Chapter 13 – Heteroskedasticity

\chapter{Heteroskedasticity\label{het.chap}}
In previous chapters it was assumed that the residuals in the regression model have a constant variance. In mathematical terms, this condition can be stated as:
\begin{equation*}
E(u_i^2)=\sigma ^2\text{ }(i=1,\ldots ,N)
\end{equation*}
This condition, however, is violated when \textbf{heteroskedasticity} is present. As noted in earlier chapters, residuals are heteroskedastic if the variance of the error terms is not constant across observations.
The causes and consequences of heteroskedasticity are discussed in this chapter. This is followed by a discussion of alternative methods of detecting and correcting for the presence of heteroskedasticity.
\section{Overview}
One of the assumptions of the classical regression model is that the variance of the residuals is the same for all observations. In particular, Assumption \ref{ind.dis} states:
\begin{quotation}
\textbf{Assumption \ref{ind.dis}}: The error terms are identically distributed with a constant variance equal to $\sigma ^2$ at all possible combinations of $X_{1i}$, $X_{2i}$, …, $X_{ki}$.
\end{quotation}
In practice, however, this homoskedasticity assumption is not always satisfied. Heteroskedasticity is present if the residuals have a variance that is not constant for all observations. In many cases, the variance of the error terms appears to be a function of one or more of the independent variables in a regression model. For example, consider the relationship that exists between household consumption expenditures ($Y$) and the level of household income ($X$). Figure~\ref{hetero_g_hc} illustrates a possible relationship between these two variables.
\begin{center}
\FRAME{ftbpFU}{4.6259in}{3.4108in}{0pt}{\Qcb{Household consumption expenditure as a function of household income.}}{\Qlb{hetero_g_hc}}{% fig13-1.gif}{\special{language “Scientific Word”;type “GRAPHIC”;maintain-aspect-ratio TRUE;display “USEDEF”;valid_file “F”;width 4.6259in;height 3.4108in;depth 0pt;original-width 4.5731in;original-height 3.365in;cropleft “0”;croptop “1”;cropright “1”;cropbottom “0”;filename ‘GRAPHS/Fig13-1.gif’;file-properties “XNPEU”;}}
\end{center}
The relationship between household consumption expenditures and household income may be expressed as:
\begin{equation}
Y_{i}=\beta _{o}+\beta _{1}X_{i}+u_{i} \label{init.het.hc} \end{equation}%
In this case, however, the variance of the residual is not constant; instead, it increases as the level of household income ($X$) increases. The reason for this is fairly simple. When the level of annual household income is \$10,000 per year, a \$500 change in consumption spending accounts for 5\% of total income. In households that have an annual income of \$100,000, this same change in spending constitutes only 0.5\% of income. Low-income households tend to have a relatively small variance in consumption expenditures (in large part because they cannot afford a wide variety of alternative consumption levels). The variance in spending among high-income households is likely to be substantially greater than the variance of spending among low-income households.\footnote{%
Evidence in support of this argument may be found in Prais and Houthakker (1955).}
In the case illustrated in Figure~\ref{hetero_g_hc}, the variance of the residuals is a function of household income ($X_{i})$. Since the variance appears to increase in direct proportion to the level of $X_{i}$, this relationship can be represented as:
\begin{equation}
\text{var}(u_{i})=X_{i}\sigma ^{2} \label{init.het.hca} \end{equation}
Heteroskedasticity of this sort is most likely to occur in cross-sectional or longitudinal studies. While time-series models may exhibit heteroskedasticity, this is somewhat less common.\footnote{% While it is rare to find that the variance of the error term in a time-series model varies with the level of one or more independent variables, it is sometimes found that the variance of the error term at time $t$ is affected by the levels of the variance in previous time periods. In this case, an autoregressive conditional heteroskedasticity (ARCH) model is often used to model the error process. ARCH models are discussed in the appendix at the end of this chapter.} The reason for this is that cross-sectional studies often involve units of observations that differ substantially in terms of the \textquotedblleft size\textquotedblright\ of the unit of observation. For example, cross-sectional studies often involve units of observation that consist of:
\begin{itemize}
\item households that have substantially different income levels; \item states with substantially different levels of population and income; and
\item countries that differ substantially in population, income, and level of economic development.
\end{itemize}
In these cases, those observations that correspond to \textquotedblleft larger\textquotedblright\ units of observation will often have a larger residual variance than those observations consisting of \textquotedblleft smaller\textquotedblright\ units of observation. This relationship may result in a form of heteroskedasticity in which the variance of the error terms varies directly with one of the independent variables in the model. Of course, more complex forms of heteroskedasticity may occur. A more general form of heteroskedasticity can be specified as:\footnote{% Note that the range of the function $f\left( Z_{1i},Z_{2i},\ldots ,Z_{mi}\right) $ must include only positive values. (A negative value for the variance is nonsensical.)}
\begin{equation}
\text{var}(u_{i})=f(Z_{1i},Z_{2i},\ldots ,Z_{mi})\sigma ^{2} \label{het.more.general}
\end{equation}%
Under this general form of heteroskedasticity, the variance of the residuals is a function of $m$ variables: $Z_{1},Z_{2},\ldots ,Z_{m}$. (These variables may contain some or all of the variables included as independent variables in the regression equation.) The simpler form of heteroskedasticity described in equation \ref{init.het.hca} is a special case of the more general relationship described by equation \ref% {het.more.general}.
\section{Consequences of Heteroskedasticity}
When heteroskedasticity is present in a regression model, OLS estimators still provide unbiased and consistent parameter estimates. These estimators, however, are not efficient. An inspection of Figure~\ref{hetero_g_hc} above suggests the reason for this inefficiency.\footnote{% A good demonstration of the inefficiency of OLS estimators under this form of heteroskedasticity is contained in Kmenta (1986), pp. 270-272. A more general proof (requiring matrix algebra) appears in Theil (1971), pp.
237-238.}
Under an OLS estimation procedure, each sample observation is weighted equally in the computation of parameter estimates. This is appropriate if each observation contains the same amount of information about the location of the regression line. When heteroskedasticity is present, however, more information is contained in those observations in which the variance of the residual is smaller. Observations in which the residual variance is large contain less information about the location of the regression line. In Figure~\ref{hetero_g_hc}, observations in which the level of $X_{i}$ is lower contain more information about the location of the regression line than those observations corresponding to high levels of $X_{i}$.
To deal with this problem, it would be desirable to construct an estimation procedure that assigns more weight to those observations that contain more information about the value of the regression parameters. The construction of such an estimator is described below.
There is a second problem associated with the use of OLS estimation methods when heteroskedasticity is present. OLS\ estimates of the standard errors for the parameter estimates are appropriate only when the variance of the error term is constant. These estimated standard errors are biased when heteroskedasticity is present. Therefore, $t$-ratios based upon these estimated standard errors are not appropriate for use in hypothesis testing.% \footnote{%
A proof (requiring matrix algebra) is contained in Theil (1971), pp. 247-248.% }
In summary, when heteroskedasticity is present:
\begin{itemize}
\item OLS estimators are unbiased and consistent;
\item OLS estimators are inefficient; and
\item the standard errors (and $t$-ratios) generated by an OLS regression are incorrect and cannot be used for hypothesis testing purposes.
\end{itemize}
An observant reader will notice that the effects of heteroskedasticity are quite similar to the effects of autocorrelation.
\section{Tests for the presence of heteroskedasticity} There are several tests that econometricians use to detect the presence of heteroskedasticity. A common practice is to plot the residuals against one or more variables that are suspected to affect the variance of the residual.
An inspection of these graphs can sometimes provide a rough indication of the presence of heteroskedasticity. This is particularly true if the variance is a function of one of the independent variables. Figure~\ref% {hrespl_g_hc} contains a graph of a residual plot in which the variance of the error terms appears to vary with the level of $X_{j}$.
\begin{center}
\FRAME{ftbpFU}{3.934in}{2.9395in}{0pt}{\Qcb{Residual plot of a case in which the variance of the error terms is a function of $X_{j}$}}{\Qlb{hrespl_g_hc}% }{fig13-2.gif}{\special{language “Scientific Word”;type “GRAPHIC”;maintain-aspect-ratio TRUE;display “USEDEF”;valid_file “F”;width 3.934in;height 2.9395in;depth 0pt;original-width 3.8856in;original-height 2.8963in;cropleft “0”;croptop “1”;cropright “1”;cropbottom “0”;filename ‘GRAPHS/Fig13-2.gif’;file-properties “XNPEU”;}}
\end{center}
Of course, if there is any reason to suspect heteroskedasticity, a formal test for its presence is desirable. In this section, three of the most commonly used tests for the presence of heteroskedasticity are examined:% \footnote{%
Alternative tests have been provided by Park (1966) and Glesjer (1969).
These alternative tests, however, are less commonly used.} \begin{itemize}
\item the Goldfeld-Quandt test;
\item the Breusch-Pagan test; and
\item White’s test.
\end{itemize}
Let’s examine each of these tests.\
\subsection{Goldfeld-Quandt test}
The Goldfeld-Quandt test was one of the first tests proposed for the detection of heteroskedasticity.\footnote{%
For a more complete discussion of this test, see Goldfeld and Quandt (1965).} This test is most appropriately used when it is believed that the variance of the residuals varies with the level of one (and only one) of the independent variables. If it is believed that the variance varies with two or more variables, then the use of either the Breusch-Pagan test or White’s test is more appropriate.
Suppose that an econometrician suspects that the variance of the error term is a function of the level of $X_{ji}$. The Goldfeld-Quandt test consists of the following steps:
\begin{enumerate}
\item[Step 1:] Order the observations from the highest to the lowest values of $X_{ji}$.
\item[Step 2:] Separate the sample into two subsamples: the first $n_1$ observations and the final $n_2$ observations after omitting the middle $d$ observations from analysis. It is standard practice to set $n_1=n_2$ (although this is not required for the test). In this case, $n_1=n_2=\left( N-d\right) /2$. The number of observations omitted generally constitutes approximately 20-33\% of the sample.
\item[Step 3:] Estimate the parameters of the regression model using a separate OLS estimation procedure in each subsample.\footnote{% Two separate regressions are necessary so that the sample residuals in the first subsample are independent of those in the second subsample.} \item[Step 4:] Form the statistic:
\begin{equation*}
\text{Goldfeld-Quandt statistic = }\frac{\hat{\sigma}_2^2}{\hat{\sigma}_1^2} \end{equation*}
where $\hat{\sigma}_1^2$ is the sample variance of the residuals in the first subsample and $\hat{\sigma}_2^2$ is the sample variance for the residuals corresponding to the second subsample. Under the null hypothesis of a constant variance for the residual, the Goldfeld-Quandt statistic is distributed as an $F$ statistic with $n_1$ and $n_2$ degrees of freedom.
\item[Step 5:] Reject the null hypothesis and conclude that heteroskedasticity is present if the Goldfeld-Quandt statistic exceeds the critical value for an $F$ distribution at the predetermined significance level.
\end{enumerate}
One of the difficulties with the Goldfeld-Quandt procedure is that it is difficult to determine how many observations to omit from the sample.
Omitting more observations results in a larger difference in variance between the two samples, but results in a larger loss in degrees of freedom.
