ps7

site/626.bib

@@ -628,3 +628,18 @@ @article{rambachan2023
     url = {https://doi.org/10.1093/restud/rdad018},
     eprint = {https://academic.oup.com/restud/article-pdf/90/5/2555/51356029/rdad018.pdf},
 }
+
+@article{chen2023,
+author = {Chen, Weiwei and French, Michael T.},
+title = {Marijuana legalization and traffic fatalities revisited},
+journal = {Southern Economic Journal},
+volume = {90},
+number = {2},
+pages = {259-276},
+keywords = {marijuana legalization, medical marijuana, recreational marijuana, traffic fatalities},
+doi = {https://doi.org/10.1002/soej.12657},
+url = {https://onlinelibrary.wiley.com/doi/abs/10.1002/soej.12657},
+eprint = {https://onlinelibrary.wiley.com/doi/pdf/10.1002/soej.12657},
+abstract = {Abstract The legal landscape for marijuana in the United States has changed dramatically over the last three decades. While several studies have examined the relationship between marijuana legalization and traffic fatalities, some of the research is becoming outdated and existing evidence remains mixed. Our research revisits the topic with two updates. First, our study includes states that legalized marijuana more recently and provides updated evidence on the effects of marijuana legalization. Second, considering recent discussions about the limitations of difference-in-differences designs, we employ alternative estimators that are robust to heterogeneous and dynamic treatment effects. Overall, our alternative estimators suggest either a smaller reduction (i.e., 3.9\% drop in the overall fatality rate) or no change in traffic fatalities associated with legalizing marijuana for medical use, compared to the two-way fixed-effect estimator. We find no significant impact on traffic fatalities associated with legalizing marijuana for recreational use.},
+year = {2023}
+}

site/problemsets/07/ps07.qmd

@@ -1,7 +1,7 @@
 ---
 title: "ECON 626: Problem Set 7"
 #author: "Paul Schrimpf"
-date: 2023-11-27
+date: 2023-12-01
 bibliography: ../../626.bib
 ---
 
@@ -21,7 +21,50 @@ $$
 
 # Problem 1
 
+In "Marijuana legalization and traffic fatalities revisited", @chen2023 analyze how marijuana legalization in the US affected traffic fatalities.
 
+## TWFE
 
+@chen2023 begin by estimating a two-way fixed effects (TWFE) model,
+$$
+F_{it} = \alpha + \beta_1 MM_{it} + \beta_2 RM_{it} + X_{it}\gamma + \eta_i + \theta_t + \varepsilon_{it}
+$$
+where $F_{it}$ is log traffic fatalaties per 100,000 in state $i$ and year $t$, $MM_{it}$ is an indicator for medical marijuana being legal, $RM_{it}$ is an indicator for legalized recreational marijuana, and $X_{it}$ are state covariates including traffic volume, economic conditions, and demographics. @chen2023 present estimates of the TWFE model "for comparison with the literature," but also compute other estimators. Why? What is a problem with TWFE here?
+
+## Multiple Treatments
+
+Suppose $1/2$ of states legalized medical marijuana in a single year and no other changes occurred in $MM_{it}$. Also suppose $1/4$ of states legalized recretaional marijuana in a single later year and no other changes occurred in $RM_{it}$. That is, there is no variation in treatment timing, but there are multiple treatments. For simplicity, ignore covariates. Would estimating a TWFE model,
+$$
+F_{it} = \alpha + \beta_1 MM_{it} + \beta_2 RM_{it} + \eta_i + \theta_t + \varepsilon_{it}
+$$
+recover an interpretable average treatment effect on the treated?
+
+## $DID_M$, $DID_\ell$
+
+@chen2023 also report the $DID_M$ and $DID_\ell$ estimators? What are these? What sort of average treatment effect do they estimate? (You will need to read @chen2023 and perhaps also @dd2020 to answer this.)
+
+## Results
+
+Table 2 and Figure 1 show the main results of the paper. What conclusions would you draw from these?
+
+![table 2](chen-tab2.png)
+
+![figure 1](chen-fig1.png)
 
 # Problem 2
+
+Consider the following linear regression model such that
+$$
+Y_i = β_0 + X_i β_1 + u_i ,
+$$
+where $X_i$ and $Y_i$ are observed random variables. Let us assume that
+$\Er [u_i ] = 0$ but $\cov(X_i , u_i ) \neq 0$.
+Suppose that there exists a variable $Z_i$ such that
+$\cov(X_i , Z_i ) > 0$ and
+$\cov(Z_i , u_i ) > 0$.
+
+Find the asymptotic bias of the 2SLS estimator of
+$\hat{\beta}_1$. (Recall that the asymptotic bias of an estimator is
+its probability limit minus the true parameter.) Can you determine
+unambiguously whether the 2SLS estimator tends to underestimate or
+overestimate the parameter $β_1$ ? If so, give explanations how.

site/problemsets/07/ps07_2022.qmd

@@ -0,0 +1,55 @@
+---
+title: "ECON 626: Problem Set 7"
+#author: "Paul Schrimpf"
+date: 2022-12-06
+bibliography: ../../626.bib
+---
+
+$$
+\def\Er{{\mathrm{E}}}
+\def\En{{\mathbb{En}}}
+\def\cov{{\mathrm{Cov}}}
+\def\var{{\mathrm{Var}}}
+\def\R{{\mathbb{R}}}
+\newcommand\norm[1]{\left\lVert#1\right\rVert}
+\def\rank{{\mathrm{rank}}}
+\newcommand{\inpr}{ \overset{p^*_{\scriptscriptstyle n}}{\longrightarrow}}
+\def\inprob{{\,{\buildrel p \over \rightarrow}\,}}
+\def\indist{\,{\buildrel d \over \rightarrow}\,}
+\DeclareMathOperator*{\plim}{plim}
+$$
+
+# Problem 1
+
+
+Consider the following linear regression model such that
+$$
+Y_i = β_0 + X_i β_1 + u_i ,
+$$
+where $X_i$ and $Y_i$ are observed random variables. Let us assume that
+$\Er [u_i ] = 0$ but $\cov(X_i , u_i ) \neq 0$.
+Suppose that there exists a variable $Z_i$ such that
+$\cov(X_i , Z_i ) > 0$ and
+$\cov(Z_i , u_i ) > 0$.
+
+Find the asymptotic bias of the 2SLS estimator of
+$\hat{\beta}_1$. (Recall that the asymptotic bias of an estimator is
+its probability limit minus the true parameter.) Can you determine
+unambiguously whether the 2SLS estimator tends to underestimate or
+overestimate the parameter $β_1$ ? If so, give explanations how.
+
+# Problem 2
+
+In the linear model,
+$$
+Y_i = \beta_0 + X_i \beta_1 + u_i
+$$
+assume that $\Er[u_i] = 0$ and $X_i \in \R^1$. Suppose that $\Er[X_i u_i] \neq 0$, but,
+somewhat strangely, you assume $\Er[u_i^2|X_i] = \sigma^2$.
+
+1. Show that a set of two elements that contains $\beta_1$ is identified. Denote this set by $B_1$.
+**Hint: use the moment condition $\Er[u_i^2 (X_i-\Er[X_i])]$.**
+
+2. Describe an estimator for $B_1$ and show that it is consistent. State any additional assumptions needed.
+
+3. Find the asymptotic distribution of your estimator for $B_1$. State any additional assumptions needed.