Jeffrey M. Wooldridge Econometric Analysis of Cross Section and Panel Data

Chapter 4 The Single-Equation Linear Model and OLS Estimation

Section 4.2 Asymptotic Properties of OLS

Subsection 4.2.2 Asymptotic Inference Using OLS

Assumption OLS.3: $E\left(u^{2}x^{\prime}x\right)=\sigma^{2}E\left(x^{\prime}x\right)\text{ where }\sigma^{2}\equiv E\left(u^{2}\right)$

Eq. (4.10) states $\widehat{Avar\left(\hat{\beta}\right)}=\hat{\sigma}^{2}\left(X^{\prime}X\right)^{-1}$ where $\hat{\sigma}$ is a consistent estimator of $\sigma$

I think it should be $\widehat{Avar\left(\hat{\beta}\right)}=\hat{\sigma}^{2}\left(\frac{1}{N}X^{\prime}X\right)^{-1}$

which one is correct:




1 Answer 1


A copy of Wooldridge is not in proximity. But the author is correct in stating equation no. $(4.10). $

Because it seems to obfuscate the thinking in OP, it warrants a brief casting of light on the matter.

The variance of the least square coefficient vector $\bf b$ is

\begin{align} \operatorname{Var} [\mathbf b\mid \mathbf X] & =\left(\mathbf{ X^\top X}\right) ^{-1}\mathbf X^\top\mathbb E \left[\boldsymbol{\varepsilon\varepsilon}^\top\mid \mathbf X\right]\left(\left(\mathbf{ X^\top X}\right) ^{-1}\mathbf X^\top\right) ^\top\\ &= \sigma^2\left(\mathbf{ X^\top X}\right) ^{-1}.\tag 1\label 1 \end{align}

For estimation of $\eqref 1, $ one needs to estimate $\sigma^2.$ It is an easy routine to deduce based on sum of squared residuals

$$\mathbb E\left[\mathbf e^\top\mathbf e\mid \mathbf X\right] = (n - k) \sigma^2.\tag 2\label 2$$

From $\eqref 2 ,$ one gets an (unbiased) estimator of $\sigma^2$ and subsequently

$$\operatorname{Est.Var} [\mathbf b\mid \mathbf X] = s^2\left(\mathbf{ X^\top X}\right) ^{-1},\tag 3$$ where $$s^2 := \frac{\mathbf e^\top \mathbf e}{n-k}.$$

Now, to ensure the data in large samples is "well-behaved", one assumption taken is

\begin{align}\operatorname{plim} \frac{\mathbf X^\top \mathbf X}n &=: \mathbf Q \\&\equiv \textrm{a positive definite matrix};\tag{A1}\end{align} this is attainable if it is assumed $$\mathbb E\left[\mathbf x_i\mathbf x_i^\top\right] = \mathbf Q, $$ which means $\sum \mathbf x_i\mathbf x_i^\top/n \overset{\mathbb P}{\to}\mathbf Q$ and hence $\operatorname{plim}\left( \frac{\mathbf X^\top\mathbf X}n\right) ^{-1} =\mathbf Q^{-1}.$

Now, the asymptotic variance, like $\eqref 1 ,$ would be

$$\operatorname{Asy.Var}[\mathbf b] = (\sigma^2/n) \mathbf Q^{-1}.\tag 4\label 4$$ Again, to get an estimator of $\eqref 4, $ one needs to concentrate on the term in parenthesis - $s^2$ can be used with proper evaluation of its consistency:

\begin{align}s^2 &= \frac{\boldsymbol \varepsilon^\top\mathbf M\boldsymbol\varepsilon}{n-k}\\&= \frac{\boldsymbol \varepsilon^\top \boldsymbol\varepsilon - \boldsymbol\varepsilon^\top \mathbf X\left(\mathbf{ X^\top X}\right)^{-1}\mathbf X^\top\boldsymbol \varepsilon}{n-k}\\ &= \frac{n}{n-k}\left[\frac{\boldsymbol \varepsilon^\top\boldsymbol\varepsilon}n - \left(\frac{\boldsymbol\varepsilon^\top\mathbf X}n\right)\left(\frac{\mathbf{ X^\top X}}n\right)^{-1}\left(\frac{\mathbf X^\top\boldsymbol \varepsilon}n\right)\right]; \tag 5\label 5, \end{align} where $\mathbf M:= \mathbf I - \mathbf X\left(\mathbf{ X^\top X}\right)^{-1}\mathbf X^\top$ is the residual maker. Using $(\mathrm A1) $ and the fact that $\operatorname{plim} \left(\frac{\boldsymbol\varepsilon^\top\mathbf X}n\right) = \mathbf O$ (since $\mathbb E[\mathbf x\boldsymbol{\varepsilon}] =\mathbf O$), the second term in the bracket in $\eqref 5$ converges to $0.$ The factor outside the bracket converges to $1.$ The only term left for inspection is $\sum \varepsilon_i^2/n.$ Now $\varepsilon_i^2$ are independent with same finite expectation $\sigma^2.$ To ensure the term's convergence (almost surely to $\sigma^2$), Markov's law of large numbers could be employed provided $\mathbb E\left[\left(\varepsilon_i^2\right)^{1+\delta}\right] <\infty$ for some $\delta \in (0, 1) $ - it would suffice then to assume for all $\varepsilon_i,$ there exist finite moments greater than $2.$ In any case, $$\operatorname{plim} s^2 = \sigma^2. \tag 6$$ This implies

$$ \operatorname{plim} s^2 \left( \frac{\mathbf X^\mathsf T\mathbf X}n\right) ^{-1}= \sigma^2\mathbf Q^{-1}. \tag{ 6.I}$$ Therefore, the appropriate estimator of $(4)$ would be $$\operatorname{Est.Asy.Var}[\mathbf b] = s^2\left(\mathbf X^\top\mathbf X\right) ^{-1}.\tag 7$$



Econometric Analysis, William H. Greene, Pearson Education, 2018.


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