Many in the profession know that I am a fan of statistics. As a matter of routine, I have been using linear regression analysis in every appraisal for many years, either as primary or secondary support for at least one adjustment. Lately, there has been plenty of talk in Portland, Oregon and elsewhere about using linear regression analysis in real estate appraisal to support adjustments. In the process, it seems that a lot of misinformation about statistics and regression is also being circulated. Consequently, I will attempt to set the record straight by discussing the proper use of R squared in real estate appraisal linear regression.
R squared, also known as the coefficient of determination, is a measure (between zero and one) of how well a regression line fits the data points. An R squared of numerical one means that the model has perfect correlation and predicts every outcome. Here is a table of data and a very simple regression example with perfect correlation.
Reported Sales Price
All of these sales have been hand selected for small sites and are similar in all other ways, other than GLA.
In the above regression chart, it is easy to see that prices are increasing at $75 per square foot as indicated by the slope (75x in the regression formula). The R squared value of one says that these sales fit the line perfectly. An appraiser can also pair any of the sales in the above table and the result will also be exactly $75 per square foot. The next chart and regression graph does not have perfect correlation due to large and small site sizes mixed in the data. Remember that the large and small site sizes are just an example. In reality, site size differences could represent any type of variation commonly found in sales of homes (e.g. condition, features, etc.).
In the above regression chart, I have introduced three new sales with large site sizes. If an appraiser pairs the large site sales with the small site sales, the adjustment is $25,000. If an appraiser pairs any of the sales for GLA, the answer will also be exactly $75 per square foot. R squared is less than the numeral one, but the slope and the adjustment are still $75 per square foot. This is because the data are no longer a perfect fit along the line, but the sales all still increase at $75 per square foot. The next chart has a lower R squared value because the adjustment for site size is $45,000, rather than $25,000 and the adjustment per square foot remains the same.
In the above chart, the larger variation for sales price results in a smaller R squared value but the adjustment for square footage remains the same. R squared is only a measure of how well the data points fit the line. In real estate appraisal, the fit of R squared will usually be much less than ideal. An R squared that is low does not mean that the adjustment provided by regression is less accurate or less valid. An R squared that is low does not change the adjustment that the appraiser should apply for that factor being measured. In the above examples, we are only solving for GLA and it does not matter that there are other factors of variation (in this case site size), as the other variables are evenly dispersed along the trend line.
The appraiser should not rely on R squared as an indicator of reliability in the regression adjustment. The appraiser should examine the raw data or the scatter chart and look for factors that might be skewing the data or pushing the line in one direction or another. Common factors that can skew a GLA regression line is sometimes larger homes also have larger sites or higher quality. It is essential for the appraiser to carefully control the search parameters of sales data in ways that avoid skewing and collect large enough samples that normal variation of other factors can balance out and not skew the results. Appraisers should ask themselves, “If I remove just one data point from this scatter chart, will the trend change dramatically?” If the answer is yes, then maybe the regression model is too small. In that case, a larger sample or a more controlled sample might be necessary. I recommend leaving the R squared off the trend line chart and out of the appraisal report. R squared values will only confuse the reader of the appraisal report and will not strengthen the appraiser’s argument for or against the regression results.
The following is my most popular video on YouTube that gives an example of how to support a GLA adjustment using simple linear regression.
Did I leave anything out or do you want to join in the conversation? Let me know by commenting below.
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Thanks for reading,
We are now officially in the holiday season — the season of giving. As an appraiser, I am asked frequently by homeowners, real estate agents, and even by other appraisers, “What do you give for BLANK value?” (“BLANK” in this context can be anything related to real estate from a window, to a garage, to vacant land, to an entire structure.) The question is one that troubles me because of the word “give.” My response to such a query is always, “Appraisers do not give value; we estimate value.”
Each property and every locale are different. There is no set value for any specific real estate or property feature. The Super Appraisal Blogger, Ryan Lundquist put it best, “A Little Black Book of Value Doesn’t Exist in Real Estate.” When an appraiser makes value adjustments, each must be supported using appraisal techniques that can withstand peer review. For instance, garage space in a high-rise downtown Portland condominium likely has a much different value than garage space in a suburban Portland location.
The three tools that an appraiser has to determine the value of anything (including adjustments for individual features or the value of the entire property) are the cost, the income, and the sales comparison approaches. When applied correctly, all of these methods are market derived and credible. Each of the three approaches has many variations that the appraiser can use to estimate value rather than give value.
The Cost Approach uses estimates of new home cost (or features) and depreciates that cost based on condition or the market acceptance. A garage might have a cost new of $20,000 and the market-derived depreciation data (including condition and market response for other factors) suggest a value of something different. Appraisers are uniquely aware that cost often does not equal value; therefore, we consider depreciation carefully.
