Standard Error of Intercept in Multiple Linear Regression

Question

How do I calculate the standard error of the intercept (b0) when the model has two explanatory variables (say x1 and x2) for y = b0 + b1x1 + b2x2?

Thanks!

I am not sure that it is going to work but you can consider a constant variable $x_0=1$. The intercept will be the regression coefficient of this variable and so you can use the same method that you used to compute the variance of the other coefficients — Donbeo, Mar 04 '15 at 23:44
The multiple regression result $Var(\widehat{\beta})=\sigma^2(X'X)^{-1}$ is given in numerous answers on site. It's estimated by replacing $\sigma^2$ by $s^2$. To get the s.e. of the intercept you simply choose the diagonal element corresponding to the intercept (conventionally the first element) and take its square root. — Glen_b, Mar 05 '15 at 01:04
It's given for example, [here](http://stats.stackexchange.com/questions/93303/variance-covariance-matrix-interpretation/93311#93311) and that variance is shown to be the best [here](http://stats.stackexchange.com/questions/131431/best-ols-estimators/131461#131461) — Glen_b, Mar 05 '15 at 02:36

score 1 · Accepted Answer · answered Mar 05 '15 at 00:59

Under the Gauss-Markov assumptions, if $C\beta$ is estimable, then

$\hat{Var}$($C\hat{\beta}$) = $\hat{\sigma}^2C(X^{\prime}X)^{-}C^{\prime}$,

where $\hat{\sigma}^2$ is simply the $SSE\over{(n-r)}$ where $n$ is the number of observations and $r$ is the rank of $C$.

In your case, to find "b0", $C=[1, 0, 0]$ and $r = 1.$

You can then find the standard error by taking $\sqrt{\hat{Var}(C\hat{\beta}) }$.

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