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Chapter 3: Problem 11

Suppose that the population model determining \(y\) is $$ y=\beta_{0}+\beta_{1} x_{1}+\beta_{2} x_{2}+\beta_{3} x_{3}+u $$ and this model satisfies Assumptions MLR.1, MLR.2, MLR.3 and MLR.4. However, we estimate the model that omits \(x_{3} .\) Let \(\bar{\beta}_{0}, \bar{\beta}_{1},\) and \(\bar{\beta}_{2}\) be the OLS estimators from the regression of \(y\) on \(x_{1}\) and \(x_{2}\) Show that the expected value of \(\tilde{\beta}_{1}\) (given the values of the independent variables in the sample) is $$\mathbf{E}\left(\tilde{\beta}_{1}\right)=\beta_{1}+\beta_{3} \frac{\sum_{i=1}^{n} \hat{r}_{i 1} x_{i 3}}{\sum_{i=1}^{n} \hat{r}_{i 1}^{2}}$$ where the \(\hat{r}_{i 1}\) are the OLS residuals from the regression of \(x_{1}\) on \(x_{2}\). [Hint: The formula for \(\tilde{\beta}_{1}\) comes from equation \((3.22) .\) Plug \(y_{i}=\beta_{0}+\beta_{1} x_{11}+\beta_{2} x_{12}+\beta_{3} x_{13}+u_{i}\) into this equation. After some algebra, take the expectation treating \(x_{i 3}\) and \(\tilde{r}_{i 1}\) as nonrandom.]

Short Answer

Expert verified
E(\tilde{\beta}_{1}) = \beta_{1} + \beta_{3} (\sum \hat{r}_{i1} x_{i3} / \sum \hat{r}_{i1}^2).

Step by step solution

01

Understand the Given Model

The given population model is: \[ y = \beta_{0} + \beta_{1} x_{1} + \beta_{2} x_{2} + \beta_{3} x_{3} + u \] We estimate this model omitting \(x_3\), producing the estimators \(\bar{\beta}_{0}, \bar{\beta}_{1}, \bar{\beta}_{2}\) for the regression of \(y\) on \(x_1\) and \(x_2\).
02

Write the Reduced Model

The reduced model without \(x_3\) is: \[ y_i = \bar{\beta}_{0} + \tilde{\beta}_{1} x_{i1} + \bar{\beta}_{2} x_{i2} + \tilde{u}_i \] where \(\tilde{u}_i = \beta_{3} x_{i3} + u_i\).
03

Rewriting Based on the Full Model

Substitute \(y_i\) from the full model into the reduced model equation: \[ y_i = \beta_{0} + \beta_{1} x_{i1} + \beta_{2} x_{i2} + \beta_{3} x_{i3} + u_i = \bar{\beta}_{0} + \tilde{\beta}_{1} x_{i1} + \bar{\beta}_{2} x_{i2} + \tilde{u}_i \] Reorganize this to focus on \(\tilde{\beta}_{1}\): \[ \tilde{\beta}_{1} x_{i1} = (\beta_{1} + \beta_{3} x_{i3} - \beta_{2} x_{i2}) x_{i1} + (u_i - \tilde{u}_i) + \bar{\beta}_{2} x_{i2} - \bar{\beta}_{0} \] This indicates a relationship that allows us to find the expectation of \(\tilde{\beta}_{1}\).
04

Find Expected Value of \(\tilde{\beta}_{1}\)

The formula for \(\tilde{\beta}_{1}\) involves projecting \(y\) onto \(x_1\) and computing an expectation: \[ \mathbf{E}(\tilde{\beta}_{1}) = \beta_{1} + \beta_{3} \frac{\sum_{i=1}^{n} \hat{r}_{i1} x_{i3}}{\sum_{i=1}^{n} \hat{r}_{i1}^{2}} \] where \(\hat{r}_{i1}\) are the OLS residuals obtained from the regression of \(x_1\) on \(x_2\). This step follows by recognizing the omitted variable bias represented by the correlation of \(x_1\) and \(x_3\) mediated by \(\hat{r}_{i1}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Ordinary Least Squares (OLS)
Ordinary Least Squares (OLS) is a method used in linear regression to estimate the parameters of a linear model. The goal of OLS is to minimize the sum of the squared differences between the observed values and the values predicted by the model. This method is widely used because it provides the best linear unbiased estimators (BLUE) when certain assumptions are met.

