On the representation of the nested logit model

TL;DR

This paper offers a simple proof of a longstanding conjecture on the nested logit model's representation, and derives a closed-form formula for correlation in Gumbel copulas, enhancing theoretical understanding.

Contribution

It provides a concise proof of the nested logit model's representation and introduces a closed-form correlation formula for Gumbel copulas, simplifying existing approaches.

Findings

A two-line proof of the nested logit model's representation.

A closed-form formula for correlation between Fréchet variables with Gumbel copula.

Enhanced theoretical clarity and simplicity in nested logit modeling.

Abstract

We give a two-line proof of a long-standing conjecture of Ben-Akiva and Lerman (1985) regarding the random utility representation of the nested logit model, thus providing a renewed and straightforward textbook treatment of that model. As an application, we provide a closed-form formula for the correlation between two Fr\'echet random variables coupled by a Gumbel copula.

Equations

$$F_{\epsilon }\left(\left(a_j\right)\right)=\exp \left(-\sum _{n\in \mathcal{N}}{\left(\sum _{j\in {\mathcal{N}}_n}{e}^{-\frac{a_j}{{\lambda }_n}}\right)}^{{\lambda }_n}\right),$$

$$\frac{{\alpha }_1{Z}_1+...+{\alpha }_N{Z}_N}{{\left({\alpha }_1^{\lambda }+...+{\alpha }_N^{\lambda }\right)}^{1/\lambda }}\sim \mathcal{P}\left(\lambda \right),$$

$$\mathrm{Pr}\left(ϵ+\log Z\le x\right)=\mathbb{E}\left[\mathbb{E}\left[1\left\{ϵ\le x-\log Z\right\}|Z\right]\right]=\mathbb{E}\left[F_{ϵ}\left(x-\log Z\right)\right]$$

$$\mathbb{E}\left[Z^{\kappa }\right]=\frac{\Gamma \left(1-\kappa /\lambda \right)}{\Gamma \left(1-\kappa \right)}.$$

$$f_{1/3}\left(x\right)$$

$$f_{1/2}\left(x\right)$$

$$f_{2/3}\left(x\right)$$

$$f_{\lambda }\left(x\right)=\frac{-1}{\pi }\sum _{k=0}^{\infty }\frac{{\left(-1\right)}^k}{k!}\sin \left(k\pi \lambda \right)\frac{\Gamma \left(\lambda k+1\right)}{x^{\lambda k+1}},$$

$$\Gamma \left(z\right)={\int }_0^{+\infty }{t}^{z-1}{e}^{-t}dt.$$

$$\lambda \log \left(\sum _{j\in \mathcal{J}}\exp \left(\frac{u_j+{\eta }_{\lambda ,j}}{\lambda }\right)\right){=}_D\log \left(\sum _{j\in \mathcal{J}}{e}^{u_j}\right)+{\eta }_{\lambda },$$

$${\lambda }_2{\lambda }_1{ϵ}_1+{\lambda }_2{\lambda }_1\log Z_1+{\lambda }_2\log Z_2\sim \mathcal{G}\left(0,1\right).$$

$$Z_1{Z}_2^{1/{\lambda }_1}\sim \mathcal{P}\left({\lambda }_1{\lambda }_2\right).$$

$${\pi }_j=\frac{{\left({\sum }_{j^{\prime }\in {\mathcal{N}}_n}{e}^{\frac{U_{j^{\prime }}}{{\lambda }_n}}\right)}^{{\lambda }_n}}{{\sum }_{\begin{array}{c}{n}^{\prime }\in \mathcal{N}\\ j^{\prime }\in {\mathcal{N}}_{n^{\prime }}\end{array}}{\left({\sum }_{j^{\prime }\in {\mathcal{N}}_{n^{\prime }}}{e}^{\frac{U_{j^{\prime }}}{{\lambda }_{n^{\prime }}}}\right)}^{{\lambda }_{n^{\prime }}}}\frac{e^{\frac{U_j}{{\lambda }_n}}}{{\sum }_{j^{\prime }\in {\mathcal{N}}_n}{e}^{\frac{U_{j^{\prime }}}{{\lambda }_n}}},$$

$$G\left(U\right)=\log \sum _{n\in \mathcal{N}}{\left(\sum _{j\in {\mathcal{N}}_n}{e}^{\frac{U_j}{{\lambda }_n}}\right)}^{{\lambda }_n},$$

