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Canvas Example MCQ — Q77

• A) requires thinking through the application and using judgment.
• B) depends on whether robust and homoskedastic-only standard errors differ significantly.
• C) is determined mechanically by comparing regression fit statistics.
• D) is dictated entirely by economic theory.

Control selection requires knowledge of the causal/economic setting; it is not mechanically dictated by R² or robust-SE differences.

Rule: Controls help address plausible confounding; they do not automatically establish causality.

Canvas Example MCQ — Q84

• A) 60% of the explanatory variables are statistically significant.
• B) Approximately 60% of the variation in the dependent variable is explained by the regression model.
• C) The model predicts the dependent variable correctly for 60% of the observations.
• D) 60% of the variation in the explanatory variables is explained by the dependent variable.
• E) None of the above.

R² measures explained sample variation in Y, not significance and not prediction accuracy.

Rule: R² answers “how much sample variation is explained?”

Canvas Example MCQ — Q85

• A) It tests the statistical significance of all explanatory variables.
• B) It adjusts R-squared for the number of parameters included in the regression model.
• C) It identifies which individual explanatory variables are statistically insignificant.
• D) It determines the F-statistic used to test overall significance.
• E) None of the above.

Ordinary R² weakly increases as regressors are added. Adjusted R² penalizes model size and can fall if added variables contribute too little.

Rule: Use adjusted R² when comparing fit across models with different numbers of regressors.

Canvas Example MCQ — Q86

• A) A lower adjusted R-squared always indicates a better regression model.
• B) A higher adjusted R-squared generally indicates more explained variation after adjusting for the number of estimated parameters.
• C) Adjusted R-squared can only be interpreted as the percentage of statistically significant explanatory variables.
• D) Adjusted R-squared is always larger than ordinary R-squared.
• E) None of the above.

This is the intended exam interpretation. Adjusted R² rewards explanatory power but penalizes unnecessary parameters.

Rule: Higher is generally preferred for comparable models with the same dependent variable.

Canvas Example MCQ — Q87

• A) Approximately 68% of the variation in the dependent variable is explained by the model, after adjusting for the number of estimated parameters.
• B) 68% of the explanatory variables are statistically significant.
• C) The model predicts the dependent variable correctly for exactly 68% of observations.
• D) The remaining 32% must be caused by omitted-variable bias.
• E) None of the above.

This is the source’s approximate interpretation. It is not 68% significant variables and not 68% correctly predicted observations.

Rule: Do not confuse fit with significance or classification accuracy.

Computer Lab 3 — Exercise 2, interpretation

• A) NVIDIA is a high-growth technology company whose earnings and valuation are more sensitive to the business cycle and growth expectations, making it far more market-sensitive than stable consumer-staples company KO
• B) Beta measures total volatility, and KO simply has lower total stock-price variance, unrelated to the market
• C) R² explains the beta difference directly, since a higher R² always implies a higher beta
• D) Beta values below 1 are statistically invalid, so KO's true beta must also be near NVDA's level

Beta reflects systematic (market-related) risk. Coca-Cola is a mature consumer-staples company with stable demand and cash flows (low beta), while NVIDIA's earnings/valuation are far more cyclical and growth-dependent, making it much more sensitive to market-wide movements (high beta).

Rule: Beta measures a stock's systematic risk/sensitivity to market-wide movements, not total volatility; cyclical/growth stocks tend to have higher betas than stable defensive stocks.

Computer Lab 3 — Exercise 2, important distinction

• A) R² measures the proportion of the stock's return variation explained by the market, while beta is the slope coefficient (market sensitivity)
• B) R² and beta measure exactly the same thing and can be used interchangeably
• C) R² is the stock's total risk, while beta is its unsystematic (idiosyncratic) risk
• D) R² is always equal to beta squared

Beta is the regression slope, quantifying how much the stock return moves per unit of market return. R² is the goodness-of-fit measure, indicating what fraction of the stock's return variance is explained by the market factor — two conceptually distinct statistics.

Rule: Beta = sensitivity to the market (slope); R² = how much of the variance is explained by that market relationship (fit quality). A high beta does not imply a high R², and vice versa.

