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Chapter 8: Problem 1

Which of the following are consequences of heteroskedasticity? i. The OLS estimators, \(\hat{\beta}_{j},\) are inconsistent. ii. The usual \(F\) statistic no longer has an \(F\) distribution. iii. The OLS estimators are no longer BLUE.

Short Answer

Expert verified
The consequences of heteroskedasticity are (ii) and (iii): the F statistic no longer has an F distribution, and OLS estimators are not BLUE.

Step by step solution

01

Understand Heteroskedasticity

Heteroskedasticity in a regression model occurs when the variance of the error terms is not constant across observations. This violates one of the assumptions of the ordinary least squares (OLS) method.
02

Evaluate Consistency of OLS Estimators

Evaluate point (i): The consistency of OLS estimators refers to their ability to converge to the true parameter values as the sample size increases. Heteroskedasticity does not affect the consistency of OLS estimators, provided other assumptions (such as no omitted variables) are met. Therefore, OLS estimators remain consistent under heteroskedasticity.
03

Assess the F Statistic's Distribution

Evaluate point (ii): The usual F statistic tests for overall significance of the regression model. Under heteroskedasticity, the errors do not follow a simple form necessary for the F statistic to maintain an F distribution. Thus, the presence of heteroskedasticity can lead to an invalid F statistic because its distribution is altered.
04

Discuss BLUE Property of OLS Estimators

Evaluate point (iii): BLUE refers to Best Linear Unbiased Estimators. When heteroskedasticity is present, OLS estimators are still unbiased and consistent, but they are no longer the best because they do not have minimum variance. This means OLS is not the most efficient among linear estimators when heteroskedasticity exists.

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Key Concepts

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

OLS Estimators
Ordinary Least Squares (OLS) estimators are a fundamental tool in regression analysis. They are used to determine the relationship between dependent and independent variables. In statistical terms, OLS estimators find the regression coefficients that minimize the sum of squared residuals (errors) between observed and predicted values. This minimization provides the line of best fit for the given data.

OLS estimators work well under certain assumptions, one of which is homoskedasticity – constant variance in the error terms. When this assumption is violated, leading to heteroskedasticity, the variability of the error terms across observations changes. Despite this, OLS estimators remain consistent; they still converge to the true parameter values if the sample size is large enough. Therefore, heteroskedasticity does not make the OLS estimators inconsistent, but it does affect other desirable properties, which we'll explore further.

It's crucial to check for heteroskedasticity in your data. If present, using heteroskedasticity-consistent standard errors is advised to get reliable estimates.
Consistency of Estimators
Consistency is a key property of estimators in statistics. An estimator is consistent if it converges to the true parameter value of the population as the sample size increases towards infinity. In simple terms, consistency means that more data leads to more accurate estimates.

For the OLS estimator, even if heteroskedasticity is present, it remains consistent. This is because consistency mainly depends on other factors, such as having no omitted variables and correctly specifying the relationship between variables.

So, while heteroskedasticity can cause inefficiency (which impacts the precision and variance of estimators), it does not undermine their ability to consistently estimate the true parameters.
F Statistic
The F statistic is used to test the overall significance of a regression model. It helps evaluate whether the model as a whole explains a significant portion of the variability in the data.

Under the assumption of homoskedasticity, the F statistic follows a specific distribution known as the F distribution. However, if heteroskedasticity is present, this assumption breaks down, and the traditional F statistic may no longer follow the F distribution. This misalignment can lead to incorrect conclusions regarding the significance of the model.

To address this problem, one can use robust statistical techniques that adjust for heteroskedasticity, allowing for more accurate hypothesis testing.
BLUE Property
The BLUE property stands for Best Linear Unbiased Estimators. When estimators are BLUE, it means they are the best (most efficient) among all unbiased linear estimators, provided certain assumptions hold.

OLS estimators are BLUE if the errors are homoskedastic and uncorrelated. Under heteroskedasticity, while OLS estimators remain linear and unbiased, they lose the property of being the 'best' because they no longer have the smallest variance among all unbiased estimators.

This inefficiency may not heavily impact the estimates themselves but can affect the confidence interval's width and hypothesis testing about the coefficients. Using methods like Weighted Least Squares (WLS) or heteroskedasticity-robust standard errors can recover efficiency.
Regression Analysis
Regression analysis is a statistical method for exploring and describing relationships among variables. The primary purpose of regression analysis is to model the relationship between a dependent variable and one or more independent variables.

You start by positing a tentative relationship and using data to estimate the parameters of this relationship. In simple regression, you typically look for how a single predictor affects an outcome. In multiple regression, you can see how several factors collectively influence the outcome.

Performing regression analysis requires making certain assumptions about the data, such as linearity, independence, homoskedasticity, and normality of errors. Violation of these assumptions can lead to inaccurate results, which is why diagnostics like checking for heteroskedasticity are essential.

By understanding each component and ensuring that assumptions are met, regression analysis can provide powerful insights into data behavior and relationships.

