---
title: "Endogenous switching or sample selection models for count data"
date: today
date-format: long
bibliography: ../inst/REFERENCES.bib
vignette: >
  %\VignetteIndexEntry{Endogenous switching or sample selection models for count data}
  %\VignetteEngine{quarto::pdf}
  %\VignetteEncoding{UTF-8}
---

# Introduction

In two seminal papers, @HECK:76 and @HECK:79 considered the case where
two variable are jointly determined: a binomial variable and an
outcome continuous variable. For example, the continuous variable can
be the wage and the binomial variable the labor force
participation. In this case, the wage is observed only for the
sub-sample of the individuals who works. If the unobserved
determinants of labor force participation are correlated with the
unobserved determinants of wage, estimating the wage equation only on
the subset of individuals who work will result in an inconsistent
estimator. This case is called the **sample selection** model.

An other example is the case where the binomial variable is a a dummy
for private vs public sector. In this case the wage is observed for
the whole sample (for the individuals in the public and in the private
sector). But, once again, if the unobserved determinants of the
chosen sector are correlated with those of the wage,
estimating the wage equation only will leads to inconsistent
estimators. This case is called the **endogenous switching** model. 

Two consistent method of estimation can be used in this context:

- the first one is a two-step method where, in a first step, a probit
  model for the binomial variable is estimated and, in a second step,
  the outcome equation is estimated by OLS with a supplementary
  covariate which is a function of the linear predictor of the
  probit^[More precisely the inverse Mills ratio.],
- the second one is the maximum likelihood estimation of the system of
  the two equations, assuming that the error terms are jointly
  normally distributed.


# Sample selection and endogenous switching for count data

Let $y$ be a count response (for sake of simplicity a Poisson
variable) and $z$ a binomial variable. The value of $z$ is given by
the sign of $\alpha ^ \top z + \nu$, where $\nu$ is a standard normal
deviate, $z$ a vector of covariates and $\alpha$ the associated vector
of unknown parameters. The distribution of $y_n$ is Poisson with
parameter $\lambda_n$, given by $\ln \lambda_n = \beta ^ \top x_n +
\epsilon_n$ where $x$ is a second set of covariates (which can overlap
with $z$), $\beta$ is the corresponding set of unknown parameters and
$\epsilon$ is a random normal deviate with 0 mean and a standard
deviation equal to $\sigma$. $\epsilon$ and $\nu$ being potentially
correlated, their joint distribution has to be considered:

$$
\begin{array}{rcl}
f(\epsilon, \nu ; \sigma, \rho) &=&
\displaystyle \frac{1}{2\pi\sqrt{1 - \rho ^ 2}\sigma}
e^{\displaystyle-\frac{1}{2} \frac{\left(\frac{\epsilon}{\sigma}\right) ^ 2 + \nu ^
2 - 2 \rho \left(\frac{\epsilon}{\sigma}\right)\nu}{1 - \rho ^2}} \\
&=& \displaystyle \frac{1}{\sqrt{2\pi}}\sigma e ^
{-\frac{1}{2}\left(\frac{\epsilon}{\sigma}\right) ^ 2}
\frac{1}{\sqrt{2\pi}\sqrt{1 - \rho ^ 2}}
e^{-\frac{1}{2}\left(\frac{(\nu - \rho \epsilon / \sigma)}{\sqrt{1 -
\rho ^ 2}}\right) ^ 2}
\end{array}
$$

$$
\begin{array}{rcl}
f(\epsilon, \nu ; \sigma, \rho) &=&
\displaystyle \frac{1}{2\pi\sqrt{1 - \rho ^ 2}\sigma}
e^{\displaystyle-\frac{1}{2} \frac{\left(\frac{\epsilon}{\sigma}\right) ^ 2 + \nu ^
2 - 2 \rho \left(\frac{\epsilon}{\sigma}\right)\nu}{1 - \rho ^2}} \\
&=& \displaystyle \frac{1}{\sqrt{2\pi}}\sigma e ^
{-\frac{1}{2}\left(\frac{\epsilon}{\sigma}\right) ^ 2}
\frac{1}{\sqrt{2\pi}\sqrt{1 - \rho ^ 2}}
e^{-\frac{1}{2}\left(\frac{(\nu - \rho \epsilon / \sigma)}{\sqrt{1 -
\rho ^ 2}}\right) ^ 2}
\end{array}
$$

The second expression classically gives the joint distribution as the
product of the marginal distribution of $\epsilon$ and the conditional
distribution of $\nu$.