A second problem with the Goldfeld-Quandt test is that it does not perform as well when the variance of the error terms is a function of two or more of the independent variables in the regression equation. Unless the exact functional form for the heteroskedasticity is known, it is impossible to order the observations so that the variance of the residuals increases monotonically from the first to the last observation. (If the exact functional form were known, of course, there would be no need for a test for the presence of heteroskedasticity!) Because of this problem, econometricians generally rely on either the Breusch-Pagan test or White’s test unless there is a strong reason to believe that the heteroskedasticity is related to a single variable.
\subsection{Breusch-Pagan test}
The Breusch-Pagan test is a variation of the Lagrange multiplier test discussed in Chapter \ref{spec.chap}.\footnote{%
A more complete (but mathematically sophisticated) discussion of this test appears in Breusch and Pagan (1979). An equivalent test for the special case of multiplicative heteroskedasticity was developed independently by Godfrey (1978b).} The Breusch-Pagan test is appropriate when it is believed that the variance of the errors is a linear function of a set of variables $% Z_{1},Z_{2},\ldots ,Z_{m}$. These $Z_{i}$’s may include some or all of the independent variables in the original regression (other variables may be included as well). The basic model is given by:
\begin{equation}
Y_{i}=\beta _{o}+\beta _{1}X_{1i}+\beta _{2}X_{2i}+\cdots +\beta _{k}X_{ki}+u_{i} \label{b-p.reg.hc}
\end{equation}%
\begin{equation*}
\text{where: var(}u_{i}\text{) = }\left( \alpha _{o}+\alpha _{1}Z_{1i}+\cdots +\alpha _{m}Z_{mi}\right) \sigma ^{2} \end{equation*}%
The Breusch-Pagan test involves the following steps: \begin{enumerate}
\item[Step 1:] Estimate the parameters of the original regression model (equation \ref{b-p.reg.hc}) through the use of an OLS estimation technique.
Use the estimated parameters from this equation to generate the estimated residuals:
\begin{equation}
\hat{u}_i=Y_i-\hat{\beta}_o-\hat{\beta}_1X_{1i}-\cdots -\hat{\beta}_kX_{ki} \label{res.BP.hc}
\end{equation}
\item[Step 2:] Estimate the parameters of the auxiliary regression given by: \begin{equation*}
\frac{\hat{u}_i^2}{\hat{\sigma}^2}=\gamma _o+\gamma _1Z_{1i}+\gamma _2Z_{2i}+\cdots +\gamma _mZ_{mi}+\epsilon _i
\end{equation*}
where $\hat{\sigma}^2$ is the sample variance of the residuals in the first-stage regression equation. Define RSS$_{aux}$ as the regression sum of squares in the estimated version of this equation.
\item[Step 3:] Use the results from Step 2 to form the Breusch-Pagan statistic defined as:
\begin{equation*}
\text{Breusch-Pagan statistic = }\frac{\text{RSS}_{aux}}2 \end{equation*}
Under the null hypothesis of homoskedastic residuals, the Breusch-Pagan statistic is distributed as a $\chi ^2$ statistic with $m$ degrees of freedom.
\item[Step 4:] Reject the null hypothesis and assume that heteroskedasticity is present if the estimated Breusch-Pagan statistic exceeds the critical value for a $\chi ^2$ statistic at the preselected significance level.
\end{enumerate}
One advantage of the Breusch-Pagan test over the Goldfeld-Quandt test is that it is not necessary to specify the exact form of the heteroskedasticity to implement the Breusch-Pagan test. In this test, all that is necessary is that the econometrician be able to select a set of variables that may be related to the heteroskedasticity in the error process.
\subsection{White’s test}
An alternative test for heteroskedasticity has been suggested by White.% \footnote{%
For a complete, but more mathematically sophisticated, discussion see White (1980). As noted by Waldman (1983), White’s test is equivalent to a slightly modified version of the Breusch-Pagan test.} Once again, the model is given by:
\begin{equation}
Y_i=\beta _o+\beta _1X_{1i}+\beta _2X_{2i}+\cdots +\beta _kX_{ki}+u_i \label{White.hc}
\end{equation}
\begin{equation*}
\text{where: var(}u_i\text{) = }f(Z_{1i},Z_{2i},\ldots ,Z_{mi})\sigma ^2 \end{equation*}
White’s test involves a form of the Lagrange multiplier test that is closer in style to the test discussed in Chapter \ref{spec.chap}. White’s test consists of the following steps:
\begin{enumerate}
\item[Step 1:] Estimate the parameters of the original regression model (equation \ref{b-p.reg.hc}) through the use of an OLS estimation technique.
Use the estimated parameters from this equation to generate the estimated residuals:
\begin{equation}
\hat{u}_i=Y_i-\hat{\beta}_o-\hat{\beta}_1X_{1i}-\cdots -\hat{\beta}_kX_{ki} \label{res.White.hc}
\end{equation}
\item[Step 2:] Estimate the parameters of the auxiliary regression given by: \begin{equation}
\hat{u}_{i}^{2}=\gamma _{o}+\gamma _{1}Z_{1i}+\gamma _{2}Z_{2i}+\cdots +\gamma _{m}Z_{mi}+\epsilon _{i} \label{aux.White.hc} \end{equation}
White suggests that the variables $Z_{1},Z_{2},\ldots ,Z_{m}$ should consist of:
\begin{itemize}
\item all variables in the original regression;
\item the squares of all variables in the original regression; and \item all possible interactions among the variables in the original regression.
\end{itemize}
\item[Step 3:] Use the results from Step 2 to form the Lagrange-multiplier statistic defined as:
\begin{equation*}
\text{Lagrange-multiplier statistic = }N\times \text{R}^{2} \end{equation*}%
where R$^{2}$ is the value of R$^{2}$ in the auxiliary regression (equation % \ref{aux.White.hc}). Under the null hypothesis of homoskedastic residuals, this Lagrange-multiplier statistic is distributed as a $\chi ^{2}$ statistic with $m$ degrees of freedom (where $m$ is the number of variables included in the auxiliary regression).
\item[Step 4:] Reject the null hypothesis and assume that heteroskedasticity is present if the estimated Lagrange-multiplier statistic exceeds the critical value for a $\chi ^2$ statistic at the predetermined significance level.
\end{enumerate}
One problem with the use of White’s test is that the form of the auxiliary regression equation suggested by White may result in an equation with a low (or negative) degrees of freedom. For example, if the original regression model contains 10 variables, the auxiliary regression equation would include up to 60 variables.\footnote{%
The number of original variables + squared variables + interaction terms = \begin{equation*}
N+N^{2}/2
\end{equation*}%
since there are $N$ original variables and $N^{2}/2$ possible squared terms and interaction terms.} In a sample with a limited number of observations, this may not be feasible. In this case, it might be necessary to drop some or all of the interaction terms from the auxiliary regression equation.
The inclusion of squared terms and interaction terms may appear to provide White’s test with an advantage over the Breusch-Pagan procedure since it allows for the possibility that the $X_{j}$’s may have a nonlinear effect on the magnitude of the variance for the error terms. It should be noted, though, that the $Z_{j}$ terms used in the Breusch-Pagan procedure may include the same squared terms and interaction terms.
When implementing White’s test, be careful to exclude squared dummy variable terms in the auxiliary regression equation since the square of a dummy variable equals the original variable (since 1$^2=1$ and $0^2=0$). Thus, a perfect multicollinearity problem would result if a dummy variable and the squared value of the same dummy variable were both included in the regression equation.
Of course, White’s test can always be based upon a simpler functional form for the auxiliary regression equation. If, for example, it is believed that the residual variation is proportional to the variable $X_{j}$, then the auxiliary regression may be formulated as:
\begin{equation*}
\hat{u}_{i}^{2}=\gamma _{o}+\gamma _{1}X_{ji}+\epsilon _{i} \end{equation*}%
Thus, White’s test provides a convenient alternative to the Goldfeld-Quandt test when it is believed that the residual variance is proportional to a single regressor.
\section{Correction for heteroskedasticity: known variances} \subsection{Estimation when var($u_{i}$) is proportional to $X_{j}\label% {simple.het}$}
If an econometrician knows the specific functional form for the variance of the residual, it is relatively easy to correct for the presence of heteroskedasticity.\footnote{%
In practice, the functional form for the process generating the heteroskedasticity is not generally known by the researcher. This assumption is made initially to simplify the exposition. The more realistic situation in which the functional form is not known is discussed below.} For example, let’s consider a simple model in which the variance of the residual is proportional to the level of one of the independent variables, $X_{j}$.% \footnote{%
It is assumed that all of the other conditions of the classical regression model are satisfied in this model. For this procedure to be used, it is also required that the level of $X_{ji}$ be positive for all realizations. While some economic variables can take on negative values (such as real interest rates and the level of profits), the variables that are likely \textquotedblleft sources\textquotedblright\ for a heteroskedasticity problem typically take on only positive values.} In this case, the regression model is given by:
\begin{equation}
Y_{i}=\beta _{o}+\beta _{1}X_{1i}+\beta _{2}X_{2i}+\cdots +\beta _{k}X_{ki}+u_{i} \label{reg.het.hc}
\end{equation}%
\begin{equation*}
\text{where: var(}u_{i}\text{) = }X_{ji}\sigma ^{2}
\end{equation*}%
As noted above, OLS estimation of equation \ref{reg.het.hc} would result in inefficient parameter estimates. To eliminate this problem, it would be desirable to transform the regression model so that the residual in the transformed model is homoskedastic. Let’s consider such a transformation.
Suppose that we multiply both sides of equation \ref{reg.het.hc} by $1/\sqrt{% X_{ji}}$. The resulting equation becomes:
\begin{equation*}
\frac{Y_{i}}{\sqrt{X_{ji}}}=\beta _{o}\frac{1}{\sqrt{X_{ji}}}+\beta _{1}% \frac{X_{1i}}{\sqrt{X_{ji}}}+\beta _{2}\frac{X_{2i}}{\sqrt{X_{ji}}}+\cdots +\beta _{k}\frac{X_{ki}}{\sqrt{X_{ji}}}+\frac{u_{i}}{\sqrt{X_{ji}}} \end{equation*}%
Defining new variables as:
\begin{equation}
\begin{array}{c}
\tilde{Y}_{i}=\frac{Y_{i}}{\sqrt{X_{ji}}} \\
\tilde{X}_{oi}=\frac{1}{\sqrt{X_{ji}}} \\
\tilde{X}_{1i}=\frac{X_{1i}}{\sqrt{X_{ji}}} \\
\tilde{X}_{2i}=\frac{X_{2i}}{\sqrt{X_{ji}}} \\
\vdots \\
\tilde{X}_{ki}=\frac{X_{ki}}{\sqrt{X_{ji}}} \\
\epsilon _{i}=\frac{u_{i}}{\sqrt{X_{ji}}}%
\end{array}
\label{trans.het.hc}
\end{equation}%
The transformed equation becomes:
\begin{equation}
\tilde{Y}_{i}=\beta _{o}\tilde{X}_{oi}+\beta _{1}\tilde{X}_{1i}+\beta _{2}% \tilde{X}_{2i}+\cdots +\beta _{k}\tilde{X}_{ki}+\epsilon _{i} \label{het.trans.eq}
\end{equation}%
The variance of the transformed residual, $\epsilon _{i}$, equals: \begin{equation}
\text{var}(\epsilon _{i})=\text{var}\left( \frac{1}{\sqrt{X_{ji}}}% u_{i}\right) \label{het.vat.hc}
\end{equation}%
\begin{equation*}
=\left( \frac{1}{\sqrt{X_{ji}}}\right) ^{2}\text{var}(u_{i}) \end{equation*}%
\begin{equation*}
=\left( \frac{1}{X_{ji}}\right) \left( X_{ji}\sigma ^{2}\right) \end{equation*}%
\begin{equation*}
=\sigma ^{2}
\end{equation*}%
Thus, the variance of the residual in this transformed equation is constant.
