Consequences of Heteroscedasticity for OLS

When heteroscedasticity is present in data, then estimates based on Ordinary Least Square (OLS) are subjected to following consequences:

1. We cannot apply the formula of the variance of the coefficients to conduct tests of significance and construct confidence intervals.
2. If error term ($\mu_i$) is heteroscedastic, then the OLS estimates do not have the minimum variance property in the class of unbiased estimators, i.e. they are inefficient in small samples. Furthermore they are asymptotically inefficient.
3. The estimated coefficients remain unbiased statistically. That means the property of unbiasedness of OLS estimation is not violated by the presence of heteroscedasticity.
4. The forecasts based on the model with heteroscedasticity will be less efficient as OLS estimation yield higher values of the variance of the estimated coefficients.

All this means the standard errors will be underestimated and the t-statistics and F-statistics will be inaccurate, caused by a number of factors, but the main cause is when the variables have substantially different values for each observation. For instance GDP will suffer from heteroscedasticity if we include large countries such as the USA and small countries such as Cuba. In this case it may be better to use GDP per person. Also note that heteroscedasticity tends to affect cross-sectional data more than time series.

Consider the simple linear regression model (slrm)

The OLS estimate of $\hat{\beta}$ and $\alpha$ are

\begin{align*}
\hat{\beta}&=\frac{\sum x_i y_i}{\sum x_i^2}=\frac{\sum x_i (\beta x_i +\epsilon_i)}{\sum x_i^2}\\
&=\beta\frac{\sum x_i^2}{\sum x_i^2}+\frac{\sum x_i \epsilon_i}{\sum x_i^2}\\
&=\beta + \frac{\sum x_i \epsilon_i}{\sum x_i^2}
\end{align*}

Applying expectation on both sides we get:

\[E(\hat{\beta}=\beta+\frac{\sum E(x_i \epsilon_i)}{\sum x_i^2}=\beta \qquad E(\epsilon_i x_i)=0\]

Similarly

\begin{align*}\hat{\alpha}&=\overline{y}-\hat{\beta}\overline{X}\\
&=\alpha+\beta\overline{X}+\overline{\epsilon}-\hat{\beta}\overline{X}\\
&=\alpha+\beta\overline{X}+0-\overline{X}\beta=\alpha
\end{align*}

Hence, unbiasedness property of OLS estimation is not affected by Heteroscedasticity.

References:

• https://en.wikipedia.org/wiki/Heteroscedasticity
• http://www3.wabash.edu/econometrics/econometricsbook/chap19.htm