6.1.8 Justification

hy don’t we just use indexing or numbers in the notation?"We have two requirements for our notation:

1.

Subsets, e.g., the analyst frequently selecting subsets of a list, such as subsets of variables, are used. For instance, there may be 35-40 variables in the MLSData sheet but only 10-20 of them. And, from one run to the next, The VE will very likely change the variables submitted to MARS/earth.

2.

Iteration, e.g., $i\; =\; 1,3\dots ,n_1$
Our algorithms, protocols, and proofs usually need to be able to iterate through all items in an ordered list, from the first to the last - or pick out some particular item.

So, we need a letter to represent the subset or list of items, and that must have another subscript that we can use to iterate over the items of the list to give is the strings that are used to index into the data frame rows or columns.

6.1.9 Matrices?

And what about the matrices of Linear Algebra? Matrices work only with numbers, either integer or real. Matrices as a data structure are supported in both R and Python. However, many of the variables we work with in MLS data are called "trings"or character data. Even if some variables are only numbers, they are often not real numbers but just names. For example, MLS Area Codes for a city may be ("60""61""62""63" ....,"72). Earth regression allows variables to be specified as "actor"variables, in which case the numbers are treated simply as names, without any sequence. In such cases, the model produced will merely provide a value contribution for each number, rather than a function that runs over the sequence. Other examples include "1"...,"6"for Condition and "1"...,"6"for quality.

6.1.10 First Simplification

The above notation for the core Excel project workbook is nice to tie things together, but to heavy to carry around for explaining algorithms or mathematical proofs. When we are doing the tedious work, we can assume it is within the given project, version, stage and sheet. We just say what we are working with, if it is not already implicit. All we really need from the above, once we indicate what we are working with is:

6.1.11 Second Simplification

It is more convenient to rename the sheets to specific kinds of data frames they support rather than use indexes. For each sheet, the rows and columns have different functionalities. So, the W set of data frames is broken up into M, V, A, I, and C data frames as follows:

• MLS Data

  $$$$
  M vj pi	=	W0vjpi
  i = 0,1,…,np
  j = 0,1,…,nv
  (6)

  where:	$n_p$	$=$	# properties
  	$n_v$	$=$	# input variables

  Note: Subject Is By Default Always First in Property Lists

  In the MLSData Excel spreadsheet and associated data frame, the first row is always reserved for the subject property. It is never sorted with the other MLS properties below it. It is never entered into the MARS regression as independent data, it is entered rather as target data, or as the dependent variable. The sames holds for property lists. In all property lists, the first element with index 0, is always assumed to be the subject property ID, unless stated otherwise.

  Even if the task is to create a model for a market area without any subject property, the first row is nonetheless loaded with some dummy property data. This makes it far more convenient to write programs, specify algorithms, or do proofs. So, if we are using c to represent the list of properties, c0 is the subject property and ci, i = 1,...,n1 are the comparable listings that will be input to MARS/earth regression.

• Regression Variable Specifications

  $$$$
  V sj vi	=	W1sjvi
  i = 1,3…,nv
  j = 1,3…,ns
  (7)

  where:	$n_v$	$=$	# available variables
  	$n_s$	$=$	# of specifications

  Note, $n_s\ge n_v$, as we usually select a subset of available variables for input to regression.

• Calculations

  $$$$
  Ccj vi	=	W2cjvi
  i = 1,3…,nv
  j = 1,3…,nc
  (8)

  where:	$n_v$	$=$	# of variables
  	$n_c$	$=$	# of calc specs

• Interactions

  $$$$
  Itj fi	=	W3tjfi
  i = 1,3…,nf
  j = 1,3…,nt
  (9)

  where:	$n_f$	$=$	# from variables
  	$n_t$	$=$	# to variables

  Note that Data Frame I is a symmetric data frame, with variables alphabetically listed for the from rows and to columns. The cells are marked with a 1 to indicate an interaction between the associated pair of variables is allowed. Otherwise, 0 indicates that no interaction is permitted. For a second-degree interaction, this is relatively simple, with only one interaction possible. With a third-degree regression, we have three pair interactions between all three variables, where every pair of interactions must be allowed to generate the three variable interaction term (or variable).

• Aggregations

  $$$$
  Avj vi	=	W4vjvi
  i = 1,3…,nv
  j = 1,3…,nv
  (10)

  where:

  $n_v$

  $=$

  # of variables

7 The RCA Sales Grid

Keep in mind that the RCA protocol requires an R or Python program that can access R/earth and perform all of the detailed steps. It is data intensive and the work could not possibly be done manually. At the same, time, assuming the program is bug free, you need a software program to ensure that the required work is done precisely and without human error.

It is also a necessity for VE’s who do this work to be able to modify such programs as needed, especially when they are dealing with new unforeseen market areas, neighborhoods, properties or other new requirements or problems.

Within RCA processing we have three types of Sales Grids:

1.

The RCA Processing Sales Grid, labeled in this paper as "LSData,"which is a very large data frame and associated spreadsheet that may contain many intermediate details that eventually go through an aggregation process, to arrive at the RCA Sales Grid mentioned below. An example would be too large to display here. Most likely valuation engineers will tailor this spreadsheet to their preferences. I intend to write a separate paper on developing such a grid at some future date.

2.

The RCA Final Sales Grid which is used in non-GSE reports that looks something like ??. This does mimic for example Fannie Mae’s form, enough to provide some base of familiarity to the reader. However, notice that it has columns for "alue Contributions"and other minor changes. A simpler form is also the 1.

3.

The GSE Upload Sales Grid that is used to upload the Sales Grid into software such as Alamode. Note however, that not all appraisal form software packages provide the ability to upload sales grid data from a spreadsheet. Out of necessity, such grids cannot include a separate column for valuation contributions, of which the GSEs like Fannie Mae have no comprehension. The strategy is to give them their required form, but back it up with an RCA Sales Grid, as well as other supporting documentation such as details on any variable aggregations.