OLS works by fitting a line through data points in a way that the distance from the points to the line is as small as possible. This is often visualized in simple linear regression, where a line is drawn to represent the relationship between a dependent variable and an independent variable.
  • The basic idea is to find coefficients that minimize the discrepancy (residuals) between the actual data values and the values predicted by the linear model.
  • In the context of multiple regression, OLS extends to fitting a plane or a hyperplane in higher dimensions, attempting to capture relationships between one dependent variable and multiple independent variables.
OLS is also foundational to understanding concepts like omitted variable bias, as in the exercise, where one predictor variable is left out of the model.
Linear Regression Assumptions
For Ordinary Least Squares (OLS) estimators to be considered the best linear unbiased estimators (BLUE), certain assumptions about the linear regression model must be satisfied. These assumptions ensure that the OLS method produces reliable and valid results.
  • Linearity: The relationship between the independent and dependent variables should be linear.
  • Independence: The residuals (errors) should be independent across observations.
  • Homoscedasticity: The variance of the residuals should be constant across all levels of the independent variables.
  • No perfect multicollinearity: The independent variables should not be too highly correlated.
In the given exercise, the regression model originally satisfied these assumptions (MLR.1 to MLR.4). However, omitting one variable, such as \(x_3\), can lead to biases in estimators due to the violation of these assumptions. The omitted variable may cause omitted variable bias if it is correlated with both the dependent variable and one or more included variables, distorting the estimation of coefficients.
Expected Value of Estimators
The expected value of an estimator is one of the key properties that help us understand its effectiveness. In simple terms, an estimator is a rule or formula that tells us how to calculate an estimate from a given set of data. The expected value of an estimator is its average value over a large number of samples. For an unbiased estimator, this expected value equals the true parameter it estimates.

In the exercise, you were shown how the expected value of \(\tilde{\beta}_1\) is affected by the omission of \(x_3\). Instead of simply representing \(\beta_1\), the expected value of the estimator is shifted by a term that represents the influence of the omitted variable. This is an example of omitted variable bias:
  • Omitted Variable Bias occurs when a relevant variable is left out of the model, causing the estimator to capture not only the effect of the included variable but also the effect of the omitted variable.
  • The formula for \(\mathbf{E}(\tilde{\beta}_1) = \beta_1 + \beta_3 \frac{\sum_{i=1}^{n} \hat{r}_{i1} x_{i3}}{\sum_{i=1}^{n} \hat{r}_{i1}^{2}}\) shows how the bias is directly related to the residuals and the omitted variable.
Understanding the expected value of estimators helps in diagnosing and correcting for biases, leading to more accurate and truthful interpretations of regression analysis results.

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Most popular questions from this chapter

Suppose that you are interested in estimating the ceteris paribus relationship between \(y\) and \(x_{1}\). For this purpose, you can collect data on two control variables, \(x_{2}\) and \(x_{3}\). (For concreteness, you might think of \(y\) as final exam score, \(x_{1}\) as class attendance, \(x_{2}\) as GPA up through the previous semester, and \(x_{3}\) as SAT or ACT score. Let \(\tilde{\beta}_{1}\) be the simple regression estimate from \(y\) on \(x_{1}\) and let \(\hat{\beta}_{1}\) be the multiple regression estimate from \(y\) on \(x_{1}, x_{2}, x_{3}\) i. If \(x_{1}\) is highly correlated with \(x_{2}\) and \(x_{3}\) in the sample, and \(x_{2}\) and \(x_{3}\) have large partial effects on \(y,\) would you expect \(\bar{\beta}_{1}\) and \(\hat{\beta}_{1}\) to be similar or very different? Explain. ii. If \(x_{1}\) is almost uncorrelated with \(x_{2}\) and \(x_{3},\) but \(x_{2}\) and \(x_{3}\) are highly correlated, will \(\tilde{\beta}_{1}\) and \(\hat{\beta}_{1}\) tend to be similar or very different? Explain. iii. If \(x_{1}\) is highly correlated with \(x_{2}\) and \(x_{3}\), and \(x_{2}\) and \(x_{3}\) have small partial effects on \(y\), would you expect \(\operatorname{se}\left(\tilde{\beta}_{1}\right)\) or \(\operatorname{se}\left(\hat{\beta}_{1}\right)\) to be smaller? Explain. iv. If \(x_{1}\) is almost uncorrelated with \(x_{2}\) and \(x_{3}, x_{2}\) and \(x_{3}\) have large partial effects on \(y,\) and \(x_{2}\) and \(x_{3}\) are highly correlated, would you expect \(\operatorname{se}\left(\tilde{\beta}_{1}\right)\) or \(\operatorname{se}\left(\hat{\beta}_{1}\right)\) to be smaller? Explain.

