In a regression model, what is a residual?
Correct Answer
A) Observed response minus its fitted response
A regression residual is observed response minus fitted response: e = y − yhat. A positive sale-price residual means the observed price exceeds the model prediction. A coefficient is a model parameter, not an observation’s residual. Income remaining after building requirements belongs to another method. Unanalyzed observations are not necessarily a deliberately selected validation or holdout sample.
Why This Is the Correct Answer
Option A is correct because a regression residual is the observed response minus the fitted response, e = y − ŷ. For a sale-price model, a positive residual means the actual price was higher than the model predicted. Residuals are examined to judge model fit and spot problem observations.
Why the Other Options Are Wrong
Option B: Income left after building requirements
Option B, income left after building requirements, describes a residual income idea from income-based land valuation techniques, not a regression residual. A regression residual compares an observed value with a fitted value.
Option C: The portion of the sample left unanalyzed
Option C, the portion of the sample left unanalyzed, describes leftover data, not a residual. Unused observations are also not the same as a deliberately chosen validation or holdout sample.
Option D: The coefficient attached to each predictor
Option D, the coefficient attached to each predictor, describes a model parameter. A residual belongs to an individual observation and measures how far its actual response is from the model's prediction.
Actual minus predicted
Residual equals what really happened minus what the model guessed: actual minus predicted.
How to use: Use this order to get the sign of a residual right on calculation questions.
Exam Tip
Remember the order: observed minus fitted. A positive residual means the model underpredicted that sale.
Common Mistakes to Avoid
- -Reversing the order and computing fitted minus observed
- -Confusing regression residuals with residual income techniques
- -Treating a coefficient as a residual
Concept Deep Dive
Analysis
This question tests the definition of a residual in regression. Each observation has an actual response value and a value predicted by the fitted model, and the residual is the difference between the two. Candidates must not confuse the term with residual income concepts from other appraisal methods.
Background Knowledge
Regression fits an equation that predicts the response from the explanatory variables. The residuals are the gaps between each actual response and its prediction. Patterns in the residuals, such as larger errors at higher prices, can reveal model problems.
Real-World Application
An appraiser reviewing an automated valuation model sees a sale with a large positive residual. He investigates and finds the house had a recent full renovation that the model's variables did not capture.
More Statistics Questions
Sale prices are $270,000, $290,000, $300,000, $310,000, $330,000. Treat the listed observations as the complete population and divide by n when computing the standard deviation. Report to the nearest dollar.
A sales sampling frame systematically excludes a relevant part of the target market. What is the principal risk?
An appraiser estimates a market-wide price trend using only sales data from one brokerage that specializes in luxury homes, without checking how other segments are represented. What is the principal concern?
Compared with analyzing one matched sale pair for one feature, what can a suitably specified multiple regression do?
The base-period median sale price is $400,000 and the current-period median is $460,000. Compute the price relative defined as current ÷ base, as a decimal to three places.
The assessment-to-sale ratios are 0.8, 0.85, 0.9, 0.92, 0.95, 0.98, 1.02, 1.05, 1.08, 1.1, 1.15, 1.2. Calculate COD as [average absolute deviation from the median ÷ median]×100. For even n, average the two middle ratios. Report the COD as a percentage to two decimals.
Assessment-to-sale ratios are 0.94, 0.96, 0.98, 1, 1.01. Compute the arithmetic mean. Report as a decimal ratio to three places.
Market data show a first quartile sale price of $405,000 and a third quartile sale price of $495,000. What is the interquartile range, to the nearest dollar?
Assessment-to-sale ratios are 0.96, 0.98, 0.99, 1.01. Compute the arithmetic mean. Report as a decimal ratio to three places.
Given first quartile $400,000, median $440,000 and third quartile $500,000, compute the trimean (Q1+2×median+Q3)/4. Report to the nearest dollar.
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