Computer Lab 4 — Exercise 2, Task 2

• A) False — it is a conditional association and is causal only under additional identification assumptions
• B) True — any regression coefficient is causal by construction
• C) True, but only when R² exceeds 0.70
• D) False — coefficients from regression are always meaningless

Holding the other included regressors constant removes their confounding influence only, not all possible confounding from omitted variables; causal interpretation requires further assumptions (e.g., no omitted confounders, correct functional form).

Rule: Partial association ≠ causal effect unless identification assumptions hold.

Computer Lab 5 — Exercise 4, Robust SE

• A) Coefficients and fitted values stay the same; standard errors, t-statistics and p-values can differ
• B) Everything, including coefficients, changes
• C) Only the R² changes
• D) Coefficients change but standard errors stay identical

OLS point estimates and fitted values depend only on the OLS fitting procedure, which is unaffected by the choice of variance estimator; only the estimated standard errors (and resulting t-stats/p-values) can change with HC1.

Rule: HC1 robust SEs change only the measure of estimation uncertainty, not the OLS coefficients themselves.

Computer Lab 6 — Exercise 1, Task 1

• A) R² never decreases when more regressors are added, even if they add no real explanatory power, so it can reward unnecessary complexity
• B) R² always decreases when regressors are added
• C) R² is undefined for models with more than 3 regressors
• D) R² automatically accounts for overfitting

Ordinary R² is non-decreasing in the number of regressors by construction, so adding irrelevant variables can never lower it, making it a poor tool for comparing models of different sizes. Adjusted R² corrects for this with a penalty for extra regressors.

Rule: Use adjusted R², not plain R², to compare models with differing numbers of regressors.

Computer Lab 6 — Exercise 1, Task 1

• A) ≈0.7030
• B) ≈0.7041
• C) ≈0.6950
• D) ≈0.7198

Adj R² = 1 − (1−0.704134)×(799/796) = 1 − 0.295866×1.00377 ≈ 1 − 0.29698 ≈ 0.7030.

Rule: Adjusted R² = 1 − (1−R²)(n−1)/(n−k−1); it is always ≤ R².

Computer Lab 6 — Exercise 1, Task 4

• A) Best fit: m11 (highest adjusted R² = 0.7408); best prediction: m10 (lowest hold-out RMSE = 0.2590)
• B) Best fit: m10; best prediction: m11
• C) Best fit and best prediction are both m12
• D) Best fit: m2; best prediction: m12

Among these models, m11 has the highest adjusted R² (0.7408), indicating the best in-sample fit, while m10 has the lowest hold-out RMSE (0.2590), indicating the best out-of-sample predictive accuracy; these need not be the same model.

Rule: Choose the model objective (fit vs. prediction) first: best in-sample fit → highest adjusted R²; best prediction → lowest hold-out RMSE.

Computer Lab 6 — Exercise 1, Task 5

• A) Cross-validation and model selection occurred strictly within the training data; the hold-out set was used only once for final evaluation
• B) Using more variables automatically guarantees better out-of-sample prediction
• C) The hold-out RMSE was used repeatedly to pick the best of many candidate models
• D) Adding interactions always improves prediction regardless of how the model is selected

To avoid overfitting to the test data, all model search and tuning (cross-validation) was confined to the 800 training observations; the 100 hold-out observations were used only once, to report final, honest performance.

Rule: Model search/tuning must happen within training data only; the test/hold-out set should be touched only once, at the end.

Computer Lab 6 — Exercise 4, LPM weaknesses

• A) Predicted probabilities can fall outside [0,1], and errors are heteroskedastic by construction, so robust standard errors are generally preferred
• B) LPM cannot estimate coefficients at all for binary outcomes
• C) LPM coefficients cannot be interpreted in percentage-point terms
• D) LPM always has a lower R² than Probit or Logit

Because LPM is linear, predicted probabilities are not constrained to [0,1], and the binary nature of Y produces heteroskedastic errors, making robust standard errors advisable, even though the coefficients remain directly interpretable in probability terms.

Rule: Key LPM limitations: possible out-of-range probabilities and heteroskedastic errors; use robust SEs.

Formula Battle — method recognition (built from the Formula Book)

• A) Highest R² in training sample
• B) Lowest hold-out RMSE
• C) Highest adjusted R²
• D) Most significant coefficients

Out-of-sample matters. Correct method: Lowest hold-out RMSE.

Rule: Lower hold-out RMSE → better prediction