Linear Regression Technical Screen: OLS Assumptions and Multicollinearity

Context: You are asked to summarize core OLS assumptions, explain multicollinearity, and discuss diagnostics, remedies, and implications for the design matrix.

Tasks

1. List and explain the standard OLS assumptions:
   • Linearity in parameters
   • Independence / no autocorrelation
   • Homoscedasticity
   • Normality of errors (for exact small-sample inference)
   • No perfect multicollinearity
   • Correct specification (including exogeneity)
2. Define multicollinearity and describe its effects on:
   • Coefficient variance and stability
   • Confidence intervals and p-values
   • Note that OLS point estimates remain unbiased under exogeneity
3. Show how to diagnose multicollinearity:
   • Correlation matrix
   • Variance Inflation Factors (VIF) and common thresholds
   • Condition number
   • Eigenvalue-based analysis
4. Propose remedies and discuss trade-offs:
   • Collect more data
   • Drop/combine features
   • Center variables and interaction terms
   • Regularization (ridge/LASSO/elastic net)
   • Dimension reduction (PCA/PLS)
5. If two predictors are perfectly collinear, what happens to X'X and how do implementations typically handle it?