Question

The potential outcomes framework in Section 3 - 7 e can be extended to more than two potential outcomes. In fact, we can think of the policy variable, \(w\), as taking on many different values, and then \(y(w)\) denotes the outcome for policy level \(w\). For concreteness, suppose \(w\) is the dollar amount of a grant that can be used for purchasing books and electronics in college, \(y(w)\) is a measure of college performance, such as grade point average. For example, \(y(0)\) is the resulting GPA if the student receives no grant and \(y(500)\) is the resulting GPA if the grant amount is \(\$ 500\). For a random draw \(i\), we observe the grant level, \(w_{i} \geq 0\) and \(y_{i}=y\left(w_{i}\right)\). As in the binary program evaluation case, we observe the policy level, \(w_{i}\), and then only the outcome associated with that level. i. Suppose a linear relationship is assumed: $$ y(w)=\alpha+\beta w+v(0) $$ where \(y(0)=\alpha+v .\) Further, assume that for all \(i, w_{i}\) is independent of \(v_{i}\). Show that for each \(i\) we can write $$ \begin{aligned} y_{i} &=\alpha+\beta w_{i}+v_{i} \\ \mathrm{E}\left(v_{i} | w_{i}\right) &=0 \end{aligned} $$ ii. In the setting of part (i), how would you estimate \(\beta\) (and \(\alpha\) ) given a random sample? Justify your answer: iii. Now suppose that \(w_{i}\) is possibly correlated with \(v_{i},\) but for a set of observed variables \(x_{y,}\) $$ \mathbf{E}\left(v_{i} | w_{i}, x_{i 1}, \ldots, x_{i k}\right)=\mathrm{E}\left(v_{i} | x_{i 1}, \ldots, x_{i k}\right)=\eta+\gamma_{1} x_{i 1}+\cdots+\gamma_{k} x_{i k} $$ The first equality holds if \(w_{i}\) is independent of \(v_{i}\) conditional on \(\left(x_{i}, \ldots, x_{i k}\right)\) and the second equality assumes a linear relationship. Show that we can write $$ \begin{aligned} & y_{i}=\psi+\beta w_{i}+\gamma_{1} x_{i 1}+\cdots+\gamma_{k} x_{i k}+u_{i} \\\ \mathrm{E}\left(u_{i} | w_{i}, x_{i 1}, \ldots, x_{i k}\right) &=0 \end{aligned} $$ What is the intercept \(\psi ?\) iv. How would you estimate \(\beta\) (along with \(\psi\) and the \(\gamma_{j}\) ) in part (iii)? Explain.

Short answer

i. Use the equation \( y_i = \alpha + \beta w_i + v_i \). ii. Use OLS to estimate \( \alpha \) and \( \beta \). iii. Formulate \( y_i = \psi + \beta w_i + \gamma_j x_{ij} + u_i \) with \( \psi = \alpha - \eta \). iv. Use multiple regression to estimate \( \beta \), \( \psi \), and \( \gamma_j \).

Step-by-step solution

Understanding Initial Conditions (i)

We start with the equation given: \[ y(w) = \alpha + \beta w + v(0) \]Given that \( y(0) = \alpha + v \), it implies that \( v(0) \) is simply \( v \) as \( w = 0 \). Therefore, for a random individual \( i \):\[ y_i = \alpha + \beta w_i + v_i \]The expected error term conditioned on the policy variable \( w_i \) is zero because \( v_i \) is independent of \( w_i \):\[ \mathrm{E}(v_i \mid w_i) = 0 \]

Estimating Parameters Under Independence (ii)

Since \( v_i \) is independent of \( w_i \) and has an expected value of zero, the linear relationship holds and we can use ordinary least squares (OLS) to estimate \( \beta \) and \( \alpha \). OLS provides unbiased and consistent estimates in this setting.