The RCA Processing Sales Grids do not look like Fannie Mae Sales Grids but are, roughly, a transpose of the Fannie Mae Sales Grid, with rows and columns switched: Each row contains data for some property, and each column contains data for some property variable. There can easily be hundreds or thousands of property rows and 200 or more columns. The first row is reserved for variable names, and the second for a subject property, even if we are only doing a market analysis. When the spreadsheet is read into a data frame, then the first row of the sheet becomes the names attribute, and the second row for the subject property becomes the first row of the data frame.

The columns in the RCA Sales Grid include identification columns, then primary, external, and synthetic variables combined and fed as the independent variables into MARS. MARS outputs from these variables those that it thinks have a significant impact on the Sale Price, also known as the dependent or target variable. In second-degree regression, we may get pairs of these variables that act as new variables. With third-degree regression, we may also get variable triplets that act as so-called independent variables, although they are, of course, highly correlated with the underlying variables.

8 Variables

On the input side of running MARS, we have only simple variables which are:

1.

Property features loaded from the MLS spreadsheet.

2.

Derived variables created by the VE to refine the MLS variables or add transformations of existing variables. For example, the VE may prefer to transform Date of Sale into "aysOffMarket"which is equal to the effective date of the appraisal minus the Date of Sale.

3.

The VE may import data from 3rd party sources, such as mortgage interest rates. Or, if latitude, longitude or elevation data are missing from the MLS data, he may import it from other sources. These variables could be called "xternal variables."/dd>

On the output side of MARS, we work with price model that is a combinations of terms that either:

1.

Contain only one of the input variable and provide the equation for the value contribution of that variable. These are called first degree terms or (output) variables. The term itself is treated as a value contribution value and given a name.

2.

Contain two of the input variables and are the equation for the value contribution of the interaction of the given two variables. These terms maybe called second degree terms or (output) variables.

3.

Contain three of the input variables and are the equation for the value contribution of the interaction between the three variables. These terms may be called 3rd degree terms or (output) variables in this paper.

4.

The output variables of MARS are then value contributions, including the residual value contribution. They are in turn treated as variables in Sales Grid calculations, such as calculating adjustments or creating graphs.

We can have different types of variables that fall into two basic ontologies: (1) Variable Origin and (2) Variable Application, i.e. Non-residual vs Residual

8.1 Variable Origin Ontology

1.

First Degree Variables

1.1

MLS Regression Variables

1.2

External Regression Variables

1.3

Derived Regression Variables

2.

Factor Variables:

To handle factor variables, a function is needed that can identify them. Since they are specified in the Excel project configuration workbook, a function is created that returns true if a variable passed to it has been classified as a factor type.

3.

Interaction Variables: These are created by MARS in generating the model, if the degree is set to 2 or higher. However, 3 interacting variables is the most that could be recommended due to the difficulty of interpretation, the probability of overfitting and the impact on Degrees of Freedom and the resultant increase in property records to ensure a robust model.

3.1

Second Degree Variables: These are common and a good example would be GLA:BathRms, GLA:LotSize, DateOfSale:BathRms, Latitude _Longitude.

3.2

Third Degree Variables: Less common interactions that suggest caution and possible over-fitting.

4.

Residual Breakdown Variables

8.2 Variable Application Ontology

1.

MARS Generated Value Contributions

1.1

First Degree Variables

1.1.1.

MLS Regression Variables

1.1.2.

External Regression Variables

1.1.3.

Derived Regression Variables

1.2

Interaction Variables

1.2.1.

Second Degree Variables

1.2.2.

Third Degree Variables

1.3

Residual Breakdown Variables

1.4

Aggregation Variables

2.

Engineer Generated Variables &Values

2.1

Residual Breakdown Variables

2.1.1.

Condition

2.1.2.

Quality

2.1.3.

Design/Aesthetics?

2.1.4.

Landscape?

2.1.5.

Functional Utility?

2.1.6.

View?

2.1.7.

...

These variable groups will be given their notation and described in more detail in the next section

8.3 Second- and Third-Degree Terms

With second-degree or third-degree regression, some of terms in the output model, may include two or respectively three variables. For example:

Notice that we name these by alphabetically sorting the variable names and concatenating them, perhaps with an underscore. We do the same with third degree terms on the variable name triples produced in the model. For example, "athRms__GLA__LotSz."

These variables get added to the primary input variables in Stage 2, when the RCA program calculates the contributions of all variables for each property, including the interaction variables.

8.3.1 Adjustment Variables

The RCA program will also calculate the adjustments for all variables and properties in Stage 2, after value contributions are calculated, by subtracting the corresponding comparable property value contribution for a feature from the subject contribution for that feature.

8.3.2 Aggregation Variables

Aggregation variables are defined to replace subsets of the variables, adding all their value contributions and adjustments into a common variable to be used in the report. For example, we may have split both finished and unfinished living area into finer detail variables:

1.

Above Grade Legal Finished Area

2.

Below Grade Legal Finished Area

3.

Above Grade Illegal Finished Area

4.

Below Grade Illegal Finished Area

5.

Above Grade Unfinished Area

6.

Below Grade Unfinished Area

Yes there are areas where this does make sense.

The MARS regression will regress on all of the variables and generate a model that will provide contributions and adjustments for them. But for a form report, these different contributions will probably have to be combined or aggregated into fewer variables. The details of the aggregation should be supplied in an addenda. A more frequent example, would be aggregating interaction adjustments into one of the primary variable adjustments. For example, we might aggregate the adjustment for GLA_Lotsize to GLA or to LotSize, or even split them 50:50 between both, as determined by the Aggregation sheet.