In this case, OLS estimation of the parameters of equation \ref{het.trans.eq} will result in estimators that are BLUE.
OLS estimation of the transformed equations results in the selection of the values of $\hat \beta _o,\hat \beta _1,\ldots ,\hat \beta _k$ that minimize the sum of squares of the transformed sample residuals ($\hat \epsilon _i$).
This minimization problem can be expressed as:
\begin{equation*}
\text{Select the values of }\hat \beta _o,\hat \beta _1,\ldots ,\hat \beta _k \text{ that minimize: }\sum_{i=1}^N\hat \epsilon _i^2 \end{equation*}
Using the definitions in \ref{trans.het.hc}, this is equivalent to the minimization problem:
\begin{equation} \label{wls.hc}
\text{Select the values of }\hat \beta _o,\hat \beta _1,\ldots ,\hat \beta _k% \text{ that minimize: }\sum_{i=1}^N\left( \frac 1{\sqrt{X_{ji}}}\hat u_i\right) ^2=\sum_{i=1}^N\frac 1{X_{ji}}\hat u_i^2
\end{equation}
An examination of the minimization problem in \ref{wls.hc} indicates that the estimated intercept and slope parameters are chosen to minimize a weighted sum of the squared values of the original error terms $\hat u_i$.
Thus, OLS\ estimation of the transformed equation is equivalent to the minimization of a weighted sum of squares for the original residuals. For this reason, the estimation procedure described above is called a \textbf{% weighted least-squares estimation procedure}. The weight assigned to each squared residual term in this procedure equals: $1/X_{ji}$. This estimator is efficient because it places greater weight on those observations that contain more information about the regression parameters (since the variance of the residual is lower when $X_{ji}$ is lower).
This weighted least-squares estimator is another special case of the generalized least-squares (or GLS) introduced in Chapter \ref{auto.chap}. As noted earlier, a GLS estimation procedure is used when a model that violates one of the assumptions of the classical regression model can be transformed so that the transformed model satisfies all of the assumptions of the classical regression model. To correct for heteroskedasticity, the appropriate transformation involves weighting each observation so that the errors for the transformed variables are homoskedastic.
Once again, this GLS estimation procedure results in estimators that are BLUE (since the conditions of the classical regression model are satisfied for the transformed regression model). Furthermore, OLS estimation of the transformed equation provides estimated standard errors that may be used for the construction of hypothesis tests involving the intercept and slope parameters.
\subsection{Estimation under more general forms of heteroskedasticity\label% {complex.het}}
More generally, suppose that the variance of the residual is a known function of a set of $m$ variables, $Z_1,Z_2,\ldots ,Z_m$. This set of variables may contain some or all of the independent variables $% X_1,X_2,\ldots ,X_k$. There is, however, no necessary relationship between these two sets of variables. The $Z_j$ variables may contain factors that affect the size of the residual variance that do not directly affect the level of the dependent variable. It is also possible that some of the factors that affect the level of the dependent variable have no effect on the variance of the error terms.
Under this more general form of heteroskedasticity, the regression model is given by:
\begin{equation*}
Y_{i}=\beta _{o}+\beta _{1}X_{1i}+\beta _{2}X_{2i}+\cdots +\beta _{k}X_{ki}+u_{i}
\end{equation*}%
\begin{equation*}
\text{where: var(}u_{i}\text{) = }f(Z_{1i},Z_{2i},\ldots ,Z_{mi})\sigma ^{2} \end{equation*}%
If the econometrician knows the specific form of the function $% f(Z_{1i},Z_{2i},\ldots ,Z_{mi})$, a generalization of the procedure described above can be used.\footnote{%
One special case of this function involves a form of heteroskedasticity known as \textbf{multiplicative heteroskedasticity}. In this case, the variance of the error terms is given by:
\begin{equation*}
var(u_{i})=\sigma ^{2}Z_{1}^{\gamma _{1}}Z_{2}^{\gamma _{2}}\cdots Z_{m}^{\gamma _{m}}
\end{equation*}%
or alternatively:
\begin{equation*}
\ln (var(u_{i}))=\ln \left( \sigma ^{2}\right) +\gamma _{1}Z_{1_{i}}+\gamma _{2}Z_{2i}+\cdots +\gamma _{m}Z_{mi}
\end{equation*}%
A good discussion of the multiplicative heteroskedasticity models appears in Harvey (1976).
\par
More typically, however, econometricians assume that heteroskedasticity occurs as \textit{additive heteroskedasticity} in which the variance of the error term is a linear function of $Z_{1},Z_{2},\ldots ,Z_{m}$: \begin{equation*}
var(u_{i})=\gamma _{o}+\gamma _{1}Z_{1}+\gamma _{2}Z_{2}+\cdots +\gamma _{m}Z_{m}
\end{equation*}%
} In this general case, each variable may be transformed using the relationships:
\begin{equation}
\begin{array}{c}
\tilde{Y}_{i}=\frac{Y_{i}}{\sqrt{f(Z_{1i},Z_{2i},\ldots ,Z_{mi})}} \\ \tilde{X}_{oi}=\frac{1}{\sqrt{f(Z_{1i},Z_{2i},\ldots ,Z_{mi})}} \\ \tilde{X}_{1i}=\frac{X_{1i}}{\sqrt{f(Z_{1i},Z_{2i},\ldots ,Z_{mi})}} \\ \tilde{X}_{2i}=\frac{X_{2i}}{\sqrt{f(Z_{1i},Z_{2i},\ldots ,Z_{mi})}} \\ \vdots \\
\tilde{X}_{ki}=\frac{X_{ki}}{\sqrt{f(Z_{1i},Z_{2i},\ldots ,Z_{mi})}} \\ \epsilon _{i}=\frac{u_{i}}{\sqrt{f(Z_{1i},Z_{2i},\ldots ,Z_{mi})}}% \end{array}
\label{trans.het.hca}
\end{equation}%
These transformed variables are used to form the regression equation: \begin{equation}
\tilde{Y}_{i}=\beta _{o}\tilde{X}_{oi}+\beta _{1}\tilde{X}_{1i}+\beta _{2}% \tilde{X}_{2i}+\cdots +\beta _{k}\tilde{X}_{ki}+\epsilon _{i} \label{het.trans.eq.a}
\end{equation}%
In this equation, the variance of the transformed residual, $\epsilon _{i}$, equals:
\begin{equation*}
var(\epsilon _{i})=var\left( \frac{u_{i}}{\sqrt{f(Z_{1i},Z_{2i},\ldots ,Z_{mi})}}\right)
\end{equation*}%
\begin{equation*}
=\frac{1}{f(Z_{1i},Z_{2i},\ldots ,Z_{mi})}var(u_{i}) \end{equation*}%
\begin{equation*}
=\frac{1}{f(Z_{1i},Z_{2i},\ldots ,Z_{mi})}\left( f(Z_{1i},Z_{2i},\ldots ,Z_{mi})\sigma ^{2}\right)
\end{equation*}%
\begin{equation*}
=\sigma ^{2}
\end{equation*}%
Since the variance of the residual is constant in this transformed equation, OLS\ estimators are BLUE. This is a generalization of the GLS estimation procedure described above. The simple case considered in Section \ref% {simple.het} is a special case of this more general formulation.
Thus, efficient estimates of all intercept and slope parameters can be derived when heteroskedasticity is present if the form of the heteroskedasticity is known. In practice, however, this information is not known by the econometrician. Instead, it is necessary to test for the presence of heteroskedasticity and to apply appropriate corrective measures.
\section{Correction for heteroskedasticity: unknown form} In most practical applications, an econometrician does not know the exact functional form for the variance of the residuals when heteroskedasticity is present. If one of the tests described above indicate that heteroskedasticity is present, however, OLS estimators are inefficient and provide biased estimates of the standard errors for the intercept and slope parameters. Thus, some corrective procedure is indicated. There are two options available in this case:
\begin{enumerate}
\item The econometrician may attempt to provide a reasonable specification for the functional form governing the variance of the residuals. The selection of such a model may be based, for example, on the results from the auxiliary regression equation estimated under either the Breusch-Pagan test or White’s test. This equation may be used to construct a weighted-least squares estimator that provides efficient estimates and asymptotically correct standard errors.
\item The econometrician may choose to leave the form of the heteroskedasticity unspecified. In this case, it is possible to use a method developed by White to derive asymptotically correct standard errors for the OLS estimates of the intercept and slope parameters. This makes it possible to construct hypothesis tests based upon the least-squares estimators. (This method is briefly discussed in section \ref{hccv.hc}.) \end{enumerate}
Let’s discuss each of these procedures.
\subsection{Weighted least-squares}
The correction for heteroskedasticity with an unknown functional form is essentially equivalent to the procedure described above in the case when the functional form is known. The only essential difference is that the functional form of the heteroskedasticity relationship must be estimated empirically before a weighted least-squares estimation procedure can be implemented.
If it is believed that the variance of the residual is proportional to one of the $X_{j}$’s, then it is possible to test this hypothesis using the Goldfeld-Quandt test, the Breusch-Pagan test or White’s test. If it is believed that the residual variance is a more complex function of a set of variables, then either the Breusch-Pagan test or White’s test may be used to test the hypothesis.
In practice, the researcher will not generally know the functional form of the heteroskedasticity relationship. In some cases, economic theory or knowledge of institutional detail will provide some suggestions concerning which variables are related to the variance of the error terms. When in doubt, it is best to err on the side of caution by including too many (rather than too few) variables in this equation. This argument is the basis for White’s suggestion of including all variables, squares of variables, and interaction terms as possible source of heteroskedasticity.
Suppose, for example, that the regression function is given by: \begin{equation}
Y_i=\beta _o+\beta _1X_{1i}+\beta _2X_{2i}+u_i \label{het.ex.hc} \end{equation}
If one of the test procedures discussed above indicates that the variance of $u_i$ varies directly with the level of $X_{1i}$, then the procedure discussed in Section \ref{simple.het} may be used. In this case, both sides of equation \ref{het.ex.hc} can be multiplied by $1/\sqrt{X_{1i}}$ to form: \begin{equation*}
\frac{Y_i}{\sqrt{X_{1i}}}=\beta _o\left( \frac 1{\sqrt{X_{1i}}}\right) +\beta _1\left( \frac{X_{1i}}{\sqrt{X_{1i}}}\right) +\beta _2\left( \frac{% X_{2i}}{\sqrt{X_{1i}}}\right) +\frac{u_i}{\sqrt{X_{1i}}} \end{equation*}
As noted above, OLS estimation of this equation will result in estimates that are efficient and will provide asymptotically correct standard errors.
