Question

Consider the potential outcomes framework, where \(w\) is a binary treatment indicator and the potential outcomes are \(y(0)\) and \(y(1) .\) Assume that \(w\) is randomly assigned, so that \(w\) is independent of \([y(0), y(1)] .\) Let \(\mu_{0}=\mathrm{E}[y(0)], \mu_{1}=\mathrm{E}[y(1)], \sigma_{0}^{2}=\operatorname{Var}[y(0)],\) and \(\sigma_{1}^{2}=\operatorname{Var}[y(1)]\). i. Define the observed outcome as \(y=(1-w) y(0)+w y(1) .\) Letting \(\tau=\mu_{1}-\mu_{0}\) be the average treatment effect, show you can write $$y=\mu_{0}+\tau w+(1-w) v(0)+w v(1)$$ where \(v(0)=y(0)-\mu_{0}\) and \(v(1)=y(1)-\mu_{1}\) ii. Let \(u=(1-w) v(0)+w v(1)\) be the error term in $$y=\mu_{0}+\tau w+u$$ Show that $$\mathbf{E}(u | w)=0$$ What statistical properties does this finding imply about the OLS estimator of \(\tau\) from the simply regression \(y_{i}\) on \(w_{i}\) for a random sample of size \(n\) ? What happens as \(n \rightarrow \infty ?\) iii. Show that $$\operatorname{Var}(u | w)=\mathrm{E}\left(u^{2} | w\right)=(1-w) \sigma_{0}^{2}+w \sigma_{1}^{2}$$ Is there generally heteroskedasticity in the error variance? iv. If you think \(\sigma_{1}^{2} \neq \sigma_{0}^{2},\) and \(\hat{\tau}\) is the OLS estimator, how would you obtain a valid standard error for \(7 ?\) v. After obtaining the OLS residuals, \(\widehat{u}_{i}, i=1, \ldots, n,\) propose a regression that allows consistent estimation of \(\sigma_{0}^{2}\) and \(\sigma_{1}^{2}\). [Hint: You should first square the residuals.]

Short answer

The average treatment effect can be estimated as \( y = \mu_0 + \tau w + u \) with zero mean error, suggesting unbiased OLS. Use robust errors for heteroscedasticity.

Step-by-step solution

Express the Observed Outcome

Given the potential outcomes framework, the observed outcome for each individual is described as \( y = (1-w) y(0) + w y(1) \). To express \( y \) in terms of \( \mu_0 \) and \( \tau \), we start by recognizing that \( \tau = \mu_1 - \mu_0 \). Therefore, substituting \( y(0) = \mu_0 + v(0) \) and \( y(1) = \mu_1 + v(1) \), we rewrite the observed outcome as:\[ y = \mu_0 + \tau w + (1-w) v(0) + w v(1) \] where \( v(0) = y(0) - \mu_0 \) and \( v(1) = y(1) - \mu_1 \).

Understand the Error Term

Define the error term \( u \) in the regression as \( u = (1-w) v(0) + w v(1) \). This term captures the deviation of the individual's potential outcome from the average potential outcome based on their treatment status.

Expectation of the Error Given Treatment

To demonstrate that \( \mathbf{E}(u | w) = 0 \), consider\[ \mathbf{E}(u | w = 0) = \mathbf{E}((1-0)v(0) + 0 \cdot v(1) | w=0) = \mathbf{E}(v(0) | w=0) = \mathbf{E}(v(0)) = 0 \] and similarly,\[ \mathbf{E}(u | w = 1) = \mathbf{E}(0 \cdot v(0) + 1 \cdot v(1) | w=1) = \mathbf{E}(v(1)) = 0 \]. This implies that the error term has zero conditional mean given \( w \), indicating that the OLS estimator of \( \tau \) is unbiased. As \( n \to \infty \), the estimator \( \hat{\tau} \) is consistent.

Variance of the Error Conditional on Treatment

Calculate the conditional variance of \( u \):\[ \operatorname{Var}(u | w=0) = \operatorname{Var}(v(0)) = \sigma_0^2 \] and \[ \operatorname{Var}(u | w=1) = \operatorname{Var}(v(1)) = \sigma_1^2 \]. Therefore, \( \operatorname{Var}(u | w) = (1-w) \sigma_0^2 + w \sigma_1^2 \), suggesting there is heteroscedasticity in the error term, as variance changes with \( w \).

Correcting Standard Errors for Heteroscedasticity

Since there is heteroscedasticity and \( \sigma_1^2 eq \sigma_0^2 \), use robust standard errors to obtain valid estimates of the variance for the OLS estimator \( \hat{\tau} \). These robust errors account for the varying \( u \)'s variance based on treatment assignments.

