Question

For a child \(i\) living in a particular school district, let voucher, be a dummy variable equal to one if a child is selected to participate in a school voucher program, and let \(s\)core\(_{i}\) be that child's score on a subsequent standardized exam. Suppose that the participation variable, voucher, is completely randomized in the sense that it is independent of both observed and unobserved factors that can affect the test score. i. If you run a simple regression score, on voucher, using a random sample of size \(n\), does the OLS estimator provide an unbiased estimator of the effect of the voucher program? ii. Suppose you can collect additional background information, such as family income, family structure (e.g. whether the child lives with both parents), and parents' education levels. Do you need to control for these factors to obtain an unbiased estimator of the effects of the voucher program? Explain. iii. Why should you include the family background variables in the regression? Is there a situation in which you would not include the background variables?

Short answer

i. Yes, OLS is unbiased due to randomization. ii. No need to control for observed variables for unbiasedness. iii. Control variables increase precision unless they do not help.

Step-by-step solution

Understand the Setup and Question i

The problem involves determining if the OLS (Ordinary Least Squares) estimator provides an unbiased estimator of the effect of a school voucher program on standardized test scores when the voucher is a completely randomized variable. Since the voucher assignment is completely independent of both observed and unobserved variables, the correlation between voucher and any error term is zero. This means the OLS estimator is unbiased in this setting.

Analyze Question i

Since the voucher variable is randomized, the only difference between the treated (voucher = 1) and untreated (voucher = 0) groups can be attributed to the voucher itself, eliminating any confounding variables. Thus, the OLS estimator is unbiased due to the random assignment of the voucher.

Evaluate Question ii

The prompt asks if additional background information like family income, structure, or parents' education need to be controlled for to achieve an unbiased estimator. Since the participation variable (voucher) is randomized and thus independent of all these background variables, there's no need to control for them to achieve unbiasedness. The randomization ensures any differences between groups are due to the voucher program.

Discuss Importance for Question iii

Even though controlling for family background variables isn't necessary for unbiasedness due to randomization, including them can increase the precision of the estimates by reducing the variance of the error term. In a situation where these controls do not improve precision or if randomization is compromised, one might not include them.

Key concepts

OLS Estimator

The Ordinary Least Squares (OLS) estimator is a tool used in linear regression analysis to estimate the relationship between variables. In the context of our school voucher program, the OLS estimator helps us understand how a voucher, as an independent variable, impacts the test score, which is the dependent variable.
OLS works by minimizing the sum of the squared differences between the actual data points and the estimated values generated by the regression line. This method ensures that the line of best fit is achieved, providing an estimate of the effect of the voucher on the test scores.
When the relationship between the independent and dependent variables is truly linear, and there is no perfect multicollinearity, the OLS estimator gives the Best Linear Unbiased Estimator (BLUE). It means OLS is the most precise estimation method possible under certain conditions. Some of these conditions include a linear relationship between variables, random sampling, and no perfect correlation among explanatory variables.

• Linear Relationship: Ensures the effects of variables are directly proportional.
• Random Sampling: The sample is randomly selected from the population.
• No Perfect Multicollinearity: Independent variables are not perfectly linear functions of each other.

In the case of the school voucher program, because the voucher allocation is randomized and independent of other factors, this ensures the OLS estimator is unbiased.

Randomized Experiments

Randomized experiments are an essential tool in statistical research for establishing causal relationships while minimizing bias. In the given exercise, the school voucher program is a perfect example of a randomized experiment since the method of assigning vouchers eliminates any influence of pre-existing differences among participants.
By randomized assignment, each child has an equal chance of receiving a voucher, and this randomness means that any variations in test scores can be directly attributed to the voucher program itself. Randomized experiments are powerful because they prevent selection bias and control confounding variables from affecting the results.
This approach is particularly advantageous for ensuring that the results are due to the treatment (in this case, the school vouchers) rather than other factors. Since the voucher assignments do not correlate with any observed or unobserved influences on test scores, any differences in outcomes can be confidently linked to the program itself.
Overall, randomized experiments like these provide:

• Elimination of Selection Bias: Ensures participants are randomly assigned to treatment or control groups.
• Control of Confounding Variables: Reduces the impact of external factors unrelated to the treatment.
• Facilitation of Causal Inference: Strengthens the ability to make causal claims regarding the treatment's effect.

Confounding Variables

Confounding variables are external influences that can skew the results of a study by being associated with both the independent and dependent variables. They are critical to consider in any study design, as they can lead to incorrect conclusions.
In the context of the given school voucher exercise, if the vouchers were not randomly assigned, various socio-economic factors like family income or parental education could act as confounding variables. These variables might affect both the likelihood of receiving a voucher and the children's performance on standardized tests, thus muddling the true impact of the voucher program.
However, the randomization in voucher assignment inherently controls for these confounding variables. Because of this randomization, any differences in test scores can be attributed directly to the effect of receiving the voucher. In situations where randomization isn't possible, researchers often need to use statistical controls or additional modeling techniques to adjust for the influence of potential confounders.
Key characteristics of confounding variables include:

• They make it difficult to ascertain the direct cause-and-effect relationship between treatment and outcome.
• They result in spurious associations if not properly controlled.
• They need adjustment through experimental design or statistical methods when randomization isn't feasible.

Unbiased Estimation

Unbiased estimation implies that the expected value of the estimated parameter is equal to the true parameter value of the population. In simple terms, an estimator is unbiased if, across many samples, it correctly estimates the population parameter without systematic errors or bias.
In the case of the school voucher program, since the voucher assignment is randomized, the OLS estimator remains unbiased. This is because randomization ensures there's no correlation between the voucher variable and unmeasured factors affecting the test score. Thus, the estimation correctly reflects the true impact of the voucher, free from biases.
It's crucial to have unbiased estimations in econometrics because they provide credible, reliable information necessary for policy and decision-making. Without bias, estimates reflect true relationships without distortions from confounding factors.
To ensure unbiased estimation, particularly in observational studies where randomization is not possible, controlling for confounding variables and using robust estimation techniques become vital. In randomized controlled trials, like our exercise scenario, unbiased estimation comes almost naturally due to the way randomization neutralizes potential biases, leading to:

• Accurate Reflection of the True Population Parameter: The estimated value matches the true value.
• Free from Systematic Error: No consistent bias in the estimation process.
• Ensures Reliable Policy Implications: Provides dependable data for informing policy and decision-making.