## If y

Where ) is a bivar ate normal density function, and In L2 X1 1 X2b2 d2 - -, 11 + 22 + M 2 _ ln 4 - - X& p - JT2 2 The notation is the same as that in Hausman and Wise (1977a), and the development leading to equation (10.29) is analogous to the approach followed there. Again the resulting likelihood function has a rather simple form. Finally, extension to two equations and two time periods is also straightforward but somewhat tedious and therefore not carried out here. Amemiya, T. 1973....

## Simultaneous Equations Models With Discrete Endogenous Variables

Recently a class of econometric models involving dichotomous, limited, and censored dependent variables was introduced by Amemiya (1974), Heckman (1976a, 1976b, 1977), Lee (1976,1977), Nelson and Olsen (1977), and others in econometrics literature. In this chapter we will investigate the estimation principle posed by Amemiya, using a unified general simultaneous equation model. The simultaneous equation model includes censored simultaneous equation models, switching simultaneous equation...

## Discrete Choice Analysis

The canonical discrete choice model has the form P(i z), where i is an alternative in a finite choice set C, z is a real vector characterizing the choice set and decision maker, and P gives the conditional probability that in the choice context characterized by z alternative will be selected. The econometric literature on discrete choice generally assumes that P has been specified up to a real parameter vector 0, in which case we write P(i z, 6). The concerns of the literature are (1)...

## Stochastic Revealed Preference

Does the hypothesis of a population of utility-maximizing consumers imply any restrictions on the distributions of observed demands An affirmative answer was given by Marschak 1960 and Block and Marschak 1960 , who established the necessity of conditions such as regularity and the triangle inequality.7 A necessary and sufficient condition for consistency with random preference maximization, analogous to the strong axiom of revealed preference for the individual consumer, has been established by...

## Simpler Solutions and the Problem of Incidental Parameters

The preceding section presents the problem of initial conditions and sketches some formal solutions to it. The solutions presented there are somewhat computationally forbidding. In this section some alternative, simply computed estimators are considered. Following a suggestion made by Mundlak 1978 for a linear regression model, it is possible to estimate a model with a components of variance structure conditional on error component r z and to estimate the t z , z 1, . . . , . The advantage of...

## M

This equivalence is easily verified by comparing the first-order conditions for the two minimization problems.19 Thus maximum likelihood estimation when the Qi are known reduces to finding 0V and kN, such that lt v2 gt amp , max min D2 gt 8, k , 2.69 where the pseudolikelihood L iQ, k is given by equation 2.67 and A 2 k e A 1 and '- OG,- 2-70 2.17 Consistency of the Constraint Equations Let 0 1 be the set of 8 for which the population constraint equations 2.60 are satisfied by some probability...

## The Continuous Logit Model

The definition of an aggregate spatial choice probability can be rewritten as where the subscript t is used to denote an individual type k at location x, y . The continuous logit model is obtained by assuming the independence from irrelevant alternatives, IIA, property McFadden 1976 .8 It implies that the spatial choice probabilities for any feasible subset of spatial alternatives, M1, can be written as where Kt p, q is a spatial choice function defined in terms of attributes at p, q . The...