FAST Regression Analysis 3 — Questions and Answers
Question 1: Which of the following is the primary goal of the least squares method in regression?
- Maximize the correlation coefficient
- Minimize the sum of squared residuals (Correct answer)
- Minimize the number of predictors
- Maximize the explained variance ratio
Correct answer: Minimize the sum of squared residuals
OLS finds the line (or hyperplane) that minimizes the sum of squared differences between observed and predicted values.
Question 2: A leverage point in regression is best described as an observation that:
- Has a large residual
- Has an unusual value on the predictor(s) (Correct answer)
- Causes heteroscedasticity
- Inflates the R² artificially
Correct answer: Has an unusual value on the predictor(s)
High-leverage points have extreme x values and can exert strong influence on the estimated regression line.
Question 3: Cook's Distance is used in regression to measure:
- The degree of multicollinearity
- The overall influence of an observation on all fitted values (Correct answer)
- The normality of residuals
- The strength of the linear association
Correct answer: The overall influence of an observation on all fitted values
Cook's D combines each observation's leverage and residual to quantify how much it shifts the entire set of fitted values if removed.
Question 4: When using a log transformation on the dependent variable (ln Y), the regression coefficient b₁ is best interpreted as:
- The absolute change in Y per unit increase in X
- The approximate percentage change in Y per unit increase in X (Correct answer)
- The log-odds of Y
- The elasticity of X with respect to Y
Correct answer: The approximate percentage change in Y per unit increase in X
In a log-linear model, multiplying b₁ by 100 gives the approximate percentage change in Y for a one-unit increase in X.
Question 5: The Durbin-Watson statistic is primarily used to detect:
- Multicollinearity among predictors
- Autocorrelation in the residuals (Correct answer)
- Non-normality of the error term
- Heteroscedasticity in residuals
Correct answer: Autocorrelation in the residuals
The Durbin-Watson test checks whether consecutive residuals are correlated, a common violation in time-series data.
Question 6: In simple linear regression, if r = –0.85, which statement is correct?
- 85% of variance in Y is explained by X
- As X increases, Y tends to decrease strongly (Correct answer)
- The regression slope is positive
- There is no meaningful relationship between X and Y
Correct answer: As X increases, Y tends to decrease strongly
r = –0.85 indicates a strong negative linear relationship: higher X values are associated with lower Y values.
Question 7: Ridge regression differs from ordinary least squares by:
- Using a different loss function based on absolute errors
- Adding a penalty term (λ) that shrinks coefficients toward zero (Correct answer)
- Removing predictors with low t-statistics
- Fitting a non-linear curve to the data
Correct answer: Adding a penalty term (λ) that shrinks coefficients toward zero
Ridge regression adds an L2 penalty (λ∑β²) to the OLS cost function, stabilizing estimates when multicollinearity is present.
FAST Regression Analysis 3
Which of the following is the primary goal of the least squares method in regression?