rcrologit

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The package provides estimation and inferential procedures for rank-ordered logit model with agents with heterogeneous taste preferences.

Setup

We have $n$ i.i.d. random draws

$$\mathcal{D}:=(Y_i,X_i{)}_{i=1}^n$$

where $Y_i:=(Y_{i0},Y_{i1},\dots ,Y_{iJ}{)}^{\mathrm{\top }},Y_{i\ell }=0,1,\dots ,J,$ is a vector of ranks and $C_i:=(X_{i0},C_{i1},\dots ,C_{iJ}{)}^{\mathrm{\top }}\in {\mathbb{R}}^{(J+1)\cdot K},$ $C_{i\ell }\in {\mathbb{R}}^K$. Let the latent utility model (McFadden, 1974) be

$$U_{ij}^{\star }=u_{ij}+{ϵ}_{ij},{ϵ}_{ij}\stackrel{\mathtt{iid}}{\sim }\mathsf{Gu}(0,1).$$

For notational convenience, define the functions $r_i:0,1,\dots ,J\to 0,1,\dots ,J,i=1,\dots ,n$. Such functions map the rank $j\in 0,1,\dots ,J$ into the corresponding item $r(j)\in 0,1,\dots ,J$ according to individual $i$'s preferences. To clarify

$$Y_{ij}=k⟺r_i(k)=j$$

An observed ranking for a respondent implies a complete ordering of the underlying utilities. An individual will prefer an item with a higher utility over an item with a lower utility. If we observe a full ranking $r_i:=(r_i(0),r_i(1),\dots ,r_i(J){)}^{\mathrm{\top }}$, we know that

$$U_{i{r}_i(0)}^{\star }>U_{i{r}_i(1)}^{\star }>\cdots >U_{i{r}_i(J)}^{\star }.$$

Therefore, the probability of observing a particular ranking $r_i$ is given by

$$\mathbb{P}[r_i\mid \mathcal{D}]=\mathbb{P}[U_{i{r}_i(0)}^{\star }>U_{i{r}_i(1)}^{\star }>\cdots >U_{i{r}_i(J)}^{\star }\mid \mathcal{D}]=\prod _{j=0}^{J-1}\frac{\exp \left(u_{i{r}_i(j)}\right)}{\sum _{j\le \ell \le J}\exp \left(u_{i{r}_i(\ell )}\right)}.$$

In light of this, we can see that the rank-ordered logit is nothing else than a series of multinomial logit (MNL) models: when $j=0$ we considered a MNL the most preferred item; another MNL for the second-ranked item to be preferred over all items except the one with rank 1, and so on. Finally, the probability of a complete ranking is made up of the product of these separate MNL probabilities. The product contains only $J$ probabilities, because ranking the least preferred item is done with probability 1.

Modelling

In its most general form, we allow the user to model $u_{i\ell }$ in the latent utility model as

$$ u_{i\ell} = X_{i\ell}^{\top} \boldsymbol{\beta}{\mathtt{F}} + Z_i^{\top} \boldsymbol{\alpha}{\ell,\mathtt{F}} + W_{i\ell}^{\top} \boldsymbol{\beta}i + V_i^{\top} \boldsymbol{\alpha}{i\ell} + \delta_{\ell} $$

An alternative, handier way to rewrite the model above is to define $Z_{i\ell }:=\sum _{j=1}^J{Z}_i\times \mathbf{1}(j=\ell )$ and $V_{i\ell }:=\sum _{j=1}^J{V}_i\times \mathbf{1}(j=\ell ),\ell =1,2,\dots ,J$, and consider the equivalent model

$$ u_{i\ell}=X_{i\ell}^\top\boldsymbol{\beta}{\mathtt{F}} + Z{i\ell}^\top\boldsymbol{\alpha}{\mathtt{F}} + W{i\ell}^\top\boldsymbol{\beta}i + V{i\ell}^\top\boldsymbol{\alpha}{i} + \delta\ell, $$

where:

• $X_{i\ell }$ are covariates varying at the unit-alternative level whose coefficients are modelled as fixed
• $Z_i$ are covariates varying at the unit level whose coefficients are modelled as fixed
• $W_{i\ell }$ are covariates varying at the unit-alternative level whose coefficients are modelled as random
• $V_i$ are covariates varying at the unit level whose coefficients are modelled as random the random coefficients
• The heterogeneous taste coefficients are modeled as a joint multivariate normal and are i.i.d. across units with mean ${[{\mathit{\alpha }}_{\mathtt{R}}^{\mathrm{\top }},{\mathit{\beta }}_{\mathtt{R}}^{\mathrm{\top }}]}^{\mathrm{\top }}$ and variance $\mathbf{\Sigma }$.
• ${\delta }_{\ell }$ are alternative-specific fixed effects
• ${ϵ}_{i\ell }\sim \mathsf{Gu}(0,1)$ are idiosyncratic i.i.d. shocks

Note that whenever $W_{i\ell }$ and $V_i$ are not specified estimates a standard rank-ordered logit with no heterogeneous preferences and the conditional choice probabilities are given by

$$\mathbb{P}[r_i\mid \mathcal{D}]=\prod _{j=0}^{J-1}\frac{\exp \left(u_{i{r}_i(j)}\right)}{\sum _{j\le \ell \le J}\exp \left(u_{i{r}_i(\ell )}\right)}.$$

If instead agents are allowed to have heterogeneous taste, then

$$\mathbb{P}[r_i\mid \mathcal{D}]=\int \prod _{j=0}^{J-1}\frac{\exp (u_{i{r}_i(j)}^{\mathrm{\top }}({\beta }_i))}{\sum _{j\le \ell \le J}\exp (u_{i{r}_i(\ell )}^{\mathrm{\top }}({\beta }_i))}\varphi ({\beta }_i;\beta ,\mathrm{\Sigma })\mathrm{d}{\beta }_i.$$

The parameter vector to be estimated is thus

$$\theta ={({\mathit{\beta }}_{\mathtt{F}}^{\mathrm{\top }},{\mathit{\beta }}_{\mathtt{R}}^{\mathrm{\top }},{\mathit{\alpha }}_{\mathtt{F}}^{\mathrm{\top }},{\mathit{\alpha }}_{\mathtt{R}}^{\mathrm{\top }},\mathrm{vech}(\mathbf{\Sigma }{)}^{\mathrm{\top }},{\delta }_{j=1}^J)}^{\mathrm{\top }}.$$

Estimation

The ideal maximum likelihood estimator is defined as

$$ \widehat{\theta}{\mathtt{ML}}:=\mathrm{arg}\max{\theta} \sum_{i=1}^n\log\int \prod_{j=0}^{J-1} \frac{\exp \left(u_{ir_i(j)}(\theta)\right)}{\sum\limits_{j\leq \ell\leq J} \exp \left(u_{ir_i(\ell)}(\theta)\right)} \phi(\beta_i;\beta_{\mathtt{R}},\Sigma) \mathrm{d} \beta_i. $$

We approximate the integral via montecarlo as

$$ \widehat{\mathbb{P}}{(\widehat{\beta},\widehat{\Sigma})}[r_i\mid \mathcal{D}]=\frac{1}{S}\sum{i=1}^S \prod_{j=0}^{J-1} \frac{\exp \left(u_{ir_i(j)}(\theta,\beta_i^{(s)})\right)}{\sum\limits_{j\leq \ell\leq J} \exp \left(u_{ir_i(\ell)}(\theta,\beta_i^{(s)})\right)}, $$

where $\boldsymbol{\beta}i\overset{\mathtt{iid}}{\sim}\mathsf{N}(\widehat{\boldsymbol\beta}{\mathtt{R}},\widehat{\boldsymbol{\Sigma}})$.

Installation

You can install the development version of rcrologit from GitHub with:

# install.packages("devtools")
devtools::install_github("filippopalomba/rcrologit")

Basic Usage

library(rcrologit)

data <- rcrologit_data

# Rank-ordered logit
dataprep <- dataPrep(data, idVar = "Worker_ID", rankVar = "rank",
                    altVar = "alternative",
                    covsInt.fix = list("Gender"),
                    covs.fix = list("log_Wage"), FE = c("Firm_ID"))
    
rologitEst <- rcrologit(dataprep)

# Rank-ordered logit
dataprep <- dataPrep(data, idVar = "Worker_ID", rankVar = "rank",
                    altVar = "alternative",
                    covsInt.het = list("Gender"),
                    covs.fix = list("log_Wage"), FE = c("Firm_ID"))
    
rologitEst <- rcrologit(dataprep, stdErr="skip")