8.3.3 Special Variables &Notation

where

8.3.4 Breaking Down Residuals

Once the MARS regression has created a model, it generates the estimated sale prices from the MARS-selected input variables. It subtracts that value from the corresponding Net Sale Price to get a residual value for each property. Now, the RCA protocol stipulates that once the VE has decided on the final property comparables, usually 6-12 property listings, he needs to go through each property listing and breakdown the calculated residual into the most likely features he thinks impact value, ignoring the possibility of random data errors. He might, for example, have these breakdown variables for a given property:

1.

Condition

2.

Quality

3.

Design

4.

Functional Utility

5.

Over/Under Market Sale

Different properties will likely have some different residual breakdown variables, which need to be collected in one ordered set.

Notation: We use the notation in the above table to arrive at:

where $n_c$ is the number of residual breakdown components.

The VE must use his judgment to break down the residual into the given components. One might think this would allow bias to enter the final value conclusion. However, that is not the case because we have mathematical proof that if the breakdown adjustments total the residual, they will not impact the value conclusion. The purpose of the breakdown is not to establish value, but to explain why the residual is what it is in comparison to other property sale residuals. The only spot where the engineers bias can impact the final value conclusion is the estimate of the residual for the subject since, of course, there is no actual sale price for the subject needed to calculate a residual for it.

Each property has its set of residual value contributions, but these need to be merged into one set C over all properties:

where $[{𝜖}_{i,c}]$ is the variable name for the value ${𝜖}_{i,c}$ and $A_i$ is the set of such variables and $n_c$ is the number of values in each ${𝜖}_i$.

So, with the subject and comparable property data in separate columns, then under all the regression model variable value contributions for each property, we should see the set C of nc corresponding residual value contributions.

9 Core RCA Proofs

  Let:

Proof I: Each and every comparable adjusted sale price equals the estimated subject sale price.

For any comparable i, the adjusted sale price ${\alpha }_i$ is the sum of the net sale price ${\pi }_i$, the base value $\beta$, the value contributions ${\phi }_{i,j},j\; =\; 1,2,\dots ,n_r$ plus the comparable residual ${𝜖}_i$ or:

Proof II: Altering the residual breakdown under the constraint that all residual component value contributions sum to the comparable residual, has no impact on the final value conclusion of the RCA protocol.

This is almost trivial. Yet without thinking about the math, it is easy to overlook this because traditional appraisers focus on the adjustments rather than the value contributions. It is the value contributions that cancel out here. - A decades long oversight. For some comparable k, $k\; >0$, the total residual adjustment is equal to sum of the $n_c$ breakdown residual components which in turn is equal to sum of the differences of value contributions, which is equal subject total residual minus the comparable total residual. And this is independent of the number of residual components created or what they represent. As long as the value of the residual components equals the total residual for both the subject and chosen comparable ${\xi }_k$ will not change

10 MARS Model

10.1 Model Variable and Interaction "!--l. 1421-->$g$"Functions

A core principle of the RCA method, is that "xplanatory reasoning"is of utmost importance, Unfortunately providing meaningful graphs for the various value contribution equations, especially when interactions and factor variables are used, can be challenging. It is therefore considered imperative to provide a clear and strict protocol for creating these graphs. This is done herewith, but understand the protocol is aimed at the developer who is writing the R or Python code to generate the graphs, and might be only superficially of interest to those not involved with the coding.

The MARS-generated price model is a sum of a base constant plus additional terms, each involving 1 to 3 variables. To analyze it, we first split the model into individual terms. Then, mutiple terms sharing the same set of variables, representing a single interaction, are combined into single interaction expressions. Assume the maximum degree of regression is 3. Consider a third-degree regression model consisting of a base constant, $n_1$ first-degree expressions, $n_2$ second-degree ineraction expressions, and $n_3$ third-degree interaction expressions. This gives us four distinct groups of expressions, which are either single variable or mutiple variable interactions, each graphed differently based on two factors: (1) whether they include a factor variable (a categorical variable), and (2) the number of unique variables they contain. Specifically:

• First-degree expressions (1 variable) are graphed in 2D.
• Second-degree expressions (2 variables) are graphed in 3D.
• Third-degree expressions (3 variables) are graphed as a set of 3D graphs.
• If an expression contains a factor variable, each unique factor reduces the graphs dimensionality by one (e.g., a 3D graph becomes 2D if one variable is a factor).

The indexing of each function $g$ provides the information needed to determine how it should be graphed, reflecting the number of unique variables and the presence of factor variables.

10.2 Indexing of the $g$ Functions

Each function $g$ is indexed as ${f_{j,k}}^{g_k^j}$, where:

• $f_{j,k}$ is the first index (a superscript before $g$):

  • $j$ ranges from 0 to 3 and identifies the group:

    • $j\; =\; 0$: the base constant (an exception to the usual indexing).
    • $j\; =\; 1$: first-degree expressions.
    • $j\; =\; 2$: second-degree expressions.
    • $j\; =\; 3$: third-degree expressions.
  • $k$ is the index of a specific function within its group (e.g., $k\; =\; 1$ for the first function in group $j$).

  • The value of $f_{j,k}$ indicates the number of unique factor variables in the function:

    • $f_{j,k}=\; 0$: no factor variables.
    • $f_{j,k}>0$: the number of factor variables (e.g., 1, 2, or 3).

• Trailing indices on $g$:

  • The superscript $j$ after $g$ (e.g., $g^j$) repeats the group number for clarity.
  • The subscript $k$ after $g$ (e.g., $g_k$) matches the $k$ in $f_{j,k}$, identifying the function within its group.

