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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 $\theta$.

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