lower and upper bounds of a two tailed T-test for the 200 random samples. Where reported results exceed the critical value, i.e. more than 2.5% rejection at the upper or lower bounds at a 55 significance, numbers are bolded. This robustness test evaluates the model’s ability to identify differences between properties.
Table 9. This table shows results from t-tests of each market for all the of the different methods analyzed in the paper—both the Basic and Hedonic models in each iteration. Bolded numbers demonstrate where the model failed to reject the null of no difference. Virtually every hedonic model rejected the null of no difference between observed and estimated rent.
Basic Hedonic All Coef All Coef No Green Sig Coef Sig Coef No Green All Coef All Coef No Green Sig Coef Sig Coef No Green N Difference t
i Source: CoStar Group, Inc. ii A cap rate, or capitalization rate is the discount rate used in valuing commercial real estate’s current or predicted income as a
perpetuity to establish current value. iii The Mahalanobis distance Dij is represented by the formula:
Dij= (xi−xj) ∑ −1(xi−xj)'
where xi and xj represent the vectors of standardized amenity coordinates for properties i and j and ∑ −1 is the variance-covariance matrix of the amenity coordinates for all properties.
iv The stepwise method used a forward selection method that adds variables to the model one by one, subject to an F test for variable significance. Potential issues with this method include some reliance on the order of inclusion. However, this method was used as a secondary method for variable inclusion, with primary focus on literature and practitioner guidance.
v NNN leases are lease types where the tenant pays a base rent and incurs all expenses. FSG or Full Service Gross leases are where the tenant pays a flat rent and the owner incurs all expenses. There is a spectrum of leases in between the two. Obviously, NNN rent would be less than FSG rent, ceteris paribus.
vi Hedonic coeficient results are available by request. vii The five are an absolute value weighting, quadratic or squared weighting, statistical reliability, distance-based weights, and a
minimum or “no zero” weight technique. viii The sum of squares method as described in Colwell et al. (1983) is:
w*j =
∑k=1
n ∑i=1
m (βi(xis−xik))2− ∑
i=1
m (βi(xis−xij))
2
(n−1) ∑k=1
n ∑
i=1
m (βi(xis−xik))2
(5)
where n is number of comps and m is adjustment factors. ix As defined by The CoStar Group.
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x Each full set of data took 5-7 days to complete. Each full market sample is comprised of roughly 2,000 randomly drawn properties, tested 500 times, for a total of 1 million tests. Each property averages over 8 comparables, using roughly 8 million observations for each full set of 500 random draws. In addition, each full market samples is executed in four ways, for an approximate total of 4 Million subject properties compared to 32 Million comps.
xi Recent research by Robinson and Sanderford (2015) shows green variables may not be reliable indicators.