While the indices of $f$ and $g$ align numerically, their roles differ: $f_{j,k}$ counts factor variables, which is critical for graphing decisions, while $j$ and $k$ on $g$ define its group and position.

10.3 Graphing Dimensions

The dimension $d$ of the graph for a function ${f_{j,k}}^{g_k^j}$ is calculated as:

where:

• $j$ is the number of unique variables (or group number, ignoring the constant case).
• $f_{j,k}$ is the number of factor variables.
• $d$ is the resulting graph dimension, with a maximum of 3 (since the max degree is 3).

For example:

• If $j\; =\; 3$ (3 variables) and $f_{j,k}=\; 0$ (no factor variables), then $d\; =\; 3\; -\; 0\; =\; 3$, requiring a 3D graph.
• If $j\; =\; 3$ and $f_{j,k}=\; 1$ (one factor variable), then $d\; =\; 3\; -\; 1\; =\; 2$, requiring a 2D graph.
• If $j\; =\; 0$ (the constant), graphing isnt typically needed, as its a single value.

When $d\; =\; 3$ (e.g., a third-degree expression with no factor variables), visualizing the full function may require multiple 3D graphs, each fixing one variable at different values to show a subset of the functions behavior.

10.4 The Final Model Equation

The MARS model is broken into four groups of expressions (constant, first-, second-, and third-degree). The indexing ${f_{j,k}}^{g_k^j}$ tells us:

• $j$: the group and number of variables.
• $f_{j,k}$: the number of factor variables.
• $d\; =\; j\; -f_{j,k}$: the graph dimension.

This structure guides how each $g$ function is visualized, ensuring the graphing method matches the expressions complexity and variable types.

The sale price estimate $\widehat{P}$ provided by the MARS model can be expressed as:

10.5 Number of Charts Needed to Graph a Model

How many graphs are required to visualize all functions in a model? The answer depends on the variables, their interactions, and whether they are factor or non-factor variables.

The VE can limit variables and interactions in the Excel configuration file for R/earth, influencing which interactions are allowed. A practical approach is to let R/earth evaluate all possible interactions and select those with significant contributions to the sale price estimate, while checking for issues like over-fitting or collinearity. Some interactions may be disallowed, affecting higher-degree interactions (e.g., if a pair is excluded in a third-degree interaction, the entire combination is excluded).

For factor variables (e.g., MLS Area with 12 values interacting with GLA), graphing becomes more complex. This could require 12 graphs—one per area. If another factor like Architectural Style (6 values) interacts with MLS Area and GLA, up to $12\; \cdot \; 6\; =\; 60$ graphs might be needed. However, R/earth rarely finds all combinations significant; typically, only a few key interactions (e.g., specific areas and styles) warrant graphing, as identified by the model.

Heres a simplified guide to the number of graphs needed:

1.

Single-variable functions (non-factor): 1 graph.

2.

Single-variable functions (factor): 1 bar graph combining all factor values.

3.

Base function: No graph needed.

4.

2nd-degree interactions (no factors): 1 graph (e.g., 2D heat map with colors for sale price).

5.

2nd-degree interactions (1 factor): 1 graph per factor value in the model (e.g., 2 values = 2 graphs).

6.

2nd-degree interactions (2 factors): 1 graph with a matrix of factor combinations (e.g., 3 areas $\times$ 2 styles = 6 cells).

7.

3rd-degree interactions (no factors): Multiple graphs (e.g., 4 graphs for Bath count = 1, 2, 3, 4).

8.

3rd-degree interactions (1 factor): 1 graph per factor value in the model.

9.

3rd-degree interactions (2 factors): 1 graph per combination of factor values in the model.

10.

3rd-degree interactions (3 factors): 1 graph per value of the factor with the fewest distinct values, using a color matrix for the other two.

More complex scenarios may require advanced graphing techniques.

There are other possibilities with likely more sophisticated graphs.

An example of a complete model generated by MARS is in 1.

11 Data Characteristics

The following section discusses MARS. MARS is entirely dependent on the data it receives, and most of this data will come from some MLS (Multiple Listing Service) or tax assessors. However, it may be routed through other data providers who maintain and improve data and then sell at some profit. In some areas, it is fairly reliable; in others, it is not so reliable. In critical instances, the property inspector has to make his observations and measurements, and their accuracy can be questioned.

The valuation engineer needs to understand what the data represents and how good of a job it does representing what it is supposed to represent. This invites a whole list of issues:

• How do you classify a variable as being a measurement?
• If a variable is not a measurement, what else can it be?
• How can regression be useful if the data it receives is not accurate?

Most of the data we get from MLS listing services and tax assessor data for regression is numerical data that implies measurements or counts. Some of it is logical, True or False.

11.1 Purpose of the Residual

The residual serves as an indirect gauge of the value added by variables not included in the regression model. By ordering properties based on their residual or residual per square foot, we achieve an objective ranking from properties with the least appeal to those with the most. This ranking method is effective, provided a skilled analyst with a deep understanding of the local market crafts the MARS (Multivariate Adaptive Regression Splines) model. The following are some additional requirements:

• Quality of Data and Model: A well-constructed regression model and high-quality data are prerequisites for reliable residuals. In the San Francisco Bay Area, my goal is to develop a model with an R2 of approximately 75-80% and a cross-validation R2 (CV R2) of 55-70%.
• Avoid Over-fitting, Limit the R2: The valuation engineer should aim for a maximum R2 of about 80% in a most mature markets in higher priced areas like the SF Bay Area because much of a residential propertys competitive value lies in its subjective appeal, which isnt quantifiable and thus cant be directly included in regression models. An R2 significantly above 80% might indicate an over-fitted model, prompting further scrutiny. Yet a word of caution; There are market areas with R2 values above 90% without over-fitting and there are difficult to appraise areas, where the best you might do is 50-60% or lower.
• High R2 in Specific Areas: However, there are regions, like subdivisions or other uniform residential areas, where higher R2 values are surprisingly common due to the homogeneity of the properties.

