Assessing Linear Regression Assumptions and Fitting Three Points

Explain the assumptions that underlie ordinary least squares, the way multicollinearity influences a fitted model, and practical approaches for handling multicollinearity and overfitting. Next, fit a simple linear regression with an intercept to the three points (0, 0), (0, 1), and (1, 1), and report the fitted line together with its residual sum of squares.

Constraints & Assumptions

• Fit an ordinary least squares model with a single predictor and an intercept.
• Use the residual sum of squares as the requested least-squares error.
• Separate the assumptions required for coefficient interpretation or classical inference from issues that mostly affect predictive performance.

Clarifying Questions to Ask

• Is the objective causal interpretation, statistical inference, or prediction?
• Should “error” refer to residual sum of squares, mean squared error, or some other normalized measure? Use residual sum of squares for this problem.

Hint: Separate the two goals. Structure the discussion around what makes coefficients meaningful and what makes uncertainty estimates reliable before carrying out the three-point arithmetic.

Part 1 — Assumptions and Remedies

Explain the linear conditional-mean assumption, exogeneity, dependence among errors, error variance, multicollinearity, and the role of normality. Describe diagnostics and remedies for multicollinearity and overfitting, without treating all of these assumptions as interchangeable.

What This Part Should Cover

• Which violations bias the coefficients, which increase uncertainty, and which invalidate common standard errors.
• Removing or combining features, collecting more data, regularization, and validation-based model selection.

Part 2 — Fit the Three Observations

Derive the slope and intercept for the given points, list the three fitted values and residuals, and compute the residual sum of squares.

What This Part Should Cover

• A reproducible least-squares calculation and an exact final error value.
• A verification that the intercept and slope satisfy the normal equations.

What a Strong Answer Covers

• A clear separation of modeling assumptions, diagnostics, remedies, and exact arithmetic for the fitted line.

Follow-up Questions

• What fitted line is obtained if the intercept is forced to zero?
• How does ridge regression alter the handling of correlated predictors?

Overview: Discuss the assumptions underlying ordinary least squares, the effect of multicollinearity on a fitted model, and practical methods for addressing multicollinearity and overfitting. Cover data and labels, leakage-safe features, baselines and model choice, offline evaluation, deployment constraints, monitoring, and drift.