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I am studying the factors influencing the annual salary for employees at a undisclosed bank. The regression model that I have decided to employ is as follows:

\begin{equation} Y_{k}=\beta_{1}+\beta_{2}E_{k}+\beta_{3}D_{gk}+\beta_{4}D_{mk}+\beta_{5}D_{2k}+\beta_{6}D_{3k}+\varepsilon_{k} \end{equation} where $Y_{k}$ is the logarithm of annual salary, $E$ is the number of years of education, $D_{g}$ is a gender dummy, $D_{m}$ is a minority dummy, and where

\begin{equation} D_{2}=\begin{cases} 1 &\text{Custodial job} \\ 0 & \text{Otherwise} \end{cases} \end{equation}
and \begin{equation} D_{3}=\begin{cases} 1 &\text{Management job} \\ 0 & \text{Otherwise} \end{cases} \end{equation} As you know, whenever one deals with GLS, $\Omega$ will almost surely be unknown and thus have to be estimated. In general there are $\frac{n(n+1)}{2}$ parameters to be estimated, which makes it pretty impossible to come up with a viable estimation out of $n$ observations. This is usually counteracted by imposing some structure on $\Omega$.

In my case, I would like to make the assumption that the disturbance terms $\varepsilon_{k}$ in the above regression model have variance $\sigma_{i}^{2}$ for $i=1,2,3$, according to whether the $i$-th employee has a job in category 1,2, or 3 respectively. Now, we may introduce the transformations $\gamma_{1}=\log (\sigma_{1}^{2}),\gamma_{2}=\log(\sigma_{2}^{2}/\sigma_{1}^{2})$, and $\gamma_{3}=\log(\sigma_{3}^{2}/\sigma_{1}^{2})$ so as to enable us to formulate the following model for

\begin{equation} \sigma_{k}^{2}= \exp \{ \gamma_{1}+\gamma_{2}D_{2k}+\gamma_{3}D_{3k} \} \end{equation} Since $\hat{\beta}_\rm{OLS}$ is a consistent estimate of $\beta$, even under the assumption of heteroscedasticity, we have that $\hat{\beta}_{\rm OLS} \xrightarrow[]{p}\beta$ as the number of observations increase. We may therefore argue that $e_{k}^{2} \approx \sigma_{k}^{2}$, and so we can regress upon information that we already possess.

Summary of procedure

(1) Calculate the OLS estimate.

(2) Calculate the OLS residual $\textbf{e}=\textbf{Y}-\textbf{X}\hat{\beta}$

(3) Calculate the OLS estimate of $\gamma$ from $e_{k}^{2}=f_{\gamma}(Z_{k})+\overline{\varepsilon}_{k}$.

(4) Calculate the FGLS estimate as the GLS estimate with $\hat{\Omega}=\Omega(\hat{\gamma})$ in place of $\Omega$.

What I would like to know is whether or not one can perform this estimation using a known function in R, say gls? If the answer is yes, then how exactly should I write to ensure that that my heteroscedasticity assumption is taken into account? Thanks for taking the time! Have a great day!

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2 Answers 2

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Estimating Regression Models with Multiplicative Heteroscedasticity

The model that you have described is discussed in Harvey (1976).

Let me rewrite the model $$ \begin{align} \mathbb{E}(Y_i \mid \mathbf{X}_i, \mathbf{Z}_i) &= \mathbf{X}_i'\boldsymbol{\beta} \\ \mathbb{V}(Y_i \mid \mathbf{X}_i, \mathbf{Z}_i) &\equiv \sigma^2_i \\ &= \exp(\mathbf{Z}_i'\boldsymbol{\alpha}) \\ \end{align} $$ Note that it is possible that $\mathbf{X}_i$ and $\mathbf{Z}_i$ have common elements.

Rewriting the conditional mean equation equivalently as $$ \begin{align} Y_i &= \mathbf{X}_i'\boldsymbol{\beta} + \varepsilon_i \\ \mathbb{E}(\varepsilon_i \mid \mathbf{X}_i, \mathbf{Z}_i) &= 0 \end{align} $$

Two-step estimation

The main aim of Harvey (1976) is to provide (efficient) estimates of $\boldsymbol{\alpha}$, rather than to use that estimate to compute a WLS estimator. However, once the parameters $\boldsymbol{\alpha}$ are computed, they can be so used.