11.2 Data Errors

Based on social media comments, the California Multiple Listing Service (MLS) data is generally of higher quality than many other United States regions. Typically, MLS data should be expected to be of good quality in most mature metropolitan areas but poorer in rural areas.

While there are occasional errors in the data for various reasons, these errors should generally cancel each other out and have minimal impact on the averages used in price models if sufficient data exists.

However, systematic biases can occur in specific areas. For instance, consider a neighborhood where many homes were initially constructed with full second stories, but some buyers were given the option to remove second-floor rooms to create vaulted ceilings on one or more first-floor rooms, significantly reducing the gross living area (GLA). The changes were not subsequently updated in the assessors records for one reason or another. Consequently, MLS might show 2,400 square feet when the measured value is only 2,000 square feet. In such scenarios, the MLS data should be corrected before performing MARS (Multivariate Adaptive Regression Splines) regression. It is important to note that obtaining this information for comparables may require some investigative effort. Discrepancies should be noted for future reference.

Given the foregoing, it is reasonable to assume that in high-quality MLS areas, the errors that spill over into the residual are minimal compared to the impact of subjective and unmeasured features such as condition, quality, and design. This is an important consideration when estimating the accuracy of the RCA method. For example, if the $R^2$ value for a given market area is $80\%$, then approximately $20\%$ of the price is attributed to the CQA residual. A $\pm \; 10\%$ error in placing the subject within the residual ranking therefore propagates to a $\pm \; 10\%\; \cdot \; 20\%\; =\; \pm 2\%$ error in the final value conclusion. This is how you improve accuracy! In many cases, accurate value estimates well below $\pm$ 1% are possible.

Therefore, treating residuals as indirect indicators of the collective value of unmodeled variables in MARS regression is reasonable, provided that the valuation engineer has sufficient local market experience to identify where and why systemic measurement biases exist. This approach ensures that the analysis accurately reflects actual property values, acknowledging the limitations and nuances of real estate data.

Worked Example: Measuring Unmeasured Drivers at Sea Ranch

When the achievable $R^2$ in a market area is lower than the market’s apparent homogeneity would lead one to expect, the framework’s response is to identify and measure the missing variable rather than to accept the limitation. The Sea Ranch community in Sonoma County illustrates this directly.

A concrete instance of the measure-don’t-accept response. The Sea Ranch community in Sonoma County is a coastal development where distance to the ocean is a primary value driver: a home several hundred feet from the bluff commands a substantially different price from one a quarter-mile inland, holding other features constant. The MLS data for the community does not record ocean distance as a variable. An initial MARS model fit on the available MLS variables produced lower explanatory power than the market’s apparent homogeneity should have supported, with a residual structure that did not sufficiently order properties by any feature visible in the data — a signal that an important driver was missing from the model rather than that the market was genuinely noisy.

The remedy was direct measurement. The author obtained a community map showing the property layout relative to the coastline and measured the ocean distance for several hundred homes. The measured distances were added as a variable to the MLS dataset and the MARS regression was rerun. The new variable was selected by the model as significant, the explanatory power increased substantially, and the residual ranking subsequently ordered properties by patterns consistent with the remaining subjective features — condition, quality, view — rather than by the latent ocean-distance signal that had previously contaminated it. Elevation has been addressed similarly in hillside neighborhoods where the MLS variable is missing or unreliable.

The methodological principle is general. When the achievable $R^2$ falls below what the market’s apparent homogeneity should support, the first hypothesis is that an important variable is missing from the model, and the appropriate response is to identify what it is and to collect the measurements directly. Sources include community maps, GIS data, tax assessor records, building department records, satellite imagery, and field inspection. The cost of such measurement work is real but bounded; once the variable is collected for a given market area, it can be carried forward into future assignments in that area. RCA absorbs new variables cleanly because the framework is indifferent to which variables enter the MARS regression — the residual structure tells the VE whether the new variable captures real signal, and the value contributions adjust accordingly.

The alternative response — accepting a low $R^2$ and reporting wider error bands — is defensible when the missing drivers are genuinely unmeasurable (buyer-specific preferences, transient market sentiment, idiosyncratic transaction circumstances) but not when they are merely uncollected. The VE’s first move on a weak model should be to ask whether the missing signal is the latter.

11.3 What Is Accuracy?

Perfect accuracy for a valuation requires:

1.

Conformity to Market Value: The hypothetical sales transaction of the subject property adheres 100% to the given definition of Market Value

2.

Accurate and Robust Model Development: A high-quality MARS model is developed with an R2 of about 75-80% and a CVR2 of roughly 55-65% depending on location.

3.

Good Data Error Analysis: All likely data errors for measured variables are understood and contained, with little spill over into residuals.

4.

Systemic Bias Accounting: Any systemic bias in the data should be accounted and adjusted for.

5.

Residual Rank Analysis: Ensure that residual rank properties follow a reasonable pattern of features from lowest to highest rank. Anomalies such as probate sales, shorts, and auction sales should be investigated, explained, and potentially removed.

6.

Subject Placement: With a thorough understanding of residual ranking, the subject property should be accurately positioned. A property with a significantly higher appeal and another with a significantly lower appeal should be identified, and the subject placed between them and scored. Then attempt to narrow the ranking between the higher and lower properties as much as possible.

7.

Explanatory Reasoning: Never sacrifice explanatory reasoning for accuracy. The appraiser must be able to explain the differences in value between the subject and comparable properties, at least to someone with reasonable intelligence. For example, a good MARS model with an optimal R2 and CVR2 can be used to explain value contributions for measured features, while the breakdown of the residual by the valuation engineer can be used to explain the value contributions of other features.

