Question

Consider the equation $$\begin{aligned}y &=\beta_{0}+\beta_{1} x+\beta_{2} x^{2}+u \\\E(u | x) &=0\end{aligned}$$ where the explanatory variable \(x\) has a standard normal distribution in the population. In particular, \(\mathrm{E}(x)=0, \mathrm{E}\left(x^{2}\right)=\operatorname{Var}(x)=1,\) and \(\mathrm{E}\left(x^{3}\right)=0 .\) This last condition holds because the standard normal distribution is symmetric about zero. We want to study what we can say about the OLS estimator of \(\beta_{1}\) we omit \(x^{2}\) and compute the simple regression estimator of the intercept and slope. i. Show that we can write $$y=a_{0}+\beta_{1} x+v$$ where \(\mathbf{E}(v)=0 .\) In particular, find \(v\) and the new intercept, \(a_{0}.\) ii. Show that \(\mathrm{E}(v | x)\) depends on \(x\) unless \(\beta_{2}=0\). iii. Show that \(\operatorname{Cov}(x, v)=0\). iv. If \(\hat{\beta}_{1}\) is the slope coefficient from regression \(y_{i}\) on \(x_{i}\), is \(\widehat{\beta}_{1}\) consistent for \(\beta_{1}\) ? Is it unbiased? Explain. v. Argue that being able to estimate \(\beta_{1}\) has some value in the following sense: \(\beta_{1}\) is the partial effect of \(x\) on \(y\) evaluated at \(x=0,\) the average value of \(x\). vi. Explain why being able to consistently estimate \(\beta_{1}\) and \(\beta_{2}\) is more valuable than just estimating \(\beta_{1}\).

Short answer

\(\hat{\beta}_{1}\) is inconsistent and biased unless \(\beta_{2} = 0\). Estimating both \(\beta_{1}\) and \(\beta_{2}\) captures the true relationship.

Step-by-step solution

Expressing y in terms of a new intercept and error term

We start by considering the given quadratic equation: \[ y = \beta_{0} + \beta_{1}x + \beta_{2}x^2 + u \] Since the simple regression omits \(x^2\), we substitute back to express \(y\) as a linear function of \(x\) and a new error term. We define \(v = \beta_{2}x^2 + u\), where the mean \(E(v|x) = \beta_{2}x^2\) and the new intercept \(a_0 = \beta_{0}\). Thus, we can write:\[ y = a_0 + \beta_{1}x + v \] Here, \(a_0 = \beta_{0}\) and the error term \(v\) includes the omitted \(x^2\).

Checking if E(v|x) depends on x

Since \( v = \beta_{2}x^2 + u \) and \( u \) has \(E(u|x) = 0\), we find \(E(v|x) = \beta_{2}x^2\), which depends on \(x\) unless \(\beta_{2} = 0\). If \(\beta_{2} = 0\), \(v = u\), and thus \(E(v|x) = 0\). Therefore, \(E(v|x)\) depends on \(x\) if \(\beta_{2} eq 0\).

Showing Cov(x, v) = 0

The covariance \(\operatorname{Cov}(x, v)\) is calculated as \(\operatorname{Cov}(x, \beta_{2}x^2 + u)\). Since \(\operatorname{Cov}(x, u) = 0\) due to \(E(u|x) = 0\), we only need to find \(\operatorname{Cov}(x, \beta_{2}x^2)\). With \(\beta_{2}\) as a constant, \(\operatorname{Cov}(x, x^2) = \beta_{2}E(x^3) = 0\) as \(E(x^3) = 0\). Hence, \(\operatorname{Cov}(x, v) = 0\).

Evaluating whether \(\hat{\beta}_{1}\) is consistent and unbiased

For \(\hat{\beta}_{1}\) to be consistent, \(E(v|x) = E(\beta_{2}x^2 + u|x)\) should be zero, which happens if \(\beta_{2} = 0\). Thus, \(\hat{\beta}_{1}\) is inconsistent when \(\beta_{2} eq 0\) and is unbiased only if \(\beta_{2} = 0\). Otherwise, it captures both \(\beta_{1}\) and a bias due to omitted \(x^2\).