$${\epsilon }_j={\lambda }_n\left({ϵ}_j+\log Z_n\right)$$

$$\mathrm{Pr}\left({\lambda }_n{ϵ}_j+{\eta }_n\le a_j\forall n\in \mathcal{N},\forall j\in {\mathcal{N}}_n\right)=\mathrm{Pr}\left({\eta }_n\le -\underset{j\in {\mathcal{N}}_n}{\max }\left\{-a_j+{\lambda }_n{ϵ}_j\right\}\forall n\in \mathcal{N}\right).$$

$$\mathrm{Pr}\left({\lambda }_n{ϵ}_n+{\eta }_n\le -{\lambda }_n\log \sum _{j\in {\mathcal{N}}_n}\exp \left(\frac{-a_j}{{\lambda }_n}\right)\forall n\in \mathcal{N}\right)$$

$$\mathrm{Pr}\left(U_j+{\lambda }_n\left({ϵ}_j+\log Z_n\right)\ge U_{j^{\prime }}+{\lambda }_{n^{\prime }}\left({ϵ}_{j^{\prime }}+\log Z_{n^{\prime }}\right)\forall n^{\prime }\in \mathcal{N},j^{\prime }\in {\mathcal{N}}_{n^{\prime }}\right)$$

$$\sum _{k=1}^K\frac{Z_n^k\exp \left(U_j/{\lambda }_n\right)}{{\sum }_{k^{\prime }=1}^K{\sum }_{n^{\prime }\in \mathcal{N},j^{\prime }\in {\mathcal{N}}_{n^{\prime }}}{Z}_{n^{\prime }}^{k^{\prime }}\exp \left(U_{j^{\prime }}/{\lambda }_{n^{\prime }}\right)}.$$

$${\Lambda }_n=\prod _{t=1}^{d_n}{\lambda }_{n\left(t\right)}.$$

$$\left\{\begin{array}[]{l}u_{z}=U_{z}\text{~{}for }z\in\mathcal{J}\text{ (i.e. }h_{z}=0\text{)}\\ u_{n}=\Lambda_{n}\log\left(\sum_{z\in\mathcal{N}_{n}\cup\mathcal{J}_{n}}\exp\left(u_{z}/\Lambda_{n}\right)\right)~{}\text{for }h_{n}>1\text{.}\end{array}\right.$$

$$\left\{\begin{array}[]{l}\pi_{0}=1\\ \pi_{z}=\pi_{n_{z}}\frac{\exp\left(u_{z}/\Lambda_{n_{z}}\right)}{\sum_{z^{\prime}\in\mathcal{N}_{n_{z}}\cup\mathcal{J}_{n_{z}}}\exp\left(u_{z^{\prime}}/\Lambda_{n_{z}}\right)}\text{ for }z\neq 0.\end{array}\right.$$

$$\frac{{\pi }_z}{{\pi }_n}=\exp \left(\frac{u_z-u_n}{{\Lambda }_n}\right),\text{for }z\in {\mathcal{N}}_n\cup {\mathcal{J}}_n.$$

$$\mathrm{Pr}\left({\epsilon }_j\le A_j\forall j\in \mathcal{J}\right)=\exp \left(-\exp \left(-a_0\right)\right),$$

$$\text{where }a\in {\mathbb{R}}^{\mathcal{Z}}\text{ is defined recursively by}$$

$$\displaystyle\left\{\begin{array}[]{l}a_{z}=A_{z}\text{~{}for }z\in\mathcal{J}\text{ (i.e. }h_{z}=0\text{)}\\ a_{n}=-\Lambda_{n}\log\left(\sum_{z\in\mathcal{N}_{n}\cup\mathcal{J}_{n}}\exp\left(-a_{z}/\Lambda_{n}\right)\right)~{}\text{for }h_{n}>1\text{.}\end{array}\right.$$

$${\pi }_j=\mathrm{Pr}\left(U_j+{\epsilon }_j\ge U_{j^{\prime }}+{\epsilon }_{j^{\prime }}\forall j^{\prime }\in \mathcal{J}\right).$$

$$G\left(U\right):=G_0\left(U\right):=\mathbb{E}\left[\underset{j\in \mathcal{J}}{\max }\left\{U_j+{\epsilon }_j\right\}\right],$$

