I am implementing an R-type summary() function in python with the restriction to exclude use of scientific libraries. (assignment) I found this https://www.nd.edu/~rwilliam/stats1/x91.pdf material explaining the calculation of standard errors of coefficients. As I already had a multiple linear model coded with gradient descent, I was able to code the standard error of coefficients, except the intercept. I just do not understand how to calculate $S{_b}_0$ based on the below equation:

Standard error of regression coefficient I can calculate $R^{2}{_X}_k{_G}_k$ for k>0, but not for k=0

The following code piece works already, but I need something for the (z=-1) below, which would fill the coeff_stderr[0] element:

        for z in range(len(X[0])):
            xk_vector = get_matrix_column(X, z)
            var_xk = variance(xk_vector)
            RRxk = coefficient_of_determination_Xk(X, z, num_iters, alpha)
            coeff_stderr[z+1] = RSE / sqrt((1-RRxk)*var_xk*(len(X)-1))

what is $X_0$? The intercept term - the column with 1s? Then it's variance is 0, and I would divide with 0...

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    $\begingroup$ @Isvan Orosz you might find this useful stats.stackexchange.com/questions/439966/… $\endgroup$ – PsychometStats Dec 9 '19 at 22:24
  • $\begingroup$ As I understand you are suggesting a matrix inversion along the line? I can't do that the moment with my program. $\endgroup$ – István Orosz Dec 10 '19 at 14:13
  • $\begingroup$ Yes, you will need to invert covariance matrix to calculate parameters $\endgroup$ – PsychometStats Dec 10 '19 at 14:22
  • $\begingroup$ if you check out the formula in the linked .pdf, it does not mention inverting covariance matrix. And as I mentioned, the formula works for the coefficients, except the intercept. My solution coefficient_of_determination_Xk() works with gradient descent. $\endgroup$ – István Orosz Dec 10 '19 at 14:59
  • $\begingroup$ You don't need to invert the matrix: you only need to compute one entry in its inverse. (That's where the formula you quote comes from, by the way: it gives diagonal entries of the inverse.) $\endgroup$ – whuber Dec 10 '19 at 19:02

I gave up finding the magic formula and coded the matrix inversion routine. this is the final piece, I calculate all the standard errors of the coefficients with this:

        # add intercept term to X (to the left)
        X = concatenate_matrices_by_column([[1.0]] * len(X), X)
        XX = multiply_matrices(transpose(X),X)
        vc = invert_matrix(XX)
        for k in range(len(vc)):
            coeff_stderr[k] = sqrt(vc[k][k] * SSE / df_residual)

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