Bias and variance of coefficient estimation of logistic regression

For a linear regression problem $$y=X\beta + \epsilon$$, I think we know very well that the estimated $$\hat{\beta} = \dfrac{X^Ty}{X^TX}$$ is unbiased, and has the variance introduced by $$\epsilon$$.

It sounds reasonable to me that over the years, we might have a good understanding of this same question for logistic regression also, but I cannot find any.

I wonder if we have these studies for Logistic regression, or maybe it's not even possible to study these questions because Logistic regression does not have a closed-form solution of $$\beta$$?

• Check the MLE (maximum likelihood estimate). Estimate of regression coefficients in logistic regression is MLE, so the properties of MLE is applicable for logistic regression. Oct 19 '18 at 15:34

• Thank you, but there is still one thing I'm confused: When we talk about the variance of estimated $\beta$, it's usually just number. Therefore, we can calculate the MSE as squared bias plus variance, but in Alecos's answer, the variance derived is a matrix. Could you please help me understand what happened? Oct 19 '18 at 21:30