Using a regression-based adjustment matrix in comparable sales analysis is a common and sound practice in real estate valuation. Regression analysis can provide insights into the factors influencing property values by examining the relationship between property sale prices and various independent variables. It enables the creation of an adjustment matrix with uniform coefficients to align comparable sales with the subject properties, ultimately leading to more accurate valuations.
Dataset and Variables
This regression output was developed using a home sales dataset from a specific town for eighteen months, from January 2023 to June 2024, generating coefficients to help create a statistically significant adjustment matrix with uniform coefficients, aiding in adjusting the comparables to evaluate a series of subject properties with a targeted July 1, 2024, valuation date.
The
regression analysis uses the sale price as the dependent variable and six
independent variables. One of the variables, "Months Since,"
represents the number of months since the sale. For example, a sale
in January 2023 will be assigned a value of 18 (July 2024 minus January 2023),
while a sale in June 2024 will be given a value of 1. By applying the
coefficient value to the months, sales from January 2023 will receive an upward
adjustment of $3,908 ($217.11 multiplied by 18), and sales from June 2024 will
be adjusted by $217.11 ($217.11 multiplied by 1).
The
"Exterior Wall" variable has been effect-coded to focus on each category's deviation from the town's median sale price. The effect-coded
Exterior Wall values range between +$93,100 (Stone) and -$30,000 (Concrete Block).
Bldg
Age is a synthetic variable calculated by subtracting the property's year built from the prediction year 2024. The other variables are quantitative data
variables obtained from public records.
Since
all properties and comparables will come from specific neighborhoods within
this town, no location variable has been introduced. Finally, the intercept was
forced to zero, as this regression was conducted to produce coefficients to
adjust comparables externally rather than producing competing regression
values, per se.
Analysis
Now, let's analyze the regression output:
1. Multiple
R and R Square: Multiple R of 0.981996 indicates a strong
positive relationship between the independent variables and the dependent
variable (sale price). The R-squared value of 0.964316 indicates that approximately 96.43% of the variance in the dependent variable is explained
by the independent variables in the model.
2. ANOVA: The
ANOVA table shows that the regression model is statistically significant. The
F-statistic of 4,400.37 with a p-value of 0 indicates that the regression model
as a whole is a good fit for the data.
3. Coefficients:
o
The coefficient for "Months Since"
is 217.11, implying that for each additional month since the sale, the sale
price increases by $217.11.
o
The coefficient for "Land SF" is
2.08, indicating that for every additional square foot of land, the sale price
increases by $2.08.
o
The negative coefficient for "Bldg
Age" (-263.638) signifies that as the age of the building increases, the
sale price decreases by $263.638.
o
Heated SF: An increase of 1 unit in Heated SF
leads to an increase of $145.43 in sale price.
o
Bathroom: Each additional bathroom is
associated with an increase of $33,300.96 in sale price.
o
Exterior Wall: The effect-coded variable indicates
how the specific type of exterior wall affects the sale price.
Overall, the regression output suggests that the model
with these independent variables can explain a substantial portion of the
variation in sale prices. The coefficients provide insights into the
relationships between the independent variables and the sale price, which can
be used to adjust comparable sales for more accurate valuations. This approach
provides a data-driven method for ensuring fair and accurate valuations using
comparable sales analysis.
It's important to note that the soundness of the
experiment also depends on the quality and representativeness of the data used,
the appropriateness of the variables selected, and the assumptions made in the
regression analysis. Conducting further validation and sensitivity analyses can
help ensure the reliability of the regression-based adjustment matrix for
property valuations.
Important to Know (for New Analysts)
1. Months Since variable
Given that "Months
Since" has passed the multicollinearity test (discussed in Blog Post 1)
and the importance of its explainability, here is why you might consider
keeping the variable in the model despite its weak p-value:
Multicollinearity
is not a concern: Since "Months Since" doesn't
correlate highly with other independent variables, it provides unique
information about the time-based adjustments, strengthening the argument for
keeping it in the model.
