Module M10 Quiz: IV, 2SLS & Causal Inference

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

How to use

Attempt each question before clicking Show Answer. IV questions require careful economic reasoning, not just algebra.

Question 1

Lesson L01 · Endogeneity Definition

A regressor $x$ is endogenous if:

Type: Multiple Choice

A) It has many missing values
B) $\text{Cov}(x, u) \neq 0$ — the regressor is correlated with the error term
C) It is correlated with other regressors (multicollinearity)
D) Its distribution is skewed

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Answer: B

Endogeneity: $\text{Cov}(x, u) \neq 0$, which means $E(u \mid x) \neq 0$ — SLR.4/MLR.4 is violated. OLS is then biased and inconsistent: $\text{plim}(\hat{\beta}_1) \neq \beta_1$ even in infinitely large samples.

Three sources of endogeneity: 1. Omitted variables correlated with $x$ (most common) 2. Measurement error in $x$ (attenuation bias) 3. Simultaneity — $y$ causes $x$ and $x$ causes $y$

Question 2

Lesson L01 · Sources of Endogeneity

A researcher regresses crime rates on police presence. Police are deployed in response to crime. This creates endogeneity because:

Type: Multiple Choice

A) Police data is measured with error
B) Simultaneity — crime causes more police AND police (may) reduce crime; the causal direction runs both ways
C) Poverty is an omitted variable
D) Both B and C are valid sources of endogeneity here

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Answer: D — Both B and C

Two separate endogeneity problems:

Simultaneity (B): High crime → government deploys more police → $\text{Cov}(\text{police}, u_{\text{crime}}) > 0$ → OLS coefficient on police is upward biased (appears that more police = more crime)
OVB (C): Poverty increases crime AND is correlated with police deployment (police go to poorer, higher-crime areas) → even in a cross-section without simultaneous causality, omitting poverty biases the police coefficient

Both require IV or quasi-experimental methods to identify the causal effect of police on crime. Levitt (1997) used electoral cycles (politicians hire more police before elections) as an instrument.

Question 3

Lesson L02 · IV Estimator

The IV estimator with a single instrument $z$ is:

Type: Multiple Choice

A) $\hat{\beta}_{IV} = \text{Cov}(x,y)/\text{Var}(x)$
B) $\hat{\beta}_{IV} = \text{Cov}(z,y)/\text{Cov}(z,x)$
C) $\hat{\beta}_{IV} = \text{Cov}(z,x)/\text{Cov}(z,y)$
D) $\hat{\beta}_{IV} = \text{Var}(z)/\text{Cov}(z,y)$

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Answer: B

\[\hat{\beta}_{IV} = \frac{\text{Cov}(z,y)}{\text{Cov}(z,x)} = \frac{\sum (z_i - \bar{z})(y_i - \bar{y})}{\sum (z_i - \bar{z})(x_i - \bar{x})}\]

Intuition: $\text{Cov}(z,y)$ captures how much $y$ varies with the instrument; $\text{Cov}(z,x)$ scales this by how much $x$ varies with the instrument. The ratio gives the effect of $x$ on $y$ using only the exogenous (instrument-driven) variation in $x$.

Question 4

Lesson L02 · Instrument Validity

A valid instrument $z$ requires two conditions. Which correctly states both?

Type: Multiple Choice

A) $\text{Cov}(z,y) \neq 0$ (relevance) and $\text{Cov}(z,x) = 0$ (exclusion)
B) $\text{Cov}(z,x) \neq 0$ (relevance) and $\text{Cov}(z,u) = 0$ (exclusion)
C) $z$ is normally distributed and uncorrelated with $y$
D) $\text{Var}(z) > \text{Var}(x)$ and $z$ is measured without error

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Answer: B

Relevance: $\text{Cov}(z,x) \neq 0$ — the instrument must actually move the endogenous regressor. Testable via the first-stage $F$-statistic ($F > 10$ rule of thumb).
Exclusion restriction: $\text{Cov}(z,u) = 0$ — the instrument affects $y$ only through $x$, not directly. Untestable — requires economic reasoning. This is always an assumption, never a fact.

Note: relevance means $z$ is correlated with $x$ (not $y$ directly); exclusion means $z$ is uncorrelated with $u$ (the error in the structural equation for $y$).

Question 5

Lesson L02 · Card (1995) Instrument

Card (1995) uses distance to nearest college as an instrument for years of education in a wage equation. Evaluate the two IV conditions.

Type: Short Answer

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Relevance: Students growing up near a college face lower cost of attendance (time, money, information) → more likely to attend → more years of education. First-stage: regress education on distance-to-college → coefficient should be negative and significant. $F$-statistic typically exceeds 10. ✅ Reasonably satisfied.

