DiSSMod {DiSSMod}

Fitting Sample Selection Models for Discrete Response Variables

Description

Function DiSSMod fits sample selection models for discrete random variables, by suitably extending the formulation of the classical Heckman model to the case of a discrete response, but retaining the original conceptual framework. Maximum likelihood estimates are obtained by Newton-Raphson iteration combined with use of profile likelihood.

Usage

DiSSMod(response, selection, data, resp.dist, select.dist, alpha,
  trunc.num, standard = FALSE, verbose = 1, eps = 1e-07,
  itmax = 1000)

Arguments

response

a formula for the response equation.

selection

a formula for the selection equation.

data

a data frame and data has to be included with the form of data.frame.

resp.dist

a character for the distribution choice of the response variable, "bernoulli" for Bernoulli distribution, "poisson" for Poisson distribution, and "negbinomial" for Negative binomial distribution. Also, the character strings can be abbreviated and can be upper or lower case as preferred.

select.dist

a character for the distribution choice of the selection variable, "gumbel" for Gumbel distribution, "normal" for Normal distribution, and "logistic" for Logistic distribution. Also, the character strings can be abbreviated and can be upper or lower case as preferred.

alpha

a vector of alpha values on which the profile log-likelihood function is evaluated; if the argument is missing, a set of values in the interval (-10, 10) is used for the initial search, followed by a second search on a revised interval which depends on the outcome from the first search.

trunc.num

an integer numeric constant used as the truncation point of an infine summation of probabilities involved when resp.dist equals "Poisson" or "NegBinomial"; if the argument is missing, a default choice is made, as described in the ‘Details’ section. Notice: this default choice of trunc.num may be subject to revision in some future version of the package, and the argument trunc.num itselt may possibly be replaced by some other ingredient.

standard

a logical value for the standardizing explanatory variables, if TRUE two types of values (standardized and not) will be returned.

verbose

an integer value for the level of printed details (values: 0|1|2); the default value is 1 which stands for shortly printed details. If the value is 2, more details are viewed such as values of the log likelihood functions and iteration numbers. If the value is 0, there is no printed detail.

eps

a numeric value for the estimating parameters, which is needed for the step of the optimization. If the sum of absolute differences between present step estimated parameters and former step estimated parameters is smaller than eps, we assume that estimated parameters are optimized.

itmax

an integer stands for maximum number for the iteration of optimizing the parameters.

Details

The specification of the two linear models regulating the response variable and the selection mechanism, as indicated in the ‘Background’ section, is accomplished by two arguments of formula type, denoted response and selection, respectively. Each formula is specified with the same syntax of similar arguments in standard functions such as lm and glm, with the restriction that the intercept term (which is automatically included) must not be removed.

The distributional assumptions associated to the response and selection components are specified by the arguments resp.dist and select.dist, respectively. Argument select.dist refers to the unobservable continuous variable of which we observe only the dichotomous outcome Yes-No.

In this respect, a remark is appropriate about the option "Gumbel" for select.dist. This choice is equivalent to the adoption of an Exponential distribution of the selection variables combined an exponential transformation of the linear predictor of the selection argument, as it is presented in Section 3.2 of Azzalini et al. (2019). Also, it corresponds to work with the log-transformation of an Exponential variable, which is essentially a Gumbel type of variable, up to a linear transformation with respect to its more commonly employed parameterization.

When resp.dist is "Poisson" or "NegBinomial" and trunc.num is missing, a default choice is made; this equals 1.5*m or 2*m in the two respective cases, where m denotes the maximum observed value of the response variable.

Function DiSSMOd calls lower level functions, nr.bin, nr.nbinom, nr.pois and the others for the actual numerical maximization of the log-likelihood via a Newton-Raphson iteration.

Notice that the automatic initialization of the alpha search interval, when this argument is missing, may change in future versions of the package.

Value

DiSSMod returns an object of class "DiSSMod", which is a list containing following components:

call

a matched call.

standard

a logical value, stands for standardization or not.

st_loglik

a vector containing the differences between log likelihoods and maximized log likelihood.

max_loglik

a maximized log likelihood value.

mle_alpha

a maximized likelihood estimator of alpha.

alpha

a vector containing grids of the alpha

Nalpha

a vector containing proper alpha, which does not have NA value for corresponding log likelihood.

num_NA

a number of NA values of log likelihoods.

n_select

a number of selected response variables.

n_all

a number of all response variables.

estimate_response

estimated values for the response model.

std_error_response

estimated standard errors for the response model.

estimate_selection

estimated values for the selection model.

std_error_selection

estimated standard errors for the selection model.

Background

Function DiSSMod fits sample selection models for discrete random variables, by suitably extending the formulation of the classical Heckman model to the case of a discrete response, but retaining the original conceptual framework. This logic involves the following key ingredients: (1) a linear model indicating which explanatory variables influence the response variable; (2) a linear model indicating which (possibly different) explanatory variables, besides the response variable itself, influence a ‘selection variable’, which is intrinsically continuous but we only observe a dichotomous outcome from it, of type Yes-No, which selects which are the observed response cases; (3) distributional assumptions on the response and the selection variable.

The data fitting method is maximum likelihood estimation (MLE), which operates in two steps: (i) for each given value of parameter alpha which regulates the level of selection, MLE is performed for all the remaining parameters, using a Newton-Raphson iteration; (ii) a scan of the alpha axis builds the profile log-likelihood function and its maximum point represents the overall MLE.

A detailed account of the underlying theory and the operational methodology is provided by Azzalini et al. (2019).

References

Azzalini, A., Kim, H.-M. and Kim, H.-J. (2019) Sample selection models for discrete and other non-Gaussian response variables. Statistical Methods & Applications, 28, 27–56. First online 30 March 2018. https://doi.org/10.1007/s10260-018-0427-1

See Also

The functions summary.DiSSMod, coef.DiSSMod, confint.DiSSMod, plot.DiSSMod are used to obtain and print a summary, coefficients, confidence interval and plot of the results.

The generic function logLik is used to obtain maximum log likelihood of the result.

See also lm, glm and formula.

Examples

set.seed(45)
data(DoctorRWM, package = "DiSSMod")
n0 <- 600
set.n0 <- sample(1:nrow(DoctorRWM), n0)
reduce_DoctorRWM <- DoctorRWM[set.n0,]
result0 <- DiSSMod(response = as.numeric(DOCVIS > 0) ~ AGE + INCOME_SCALE + HHKIDS + EDUC + MARRIED,
                   selection = PUBLIC ~ AGE + EDUC + FEMALE,
                   data = reduce_DoctorRWM, resp.dist="bernoulli", select.dist = "normal",
                   alpha = seq(-5.5, -0.5, length.out = 21), standard = TRUE)

print(result0)

data(CreditMDR, package = "DiSSMod")
n1 <- 600
set.n1 <- sample(1:nrow(CreditMDR), n1)
reduce_CreditMDR <- CreditMDR[set.n1,]
result1 <- DiSSMod(response = MAJORDRG ~ AGE + INCOME + EXP_INC,
                   selection = CARDHLDR ~ AGE + INCOME + OWNRENT + ADEPCNT + SELFEMPL,
                   data = reduce_CreditMDR, resp.dist="poi", select.dist = "logis",
                   alpha = seq(-0.3, 0.3,length.out = 21), standard = FALSE, verbose = 1)

print(result1)