Explainability
is crucial: If reviewers and taxpayers need to
understand the rationale behind the adjustments, "Months Since" clarifies why older sales receive larger adjustments to reflect market
changes.
Here
are some strategies to address the weak p-value:
Explaining
the context: It must be clearly stated in the report that
"Months Since" is included for explanatory purposes, even though its
p-value is not statistically significant at the usual 0.05 level. Additionally, mention that it passed the multicollinearity test and emphasize its role
in improving model interpretability.
Focusing
on coefficient direction and magnitude: While the p-value suggests a
lack of strong statistical evidence, it's worth noting the positive coefficient on "Months Since" and its implication that older sales receive larger adjustments, aligning with market trends, where recent sales
likely reflect more current prices.
Exploring
alternative presentations: It's worth presenting
the adjustment matrix alongside a graph that visually depicts the impact of
"Months Since" on the adjustment, providing a clearer picture of the
time-based adjustments.
Overall,
retaining “Months Since” seems reasonable in this case. The lack of
multicollinearity and the importance of explainability outweigh concerns about p-values. However, it's crucial to ensure clear communication
regarding the variable's limitations in terms of statistical significance,
maintaining transparency and trust in the analysis.
Forcing the Intercept to
Zero
Forcing
the intercept term to zero in a regression model is common in some contexts,
especially when the focus is on estimating coefficients for adjustment purposes
rather than predicting absolute values. In this case, the regression was
conducted to produce coefficients to adjust comparable sales externally, not to
predict the actual sale prices of properties.
Statistically,
forcing the intercept to zero can be sound if there is strong support from the
underlying theory or domain knowledge. This suggests that the relationship
between the independent variables and the dependent variable passes through the
origin (i.e., when all independent variables equal zero, the dependent variable
should also equal zero).
Since the regression was
explicitly conducted to produce coefficients for adjustment rather than to predict absolute values, forcing the intercept to zero is justifiable.
However, it's essential to ensure that the statistical soundness of this
decision aligns with the specific requirements and goals of the analysis.
Conducting sensitivity analyses or comparing results with and without forcing the intercept term to zero could provide further insights into the
robustness of the regression model.
Regression-based Adjustment Matrix
Important to Know (for New Graduates)
When
performing a comparable sales analysis, adjustments are typically made to the
sale prices of the comparable properties to account for differences in their
characteristics relative to the subject property. The idea is to estimate how
much the comparable properties would have sold for if they had the exact same
characteristics as the subject property.
In
this context, since the subject property serves as the baseline or reference
point against which adjustments are made, no adjustment needs to be applied
directly to the subject property itself. Instead, adjustments are calculated
for each comparable property based on the differences in its characteristics
compared to the subject property. These adjustments are then applied to the
sale prices of the comparable properties to derive adjusted sale prices that reflect the comparables' value as if they were similar to the subject
property.
Therefore,
in the absence of a sale price for the subject property, the focus is on
adjusting the sale prices of comparable properties relative to the subject property to accurately estimate the subject's value.
Here is a
comprehensive analysis of the adjustment process:
1. Sale
Month Adjustment:
- The adjustment for the month of sale is based on the difference in months between the comparables' sale dates and the valuation date (July 2024). A greater number of months since the sale leads to a larger adjustment.
- For example, Comp-1 was sold in January
2023, 18 months before the valuation date. The adjustment for Comp-1 is
calculated as 18 * 217.11 = 3,908.
- The adjustment for Comp-2 is smaller as
it was sold in June 2024, just one month before the valuation date. The
adjustment for Comp-2 is 1 * 217.11 = 217.
- These adjustments account for the time
difference in property sales.
2. Land
SF Adjustment:
- The adjustment for Land Square Footage is
based on the differences between the subject and comparable properties. A
larger size receives a negative adjustment, while a smaller size receives
a positive adjustment.
