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# 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(

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.