Exercises: Panel Data
Theoretical Exercises
Event Study Estimator as Linear Regression
Consider the setting of the lecture on event studies under an assumption of no trends.
Show that the estimator \(\widehat{AE}_{ES}\) is equal to the OLS estimator for regressing \(Y_{it}\) on \((1, D_{it})\) (proposition 3).
Consider the following representation for the realized outcomes \(Y_{it}\): \[ \begin{aligned} Y_{it} & = \beta_0 + \beta_1 D_{it} + U_{it}, \\ \beta_0 & = \E[Y_{i1}^0], \quad \beta_1 = \E[Y_{i2}^1- Y_{i2}^0] . \end{aligned} \] Express \(U_{it}\) in terms of \(\beta_0, \beta_1\) and the potential outcomes \(Y_{it}^0, Y_{it}^1\).
Recall that in a linear causal model OLS is consistent for the coefficient vector provided the regressors \(\bX_{it}\) are orthogonal to the residuals in the sense \(\E[\bX_{it}U_{it}]=0\). Does this condition hold in the above regression? What about \(\E[U_{it}|\bX_{it}]=0\)? Connect to the assumption of no trends.
Allowing Trends in Event Studies
When talking about event studies in the lecture, we have made the assumption of no trends in the untreated outcomes: that \(\E[Y_{it}^0]\) does not depend on \(t\). This problem is about relaxing this assumption to allow some dynamics in outcomes.
Consider the multiple period framework in the slides on event studies. Suppose that we replace the assumption of no trends with an assumption of linear average growth in the untreated outcomes: \[ \E[Y_{it}^0] - \E[Y_{i1}^0] = \gamma(t-1), \quad t= 1, \dots, T \tag{1}\] We are interested in estimating \(\beta_{\tau} = \E[Y_{i\tau}^1-Y_{i\tau}^0]\) — the average treatment effects in periods \(\tau\geq t_0\).
Propose a consistent estimator for \(\E[Y_{it}^1-Y_{it}^0]\). Show its consistency. You may freely use the consistency results proved in the asymptotic theory part of the class.
Difference-in-Differences as Two-Way Fixed Effect Regression
Consider the setting of the lecture on difference-in-differences with two groups and two periods.
- Show that the DiD estimator is a two-way fixed effect estimator (prove proposition 2 in the slides).
- Prove that the parallel trends assumption implies strict exogeneity in Equation 4 in the slides.
- Conclude that the DiD estimator is consistent and asymptotically normal for the ATT. You may freely use the consistency results proved in the asymptotic theory part of the class.
Asymptotic Properties of Fixed Effects Estimator
Consider the setting of the lecture on fixed effect estimation. Suppose that \(\bX_{it}\) is some vector of treatments, and that the outcomes follow the two-way fixed effect potential outcome model \[ Y_{it}^{\bx} = \alpha_i + \gamma_t + \bx'\bbeta+ U_{it}. \]
- Show that the corresponding TWFE estimator of \(\bbeta\) is consistent and asymptotically normal under the assumptions of Proposition 4.
- Let the true potential outcomes model only have individual random intercepts \(\alpha_i\): \[ Y_{it}^{\bx} = \alpha_i + \bx'\bbeta+ U_{it}. \] Suppose that you use the two-way FE estimator regardless. Is it consistent for \(\bbeta\)?
Asymptotic Properties of the Mean Group Estimator
Let \(Y_{it}, \bX_{it}\), \(\bbeta_i\), and \(U_{it}\) be linked through the linear potential outcomes model with unit-specific coefficients: \[ Y_{it}^{\bx} = \bx'\bbeta_i + U_{it}. \] We observe \((Y_{it}, \bX_{it})_{i=1, \dots, N}^{t=1, \dots, T}\). We assume that \(T\geq p\), where \(p\) is the number of coordinates of \(\bbeta\). We also assume that \((\bX'\bX)\) is invertible for each unit \(i\).
We are interested in learning \(\E[\bbeta_i]\). We estimate it using the mean group estimator: \[ \begin{aligned} \hat{\bbeta}^{MG} & = \dfrac{1}{N}\sum_{i=1}^N \hat{\bbeta}_i, \\ \hat{\bbeta}_i & = (\bX_i'\bX_i)^{-1}\bX_i'\bY_i \end{aligned} \]
- Show that \(\hat{\bbeta}^{MG}\) is consistent for \(\E[\bbeta_i]\) under the assumptions of proposition 1 in the slides on mean group estimation.
- Show that \(\hat{\bbeta}^{MG}\) is asymptotically normal under the the assumptions of proposition 2 in the slides on mean group estimation.
- Suppose that \(p=1\) (scalar case). Propose a confidence interval for \(\E[\beta_i]\) with asymptotic coverage \(1-\alpha\).
Applied Exercises
Event Study: Brexit
On June 23, 2016 the UK held the Brexit referendum. The results of the referendum were broadly unexpected, leading to a strong negative movement in stock prices on June 24. Conduct a financial event study to quantify the effect of the referendum on a selection of leading British companies (say, the companies in the FTSE 100 Index). Proceed as in the lecture to compute the abnormal returns.
DiD: Police and Crime
Does police presence reduce crime? A famous article by Di Tella and Schargrodsky (2004) analyzes this using exogenous variation in police presence created by a terrorist attack in Buenos Aires. Read their article and replicate the results of their difference-in-differences estimation (table 3). You can download their data from the article’s page on the website of the American Economic Association.
FE and MG: Labor Markets and Pollution
Consider again the paper by Borgschulte, Molitor, and Zou (2024). In the lectures we have focused on the effect of pollution on earnings. However, Borgschulte, Molitor, and Zou (2024) also consider the effects on total employment and labor force participation. Replicate their results for these two outcomes — columns (3) and (4) in their table 1. To do so, download their data from the Harvard Dataverse and suitably modify the lecture code.