Module M03 Quiz: Multiple Regression

16 questions · Graduate (ECMT5001) · MCQ, calculation, short answer

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Question 1

Lesson L01 · Interpretation

In the multiple regression $\ln(\text{wage}) = \beta_0 + \beta_1 \text{educ} + \beta_2 \text{exper} + u$, the coefficient $\beta_1$ represents:

Type: Multiple Choice

A) The simple correlation between education and log-wages
B) The change in log-wages for one additional year of education, holding experience constant (ceteris paribus)
C) The total effect of education on wages, including its effect through experience
D) The average log-wage for a worker with zero education

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Answer: B

The ceteris paribus interpretation: $\beta_1$ measures the effect of education on log-wages holding all other regressors (here, experience) fixed. This is the whole point of multiple regression — isolating the partial effect of one variable while controlling for others. The Frisch-Waugh-Lovell theorem shows this is literally the slope from regressing wages (residualised on experience) on education (residualised on experience).

Question 2

Lesson L04 · Omitted Variable Bias

A researcher regresses wages on education only, omitting ability. Ability is positively correlated with education and positively affects wages. The bias in $\hat{\beta}_\text{educ}$ is:

Type: Multiple Choice

A) Downward — the estimated return to education is too low
B) Upward — the estimated return to education is too high
C) Zero — omitting variables never causes bias if the sample is large
D) Ambiguous — we cannot determine the direction without data

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Answer: B

OVB formula: $\tilde{\beta}_1 = \beta_1 + \beta_2 \cdot \delta_{12}$ where $\delta_{12} = \text{Cov}(\text{ability}, \text{educ})/\text{Var}(\text{educ}) > 0$ (ability positively correlated with education) and $\beta_2 > 0$ (ability raises wages). So $\tilde{\beta}_1 > \beta_1$ — the estimated return to education is upward biased. Part of what looks like a "return to education" is actually the return to unobserved ability.

Question 3

Lesson L04 · OVB Formula

[Calculation] The OVB formula is $\tilde{\beta}_1 = \beta_1 + \beta_2 \cdot \delta_{12}$. Given: true $\beta_1 = 0.08$ (education effect), $\beta_2 = 0.05$ (ability effect), $\delta_{12} = 0.6$ (slope from regressing ability on education). What is $\tilde{\beta}_1$?

Type: Calculation

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\[\tilde{\beta}_1 = \beta_1 + \beta_2 \cdot \delta_{12} = 0.08 + 0.05 \times 0.6 = 0.08 + 0.03 = \mathbf{0.11}\]

The simple regression overstates the education return by 0.03 — about 38% inflation. This is a substantial bias in practice, explaining why education returns estimated without ability controls tend to be higher than those from natural experiments (IV) that arguably hold ability constant.

Question 4

Lesson L04 · OVB Direction

In the OVB formula $\tilde{\beta}_1 = \beta_1 + \beta_2 \delta_{12}$, if $\text{Corr}(x_2, x_1) < 0$ and $\beta_2 > 0$, the bias is:

Type: Multiple Choice

A) Upward ($\tilde{\beta}_1 > \beta_1$)
B) Downward ($\tilde{\beta}_1 < \beta_1$)
C) Zero
D) Indeterminate

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Answer: B

$\delta_{12} < 0$ (negative correlation between $x_2$ and $x_1$) and $\beta_2 > 0 \Rightarrow \beta_2 \cdot \delta_{12} < 0 \Rightarrow \tilde{\beta}_1 < \beta_1$: downward bias. Full OVB direction table:

$\text{Corr}(x_2, x_1)$	$\beta_2 > 0$	$\beta_2 < 0$
Positive	Upward bias	Downward bias
Negative	Downward bias	Upward bias

Question 5

Lesson L01 · Multicollinearity

If two regressors are perfectly collinear (e.g. $x_2 = 3x_1$), OLS:

Type: Multiple Choice

A) Produces estimates with very large standard errors
B) Is undefined — $(X'X)$ is singular and cannot be inverted
C) Produces unbiased but imprecise estimates
D) Automatically drops the collinear regressor

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Answer: B

Perfect multicollinearity violates MLR.3 (no perfect linear relationships among regressors). The matrix $X'X$ becomes singular (rank-deficient) and cannot be inverted, so $\hat{\beta} = (X'X)^{-1}X'y$ is undefined. Statistical software will typically drop one of the collinear columns and warn you. Near perfect multicollinearity (high but not perfect correlation) is handled differently — OLS works but estimates have very large standard errors.

Question 6

Lesson L03 · MLR.3 Assumption

Give a concrete example where perfect multicollinearity would arise due to the dummy variable trap.

Type: Short Answer

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Suppose you include dummies for all seasons: $D_\text{Spring}$, $D_\text{Summer}$, $D_\text{Autumn}$, $D_\text{Winter}$, and also an intercept. Since exactly one season dummy = 1 for every observation: $D_\text{Spring} + D_\text{Summer} + D_\text{Autumn} + D_\text{Winter} = 1$ for all $i$. This is a perfect linear relationship with the intercept column (constant 1). Solution: drop one dummy (the "base group") — include only 3 of 4 season dummies. The omitted category becomes the reference level captured by the intercept.

