Category Archives: Remedial Measures of Heteroscedasticity

Unbiased and inconsistent estimates in presence of Heteroscedasticity need some remedial measures

Remedial Measures for Heteroscedasticity

The OLS estimators remains unbiased and consistent in the presence of Heteroscedasticity, but they are no longer efficient not even asymptotically. This lack of efficiency makes the usual hypothesis testing procedure of dubious value. Therefore remedial measures may be called. There are two approaches for remedial measures of heteroscedasticity

(i) $\sigma_i^2$ is known

Consider the simple linear regression model Yi=α+βXii.

If $V(\mu_i)=\sigma_i^2$ then heteroscedasticity is present. Given the values of $\sigma_i^2$ heteroscedasticity can be corrected by using weighted least squares (WLS) as a special case of Generalized Least Square (GLS). Weighted least squares is the OLS method of estimation applied to the transformed model.

When heteroscedasticity is detected by any appropriate statistical test, then appropriate solution is transform the original model in such a way that the transformed disturbance term has constant variance. The transformed model reduces to the adjustment of the original data. The transformed error term μi has a constant variance i.e. homoscedastic. Mathematically

\begin{eqnarray*}
V(\mu_i^*)&=&V\left(\frac{\mu_i}{\sigma_i}\right)\\
&=&\frac{1}{\sigma_i^2}Var(\mu_i)\\
&=&\frac{1}{\sigma_i^2}\sigma_i^2=1
\end{eqnarray*}

This approach has its limited use as the individual error variance are not always known a priori. In case of significant sample information, reasonable guesses of the true error variances can be made and be used for $\sigma_i^2$.

(ii) When $\sigma_i^2$ is unknown

If $\sigma_i^2$ is not known a priori, then heteroscedasticity is corrected by hypothesizing a relationship between the error variance and one of the explanatory variables. There can be several versions of the hypothesized relationship. Suppose hypothesized relationship is $Var(\mu)=\sigma^2 X_i^2$ (error variance is proportional to $X_i^2$). For this hypothesized relation we will use the following transformation to correct for heteroscedasticity for the following simple linear regression model Yi=α+βXii.
\begin{eqnarray*}
\frac{Y_i}{X_i}&=&\frac{\alpha}{X_i}+\beta+\frac{\mu_i}{X_i}\\
\Rightarrow \quad Y_i^*&=&\beta +\alpha_i^*+\mu_i^*\\
\mbox{where } Y_i^*&=&\frac{Y_i}{X_i}, \alpha_I^*=\frac{1}{X_i} \mbox{and  } \mu_i^*=\frac{\mu}{X_i}
\end{eqnarray*}

Now the OLS estimation of the above transformed model will yield the efficient parameter estimates as $\mu_i^*$’s have constant variance. i.e.

\begin{eqnarray*}
V(\mu_i^*)&=&V(\frac{\mu_i}{X_i})\\
&=&\frac{1}{X_i^2} V(\mu_i^2)\\
&=&\frac{1}{X_i^2}\sigma^2X_i^2\\
&=&\sigma^2=\mbox{ Constant}
\end{eqnarray*}

For correction of heteroscedasticity some other hypothesized relations are

  • Error variance is proportional to Xi (Square root transformation) i.e $E(\mu_i^2)=\sigma^2X_i$
    The transformed model is
    \[\frac{Y_i}{\sqrt{X_i}}=\frac{\alpha}{\sqrt{X_i}}+\beta\sqrt{X_i}+\frac{\mu_i}{\sqrt{X_i}}\]
    It (transformed model) has no intercept term. Therefore we have to use the regression through the origin model to estimate $\alpha$ and β.To get original model, multiply $\sqrt{X_i}$ with transformed model.
  • Error Variance is proportional to the square of the mean value of Y. i.e. $E(\mu_i^2)=\sigma^2[E(Y_i)]^2$
    Here the variance of $\mu_i$ is proportional to the square of the expected value of Y, and E(Yi)=α+βxi.
    The transformed model will be
    \[\frac{Y_i}{E(Y_i)}=\frac{\alpha}{E(Y_i)}+\beta\frac{X_i}{E(Y_i)}+\frac{\mu_i}{E(Y_i)}\]
    This transformation is not appropriate because E(Yi) depends upon $\alpha$ and β which are unknown parameters. $\hat{Y_i}=\hat{\alpha}+\hat{\beta}$ is an estimator of E(Yi), so we will proceed in two steps

    1. We run the usual OLS regression dis-regarding the heteroscedasticity problem and obtain $\hat{Y_i}$
    2. We will transform the model by using estimated $\hat{Y_i}$ i.e. $\frac{Y_i}{\hat{Y_i}}=\alpha\frac{1}{\hat{Y_i}}+\beta_1\frac{X_i}{\hat{Y_i}}+\frac{\mu_i}{\hat{Y_i}}$ and run the regression on transformed model.

    This transformation will perform satisfactory results only if the sample size is reasonably large.

  • Log transformation such as ln Yi=α+β ln Xii
    Log transformation compresses the scales in which the variables are measured. But this transformation is not applicable if some of the Y and X values are zero or negative.

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