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
The 4,000-square-foot subject lies outside the 1,200–2,400-square-foot training range. This is extrapolation: the fitted relationship may not continue there, so additional evidence and validation are needed. The error could be upward or downward. Adding an unknown subject price to the training data would not supply independent validation, and R-squared on the original sample does not settle applicability.
Why This Is the Correct Answer
Option D is correct because the 4,000 square foot subject lies outside the 1,200 to 2,400 square foot range used to fit the model. This is extrapolation, and the fitted relationship may not continue beyond the data, so the estimate could be too high or too low. Additional evidence and validation are needed before relying on it.
Why the Other Options Are Wrong
Option A: The model has too few independent variables in it
Option A says the model has too few independent variables. The concern described is applying the model outside its data range, which adding more variables does not resolve.
Option B: R-squared cannot be computed outside the sample
Option B says R-squared cannot be computed outside the sample. R-squared is a measure of in-sample fit, and its value on the original data does not settle whether the model applies to a much larger house.
Option C: The subject must be added to the data set first
Option C says the subject must be added to the data set first. The subject's price is unknown, so adding it would not supply independent validation of the model at 4,000 square feet.
Stay inside the fence
A regression is reliable only inside the fence of the data it learned from; outside the fence it is guessing.
How to use: Use this image when a subject's characteristics fall outside the model's data range.
Exam Tip
When the subject is outside the range of the sample data, look for the answer about extrapolation.
Common Mistakes to Avoid
- -Applying a model to a subject outside its data range without added support
- -Relying on in-sample R-squared to justify extrapolation
- -Assuming extrapolation errors only run in one direction
Concept Deep Dive
Analysis
This question tests the danger of extrapolation. A regression is fitted to homes of 1,200 to 2,400 square feet, so applying it to a 4,000 square foot home goes well beyond the data. The fitted relationship may not continue outside the range it was built on.
Background Knowledge
Regression estimates relationships only within the range of the observed data. Beyond that range, the relationship between size and price may change shape, for example through diminishing returns for very large homes. Using a model outside its data range is extrapolation and calls for added support.
Real-World Application
An appraiser is asked to value a 4,000 square foot estate using a model built on typical tract homes. She recognizes the extrapolation problem and searches for sales of other large homes to support the value.
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