Practice: Simple Linear Regression

Quantitative Methods. 10 question(s) in this unit's pool (0 above the exam). Free up to ten a day; the coach picks which ones based on what you have already answered and when each is next due.

Quantitative MethodsSimple Linear Regression
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Today's practice

Pick an answer, say how sure you are, then reveal. Every wrong choice gets its own explanation. Questions you have already answered correctly and confidently stay out of the way until they are due for review again.

Question 1Exam level

A regression of quarterly portfolio returns (Y) on market returns (X) produces the following output: Intercept = 0.50, Slope = 1.20, R-squared = 0.72. Which of the following best interprets the slope coefficient of 1.20, most likely?

How sure are you?

Correct: A. The correct answer is For every 1% increase in market returns, portfolio returns increase by 1.20%..
B. You might confuse the slope with the intercept. The 0.50 intercept is the expected Y when X = 0. The intercept (b0 = 0.50) answers 'what is Y when X is zero.' The slope answers 'by how much does Y change per unit of X.'.
C. R-squared = 0.72 is visible in the output and 72% is a plausible-sounding interpretation. R-squared (0.72) measures proportion of variation explained. It is a model fit statistic, not a coefficient interpretation.

Unit: simple-linear-regression

Question 2Harder

A regression output shows R-squared = 0.85 and reports that the p-value on the slope coefficient is 0.42. Which conclusion is most appropriate?

How sure are you?

Correct: B. The correct answer is The high R-squared combined with insignificant coefficient is a warning sign of multicollinearity..
A. R-squared = 0.85 looks impressive. You might are trained to associate high R-squared with a 'good model.'. R-squared measures proportion of variation explained. It says nothing about whether the relationship is statistically significant or causal. A significant F-test and significant t-tests are required to validate the model.
C. You might assume that if the model fits well (high R-squared), the slope must be significant. R-squared and coefficient significance are separate tests. In a multiple regression with correlated predictors, R-squared can be high while all individual t-stats are insignificant. Exactly the multicollinearity symptom.

Unit: simple-linear-regression

Question 3Exam level

An analyst estimates the regression: Sales = 5.2 + 3.1(Advertising). A 95% confidence interval for the slope coefficient is [1.8, 4.4]. What is the most appropriate interpretation?

How sure are you?

Correct: B. The correct answer is In repeated sampling, 95% of such intervals would contain the true slope coefficient..
A. Saying '95% probability the true value is in this interval' is intuitive and what most people mean when they use confidence intervals colloquially. The true parameter is fixed (not random). The interval is random. The correct interpretation applies to the long-run frequency of the procedure, not the probability that any single interval contains the parameter.
C. The test for statistical significance checks whether the CI includes zero. Which it does not here. But the significance test is about whether the interval includes zero, not whether it includes the point estimate. Statistical significance is confirmed by checking if the CI excludes zero (which it does. [1.8, 4.4] is entirely positive).

Unit: simple-linear-regression

Question 4Exam level

A Durbin-Watson statistic of 0.85 is calculated for a regression of monthly stock returns on interest rate changes. The DW critical values are dL = 1.27 and dU = 1.45. What does this most likely indicate?

How sure are you?

Correct: B. The correct answer is Positive serial correlation in the residuals..
A. You might memorize 'DW ≈ 2 means no serial correlation' but misapply this as 'any DW is fine if the model fits well.'. DW = 0.85 is far from 2. Values close to 0 indicate strong positive serial correlation in the residuals. Consecutive residuals have the same sign.
C. The inconclusive zone exists (dL < DW < dU). You might apply this concept even when the DW is clearly below dL. The inconclusive zone applies when dL < DW < dU (here, 1.27 to 1.45). DW = 0.85 is below dL = 1.27, placing it firmly in the 'positive serial correlation' zone.

Unit: simple-linear-regression

Question 5Exam level

A multiple regression model produces an R-squared of 0.68 and an adjusted R-squared of 0.59. An analyst adds two more independent variables that have near-zero correlations with the dependent variable. What will happen to R-squared and adjusted R-squared, most likely?

How sure are you?