Estimating Variances from Residuals

To estimate \( \sigma_0^2 \) and \( \sigma_1^2 \), after estimating \( \hat{u}_i = y_i - \hat{\mu}_0 - \hat{\tau} w_i \), we regress the squared residuals \( \hat{u}_i^2 \) on a constant and the treatment indicator \( w \). The constant term provides an estimate of \( \sigma_0^2 \), while the coefficient of \( w \) gives \( \sigma_1^2 - \sigma_0^2 \).

Key concepts

Binary Treatment Indicator

In the potential outcomes framework, a binary treatment indicator, often denoted as \(w\), plays a crucial role. It simply indicates whether an individual has received a particular treatment (\(w = 1\)) or not (\(w = 0\)). This variable helps us distinguish between the two potential states for each individual:

• \(y(0)\) – the potential outcome without treatment
• \(y(1)\) – the potential outcome with treatment

Binary treatment indicators make it straightforward to determine which scenario applies to an individual. When \(w\) is independent of potential outcomes, it implies a random treatment assignment. This randomness is crucial for identifying causal effects. It ensures the differences in outcomes between treated and untreated groups are due to the treatment itself and not other factors.

A good analogy for understanding \(w\) is to think of it like a switch: if the switch is on (\(w = 1\)), the treatment is administered and the potential outcome with treatment, \(y(1)\), is observed. If the switch is off (\(w = 0\)), we observe \(y(0)\), the potential outcome without treatment.

Average Treatment Effect

The Average Treatment Effect (ATE) is a key concept in causal inference within the potential outcomes framework. Represented by \(\tau\), the ATE measures the average difference between potential outcomes with and without treatment across a population. Mathematically, it can be expressed as:\[ \tau = \mu_{1} - \mu_{0} \]Here, \(\mu_{1}\) is the expected value of the potential outcome with treatment, and \(\mu_{0}\) is the expected value without treatment.

ATE tells us how much, on average, a treatment affects an outcome. It is particularly important in policy and clinical settings, as it provides insights into the effectiveness of interventions. Understanding ATE allows for an objective assessment of an intervention's impact, helping decision-makers evaluate the merits of implementing a treatment.

In practice, estimating \(\tau\) accurately depends on proper recognition of the framework's assumptions, including the independence of the treatment assignment from potential outcomes. When these assumptions hold true, the Simple Linear Regression model serves as a reliable tool for estimating the ATE.

OLS Estimator Properties

Ordinary Least Squares (OLS) is a method commonly used in estimating treatment effects due to its appealing properties when certain conditions are met. When using the simple regression of outcome \(y_i\) on the binary treatment indicator \(w_i\), the OLS estimator for the treatment effect \(\tau\) has several important properties:

• Unbiasedness: Provided the treatment indicator \(w\) is truly random, the OLS estimator of \(\tau\) is unbiased. This means it correctly reflects the average treatment effect in the population.
• Consistency: As the sample size \(n\) increases towards infinity, the estimator \(\hat{\tau}\) converges to the true population parameter \(\tau\).

These properties hinge on the assumption that the error term \(u\) in the regression model has an expected value of zero given \(w\), i.e., \(\mathbf{E}(u \mid w) = 0\). This zero conditional mean guarantees that the OLS estimator is not systematically biased by errors in prediction.

These characteristics of OLS make it a powerful and reliable method for estimating causal effects when the necessary assumptions hold.

Heteroscedasticity in Error Terms

In regression analysis, heteroscedasticity refers to the condition where the variance of the error terms varies across observations. In the context of the potential outcomes framework, the variance of the error term \(u\), conditional on treatment, can be expressed as:\[ \operatorname{Var}(u \mid w) = (1-w)\sigma_{0}^{2} + w\sigma_{1}^{2} \]This expression indicates that the variance of \(u\) may not be constant because it depends on \(w\).

• If \(\sigma_{0}^{2} eq \sigma_{1}^{2}\), this results in heteroscedasticity, as the error variance changes with the treatment assignment.
• Heteroscedasticity violates the homoscedasticity assumption of constant variance in classic regression models, potentially leading to inefficient estimates and incorrect standard errors.

To address heteroscedasticity, robust standard errors can be used in OLS estimation. This approach adjusts the confidence intervals and hypothesis tests to account for varying error variances, ensuring valid inference despite heteroscedasticity.

It's important to diagnose and correct for heteroscedasticity to maintain the reliability and validity of estimations in regression analysis.