$$G_n\left(U\right):=\mathbb{E}\left[\underset{j\in {\tilde{\mathcal{J}}}_n}{\max }\left\{U_j+{\epsilon }_j\right\}\right]$$

$$u_n=G_n\left(U\right)$$

$$\frac{\partial G_n\left(U\right)}{\partial U_j}=\frac{{\pi }_j}{{\pi }_n}1\left\{j\in {\tilde{\mathcal{J}}}_n\right\},$$

$${\epsilon }_j=\sum _{t=1}^{d_{n_j}}{\Lambda }_{n_j\left(t\right)}\log Z_{n_j\left(t\right)}+{\Lambda }_{n_j}{ϵ}_j$$

Full text

Keywords: random utility models, multinomial choice, nested logit model

JEL Classification: C51, C60

1. Positive stable distributions and Gumbel distributions

Ben-Akiva (1973) introduced the nested logit model in his PhD thesis. In the light of McFadden’s (1978) theory of the generalized extreme value distribution, one may view the nested logit model as a choice model over various alternatives $j\in\bigcup_{n\in\mathcal{N}}\mathcal{N}_{n}$, where $\mathcal{N}_{n}$ is a nest, a subset of alternatives. The utility associated with alternative $j$ in nest $n$ will be $U_{nj}+\varepsilon_{nj}$, where $U_{nj}$ is the deterministic (“systematic”) part of the utility and $\varepsilon_{nj}$ is a random utility term. The nested logit model specifies that the random vector $\left(\varepsilon_{nj}\right)$ has cumulative distribution function (c.d.f.) $F_{\varepsilon}$ given by

[TABLE]

where $\lambda_{n}\in(0,1]$ is a measure of how correlated the random utility terms are within nest $n$, $\lambda_{n}=1$ meaning full independence, and the limit $\lambda_{n}\rightarrow 0$ meaning perfect correlation within the nest. As pointed out in McFadden (1978), the correlation between random utility terms within nest $n$ is not given by $1-\lambda_{n}$. The correct formula for the correlation ($1-\lambda_{n}^{2}$) had been obtained by Tiago de Oliveira (1958) from a non-trivial analytic derivation.

In their 1985 book, Ben-Akiva and Lerman proposed (section 10.3) a heuristic interpretation of the nested logit model. Calling $\mathcal{G}\left(\mu,\beta\right)$ the distribution of the Gumbel random variable with location-scale parameters $\left(\mu,\beta\right)$, whose cumulative distribution function (c.d.f.) is thus $F_{\epsilon}\left(z\right)=\exp\left(-\exp\left(-\left(z-\mu\right)/\beta\right)\right)$, Ben-Akiva and Lerman (p. 282)111Cardell (1997) mentions that a similar conjecture appeared in a 1975 manuscript of his, but we haven’t been able to access that manuscript. conjectured that if $\epsilon$ has distribution $\mathcal{G}\left(0,1\right)$, there is a random variable $\eta$ independent from $\epsilon$ such that $\lambda\epsilon+\eta$ has distribution $\mathcal{G}\left(0,1\right)$. This conjecture allowed them to derive the correlation structure between the random utilities within a nest. The existence of $\eta$ was the object of work by Cardell (1997) who provides a formula for the density of $\eta$ as the sum of an alternating series by the inversion of its characteristics function. However, Cardell’s proof of existence is incomplete, as it omits to show that the candidate density obtained as the sum of this series is indeed nonnegative. In the present paper we provide a complete, very short proof involving “positive stable distributions”, which are defined as follows in Feller (1971), sections VI.1 and XIII.6222The existence of positive stable distributions was established in Pollard (1948), using a lemma in Bochner (1937).:

Definition**.**

For $0<\lambda\leq 1$, there is a probability distribution $\mathcal{P}\left(\lambda\right)$ over $\mathbb{R}_{+}$, called positive stable distribution with parameter $\lambda$, such that if $\left(Z_{1},...Z_{N}\right)$ are i.i.d. draws from that distribution, then for positive reals $\alpha_{1},...,\alpha_{N}$,

[TABLE]

and the Laplace distribution of $\mathcal{P}\left(\lambda\right)$ is $\mathbb{E}_{\mathcal{P}\left(\lambda\right)}\left[\exp\left(-tZ\right)\right]=\exp\left(-t^{\lambda}\right)$.