Question 7

Lesson L05 · Variance Formula

The variance of $\hat{\beta}_j$ is $\text{Var}(\hat{\beta}_j) = \sigma^2 / [SST_j(1-R_j^2)]$. What is $R_j^2$ and what does $\text{VIF}_j = 1/(1-R_j^2)$ measure?

Type: Short Answer

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$R_j^2$ is the $R^2$ from regressing $x_j$ on all other regressors in the model. It measures how much of $x_j$'s variation is explained by the other regressors.

$\text{VIF}_j$ (Variance Inflation Factor) measures how much the variance of $\hat{\beta}_j$ is inflated relative to the case where $x_j$ is uncorrelated with all other regressors ($R_j^2 = 0 \Rightarrow \text{VIF} = 1$).

$\text{VIF} = 1$: no multicollinearity
$\text{VIF} = 5$: $\text{Var}(\hat{\beta}_j)$ is 5× what it would be without collinearity; $R_j^2 = 0.8$
$\text{VIF} > 10$: rule-of-thumb threshold for "serious" multicollinearity ($R_j^2 > 0.9$)

Question 8

Lesson L05 · Adding Variables

If you add an irrelevant variable $x_3$ (truly $\beta_3 = 0$ in the population) to a regression, what happens to $\hat{\beta}_1$?

Type: Multiple Choice

A) $\hat{\beta}_1$ becomes biased upward
B) $\hat{\beta}_1$ remains unbiased but its variance increases (or at best stays the same)
C) $\hat{\beta}_1$ becomes more precise because we have more information
D) $\hat{\beta}_1$ is inconsistent

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Answer: B

Including an irrelevant variable does not cause bias (since the true $\beta_3 = 0$, there is no OVB from including it). However, $\text{Var}(\hat{\beta}_1)$ increases because: (1) we lose a degree of freedom, and (2) if $x_3$ is correlated with $x_1$, VIF$_1$ rises. This is the bias-variance trade-off in variable selection: omitting a relevant variable causes bias; including an irrelevant one inflates variance. The Gauss-Markov Theorem says OLS is efficient among correctly specified models.

Question 9

Lesson L03 · Gauss-Markov

What do the five MLR assumptions (MLR.1–MLR.5) together guarantee?

Type: Short Answer

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Under MLR.1–MLR.5 (linearity, random sampling, no perfect multicollinearity, zero conditional mean $E(u|\mathbf{x})=0$, homoskedasticity $\text{Var}(u|\mathbf{x})=\sigma^2$), the Gauss-Markov Theorem guarantees that OLS is BLUE: the Best Linear Unbiased Estimator. "Best" means minimum variance among all linear unbiased estimators. Adding MLR.6 (normality: $u|\mathbf{x} \sim N(0,\sigma^2)$) upgrades OLS to BUE (Best Unbiased Estimator, not just among linear ones) and validates exact $t$ and $F$ tests in finite samples.

Question 10

Lesson L04 · Adjusted R²

Adjusted $R^2 = 1 - \frac{SSR/(n-k-1)}{SST/(n-1)}$. Unlike $R^2$, adjusted $R^2$ can decrease when a variable is added. Why?

Type: Multiple Choice

A) Because adjusted $R^2$ is always smaller than $R^2$
B) Because adding a variable always reduces SSR, but the penalty for the lost degree of freedom ($n-k-1$ shrinks) may outweigh the SSR reduction if the variable adds little explanatory power
C) Because adjusted $R^2$ is only valid for large samples
D) It cannot decrease — adding variables always improves model fit

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Answer: B

$R^2$ never decreases when a variable is added (SSR can only fall or stay flat). Adjusted $R^2$ imposes a degrees-of-freedom penalty: the denominator $n-k-1$ shrinks by 1 each time a variable is added. If the reduction in SSR is small (the variable is nearly irrelevant), the penalty outweighs the gain and adjusted $R^2$ falls. Adjusted $R^2$ provides a better criterion for model selection than $R^2$ because it does not reward adding noise variables.

Question 11

Lesson L05 · FWL Theorem

The Frisch-Waugh-Lovell (FWL) theorem says $\hat{\beta}_1$ from regressing $y$ on $x_1, x_2$ equals the slope from regressing $\tilde{y}$ on $\tilde{x}_1$. What is $\tilde{x}_1$?

Type: Multiple Choice

A) $x_1$ standardised to have mean zero and variance one
B) The residuals from regressing $x_1$ on $x_2$ — the part of $x_1$ not explained by $x_2$
C) $x_1$ minus its sample mean
D) The fitted values from regressing $x_1$ on $x_2$

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Answer: B

$\tilde{x}_1 = M_2 x_1$ where $M_2$ is the residual maker for $x_2$. Specifically: regress $x_1$ on $x_2$ and take the residuals. These residuals are the variation in $x_1$ orthogonal to $x_2$ — the part of $x_1$ that cannot be predicted by $x_2$. Similarly $\tilde{y} = M_2 y$ are the residuals from regressing $y$ on $x_2$. The FWL slope $\hat{\beta}_1 = \text{Cov}(\tilde{y}, \tilde{x}_1)/\text{Var}(\tilde{x}_1)$ is identical to the multiple regression slope.