Suppose instead, that the variance of $u_{i}$ is believed to be a function of $Z_{1}$ and $Z_{2}$ according to the relationship: \begin{equation}
var(u_{i})=\gamma _{o}+\gamma _{1}Z_{1i}+\gamma _{2}Z_{2i} \label{het.func.form.hc}
\end{equation}%
Then, consistent estimates of the parameters in equation \ref% {het.func.form.hc} may be provided by an OLS regression of the auxiliary regression equation:\footnote{%
A proof is contained in Amemiya (1985), pp. 203-207.} \begin{equation}
\hat{u}_{i}^{2}=\gamma _{o}+\gamma _{1}Z_{1i}+\gamma _{2}Z_{2i}+\epsilon _{i} \label{het.func.form.hca}
\end{equation}%
In this case, the transformed regression equation is given by:\footnote{% A problem occurs if some of the predicted values in equation \ref% {het.func.form.hca} are negative or equal to zero. If this occurs, the usual procedure is to use a weight of $1/\sqrt{\hat{u}_{i}^{2}}$ instead of $1/% \sqrt{\hat{\gamma}_{o}+\hat{\gamma}_{1}Z_{1i}+\hat{\gamma}_{2}Z_{2i}}$ as the weight for the affected observations.}
\begin{equation}
\frac{Y_{i}}{\sqrt{\hat{\gamma}_{o}+\hat{\gamma}_{1}Z_{1i}+\hat{\gamma}% _{2}Z_{2i}}}=\beta _{o}\left( \frac{1}{\sqrt{\hat{\gamma}_{o}+\hat{\gamma}% _{1}Z_{1i}+\hat{\gamma}_{2}Z_{2i}}}\right) +\beta _{1}\left( \frac{X_{1}}{% \sqrt{\hat{\gamma}_{o}+\hat{\gamma}_{1}Z_{1i}+\hat{\gamma}_{2}Z_{2i}}}\right) \label{het.func.form.hca1}
\end{equation}%
\begin{equation*}
+\beta _{2}\left( \frac{X_{2}}{\sqrt{\hat{\gamma}_{o}+\hat{\gamma}_{1}Z_{1i}+% \hat{\gamma}_{2}Z_{2i}}}\right) +\frac{u_{i}}{\sqrt{\hat{\gamma}_{o}+\hat{% \gamma}_{1}Z_{1i}+\hat{\gamma}_{2}Z_{2i}}}
\end{equation*}%
Where the parameters $\hat{\gamma}_{o},\hat{\gamma}_{1}$, and $\hat{\gamma}% _{2}\,$ are the OLS estimates of the parameters in equation \ref% {het.func.form.hca}.
In most econometric software packages, this procedure can be easily accomplished by saving the predicted values from the auxiliary equation (equation \ref{het.func.form.hca}) and using these values to constructs weights for a GLS\ estimator. A potential problem with this procedure is that the predicted value of $\hat{u}_{i}^{2}$ may be negative or equal to zero. A negative value, of course, is nonsensical for a squared error term and negative or zero values for the predicted squared error term would result in a problem when you attempt to take the square root of the reciprocal to construct the weights. For observations for which the predicted squared error term ($\hat{\gamma}_{o}+\hat{\gamma}_{1}Z_{1i}+\hat{% \gamma}_{2}Z_{2i}$) is less than or equal to zero, the actual value of $\hat{% u}_{i}^{2}$ should be used in place of the predicted value\ in computing the transformation in equation \ref{het.func.form.hca1}. In most econometrics packages, this process involves:
\begin{enumerate}
\item Estimate the parameters of the original equation (equation \ref% {het.ex.hc}) and store the predicted residuals, $\hat{u}_{i}^{2}$.
\item Use the squared values of these residuals as a dependent variable in the auxiliary equation (equation \ref{het.func.form.hca}) and save the predicted values from this equation.
\item Use a conditional create command\footnote{%
Conditional create commands often have a syntax similar to:% \begin{equation*}
\text{If }(x>0)\text{ }q=a\text{ else }q=b
\end{equation*}%
\par
This creates a new variable $q$ that equals the variable $a$ when $x$ is greater than 0 but will equal the variable $b$ when $x$ is less than or equal to zero.} to create a new variable that equals the predicted variable from the auxiliary regression computed in Step (2) for those observations for which the predicted value is greater than zero; the actual squared residual ($\hat{u}_{i}^{2}$) should be used for those observations for which the predicted squared residual from equation \ref{het.func.form.hca} is less than or equal to zero.
\item Rescale the dependent and independent variables by computing the reciprocal of the square root of the variable created in step (3) to form equation \ref{het.func.form.hca1}. Estimate the parameters of this transformed equation using an OLS regression procedure. (Alternatively, most econometric regression packages provide an option of directly estimating a GLS estimator by using a weighting option in their\ regression procedure.% \footnote{%
If you use a weighted least squares regression option, be sure that you understand how the weight is interpreted by the regression software. (Some packages compute the reciprocal of the square root of the weighting variable as weights for the error terms, others assume that you have done this and simply multiply by sample error terms by the.weights you specify.) Read the documentation for this procedure in your software package before specifying weights.})
\end{enumerate}
In summary, the estimated regression equation in White’s auxiliary regression equation may be used directly to provide the weights for a GLS estimation procedure of the sort described in Section \ref{complex.het}.
Once the appropriate weights are determined, the weighted-least squares estimator described above will provide efficient estimates and asymptotically correct standard errors.
\subsection{White’s heteroskedasticity consistent variance-covariance estimator\label{hccv.hc}}
In many situations, it is possible that the variance of the residuals may vary in a manner that cannot be easily specified. If the econometrician is not willing to specify a functional form for the variance of the residuals, it is possible to generate consistent estimates of the standard errors using a procedure devised by White.\footnote{%
This procedure was developed in White (1980). A full discussion of White’s procedure requires mathematical tools beyond the scope of this text.} In this case, the parameters of the original regression equation are estimated by OLS. Using White’s procedure, corrected standard errors can be derived that make it possible to use the OLS parameter estimates for hypothesis tests. Under White’s procedure, the squared sample residuals are used in place of the unknown $\sigma _i^2$ to construct estimates of the variances and covariances for all estimated parameters. Of course, OLS estimators are still inefficient in this case.\footnote{%
Cragg (1983) has developed a variation of White’s procedure that makes it possible to construct estimators that are more efficient than the OLS estimators.}
Several econometrics packages provide an option that computes estimated standard errors that are corrected for heteroskedasticity using White’s estimator. If heteroskedasticity is suspected, but you are unsure of the form of the heteroskedasticity, this option may provide a desirable alternative. The advantage of this procedure is that it does not require a specification of the functional form for the heteroskedasticity. As compared to the GLS correction procedure described above, however, this procedure will provide parameter estimates that are less efficient (assuming that the functional form for the heteroskedasticity is appropriately chosen in the GLS specification).
\section{Example: Automotive fatalities}
To illustrate how to deal with heteroskedasticity, let’s consider a simple model of highway fatalities. The regression model is given by: \begin{equation}
\text{Deaths}_{i}=\beta _{o}+\beta _{1}\text{Congestion}_{i}+\beta _{2}\text{% Drivers}_{i}+\beta _{3}\text{MPH65}_{i}+u_{i} \label{deaths.hc} \end{equation}%
\begin{equation*}
\begin{array}{ll}
\text{where:} & \text{Deaths}_{i}=\text{ \# of highway fatalities in state }i% \text{ in 1991} \\
& \text{Congest}_{i}=\text{ vehicle miles of travel per mile of road (in thousands)} \\
& \text{Drivers}_{i}=\text{ \# of licensed drivers in state }i \\ & \text{MPH65}_{i}=\text{ 1 if a 65 MPH speed limit was in effect in parts of state }i\text{ in 1991}%
\end{array}%
\end{equation*}%
The parameters of equation \ref{deaths.hc} were estimated using cross-sectional data on the 50 U.S. states and the District of Columbia The data for this model appears in Table~\ref{hdeaths.dat} in Appendix \ref% {data.appendix}.\footnote{%
A copy of this data also appears in the file \textquotedblleft hdeaths.dat\textquotedblright\ on the data disk that accompanies this text.} The estimated equation is given by:
\begin{equation}
\widehat{\text{Deaths}}_{i}=-\underset{(-1.54)}{152.53}+\underset{(0.194)}{% 0.012}\text{Congest}_{i}+\underset{(29.92)}{0.23}\text{Drivers}_{i} \label{est.deaths.hc}
\end{equation}%
\begin{equation*}
+\underset{(3.09)}{262.36}\text{MPH65}_{i}
\end{equation*}%
\begin{equation*}
\text{(}t\text{-statistics in parentheses)}
\end{equation*}
To investigate the possibility that the variance of the error term might increase with the size of the state, the estimated residuals from equation % \ref{est.deaths.hc} were plotted against the variable Drivers$_{i}$. Figure~% \ref{auto_g_hc} contains a graph of this relationship. In this graph, it appears that the variance of the error terms may increase with the level of this variable. The Goldfeld-Quandt test may be used to test this hypothesis.
\begin{center}
\FRAME{ftbpFU}{6.4377in}{4.6985in}{0pt}{\Qcb{Sample residuals in auto fatality regression}}{\Qlb{auto_g_hc}}{fig13-3.gif}{\special{language “Scientific Word”;type “GRAPHIC”;maintain-aspect-ratio TRUE;display “USEDEF”;valid_file “F”;width 6.4377in;height 4.6985in;depth 0pt;original-width 6.3754in;original-height 4.6458in;cropleft “0”;croptop “1”;cropright “1”;cropbottom “0”;filename
‘GRAPHS/Fig13-3.gif’;file-properties “XNPEU”;}}
\end{center}
\subsubsection{\protect\bigskip Goldfeld-Quandt test} To conduct the Goldfeld-Quandt test, the data is first ordered according to the magnitude of the variable Drivers$_{i}$. Then the middle 11 observations were omitted from the analysis (leaving 20 states in each subsample). After estimating the equations separately in each subsample, the Goldfeld-Quandt statistic is measured as:
\begin{equation*}
\text{Goldfeld-Quandt statistic = }\frac{\hat{\sigma}_{2}^{2}}{\hat{\sigma}% _{1}^{2}}
\end{equation*}%
\begin{equation}
=\frac{55958.8}{4931.4}=11.35 \label{GQ.example}
\end{equation}%
Under the null hypothesis of a constant error variance, this statistic is distributed as an $F$ distribution with 20 degrees freedom in both the numerator and denominator. At a 1\% significance level, the critical value for an $F(20,20)$ distribution equals 2.94. Since the estimated value of the Goldfeld-Quandt statistic exceeds this level, the null hypothesis is rejected. Thus, it appears that the variance of the error term is related to the number of licensed drivers.
A more general test for heteroskedasticity can be conducted using either the Breusch-Pagan test or White’s test. With either of these tests, we can examine the possibility of more complex forms of heteroskedasticity relationships. Let’s first examine the application of the Breusch-Pagan test.
