Assignments: Problem Sets and Solutions

1.1 \(\beta_1=\operatorname{Cov}(X,\log(1+X))/\operatorname{Var}(X)\), where \(\operatorname{Var}(X)=1\) and \(E[\log(1+X)]=e\,E_1(1)\approx0.596\) (\(E_1\) the exponential integral). Evaluate \(E[X\log(1+X)]\) numerically, giving \(\beta_1\approx0.40\). (c) OLS estimates the best linear approximation to a concave CEF. It is useful as a summary of the average slope, weighted by the design, but not as a structural marginal effect at a particular \(x\).

1.2 By FWL, partialling out the dummy \(e_i\) deletes row \(i\) from every other regressor. The dummy's coefficient absorbs \(y_i-X_i'\widehat\beta_{(i)}\), so \(\widehat u_i=0\). The leave-one-out formula follows from the Sherman–Morrison identity applied to \((X'X-X_iX_i')^{-1}\).

1.3 With the delta method, the interval is symmetric by construction. The percentile-\(t\) interval is typically asymmetric and wider when \(\widehat\beta_2\) is imprecise, reflecting the ratio form of the turning point.

2.1 Human capital is positively related to both growth and investment, giving upward bias. Measurement error attenuates. With reliability \(\lambda\) of investment after partialling out initial income, \(\operatorname{plim}\widehat\beta=\lambda(\beta+\gamma\delta)\), where \(\gamma\delta\) is the OVB term. It is accidentally unbiased when \(\lambda(\beta+\gamma\delta)=\beta\).

2.2 Treatment was randomized within schools, across classrooms, so the assignment unit is the classroom: cluster there and include school fixed effects. School-level clustering is not required by the design and leaves about 79 clusters with much noisier variance estimates (Abadie et al., 2023).

2.3 With one point at \(h_{ii}\approx0.6\), HC0 coverage is typically about 0.85–0.88, while HC3 is about 0.93–0.95.

3.1 The IV estimate exceeds OLS (roughly 0.13 versus 0.07–0.08 in Card's specification), with a much larger standard error. The first stage is significant but not overwhelming. Compliers are men whose schooling depends on living near a college, likely from lower-income families with higher marginal returns. That is one explanation for IV > OLS, alongside measurement error. Exclusion concerns: college proximity correlates with urban labor markets and family background.

3.2 With many instruments, 2SLS moves toward OLS; LIML and Fuller stay closer to the just-identified estimate, with wider dispersion.

3.3 Example: the excluded variable enters the reduced form of the endogenous regressor with coefficient zero. The 2SLS estimate is then centered near the OLS limit, with an erratic distribution, and the first-stage \(F\) is approximately \(\chi^2_\ell/\ell\).

4.1 \(\widehat\beta_{RE}=W\widehat\beta_B+(I-W)\widehat\beta_W\), with \(W=[X_W'X_W+\psi X_B'X_B]^{-1}\psi X_B'X_B\), \(\psi=\sigma_u^2/(\sigma_u^2+T\sigma_c^2)\), where \(X_W\) and \(X_B\) are the within and between design matrices.

4.2 Expect pooled > RE > FE ≈ FD (FD smallest if measurement error dominates). A Mundlak rejection indicates correlation between the unit effect and union status.

4.3 Results should bracket the lagged-employment coefficient between FE and pooled OLS. Estimates that move substantially when instruments are collapsed signal proliferation.

5.1 \(\rho_1=(1+\phi\theta)(\phi+\theta)/(1+2\phi\theta+\theta^2)\) and \(\rho_j=\phi\rho_{j-1}\) for \(j\ge2\).

5.2 Since the mid-1980s, a random walk or IMA(1,1) is very hard to beat at a 4-quarter horizon. AR models do relatively better in high-volatility periods.

5.3 Typical half-lives are 3–5 years, with wide intervals: unit-root tests rarely reject in post-Bretton Woods samples, while panel or long-span data more often do.

6.1 \(\Theta(1)=\operatorname{chol}(\Psi(1)\Sigma_u\Psi(1)')\) and \(B_0^{-1}=\Psi(1)^{-1}\Theta(1)\). Too few lags bias \(\Psi(1)\), and long-run identification is very sensitive to it (Faust and Leeper, 1997).

6.2 Without commodity prices, the price level typically rises for several quarters after a tightening. LP responses are noisier but have similar output troughs.

6.3 The spread is usually found to be stationary. The short rate tends to adjust more, but results depend on the sample and on regime shifts in monetary policy.

7.1 With effects that grow after adoption, TWFE is attenuated relative to heterogeneity-robust estimators. The Bacon decomposition shows the weight on "later versus earlier treated" comparisons.

7.2 The estimated incumbency effect on the next vote share is about 7–8 percentage points, with no evidence of manipulation in the density of the margin.

7.3 FWER control (Holm, Romano–Wolf) suits confirmatory claims about each outcome; FDR suits exploratory screening; the index answers "did the program improve this domain overall?"