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Most popular questions from this chapter

Consider a model at the employee level, $$y_{i, e}=\beta_{0}+\beta_{1} x_{i, e, 1}+\beta_{2} x_{i, e, 2}+\cdots+\beta_{k} x_{i, e, k}+f_{i}+v_{i, e}$$ where the unobserved variable \(f_{i}\) is a "firm effect" to each employee at a given firm \(i .\) The error term \(v_{i, e}\) is specific to employee \(e\) at firm \(i .\) The composite error is \(u_{i, e}=f_{i}+v_{i, e},\) such as in equation (8.28) i. Assume that \(\operatorname{Var}\left(f_{i}\right)=\sigma_{f}^{2}, \operatorname{Var}\left(v_{i, e}\right)=\sigma_{v}^{2},\) and \(f_{i}\) and \(v_{i, e}\) are uncorrelated. Show that \(\operatorname{Var}\left(u_{i, e}\right)=\sigma_{f}^{2}+\sigma_{v}^{2} ;\) call this \(\sigma^{2}\) ii. Now suppose that for \(e \neq g, v_{i, e}\) and \(v_{i, g}\) are uncorrelated. Show that \(\operatorname{Cov}\left(u_{i, e}, u_{i, g}\right)=\sigma_{f}^{2}\) iii. Let \(\bar{u}_{i}=m_{i}^{-1} \sum_{i=1}^{m i} u_{i, e}\) be the average of the composite errors within a firm. Show that $$\operatorname{Var}\left(\bar{u}_{i}\right)=\sigma_{f}^{2}+\sigma_{v}^{2} / m_{i}$$. iv. Discuss the relevance of part (iii) for WLS estimation using data averaged at the firm level, where the weight used for observation \(i\) is the usual firm size.

There are different ways to combine features of the Breusch-Pagan and White tests for heteroskedasticity. One possibility not covered in the text is to run the regression $$\widehat{u}_{i}^{2} \text { on } x_{i 1}, x_{i 2}, \dots, x_{i k}, \hat{y}_{i}^{2}, i=1, \dots, n$$ where the \(\hat{u}_{i}\) are the OLS residuals and the \(\hat{y}_{i}\) are the OLS fitted values. Then, we would test joint significance of \(x_{11}, x_{i 2}, \ldots, x_{i k}\) and \(\hat{y}_{i}^{2}\) (of course, we always include an intercept in this regression.) i. What are the \(d f\) associated with the proposed \(F\) test for heteroskedasticity? ii. Explain why the \(R\) -squared from the regression above will always be at least as large as the \(R\) squareds for the BP regression and the special case of the White test. iii. Does part (ii) imply that the new test always delivers a smaller \(p\) -value than either the BP or special case of the White statistic? Explain. iv. Suppose someone suggests also adding \(\hat{y}_{i}\) to the newly proposed test. What do you think of this idea?

Consider the potential outcomes framework, where \(w\) is a binary treatment indicator and the potential outcomes are \(y(0)\) and \(y(1) .\) Assume that \(w\) is randomly assigned, so that \(w\) is independent of \([y(0), y(1)] .\) Let \(\mu_{0}=\mathrm{E}[y(0)], \mu_{1}=\mathrm{E}[y(1)], \sigma_{0}^{2}=\operatorname{Var}[y(0)],\) and \(\sigma_{1}^{2}=\operatorname{Var}[y(1)]\). i. Define the observed outcome as \(y=(1-w) y(0)+w y(1) .\) Letting \(\tau=\mu_{1}-\mu_{0}\) be the average treatment effect, show you can write $$y=\mu_{0}+\tau w+(1-w) v(0)+w v(1)$$ where \(v(0)=y(0)-\mu_{0}\) and \(v(1)=y(1)-\mu_{1}\) ii. Let \(u=(1-w) v(0)+w v(1)\) be the error term in $$y=\mu_{0}+\tau w+u$$ Show that $$\mathbf{E}(u | w)=0$$ What statistical properties does this finding imply about the OLS estimator of \(\tau\) from the simply regression \(y_{i}\) on \(w_{i}\) for a random sample of size \(n\) ? What happens as \(n \rightarrow \infty ?\) iii. Show that $$\operatorname{Var}(u | w)=\mathrm{E}\left(u^{2} | w\right)=(1-w) \sigma_{0}^{2}+w \sigma_{1}^{2}$$ Is there generally heteroskedasticity in the error variance? iv. If you think \(\sigma_{1}^{2} \neq \sigma_{0}^{2},\) and \(\hat{\tau}\) is the OLS estimator, how would you obtain a valid standard error for \(7 ?\) v. After obtaining the OLS residuals, \(\widehat{u}_{i}, i=1, \ldots, n,\) propose a regression that allows consistent estimation of \(\sigma_{0}^{2}\) and \(\sigma_{1}^{2}\). [Hint: You should first square the residuals.]
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