I found the coefficients of a simple regression Y = aX1+bX2 using a maximum likelihood optimization. Now I would like to find the t-statistics of coefficient a and b.
The normal regress functions don't allow me to give them as an input though.
Any suggestions how I can do this?
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If you use regstats to estimate the coefficient standard errors, here's what you get using the Hald data:
load hald s = regstats(heat,ingredients,'linear'); sqrt(diag(s.covb * (8/13)))' % 8/13 converts unbiased variance to mle
If you have another estimator that is the mle for some probability model, you could compute the second derivative of the log likelihood function and use that to estimate the standard errors. Here's an example using the normal distribution so it gives roughly the same answer as above:
loglik = @(b) sum(log(normpdf(heat,b(1)+ingredients*b(2:end),sqrt(8/13)*sqrt(s.mse)))); H = zeros(5); b = s.beta; db = 0.001; for j=1:5, ej = j==(1:5)'; H(j,j) = (loglik(b+ej*db)-2*loglik(b)+loglik(b-ej*db))/db^2; for k=j+1:5 ek = k==(1:5)'; H(j,k) = ( loglik(b+ej*db+ek*db) + loglik(b) ... - loglik(b+ej*db) - loglik(b+ek*db))/db^2; H(k,j) = H(j,k); end end sqrt(diag(inv(-H)))'
The t statistic is the ratio of the estimate to its standard error. If you can determine the standard error, you can take this ratio yourself.
However, least squares is the maximum likelihood method for a regression if the residuals are normally distributed. In that case you can let regress (or regstats or LinearModel) compute the coefficients and t statistics for you. If you have some other estimation method that is not least squares, you would need to determine the standard errors of those estimators.
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