Exclusion restriction: Distance to college should affect wages only through education — not directly. Potential violations: - Areas near colleges may have different labour market conditions (higher demand for educated workers, agglomeration effects) - Parental characteristics (more educated parents live near colleges) directly affect wages

Card addresses this partly by adding family background controls. The exclusion restriction is debated — it is the standard validity concern for this instrument. ⚠️ Arguable but widely used.

Question 6

Lesson L03 · 2SLS Procedure

[Short Answer] Describe the Two-Stage Least Squares (2SLS) procedure step by step, including what goes into each stage.

Type: Short Answer

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Stage 1 — First Stage:

Regress the endogenous variable $x$ on the instrument $z$ and all other exogenous controls $\mathbf{w}$: $$x_i = \pi_0 + \pi_1 z_i + \pi_2 w_{2i} + \cdots + v_i$$

Save fitted values $\hat{x}_i = \hat{\pi}_0 + \hat{\pi}_1 z_i + \hat{\pi}_2 w_{2i} + \cdots$

$\hat{x}_i$ is the exogenous component of $x$ — the part driven by the instrument $z$, which is uncorrelated with $u$ by the exclusion restriction.

Stage 2 — Second Stage:

Regress $y$ on $\hat{x}$ (the fitted values from Stage 1) and all exogenous controls: $$y_i = \beta_0 + \beta_1 \hat{x}_i + \beta_2 w_{2i} + \cdots + u_i$$

$\hat{\beta}_1$ from Stage 2 is the 2SLS estimate — consistent under valid instruments.

Critical warning: Never compute Stage 2 manually using OLS — the resulting standard errors are wrong (they don't account for Stage 1 uncertainty). Always use software's built-in IV/2SLS estimator (ivreg in R, ivregress in Stata, linearmodels.IV2SLS in Python).

Question 7

Lesson L04 · Weak Instruments

[Calculation] A first-stage regression of $x$ on instrument $z$ and controls gives $F = 7.2$. Is this a strong or weak instrument?

Type: Calculation

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$F = 7.2 < 10$ → Weak instrument by the Staiger & Stock (1997) rule of thumb.

With a weak instrument: - 2SLS is severely biased (toward the inconsistent OLS estimator) - Standard confidence intervals are unreliable (too narrow) - IV estimates are very sensitive to small violations of the exclusion restriction

The $F$-statistic here is the test of $H_0: \pi_1 = 0$ in the first stage. $F < 10$ suggests $z$ moves $x$ only weakly — there is not enough signal in the instrument to identify $\beta_1$ reliably.

Remedy: Find a stronger instrument ($F > 10$), use more instruments, or report weak-instrument-robust confidence intervals (Anderson-Rubin).

Question 8

Lesson L03 · 2SLS Standard Errors

If you manually run Stage 2 OLS using $\hat{x}$ from Stage 1, the standard errors reported by the Stage 2 regression are:

Type: Multiple Choice

A) Correct — 2SLS standard errors are the same as Stage 2 OLS standard errors
B) Too small — they ignore the sampling uncertainty in $\hat{x}$ from Stage 1 estimation
C) Too large — they overstate the true precision
D) Correct only if the first-stage $F > 10$

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Answer: B — Too small

Stage 2 OLS treats $\hat{x}$ as if it were the true (non-estimated) regressor. But $\hat{x}$ has its own sampling uncertainty from Stage 1. The true 2SLS variance formula accounts for this additional source of uncertainty, producing larger standard errors than naive Stage 2 OLS.

Using wrong SEs leads to over-rejection of $H_0$ (liberal inference). This is why software implements the correct 2SLS variance formula directly — never manually chain two OLS regressions for IV.

Question 9

Lesson L05 · DiD Estimator

[Calculation] Using Card & Krueger (1992) data: NJ (treated) employment pre = 20.44, post = 21.03; PA (control) employment pre = 23.33, post = 21.17. Calculate $\hat{\beta}_{DD}$ and interpret.

Type: Calculation

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\[\hat{\beta}_{DD} = (\bar{Y}_{NJ,\text{post}} - \bar{Y}_{NJ,\text{pre}}) - (\bar{Y}_{PA,\text{post}} - \bar{Y}_{PA,\text{pre}})\]

\[= (21.03 - 20.44) - (21.17 - 23.33) = 0.59 - (-2.16) = \mathbf{2.75}\]

Interpretation: The NJ minimum wage increase was associated with approximately 2.75 more full-time equivalent employees per restaurant relative to what would have been expected based on PA's trend. Employment in NJ rose relative to the PA counterfactual — contrary to the standard competitive model prediction that minimum wages reduce employment.