For $z=1$, the unconditional distribution of $y$ is obtained by
integrating out $g(y_n \mid x_n, \epsilon_n, \nu_n, z_n = 1)$ with
respect with the two random deviates:

$$
P(y_n \mid x_n, z_n = 1) = \int_{-\infty} ^ {+\infty}\int_{-\alpha ^ \top
z_n} ^ {+ \infty} g(y_n \mid x_n, \epsilon_n, z_n = 1)
f(\epsilon, \nu) d\epsilon d\nu
$$

$$
\begin{array}{rcl}
P(y_n \mid x_n, z_n = 1) &=& \displaystyle \int_{-\infty} ^ {+\infty}
g(y_n \mid x_n, \epsilon_n, z_n = 1)
\left(\int_{-\alpha ^ \top z_n} ^ {+ \infty}
\frac{1}{\sqrt{2\pi}\sqrt{1 - \rho ^ 2}}
e^{-\frac{1}{2}\left(\frac{(\nu - \rho \epsilon / \sigma)}{\sqrt{1 -
\rho ^ 2}}\right) ^ 2}
d\nu\right)\\
&\times& \displaystyle \frac{1}{\sqrt{2\pi}\sigma} e ^
{-\frac{1}{2}\left(\frac{\epsilon}{\sigma}\right) ^ 2}
d\epsilon
\end{array}
$$

By symetry of the normal distribution, the term in braquet is: 

$$
\Phi\left(\frac{\alpha ^ \top z_n + \rho / \sigma \epsilon}{\sqrt{1 -
\rho ^ 2}}\right)
$$

which is the probability that $z = 1$ for a given value of
$\epsilon$. The density of $y$ given that $z = 1$ is then:

$$
P(y_n \mid x_n, z_n = 1) = \int_{-\infty} ^ {+\infty} \frac{e ^
{-\mbox{exp}(\beta ^ \top x_n + \epsilon_n)}e^{y_n(\beta^\top x_n +
\epsilon_n)}}{y_n!}\Phi\left(\frac{\alpha ^ \top z_n + \rho / \sigma \epsilon}{\sqrt{1 -
\rho ^ 2}}\right)\frac{1}{\sqrt{2\pi}\sigma} e ^
{-\frac{1}{2}\left(\frac{\epsilon}{\sigma}\right) ^ 2} d\epsilon
$$ {#eq-probyz}

By symmetry, it is easily shown that $P(y_n \mid x_n, z_n = 0)$ is
similar except that $\Phi\left( (\alpha ^ \top z_n + \rho / \sigma
\epsilon) / \sqrt{1 - \rho ^ 2}\right)$ is replaced by $1 - \Phi\left(
(\alpha ^ \top z_n + \rho / \sigma \epsilon) / \sqrt{1 - \rho ^
2}\right) = \Phi\left( -(\alpha ^ \top z_n + \rho / \sigma \epsilon) /
\sqrt{1 - \rho ^ 2}\right)$, so that a general formulation of the
distribution of $y$ is, denoting $q = 2 z -1$:

$$
P(y_n \mid x_n) = \int_{-\infty} ^ {+\infty} \frac{e ^
{-\mbox{exp}(\beta ^ \top x_n + \epsilon_n)}e^{y_n(\beta^\top x_n +
\epsilon_n)}}{y_n!}\Phi\left(q_n\frac{\alpha ^ \top z_n + \rho / \sigma \epsilon}{\sqrt{1 -
\rho ^ 2}}\right)\frac{1}{\sqrt{2\pi}\sigma} e ^
{-\frac{1}{2}\left(\frac{\epsilon}{\sigma}\right) ^ 2} d\epsilon
$$ {#eq-proby}

There is no closed form for this integral but, using the change of
variable $\eta = \epsilon / \sqrt{2} / \sigma$, we get:

$$
P(y_n \mid x_n) = \int_{-\infty} ^ {+\infty} \frac{e ^
{-\mbox{exp}(\beta ^ \top x_n + \sqrt{2} \sigma  \eta)}e^{y_n(\beta^\top x_n +
\sqrt{2}\sigma\eta)}}{y_n!}\Phi\left(q_n\frac{\alpha ^ \top z_n + \sqrt{2} \rho  \eta}{\sqrt{1 -
\rho ^ 2}}\right)\frac{1}{\sqrt{\pi}} e ^
{-\eta ^ 2} d\eta
$$

which can be approximated using Gauss-Hermite quadrature. Denting
$\eta_r$ the nodes and $\omega_r$ the weights:

$$
P(y_n \mid x_n) \approx 
\sum_{r = 1} ^ R
\omega_r \frac{e ^
{-\mbox{exp}(\beta ^ \top x_n + \sqrt{2} \sigma  \eta_r)}e^{y_n(\beta^\top x_n +
\sqrt{2}\sigma\eta_r)}}{y_n!}\Phi\left(q_n\frac{\alpha ^ \top z_n + \sqrt{2} \rho  \eta_r}{\sqrt{1 -
\rho ^ 2}}\right)\frac{1}{\sqrt{\pi}} e ^
{-\eta_r ^ 2}
$$