\subsubsection{Breusch-Pagan test\label{bp_drivers_hc}} To conduct a Breusch-Pagan test, it is first necessary to decide which variables might be the source of heteroskedastic error terms. While the Goldfeld-Quandt test suggests that the variance of the error terms may be a function of the number of drivers, it is possible that a more elaborate functional form might be appropriate. Following White’s suggestion, we will include all of the original independent variables, the squares of the independent variables,\footnote{%
Note that the square of MPH65$_{i}$ is not included in this equation since the square of a dummy variable is equal to the original variable. If such a variable is inadvertently included, perfect multicollinearity would occur and the auxiliary regression equation could not be estimated.} and all possible interaction terms in the auxiliary regression equation. To conduct this test, the squared sample error terms from equation \ref{est.deaths.hc} are used to form the dependent variable in the auxiliary regression equation:% \footnote{%
The denominator in this term is the sample variance of the residual terms:% \begin{equation*}
\hat{\sigma}^{2}=\frac{\dsum\limits_{i=1}^{N}\hat{u}_{i}^{2}}{N-(k+1)} \end{equation*}%
In this case, $\hat{\sigma}^{2}=33,331.3193.$}%
\begin{equation}
\widehat{\frac{\hat{u}_{i}^{2}}{\hat{\sigma}^{2}}}=\underset{(1.89)}{2.40}-% \underset{(-1.31)}{0.002}\text{Congest}_{i}+\underset{(2.28)}{0.0003}\text{% Drivers}_{i} \label{deaths_bp_hc}
\end{equation}%
\begin{equation*}
-\underset{(-1.92)}{2.38}\text{MPH65}_{i}+\underset{(1.05)}{0.0000039}\text{% Congest}_{i}^{2}
\end{equation*}%
\begin{equation*}
-\underset{(-2.67)}{0.000000032}\text{Drivers}_{i}^{2}+\underset{(0.46)}{% 0.000000069}\text{Congest}_{i}\times \text{Drivers}_{i} \end{equation*}%
\begin{equation*}
+\underset{(0.29)}{0.00041}\text{Congest}_{i}\times \text{MPH65}_{i} \end{equation*}%
\begin{equation*}
+\underset{(2.26)}{0.00033}\text{Drivers}_{i}\times \text{MPH65}_{i} \end{equation*}%
\begin{equation*}
\text{RSS}_{aux}=28.692
\end{equation*}%
The Breusch-Pagan statistic is defined as:%
\begin{equation*}
\text{Breusch-Pagan statistic = }\frac{\text{RSS}_{aux}}{2} \end{equation*}%
\begin{equation*}
=\frac{28.692}{2}=14.346
\end{equation*}%
Under a null hypothesis of no heteroskedasticity, the Breusch-Pagan statistic is distributed as a $\chi ^{2}(8)$ (since there are 8 independent variables in the right-hand side of equation \ref{deaths_bp_hc}). At a 5\% significance level, the critical value for a $\chi ^{2}(8)$ equals 15.5073.
Since the estimated Breusch-Pagan statistic is less than the critical value, the hypothesis of no heteroskedasticity cannot be rejected.
\subsubsection{White’s test\label{white_dr_hc}}

To conduct White’s test, the following auxiliary regression equation is estimated:
\begin{equation*}
\hat{u}_{i}^{2}=\underset{(1.89)}{79937.3}-\underset{(-1.31)}{65.99}\text{% Congest}_{i}+\underset{(2.28)}{9.93}\text{Drivers}_{i} \end{equation*}%
\begin{equation*}
-\underset{(-1.92)}{79320.0}\text{MPH65}_{i}+\underset{(1.05)}{0.013}\text{% Congest}_{i}^{2}
\end{equation*}%
\begin{equation*}
-\underset{(-2.67)}{0.0011}\text{Drivers}_{i}^{2}+\underset{(0.46)}{0.0023}% \text{Congest}_{i}\times \text{Drivers}_{i} \end{equation*}%
\begin{equation*}
+\underset{(0.29)}{13.56}\text{Congest}_{i}\times \text{MPH65}_{i} \end{equation*}%
\begin{equation*}
+\underset{(2.26)}{11.06}\text{Drivers}_{i}\times \text{MPH65}_{i} \end{equation*}%
\begin{equation*}
R^{2}=0.419
\end{equation*}%
\begin{equation*}
\text{(}t\text{-ratios in parentheses)}
\end{equation*}%
where $\hat{u}_{i}^{2}$ is the estimated squared residual from equation \ref% {est.deaths.hc}. Note that the only difference between White’s test and the Breusch-Pagan test described above is in the scaling of the dependent variable. Therefore, the $t$-ratios for all of the independent variables remain the same.
White’s test relies on the Lagrange multiplier statistic defined as: \begin{equation*}
\text{Lagrange-multiplier statistic = }N\times \text{R}^{2} \end{equation*}%
\begin{equation*}
=51(0.419)=21.37
\end{equation*}%
The critical value for a $\chi ^{2}$ distribution with 8 degrees of freedom is 20.09 at a 1\% significance level. Since the estimated statistic exceeds this critical value, White’s test suggests that heteroskedasticity is present.
\subsubsection{Correction for heteroskedasticity} Since the White and Breusch-Pagan tests provide conflicting results, the test for heteroskedasticity is somewhat inconclusive. Since the consequences of ignoring heteroskedasticity when it is present, however, is more severe than the consequences of an unnecessary correction for heteroskedasticity, it is best to err on the side of caution and correct for the possible presence of heteroskedasticity. Notice that the only significant $t$-ratios in the auxiliary regression equations involve terms in which the variable Drivers$_{i}$ appears. This suggests that the heteroskedasticity problem might be eliminated by specifying a model in which the variance of the error terms is proportionate to the level of the variable Drivers$_{i}$. In this case, the appropriate estimation procedure involves transforming the regression equation (equation \ref{deaths.hc}) by multiplying both sides of the equation by 1/$\sqrt{\text{Drivers}_{i}}$.\footnote{% Alternatively, one could use the estimated results from the auxiliary regression equation to correct for heteroskedasticity, by multiplying it by: $1/\sqrt{\widehat{\hat{u}_{i}^{2}}}$, where $\widehat{\hat{u}_{i}^{2}}$ is the predicted squared residual term from the auxiliary equation estimated while performing White’s test. As noted above, negative predicted values of the squared residual term should be replaced with the observed squared residual before computing the reciprocal of the square root of this variable.% } Thus, the transformed equation becomes:% \begin{equation}
\frac{\text{Deaths}_{i}}{\sqrt{\text{Drivers}_{i}}}=\beta _{o}\frac{1}{\sqrt{% \text{Drivers}_{i}}}+\beta _{1}\frac{\text{Congestion}_{i}}{\sqrt{\text{% Drivers}_{i}}}+\beta _{2}\frac{\text{Drivers}_{i}}{\sqrt{\text{Drivers}_{i}}} \label{deaths_trans.hc}
\end{equation}%
\begin{equation*}
+\beta _{3}\frac{65\text{MPH}_{i}}{\sqrt{\text{Drivers}_{i}}}+\frac{u_{i}}{% \sqrt{\text{Drivers}_{i}}}
\end{equation*}
A GLS estimation procedure simply involves the application of an OLS estimation procedure to the transformed version of this model appearing in equation \ref{deaths_trans.hc}. The estimated values of the intercept and slope terms in this transformed equation are the GLS estimates of the intercept and slope terms in the original equation. This GLS estimate is given by:
\begin{equation}
\widehat{\text{Deaths}}_{i}=-\underset{(-0.93)}{62.52}+\underset{(0.162)}{% 0.005}\text{Congest}_{i}+\underset{(23.26)}{0.23}\text{Drivers}_{i} \label{est.deaths.a.hc}
\end{equation}%
\begin{equation*}
+\underset{(1.98)}{121.30}\left( \text{65MPH}_{i}\right) \end{equation*}%
\begin{equation*}
\text{(}t\text{-statistics in parentheses)} \end{equation*}%
A comparison of the GLS estimates appearing in equation \ref{est.deaths.a.hc} with the uncorrected OLS estimates indicates that when heteroskedasticity is taken into account, the $t$-ratios are reduced on each of the estimated slope coefficients. The estimated effect of the 65 MPH speed limit is also substantially smaller under the GLS estimation procedure (and is not significant at the .01 level).
As this result indicates, a GLS\ correction for heteroskedasticity can have a substantial effect on both the magnitude of estimated regression coefficients and hypothesis tests involving these coefficients. Thus, a correction for heteroskedasticity should be made whenever there is evidence of its presence.
\subsubsection{White’s heteroskedasticity consistent variance-covariance estimator\ }
If an econometrician is reluctant to specify a functional form describing the heteroskedastic error process, the use of White’s heteroskedasticity consistent variance-covariance estimator provides an alternative option. In this case, parameter estimates are provided from an OLS regression procedure (since these are unbiased and consistent even when heteroskedasticity is present), but the estimated standard errors are corrected using a procedure that requires no knowledge of the functional form of the error process. Many modern econometric software packages provide this correction. In this case, this procedure results in the following estimated equation:% \begin{equation}
\widehat{\text{Deaths}}_{i}=-\underset{(-1.35)}{152.53}+\underset{(0.208)}{% 0.012}\text{Congest}_{i}+\underset{(30.52)}{0.23}\text{Drivers}_{i} \label{whitehcvc.hc}
\end{equation}%
\begin{equation*}
+\underset{(2.62)}{262.36}\text{MPH65}_{i} \end{equation*}%
\begin{equation*}
\text{(}t\text{-statistics in parentheses)} \end{equation*}%
In this particular case, White’s correction procedure has a relatively minor effect on the estimated $t$-statistics.
\section{Heteroskedasticity and specification error\label{het_func_form_hc}} In some cases, specification error may result in a model in which the estimated residuals appear to be heteroskedastic even though the residuals in a correctly specified model are homoskedastic. An inappropriate finding of heteroskedasticity may result from the omission of relevant variables from a regression equation or the choice of an incorrect function form.
Let’s consider both of these possibilities.
\subsection{Omitted relevant variables}
Suppose the true model is given by:
\begin{equation}
Y_{i}=\beta _{o}+\beta _{1}X_{1i}+\beta _{2}X_{2i}+u_{i} \label{correct.hc} \end{equation}%
but an econometrician mistakenly specifies the model as: \begin{equation}
Y_{i}=\beta _{o}+\beta _{1}X_{1i}+\epsilon _{i} \label{incorrect.hc} \end{equation}%
A comparison of these two specifications indicates that the residual in the misspecified model can be expressed as:
\begin{equation*}
\epsilon _{i}=\beta _{2}X_{2i}+u_{i}
\end{equation*}%
Thus, under the misspecified model, the variance of the residual is equal to:
\begin{equation*}
\text{var}(\epsilon _{i})=\beta _{2}^{2}\text{var}(X_{2i})+\text{var}(u_{i}) \end{equation*}%
(since it is assumed that $X_{2i}$ and $u_{i}$ are independent). It is quite possible that the variance in $X_{2i}$ may vary with the level of $X_{1i}$.
Suppose, for example, that the variance of $X_{2i}$ rises as the level of $% X_{1i}$ increases. In this case, the variance of the error term in this model will increase with the level of $X_{1i}$ even if the variance of $% u_{i} $ is constant.
Thus, when an econometrician finds evidence of heteroskedasticity, he or she should carefully analyze the specification of the model to be sure that the apparent heteroskedasticity is not the result of model misspecification.
\subsection{Incorrect functional form}
The discussion in the this chapter up to this point has been based on the assumption that the researcher has selected the correct functional form for the regression model. As noted in earlier chapters, however, economic theory does not always provide clear guidance on the appropriate choice of functional form. Suppose, for example, that the true relationship between two variables is linear in the logs of the variables, but a linear model is estimated instead. In this case, an incorrectly specified linear model would generally exhibit heteroskedasticity even though the error process for the true log-linear model is homoskedastic. Since heteroskedasticity may be the result of model misspecification, it may be useful to explore alternative functional forms when heteroskedasticity is found. Specification tests of the sort discussed in Chapter \ref{spec.chap} may be used to guide this selection process.
In practice, there are two common situations in which an alternative model specification may eliminate heteroskedasticity: \begin{itemize}
\item for models in which the dependent variable involves the level of spending or income, a log-linear or double-log specification often results in a homoskedastic error process when a linear model exhibits heteroskedasticity,\footnote{%
Log transformations, of course, may not be used for variables that take on values that are less than or equal to zero.} \item when aggregate cross-sectional data is used (such as data in which the units of observations are states or countries), heteroskedasticity is often eliminated when \textit{per capita} measures of the dependent and independent variables are used in place of aggregate total measures.
