The Art and Science of Comparable Sales Analysis (Part 2 of 3)

Target Audience: New Graduates/Analysts

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 that influence property values by analyzing the relationship between the sale price of properties and various independent variables. It allows for the creation of an adjustment matrix with uniform coefficients to adjust comparable sales to the subject properties, ultimately leading to more accurate valuations.

Dataset and Variables

This regression output was developed using 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 took place. 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 the deviation of each category 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 year the property was 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 Square value of 0.964316 suggests that approximately 96.43% of the variance in the dependent variable can be 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" explains why older sales receive higher adjustments to account for 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 mentioning that it passed the multicollinearity test and emphasizing 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 wise to discuss the positive coefficient of "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 the concern about the p-value. 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 purposes rather than predicting 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 the results with and without the intercept term forced 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 their characteristics compared to the subject property. These adjustments are then applied to the sale prices of the comparable properties to arrive at adjusted sale prices that reflect the value of the comparables 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 the comparable properties relative to the subject property to estimate the subject's value accurately.

Here is a comprehensive analysis of the adjustment process:

1. Sale Month Adjustment:

• The adjustment for the month of sale is made based on the difference in months between the sale date of the comparables and the valuation date (July 2024). A higher number of months since the sale leads to a more significant 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 age of the buildings 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. 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 up all the individual adjustments for each comparable property, we arrive at the total adjustments made to adjust 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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