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

OLS Estimation in the Presence of Heteroscedasticity, the variance of the 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 the case of heteroscedasticity, the OLS estimators

Heteroscedasticity: OLS Estimation
  • $\hat{\beta_2}$ 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 the disturbances ($u_i$) to be homoscedastic.
  • 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.

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