For the exogenous switching model, the contribution of one observation
to the likelihood is given by @eq-proby). 

For the sample selection model, the contribution of one observation to
the likelihood is given by @eq-probyz if $z_n = 1$. If $z_n =
0$, $y$ is unobserved and the contribution of such observations to the
likelihood is the probability that $z_n = 0$, which is:

$$
P(z_n = 0 \mid x_n) = \int_{-\infty} ^ {+\infty}
\Phi\left(q_n\frac{\alpha ^ \top z_n + \rho / \sigma \epsilon}{\sqrt{1 -
\rho ^ 2}}\right)\frac{1}{\sqrt{2\pi}\sigma} e ^
{-\frac{1}{2}\left(\frac{\epsilon}{\sigma}\right) ^ 2} d\epsilon
$$


The ML estimator is computing intensive as the integral has no closed
form. One alternative is to use non-linear least squares, by first
computing the expectation of $y$. @TERZ:98 showed that, in the
endogenous switching case:

$$
\mbox{E}(y_n\mid x_n) = \mbox{exp}\left(\beta ^ \top x_n + \ln \frac{\Phi\left(q_n(\alpha ^\top
z_n + \theta)\right)}{\Phi\left(q_n(\alpha ^\top z_n)\right)}\right)
$$

For the sample selection case, we have:

$$
\mbox{E}(y_n\mid x_n, z_n = 1) = \mbox{exp}\left(\beta ^\top x_n + \ln \frac{\Phi(\alpha ^\top
z_n + \theta)}{\Phi(\alpha ^ \top z_n)}\right)
$$


@GREE:01 noted that, taking a first order Taylor series of $\ln
\frac{\Phi(\alpha ^\top z_n + \theta)}{\Phi(\alpha ^ \top z_n)}$
around $\theta = 0$ gives: $\theta \phi(\alpha ^ \top z_n) /
\Phi(\alpha ^ \top z_n)$, which is the inverse mills ratio that is
used in the linear case in order to correct the inconsistency due to
sample selection. As $\alpha$ can be consistently estimated by a
probit model, the NLS estimator is obtained by minimizing with respect
to $\beta$ and $\theta$ the sum of squares of the following residuals:

$$
y_n - e^{\displaystyle \beta ^ \top x_n + \ln
\frac{\Phi(\hat{\alpha} ^ \top z_n + \theta)}{\Phi(\hat{\alpha}
^ \top z_n)}}
$$

As it is customary for these two-step estimator, the covariance
matrix of the estimators should take into account the fact that
$\alpha$ has been estimated in the first step. Moreover, only $\theta
= \rho \sigma$ is estimated. To retrieve an estimator of $\sigma$,
@TERZ:98 proposed to insert into the log-likelihood function the
estimated values of $\alpha$, $\beta$ and $\theta$ and then to
maximize it with respect to $\sigma$^[These once again requires to use
Gauss-Hermite quadrature, but the problem is considerably simpler as
the likelihood is maximized with respect with only one parameter.].

# The `escount` function


```{r }
library(micsr)
```

The `escount` function estimates the endogenous switching and the
sample selection model for count data. The first one is obtained by
setting the `model` argument to `'es'` (the default) and the second
one to `'ss'`. The estimation method is selected using the `method`
argument, which can be either `'twostep'` for the two-step non-linear
least squares model (the default) or `'ML'` for maximum
likelihood. The model is described by an extended formula (using the
**Formula** package) of the form:

`y | d ~ x + y + z + d | x + y + w`

which indicates that the two responses are `y` (the count) and `d` (the
binomial variable), that the covariates are `x`, `y` and `z`
for the count equation and `x`, `y` and `w` for the
switching/selection equation. When there are two large sets of
covariates that overlap, it is possible to define the second set of
covariates by updating the first one:

`y | d ~ x + y + z + d | . - d - z + w`

`R` is an integer that indicates the number of points used for the
Gauss-Hermite quadrature method. Relevant only for the ML method, the
`hessian` argument is a boolean: if `TRUE`, the covariance matrix of
the coefficients is estimated using the numerical hessian, which is
computed using the `hessian` function of the **numDeriv**
package, otherwise, the outer product of the gradient is used.

