---
title: "Estimating the Tobit-1 model with the charitable data set"
header-includes:
  - \usepackage{xcolor}
output: 
  rmarkdown::html_vignette
toc: false
bibliography: ../inst/REFERENCES.bib
vignette: >
  %\VignetteIndexEntry{Estimating the Tobit-1 model with the charitable data set}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---

We'll reproduce here some results obtained by @WILH:08 using a data
set which deals with charitable giving. The `charitable` data set is shiped with
the `micsr`.

```{r setup, include = FALSE}
knitr::opts_chunk$set(message = FALSE, warning = FALSE, out.width = "100%",
                      fig.align = "center", cache = FALSE, echo = TRUE)
options(knitr.table.format = "latex", knitr.booktabs = TRUE)
```

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

```{r }
print(charitable, n = 5)
```

The response is called `donation`, it measures annual charitable
givings in $US. This variable is left-censored for the value of 25, as
this value corresponds to the item "less than 25 $US
donation". Therefore, for this value, we have households who didn't
make any charitable giving and some which made a small giving (from 1
to 24 $US).

The covariates used are the donation made by the parents
(`donparents`), two factors indicating the educational level and
religious beliefs (respectively `education` and `religion`), annual
income (`income`) and two dummies for living in the south (`south`)
and for married couples (`married`). 

@WILH:08 consider the value of the donation in logs and substract $\ln
25$, so that the response is 0 for households who gave no donation or
a small donation.


```{r }
charitable$logdon <- with(charitable, log(donation) - log(25))
```

The tobit model can be estimated by maximum likelihood using
`AER::tobit`, `censReg::censReg` or with the `tobit1` package. 

```{r }
char_form <- logdon ~ log(donparents) + log(income) +
    education + religion + married + south
if (requireNamespace("AER")){
    library("AER")
    ml_aer <- tobit(char_form, data = charitable)
}
if (requireNamespace("censReg")){
    library("censReg")
    ml_creg <- censReg(char_form, data = charitable)
}
ml <- tobit1(char_form, data = charitable)
```

`tobit1` provide a rich set of estimation methods, especially the
**SCLS** (symetrically censored least squares) estimator proposed by
@POWE:86. We also, for pedagogical purposes, estimate the OLS
estimator although it is known to be unconsistent.


```{r }
scls <- update(ml, method = "trimmed")
ols <- update(ml, method = "lm")
```

The results of the three models are presented in table below

```{r models, echo = FALSE}
if (requireNamespace("modelsummary")){
    modelsummary::msummary(list("OLS" = ols, "maximum likehihood" = ml, "SCLS" = scls),
                           title = "Estimation of charitable giving models",
                           label = "tab:models", single.row = TRUE, digits = 3)
}
```

The last two columns of the  match exactly the
first two columns of [@WILH:08, table 3 page 577]. Note that the OLS
estimators are all lower in absolute values than those of the two
other estimators, which illustrate the fact that OLS estimators are
biased toward zero when the response is censored. The maximum
likelihood is consistent and asymtotically efficient if the
conditional distribution of $y^*$ (the latent variable) is
homoscedastic and normal.

Specification tests for the maximum likelihood estimator can  be conducted
using conditional moments tests. This can easily be done using the
`micsr::cmtest` function, which can take as input a model fitted by
either `AER::tobit`, `censReg::censReg` or `tobit1::tobit1`:


```{r }
cmtest(ml)
```

```{r include = FALSE}
cmtest(ml_aer)
cmtest(ml_creg)
```

`cmtest` has a `test` argument with default value equal to
`normality`. To get a heteroscedasticity test, we would use:

```{r }
cmtest(ml, test = "heterosc")
```
Normality and heteroscedasticity are strongly rejected. The values are
different from @WILH:08 as he used the "outer product of the gradient"
form of the test. These versions of the test can be obtained by
setting the `OPG` argument to `TRUE`.

```{r }
cmtest(ml, test = "normality", opg = TRUE)
cmtest(ml, test = "heterosc", opg = TRUE)
```

Non-normality can be further investigate by testing separately
the fact that the skewness and kurtosis indicators are respectively
different from 0 and 3.

```{r }
cmtest(ml, test = "skewness")
cmtest(ml, test = "kurtosis")
```

The hypothesis that the conditional distribution of the response is
mesokurtic is not rejected at the 1% level and the main problem seems
to be the asymetry of the distribution, even after taking the
logarithm of the response. 

This can be illustrated by plotting the
(unconditional) distribution of the response (for positive values) and
adding to the histogram the normal density curve.


```{r histnorm, fig.cap = "Empirical distribution of the response and normal approximation", echo = FALSE}
library("ggplot2")
library("dplyr")
moments <- charitable %>% filter(logdon > 0) %>%
    summarise(mu = mean(logdon), sigma = sd(logdon))
ggplot(filter(charitable, logdon > 0), aes(logdon)) +
    geom_histogram(aes(y = ..density..), color = "black",
                   fill = "white", bins = 10) +
    geom_function(fun = dnorm, args = list(mean = moments$mu,
                                           sd = moments$sigma)) +
    labs(x = "log of charitable giving", y = NULL)
```

# References
