MFIN6201 Chap.6 Multiple Regression and Multicollinearity
Multiple Regression and Multicollinearity
This chapter interprets coefficients in a multiple-regression model as partial effects with included regressors held fixed. It distinguishes perfect multicollinearity, which prevents separate estimation, from imperfect multicollinearity, which raises uncertainty. Dummy-variable coding, reference categories, specification changes and the limited role of R-squared are developed through applied decisions.
A partial effect compares observations after accounting for the linear projection of the other included variables. That conditional phrase belongs in every coefficient interpretation. Controls can improve comparability when they capture common causes, but their inclusion does not certify exogeneity and their own coefficients need not be causal.
Perfect multicollinearity is exact algebraic redundancy, as when an intercept accompanies indicators for every exhaustive category. Omitting one category establishes a reference without changing fitted group comparisons. Imperfect multicollinearity leaves estimation possible but reduces independent regressor variation, often increasing standard errors and making estimates unstable.
Adding regressors cannot reduce ordinary R-squared mechanically, so fit maximisation is a poor specification rule. The target estimand and causal story should decide which variables enter before the output is inspected.
What this chapter covers
- 01
Multiple regression model
- 02
Ceteris-paribus coefficients
- 03
Dummy-variable coding
- 04
Perfect multicollinearity
- 05
Imperfect multicollinearity
- 06
Single-coefficient inference
Worked example · free
A partial-effect comparison
- +1Use the leverage coefficient because size is held fixed.
- +1Multiply 1.2 by 0.25 to obtain 0.30.
- +1Interpret 0.30 in the outcome's units, conditional on equal included size.
Key terms
- Partial effect
- A coefficient interpretation that changes one regressor while holding included regressors fixed.
- Perfect multicollinearity
- Exact linear redundancy among included regressors that blocks separate estimation.
- Reference category
- The omitted group against which included category indicators are interpreted.
Multiple Regression and Multicollinearity FAQ
What changes in a multiple-regression slope?
The coefficient becomes a conditional comparison: predicted outcome change for one unit of its regressor while the other included variables remain fixed. This can improve comparability, but it does not guarantee that unobserved confounding is gone.
Why does the dummy-variable trap occur?
With an intercept and indicators for every mutually exclusive category, the indicators sum exactly to the intercept. One column is a linear combination of the others, so software cannot estimate all coefficients separately. Omitting one category creates the reference group.
Does high multicollinearity cause bias?
High but imperfect correlation among regressors mainly reduces precision and can make estimates unstable because little independent variation remains. It does not mechanically violate conditional mean zero or create bias, although it can make useful effects difficult to distinguish.
Why can a control coefficient be misleading?
A control may be included to isolate the target coefficient rather than estimate its own causal effect. Omitted determinants related to that control can still bias its displayed slope even when the target comparison improves.
Should I maximise adjusted fit?
Fit criteria can compare predictive models but cannot decide whether a variable is a confounder, mediator or collider. For explanation, choose the conditioning set from the estimand and causal timing, then treat fit as secondary evidence.
What changes when the reference category changes?
The intercept and displayed dummy coefficients are re-expressed relative to the new baseline, but fitted values and contrasts between actual categories remain unchanged. Recomputing one pairwise contrast is a useful coding check after changing the reference.
Can adding a regressor lower ordinary R-squared?
No. OLS can assign the added coefficient zero and reproduce the previous fit, so ordinary R-squared cannot fall. That mechanical property is precisely why a higher value cannot determine whether the added variable improves the causal specification. Compare adjusted fit and identification logic separately.
Exam move
Practise interpreting slopes with an explicit held-fixed clause. Build dummy-variable models by naming the omitted reference category before reading coefficients. Diagnose whether a collinearity problem is exact redundancy or merely high correlation, then state the correct consequence. Compare specifications by causal logic rather than by R-squared alone.
Recode the same categorical variable with a different reference group and verify that fitted values and pairwise category contrasts do not change. For continuous regressors, calculate a partial-effect difference for several realistic unit changes and check support. Create one model with exact redundancy and another with near redundancy, then predict the different software and standard-error symptoms.
When a coefficient moves after adding a control, list both a confounding explanation and a bad-control possibility. Keep the coefficient of interest separate from coefficients included only to improve the comparison. End each model-building exercise by writing the estimand, held-fixed set and strongest remaining threat in a single sentence.
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