Question

Let kids denote the number of children ever born to a woman, and let \(e d u c\) denote years of education for the woman. A simple model relating fertility to years of education is $$k i d s=\beta_{0}+\beta_{1} e d u c+u,$$ where \(u\) is the unobserved error. i. What kinds of factors are contained in \(u\) ? Are these likely to be correlated with level of education? ii. Will a simple regression analysis uncover the ceteris paribus effect of education on fertility? Explain.

Short answer

Factors in \(u\) include cultural norms and economic status, which may be correlated with education. Simple regression may not reveal the effect of education on fertility due to these correlations.

Step-by-step solution

Identify Factors in u

The error term \(u\) includes factors affecting fertility (number of children) that are not accounted for by education. These factors might include cultural beliefs about family size, economic status, access to family planning, personal or family health issues, and possible regional or community norms that influence family size.

Analyze Correlation with Education

Some factors in \(u\), such as economic status and cultural norms, may be correlated with education level. For example, families with higher economic status may have better access to education and may also choose to have fewer children due to career commitments, leading to a potential correlation between \(u\) and education.

Assess Simple Regression Analysis

A simple regression assumes that \(u\) is uncorrelated with the explanatory variable, education. If factors such as economic status or cultural norms are correlated with education, the simple regression may not isolate the ceteris paribus effect of education on fertility. The effect captured would include the influence of education as well as the correlated unobserved factors.

Key concepts

Ceteris Paribus

In the context of statistical studies like regression analysis, the term "ceteris paribus" is used to describe the effect of one variable on another while keeping all other influencing factors constant. Imagine you want to understand the true impact of education (\(educ\)) on fertility (the number of children, \(kids\)). You are interested in knowing how fertility changes with education when no other circumstances are changing. However, in the real world, achieving a true ceteris paribus condition is tricky. This is because numerous other factors can affect fertility choices, and they might also interact with a woman’s level of education. Some of these factors could be societal norms or economic situations.

A simple regression, as noted in the solution, might fail to capture this pure effect because it doesn't control these other influences. So, even if education appears to have a direct effect on fertility, without ceteris paribus, you cannot be completely confident about this relationship being direct.

Error Term in Regression

In a regression model, the error term \(u\) represents all other unobserved factors that influence the dependent variable, after accounting for the predictors in the model. In the fertility example, \(u\) captures all aspects affecting the number of children a woman has that are not explained purely by her years of education. This might include:

• Cultural beliefs and family traditions
• Economic status and personal finances
• Access to healthcare and family planning
• Personal health conditions
• Community or societal norms and values

These factors can complicate the relationship between education and fertility.

If any factors in the error term are correlated with education, this can lead to biased estimates in a regression analysis. For example, wealthier families might both have higher education levels and a preference for fewer children, meaning that economic status could distort the perceived effect of education on fertility. Identifying and addressing these correlations is crucial for accurate analysis.

Factors Affecting Fertility

Fertility decisions are influenced by a multitude of factors beyond just education level. Understanding these various elements helps provide a more comprehensive picture of what affects the number of children a woman decides to have:

• Economic Factors: Economic security and wealth can influence family size, often with wealthier individuals choosing to have fewer children.
• Cultural and Social Norms: In many societies, cultural expectations and traditions play a significant role in determining family size. Some cultures may encourage larger families, while others might lean toward smaller ones.
• Access to Healthcare: Adequate access to healthcare and family planning resources can directly affect fertility, as these provide more reproductive choices.
• Employment and Career Goals: Women's career ambitions might impact their fertility decisions, sometimes delaying having children or reducing the overall number of children.
• Education Level: Higher education levels often lead to later family formation and potentially smaller families, due to factors such as career focus and delayed marriage.
• Personal Preferences and Health: Personal desires and health conditions can also shape family planning and fertility outcomes.

These factors, potentially interacting with one another, create complex patterns in fertility outcomes. Addressing these in regression models is essential for capturing the real effect of each, particularly when trying to isolate the impact of education.