COMM5000 Chap.7 Multiple Regression and Robustness
Multiple Regression and Robustness
Interpret ceteris paribus coefficients, reason about omitted variables and indicators, and design robustness checks that challenge rather than decorate a model. In multiple regression, a coefficient compares observations that share the included values of other regressors.
This conditional comparison can reduce confounding when appropriate variables are measured and modelled, but holding a variable constant in an equation does not create experimental control. Omitted-variable bias can arise when an excluded factor affects the outcome and is related to an included regressor.
Adding every available variable is not a solution: mediators, colliders, poorly measured controls and post-outcome information can distort the target. Dummy variables require a reference category, interaction terms make effects depend on another variable, and nonlinear terms change marginal effects across the regressor range.
Exact multicollinearity prevents separate estimation, while strong multicollinearity can inflate coefficient standard errors. Robustness analysis should begin with a named threat and a justified alternative specification. Compare coefficient direction, magnitude and uncertainty, or compare out-of-sample prediction performance when prediction is the goal.
Report instability instead of selecting the version that tells the preferred story. Document units, transformations, reference groups, validation design and the business purpose that determines which model evidence matters.
What this chapter covers
- 01
Conditional coefficients
- 02
Confounding and endogeneity
- 03
Dummy variables
- 04
Nonlinearities and multicollinearity
- 05
Robustness analysis
Worked example · free
Controlled sales comparison
- +1The four marks used here are not an official university marking scheme. Interpret price and advertising coefficients holding other regressors constant.
- +1Interpret suburban relative to the urban reference category.
- +1For price 20, advertising 6 and suburban status, predict 191 thousand dollars.
- +1For price 22, advertising 5 and urban status, predict 162.1 thousand dollars.
Key terms
- Ceteris paribus
- A comparison of one regressor while holding the model's other included regressors constant.
- Reference category
- The omitted group against which dummy-variable coefficients are interpreted.
- Multicollinearity
- Strong linear dependence among regressors that can inflate coefficient standard errors.
Multiple Regression and Robustness FAQ
Should I add every available control?
No. Controls should follow the question and causal logic. Mediators and colliders can distort the target comparison, while irrelevant variables can add noise or reduce interpretability.
What makes a robustness check useful?
A useful check addresses a named threat, uses a defensible alternative specification, compares the relevant estimate or prediction, and reports instability instead of hiding it.
How should I document alternative regression models?
Create a table that links each specification to the threat it addresses, such as omitted confounding, functional form, influential cases or validation performance. Record units, transformations, reference categories and interaction terms so coefficients remain interpretable. Compare the target estimate's sign, magnitude and uncertainty across models, including versions that weaken the preferred conclusion.
For prediction, report results on relevant unseen cases and subgroup errors rather than training fit. End with a reasoned model choice tied to explanation, prediction or decision needs, plus the limits and monitoring conditions that apply in use.
Assessment move
Write the conditional comparison represented by each coefficient and list the regressors held constant. For indicators, code the reference category explicitly and interpret every coefficient against it. Recalculate two fitted profiles on separate lines so changes in price, advertising or category are not lost in one mental subtraction.
Draw a causal sketch before choosing controls and label candidate confounders, mediators and outcomes. For every additional specification, write the threat it addresses before running it. Compare coefficient sign, magnitude and interval across credible models, and explain which added variable or functional form drives any change.
If prediction is the objective, reserve relevant unseen cases, compare against a simple baseline and check subgroup errors rather than relying on training fit. Practise detecting the dummy-variable trap and recognising when an interaction makes a main coefficient conditional. In the report, state whether the model is intended to explain, predict or support a decision.
That goal should determine the evaluation metrics, documentation, monitoring and tolerance for complexity. Create a specification table with one column for the threat, one for the model change and one for what happened to the target estimate. Include results that weaken the preferred conclusion. For a dummy-variable model, write the full coding scheme and verify that one category is omitted when an intercept is present.
For an interaction, calculate the effect at two meaningful moderator values so the conditional nature of the coefficient becomes concrete. Check whether a high variance inflation signal reflects redundant measurement or a design that lacks independent variation. For predictive models, compare validation performance across relevant time periods and groups, not only an overall average.
Finish with a model card that states inputs, units, reference categories, fitting period, intended use, prohibited extrapolations and review owner.
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