The Income Approach is based around many different ways to assign present value to the likelihood of future income. For example, a garage might add $100 per month in rent to a particular property. Using the simplest of income approach methods, if similar sales suggest value is roughly one hundred times the gross monthly rent, we might conclude that the income approach suggests that the garage value is $10,000.
The Sales Comparison Approach uses similar sales of carefully-selected properties to estimate value. To isolate the value of a specific feature, appraisers might use paired sales of properties that are similar except for that feature. Since strong individual paired sales are often unavailable, an appraiser might use several adjusted pairs, statistical regression models, or groups of sales to isolate a particular feature. A sample of similar properties without a garage compared to a sample of similar properties with a garage might provide the evidence that an appraiser needs to strongly and quickly support a garage adjustment. In this example, appraisers need to be careful that the properties with garages do not tend to have other value-added features (that the properties without garages do not tend to have) that were not controlled for in the sample.
Did I leave anything out or do you want to join in the conversation? Let me know in the comments below.
If you find this information interesting or useful, please subscribe to our blog and like us on Facebook. Also, please support us by making Portland real estate appraisal related comments on our blogs and YouTube videos. If you need Portland, Oregon area residential real estate appraisal services for any reason, please request appraisal fee quote or book Gary F. Kristensen to speak at your next event. We will do everything possible to assist you.
I apologize to my loyal subscribers who have recently tried (unsuccessfully) to comment on my blog postings through the AQA website. The website software company, A La Mode Software, has been working (unsuccessfully) on a fix for over a month now. Needless to say, I am getting impatient.
Last week’s blog, Statistics Work for Real Estate Appraisers, Even with Few Sales, did generate quite a bit of interest; consequently, I received many appraiser questions about it thru our email and social media links. Here are some great comments that I received along with my responses to them.
Question: Several people asked about how to make a simple regression chart using Excel and what the formula on the graph indicates.
Response: To make a simple regression chart, all you need to do is make a scatter chart of the data with the Y-axis being Reported Sales Price and X-axis being the variable that you want to solve for. In this example, we want to solve for a living area adjustment, but we could also solve for sales date, site size, or any variable that we think might have some linear relationship to sales price. Here is a video showing how a simple regression analysis can be added to an appraisal report in about one minute (the video takes longer because I’m talking).
Question: Edd asked, “I also use simple regression for many of the same reasons and in similar circumstances as you do. The problem I am concerned about is that statisticians (not appraisers) say small samples are not statistically significant. What if a sophisticated attorney were to ask on cross about the reliability of small sample or populations?”
Response: I would simply tell that attorney that we have to use what we have. Statisticians analyze data in crash testing of automobiles, but they do not use large samples for good reason. I don’t hear anyone saying that the small samples in crash testing are statistically insignificant. With that said, I do not solely rely on statistics and I always make sure that my sample sizes are large enough for a reasonable conclusion (little variation requires fewer sales than large variation). After I have developed an adjustment using statistics, I test the reasonableness using the cost approach (an adjustment of $20 per sf might not be reasonable for a like-new building that costs $150 per sf to build) and a test of paired sales when it is applied to the adjustment grid. If the adjustment is moving the adjusted indicators closer together than a higher or lower adjustment would (after other market derived adjustments have been applied), then the paired sales are validating the statistically derived adjustment. If not, then maybe we need to look at factors that might be skewing our sample.
Question: David asks, “What was your R-Square on this graph? Doesn’t look like it would be much more than 50-60% from the scatter graph. This could suggest that there are more variables at play in your data set than just size of the sales.”
Response: I do not know what the R-Squared is for the above data. It is not ideal. However, we have to work with what we have. R-Squared for regression are typically high when used in real estate, because as you point out, there are many variables. The thing to understand about the use of a simple regression in an appraisal is that we are not trying to solve for all of the variables in one simple regression analysis. A high R-Squared is not necessarily a bad thing in that context unless the outliers are clustered toward one end of the scatter chart, then they will be much more likely to change the slope of the trend, thereby skewing the adjustment.
If you find this information interesting or useful, please subscribe to our blog or like us on Facebook. Also, please support us by making Portland real estate appraisal related comments on our blogs and YouTube videos. If you need Portland, Oregon area residential real estate appraisal services for any reason, please request appraisal fee quote or book Gary F. Kristensen to speak at your next event. We will do everything possible to assist you.
If you are personally acquainted with me or merely follow my blog, you probably know that I am an appraiser who is a fan of using statistical regression to support adjustments and to analyze trends in the appraisal process. In fact, I recently made a video on the topic.