The two-step estimator that you have described, follows the procedure:
1. Compute the OLS estimator $\hat{\boldsymbol{\beta}}$, and the OLS residuals $\hat{\varepsilon}_i^2$,
2. Compute the estimates $\hat{\boldsymbol{\alpha}}$, from the regression
$$ \log \hat{\varepsilon}_i^2 = \mathbf{Z}_i'\boldsymbol{\alpha} + \nu_i $$

Maximum likelihood estimation

The MLE is claimed to be up to 60% more efficient than the two-step estimator above. There are some other advantages that are apparent to me including that the MLE would be the pseudo- maximum likelihood under distributional misspecification, and that there is no need to recompute the WLS after computing $\hat{\boldsymbol{\alpha}}$. However, this does not seem to be borne out by my calculations below.

Under conditional normality of the response, the likelihood is very simple to write down and optimize

$$ \begin{align} \log L_i(\boldsymbol{\beta}, \boldsymbol{\alpha}) &= -\frac{1}{2}(\log2 \pi + \mathbf{Z}_i'\boldsymbol{\alpha})\\ &\qquad -\frac{1}{2}\left(\dfrac{(Y_i - \mathbf{X}_i'\boldsymbol{\beta})^2}{\exp(\mathbf{Z}_i'\boldsymbol{\alpha})}\right) \end{align} $$

Two-step estimation: R

To implement this, first let us simulate some heteroskedastic data using the model given, and estimate it using OLS.

#==========================================================
# simulate the heteroskedastic data
#==========================================================
iN = 1000
iK1 = 7
iK2 = 4

mX = cbind(1, matrix(rnorm(iN*iK1), nrow = iN, ncol = iK1))
mZ = cbind(1, matrix(rnorm(iN*iK2), nrow = iN, ncol = iK2))

vBeta = rnorm(1 + iK1)
vAlpha = rnorm(1 + iK2)

vY = rnorm(iN, mean = mX %*% vBeta, sd = sqrt(exp(mZ %*% vAlpha)))

#==========================================================
# fit the data using OLS
#==========================================================
vBetaOLS = coef(lmHetMean <- lm.fit(y = vY, x = mX))

Next, we can get the results using the two-step procedure:

#==========================================================
# two-step estimation
#==========================================================
residHet = resid(lmHetMean)
vVarEst = exp(fitted(lmHetVar <- lm.fit(y = log(residHet^2), x = mZ)))

vBetaTS = coef(lm.fit(y = vY/vVarEst, x = apply(mX, 2, function(x) x/vVarEst)))

Maximum likelihood estimation: R

#==========================================================
# likelihood function
#==========================================================
fnLogLik = function(vParam, vY, mX, mZ) {
  vBeta = vParam[1:ncol(mX)]
  vAlpha = vParam[(ncol(mX)+1):(ncol(mX)+ncol(mZ))]

  negLogLik = -sum(0.5*(log(2*pi) - mZ %*% vAlpha - 
                           (vY - mX %*% vBeta)^2/(exp(mZ %*% vAlpha))))
  return(negLogLik)
}

# test the function
# debugonce(fnLogLik)
fnLogLik(c(vBeta, vAlpha), vY, mX, mZ)

#==========================================================
# MLE
#==========================================================
vParam0 = rnorm(13)
optimHet = optim(vParam0, fnLogLik, vY = vY, mX = mX, mZ = mZ)
vBetaML = optimHet$par

#==========================================================
# collect all the results
#==========================================================
cbind(vBeta, vBetaOLS, vBetaTS, vBetaML = vBetaML[1:8])

I am not entirely sure why the ML results are a bit farther off than even the OLS, and I am not ruling out a coding mistake.

But as you can see, the 2-step estimator seems to do better than the OLS estimator.

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There should also be a minus sign before $\log(2\pi)$ in negLogLik since this term appears in the denominator of the likelihood function, see https://en.wikipedia.org/wiki/Maximum_likelihood_estimation#Non-independent_variables. However, this change is a neutral operation in a minimization problem since $\log(2\pi)$ is a constant term.

When you replace the optim function by the nlminb function (arguments do not have to be altered), the ML estimator of beta will be much closer to the original beta vector. Probably, optim just does not work as well as nlminb in this situation. For nlminb see https://stat.ethz.ch/R-manual/R-devel/library/stats/html/nlminb.html.

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