GMM regarding logistic regression

Consider the binary logit model for the choice between two alternatives, 0 and 1. We take the simplest case with a single regressor, so the probability that individual i chooses alternative 1 is πi(β) = exi β/(1 + exi β), with β to be estimated. We observe the xi and di , with di = 1 if individual i chooses alternative 1 and di = 0 otherwise. A little useful result is ∂π/∂z = π(1 − π) when π = ez/(1 + ez) (check). And don’t forget the chain rule. We condition on x (as usual in regression although often not stated explicitly) so we have E(di − πi | xi ) = 0.

a)  Give two moment conditions of your choice for estimating β. Why are they valid? What is arguably the simplest moment condition?
(b)  With two moment conditions we do GMM for estimating β. Choose the unit matrix for your weight matrix. What is the first-order condition from which βˆ follows?
(c)  Elaborate the general formula for the estimated asymptotic variance of βˆ for the specific case of your estimator.
(d)  Adapt your results for the case of optimal weighting.
(e)  Elaborate the general form of the J-test for this case.

hi I also want to ask why the following condition is considered in the linear regression

Consider adding the moment condition based on E(x_i^2*yi). What is it? Discuss how useful adding this moment condition is.