Correct: B. The correct answer is R-squared will increase or stay the same; adjusted R-squared will decrease..
A. More variables intuitively feel like they should improve the model. If R-squared improves, candidates assume adjusted R-squared also improves. Adjusted R-squared penalizes for additional parameters. Adding variables with no explanatory power hurts the adjusted R-squared penalty term more than the marginal gain.
C. Near-zero correlation with the dependent variable sounds like it would reduce fit. R-squared cannot decrease when variables are added. Mathematically, OLS always finds the RSS-minimizing coefficients including the option of setting new coefficients to near-zero.

Unit: simple-linear-regression

Question 6Exam level

In a simple linear regression, the total sum of squares (TSS) = 500, and the regression sum of squares (RSS) = 350. What is R-squared and what does it mean, most likely?

How sure are you?

Correct: A. The correct answer is R-squared = 0.70; 70% of variation in the dependent variable is explained by the independent variable..
B. 0.70 is computed correctly but the probability interpretation is wrong. R-squared sounds like it could be a probability measure. R-squared is a proportion of variance explained, not a probability. A 'probability the model is correct' has no statistical meaning in this context.
C. Candidates who confuse the formula might compute TSS/RSS = 500/350 = 1.43 instead of RSS/TSS. R-squared = RSS/TSS, not TSS/RSS. R-squared is bounded [0, 1]. A value of 1.43 is mathematically impossible.

Unit: simple-linear-regression

Question 7Harder

A regression of bond yields (Y) on inflation (X1) and GDP growth (X2) is estimated. The coefficient on inflation is 0.85 with a t-statistic of 0.62. The coefficient on GDP growth is 1.10 with a t-statistic of 4.21. The F-statistic is 18.3 (p-value = 0.0001). Which statement is most accurate?

How sure are you?

Correct: B. The correct answer is The model is statistically significant overall, but the inflation coefficient is not individually significant..
A. The significant F-statistic implies the model works overall, so candidates assume all variables must be contributing. F-test and t-tests are separate. A significant F-test means at least one coefficient matters. Not all of them. Individual significance requires individual t-tests.
C. Candidates who equate overall model significance with individual coefficient significance select this distractor. The F-test and t-tests can and do disagree. A model can have a significant F-test (driven by one powerful variable) while other variables have insignificant t-statistics.

Unit: simple-linear-regression

Question 8Exam level

An analyst suspects multicollinearity in a multiple regression. Which of the following is the most reliable indicator of multicollinearity?

How sure are you?

Correct: B. The correct answer is High R-squared and high F-statistic but individually insignificant t-statistics on most coefficients..
A. Low R-squared sounds like the model is not working, which might suggest a problem with the variables. Low R-squared + significant F-stat is not a multicollinearity pattern. Multicollinearity is associated with HIGH R-squared combined with insignificant t-stats.
C. DW significantly below 2 is a diagnostic test for regression problems. But candidates confuse which problem it diagnoses. DW statistic diagnoses serial correlation (autocorrelation), not multicollinearity. These are different violations of different regression assumptions.

Unit: simple-linear-regression

Question 9Exam level

Which assumption of the classical linear regression model (CLRM) is violated when the variance of the error term is most likely NOT constant across all observations?

How sure are you?

Correct: B. The correct answer is Heteroskedasticity.
A. Serial correlation and heteroskedasticity are both violations involving the error term. You might sometimes confuse them. Serial correlation involves the error terms being correlated with each other across time (cov(εi, εj) ≠ 0 for i ≠ j). Heteroskedasticity involves non-constant variance of each individual error term.
C. Multicollinearity also causes problems with significance tests, and candidates who memorize only 'multicollinearity causes insignificant t-stats' may select this. Multicollinearity is a problem with the independent variables (they are correlated with each other), not with the error term. Non-constant error variance = heteroskedasticity.

Unit: simple-linear-regression

Question 10Exam level

An analyst runs a simple regression of stock returns on earnings growth. The regression shows a high R-squared of 0.82. The analyst concludes that earnings growth causes stock returns. Which reasoning error most likely has the analyst made?

How sure are you?

Correct: A. The correct answer is The analyst confused correlation with causation..
B. Adjusted R-squared is mentioned as an improvement over R-squared, so candidates think the error was using the wrong metric. Whether adjusted or unadjusted R-squared is used makes no difference for the causation error. Neither R-squared measure ever implies causation.
C. Significance testing is important in regression and candidates may think the error was skipping a statistical test. R-squared itself is not subjected to a significance test. The F-statistic tests overall model significance. But even if the F-test is passed, causation is still not established. Correlation ≠ causation.

Unit: simple-linear-regression