Diagnose Violations of Linear Regression Assumptions

State the assumptions needed to interpret ordinary least squares estimates and common inferential statistics. For each important assumption, explain how you would diagnose a possible violation and what you would change if the assumption does not hold. ### Constraints & Assumptions - Distinguish assumptions needed for unbiased or consistent coefficient estimates from assumptions needed for conventional standard errors and exact small-sample tests. - Do not treat a single diagnostic test as proof that an assumption is true. - Include prediction concerns as well as coefficient interpretation. ### Clarifying Questions to Ask - Is the goal causal interpretation, descriptive association, or out-of-sample prediction? - Are observations cross-sectional, clustered, or ordered in time? - Is the model intended to support confidence intervals and hypothesis tests? - How large is the sample relative to the number of predictors? ```hint Organize by failure mode Separate functional-form problems, dependence, unequal variance, collinearity, influential observations, and distributional assumptions. ``` ### What a Strong Answer Covers - Linearity in parameters and an adequate conditional-mean specification. - Exogeneity and why residual plots cannot establish it. - Independence or a correctly modeled dependence structure. - Homoskedasticity, multicollinearity, influence, and residual normality in their proper roles. - Diagnostics, remedies, and the limitations of each remedy. ### Follow-up Questions - Which assumption is most important for a causal interpretation of one coefficient? - When are heteroskedasticity-robust standard errors insufficient? - How would your workflow change for a time series with autocorrelated errors?