One comment often heard from other appraisers is that one needs large samples of data for statistical regression to work. This statement is only partially true. Large samples are required when there is great variation among the comparable sales or many outliers, as appraisers are familiar with when filling the 1004MC Market Conditions Addendum. However, if the sales are similar in most ways, the sample size may not need to be large. For example, I performed a regression analysis of land sales within a single development to determine the contributory value of each additional square foot of site size. Since all of the sales were very similar in terms of most factors (except for the size), only six sales were necessary to produce a strong estimate with near perfect linear correlation.
When working in rural areas or areas that have less comparable sales data, I actually use regression analysis more often than in urban areas. This is because when comparable sales are less than ideal, one needs to spend more time carefully supporting the adjustments to come to a credible opinion of value. On the other hand, if comparable sales are almost exactly like the subject, ranging little in sales price before adjustments, it is easy for appraisers to come to the most reasonable value opinion thru proper weighting in reconciliation, regardless of how large or small the individual adjustments are for each factor on each comparable sale. When comparable properties differ a great deal in terms of location, date of sale, site size, living space, or other factors, statistics can be used to better support these important quantifiable adjustments and to yield a more credible final opinion of value.
I recently appraised a rural manufactured home on 40 acres. There are few sales of similar properties, but I was able to take a sample of similar size and quality manufactured homes between one and two acres (there are lots of sales of these in the competitive market area) to support adjustments for the living area. In the absence of other data, I made a strong case that each additional square foot of living space on similar manufactured improvements with smaller sites is consistent with the adjustment for a property with 40 acres. Even if one could argue that the buyers of the 40-acre property would be willing to pay more or less per square foot of living space than the buyers of one and two-acre manufactured homes, analysis of the statistical data helps me have a starting point to make a more reasoned adjustment estimate.
For this same rural property, I also used statistical regression to support time adjustments (using market data from the entire competitive market area trended over time), site size adjustments (controlling my data by looking only at vacant land), and location adjustments (comparing samples of similar properties from different areas). The moral to the story is that appraisers should embrace statistics for help when little data exists, not pull away.
As a local appraiser, people are always asking me, “How much is a square foot of land worth in Portland, Oregon?” This is a difficult question to answer in general terms because every piece of land is different and Portland represents a large metropolitan area. However, we can explore some relationships that relate to property and its value, as well as examine the average trend in median land value as site size increases.
Properties with smaller lots that are closer to the center of Portland (or that are located in desirable parts of the City) will generally have a higher price per square foot. As the lot gets larger or the location moves out from the center of Portland, the value of each additional square foot of land will typically become less valuable, unless there are other factors like usability issues or external influences. The relationship holds true as long as the lots are just becoming larger yards or “surplus land” and do not represent additional building lots or “excess land”.
When we do a typical home appraisal in Portland, we estimate the vacant land value and then we also estimate the contributory value of each additional square foot of surplus land so that we can make adjustments to the comparable sales. This process is done on a market area and property-specific basis. For example, if our comparable sales only range from 5,000 to 8,000 square feet in site size, we then focus only on sales within that range to estimate our site size adjustments.
So, you still want to know a general answer to how much is a square foot of land worth in Portland, OR? Okay, here we go, but the following information is in general terms and should not be applied to any one particular property.
The following search was performed on the RMLS for vacant land in Portland from 3,000 square feet to one acre:
The range of the search results are from $28,000 to $575,000. The median land sales price is $120,000 and the median site size is 0.17 acres (7,405 sf). This is roughly $16 per square foot. Remember that this figure is the total median price of a lot and if we added or subtracted a small amount of site area (added yard space or took away yard space), that would not change the value by $16 per square foot.
Simple linear regression suggests that for each square foot of site size, value increases on average by $1.65 per square foot. Remember that this is just an average and does not necessarily apply to one particular property. For some of the higher priced, close-in neighborhoods with small sites, our appraisers have seen land value increase by as much as $10 per square foot of surplus land and other less desirable areas where there is almost no measurable difference in small changes in site size.
A straight trend line is shown in this site size regression for simplicity, but it would be more accurate to show a curved line. This is because each additional square foot of land is usually worth a little less than the one before. For example, it would be rare for a property around Portland with five acres to sell for $1.65 per square foot or $71,797 more than a property with four acres. The average relationship of $1.65 per additional square foot of land only holds true for this set of data and within this range below one acre.
If you find this information interesting or useful, please subscribe to my blog. Also, please support us by making Portland real estate appraisal related comments on our blogs and YouTube videos. If you need Portland, Oregon area residential real estate appraisal services for any reason, please contact us. We will do everything possible to assist you.