Here and throughout the paper we use $\sim$ to indicate “distributed as”. Our main result connects the Gumbel distribution and positive stable distributions:

Theorem**.**

For $0<\lambda\leq 1$, if random variables $\epsilon$ and $Z$ are independent, and $\epsilon\sim\mathcal{G}\left(0,1\right)$, and $Z\sim\mathcal{P}\left(\lambda\right)$ has a positive stable distribution with parameter $\lambda$, then $\epsilon+\log Z\sim\mathcal{G}\left(0,1/\lambda\right)$.

Proof.

Assuming $\epsilon\sim\mathcal{G}\left(0,1\right)$, and $Z\sim\mathcal{P}\left(\lambda\right)$, the c.d.f. of $\epsilon+\log Z$ is expressed as

[TABLE]

which is the c.d.f. of the $\mathcal{G}\left(0,1/\lambda\right)$ distribution.

As a result, we can deduce the moments of $\eta=\lambda\log Z$, as is done in Ben-Akiva and Lerman (1985), section 10.3. The expectation of $\eta$ is $\mathbb{E}\left[\eta\right]=\left(1-\lambda\right)\mathbb{E}\left[\epsilon\right]=\left(1-\lambda\right)\gamma$ where $\gamma$ is Euler’s constant. The variance of $\eta$ is $var\left(\eta\right)=\left(1-\lambda^{2}\right)var\left(\epsilon\right)=\left(1-\lambda^{2}\right)\pi^{2}/6$. More generally, the moment generating function of $\eta$ is given by $\mathbb{E}\left[e^{t\eta}\right]=\Gamma\left(1-t\right)/\Gamma\left(1-\lambda t\right)$, that is, for $\kappa\in\left(0,\lambda\right)$,

[TABLE]

The probability distribution function $f_{\lambda}$ associated with $\mathcal{P}\left(\lambda\right)$ is in general not available in closed form, except for a few cases, namely $\lambda=1/3,1/2$ and $2/3$, in which case one obtains

[TABLE]

where $Ai$ is the Airy function and $U$ is the confluent hypergeometric function; see Dishon and Bendler (1990). In general, $f_{\lambda}$ cannot be given an explicit expression, but can be expressed as a convergent series expansion using Humbert’s (1945) formula

[TABLE]

where $\Gamma$ is the Gamma function, classically defined as

[TABLE]

Remark 1** (Relating plus-stability and max-stability).**

Setting $\eta_{\lambda}=\lambda\log Z_{\lambda}$ for $Z_{\lambda}\sim\mathcal{P}\left(\lambda\right)$, and taking $\left(\tilde{\eta}_{\lambda,j}\right)_{j\in\mathcal{J}}$ a vector of independent copies of $\eta_{\lambda}$, it follows from (2) that we have

[TABLE]

and it follows from $\eta_{\lambda}+\lambda\epsilon\sim\mathcal{G}\left(0,1\right)$ that $\eta_{\lambda}$ converges weakly to $\mathcal{G}\left(0,1\right)$. As the log-sum-exp (a.k.a. smooth-max) converges to the max when $\lambda\rightarrow 0$, we recover the max-stability of the Gumbel distribution as a consequence.

As noted by Cardell (1997, lemma 1), another interesting consequence of the theorem is as follows.

Remark 2**.**

Take $\lambda_{1},\lambda_{2}\in\left(0,1\right)$. From the theorem, it holds that if $Z_{1}\sim\mathcal{P}\left(\lambda_{1}\right)$ and $\epsilon_{1}\sim\mathcal{G}\left(0,1\right)$ are independent, then $\epsilon_{2}=\lambda_{1}\left(\epsilon_{1}+\log Z_{1}\right)\sim\mathcal{G}\left(0,1\right)$. Thus, if $Z_{2}\sim\mathcal{P}\left(\lambda_{2}\right)$ is independent from $\epsilon_{1}$ and $Z_{1}$, then $\lambda_{2}\left(\epsilon_{2}+\log Z_{2}\right)\sim\mathcal{G}\left(0,1\right)$, then

[TABLE]

