Question

Consider the equation. $$\begin{aligned}y &=\beta_{0}+\beta_{1} x+\beta_{2} x^{2}+u \\\\\mathrm{E}(u | x) &=0\end{aligned}$$ 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 if we omit \(x^{2}\) and compute the simple regression estimator of the intercept and slope. i. Show that we can write $$y=\alpha_{0}+\beta_{1} x+v$$ where \(E(v)=0 .\) In particular, find \(v\) and the new intercept, \(\alpha_{0}\) 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 \(\hat{\beta}_{1}\) 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 \(\mathrm{E}(y | x)\) 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

The new equation is \( y = \alpha_0 + \beta_1 x + v \) with \( v = \beta_2 x^2 + u \). \( \hat{\beta}_1 \) is biased unless \( \beta_2 = 0 \). Estimating both \( \beta_1 \) and \( \beta_2 \) is more valuable for complete model specification.

Step-by-step solution

- Rearrange the Original Equation

Start with the original equation \( y = \beta_0 + \beta_1 x + \beta_2 x^2 + u \). By comparing this with the desired form \( y = \alpha_0 + \beta_1 x + v \), we can express the terms \( \alpha_0 \) and \( v \) as follows: \( \alpha_0 = \beta_0 \) and \( v = \beta_2 x^2 + u \). This is because we isolate \( \beta_2 x^2 \) and \( u \) to create \( v \). Since \( E(u|x)=0 \), we need to ensure that in this transformation, \( E(v)=0 \) as well.

- Show \( E(v|x) \) depends on \( x \) unless \( \beta_2 = 0 \)

Assuming \( v = \beta_2 x^2 + u \), calculate \( E(v|x) = \beta_2 E(x^2|x) + E(u|x) = \beta_2 x^2 \). Since \( E(u|x) = 0 \) and \( E(x^2|x) = x^2 \), \( E(v|x) \) equals \( \beta_2 x^2 \). Thus, \( E(v|x) \) is a function of \( x \) when \( \beta_2 eq 0 \). If \( \beta_2 = 0 \), then \( E(v|x) = 0 \).

- Show \( \operatorname{Cov}(x, v) = 0 \)

Calculate \( \operatorname{Cov}(x, v) = \operatorname{Cov}(x, \beta_2 x^2 + u) \). This simplifies to \( \beta_2 \operatorname{Cov}(x, x^2) + \operatorname{Cov}(x, u) \). Because \( E(x^3)=0 \) and by assumption \( E(x^2|x)=x^2 \), \( \operatorname{Cov}(x, x^2) = E(x^3) - E(x)E(x^2) = 0 \). \( \operatorname{Cov}(x, u) = E(xu) - E(x)E(u) = 0 \) since \( E(u)=0 \). So \( \operatorname{Cov}(x, v) = 0 \).

- Discuss Consistency and Unbiasedness of \( \hat{\beta}_1 \)

Because \( E(v|x) \) isn't zero when \( \beta_2 eq 0 \), the omitted variable bias arises, and thus, \( \hat{\beta}_1 \) isn't consistent for \( \beta_1 \). Since \( \hat{\beta}_1 \) is biased unless \( \beta_2 = 0 \), it is typically not unbiased.

- Partial Effect of \( x \) on \( E(y|x) \) at \( x=0 \)

Here \( \beta_1 \) approximates the effect of small changes in \( x \) around zero. This provides some insights into the local linear approximation of the relationship between \( y \) and \( x \).

- Value of Consistently Estimating Both \( \beta_1 \) and \( \beta_2 \)

Estimating both allows understanding of both linear and quadratic effects of \( x \). Only estimating \( \beta_1 \) can miss these non-linear contributions, leading to a less comprehensive understanding of the model. This full estimation also ensures unbiased estimates and correctly specifies the model, thus avoiding omitted variable biases.

Key concepts

Ordinary Least Squares (OLS)

Ordinary Least Squares (OLS) is a method used in econometrics to estimate the parameters of a linear regression model. The goal is to find the best-fitting line through a set of data points by minimizing the sum of the squared differences between the observed values and those predicted by the model.

In our original equation, we consider multiple explanatory variables, including a quadratic term, and estimate them using OLS. This technique provides a way to see how variables do contribute to the dependent variable, without making any assumptions about the distribution of errors. It's essential in ensuring a linear approximation of the relationship between the independent variables and the dependent variable, giving us the most probable estimates for the parameters if the model is correctly specified.

Omitted Variable Bias

Omitted Variable Bias arises when a model leaves out relevant variables that influence both the dependent and independent variables. In our exercise, if the quadratic term \( \beta_2 x^2 \) is omitted, the error term will absorb these effects, causing bias in the remaining estimated coefficients.

This bias occurs because the omitted variable correlates with the included variables, leading to an inaccurate estimation of the coefficients. In our context, if \( \beta_2 eq 0 \), there's a bias because the regression assumes an incorrect model form, resulting in an inconsistent estimate for the ordinary coefficient \( \beta_1 \). Thus, failing to include \( x^2 \), which contributes to the relationship, can substantially affect the accuracy of our model.

Consistency and Unbiasedness

The concepts of consistency and unbiasedness are fundamental in regression analysis. An estimator is unbiased if its expected value equals the true population parameter, whereas it is consistent if it converges to the true value as the sample size increases.

In our scenario, because the expected value of \( v \) conditional on \( x \) is not zero unless \( \beta_2 = 0 \), \( \hat{\beta}_1 \) is not unbiased and is also not consistent when \( \beta_2 eq 0 \). The omitted variable bias introduced through the exclusion of \( x^2 \) prevents consistent estimation of \( \beta_1 \). When \( \beta_2 = 0 \), however, \( \hat{\beta}_1 \) could be an unbiased and consistent estimator for the coefficient \( \beta_1 \).

Partial Effect

The partial effect of an explanatory variable, such as \( x \), on the dependent variable \( y \), tells us how changes in \( x \) affect \( y \) holding other variables constant. In our example, \( \beta_1 \) represents this partial effect evaluated at \( x = 0 \).

This concept is vital because it provides insights into the local changes around the mean of \( x \). Even if the regression model is not perfectly specified, understanding the partial effect near the mean allows us to infer behavior within a small neighborhood, making it a useful analysis tool in many practical situations.

Quadratic Regression

Quadratic Regression extends the linear regression model by including a square of an independent variable, allowing for curvature in the relationship between the variables. This is crucial when the relationship between the explanatory and response variable is not strictly linear.

In our exercise, the inclusion of \( \beta_2 x^2 \) allows the model to capture these nonlinear dynamics between \( y \) and \( x \). Estimating both \( \beta_1 \) and \( \beta_2 \) gives a more complete understanding of how \( x \) influences \( y \), whether directly or through its square term. This comprehensive model specification corrects for any omitted variable bias ensuring the reliability of inferences made from the model.