# Scale normalisations in multinomial choice models

I am looking for advice on how to impose scale/location normalisations in the following multinomial choice model:

The model $$Y=argmax_{y\in \mathcal{Y}\equiv \{0,1,2\}}U_y(X)+\epsilon_y$$ or, equivalently, $$Y=\sum_{y\in \mathcal{Y}}y \times {1}\Big\{U_y(X)+\epsilon_y \geq U_{\tilde{y}}(X)+\epsilon_{\tilde{y}} \text{ }\forall \tilde{y}\neq y\Big\}$$

where

• $$X$$ is a discrete random variable with support $$\mathcal{X}\equiv \{1,2\}$$

• $$\epsilon\equiv (\epsilon_0, \epsilon_1, \epsilon_2)$$ is continuously distributed conditional on $$X$$

• $$\forall y \in \mathcal{Y}$$, $$U_y:\mathcal{X}\rightarrow \mathbb{R}$$.

• The econometrician observes $$Y,X$$

• $$1\{A\geq 0\}=1$$ if $$A\geq 0$$ and $$0$$ otherwise

According to the model above, $$\begin{cases} \mathbb{P}(Y=1|X=x)=\mathbb{P}(\epsilon_1-\epsilon_0\geq -U_1(X), \epsilon_1-\epsilon_2\geq U_2(X)-U_1(X)|X=x)\\ \mathbb{P}(Y=2|X=x)=\mathbb{P}(\epsilon_2-\epsilon_0\geq -U_2(X), \epsilon_2-\epsilon_1\geq U_1(X)-U_2(X)|X=x)\\ \mathbb{P}(Y=0|X=x)=\mathbb{P}(\epsilon_0-\epsilon_1\geq U_1(X), \epsilon_0-\epsilon_2\geq U_2(X)|X=x) \end{cases}$$

Identification question

Let $$\theta\equiv (U_{0}(1),U_{1}(1),U_{2}(1),U_{0}(2),U_{1}(2),U_{2}(2)$$ and let $$F_{V|X}$$ be the joint cdf of $$(\epsilon_1-\epsilon_0, \epsilon_2-\epsilon_0, \epsilon_1-\epsilon_2)$$ conditional on $$X$$.

Let $$\theta \in \Theta$$ and $$F_{V|X}\in \mathcal{F}_{V|X}$$, i.e., $$\Theta\times \mathcal{F}_{V|X}$$ is the parameter space.

The identification question is: we want to find the set of $$(\theta, F_{V|X})\in \Theta\times \mathcal{F}_{V|X}$$ which can generate the empirical probability distribution of $$Y$$ conditional on $$X$$ when combined with the model's assumptions above. By empirical probability distribution of $$Y$$ conditional on $$X$$ we mean the probability distribution of $$Y$$ conditional on $$X$$ that we see in the data.

If there exists only one $$(\theta, F_{V|X})\in \Theta\times \mathcal{F}_{V|X}$$ which can generate the empirical probability distribution of $$Y$$ conditional on $$X$$ when combined with the model's assumptions above, then we say that $$\theta$$ is point identified.

If there exists more than one $$(\theta, F_{V|X})\in \Theta\times \mathcal{F}_{V|X}$$ which can generate the empirical probability distribution of $$Y$$ conditional on $$X$$ when combined with the model's assumptions above, then we say that $$\theta$$ is partially identified.

Scale and location normalisations and question

We know that identification of discrete choice models is up to scale and location. In other words, if $$(\theta, F_{V|X})\in \Theta\times \mathcal{F}_{V|X}$$ generates the empirical probability distribution of $$Y$$ conditional on $$X$$, then $$(g(\theta), F_{V|X})\in \Theta\times \mathcal{F}_{V|X}$$ generates the empirical probability distribution of $$Y$$ conditional on $$X$$ for any strictly increasing transformation $$g$$.

Question: how do we impose scale and location normalisations here?

Further thoughts

I find scale normalisations tricky in a non-parametric setting as the one above. Indeed, undergrad econometrics teaches how to impose a scale normalisation when $$F_{V|X}$$ is parametrically specified (e.g., in the multinomial logit model), and, typically, suggests to impose it on the variance-covariance matrix of $$F_{V|X}$$ (see e.g., Train, 2009). When we move to a non-parametric setting, I don't understand whether we still have to impose restrictions on $$F_{V|X}$$ or we have to impose restrictions on $$\theta$$.