Question

Suppose that the model $$\text {pctstck}=\beta_{0}+\beta_{1} \text {funds}+\beta_{2} \text {risktol}+u$$ satisfies the first four Gauss-Markov assumptions, where \(p c t s t c k\) is the percentage of a worker's pension invested in the stock market, funds is the number of mutual funds that the worker can choose from, and risktol is some measure of risk tolerance (larger risktol means the person has a higher tolerance for risk). If funds and risktol are positively correlated, what is the inconsistency in \(\tilde{\beta}_{1},\) the slope coefficient in the simple regression of \(p c t s t c k\) on funds?

Short answer

\( \tilde{\beta}_{1} \) is biased upwards due to the positive correlation between funds and risktol.

Step-by-step solution

Understanding the problem

We are given a regression model with the dependent variable \( \text{pctstck} \), the percentage of a worker's pension invested in the stock market, predicted by two independent variables: \( \text{funds} \) (number of mutual funds) and \( \text{risktol} \) (a measure of risk tolerance). We need to find the inconsistency in \( \tilde{\beta}_{1} \), the coefficient of a simple regression of \( \text{pctstck} \) on \( \text{funds} \), assuming funds and risktol are positively correlated.

Setting up the Simple Regression Model

Consider a simple regression model where \( \text{pctstck} = \alpha_0 + \tilde{\beta}_{1} \text{funds} + v \). Here, \( v \) represents the error term that captures all omitted factors, including \( \text{risktol} \).

Expressing the Error Term

Since \( \text{risktol} \) is omitted from the simple regression model, the error term \( v \) can be written as \( \beta_2 \text{risktol} + u \), where \( u \) is the original error term in the multiple regression model.

Using the Omitted Variable Bias Formula

The omitted variable bias formula states that the bias in \( \tilde{\beta}_{1} \) is given by \( \beta_2 \times \text{Cov}(\text{funds}, \text{risktol}) / \text{Var}(\text{funds}) \). Since \( \text{funds} \) and \( \text{risktol} \) are positively correlated, \( \text{Cov}(\text{funds}, \text{risktol}) > 0 \).

Determining the Sign and Magnitude of the Inconsistency

Given that \( \beta_2 > 0 \) (because \( \text{risktol} \) is expected to positively affect \( \text{pctstck} \)), and \( \text{Cov}(\text{funds}, \text{risktol}) > 0 \), the term \( \beta_2 \times \text{Cov}(\text{funds}, \text{risktol}) \) is positive. Thus, \( \tilde{\beta}_{1} \) is biased upwards in the simple regression.

Key concepts

Gauss-Markov Assumptions

The Gauss-Markov assumptions are foundational in econometrics. They ensure that estimators have desirable properties under ordinary least squares (OLS) regression. These assumptions underpin the best, linear unbiased estimator (BLUE) theorem. There are five key assumptions, but here we'll highlight the first four:

• Linearity: The relationship between dependent and independent variables is linear.
• Random Sampling: Observations are randomly drawn from the population.
• No Perfect Multicollinearity: Independent variables are not perfectly linearly related.
• Zero Conditional Mean: The expected value of the error term ( u ) given any value of the independent variables is zero.

Understanding these assumptions helps in evaluating the validity of OLS estimates.
When any assumption is violated, it may lead to biased estimators.

Omitted Variable Bias

Omitted Variable Bias (OVB) occurs when a model leaves out one or more relevant variables, causing the estimated coefficients to be biased. This bias arises because the omitted variable is correlated with both the dependent and included independent variables.

In our exercise, the variable 'risktol' is omitted, leading to bias in the estimated coefficient of 'funds'. The omitted variable bias formula is:\[ \text{Bias}(\tilde{\beta}_1) = \beta_2 \cdot \frac{\text{Cov}(\text{funds}, \text{risktol})}{\text{Var}(\text{funds})} \]Given that 'funds' and 'risktol' are positively correlated, and 'risktol' positively affects 'pctstck', the bias is upward. Recognizing and addressing OVB is crucial in producing credible econometric analyses.

Simple Regression Model

A Simple Regression Model examines the relationship between two variables, typically a dependent variable ( Y ) and one independent variable ( X ). Its formula is:\[ Y = \alpha + \beta X + \epsilon \]where:

• \( \alpha \) is the intercept.
• \( \beta \) is the slope coefficient.
• \( \epsilon \) is the random error term.

The model assesses how changes in X impact Y , holding all else constant.

In the context of our exercise, the simple regression simplifies from a multiple regression by focusing solely on 'funds'. Such simplification, however, introduces risks like omitted variable bias.

Error Term

The Error Term ( u ) in an econometric model captures all unobserved factors affecting the dependent variable that are not included in the model. In multiple regression models, the error term should be uncorrelated with the included independent variables if all Gauss-Markov assumptions hold.

In our exercise, when simplifying to a simple regression, the error term then becomes:\[ v = \beta_2 \text{risktol} + u \]This expression shows how 'risktol', being omitted, becomes part of the error term, potentially confounding the results and leading to omitted variable bias.
Careful attention to the role and composition of the error term is essential for accurate modeling and hypothesis testing in econometrics.