\end{itemize}
To illustrate the effects of alternative model specification, consider the choice between the following three model specifications:\footnote{% The data used to estimate this model appear in the file \textquotedblleft healthex.dat.\textquotedblright\ \ This data is described in Table \ref% {healthex.dat} on p. \pageref{healthex.dat}.}% \begin{equation*}
\text{Model I:\ health}_{i}=\beta _{o}+\beta _{1}\text{income}_{i}+u_{i} \end{equation*}%
and%
\begin{equation*}
\text{Model II: }\ln (\text{health}_{i})=\beta _{o}+\beta _{1}\text{income}% _{i}+v_{i}
\end{equation*}%
\begin{equation*}
\text{Model III:\ health}_{i}/\text{pop}_{i}=\beta _{o}+\beta _{1}(\text{% income}_{i}/\text{pop}_{i})+e_{i}
\end{equation*}%
\begin{equation*}
\begin{array}{ll}
\text{where:} & \text{health}_{i}\text{ = total personal health care expenditure in 1998 in state }i \\
& \text{income}_{i}\text{ = personal income in state }i% \end{array}%
\end{equation*}%
The estimated versions of these two equations are given by:% \begin{equation*}
\text{Model I:\ }\widehat{\text{health}_{i}}=\underset{(2.28)}{1263.7}+% \underset{(51.51)}{129.48}\text{income}_{i} \end{equation*}%
\begin{equation*}
\text{Model II:\ }\widehat{\ln (\text{health}_{i})}=\underset{(64.01)}{5.147}% +\underset{(54.31)}{0.959}\text{ln(income)}_{i} \end{equation*}%
\begin{equation*}
\text{Model III:\ }\widehat{\frac{\text{health}_{i}}{\text{pop}_{i}}}=% \underset{(1.14)}{0.659}+\underset{(5.44)}{119.02}\frac{\text{income}_{i}}{% \text{pop}_{i}}
\end{equation*}%
\begin{equation*}
\text{(}t\text{-statistics in parentheses)} \end{equation*}%
\FRAME{ftbpFU}{4.0145in}{7.0257in}{0pt}{\Qcb{Comparison of linear, double log, and \textit{per capita }specifications of health care expenditure function.}}{\Qlb{lin_v_loglin_graph_hc}}{fig13-4.wmf}{\special{language “Scientific Word”;type “GRAPHIC”;display “USEDEF”;valid_file “F”;width 4.0145in;height 7.0257in;depth 0pt;original-width 1.9078in;original-height 3.8649in;cropleft “0”;croptop “1”;cropright “1.1539”;cropbottom “0”;filename ‘GRAPHS/fig13-4.WMF’;file-properties “XNPEU”;}} An inspection of Figure \ref{lin_v_loglin_graph_hc} suggests that the residuals may be heteroskedastic under the linear specifications of Model I and Model III, but appear to be homoskedastic under the double-log specification in Model II. The value of White’s test statistic is: 40.01 for Model I, 4.92 for Model II,\ and 15.20 for Model II. Since the critical value for a $\chi ^{2}$ variate with 2 degrees of freedom equals 5.9914 at a 5\% significance level, White’s test indicates that there is a statistically significant finding of heteroskedasticity in Models I and III. The hypothesis of homoskedasticity in Model II, though, cannot be rejected at a 5\% significance level.
Does this mean that the double-log model appearing in Model II\ is most appropriate? This question cannot be determined solely based upon a test for heteroskedasticity. Ultimately, as noted in section \ref% {log_transform_sec_lc}, the choice between Model I and Model II depends on whether a one-unit change in the independent variable results in a constant change in the level of the dependent variable or whether a one-percent change in the level of the independent variable results in a constant percentage change in the level of the dependent variable. If economic theory and knowledge of institutional factors do not shed light on this issue, then the \textit{P-}test discussed in section \ref{p-test_spec_chap} may be used to guide the choice of appropriate model specification.
\section{Summary}
In this chapter, the effects of heteroskedasticity have been examined. In particular, the presence of heteroskedasticity causes OLS estimates to be inefficient. Furthermore, OLS standard errors are biased when this condition occurs. Thus, the effects of heteroskedasticity are quite similar to the effects resulting from the presence of autocorrelation.
An econometrician may test for the presence of heteroskedasticity by using the Goldfeld-Quandt test, the Breusch-Pagan test, or White’s test. If one of these tests indicates the presence of heteroskedasticity, efficient parameter estimates may be obtained through the use of a weighted-least squares estimation procedure. Alternatively, White’s procedure may be used to provide consistent estimates of the standard errors for the OLS parameter estimates.
Before leaving this chapter, you should be sure that you: \begin{itemize}
\item understand the effects of heteroskedasticity; \item know how to test for this violation of the classical regression model; and
\item understand how to implement a weighted-least squares estimation procedure.
\end{itemize}
\section{Key Concepts}
heteroskedasticity
weighted-least squares estimator
Goldfeld-Quandt test
Breusch-Pagan test
White’s test
White’s heteroskedasticity consistent variance-covariance estimator \section{Exercises and problems}
\begin{enumerate}
\item Provide an intuitive explanation of why a weighted least squares estimator is more efficient than an OLS estimator when heteroskedasticity is present.
\item If OLS\ estimators are still unbiased and consistent when heteroskedasticity is present, why should econometricians be concerned with heteroskedasticity?
\item Explain why either the Breusch-Pagan test or White’s test might be preferred to the Goldfeld-Quandt test.
\item Use the data in the file \textquotedblleft hdeaths.dat\textquotedblright\ (described in Table \ref{hdeaths.dat} in Appendix \ref{data.appendix}) to verify the Goldfeld-Quandt statistic appearing in equation \ref{GQ.example}. Report the results of both subsample regressions as part of your response.
\item
\begin{enumerate}
\item Verify the results for the Breusch-Pagan test reported in section \ref% {bp_drivers_hc}. Do your results match those presented in this section?
\item Consider an alternative specification of the auxiliary regression equation given by:%
\begin{equation*}
\frac{\hat{u}_{i}^{2}}{\hat{\sigma}^{2}}=\gamma _{o}+\gamma _{1}\text{Congest% }_{i}+\gamma _{2}\text{Drivers}_{i}+\gamma _{3}\text{MPH65}_{i}+\epsilon _{i} \end{equation*}%
Perform a Breusch-Pagan test using this specification at a 5\% significance level. Is the outcome of this test the same as in part (a)?
\end{enumerate}
\item
\begin{enumerate}
\item Verify the results for White’s test reported in section \ref% {white_dr_hc}. Do your estimates agree with those reported in this section?
\item Use a GLS estimation procedure to correct for heteroskedasticity in equation \ref{deaths.hc} using weights equal to the reciprocal of the square roots of the predicted values in the auxiliary regression equation. (Hint: Save the predicted values from the auxiliary regression in a variable called $Z$. If $Z$ takes on any nonpositive values for some observations, replace the affected observations with the original value of $\hat{u}_{i}^{2}$. Then use a weighted least squares estimator to re-estimate the parameters of equation \ref{deaths.hc} using weights equal to$1/\sqrt{Z}$.) Are these results similar to the GLS estimates in equation \ref{est.deaths.a.hc}?
\end{enumerate}
\item
\begin{enumerate}
\item Use the data in the file \textquotedblleft travel.dat\textquotedblright\ (described in Table \ref{travel.dat} on p. % \pageref{travel.dat}) to estimate the following model:% \begin{equation*}
\text{Travel}_{i}=\beta _{o}+\beta _{1}\text{Income}_{i}+u_{i} \end{equation*}%
(This type of relationship between expenditures on a good and income is called an \textquotedblleft Engels’ curve.\textquotedblright ) \item Provide a scatterplot of the data points in the above relationship.
Does this scatterplot suggest anything about the presence of heteroskedasticity?
\item Test for the presence of heteroskedasticity at a 5\% significance level using a Goldfeld-Quandt test. Omit the middle 11 observations when conducting this test. (Note that there are 51 observations since the District of Columbia is included in this sample.) What does this test indicate?
\item Test for the presence of presence of heteroskedasticity using a Breusch-Pagan test (at a 5\% significance level) using an auxiliary regression equation in which \textquotedblleft Income$_{i}$% \textquotedblright\ and \textquotedblleft Income$_{i}^{2}$% \textquotedblright\ are independent variables. Describe the results of this test.
\item Repeat part (d)\ using White’s test instead of the Breusch-Pagan test.
Does this test result in an equivalent outcome?
\item What can you conclude about the possibility of heteroskedasticity? If heteroskedasticity is found, apply an appropriate transformation of the variables and re-estimate the model in part (a).
\end{enumerate}
\item
\begin{enumerate}
\item Use the data in the file \textquotedblleft travel.dat\textquotedblright\ (described in Table \ref{travel.dat} on p. % \pageref{travel.dat}) to estimate the following three models:% \begin{equation*}
\text{Travel}_{i}=\beta _{o}+\beta _{1}\text{Income}_{i}+u_{i} \end{equation*}%
\begin{equation*}
\ln (\text{Travel}_{i})=\beta _{o}+\beta _{1}\ln (\text{Income}_{i})+u_{i} \end{equation*}%
\begin{equation*}
\frac{\text{Travel}_{i}}{\text{Pop}_{i}}=\beta _{o}+\beta _{1}\frac{\text{% Income}_{i}}{\text{Pop}_{i}}+u_{i}
\end{equation*}%
(Curves, such as these, that illustrate the relationship that exists between the level of expenditures on a good and income are called \textquotedblleft Engels’ curves.\textquotedblright )
\item Provide a scatterplot of the data points in each of the above relationships. For each graph, describe whether the error process appears to be heteroskedastic or homoskedastic.
\item Test for the presence of presence of heteroskedasticity in each of these equations using White’s test (at a 5\% significance level). Use the independent variable and the square of the independent variable as regressors in the auxiliary regressions (\textit{i.e.}. use \textquotedblleft Income$_{i}$ and \textquotedblleft Income$_{i}^{2}$% \textquotedblright\ in the first equation, $\ln ($Income$_{i})$ and ($\ln ($% Income$_{i}))^{2}$ in the second, \textit{etc}.). Describe the results of this test for each equation.
\item Does the use of alternative functional forms have an impact in this case?
\end{enumerate}
\item
\begin{enumerate}
\item Use the data in the file \textquotedblleft cars.dat\textquotedblright\ (described in Table \ref{cars.dat} on p. \pageref{cars.dat}) to estimate the following equation:%
\begin{equation*}
\text{City}_{i}=\beta _{o}+\beta _{1}\text{width}_{i}+\beta _{2}\text{length}% _{i}+\beta _{3}\text{height}_{i}+\beta _{4}\text{weight}_{i} \end{equation*}%
\begin{equation*}
+\beta _{5}\text{disp}_{i}+\beta _{6}\text{horse}_{i}+u_{i} \end{equation*}%
\begin{equation*}
\begin{array}{ll}
\text{where:} & \text{City}_{i}=\text{EPA city miles per gallon for car model }i \\
& \text{width}_{i}=\text{ width (in inches) of car model }i \\ & \text{length}_{i}=\text{ length (in inches) of car model }i \\ & \text{height}_{i}=\text{ height (in inches) of car model }i \\ & \text{weight}_{i}\text{ }=\text{ weight (in pounds) of car model }i \\ & \text{disp}_{i}=\text{ displacement of engine (in liters) for car model }i \\
& \text{horse}_{i}=\text{ horsepower of engine in car model }i% \end{array}%
\end{equation*}%
(Be sure to save the estimated residual terms from this equation for subsequent analysis.)