Incorporating Observed Variables (iii)

Now, assume \( w_i \) might be correlated with \( v_i \), but this can be controlled for observed variables \( x_{i1}, \ldots, x_{ik} \).Given:\[ \mathbf{E}(v_i \mid w_i, x_{i1}, \ldots, x_{ik}) = \eta + \gamma_1 x_{i1} + \cdots + \gamma_k x_{ik} \]Replacing \( v_i \) in the outcome model:\[ y_i = \psi + \beta w_i + \gamma_1 x_{i1} + \cdots + \gamma_k x_{ik} + u_i \]where \( u_i = v_i - \eta - \sum_{j=1}^k \gamma_j x_{ij} \) and \( \mathrm{E}(u_i \mid w_i, x_{i1}, \ldots, x_{ik}) = 0 \).\( \psi \) is defined as \( \alpha \ - \eta \).

Estimating Parameters with Controls (iv)

To estimate \( \beta \), \( \psi \), and the \( \gamma_j \) coefficients in the presence of control variables \( x_{ij} \), we apply multiple regression analysis using OLS. By doing so, we can control for the effects of the additional variables \( x_{ij} \) and ensure estimates are not biased by relevant observed variables. This approach provides consistent estimates when \( u_i \) meets the condition of having an expected value of zero given \( w_i \) and the \( x_{ij} \).

Key concepts

Potential Outcomes Framework

The potential outcomes framework is a powerful concept often used in econometrics to evaluate the effect of a policy intervention. Imagine a scenario where you want to determine how different levels of financial grants can impact a student's GPA.

• Each possible grant amount represents a different policy level, denoted by \(w\).
• The corresponding outcome for each level, such as the student's GPA, is represented by \(y(w)\).

This framework helps economists assess causal effects by comparing potential outcomes under different treatment scenarios. Think of it this way:- If one group of students received a \(\$500\) grant and another group received none, the framework enables comparison of the potential GPAs in both scenarios.
By considering the independent potential outcomes for every policy level, this approach provides a structured way to understand the impacts of interventions in a real-world setting.

Linear Regression

Linear regression is a statistical method that models relationships between variables using a straight line. In the context of our exercise, the relationship between the policy variable (grant amount \(w\)) and the outcome (GPA \(y(w)\)) is assumed to be linear.
The equation follows this form:\[ y(w) = \alpha + \beta w + v(0) \] Here:

• \(\alpha\) is the intercept, representing the expected GPA when no grant is given.
• \(\beta\) is the slope, indicating how much the GPA is expected to change for each additional dollar in the grant.
• \(v(0)\) represents the error term or any other factors affecting GPA that aren't explained by \(w\).

By fitting a line through collected data points, linear regression helps predict trends and make informed decisions on policy outcomes.

Ordinary Least Squares

Ordinary least squares (OLS) is a technique used to estimate the parameters of a linear regression model. The goal of OLS is to minimize the sum of the squares of the differences between the observed and predicted values.
In other words, OLS finds the line that best fits the data by reducing the error in prediction as much as possible.
Using OLS, you can estimate the coefficients \(\alpha\) and \(\beta\) by solving:\[\text{minimize} \sum (y_i - (\alpha + \beta w_i))^2\]

• Where \(y_i\) is the observed outcome.
• \(w_i\) is the policy variable for each observation.

OLS is particularly useful due to its properties of providing unbiased, consistent estimates under the assumption that the error term is uncorrelated with \(w\). This makes it a robust method for understanding the impact of policy levels on outcomes.

Policy Evaluation

Policy evaluation involves the assessment of the effect of a policy on an outcome, using econometric methods. By establishing a causal link between policy variables (like grant amounts) and the observed outcome (such as GPA), economists can determine the effectiveness of interventions.
In practice, policy evaluation looks at how different levels of an intervention impact the outcome of interest. Here’s how to approach it:

• Identify and measure the policy variable and its potential outcomes.
• Use models like linear regression to determine relationships and apply techniques like OLS for estimation.
• Control for any other variables that could affect the outcome to isolate the causal impact of the policy.

The ultimate aim of policy evaluation is to provide insights that inform decision-making, ensuring resources are invested in the most effective strategies to achieve desired outcomes.