As a result, $\lambda_{2}\lambda_{1}\log Z_{1}+\lambda_{2}\log Z_{2}$ has the same distribution as $\lambda_{1}\lambda_{2}\log Z_{12}$, where $Z_{12}\sim\mathcal{P}\left(\lambda_{1}\lambda_{2}\right)$. Thus for $\lambda_{1},\lambda_{2}\in\left(0,1\right)$, if $Z_{1}\sim\mathcal{P}\left(\lambda_{1}\right)$ and $Z_{2}\sim\mathcal{P}\left(\lambda_{2}\right)$ are independent, then

[TABLE]

2. Representation of the nested logit model

We can use the result of the previous section to provide rigorous ground for the representation of the random utility associated with the nested logit model, which we carry first in the context of a single layer of nests, before moving on to the case of multiple layers.

2.1. Single layer of nests

The model described in section 1 is a single nest model, where, as it is well-known, if $\left(\varepsilon_{nj}\right)\sim F_{\varepsilon}$ has the distribution given by (1), then the probability that $j\in\mathcal{N}_{n}$ maximizes $U_{j^{\prime}}+\varepsilon_{j^{\prime}}$ over $j^{\prime}\in\mathcal{N}_{n^{\prime}}$ and $n^{\prime}\in\mathcal{N}$, is given by

[TABLE]

which is the product of the probability that nest $n$ is chosen by the conditional probability that $j$ is chosen within nest $n$. It is also well-known that in this case, the Emax operator $G\left(U\right)=\mathbb{E}\left[\max_{n\in\mathcal{N},j\in\mathcal{N}_{n}}\left\{U_{j}+\varepsilon_{j}\right\}\right]$ has expression

[TABLE]

and we can recover that, as predicted by the Daly-Zachary-Williams theorem, the expression of $\pi_{j}$ (4) is indeed obtained as the derivative of $G$ with respect to $U_{j}$.

With these reminders in mind, we can use the results in section 1 to provide a factor representation of the random vector $\left(\varepsilon_{j}\right)$.

Proposition 1**.**

If $\left(\varepsilon_{j}\right)$ is a random vector distributed according to the nested logit distribution (1), then one can represent the random vector $\left(\varepsilon_{j}\right)$ as

[TABLE]

where $\epsilon_{j}\sim\mathcal{G}\left(0,1\right)$ for $j\in\mathcal{N}_{n},n\in\mathcal{N}$, and $Z_{n}\sim\mathcal{P}\left(\lambda_{n}\right)$ for $n\in\mathcal{N}$ are all independently distributed. In particular, the correlation between $\varepsilon_{j}$ and $\varepsilon_{j^{\prime}}$ for $j\in\mathcal{N}_{n}$ and $j^{\prime}\in\mathcal{N}_{n^{\prime}}$ is $\left(1-\lambda_{n}^{2}\right)$ if $n=n^{\prime}$, [math] otherwise.

Note that the correlation structure was first rigorously derived by Tiago de Oliveira (1958) using very different techniques, see also p. 266 of his 1997 monograph.

Proof.

Letting $\eta_{n}=\lambda_{n}\exp Z_{n}$, the c.d.f. of the vector $\left(\lambda_{n}\epsilon_{j}+\eta_{n}\right)_{j\in\mathcal{N}_{n},n\in\mathcal{N}}$ is

[TABLE]

But $\max_{j\in\mathcal{N}_{n}}\left\{-a_{nj}+\lambda_{n}\epsilon_{j}\right\}$ has the same distribution as $\lambda_{n}\log\sum_{j\in\mathcal{N}_{n}}\exp\left(\frac{-a_{j}}{\lambda_{n}}\right)+\lambda_{n}\hat{\epsilon}_{n}$ where $\hat{\epsilon}_{n}$ has a $\mathcal{G}\left(0,1\right)$ distribution and the $\hat{\epsilon}_{n}$ and the $\eta_{n}$ terms are all independent. Hence the displayed equation equals

[TABLE]

which, as \left(\lambda_{n}\epsilon_{n}+\eta_{n}\right)\are i.i.d. with distribution $\mathcal{G}\left(0,1\right)$, is the same as expression (1).

