Question

A simple model to determine the effectiveness of condom usage on reducing sexually transmitted diseases among sexually active high school students is $$\text { in frate }=\beta_{0}+\beta_{1} \text { conuse }+\beta_{2} \text { percmale }+\beta_{3} \text { avginc }+\beta_{4} \text { city }+u_{1},$$ where in frate \(=\) the percentage of sexually active students who have contracted venereal disease. conuse \(=\) the percentage of boys who claim to use condoms regularly. avginc = average family income. city = a dummy variable indicating whether a school is in a city. The model is at the school level. i. Interpreting the preceding equation in a causal, ceteris paribus fashion, what should be the sign of \(\beta_{1} ?\) ii. Why might infrate and conuse be jointly determined? iii. If condom usage increases with the rate of venereal disease, so that \(\gamma_{1}>0\) in the equation conuse \(=\gamma_{0}+\gamma_{1}\) in frate \(+\) other factors what is the likely bias in estimating \(\beta_{1}\) by OLS? iv. Let condis be a binary variable equal to unity if a school has a program to distribute condoms. Explain how this can be used to estimate \(\beta_{1}\) (and the other betas) by IV. What do we have to assume about condis in each equation?

Short answer

\( \beta_1 \) should be negative. Infrate and conuse are endogenously related. Estimating \( \beta_1 \) by OLS likely leads to positive bias. Use Condis as an IV for consistent estimation.

Step-by-step solution

Determining the Expected Sign for \( \beta_1 \)

The parameter \( \beta_1 \) shows the impact of condom use percentage on the infection rate, ceteris paribus. Since using condoms is associated with reducing the transmission of sexually transmitted diseases (STD), the expected sign for \( \beta_1 \) is negative. This means higher condom use should lower the infection rate.

Understanding Joint Determination of Variables

The variables infrate and conuse might be jointly determined because higher infection rates may lead to increased public health campaigns, which in turn encourage condom use. Conversely, more boys using condoms could directly affect the infection rate by lowering it. This interplay makes both variables potentially endogenous.

Bias in Estimating \( \beta_1 \) with Correlated Errors

If \( \gamma_1 > 0 \), an increase in infection rates causes an increase in condom use, introducing correlation between infrate and conuse. In Ordinary Least Squares (OLS) estimation, this correlation can lead to omitted variable bias or reverse causation bias, potentially causing \( \beta_1 \) to be overestimated (positive bias).

Using Instrumental Variables (IV) for Estimation

Condis, which indicates if a school has a program distributing condoms, can be used as an instrument for conuse in the model. For Condis to be a valid instrument, it must be correlated with conuse (relevance) and affect infrate only through conuse (exogeneity). If Condis satisfies these conditions, it can help in obtaining consistent estimates of \( \beta_1 \) and other parameters by accounting for the endogeneity in conuse.

Key concepts

Condom Usage

Condom usage is an important factor in the model. In essence, condoms act as a barrier to reduce the exchange of bodily fluids, which decreases the chances of sexually transmitted diseases (STDs) being contracted. When we're talking about "condom usage," we're referring to the percentage of boys who regularly use condoms. This number is crucial for understanding how behavioral practices can influence health outcomes in a school setting.
Condoms help in:

• Reducing transmission of STDs such as HIV, gonorrhea, and syphilis.
• Providing a safe method for sexually active individuals to protect themselves.
• Promoting safer sexual practices among young adults, which can have long-term cultural shifts in health behaviors.

Sexually Transmitted Diseases

Sexually transmitted diseases are infections that spread primarily through sexual contact. For high school students, awareness and prevention are key to reducing the prevalence of these diseases.
The model uses "infrate," which represents the percentage of students who have contracted an STD, to measure the severity of the issue at hand. Reducing the "infrate" is a primary goal of health initiatives implemented by schools and public health campaigns.
Key points regarding STDs include:

• Education about STDs is vital in prevention strategies.
• Encouraging the use of prevention methods such as condoms helps in controlling the spread.
• Early detection and treatment are critical.

Instrumental Variables

Instrumental variables (IV) are tools that help in uncovering causal relationships when there are potential endogeneity issues. In our exercise, the presence of school programs distributing condoms ("condis") can serve as an instrumental variable.
To use "condis" effectively, two main conditions must be met:

• Relevance: "Condis" should be correlated with "conuse," meaning schools distributing condoms would see higher condom usage rates.
• Exogeneity: "Condis" should only influence the "infrate" through its effect on "conuse," without any other pathways.

By satisfying these conditions, the use of IV allows for consistent estimation of causal effects, helping us to better understand the true impact of condom programs.

Endogeneity

Endogeneity occurs when there's a correlation between independent variables and the error term, often due to omitted variable bias or simultaneous causality. In the scenario given, there's a risk that "conuse" and "infrate" are jointly determined.
This can happen as:

• Higher disease rates might lead to increased condom usage due to behavioral changes prompted by health campaigns.
• Conversely, an increase in condom usage could lower the infection rates, forming a cyclical relationship.

Such scenarios pose challenges for obtaining unbiased estimates, making it difficult to determine the direct effect of condom usage on disease rates without properly addressing endogeneity.

Ordinary Least Squares

Ordinary Least Squares (OLS) is a method used to estimate the parameters in a linear regression model. It aims to minimize the sum of the squared differences between observed and predicted values.
However, OLS can face limitations such as:

• Bias due to Endogeneity: If there's a correlation between predictors and the error term, OLS estimates may be unreliable.
• Assumption of Ceteris Paribus: OLS assumes other variables remain constant, which might not always be true in real-world applications.

In the context of estimating the parameter \( \beta_1 \), OLS may not give a true measure of the effect of condom usage on STD rates due to potential endogeneity, requiring the use of IV for more accurate estimation.