8.

Over-fitting: Avoid over-fitting models, by not using property features that can identify either individually or in combination one or several properties. In particular avoid using factor variables whose values can specify a small number of properties. A factor variable that identifies neighborhoods in sizable data sets is appropriate, but one that identifies properties with swimming pools in an area where very few swimming pools exist, must be used with caution. A feature that identifies unique properties such as exact GIS coordinates, parcel numbers or addresses, is absolutely not allowed.

If the regression R2 is 80%, we expect about 20% of the subject value to be due to the residual. Now, this is only approximate. We know from the CQA-Residual curve that most of the distribution of residuals is at the lower and higher extremes of CQA. Based on the CQA-Residual curve and the extreme slope for the low and high CQA values, we might expect a less accurate estimate of the residual in these ares. However, that is counterbalanced by the relative scarcity of properties in these areas and the greater difference in appeal we see between adjacent properties. So , while we might talk of of the impact of a range of $\pm \; 5\%$ in the mid-range, at the extremes we would be more concerned with a somewhat narrower range of $\pm \; 2\; -\pm 3\%$

Calculating the accuracy of the RCA is, in general, somewhat difficult because of the non-linearity of the CQA-Residual curve. But it is possible on a case by case basis, given you have a reasonable residual ranking. The range of CQA scores is 0.0 to 10.0. With a good MARS model, your residual ranking should show a clear pattern of least to highest appeal and you should not find it difficult to rate within $\pm \; 5\%$ of the total range or $\pm \; 0.5$ for the CQA score. If you have 100 properties in the regression, then that would be ranking the subject property accurately within a group of 10 properties. So, referring to the Burlingame sales data in Table 2, lets calculate the approximate accuracy (or error) for a VE estimated CQA of 7.5, 5.1 and 1.9, where his the value conclusion for the subject is $3,000,000. Note that since Table 2 only has a sample of the several hundred properties used, you will need to interpolate the residual where necessary (assuming linearity). To be clear, we are assuming the VE is very sure in each case that his ranking of the subject is withing a range of $\pm \; 5$ properties in the resdiual ranking of only 100 properties.

Finally, the only objective way to verify accuracy is to establish a strict protocol (which this paper only partially hints at in broad strokes) and verify its adherence. This is necessary because actual sale prices can deviate from market value due to imperfections in the market. Comparing appraisals and subsequent sales can provide some indication of accuracy. However, the complexities and instability of the real estate market necessitate the expertise of appraisers or highly qualified valuation engineers to provide value estimates based on established standards and protocols. These should be stringent enough to ensure that competent valuation engineers will independently arrive at nearly the same value conclusion. This is should be an achievable goal, in most cases where good data is available.

11.4 The Subject Property As An Anomaly

Since estimating the Market Value of a property requires analysis of recent sales transactions, it is fair to ask if recent sales transactions do a good job of representing the various features of an older subject property that lacks modernization and repairs. This is particularly true in the San Francisco Bay Area, where recent sales transactions typically are homes that have gone through updating to improve their price for sale. So, the average quality and condition of sales comparables are generally superior to older homes that have not been on the market for quite some time. This difference in value should be picked up by the estimate of the residual for the subject property, or in other words, it’s CQA score. This is an important consideration with respect to data quality and accuracy in valuing the subject property.

11.5 The CVR2’s Importance In Dealing With Unexpected Data

MARS regression is based on multiple iterations of R/earth applied to a randomly selected subset of properties, which typically comprises 80% to 90% of the MLS data set referred to as the training set. The remaining 10% - 20% of properties serve as the test set. In this phase, the MARS model is created from the training data set and then is used to predict the sale prices of properties in the test data set, which are then compared to the actual sale prices. This methodology has been recognized by data miners as the most effective strategy to develop a robust model capable of predicting the sale prices of properties not yet observed.

However, the primary utility of the CVR2 metric lies not in its reflection of the regression model’s factual accuracy concerning actual market area sales, but rather in its optimization of the model’s predictive capabilities regarding hypothetical property sales as of a specified effective date. These properties are not included in the training data set and may exhibit new or unexpected combinations of characteristics missing in both the training and test data set property data.

Given this context, how can we estimate the sale price of a property that hasn’t been updated in the past 15 years, assuming it was listed, for example, one month prior to the effective appraisal date? A model that demonstrates a strong CVR2 is more likely to provide the most accurate estimate, rather than a model that has not gone through this repetitive cross validation. It’s also vital that the data set includes a sufficient number of comparable properties with varying degrees of updates, repairs and modernization.

Ultimately, if we possess a robust model that accurately predicts the value contributions of various properties features in our data set of comparable properties, we can effectively place the new subject property within the model’s ranking of these properties by residual. This placement allows us to assign an appropriate CQA score and residual to the property.

12 Residuals

As indicated by the name Residual Constraint Approach, residuals are central to this methodology. Here, residual refers to the difference between a propertys actual sale price and the sale price predicted by regression analysis.

A positive residual suggests that the property sold for more than its estimated price, based on typical measurable attributes like gross living area (GLA) and lot size, implying it might have more appeal than average. A negative residual indicates the property sold for less, potentially suggesting less appeal or other uncaptured negative attributes.

Given that larger homes likely have larger residuals, we might use residual per square foot based on GLA for a more normalized comparison. A more in-depth discussion of this issue is reserved for another paper.