Suppose that average worker productivity at manufacturing firms (avgprod) depends on two factors, average hours of training (avgtrain) and average worker ability (avgabil): $$\text { avgprod }=\beta_{0}+\beta_{1} \text { avgtrain }+\beta_{2} \text { avgabil }+u$$ Assume that this equation satisfies the Gauss-Markov assumptions. If grants have been given to firms whose workers have less than average ability, so that avgtrain and avgabil are negatively correlated, what is the likely bias in \(\tilde{\beta}_{1}\) obtained from the simple regression of avgprod on avgtrain?

The following model is a simplified version of the multiple regression model used by Biddle and Hamermesh (1990) to study the tradeoff between time spent sleeping and working and to look at other factors affecting sleep: $$\text { sleep }=\beta_{0}+\beta_{1} \text { totwr } k+\beta_{2} e d u c+\beta_{3} a g e+u$$ where sleep and totwrk (total work) are measured in minutes per week and educ and age are measured in years. (See also Computer Exercise \(\mathrm{C} 3\) in Chapter \(2 .\) ) i. If adults trade off sleep for work, what is the sign of \(\beta_{1} ?\) ii. What signs do you think \(\beta_{2}\) and \(\beta_{3}\) will have? iii. Using the data in SLEEP75, the estimated equation is $$\begin{aligned} \widehat{\text { sleep }} &=3,638.25-.148 \text { totwrk }-11.13 \text { educ }+2.20 \text { age } \\ n &=706, R^{2}=.113 . \end{aligned}$$ If someone works five more hours per week, by how many minutes is sleep predicted to fall? Is this a large tradeoff? iv. Discuss the sign and magnitude of the estimated coefficient on \(e d u c .\) v. Would you say totwrk, educ, and age explain much of the variation in sleep? What other factors might affect the time spent sleeping? Are these likely to be correlated with totwrk?

i. Consider the simple regression model \(y=\beta_{0}+\beta_{1} x+u\) under the first four Gauss-Markov assumptions. For some function \(g(x),\) for example \(g(x)=x^{2}\) or \(g(x)=\log \left(1+x^{2}\right),\) define \(z_{i}=g\left(x_{i}\right) .\) Define a slope estimator as $$ \tilde{\beta}_{1}=\left(\sum_{i=1}^{n}\left(z_{i}-\bar{z}\right) y_{i}\right) /\left(\sum_{i=1}^{n}\left(z_{i}-\bar{z}\right) x_{i}\right) $$ Show that \(\bar{\beta}_{1}\) is linear and unbiased. Remember, because \(\mathrm{E}(u | x)=0,\) you can treat both \(x_{i}\) and \(z_{i}\) as nonrandom in your derivation. ii. Add the homoskedasticity assumption, MLR.5. Show that $$ \operatorname{Var}\left(\tilde{\beta}_{1}\right)=\sigma^{2}\left(\sum_{i=1}^{n}\left(z_{i}-\bar{z}\right)^{2}\right) /\left(\sum_{i=1}^{n}\left(z_{i}-\bar{z}\right) x_{i}\right)^{2} $$ iii. Show directly that, under the Gauss-Markov assumptions, Var( \(\left.\hat{\beta}_{1}\right) \leq \operatorname{Var}\left(\tilde{\beta}_{1}\right),\) where \(\hat{\beta}_{1}\) is the OLS estimator. [Hint: The Cauchy-Schwartz inequality in Appendix B implies that $$ \left(n^{-1} \sum_{i=1}^{n}\left(z_{i}-\bar{z}\right)\left(x_{i}-\bar{x}\right)\right)^{2} \leq\left(n^{-1} \sum_{i=1}^{n}\left(z_{i}-\bar{z}\right)^{2}\right)\left(n^{-1} \sum_{i=1}^{n}\left(x_{i}-\bar{x}\right)^{2}\right) $$ notice that we can drop \(\bar{x}\) from the sample covariance.

Which of the following can cause OLS estimators to be biased? i. Heteroskedasticity. ii. Omitting an important variable. iii. A sample correlation coefficient of .95 between two independent variables both included in the model.
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