\item Conduct a Breusch-Pagan test to investigate the possibility of heteroskedasticity in this model using a specification for the auxiliary equation given by:%
\begin{equation*}
\frac{\hat{u}_{i}^{2}}{\hat{\sigma}^{2}}=\gamma _{o}+\gamma _{1}\text{width}% _{i}+\gamma _{2}\text{length}_{i}+\gamma _{3}\text{height}_{i}+\gamma _{4}% \text{weight}_{i}
\end{equation*}%
\begin{equation*}
+\gamma _{5}\text{disp}_{i}+\gamma _{6}\text{horse}_{i}+\epsilon _{i} \end{equation*}%
Using a 5\% significance level, what does this test indicate?
\item Conduct White’s test to investigate the possibility of heteroskedasticity in the model appearing in part (a) using a specification for the auxilary equation given by:%
\begin{equation*}
\frac{\hat{u}_{i}^{2}}{\hat{\sigma}^{2}}=\gamma _{o}+\gamma _{1}\text{width}% _{i}+\gamma _{2}\text{length}_{i}+\gamma _{3}\text{height}_{i}+\gamma _{4}% \text{weight}_{i}
\end{equation*}%
\begin{equation*}
+\gamma _{5}\text{disp}_{i}+\gamma _{6}\text{horse}_{i}+\epsilon _{i} \end{equation*}%
Using a 5\% significance level, what does this test indicate?
\item What can you conclude from the tests conducted in parts (b) and (c)?
If there is evidence of heteroskedasticity, apply an appropriate correction procedure to re-estimate the parameters of the equation in part (a).
\end{enumerate}
\item
\begin{enumerate}
\item Use the data in the file \textquotedblleft cars.dat\textquotedblright\ (described in Table \ref{cars.dat} on p. \pageref{cars.dat}) to estimate the following equation:%
\begin{equation*}
\text{High}_{i}=\beta _{o}+\beta _{1}\text{width}_{i}+\beta _{2}\text{length}% _{i}+\beta _{3}\text{height}_{i}+\beta _{4}\text{weight}_{i} \end{equation*}%
\begin{equation*}
+\beta _{5}\text{disp}_{i}+\beta _{6}\text{horse}_{i}+u_{i} \end{equation*}%
\begin{equation*}
\begin{array}{ll}
\text{where:} & \text{High}_{i}=\text{EPA highway miles per gallon for car model }i \\
& \text{width}_{i}=\text{ width (in inches) of car model }i \\ & \text{length}_{i}=\text{ length (in inches) of car model }i \\ & \text{height}_{i}=\text{ height (in inches) of car model }i \\ & \text{weight}_{i}\text{ }=\text{ weight (in pounds) of car model }i \\ & \text{disp}_{i}=\text{ displacement of engine (in liters) for car model }i \\
& \text{horse}_{i}=\text{ horsepower of engine in car model }i% \end{array}%
\end{equation*}%
(Be sure to save the estimated residual terms from this equation for subsequent analysis.)
\item Conduct a Breusch-Pagan test to investigate the possibility of heteroskedasticity in this model using a specification for the auxilary equation given by:%
\begin{equation*}
\frac{\hat{u}_{i}^{2}}{\hat{\sigma}^{2}}=\gamma _{o}+\gamma _{1}\text{width}% _{i}+\gamma _{2}\text{length}_{i}+\gamma _{3}\text{height}_{i}+\gamma _{4}% \text{weight}_{i}
\end{equation*}%
\begin{equation*}
+\gamma _{5}\text{disp}_{i}+\gamma _{6}\text{horse}_{i}+\epsilon _{i} \end{equation*}%
Using a 5\% significance level, what does this test indicate?
\item Conduct White’s test to investigate the possibility of heteroskedasticity in the model appearing in part (a) using a specification for the auxilary equation given by:%
\begin{equation*}
\frac{\hat{u}_{i}^{2}}{\hat{\sigma}^{2}}=\gamma _{o}+\gamma _{1}\text{width}% _{i}+\gamma _{2}\text{length}_{i}+\gamma _{3}\text{height}_{i}+\gamma _{4}% \text{weight}_{i}
\end{equation*}%
\begin{equation*}
+\gamma _{5}\text{disp}_{i}+\gamma _{6}\text{horse}_{i}+\epsilon _{i} \end{equation*}%
Using a 5\% significance level, what does this test indicate?
\item What can you conclude from the tests conducted in parts (b) and (c)?
If there is evidence of heteroskedasticity, apply an appropriate correction procedure to re-estimate the parameters of the equation in part (a).
\end{enumerate}
\item Use the data in the file \textquotedblleft crime.dat\textquotedblright\ (described in Table \ref{crime.dat} in Appendix % \ref{data.appendix}) to estimate the parameters of the following equation: \begin{equation}
\text{Crime}_{i}=\beta _{o}+\beta _{1}\text{Pov}_{i}+\beta _{2}\text{Metro}% _{i}+\beta _{3}\text{PopDens}_{i}+u_{i} \label{crime.het} \end{equation}%
\begin{equation*}
\begin{array}{ll}
\text{where:} & \text{Crime}_{i}\text{ = total crime rate per 100,000 population in state }i \\
& \text{Pov}_{i}\text{ = proportion of the population below the poverty line} \\
& \text{Metro}_{i}\text{ = metropolitan population as percentage of state population} \\
& \text{PopDens}_{i}\text{ = population per square mile}% \end{array}%
\end{equation*}
\begin{enumerate}
\item Perform either a Breusch-Pagan or White test at a 5\% significance level.
\item Do your results indicate the presence of heteroskedasticity? If so, apply an appropriate GLS\ correction technique and report the resulting estimates.
\item Would heteroskedasticity have been more or less likely if the data was measured in the form of totals (such as the number of total crimes, and the number of individuals below the poverty line), rather than as rates?
\end{enumerate}
\item Consider the following regression model of voting behavior in the 1992 U.S. Presidential election:
\begin{equation}
\text{DVOTE}_{i}\text{ = }\beta _{o}+\beta _{1}\text{UN}_{i}+\beta _{2}\text{% FedFunds}_{i}+\beta _{3}\text{Defense}_{i} \label{dvote.het} \end{equation}%
\begin{equation*}
+\beta _{4}\text{Crime}_{i}+u_{i}
\end{equation*}%
\begin{equation*}
\begin{array}{ll}
\text{where:} & \text{DVOTE}_{i}\text{ = proportion of state vote cast for the Democratic candidate} \\
& \text{UN}_{i}\text{ = unemployment rate in state }i \\ & \text{FedFunds}_{i}\text{ = \textit{per capita} federal spending in state }% i \\
& \text{Defense}_{i}\text{ = defense contract awards per 1,000 population in state }i \\
& \text{Crime}_{i}\text{ = total crime rate per 100,000 population in state }% i%
\end{array}%
\end{equation*}
\begin{enumerate}
\item Estimate the parameters of equation \ref{dvote.het} using the data in the file \textquotedblleft election.dat\textquotedblright\ (described in Table \ref{election.dat} in Appendix \ref{data.appendix} ).
\item Use a Breusch-Pagan test (at a 5\% significance level) to evaluate the possibility of heteroskedasticity. What does this test suggest?
\end{enumerate}
\item Use the data in the file \textquotedblleft inflation.dat\textquotedblright\ to estimate the parameters of the inflation-monetary growth model discussed in Chapter \ref{intro.chap}: \begin{equation*}
\widehat{\text{Inflation}}_{i}=\beta _{o}+\beta _{1}\text{MGrowth}_{i} \end{equation*}
\begin{enumerate}
\item Conduct a Goldfeld-Quandt test (at a 5\% significance level) to determine whether the variance of the residuals varies with the rate of monetary growth.
\item Conduct a Breusch-Pagan test (at a 5\% significance level) to determine whether the variance of the residuals varies with the rate of monetary growth.
\item Use White’s test (at a 5\% significance level) to determine whether the variance of the residuals varies with the rate of monetary growth.
\item If any one of these tests indicates that heteroskedasticity is present, use a GLS estimation procedure to correct for its presence.
\end{enumerate}
\item
\begin{enumerate}
\item Use the data in the file \textquotedblleft healthex.dat\textquotedblright\ (described in Table \ref{healthex.dat} on p. % \pageref{healthex.dat}) to verify the estimated equations in Models I, II, and II in section \ref{het_func_form_hc}.
\item At a 5\% significance level, use a Breusch-Pagan test to test for heteroskedasticity in each of these models (use the independent variable and the square of the independent variable in the original equation as regressors in the corresponding auxiliary equation). Do the results agree with those reported for White’s test?
\end{enumerate}
\item
\begin{enumerate}
\item Use the data in the file \textquotedblleft healthex.dat\textquotedblright\ (described in Table \ref{healthex.dat} on p. % \pageref{healthex.dat}) to estimate the parameters of the following equations:%
\begin{equation*}
\text{Hosp}_{i}=\beta _{o}+\beta _{1}\text{Income}_{i}+u_{i} \end{equation*}%
\begin{equation*}
\ln (\text{Hosp}_{i})=\beta _{o}+\beta _{1}(\text{Income}_{i})+v_{i} \end{equation*}%
\begin{equation*}
\frac{\text{Hosp}_{i}}{\text{Pop}_{i}}=\beta _{o}+\beta _{1}\frac{\text{% Income}_{i}}{\text{pop}_{i}}+w_{i}
\end{equation*}
\item At a 5\% significance level, use a Goldfeld-Quandt test to test for the presence of heteroskedasticity in each of these equations (omit the middle 5 observations). What does this test indicate in each of these models?
\end{enumerate}
\item
\begin{enumerate}
\item Use the data in the file \textquotedblleft healthex.dat\textquotedblright\ (described in Table \ref{healthex.dat} on p. % \pageref{healthex.dat}) to estimate the parameters of the following equations:%
\begin{equation*}
\text{Phys}_{i}=\beta _{o}+\beta _{1}\text{Income}_{i}+u_{i} \end{equation*}%
\begin{equation*}
\ln (\text{Phys}_{i})=\beta _{o}+\beta _{1}(\text{Income}_{i})+v_{i} \end{equation*}%
\begin{equation*}
\frac{\text{Phys}_{i}}{\text{Pop}_{i}}=\beta _{o}+\beta _{1}\frac{\text{% Income}_{i}}{\text{pop}_{i}}+w_{i}
\end{equation*}
\item At a 5\% significance level, use White’s test to test for the presence of heteroskedasticity in each of these equations. What does this test indicate in each of these models?
\end{enumerate}
\item
\begin{enumerate}
\item Use the data in the file \textquotedblleft healthex.dat\textquotedblright\ (described in Table \ref{healthex.dat} on p. % \pageref{healthex.dat}) to estimate the parameters of the following equations:%
\begin{equation*}
\text{Drugs}_{i}=\beta _{o}+\beta _{1}\text{Income}_{i}+u_{i} \end{equation*}%
\begin{equation*}
\ln (\text{Drugs}_{i})=\beta _{o}+\beta _{1}(\text{Income}_{i})+v_{i} \end{equation*}%
\begin{equation*}
\frac{\text{Drugs}_{i}}{\text{Pop}_{i}}=\beta _{o}+\beta _{1}\frac{\text{% Income}_{i}}{\text{pop}_{i}}+w_{i}
\end{equation*}
\item At a 5\% significance level, use the Breusch-Pagan test to test for the presence of heteroskedasticity in each of these equations. What does this test indicate in each of these models?