Remark 3**.**

Representation (6) allows to view the nested logit model as a mixed logit model. Indeed, it implies

[TABLE]

where $Z_{n}\sim\mathcal{P}\left(\lambda_{n}\right)$, and therefore, if the $Z_{n}^{k}$, $k\in\left\{1,...,K\right\}$ are $K$ draws from the $\mathcal{P}\left(\lambda_{n}\right)$ distribution, one can approximate the previous expression by

[TABLE]

The simulation of positive stable distribution is discussed in Ridout (2009), where an algorithm is proposed and implemented in R.

2.2. Multiple layers of nests

The nested logit model is defined in its full generality on an arborescence of nests. In order to facilitate our description of the model, we need to recall the standard mathematical terminology of trees.

2.2.1. Trees

Consider a finite set of nodes $\mathcal{Z}$ with more than one element, and call root one of its nodes, denoted [math]. An arborescence (a.k.a. “directed rooted tree”) on $\mathcal{Z}$ is a directed graph such that there is exactly one directed path from [math] to any node $z$. Nodes that have no outward arcs are called leaves – which we will identify in our case as “alternatives.” The set of leaves/alternatives is denoted $\mathcal{J}$. We shall call nests the nodes which are not alternatives, and denote by $\mathcal{N}$ the set of nests. We have therefore $\mathcal{Z}=\mathcal{N}\cup\mathcal{J}$, and $0\in\mathcal{N}$.

Each nest $n$ has a set $\mathcal{N}_{n}$ set of child nests and $\mathcal{J}_{n}$ set of child alternatives. Both $\mathcal{N}_{n}$ and $\mathcal{J}_{n}$ can be empty, but $\mathcal{N}_{n}\cup\mathcal{J}_{n}$ cannot. Alternatives have no children: they are the “leaves” of the arborescence.

The parent node of any node $z$ which is not the root is a nest denoted $n_{z}$.

The depth $d_{z}$ of a node $z$ is the number of edges from the nest to the root nest [math]. The depth of the root nest is zero.

The height $h_{z}$ of a node $z$ is the number of edges on the longest path from the nest to a leaf (alternative). In particular, alternatives have height zero, and nodes that have only alternatives (and no nests) as children, have height one. The height of the arborescence is the height of [math].

If a nest $n$ has depth $d$, the path from the root nest to $n$ along the arborescence is denoted $n\left(1\right),...,n\left(d\right)$, where $n\left(1\right)\in\mathcal{N}_{0}$ and $n\left(k+1\right)\in\mathcal{N}_{n\left(k\right)}$, and $n\left(d\right)=n$. Note that by convention, the root nest [math] is omitted from this path.

If there is a path from $z$ to $z^{\prime}$ along the arborescence, one says that $z$ is an ancestor of $z^{\prime}$ and $z^{\prime}$ is a descendant of $z$.

We denote by $\widetilde{\mathcal{J}}_{z}$ the set of alternatives that are descendant of node $z$. This is to be contrasted by $\mathcal{J}_{z}$, which is the set of alternatives that are children of $z$. Of course, $\widetilde{\mathcal{J}}_{0}=\mathcal{J}$. Note that $\widetilde{\mathcal{J}}_{j}=\left\{j\right\}$ for $j\in\mathcal{J}$.

The lowest common ancestor to two nodes is the node which is an ancestor of both of them and which has the largest depth.

2.2.2. Reminders on the nested logit model

For any nest $n\in\mathcal{N}$, we define $\lambda_{n}\in(0,1]\,$. We assume that $\lambda_{0}=1$, and we define

[TABLE]

The nested logit model takes as an input the systematic utility $U_{j}$ associated with each alternative $j\in\mathcal{J}$, and returns the vector of choice probability $\pi_{j}$ associated with each $j$, a probability vector on $\mathcal{J}$. As is well-known, $\pi$ is computed as follows:

1. Computing utilities by backward induction. One defines the payoff vector $u$ on all the nodes $z\in\mathcal{Z}=\mathcal{N}\cup\mathcal{J}$ recursively in ascending height order by

[TABLE]

This is well-defined because the children of a nest of height $t$ has height at most $t-1$.