Understanding the value of estimating \(\beta_{1}\)

The coefficient \(\beta_{1}\) represents the slope of \(y\) with respect to \(x\) when evaluated at the mean of \(x = 0\). It's a measure of the linear effect of \(x\) on \(y\) at this point, providing partial information even in the presence of omitted \(x^2\).

Value of estimating both \(\beta_{1}\) and \(\beta_{2}\)

Estimating both \(\beta_{1}\) and \(\beta_{2}\) allows for capturing the true shape of the quadratic relationship between \(x\) and \(y\), thus providing a more accurate and unbiased understanding of how \(x\) affects \(y\). It allows us to differentiate between linear and non-linear effects of \(x\) on \(y\).

Key concepts

Regression Analysis

Regression analysis is a powerful statistical tool used to understand the relationship between different variables. In the context of our exercise, we are attempting to understand how the variable \( y \) changes as \( x \) increases or decreases, specifically when the relationship includes quadratic terms.
The exercise begins with the quadratic equation \( y = \beta_{0} + \beta_{1}x + \beta_{2}x^2 + u \), where \( u \) is an error term. The objective is to understand how well the linear regression model estimates \( \beta_{1} \), especially when the quadratic term \( x^2 \) is omitted.

• Ignoring \( x^2 \) may lead to biased estimates of \( \beta_{1} \), since the omitted term could have a significant effect on \( y \).
• We are interested in testing whether the relationship is truly linear or if it requires the quadratic term for accuracy.

Changing \( v \) to account for \( \beta_{2}x^2 \) and checking \( E(v|x) \) helps us see how omitting variables impacts regression.

Quadratic Equations

Quadratic equations, like the ones used in this exercise, consist of variables raised to the power of two. In our equation, the term \( \beta_{2}x^2 \) is crucial. It captures the curvature in the relationship between \( x \) and \( y \).
Quadratic equations are often necessary when the effect of \( x \) on \( y \) isn't just a simple straight line (linear relation). When we ignore \( \beta_{2}x^2 \), we assume a linear relationship, which may not always hold true.

• In reality, many relationships are non-linear and require higher-order terms to accurately model their behavior.
• By seeing \( \beta_{1}x \) and \( \beta_{2}x^2 \) together in the equation, we acknowledge both the linear and non-linear aspects of the relationship.

For instance, if \( \beta_{2} eq 0 \), the quadratic term has a tangible effect, indicating the presence of curvature that must be part of the analysis.

Statistical Consistency

Statistical consistency relates to how well our estimators perform as the size of our data increases. An estimator is consistent if it produces closer and closer estimates to the true value, given more data.
In the problem, we examine whether the estimator \( \hat{\beta}_{1} \) is consistent when we omit \( x^2 \).

• When \( \beta_{2} = 0 \), \( \hat{\beta}_{1} \) is consistent as \( v = u \) and \( E(v|x) = 0 \).
• If \( \beta_{2} eq 0 \), \( \hat{\beta}_{1} \) is not consistent, as the omitted \( x^2 \) introduces a systematic error in \( \hat{\beta}_{1} \).

Consistency is important because it ensures that results can be trusted to be accurate reflections of the true underlying relationship, especially when using regression analysis for predictions or testing.

Estimator Bias

Estimator bias occurs when the expected value of an estimator does not equal the true value of the parameter being estimated. Bias indicates systematic error in the estimator, hindering our ability to make accurate predictions.
In our exercise, the bias in \( \hat{\beta}_{1} \) stems from omitting the \( x^2 \) term.

• If \( \beta_{2} eq 0 \), then \( E(v|x) = \beta_{2}x^2 \), causing the estimation of \( \beta_{1} \) to be biased because it doesn't fully account for the quadratic relationship.
• To obtain unbiased results, all relevant variables must be included in the regression model.

Therefore, the presence of bias means \( \hat{\beta}_{1} \) doesn't illustrate the true partial effect of \( x \) on \( y \), especially when \( x^2 \) plays a role, reinforcing the need for complete specification of the model.