\end{enumerate}
\item
\begin{enumerate}
\item Use the data in the file \textquotedblleft schools.dat\textquotedblright\ (described in Table \ref{schools.dat} in Appendix \ref{data.appendix} ) to estimate the parameters of the following equation:
\begin{equation*}
\text{RD}_{i}=\beta _{o}+\beta _{1}\text{LUN}_{i}+u_{i} \end{equation*}%
\begin{equation*}
\begin{array}{llll}
\text{where:} & \text{RD}_{i} & = & \text{percent of 3rd grade students in school }i\text{ achieving } \\
& & & \text{reading \textquotedblleft mastery\textquotedblright } \\ & \text{LUN}_{i}\text{ } & = & \text{\% of students in elementary school }i% \text{ eligible for } \\
& & & \text{school lunch program}%
\end{array}%
\end{equation*}
\item Use a Goldfeld-Quandt test (at a 5\% significance level) to examine the possibility of heteroskedasticity in the residuals of this equation.
\item Use White’s test (at a 5\% significance level) to examine the possibility of heteroskedasticity in the residuals of this equation.
\item If heteroskedasticity is found, apply an appropriate GLS correction technique.
\end{enumerate}
\item
\begin{enumerate}
\item Use the data in the file \textquotedblleft life.dat\textquotedblright\ (described in Table \ref{life.dat} in Appendix \ref{data.appendix}) to estimate the parameters of the equation:
\begin{equation*}
\text{LifeEx}_{i}=\beta _{o}+\beta _{1}\text{TV}_{i}+\beta _{2}\text{% PopPerDoc}_{i}+\beta _{3}\text{GDP}_{i}+u_{i} \end{equation*}%
\begin{equation*}
\begin{array}{ll}
\text{where:} & \text{LifeEx}_{i}\text{ = life expectancy at birth in country }i \\
& \text{TV}_{i}\text{ = televisions / 100 people in country }i \\ & \text{PopPerDoc}_{i}\text{ = population per physician in country }i \\ & \text{GDP}_{i}\text{ = real \textit{per capita }GDP\ in country }i% \end{array}%
\end{equation*}
\item Use White’s test (at a 5\% significance level) to investigate the possibility of heteroskedasticity.
\item If heteroskedasticity is found, apply an appropriate GLS estimation technique.
\end{enumerate}
\item
\begin{enumerate}
\item Use the data in the file \textquotedblleft ceo.dat\textquotedblright\ to estimate the parameters of the following equation:% \begin{equation*}
\text{Totcomp}_{i}=\beta _{o}+\beta _{1}\text{Sales}_{i} \end{equation*}
\item At a 1\% significance level, test for the presence of heteroskedasticity using White’s test. If heteroskedasticity is found, apply an appropriate GLS estimation technique and re-estimate the equation.
\end{enumerate}
\item Ramsey’s RESET test (discussed in Chapter \ref{spec.chap}) can also be used as a test for heteroskedasticity. Is there any similarity between the RESET test and White’s test? Explain. Why might White’s test be preferred to Ramsey’s test?
\end{enumerate}
\newpage\
\section{Appendix}
\subsection{ ARCH models}
Engle (1982) observed that in many time-series processes, the variance of residuals varied in a systematic manner over time. In particular, Engle noted that the forecasts of econometricians were more accurate in some time periods than others. It appears that in many cases, the variance of the error term appears to gradually increase or decrease over time.
Under an autoregressive conditional heteroskedasticity (ARCH) model, it is assumed that the variance of the residual at time $t$ is a function of lagged residual variances. In particular, a $p$th-order ARCH model is given by:
\begin{equation}
Y_{t}=\beta _{o}+\beta _{1}X_{1t}+\beta _{2}X_{2t}+\cdots +\beta _{k}X_{kt}+u_{t} \label{arch.reg.hc}
\end{equation}%
\begin{equation}
\text{where: }\sigma _{t}^{2}=\alpha _{o}+\alpha _{1}\sigma _{t-1}^{2}+\alpha _{2}\sigma _{t-2}^{2}+\cdots +\alpha _{p}\sigma _{t-p}^{2} \label{arch.hc}
\end{equation}%
An examination of equation \ref{arch.hc} provides an explanation for the name of this model. In an ARCH model, the variance of the error term at time $t$ is assumed to follow a $p$th-order autoregressive process. It has been found that many financial time-series and exchange-rate models have error terms that follow an ARCH process.
Since an ARCH model involves a special case of heteroskedastic residuals, OLS estimators are unbiased and consistent, but are inefficient.\footnote{% As noted by Greene (2000, pp. 797-9), OLS estimators are still BLUE if the unconditional variance of the error terms remains constant over time. Even in this case, however, maximum likelihood estimators provide an efficiency gain.} Engle suggests the use of a maximum likelihood estimator when an ARCH model is present.
A\ test for the presence of a $p$th-order ARCH process can be conducted by testing the null hypothesis:
\begin{equation*}
\text{H}_o\text{: }\alpha _o=\alpha _1=\cdots =\alpha _p=0 \end{equation*}
If this null hypothesis is rejected, then it may be assumed that error terms follows an ARCH process. The Lagrange multiplier test introduced in Chapter % \ref{spec.chap} provides a simple method for testing this hypothesis: \begin{enumerate}
\item[Step 1:] Estimate the parameters of the original regression equation (equation \ref{arch.reg.hc}) by OLS.
\item[Step 2:] Use the estimated residuals from this equation to formulate the auxiliary regression equation:
\begin{equation*}
\hat{u}_t^2=\gamma _o+\gamma _1\hat{u}_{t-1}^2+\gamma _2\hat{u}% _{t-2}^2+\cdots +\gamma _p\hat{u}_{t-p}^2+\epsilon _t \end{equation*}
This auxiliary regression equation is the empirical counterpart of equation % \ref{arch.hc}. This equation is estimated using $N-p$ observations due to the use of the lagged squared error terms.
\item[Step 3:] Construct the Lagrange multiplier statistic: \begin{equation*}
\text{Lagrange multiplier statistic = }\left( N-p\right) R^2 \end{equation*}
Under the null hypothesis of a constant residual variance, the Lagrange multiplier statistic is distributed as a $\chi ^2$ statistic on $p$ degrees of freedom.
\item[Step 4:] Reject the null hypothesis and assume that the error process follows an ARCH($p$) model if the Lagrange multiplier statistic exceeds the critical value for a $\chi ^2$ distribution.
\end{enumerate}
If the Lagrange multiplier test indicates that the error terms follow an ARCH process, then a maximum likelihood procedure should be used to estimate the parameters of this equation.
\subsection{Example: interest rate determination} A simple model may be used to represent the effect of fiscal and monetary policy changes on interest rates:%
\begin{equation*}
\text{interest}_{t}=\beta _{o}+\beta _{1}\text{deficit}_{t-1}+\beta _{2}% \text{deficit}_{t-2}+\beta _{3}\text{deficit}_{t-3}+\beta _{4}\text{deficit}% _{t-4}
\end{equation*}%
\begin{equation*}
+\beta _{5}\text{M2}_{t-1}+\beta _{6}\text{M2}_{t-2}+\beta _{7}\text{M2}% _{t-3}+\beta _{8}\text{M2}_{t-4}+u_{t}
\end{equation*}%
When an OLS estimation procedure is applied, the following equation is estimated:%
\begin{equation*}
\text{interest}_{t}=\underset{(16.53)}{6.01}+\underset{(0.09)}{0.00079}\text{% deficit}_{t-1}-\underset{(-0.622)}{0.0068}\text{deficit}_{t-2}-\underset{% \left( -0.26\right) }{0.0034}\text{deficit}_{t-3} \end{equation*}%
\begin{equation*}
+\underset{(1.94)}{0.02}\text{deficit}_{t-4}+\underset{(1.68)}{0.028}\text{M2% }_{t-1}-\underset{(-0.54)}{0.017}\text{M2}_{t-2} \end{equation*}%
\begin{equation*}
+\underset{(0.65)}{0.021}\text{M2}_{t-3}-\underset{(-1.97)}{0.033}\text{M2}% _{t-4}+u_{t}
\end{equation*}%
\begin{equation*}
(t\text{-statistics in parentheses})
\end{equation*}%
To test for the presence of ARCH effects, the following equation was estimated:%
\begin{equation*}
\hat{u}_{t}^{2}=\underset{(1.68)}{0.951}+\underset{(11.79)}{0.905}\hat{u}% _{t-1}^{2}-\underset{(-2.70)}{0.269}\hat{u}_{t-2}^{2}+\underset{(4.45)}{0.454% }\hat{u}_{t-3}^{2}-\underset{(-2.96)}{0.234}\hat{u}_{t-4}^{2} \end{equation*}%
\begin{equation*}
\text{R}^{2}=0.744
\end{equation*}%
\begin{equation*}
N-4=166
\end{equation*}%
\begin{equation*}
\text{Lagrange multiplier statistic = }N\times \text{R}^{2}=123.54 \end{equation*}%
Since the critical value for a $\chi ^{2}$ variate with 4 degrees of freedom equals 13.28 at a 1\% significance level, this tests indicates the presence of significant ARCH effects.
When a maximum likelihood estimator is used to estimate this model, the following equation results:%
\begin{equation*}
\text{interest}_{t}=\underset{(22.87)}{5.99}-\underset{(-1.71)}{0.0082}\text{% deficit}_{t-1}+\underset{(2.38)}{0.0148}\text{deficit}_{t-2}-\underset{% \left( -1.47\right) }{0.0118}\text{deficit}_{t-3} \end{equation*}%
\begin{equation*}
+\underset{(2.08)}{0.0132}\text{deficit}_{t-4}+\underset{(1.91)}{0.0148}% \text{M2}_{t-1}-\underset{(-0.32)}{0.00529}\text{M2}_{t-2} \end{equation*}%
\begin{equation*}
+\underset{(1.76)}{0.0338}\text{M2}_{t-3}-\underset{(-4.73)}{0.0442}\text{M2}% _{t-4}+u_{t}
\end{equation*}%
\begin{equation*}
(t\text{-statistics in parentheses})
\end{equation*}
\subsection{Extensions of the ARCH model} In recent years, there have been a wide range of extensions to the basic ARCH model proposed by Engle (1982). The generalized autoregressive conditional autoregressive heteroskedasticity (GARCH) model adds moving average terms to the ARCH specification.\footnote{% Autoregressive and moving average processes will be discussed in more detail in Chapter \ref{ARIMA.chap}.} IGARCH, EGARCH and FACTOR-ARCH models are recent further extensions of the GARCH\ model. A discussion of these models is beyond the scope of the current text. Interested readers may find a discussion of these models in Engle (1995), Gouri\'{e}roux (1997), or Mills (1999).
\subsection{Caution:\ ARCH and Durbin-Watson test} When an ARCH process is present in the error terms, Durbin-Watson statistics derived from OLS regressions are unreliable. If an ARCH\ process is suspected, it is important to test for an ARCH process before using the Durbin-Watson statistic to test for the presence of an autoregressive error process.\footnote{%
The combination of autoregressive and moving average terms to model a time-series process will be discussed in Chapter \ref{ARIMA.chap}.}