Identification of discrete choice models

Consider the classical Logit model. In particular, let $$\mathcal{Y}\equiv (0,1,...,L)$$ be the set of options available to consumers, where $$0$$ denotes the outside option. Let $$u_y\equiv \begin{pmatrix} \beta_y' X_y+\epsilon_y & \text{ if } y>0\\ \epsilon_y& \text{ if } y=0\\ \end{pmatrix}$$ be the payoff derived from choosing option $$y\in \mathcal{Y}$$. Let $$\{\epsilon_y\}_{y\in \mathcal{Y}}$$ be i.i.d. Gumbel with scale $$1$$ and location $$0$$.

Suppose that the researcher knows the probability distribution of $$Y$$ conditional on $$X\equiv (X_1,...,X_L)$$. Then, one can show that $$(\beta_1,...,\beta_L)$$ is point identified. Indeed, $$Pr(Y=y|X)=\begin{cases} \frac{\exp(\beta_y'X_y)}{\sum_{y\in \mathcal{Y}}\exp(\beta_y'X_y)}& \text{ if } y>0\\\\ \frac{1}{\sum_{y\in \mathcal{Y}}\exp(\beta_y'X_y)}& \text{ if } y=0\\\\ \end{cases}$$ Hence, for $$y>0$$, $$\log(Pr(Y=y|X))-\log(Pr(Y=0|X))=\beta_y'X_y$$ $$\Updownarrow$$ $$[\log(Pr(Y=y|X))-\log(Pr(Y=0|X))]X_y'=\beta_y'X_y X_y'$$ $$\Updownarrow$$ $$E\Big([\log(Pr(Y=y|X))-\log(Pr(Y=0|X))]X_y'\Big)=\beta_y'E\Big(X_y X_y'\Big)$$ $$\Updownarrow$$ $$E\Big([\log(Pr(Y=y|X))-\log(Pr(Y=0|X))]X_y'\Big)E^{-1}\Big(X_y X_y'\Big)=\beta_y'$$

where the left-hand-side is known and well defined provided that $$E^{-1}\Big(X_y X_y'\Big)$$ exists.

Question: I would like to replicate the same proof when $$\{\epsilon_y\}_{y\in \mathcal{Y}}$$ are i.i.d. standard normals, but I'm struggling because I get quite complicated expressions when using the cdf of the standard normal. Could you help? What if instead of the standard normal, I assume that $$(\epsilon_0,...,\epsilon_L)\sim N((0,0,...,0),\begin{pmatrix} 1& \rho & \rho & ... & \rho\\ \rho & 1 & \rho & ... & \rho\\ ... & ... & ... & ... & ...\\ \rho & \rho & \rho & ... & 1\\ \end{pmatrix})$$

I would like to replicate the same proof when {ϵy}y∈Y are i.i.d. standard normals, but I'm struggling because I get quite complicated expressions when using the cdf of the standard normal.

The choice probabilities $$\Pr(Y=y|X)$$ do not have a closed form solution for the multinomial normal error case. That they do for the Gumbel errors (which give logistic error differences) is the main reason the logistic model is used for outcomes involving many options, despite entailing implausible restrictions like independence of irrelevant alternatives.

The absence of a closed form solution is a major issue for identification and estimation of these models.

When it comes to identification, you cannot proceed as you have done, and invert the choice probabilities to obtain the parameters as a function of features of the distribution of observables $$\Pr(Y,X)$$.

Nonetheless, the model is identified, and estimable with simulation.

What if instead of the standard normal, I assume that [...]

Also identified, but it's not true that a completely unrestricted variance covariance matrix is identified.

Since identification proofs for the multinomial probit aren't given in Kenneth Train's excellent textbook (Discrete Choice Methods with Simulation), I've always assumed they're quite technical.

If you want to check them out, the reference is Bunch (1991), Estimability in the multinomial probit model.