12.1 Ranking &Scoring

After creating the MARS (Multivariate Adaptive Regression Splines) model for measured variables, residuals are computed for each property. These residuals represent the difference between the observed net sale prices and the sale prices predicted by the model. An additional column should be created for the Residual/SF, that is, the residual divided by the living area of the residence, with the understanding that generally, the residual for specific properties is more significant with the size of the living area. Other possible statistics may work better in some neighborhoods, but they are a subject for another time

Next, the properties are sorted by their residuals from the most negative to the most positive. A Condition-Quality-Appeal (CQA) score is then generated for each property, ranging from 0.0 to 10.0. This score is calculated as follows:

The CQA score corresponds to the percentile rank of a propertys residual among all properties. Specifically, the score is computed by determining the percentage of properties that have residuals lower than that of the given property and dividing that percentage by 10. For instance, if a propertys residual is higher than the residuals of 50% of properties, then its score would be 5.0.

Due to rounding, one property can receive a score of 10.0, representing the property with the highest residual. However, a perfect score of 10.0 is theoretically impossible because, logically, 100On the other hand, the lowest possible score of 0.0 is possible since at least one property has no other properties with lower residuals. Note that more than one property might have a score of 0.0 due to rounding if many properties are being compared.

Therefore, the CQA score provides an intuitive measure of how well a propertys attributes align with the models expectations, with higher scores indicating better alignment with the models predictions.

It is essential to acknowledge that the CQA score can only be considered an indirect indicator of a comparable propertys appeal when the sale conditions are commensurate with its market value. The Valuation Engineer is responsible for ascertaining whether the residual value incorporates over- or under-market elements by providing a comprehensive explanation and thorough investigation of the sales transaction. While significant over- and under-market sales are not frequent, they can occur in circumstances such as forced sales, probate, incompetence, collusion, etc.

Table 2 presents a semi-realistic extract of MLS data, along with residuals and CQA scores, both in total and per square foot. The total number of properties analyzed in this study was approximately 600. It is noteworthy that several variables that were regressed are not displayed. In other words, the residuals presented are after the impact of other variables, such as the date of sale, has been considered by MARS regression.

12.2 CQA-Residual Curve

A plot of the CQA vs. the Residual/SF for a neighborhood will invariably result in the CQA-Residual Characteristic Curve shown in Figure 2. This particular plot represents data from Burlingame, CA, circa 2022. Similar curves are generally observed when plotting CQA scores against Residual values; however, I prefer using Residual/SF for its relevance in contextualizing the CQA with respect to the Gross Living Area (GLA) of the subject property. By multiplying the Residual/SF by the GLA, one can convert the CQA of a property into a corresponding residual value.

The valuation engineer must carefully evaluate whether the size of the property, specifically the Gross Living Area (GLA), significantly influences the magnitude of residuals. This assessment is critical because it can determine the most accurate method for calculating property values. For instance, high-end kitchen appliances and luxurious bathroom fixtures can substantially enhance a home’s residual value. However, in neighborhoods where the size and layout of kitchens and bathrooms are generally uniform, these upgrades might not correlate strongly with the GLA. Their impact on the residual value may be less about the space they occupy and more about the quality they represent.

Conversely, expensive exterior features and materials, which notably improve a property’s curb appeal, may have a more direct relationship with the property’s size. Larger homes can accommodate more elaborate exteriors, such as extensive landscaping, superior roofing materials, or expansive patios, which naturally enhance the propertys overall appeal and, consequently, its residual value.

Therefore, it is essential for valuation engineers to consider these dynamics when analyzing residuals. They should not only assess the intrinsic value added by high-quality fixtures and features but also understand how these enhancements interact with the propertys physical dimensions. This comprehensive approach ensures that valuations accurately reflect both the qualitative and quantitative attributes of a property.

Consequently, it is recommended to plot both the CQA-Residual and CQA-Residual/SF foot curves and study the differences, which can indicate patterns in a market area. The valuation engineer can have his program compute both residuals and take some weighted average to calculate the residual to be assigned to the subject. It may or may not make much of a difference.

12.3 The CQA Curve Characteristics

The Composite Quality Appeal (CQA) curve offers a sophisticated model for understanding how property values fluctuate based on appeal, condition, and market dynamics within a given area. This model captures the interplay of economic and physical factors that influence the real estate market, reflecting nuanced realities of property valuation across different market segments.

12.3.1 Analysis of the CQA Curve Components

1.

Exponential Decay on the Left Side of the Curve:

As the CQA score decreases from around 2 to 0, properties exhibit characteristics of rapid deterioration. This part of the curve is well-represented by an exponential decay function, emphasizing how quickly a property can lose its appeal and value without proper maintenance and updates. Physical degradation such as rotting wood, fading colors, and rusting metal accelerates this decline, particularly in homes that are not regularly maintained.

2.

Linear Increase in the Middle Range:

The bulk of the market typically fits within this category, where the appeal increases almost linearly from scores of around 2 to 8. This section reflects properties that are generally maintained to standard but don’t necessarily feature exceptional qualities or locations. It represents the broad middle class of housing, encompassing the majority of the housing market where homes are competitively priced based on their standard features and overall condition.

3.

Exponential Growth on the Right Side of the Curve:

Here, the curve rises exponentially from scores 8 to 10, mirroring the higher appeal end of the market. Properties in this segment often belong to wealthier individuals who not only invest in superior maintenance but also in enhancements that significantly boost property appeal and value. The exponential growth reflects the scarcity of such high-quality properties and the high demand among affluent buyers. And understand, a lower priced home may rate very high in terms of appeal, while a large and expensive home may rate low. These dynamics are uncovered by expert use of MARS and usually go unnoticed by most appraisers.

12.3.2 Market Dynamics and Social Implications

The CQA curve implicitly narrates the story of relative wealth and economic disparity within different regions. In places like San Mateo County, the variation in property values between areas like Burlingame and wealthier neighborhoods such as Hillsborough, Woodside, or Atherton illustrates significant socioeconomic divides. This disparity affects everything from property maintenance to market valuations and ultimately influences buyer demographics and real estate market trends.