Question 12

Lesson L05 · FWL Application

In a wage regression of $\ln(\text{wage})$ on education and experience, FWL says the education coefficient represents:

Type: Short Answer

Show Answer

The education coefficient captures the relationship between wages and education after removing the linear influence of experience from both. Concretely:

Residualise education on experience: $\tilde{\text{educ}} = \text{educ} - \hat{\delta}_0 - \hat{\delta}_1 \cdot \text{exper}$
Residualise log-wage on experience: $\tilde{y} = \ln(\text{wage}) - \hat{\gamma}_0 - \hat{\gamma}_1 \cdot \text{exper}$
$\hat{\beta}_\text{educ} = \text{Cov}(\tilde{y}, \tilde{\text{educ}})/\text{Var}(\tilde{\text{educ}})$

$\tilde{\text{educ}}$ is the component of education not predicted by experience. The regression uses only this "clean" variation. This is precisely what "holding experience constant" means — and why adding experience to the regression changes the education coefficient (it removes the part of the education-wage correlation that was actually driven by experience).

Question 13

Lesson L02 · Matrix OLS

For OLS with $n = 100$ observations and $k = 2$ slope regressors (plus intercept), the design matrix $X$ is $100 \times 3$. What is the dimension of $(X'X)$?

Type: Multiple Choice

A) $100 \times 100$
B) $3 \times 100$
C) $3 \times 3$
D) $100 \times 3$

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Answer: C

$X$ is $n \times (k+1) = 100 \times 3$. Then $X' $ is $3 \times 100$. So $X'X$ is $(3 \times 100)(100 \times 3) = \mathbf{3 \times 3}$. The inverse $(X'X)^{-1}$ is also $3 \times 3$. This is why OLS is computationally easy even with huge $n$ — the key matrix to invert is only $(k+1) \times (k+1)$.

Question 14

Lesson L04 · Practical OVB

A study regresses crime rates on the number of police officers per capita, without controlling for poverty. The coefficient on police is positive — more police is associated with more crime. Is this causal?

Type: Short Answer

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Almost certainly not causal — this is a classic OVB problem. Poverty (the omitted variable) is likely: - Positively correlated with police presence ($\delta_{12} > 0$: governments deploy more police to high-crime/high-poverty areas) - Positively affects crime ($\beta_2 > 0$: poverty is a driver of crime)

Therefore $\beta_2 \cdot \delta_{12} > 0$ → upward bias in the police coefficient. The positive association between police and crime reflects reverse causality (more crime → more police) and confounding by poverty — not that police cause crime. To estimate the causal effect, one needs an instrument for police deployment (e.g. election cycles affecting political motivation to hire police, as in Levitt 1997) or a natural experiment.

Question 15

Lesson L02 · Calculation

[Calculation] A regression with $n = 50$, $k = 2$ slope regressors (plus intercept) gives $SSR = 400$ and $SST = 1{,}000$. Calculate: (a) $R^2$, (b) Adjusted $R^2$, (c) $\hat{\sigma}^2$.

Type: Calculation

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(a) $R^2 = 1 - SSR/SST = 1 - 400/1000 = \mathbf{0.60}$

(b) $\bar{R}^2 = 1 - \frac{SSR/(n-k-1)}{SST/(n-1)} = 1 - \frac{400/47}{1000/49} = 1 - \frac{8.511}{20.408} = 1 - 0.417 = \mathbf{0.583}$

Adjusted $R^2 < R^2$ (always true). The penalty for $k=2$ regressors is modest here.

(c) $\hat{\sigma}^2 = SSR/(n-k-1) = 400/(50-2-1) = 400/47 \approx \mathbf{8.51}$

Question 16

Lesson L02 · ECMT5001 style

[Short Answer + Calculation] A regression of $\ln(\text{salary})$ on years of experience and a female dummy gives:

Variable	Coef	Std Err
experience	0.045	0.008
female	−0.183	0.052
intercept	3.210	0.065

$n = 200$, $R^2 = 0.41$.

(a) Interpret the experience and female coefficients. (b) Test $H_0: \beta_\text{female} = 0$ at the 1% significance level. Critical value $t_{0.005, 197} \approx 2.601$.

Type: Short Answer + Calculation

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(a) Interpretation: - Experience: Each additional year of experience is associated with approximately $100 \times 0.045 = 4.5\%$ higher salary, holding gender constant. - Female: Female workers earn approximately $100 \times 0.183 = 18.3\%$ less than male workers with the same experience. More precisely: $e^{-0.183} - 1 \approx -16.7\%$.

(b) t-test for gender gap: $$t = \frac{-0.183 - 0}{0.052} = \frac{-0.183}{0.052} \approx -3.52$$

$|t| = 3.52 > t_{0.005, 197} = 2.601$, so reject $H_0$ at the 1% level. The gender pay gap is statistically significant at the 1% level. Note: this is a conditional gap (holding experience fixed); it does not account for occupational sorting, hours worked, or other confounders.