2. Computing choice probabilities by forward induction. Define $\pi_{z}$ in ascending depth order by

[TABLE]

This is well-defined because the parent of a node of depth $t$ has depth $t-1$. In particular, this procedure defines the probabilities $\pi_{j}$ on the alternatives $\mathcal{J}$, which are the leaves of the arborescence. Note that combining (7) and (8) yields the log-odds ratio formula

[TABLE]

It is well-known that the nested logit model is an additive random utility model. Consider a random vector $\varepsilon$ whose c.d.f. is defined by backward induction by

[TABLE]

Then the choice probabilities $\left(\pi_{j}\right)$ determined by the backward-forward procedure coincide with the probabilities induced by the additive random utility model associated with $\varepsilon$, that is

[TABLE]

Another well-known relation between the distribution of $\varepsilon$ and the payoff vector $u$ is given by the Emax operator defined (see McFadden, 1978) by

[TABLE]

and more generally, the restricted Emax at $n$ is defined by taking the maximum over alternatives that are descendent of $n$, that is

[TABLE]

and one has

[TABLE]

where $u_{n}$ is computed as in (7). One has

[TABLE]

and in particular setting $n=0$ one recovers $\partial G\left(U\right)/\partial U_{j}=\pi_{j}$, which is the Daly-Zachary-Williams theorem.

2.2.3. The representation result

From the main theorem in section 1, we are able to derive the following representation.

Proposition 2**.**

If $\left(\varepsilon_{j}\right)$ is a random vector distributed according to the nested logit distribution (10), then one can represent the random vector $\left(\varepsilon_{j}\right)_{j\in\mathcal{J}}$ as

[TABLE]

where $\epsilon_{j}\sim\mathcal{G}\left(0,1\right)$ and $Z_{n\left(t\right)}\sim\mathcal{P}\left(\lambda_{t}\right)$ are all independent.

Proof.

By induction on $h\geq 0$ and less than the height of the arborescence, one shows that the probability that

[TABLE]

coincides with the probability that

[TABLE]

where $\epsilon_{z}$ is defined on all the nodes $z\in\mathcal{Z}=\mathcal{N}\cup\mathcal{J}$ recursively in increasing height order by

[TABLE]

Base case: if $h_{n}=1$, then $\mathcal{N}_{n}=\emptyset$, and the base case ($h=0$) is proven.

Inductive step: assume the property proven up to rank $h$, and consider a nest $n$ of height $h_{n}=h+1$. Then the nodes that are children of $n$ are either alternatives, or other nests that have height at most $h$. Collecting all the terms pertaining to $n$, these are

[TABLE]

The conditional probability that this occurs is the probability that

[TABLE]

where we have set $\tilde{\epsilon}_{n}:=\max_{z\in\mathcal{N}_{n}\cup\mathcal{J}_{n}}\left\{-a_{z}/\Lambda_{n}+\epsilon_{z}\right\}-\sum_{z\in\mathcal{N}_{n}\cup\mathcal{J}_{n}}\exp\left(-a_{z}/\Lambda_{n}\right)$. This is the same as the probability that

[TABLE]

and setting $\epsilon_{n}:=\lambda_{n}\log Z_{n}+\lambda_{n}\tilde{\epsilon}_{n}^{1}$ yields

[TABLE]

Collecting all $n$ with height $h_{n}=h+1$ allows to complete the proof of the inductive step.

Corollary 1**.**

Under the assumptions of Proposition 2, for $j\neq j^{\prime}$, the correlation between $\varepsilon_{j}$ and $\varepsilon_{j^{\prime}}$ is $1-\Lambda_{n}^{2}$ where $n$ is the lowest common ancestor between $j$ and $j$’.

Proof.

Let $n$ be the lowest common ancestor between $j$ and $j^{\prime}$, we get that if $n=n_{j}\left(s\right)$

[TABLE]

is distributed as a $\mathcal{G}\left(0,1\right)$ distribution, which one shall denote $\epsilon_{n,j}$. The same applies to the representation of $\varepsilon_{j^{\prime}}$, and hence $(\varepsilon_{j},\varepsilon_{j^{\prime}})$ can be represented as

[TABLE]

where $\left(\epsilon_{n,j},\epsilon_{n,j^{\prime}}\right)$ are independent and are distributed according to a $\mathcal{G}\left(0,1\right)$ distribution. Therefore

[TABLE]

which established the formula.