Moreover, the anecdote about a relatively wealthy individual investing in a high-quality home within a mid-tier neighborhood they are comfortable with and where for various reasons they would prefer to live, underscores another critical aspect of real estate economicsthe concept of "verbuilt"properties. Such properties, while very appealing, may not find local buyers able to pay the price they might have fetched in a higher priced market area.

The CQA-Residual curve is characteristic to each market area, and should not be overlooked when doing your analysis.

12.4 Subject Residual

Unlike the comparables, the subject property does not possess an objectively determined residual; it must be estimated by the Valuation Expert (VE). This estimation introduces a degree of subjectivity into the Residual Constraint Approach (RCA) protocol, marking a critical juncture where personal judgment plays a pivotal role.

To estimate the Condition-Quality-Appeal (CQA) score, the VE meticulously analyzes the ranked properties, searching for discernible patterns within the rankings. This analysis often involves a detailed review of MLS photos associated with each property. An exhaustive review is impractical for a large dataset, such as 600 properties. Instead, the VE applies practical reasoning to approximate the subjects likely position within the overall ranking. Properties requiring extensive repairs, or fixers, are generally positioned near the bottom, whereas those with superior updates and appeal tend to rank near the top.

The VE then adopts a more focused approach, jumping within the ranking to identify comparables that closely match the subject’s appeal. As the range narrows, the VE pays closer attention to similarities among the properties, particularly in aspects like quality, condition, design aesthetics (including elements like paint, woodwork, stonework, and windows), functional utility, and other relevant characteristics

It is crucial to understand that the VEs judgments are fundamentally guided by the market-established rankings. This ensures that the subjective elements of the process are anchored by market realities, striving for a fair and accurate valuation of the subject property.

12.5 RCA Value Conclusion

12.5.1 General Considerations

Determining a CQA score is pivotal for establishing the estimated residual value of a property, as these elements are intrinsically linked. While it is possible to assess a property without a CQA, this metric provides valuable context by illustrating where the property stands compared to others in the market area. For instance, a CQA score of 3.2 indicates that the property has greater appeal than 32% of comparable properties in the vicinity. If objections exist to the assigned score, disputants must consult the rankings and justify any proposed adjustments based on photographic evidence or other substantial data. If the ranking system is robust and the Valuation Expert (VE) has accurately positioned the property within this framework, it is unlikely that significant discrepancies will arise.

In the reporting phase, typically, 6 to 12 recent and relevant sales comparables are selected to support the valuation conclusion. The differences between these comparables and the subject property are analyzed, with each variable being adjusted accordingly. These adjustments are then applied to the net sale prices of the comparables to derive adjusted sale prices. As demonstrated mathematically, these adjusted prices should converge with the estimated sale price of the subject property.

12.5.2 Review, Auditing, Complaints

Once the estimated residual for the subject property is determined, residual adjustments can be calculated for all comparables. These adjustments serve as constraints for further manual breakdowns by the VE, tailored to specific CQA variables deemed pertinent. It is important to emphasize that these detailed breakdowns are explanatory and do not influence the final valuation, as affirmed in Proof II.

Concerning complaints submitted by users not satisfied with the appraisal value conclusion, a traditional appraiser typically has to waste time explaining why this or that property with a higher or lower price does not have the user-expected impact on value. With RCA, all possible comparable sales (usually 100-600) are typically evaluated to precisely the same value as the price conclusion, so it takes little effort to show the adjustments for some arbitrary property. The entire RCA process is automated and objective, except for the scoring of the subject residual. And so another advantage of the RCA is that criticism for imperfections in the subjective judgment of the VE is mainly restricted to that one area - and it can be effectively and efficiently managed. Also, assuming an R2 of about 80%, the impact of a $\pm$ 10% error in estimating the subject residual amounts to a $\pm$ 2% error in the final value estimate.

In many cases, the subject fits so perfectly in the residual ranking of properties, that the accuracy can be expected to be within $\pm$ 1%.

The sum of the estimated residual and the subjects MARS estimated sale price yields the final value conclusion known as the Residual Constraint Analysis (RCA) Value Conclusion. This process ensures a comprehensive and transparent approach to property valuation, fostering confidence in the accuracy and fairness of the assessment.

13 Conclusion

The Residual Constraint Approach is a framework with three substantive contributions: a structural anti-anchoring property derived from fitting the model on the whole market rather than on a small comparable set (Proof I); an invariance result that allows residual breakdown to serve as an explanatory layer without disturbing the value conclusion (Proof II and the remark following it); and an empirical regularity — the CQA-Residual characteristic curve — that appears to recur across market areas in characteristic exponential-linear-exponential form. The framework’s limitations are equally substantive: the residual decomposition is invariant but not identifying, the CQA placement step relies on VE judgment that has not yet been validated through formal inter-rater study, and the accuracy claims await benchmarking against alternatives.

What this paper presents is a framework and a protocol. What it does not yet present, but the larger research program now in progress does, is an executable, audit-traceable pipeline that operationalizes the protocol in software with defensible defaults. That pipeline — the earthUI package on CRAN for the MARS basis-function stage, the glmnetUI package for the regularized structural model on those bases, and the mgcvUI package for spatial smoothing — is the subject of companion articles in this issue and is the bridge from RCA-as-method to RCA-as-deliverable. The doctrine-practice gap in appraisal regulation (Craytor, 2026, this issue) supplies the reason that bridge matters: appraisal outputs are economically consumed as forward-looking risk inputs, and a framework that makes the present-value opinion auditable and reproducible is a precondition for any honest engagement with that forward-looking use. RCA is one move in that larger project. It is not the whole project, and this paper does not claim it is.

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