`escount` returns an object of class `escount` which inherits from
`lm`. 

<!-- This is a list that contains usual items that can be found in -->
<!-- `lm` objects and some more specific items: -->

<!-- - `K` and `L` are the number of covariates in the count equation and -->
<!--   in the binomial equation, -->
<!-- - `sigma` and `rho` are the estimates of $\sigma$ and $\rho$ for the -->
<!--   two-steps method. -->

# Trips demand

@TERZ:98 analyzed the number of trips taken by 577 individuals in the
United States in 1978 the day before they were interviewed. The
`trips` data set is included in the **micsr** package.A major
determinant of trips demand is the availability of a car in the
household. @TERZ:98 advocates that the unobserved determinants of the
decision of having a car may be correlated with those of the trip
demand equation. In this case the estimation of the Poisson model will
lead to inconsistent estimators.  The covariates are the share of
trips for work or school (`workschl`), the number of individuals in
the household (`size`), the distance to the central business district
(`dist`) a factor (`smsa`) with two levels for small and large urban
area, the number of full-time worker in household (`fulltime`), the
distance from home to nearest transit node, household income divided
by the median income of the census tract (`realinc`), a dummy if the
survey period is Saturday or Sunday (`weeekend`) and a dummy for
owning at least a car (`car`). Although the coefficients are
identified, as in the classic Heckman model by the non-linearity of
the correction term, @TERZ:98 use a different set of variable for the
binomial and the count parts of the model. Namely, the `weekend`
covariate is removed and `adults` is added in the binomial part of the
model.

We first compute the two-steps NLS estimator. The `model` and `method`
arguments needn't to be set are the default values are `es` (endogenous
switching) and `twosteps` (two-steps NLS).

```{r}
#| warning: false
trips_2s <- escount(trips + car ~ workschl + size + dist + smsa + fulltime +
                        distnod + realinc + weekend + car | . - car -
                        weekend + adults, data = trips)
names(trips_2s)
```

The result is a list with usual items, except
<!-- `L` and `K` which are -->
<!-- the number of the covariates in the binomial and in the count -->
<!-- equations,  -->
`sigma` and `rho` which are the estimates of $\sigma$ and
$\rho$ obtained from the estimation of $\theta$.

The print of the summary method returns the usual table of
coefficients, the value of the objective function (the sum of squares
residuals) and the estimated values of $\sigma$ and $\rho$:

```{r}
summary(trips_2s)
```

```{r}
#| label: tripsest
#| warning: false
trips_pois <- glm(trips ~ workschl + size + dist + smsa + fulltime +
                      distnod + realinc + weekend + car,
                  data = trips, family = poisson)
trips_ml <- update(trips_2s, method = "ml")
```

<!-- @tbl-tripsres presents the results of a Poisson estimation -->
<!-- and of the endogenous switching model, estimated by the two-steps -->
<!-- non-linear least squares method and by maximum likelihood. -->


<!-- ```{r} -->
<!-- #| label: tbl-tripsres -->
<!-- #| warning: false -->
<!-- #| echo: false -->
<!-- #| tbl-cap: "Estimation results for the trip demand model" -->
<!-- if (requireNamespace("modelsummary")){ -->
<!--     modelsummary::msummary(list("Poisson" = trips_pois, "two-step" = trips_2s, "ML" = trips_ml)) -->
<!-- } -->
<!-- ``` -->

The coefficient of `car` is equal to 1.413 for the Poisson model,
which implies that the number of trips increases by $e^{1.413} - 1 =
311$% for individuals who belongs to households that own at least one
car. The coefficient is much higher in the selection models (2.796 for
the 2-step estimator and 2.160 for the ML estimator), which implies an
increase of 767% for the ML estimator. The Poisson estimator
is therefore downward biased, which indicates a negative correlation
between the unobserved part of the trips demand equation and the
propensity to own a car equation.

