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Given the simple linear regression model $Y = \beta_0 + \beta_1x + U$, where $U\sim N(0,\sigma^2)$, I know how to derive \begin{equation} \text{Var}[\widehat{\beta}_1] = \dfrac{\sigma^2}{\sum_{i=1}^n(x_i - \overline{x})^2} \end{equation}

However, I do not know how to prove that \begin{equation} S^2 := \dfrac{\frac{1}{n-2}\sum_{i=1}^n {\widehat{U}_i}^2}{\sum_{i=1}^n(x_i - \overline{x})^2} \end{equation} is the unbiased estimator of the variance of $\widehat{\beta}_1$, that is, $E[S^2] = \text{Var}[\widehat{\beta}_1]$.

I started \begin{align} E[S^2] &= E\left[ \dfrac{\frac{1}{n-2}\sum_{i=1}^n {\widehat{U}_i}^2}{\sum_{i=1}^n(x_i - \overline{x})^2} \right]\\[1ex] &= \frac{1}{n-2} \cdot \frac{1}{\sum_{i=1}^n(x_i - \overline{x})^2} E\left[ \sum_{i=1}^n {\widehat{U}_i}^2\right] \end{align} where we notice that we only have to show that $\displaystyle E\left[ \sum_{i=1}^n {\widehat{U}_i}^2\right] = \sigma^2(n-2)$. Thus, I started \begin{align} E\left[\sum_{i=1}^n {\widehat{U}_i}^2 \right] &= \sum_{i=1}^n E[{\widehat{U}_i}^2]\\[1ex] &= \sum_{i=1}^n E\left[(Y_i - \widehat{\beta}_0 - \widehat{\beta}_1x_i)^2\right]\\[1ex] &= \sum_{i=1}^n E\left[{Y_i}^2 + {\widehat{\beta}_0}^2 + {\widehat{\beta}_1}^2{x_i}^2 - 2\widehat{\beta}_0Y_i - 2\widehat{\beta}_1x_iY_i + 2\widehat{\beta}_0\widehat{\beta}_1x_i \right]\\[1ex] &= \sum_{i=1}^n \left(E[{Y_i}^2] + E[{\widehat{\beta}_0}^2] + {x_i}^2E[{\widehat{\beta}_1}^2] - 2E[\widehat{\beta}_0Y_i] - 2x_iE[\widehat{\beta}_1Y_i] + 2x_iE[\widehat{\beta}_0\widehat{\beta}_1]\right) \end{align} but I am not sure how to continue. Could you kindly show, preferably without matrix albegra?

EDIT: I think I made some progress. Knowing that $E[XY] = \text{cov}(X,Y) + E[X]E[Y]$ and that $\text{cov}(X,X) = \text{Var}(X)$ gets us going. In addition, we need to know that \begin{align*} Y_i & \sim N\left(\beta_0 + \beta_1x_i, \sigma^2\right)\\ \widehat{\beta}_0 & \sim N\left(\beta_0, \frac{\sum {x_i}^2}{n\sum(x_i - \overline{x})^2}\sigma^2\right)\\ \widehat{\beta}_1 & \sim N\left(\beta_1, \frac{1}{\sum(x_i - \overline{x})^2} \sigma^2 \right) \end{align*}

From these facts we get \begin{align} E[{Y_i}^2] &= \text{cov}(Y_i,Y_i) + E[Y_i]E[Y_i]\\ &= \text{Var}(Y_i) + E[Y_i]^2\\ &= \sigma^2 + (\beta_0 + \beta_1x_i)^2 \end{align} Similarly \begin{align} E[\widehat{\beta}_0^2] &= \text{Var}(\widehat{\beta}_0) + E[\widehat{\beta}_0]^2\\ &= \frac{\sum {x_i}^2}{n\sum(x_i - \overline{x})^2}\sigma^2 + {\beta_0}^2 \end{align} and \begin{align} E[\widehat{\beta}_1^2] &= \text{Var}(\widehat{\beta}_1) + E[\widehat{\beta}_1]^2\\ &= \frac{1}{\sum(x_i - \overline{x})^2}\sigma^2 + {\beta_1}^2 \end{align}

To continue, I know need to know how to handle the last three expected values. Any help on that?

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You have a typo: the numerator of $S^2$ should be $\frac{1}{n-2}\sum_{i=1}^n\hat{U}_i^2$ where $\hat{U}_i$ is the OLS residual of the $i$-th observation.

Let $X=(\iota\; x)$ be the $n\times 2$ matrix of regressors ($\iota$ is the $n\times 1$ column of $1$'s). Let $P=X(X'X)^{-1}X'$ and $M=I_n-P$. Then with $\hat{U}=(\hat{U}_1,\ldots,\hat{U}_n)'$ $$ \sum_{i=1}^n\hat{U}_i^2=\hat{U}'\hat{U}=(MY)'(MY)=(MU)'MU=U'MU. $$ Recall that $M$ is both symmetric and idempotent. Then, using the facts that a number equals its trace and that trace and expectation commute, we have \begin{align*} E\left(\sum_{i=1}^n\hat{U}_i^2\right)&=E[U'MU]=E[\text{trace}(U'MU)]=E[\text{trace}(MUU')]\\ &=\text{trace}[E(MUU')]=\text{trace}[ME(UU')]\\ &=\text{trace}[M\sigma^2I_n]=\sigma^2\text{trace}[M]. \end{align*} To find $\text{trace}[M]$, we note \begin{align*} \text{trace}[M]&=\text{trace}[I_n-X(X'X)^{-1}X']=\text{trace}(I_n)-\text{trace}[X(X'X)^{-1}X']\\ &=\text{trace}(I_n)-\text{trace}[X'X(X'X)^{-1}]=\text{trace}(I_n)-\text{trace}(I_2)=n-2. \end{align*} It remains to put these results together. I leave that to you.

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  • $\begingroup$ Fixed the typo! However, I would greatly appreciate if you could, in addition, answer without matrix algebra. $\endgroup$ – Mappi May 28 '16 at 17:41
  • $\begingroup$ @Mappi Sorry won't do that. If you want to learn linear regression models and econometrics properly, you need to learn matrix algebra eventually. But perhaps another poster may feel inclined to help you. Good luck. $\endgroup$ – yurnero May 28 '16 at 17:44
  • $\begingroup$ I studied through the proof, very nice. I think there's a typo: $My$ should be $MY$. Moreover, I think it would save future readers some time if you mentioned that $M$ is idempotent and symmetric. Btw, I was able to make some progress and edited it above. Any help on that? $\endgroup$ – Mappi May 29 '16 at 18:45
  • $\begingroup$ @Mappi Thanks. I'll switch '$y$' to '$Y$'. $\endgroup$ – yurnero May 29 '16 at 18:58

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