ETF1100 Chap.9 Multiple Regression
Multiple Regression
Multiple Regression extends the fitted line to several predictors and interprets each slope as a comparison holding the other included predictors constant. The chapter handles zero-one category indicators and reference groups, distinguishes R-squared from adjusted R-squared, and uses business logic, residual checks and validation to assess variable choice, omitted-variable risk and prediction quality.
What this chapter covers
- 01
Controlled coefficient comparisons
- 02
Numerical predictor units
- 03
Indicator variables and references
- 04
R-squared and adjusted fit
- 05
Omitted-variable risk
- 06
Residual checks and validation
Worked example · free
Interpret a category indicator
- 1Compute the premium prediction as 72 - 21.6 + 38 + 14 = 102.4.
- 1Set Premium to zero for the standard prediction of 88.4.
- 1Report a 14.0-thousand adjusted difference.
- 1Name standard sites as the reference category.
- 1Limit the comparison to included price and rating controls.
Key terms
- Multiple Regression
- A model predicting one numerical outcome from several predictors.
- Controlled Comparison
- A predicted comparison changing one predictor while holding other included predictors fixed.
- Indicator Variable
- A zero-one predictor representing membership in a named category.
- Reference Category
- The omitted category against which indicator coefficients are compared.
- Adjusted R-squared
- A fit measure that penalises adding predictors that contribute little explanatory value.
Multiple Regression FAQ
What does holding other variables constant mean?
It means comparing fitted cases with the same values of the other predictors actually included in the model. It does not control omitted factors. Name the included controls rather than claiming that every other difference has been removed.
How do I interpret an indicator coefficient?
Identify the category coded one and the reference coded zero. The coefficient is their predicted outcome difference at the same values of the other included predictors. Its unit is the outcome unit, and observational models still require causal caution.
Is a higher R-squared always a better model?
No. Ordinary R-squared cannot fall when predictors are added, even if they contribute little. Consider adjusted fit, residual patterns, business relevance and performance on new data. A high in-sample fit can still reflect overfitting or invalid specification.
Why can a coefficient change after another predictor is added?
The slope becomes a different conditional comparison because it now holds the added predictor fixed. A sign or magnitude change may reflect shared predictor information, confounding or instability. Inspect correlations, model purpose and subject knowledge before interpreting the revised coefficient; the change is not automatically proof that either model is wrong.
Exam move
Practise coefficient interpretation from a compact output table. Circle the predictor row, identify whether it is numerical or categorical, attach units and write the hold-constant phrase using only included predictors. For each indicator, name its reference category before interpreting the coefficient. Recode a category once to see that fitted predictions stay the same even though the intercept and labels change.
Compare ordinary and adjusted R-squared, but never use fit alone to justify a model. Build a diagnostic checklist for residual form, changing variance, influential cases, correlated predictors and performance on held-out data. When a claim sounds causal, rewrite it as an association unless the design supports intervention language. Choose variables from the business process and purpose, not only from small p-values.
End reports with specific, data-supported recommendations and limitations tied to sampling, measurement or omitted variables. In the exam, distinguish coefficients from their standard errors and p-values before entering an output value. Build a coefficient translation table with columns for outcome unit, predictor unit, comparison increment, direction, reference category and variables held fixed.
Complete every column before writing prose. Use the fitted equation to predict two otherwise identical cases that differ only in one predictor; their predicted difference should reproduce that coefficient on the correct scale. For a multi-level category, identify the omitted reference and interpret each indicator against it, then recode the reference and confirm that fitted predictions do not change.
Compare a predictor's overall scatterplot relationship with its multiple-regression coefficient and explain why control can alter sign or magnitude. Inspect correlations among predictors and treat unstable coefficients as a warning about overlapping information, not as proof that one variable is irrelevant.
Separate explanation from prediction: subject knowledge guides specification, while held-out error measures transfer to new cases. Examine residual form, changing spread and influential observations, and report the analysed row count after missing values are handled. Compare ordinary and adjusted R-squared only alongside purpose and validation evidence.
Rewrite any causal conclusion as a conditional association unless assignment or a credible causal design supports more. End with a recommendation that names the predicted outcome, controlled comparison, uncertainty, population scope and omitted factor most likely to change the decision.
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