3. Correlation structure of the multivariate Fréchet distribution

with Gumbel copula

We can use the previous results in order to provide an explicit formula for the correlation between two Fréchet random variables coupled with a Gumbel copula of parameter $1/\lambda$, a formula which can be useful, see Mattsson et al. (2014), Fosgerau et al. (2018), and Hårsman and Mattsson (2020)333The author thanks L.-G. Mattsson for an insightful question leading to this section.. Recall that a Gumbel copula of parameter $1/\lambda$ is expressed as $C\left(u,v\right)=\exp\left(-\left(\left(\left(-\log u\right)^{1/\lambda}+\left(-\log v\right)^{1/\lambda}\right)\right)^{\lambda}\right)$. Also recall that a Fréchet random variable of shape parameter $\alpha>0$ has c.d.f. $F\left(x\right)=\exp\left(-x^{-\alpha}\right)$; it can be represented as $\exp\left(\varepsilon/\alpha\right)$, where $\varepsilon\sim\mathcal{G}\left(0,1\right)$. It is easy to see that a pair $\left(\delta_{1},\delta_{2}\right)$ of Fréchet random variables coupled with a Gumbel copula of parameter $1/\lambda$ can be written as $\left(\delta_{1}=\exp\left(\varepsilon_{1}/\alpha\right),\delta_{2}=\exp\left(\varepsilon_{2}/\alpha\right)\right)$, where the joint c.d.f. of $\left(\varepsilon_{1},\varepsilon_{2}\right)$ is $\exp\left(-\left(\exp\left(-x/\lambda\right)+\exp\left(-y/\lambda\right)\right)^{\lambda}\right)$, falling under specification (1). By proposition 1, it follows that we can represent $\left(\delta_{1},\delta_{2}\right)$ as

[TABLE]

where $\epsilon_{1},\epsilon_{2}\sim\mathcal{G}\left(0,1\right)$, $Z\sim\mathcal{P}\left(\lambda\right)$, and $\left(\epsilon_{1},\epsilon_{2},Z\right)$ are independent. We can therefore provide a close-form formula for the correlation between $\delta_{1}$ and $\delta_{2}$.

Proposition 3**.**

Assume $\alpha>2$ and $\lambda\in\left(0,1\right)$. Then the correlation between two Fréchet random variables of shape parameter $\alpha$ coupled by a Gumbel copula of parameter $1/\lambda$ is given by

[TABLE]

Proof.

We have $\rho\left(\delta_{1},\delta_{2}\right)=\left(\mathbb{E}\left[\delta_{1}\delta_{2}\right]-\mathbb{E}\left[\delta_{1}\right]^{2}\right)/\left(\mathbb{E}\left[\delta_{1}^{2}\right]-\mathbb{E}\left[\delta_{1}\right]^{2}\right)$. Using the well-known fact that the $\mathcal{G}\left(0,1\right)$ distribution has moment generating function $\Gamma\left(1-t\right)$, where $\Gamma$, we get that $\mathbb{E}\left[\delta_{1}\right]=\Gamma\left(1-1/\alpha\right)$, and that $\mathbb{E}\left[\delta_{1}^{2}\right]=\Gamma\left(1-2/\alpha\right)$ $\mathbb{E}\left[\delta_{1}\delta_{2}\right]=\mathbb{E}\left[\exp\left(\frac{\lambda}{\alpha}\left(\epsilon_{1}+\epsilon_{2}+2\log Z\right)\right)\right]=\Gamma\left(1-\frac{\lambda}{\alpha}\right)^{2}\mathbb{E}\left[Z^{2\lambda/\alpha}\right]$. Further, one has by formula (3) that $\mathbb{E}\left[Z^{2\lambda/\alpha}\right]=\Gamma\left(1-2/\alpha\right)/\Gamma\left(1-2\lambda/\alpha\right)$, from which we deduce (16).

4. Conclusion

This paper is the first to provide a complete and rigorous random utility representation of the nested logit model, solving a four-decade old problem since Ben-Akiva’s seminal work on the topic. The representation will hopefully shed new light on the nested logit model and help formulate specifications as well as interpret estimated nested logit models. Since this logit model is popular in practical work, the representation can be useful in many applications, as in the example given in section 3. An additional contribution of the paper is the complete derivation of the general recursive nested logit model based on trees, carried out in section 2.2, which is typically not performed in full in expository texts. We feel that the standard textbook treatment of the nested logit model would benefit from adopting the present approach.