Heteroscedastic Tobit Regression

In htobit2017: Heteroscedastic Tobit Regression Models

library("htobit2017")
library("car")
library("crch")
library("lmtest")
library("memisc")

Introduction

The htobit package fits tobit regression models with conditional heteroscedasticy using maximum likelihood estimation. The model assumes an underlying latent Gaussian variable

$$y_i^* \sim \mathcal{N}(\mu_i, \sigma_i^2)$$

which is only observed if positive and zero otherwise: $y_i = \max(0, y_i^*)$. The latent mean $\mu_i$ and scale $\sigma_i$ (latent standard deviation) are linked to two different linear predictors $$ \begin{aligned} \mu_i & = x_i^\top \beta \ \log(\sigma_i) & = z_i^\top \gamma \end{aligned} $$ where the regressor vectors $x_i$ and $z_i$ can be set up without restrictions, i.e., they can be identical, overlapping or completely different or just including an intercept, etc.

See also @crch for a more detailed introduction to this model class as well as a better implementation in the package crch. The main purpose of htobit is to illustrate how to create such a package from scratch.

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Implementation

As usual in many other regression packages for R [@R], the main model fitting function htobit() uses a formula-based interface and returns an (S3) object of class htobit:

htobit(formula, data, subset, na.action,
  model = TRUE, y = TRUE, x = FALSE,
  control = htobit_control(...), ...)

Actually, the formula can be a two-part Formula [@Formula], specifying separate sets of regressors $x_i$ and $z_i$ for the location and scale submodels, respectively.

The underlying workhorse function is htobit_fit() which has a matrix interface and returns an unclassed list.

A number of standard S3 methods are provided:

| Method | Description | |:----------------|:-------------------------------------------------------------------------------------| | print() | Simple printed display with coefficients | | summary() | Standard regression summary; returns summary.htobit object (with print() method) | | coef() | Extract coefficients | | vcov() | Associated covariance matrix | | predict() | (Different types of) predictions for new data | | fitted() | Fitted values for observed data | | residuals() | Extract (different types of) residuals | | terms() | Extract terms | | model.matrix()| Extract model matrix (or matrices) | | nobs() | Extract number of observations | | logLik() | Extract fitted log-likelihood | | bread() | Extract bread for sandwich covariance | | estfun() | Extract estimating functions (= gradient contributions) for sandwich covariances | | getSummary() | Extract summary statistics for mtable() |

Due to these methods a number of useful utilities work automatically, e.g., AIC(), BIC(), coeftest() (lmtest), lrtest() (lmtest), waldtest() (lmtest), linearHypothesis() (car), mtable() (memisc), Boot() (car), etc.

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Illustration

To illustrate the package's use in practice, a comparison of several homoscedastic and heteroscedastic tobit regression models is applied to data on budget shares of alcohol and tobacco for 2724 Belgian households [taken from @Verbeek:2004]. The homoscedastic model from @Verbeek:2004 can be replicated by:

data("AlcoholTobacco", package = "htobit2017")
library("htobit2017")
ma <- htobit(alcohol ~ (age + adults) * log(expenditure) + oldkids + youngkids, data = AlcoholTobacco)
summary(ma)

This model is now modified in two directions: First, the variables influencing the location parameter are also employed in the scale submodel. Second, because the various coefficients for different numbers of persons in the household do not appear to be very different, a restricted specification for this is used. Using a Wald test for testing linear hypotheses from car [@car] yields

library("car")
linearHypothesis(ma, "oldkids = youngkids")
linearHypothesis(ma, "oldkids = adults")

Therefore, the following models are considered:

AlcoholTobacco$persons <- with(AlcoholTobacco, adults + oldkids + youngkids)
ma2 <- htobit(alcohol ~ (age + adults) * log(expenditure) + oldkids + youngkids |
  (age + adults) * log(expenditure) + oldkids + youngkids, data = AlcoholTobacco)
ma3 <- htobit(alcohol ~ age + log(expenditure) + persons | age + log(expenditure) + persons, data = AlcoholTobacco)
BIC(ma, ma2, ma3)

The BIC would choose the most parsimonious model but a likelihood ratio test would prefer the unconstrained person coefficients:

library("lmtest")
lrtest(ma, ma2, ma3)

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Replication

To assess the reliability of the htobit() implementation, it is benchmarked against the crch() function of @crch, using the same restricted model as above.

library("crch")
ca3 <- crch(alcohol ~ age + log(expenditure) + persons | age + log(expenditure) + persons,
  data = AlcoholTobacco, left = 0)

Using a model table from memisc [@memisc] it can be easily seen the results can be replicated using both packages:

library("memisc")
mtable("htobit" = ma3, "crch" = ca3)

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References

htobit2017 documentation built on May 2, 2019, 6:04 p.m.