13.12 Exercises
Show that the Average Treatment Effect (ATE) in the simple linear regression framework
\[ Y_i = \beta_0 + \tau D_i + \mu_i, \] assuming non-informative prior distributions, so that the posterior mean of the location parameter coincides with the maximum likelihood estimator, is equal to
\[ \bar{y}_1 - \bar{y}_0. \]Some readers may question the assumption that potential outcomes are normally distributed.
However, it is important to note that the normal distribution is the maximum entropy continuous distribution given a specified mean \(\mu\) and finite variance \(\sigma^2\).
In other words, among all distributions with the same mean and variance, the normal distribution represents the one with the greatest level of uncertainty or unpredictability (Cover and Thomas 2006).Show that the normal distribution is the maximum entropy continuous distribution given a specified mean \(\mu\) and finite variance \(\sigma^2\) by considering the formal definition of entropy:
\[ H(f) = - \int_{-\infty}^{\infty} f(y) \log f(y) \, dy, \] where \(f(y)\) is a probability density function.Use the package dagitty to construct the DAG implied by the Conditional Independence Assumption, verify that it is acyclic, and check whether the causal effect of \(D\) on \(Y\) is identifiable by controlling for \(\mathbf{X}\).
Use the package dagitty to construct the DAG implied by the collider bias, verify that it is acyclic, and check that the causal effect of \(D\) on \(Y\) is identifiable by controlling for \(\mathbf{X}\) but not for \(C\).
Use the package dagitty to construct the DAG implied by the instrumental variable, taking into account that \(\mathbf{U}\) is unobserved (latent). Verify that it is acyclic, check that \(Z\) is a valid instrument, and determine whether the causal effect of \(D\) on \(Y\) is identifiable.
401(k) participation on net financial assets continues I
Apply the framework from this example to compute the intention-to-treat effect, the local average treatment effect, and the effect of eligibility on participation.401(k) participation on net financial assets continues II
Perform inference in this example assuming that the stochastic errors follow a Dirichlet process, using the function rivDP from the bayesm package to analyze 401(k) participation and its effect on net financial assets under the same specification as in the main text, and plot the LATE.Difference-in-Differences simulation continues I
Perform the simulation of the DiD example, and perform inference using the specification:
\[ Y_{it} = \alpha + \alpha_i + \phi_t + \tau_2 \,\big[ D_i \cdot \mathbf{1}(t = 2) \big] + \epsilon_{it}. \]Difference-in-Differences simulation continues II
Note that another strategy to perform inference on the ATT is to estimate the saturated model
\[ Y_{it} = \sum_{t,l} \mu_{tl} \big[ D_{il} \cdot \mathbf{1}(t = t) \big] + \epsilon_{it}, \quad t = 1, 2, \ l = 1, 0, \] and then use the posterior draws to compute
\[ \tau_2 = (\mu_{21} - \mu_{11}) - (\mu_{20} - \mu_{10}). \] Explain why inference on \(\tau_2\) using this approach in the simulation setting shows that the posterior mean is similar to that from the previous approaches, but the level of uncertainty is higher.Perform a simulation exercise to assess the ability of BETEL to identify the causal effect when an instrument is used to address the omission of relevant regressors. Illustrate the consequences of varying the dependence between the omitted regressor and the observed regressor.
Simultaneous causality continues
Use the demand–supply simulation and the moment conditions to infer the causal effect of a 10% tax, implementing a BETEL algorithm from scratch.Perform a simulation of the instrumental variable quantile regression, and then perform inference using the general Bayes posterior framework.
Use the Laplace asymmetric distribution to simulate the stochastic error, such that
\[ \int_{-\infty}^{0} f_{\tau}(\mu_i) \, d\mu_i = \tau, \] which implies that the \(\tau\)-th quantile of \(\mu_i\) is 0. In addition, let
\[ q_{\tau}(Y_i \mid X_i, D_i) = \beta_0 + \beta_1 X_i + \beta_2 D_i \] denote the \(\tau\)-th quantile regression function of \(Y_i\) given \(X_i\) and \(D_i\), with \(0 < \tau < 1\).
Specifically, consider the model
\[ Y_i = \beta_{0\tau} + \beta_{1\tau} X_i + \beta_{2\tau} D_i + \mu_i, \] together with
\[ D_i = \gamma Z_i + \delta \mu_i, \quad Z_i \sim N(0,1), \] see Section 6.9.Example: Doubly robust Bayesian inference continues
Perform a simulation exercise to assess the consequences of misspecifying the propensity score function and the outcome regression in doubly robust Bayesian inference.