Question

how do we calculate the standard errors of regression coefficients in multiple linear regression, that is, the standard error of the constant(B0), the standard error of B1 and standard error of B2

Answer 1

Let there be n observations of y, x1 and x2.

Let be a vector of order n X 1 containing the values of x1, be a vector of order n X 1 containing the values of x2, be a vector of order n X 1 containing the values of y and, lastly, be a vector containing "n" number of 1's and, as a result, of order n X 1 (required for the intercept term).

Now, the design matrix is given as: , which is order n X 3.

The 3 parameters (2 regression coefficients and 1 intercept), represented by , are estimated in the following way: .

The SSE is now calculated as: .

Now, MSE = SSE/n - 3 = . This is our residual standard error.

Now, the standard errors of the regression coefficients are given in the following way:
.
This will resulting in a diagonal matrix, where the square root of the diagonals are the standard errors of the constant, B1 and B2 respectively.