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A regression fitted to homes between 1,200 and 2,400 square feet is used to value a 4,000 square foot house. What is the concern?

Correct Answer

D) The relationship may not hold beyond the data

Why this is correct: Using a regression model fitted to data from 1,200-2,400 sq ft to value a 4,000 sq ft house is extrapolation. The linear relationship (price per sq ft) observed within the data range may not hold outside it. For larger homes, the price per sq ft often decreases, so extrapolation risks significant overvaluation. Why the other choices are wrong: "The model has too few independent variables in it" is wrong; the core issue is extrapolation, not the number of variables. "R-squared cannot be computed outside the sample" is wrong; R-squared is a measure of fit for the model on the sample data, not a barrier to computation. "The subject must be added to the data set first" is wrong; you cannot add the subject's value to the data set before estimating it. Exam tip: Extrapolation (predicting outside the data range) is risky and often invalid. Stick to the range of your comparable data.

Answer Options

The model has too few independent variables in it

R-squared cannot be computed outside the sample

The subject must be added to the data set first

The relationship may not hold beyond the data

Why This Is the Correct Answer

Option D is correct because the fitted relationship is supported only within the observed range, and there is no evidence it continues at 4,000 square feet. Extrapolation assumes a linearity the data never tested, and in residential markets the price-per-square-foot curve typically flattens as homes get larger, so the model will tend to overvalue. The subject may also sit in a different market segment with its own buyer pool. The remedy is to expand the sample to include large homes or to use a technique that does not depend on out-of-range prediction.

Why the Other Options Are Wrong

Option A: The model has too few independent variables in it

The number of independent variables is a separate question from the range of the data, and adding variables would not cure extrapolation. A model with a rich set of predictors is still blind beyond the sizes it observed. The stem points specifically to a subject outside the sampled range, which is a data coverage problem rather than a specification problem.

Option B: R-squared cannot be computed outside the sample

R-squared is computed from the fitted sample and describes how much variation the model explains within that sample, so it can always be computed; the issue is that it says nothing about performance outside the range. A high R-squared on mid-size homes offers false comfort about a 4,000 square foot prediction. The statement also misstates what the measure is and where it applies.

Option C: The subject must be added to the data set first

Adding the subject to the data set would be circular, since the subject's sale price is precisely the unknown the model is meant to estimate. Regression is built from arm's-length transactions with known prices, and inserting an unsold property would corrupt the sample. What the sample actually needs is more sold comparables at the larger end of the size range.

Off the end of the ruler

A regression is a ruler marked only where you took measurements. Past the last mark there are no numbers, just the assumption that the pattern kept going. In housing it usually does not, because big homes sell for less per foot.

How to use: Compare the subject's characteristics with the sample's range before trusting any model output. If the subject sits outside the range on a major variable, the answer is about extrapolation.

Exam Tip

Note the likely direction of the error, not just its existence. Extrapolating a linear model to a much larger home almost always overstates value.

Common Mistakes to Avoid

-Applying a model outside the range of its underlying data
-Assuming price per square foot stays constant as size grows
-Trusting a high R-squared as evidence of out-of-range accuracy
-Ignoring that a much larger home may belong to a different market segment

Concept Deep Dive

Analysis

This tests extrapolation, the risk of applying a statistical model outside the range of the data that produced it. A regression estimates a relationship only where it has observations; between 1,200 and 2,400 square feet the model has evidence, and beyond that it has arithmetic. Applying it to a 4,000 square foot house extends the fitted line two thirds again beyond the largest observation in the sample. The direction of the error is predictable in housing, because price per square foot generally declines as size increases, reflecting the diminishing marginal contribution of additional area, so a straight line fitted to mid-size homes will typically overstate the value of a much larger one. There is a market reason as well as a statistical one: a 4,000 square foot house may belong to a different market segment altogether, with different buyers, lot expectations, and competing supply. The sound response is to gather data from that segment or to rely on paired sales of large homes.

Background Knowledge

You need to understand the difference between interpolation within a sample's range and extrapolation beyond it, and why regression provides no evidence outside the observed data. You should also know that price per square foot typically declines as size increases, that market segmentation can place a much larger home in a different buyer pool, and that the appraiser is responsible for the credibility of any model-derived result.

Real-World Application

A client requests an appraisal on a 4,000 square foot custom home in a tract of 1,200 to 2,400 square foot houses. Rather than running the neighborhood regression out to the subject's size, you search a wider geographic area for large-home sales and support the size adjustment with paired sales from that segment.

extrapolationregression analysisrange of datamarket segmentationprice per square foot

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