Skip to main content

Consistency of Var(b|X)

We have

σ^2=e′e(N−K)=ε′Mε(N−K)=ε′[IN−X(X′X)−1X′]ε(N−K)=ε′ε−ε′X(X′X)−1X′ε(N−K)\begin{align*} \hat{\sigma}^2=\frac{\bold{e'e}}{(N-K)}&=\frac{\boldsymbol{\varepsilon'}\bold{M}\boldsymbol{\varepsilon}}{(N-K)}\\ &=\frac{\boldsymbol{\varepsilon'}[\bold{I_N-X(X'X)^{-1}X'}]\boldsymbol{\varepsilon}}{(N-K)}\\ &=\frac{\boldsymbol{\varepsilon'}\boldsymbol{\varepsilon}-\boldsymbol{\varepsilon'}\bold{X(X'X)^{-1}X'}\boldsymbol{\varepsilon}}{(N-K)}\\ \end{align*}

Let's check the consistency of σ^2\hat{\sigma}^2 first

plim σ^2=plim {ε′ε−ε′X(X′X)−1X′ε(N−K)}=plim {N(N−K)[ε′εN−(ε′XN)(X′XN)−1(X′εN)]}=plim (NN−K) plim (ε′εN−(ε′XN)(X′XN)−1(X′εN))=1⋅ plim (ε′εN−(ε′XN)(X′XN)−1(X′εN))=plim (ε′εN)−[ plim (ε′XN) plim (X′XN)−1⏟Q~−1 plim (X′εN)⏟=0]. plim (ε′XN)=plim(1N[∑i=1Nεix1i∑i=1Nεix2i⋯∑i=1Nεixki]1×k).\begin{align*} \text{plim }\hat{\sigma}^2&=\text{plim }\Big\{ \frac{\boldsymbol{\varepsilon'}\boldsymbol{\varepsilon}-\boldsymbol{\varepsilon'}\bold{X(X'X)^{-1}X'}\boldsymbol{\varepsilon}}{(N-K)}\Big\}\\ &=\text{plim }\Big\{\frac{N}{(N-K)}\Big[\frac{\boldsymbol{\varepsilon'\varepsilon}}{N}-\Big(\frac{\boldsymbol{\varepsilon'}\bold{X}}{N}\Big)\Big(\frac{\bold{X'X}}{N}\Big)^{-1}\Big(\frac{\bold{X'}\boldsymbol{\varepsilon}}{N}\Big)\Big]\Big\}\\ &=\text{plim }\Big(\frac{N}{N-K}\Big)\text{ plim }\Big(\frac{\boldsymbol{\varepsilon'\varepsilon}}{N}-\Big(\frac{\boldsymbol{\varepsilon'}\bold{X}}{N}\Big)\Big(\frac{\bold{X'X}}{N}\Big)^{-1}\Big(\frac{\bold{X'}\boldsymbol{\varepsilon}}{N}\Big)\Big)\\ &=1\cdot\text{ plim }\Big(\frac{\boldsymbol{\varepsilon'\varepsilon}}{N}-\Big(\frac{\boldsymbol{\varepsilon'}\bold{X}}{N}\Big)\Big(\frac{\bold{X'X}}{N}\Big)^{-1}\Big(\frac{\bold{X'}\boldsymbol{\varepsilon}}{N}\Big)\Big)\\ &=\text{plim }\Big(\frac{\boldsymbol{\varepsilon'\varepsilon}}{N}\Big)-\Big[\text{ plim }\Big(\frac{\boldsymbol{\varepsilon'}\bold{X}}{N}\Big)\underbrace{\text{ plim }\Big(\frac{\bold{X'X}}{N}\Big)^{-1}}_{\bold{\tilde{Q}^{-1}}}\underbrace{\text{ plim }\Big(\frac{\bold{X'}\boldsymbol{\varepsilon}}{N}\Big)}_{=\bold{0}}\Big].\\ \text{ plim }\Big(\frac{\boldsymbol{\varepsilon'}\bold{X}}{N}\Big) &=\text{plim}\Big(\frac{1}{N} \begin{bmatrix} \sum_{i=1}^N\varepsilon_ix_{1i}&\sum_{i=1}^N\varepsilon_ix_{2i}\cdots\sum_{i=1}^N\varepsilon_ix_{ki} \end{bmatrix}_{1\times k}\Big). \end{align*}

Using Weak Law of Large Numbers

 plim (ε′XN)=0.\begin{align*} \text{ plim }\Big(\frac{\boldsymbol{\varepsilon'}\bold{X}}{N}\Big) &=\bold{0}. \end{align*}

Hence

plim σ^2=plim (ε′εN)=plim (1N∑i=1Nεi2)=E[εi2]=σ2\begin{align*} \text{plim }\hat{\sigma}^2&=\text{plim }\Big(\frac{\boldsymbol{\varepsilon'\varepsilon}}{N}\Big)\\ &=\text{plim }\Big(\frac{1}{N}\sum_{i=1}^N\varepsilon_i^2\Big)\\ &=\mathbb{E}[\varepsilon_i^2]\\ &=\sigma^2 \end{align*}

Ques: Find plim Var(b∣X).\text{plim }\mathbb{Var}\bold{(b|X)}.