How to estimate logit model I am trying to understand how to fit a logit model using maximum likelihood described in a paper:
$$p_{it}=\frac{exp(\alpha+\beta q_{it})}{1+exp(\alpha+\beta q_{it})} $$
where
$$q_{it}=\frac{q_0d^{t-1}+x_{i1}d^{t-2}+...+x_{it-2}d+x_{it-1}}{d^{t-1}+d^{t-2}+...+1}$$
My question is how to estimate $\alpha$, $\beta$, $q_0$ and $d$ and also their standard errors.
I am thinking to write out the expressions:
$$q_{it}=a+b_1x_{i1}+b_2x_{i2}+...+b_{t-1}x_{it-1}$$ such that
$$a=\alpha+\frac{\beta q_0d^{t-1}}{d^{t-1}+...+1}$$
$$b_1=\frac{\beta}{d^{t-1}+...+1}d^{t-2}$$
$$...$$
$$b_{t-1}=\frac{\beta}{d^{t-1}+...+1}$$
I kind of know how to solve the equations to get $\beta$ and $d$, but I don't know how to get $\alpha$ and $q_0$. And the model is over parameterized ($t=75$), so I do not know how to get the standard errors. So far I think of it in terms of regressions, and perhaps it is not a right way to get the parameters, but I have no clue how to do it.

It is unlikely that there is a closed form solution for the ML-estimators of the parameters. You must thus find the values numerically with an optimization (minimization/maximization) algorithm.
The outline is as follows

*

*Define the log-likelihood function as $$l = \sum_{i,t} \log p_{it}$$


*Use a software library routine, e.g. optim in standard R, to find the parameters that maximize l. Caveat: if the function minimizes the given function (like optim does), you must replace $l$ with $-l$.
To estimate standard errors for the parameters, there is a complicated method based on the inverse Hessian matrix (which I occasionally observe to be numerically unstable). A more robust and very simple method is instead to use the jacknife method, which I have described in section 5.2 of this article:

Dalitz: "Construction of confidence intervals." Technical Report No. 2017-01, pp. 15-28, Hochschule Niederrhein, Fachbereich Elektrotechnik und Informatik, 2017

