OLS Estimation in the Presence of Heteroscedasticity

For the OLS Estimation in the presence of heteroscedasticity, consider the two-variable model

Y_i &= \beta_1 +\beta_2X_i + u_i\\
\hat{\beta}_2&=\frac{\sum x_i y_i}{\sum x_i^2}\\
Var(\hat{\beta}_2)&= \frac{\sum x_i^2\, \sigma_i^2}{(\sum x_i^2)^2}

OLS Estimation in the Presence of Heteroscedasticity, the variance of OLS estimator will be

$Var(\hat{\beta}_2)$ under the assumption of homoscedasticity is $Var(\hat{\beta}_2)=\frac{\sigma^2}{\sum x_i^2}$. If $\sigma_i^2=\sigma^2$ the both $Var(\hat{\beta}_2)$ will be same.

Note that in case of heteroscedasticity, the OLS estimators

  • $\hat{\beta_2}$ is BLUE if the assumptions of the classical model, including homoscedasticity, hold.
  • To establish the unbiasedness of $\hat{\beta}_2$, it is not necessary for that the disturbances ($u_i$) be homoscedastic.
  • In fact, the variance of $u_i$, homoscedasticity, or heteroscedasticity plays no part in the determination of the unbiasedness property.
  • $\hat{\beta}_2$ will be a consistent estimator despite heteroscedasticity.
  • With the increase of sample size indefinitely, the $\hat{\beta}_2$ (estimated $\beta_2$) converges to its true value.
  • $\hat{\beta}_2$ is asymptotically normally distributed.

For AR(1) the two variable model will be $Y_t=\beta_1+\beta_2 X_2+u_t$.

The variance of $\hat{\beta}_2$ for AR(1) scheme is

$$Var(\hat{\beta}_2)_{AR(1)} = \frac{\sigma^2}{\sum x_i^2}\left[ 1+ 2 \rho \frac{\sum x_t x_{t-1}}{\sum x_t^2} +2\rho^2 \frac{\sum x_t x_{t-2}}{\sum x_t^2} +\cdots + 2\rho^{n-1} \frac{x_tx_n}{\sum x_t^2} \right]$$

If $\rho=0$ then $Var(\hat{\beta}_2)_{AR(1)} = Var(\hat{\beta}_2)_{OLS}$.

Assume that the regressors $X$ also follows the AR(1) scheme with a coefficient of autocorrelation for $r$, then

Var(\hat{\beta}_2)_{AR(1)} &= \frac{\sigma^2}{\sum x_t^2}\left(\frac{1+r\rho}{1-r \rho} \right)\\
&=Var(\hat{\beta}_2)_{OLS}\left(\frac{1+r\rho}{1-r \rho} \right)

That is, the usual OLS formula of variance of $\hat{\beta}_2$ will underestimate the variance of $\hat{\beta}_2{_{AR(1)}} $.

Note $\hat{\beta}_2$ although linear-unbiased but not efficient.

In general, in economics, negative autocorrelation is much less likely to occur than positive autocorrelation.

Higher-Order Autocorrelation

Autocorrelation can take many forms. for example,

$$u_t = \rho_1 u_{t-1} + \rho_2 u_{t-2} + \cdots + \rho_p u_{t-p} + \varepsilon_t$$

It is $p$th order autocorrelation.

If we have quarterly data, and we omit seasonal effects, we might expect to find that a 4th-order autocorrelation is present. Similarly, monthly data might exhibit 12th-order autocorrelation.

Learn about Heteroscedasticity Tests and Remedies

Muhammad Imdad Ullah

Currently working as Assistant Professor of Statistics in Ghazi University, Dera Ghazi Khan. Completed my Ph.D. in Statistics from the Department of Statistics, Bahauddin Zakariya University, Multan, Pakistan. l like Applied Statistics, Mathematics, and Statistical Computing. Statistical and Mathematical software used is SAS, STATA, GRETL, EVIEWS, R, SPSS, VBA in MS-Excel. Like to use type-setting LaTeX for composing Articles, thesis, etc.

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