- For example, Comp-1 has 11,000 SF, and
the subject has 10,454 SF. The resulting negative adjustment for Comp-1 is
(10,454 - 11,000) * 2.08 = -$1,136.
- Similarly, the adjustment for Comp-2 is
positive as it has 9,000 SF, resulting in an adjustment of (10,454 -
9,000) * 2.08 = $3,024.
3. Bldg
Age Adjustment:
- The adjustment for Building Age is based
on the buildings' age relative to the subject property. Older
properties receive a negative adjustment, while newer ones receive a
positive adjustment.
- For example, Comp-1 is 40 years old,
while the subject is 47 years old. The adjustment for Comp-1 is (47 - 40) * -263.64
= $1,845.
- On the other hand, Comp-2 is 55 years
old, resulting in an adjustment of (47 - 55) * -263.64 = -$2,109.
4. Heated
SF Adjustment:
- The adjustment for Heated Square Footage
is made with the same objectivity as the Land SF adjustment. Larger sizes receive negative adjustments, and smaller sizes receive positive adjustments. For example, Comp-1 has 1,750 SF, while the subject has 1,664 SF. The adjustment for Comp-1 is (1,664 - 1,750) * 145.43 =—$12,507.
- In contrast, Comp-2 has 1,550 SF, leading
to an adjustment of (1,664 - 1,550) * 145.43 = $16,579.
5. Bathrooms
Adjustment:
- The coefficient for Bathrooms is
33,300.96.
- The adjustment for Bathrooms is
calculated as the difference between the number of bathrooms in each
comparable property and the subject property, multiplied by the
coefficient.
- Comp-1 has three bathrooms, while
the subject property has two bathrooms.
- The adjustment for Comp-1 is (3 -
2) * 33,300.96 = -$33,300.96.
- Similarly, the adjustment for
Comp-2 can be calculated based on the number of bathrooms in Comp-2
relative to the subject property.
6. Exterior
Wall Adjustment:
· Given that the subject property's Exterior
Wall is Stone, we can calculate the adjustments for the Exterior Wall for the
comparable properties using the effect-coded values. Here is how the
adjustments for the Exterior Walls for Comp-1 and Comp-2 can be calculated
based on the effect-coded values:
- For Comp-1 with an exterior wall of
Concrete Block (effect-coded value is -$30,000): Adjustment = (-$30,000 -
$93,100) * 0.16 = -$10,096.
- For Comp-2 with an exterior wall of Stone:
Adjustment = ($93,100 - $93,100) * 0.16 = $0.
7. Total
Adjustments and Adjusted Sale Price:
- By summing the individual adjustments for each comparable property, we arrive at the total adjustments made to the comparable sale prices to reflect the
subject property's characteristics.
- The adjusted sale price for each
comparable property is obtained by adding the total adjustments to the
original sale price.
- The subject value is then calculated as
the average of the two adjusted sale prices.
In essence,
the adjustment process involves:
- Carefully
analyzing the differences in characteristics between the subject property
and the comparables.
- Applying
the respective coefficients to make adjustments.
- Arriving
at an adjusted sale price for each comparable property.
The
subject value is then determined based on the average of the adjusted sale
prices.
Conclusion
In this blog post, I used multiple regression
analysis to show how statistically sound coefficients can help adjust
comparable sales and accurately value subject properties. The regression output
provided clear evidence of the impact of these coefficients in the valuation
process, leading to more refined and reliable estimates. While the study
focused on a specific town with limited adjustment variables, the upcoming book
will explore more advanced analyses across various counties, covering the
development and application of regression-based adjustment matrices at the
county and tax district levels, including techniques such as hybrid fixed
neighborhood and location surface analysis, effect-coded categorical variables,
one-hot binary variables, and more. These advancements will undoubtedly improve
the precision and sophistication of property valuation methodologies, setting a
new standard in the field.
Coming Soon: Part 3 of 3 – Applying regression to value subjects using different methodologies.
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