Question 10

Lesson L05 · Parallel Trends

The parallel trends assumption in DiD states:

Type: Multiple Choice

A) Treated and control groups must have identical outcomes before treatment
B) In the absence of treatment, the treated group would have followed the same trend as the control group
C) Treatment and control groups must be randomly assigned
D) The treatment effect must be constant across all observations

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Answer: B

Parallel trends: $E(Y_{1,t}^{(0)} - Y_{1,t-1}^{(0)}) = E(Y_{0,t} - Y_{0,t-1})$ for all $t$ — the counterfactual trend for the treated group equals the observed trend for the control group.

Key points: - Does not require pre-treatment levels to be equal (that's answer A — too strong) - Is untestable for the post-treatment period (we never observe the treated group without treatment) - Can be partially tested using pre-treatment periods: if treated and control had parallel trends before treatment, it's more credible they would have continued - Groups need not be randomly assigned (DiD is used in observational settings)

Question 11

Lesson L05 · DiD Regression

In the DiD regression $y_{it} = \alpha + \gamma D_i^{\text{treat}} + \lambda D_t^{\text{post}} + \beta (D_i^{\text{treat}} \times D_t^{\text{post}}) + \varepsilon_{it}$, what does $\gamma$ represent?

Type: Multiple Choice

A) The treatment effect
B) The pre-treatment difference between treated and control groups
C) The time trend in the treated group
D) The post-treatment mean of the control group

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Answer: B

$\gamma$ is the coefficient on $D_i^{\text{treat}}$ alone (without the post dummy). It captures the pre-existing difference between treated and control groups before treatment occurs — the baseline gap.

Full interpretation of all coefficients: - $\alpha$: control group mean in the pre-period - $\gamma$: pre-treatment gap (treated − control, in pre-period) - $\lambda$: time trend in the control group (post − pre for controls) - $\beta$: DiD treatment effect (the interaction term)

Question 12

Lesson L05 · Pre-Trend Testing

[Short Answer] How would you formally test whether treated and control groups had parallel trends before the treatment?

Type: Short Answer

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Method — Leads/Pre-treatment interaction test:

With multiple pre-treatment periods (say $t = -3, -2, -1, 0$), define interaction terms for each pre-period: $D_i^{\text{treat}} \times \mathbf{1}[t = -2]$, $D_i^{\text{treat}} \times \mathbf{1}[t = -1]$ (using $t = -3$ as the reference period)
Add these interactions to the DiD regression alongside the main treatment interaction
Test: $F$-test of joint significance of all pre-period interactions under $H_0$: all pre-period interaction coefficients = 0
If $H_0$ is not rejected (pre-period interactions are small and jointly insignificant): parallel trends held pre-treatment → credible assumption for the post-period
If $H_0$ is rejected (pre-trends are significantly different): the groups were already diverging before treatment → DiD estimate is unreliable; may need to control for group-specific trends or find a better comparison group

Also useful: placebo tests — apply the DiD to a period with no treatment and verify $\hat{\beta}_{DD} \approx 0$.

Question 13

Lesson L01 · IV vs OLS Consistency

OLS is inconsistent when $x$ is endogenous. IV is consistent but:

Type: Multiple Choice

A) IV is unbiased in finite samples
B) IV is biased in finite samples (especially with weak instruments), but consistent — the bias vanishes as $n \to \infty$
C) IV has lower variance than OLS always
D) IV is only valid when the instrument is binary

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Answer: B

IV has a finite-sample bias that is proportional to the "weakness" of the instrument and the degree of OVB. With a strong instrument ($F \gg 10$) and large $n$, the finite-sample bias is small and the consistency advantage over OLS dominates.

Trade-offs: - IV is consistent (bias → 0 as $n \to \infty$); OLS is not - IV has larger variance than OLS — using only the $z$-driven variation in $x$ discards information - With a weak instrument, IV finite-sample bias can be so large that OLS is actually less biased — a key reason the $F > 10$ rule matters

Question 14

Lesson L02 · Overidentification

When you have more instruments than endogenous variables (overidentified model), you can test the validity of instruments using the Sargan-Hansen (J) test. The J test:

Type: Short Answer

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The J-test (Sargan-Hansen overidentification test):

With $k$ instruments and 1 endogenous regressor, you have $k - 1$ overidentifying restrictions — testable restrictions on instrument validity.

Procedure: 1. Estimate 2SLS; obtain 2SLS residuals $\hat{u}_{IV}$ 2. Regress $\hat{u}_{IV}$ on all instruments $z_1, \ldots, z_k$ and exogenous controls 3. $J = n \cdot R^2_{\text{aux}} \sim \chi^2_{k-1}$ under $H_0$

$H_0$: All instruments satisfy the exclusion restriction

Reject $H_0$: At least one instrument is correlated with $u$ (invalid)

Important caveat: The J-test tests the instruments relative to each other. If ALL instruments are invalid in the same direction, the J-test will not detect it. Passing the J-test is necessary but not sufficient for instrument validity — economic reasoning remains essential.