# Physician advice and alcohol consumption

@KENK:TERZ:01 investigate the effect of physician's advice on alcohol
consumption. The outcome variable `drinks` is the number of drinks in
the past 2 weeks and the selection variable `advice` is a dummy based
on the respondents' answer to the question "Have you ever been told
by a physician to drink less". The unobserved part of the equation
indicating the propensity to receive an advice form the physician can
obviously be correlated with the one of the alcohol consumption
equation. The data set `drinks` is part of the **micsr**
package. The covariates are monthly income in thousands of dollars
(`income`), `age` (a factor with six 10 years categories of age),
education in years (`educ`), `race` (a factor with levels `white`,
`black` and `other`), the marital status (`marital`, a factor with
levels `single`, `married`, `widow`, `separated`), the employment
status (a factor `empstatus` with levels `other`, `emp` and `unemp`)
and the region (`region`, a factor with levels `west`, `northeast`,
`midwest` and `south`). For the binomial part of the model, the same
covariates are used (except of course `advice`) and 10 supplementary
covariates indicating the insurance coverage and the health status are
added.

```{r}
#| label: drinksest
#| warning: false
#| cache: true
kt_pois <- glm(drinks ~ advice + income + age + educ + race + marital +
                   empstatus + region, data = drinks, family = poisson)
kt_ml <- escount(drinks + advice ~ advice + income + age + educ + race +
                     marital + empstatus + region | income + age + educ +
                     race + marital + empstatus + region + medicare +
                     medicaid + champus + hlthins + regmed + dri + limits +
                     diabete + hearthcond + stroke,
                 data = drinks, method = "ml")
kt_2s <- update(kt_ml, method = "twostep")
```


<!-- @tbl-drinkers presents the results of a Poisson -->
<!-- estimation and of the endogenous switching model, estimated by the -->
<!-- two-steps non-linear least squares method and maximum likelihood^[The -->
<!-- coefficients of `marital`, `empstatus` and `region` are omitted to -->
<!-- save place.]. -->

<!-- ```{r} -->
<!-- #| label: tbl-drinkers -->
<!-- #| warning: false -->
<!-- #| echo: false -->
<!-- #| tbl-cap:  "Poisson and endogenous switching models for alcohol demand" -->
<!-- if (requireNamespace("modelsummary")){ -->
<!--     modelsummary::msummary(list("Poisson" = kt_pois, "two-step" = kt_2s, "ML" = kt_ml), -->
<!--                            gof_omit = "AIC|BIC|Log.Lik.", -->
<!--                            coef_omit = "region|marital|empstatus|age") -->
<!-- } -->
<!-- ``` -->



The coefficient of `advice` in the alcohol demand equation is positive
in the Poisson model, which would imply that a positive effect of
physical advice on alcohol consumption. The estimation of the
endogenous switching model shows that this positive coefficient is
due to the positive correlation between the error terms of the two
equations (the unobserved propensities to drink and to receive an
advice from a physician are positively correlated).

<!-- # Job changes -->

<!-- @WYSZ:MARR:18 analyzed the determinants of the number of job -->
<!-- changes. The data set `soep` (included in the **escount** package) is -->
<!-- an extract of the German Socio-Economic Panel survey of 1984. It -->
<!-- concerns 2651 individuals and the response `djc` is the number of -->
<!-- direct job changes (changes without an intervening spell of -->
<!-- unemployment). Obviously this variable is observed only for -->
<!-- individuals who are part of the job market, which corresponds to a -->
<!-- value equal to 1 for the selection variable `lfp`. The covariates for -->
<!-- the count equation are `spd` (a factor indicating if the individual is -->
<!-- a strong or a very strong supporter of the social-democrat party in -->
<!-- Germany), `whitecol` (a factor indicating it the individual is a white -->
<!-- collar) and `educ` (number of years of education). For the selection -->
<!-- equation, the covariates are `single` (a factor indicating if the -->
<!-- individual is single), `whitecol` and `exper` (full-time employment in -->
<!-- years). The results are presented in table \@ref(tab:soepres). -->

<!-- ```{r soepest, warning = FALSE, cache = TRUE} -->
<!-- soep_pois <- glm(djc ~ spd + whitecol + educ, data = soep, family = poisson) -->
<!-- soep_2s <- escount(djc | lfp ~ spd + whitecol + educ | single + whitecol + exper, -->
<!--                    data = soep, model = "ss") -->
<!-- soep_ml <- update(soep_2s, method = "ml") -->
<!-- ``` -->

<!-- ```{r soepres, warning = FALSE, eval = FALSE} -->
<!-- msummary(list("Poisson" = soep_pois, "2 steps" = soep_2s, "ML" = soep_ml), -->
<!-- #         gof_omit = "AIC|BIC|Log.Lik.", -->
<!--          label = "tab:soepres", -->
<!--          title = "Poisson and selection models for the job change model") -->
<!-- ``` -->


# References

