rcbr {RCBR}

Estimation of Random Coefficient Binary Response Models

Description

Two methods are implemented for estimating binary response models with random coefficients: A nonparametric maximum likelihood method proposed by Cosslett (1986) and extended by Ichimura and Thompson (1998), and a (hemispherical) deconvolution method proposed by Gautier and and Kitamura (2013). The former is closely related to the NPMLE for mixture models of Kiefer and Wolfowitz (1956). The latter is an R translation of the matlab implementation of Gautier and Kitamura.

Usage

rcbr(formula, data, subset, offset, mode = "GK", ...)

Arguments

formula

an expression of the generic form y ~ z + v where y is the observed binary response, z is an observed covariate with a random coefficient, and v is an observed covariate with coefficient normalize to be one. If z is not present then the model has only a random "intercept" coefficient and thus corresponds to the basic model of Cosslett (1983); this model is also referred to as the current status model in the biostatistics literature, see Groeneboom and Hendrikx (2016). When z is present there are random coefficients associated with both the intercept and z.

data

is a data.frame containing the data referenced in the formula.

subset

specifies a subsample of the data used for fitting the model

offset

specifies a fixed shift in v representing the potential effect of other covariates having fixed coefficients that may be useful for profile likelihood computations. (Should be vector of the same length as v.

mode

controls whether the Gautier and Kitamura, "GK", or Kiefer and Wolfowitz, "KW" methods are used.

...

miscellaneous other arguments to control fitting. See GK.control and KW.control for further details.

Details

The predict method produces estimates of the probability of a "success" (y = 1) for a particular vector, (z,v), when aggregated over the estimated distribution of random coefficients.

The logLik produces an evaluation of the log likelihood value associated with a fitted model.

Value

of object of class GK, KW1, with components described in further detail in the respective fitting functions.

Author(s)

Jiaying Gu and Roger Koenker

References

Kiefer, J. and J. Wolfowitz (1956) Consistency of the Maximum Likelihood Estimator in the Presence of Infinitely Many Incidental Parameters, Ann. Math. Statist, 27, 887-906.

Cosslett, S. (1983) Distribution Free Maximum Likelihood Estimator of the Binary Choice Model, Econometrica, 51, 765-782.

Gautier, E. and Y. Kitamura (2013) Nonparametric estimation in random coefficients binary choice models, Ecoonmetrica, 81, 581-607.

Gu, J. and R. Koenker (2020) Nonparametric Maximum Likelihood Methods for Binary Response Models with Random Coefficients, J. Am. Stat Assoc

Groeneboom, P. and K. Hendrickx (2016) Current Status Linear Regression, preprint available from https://arxiv.org/abs/1601.00202.

Ichimura, H. and T. S. Thompson, (1998) Maximum likelihood estimation of a binary choice model with random coefficients of unknown distribution," Journal of Econometrics, 86, 269-295.

Examples

{
if(packageVersion("Rmosek") > "8.0.0"){
    # Simple Test Problem for rcbr
    n <- 60
    B0 = rbind(c(0.7,-0.7,1),c(-0.7,0.7,1))
    z <- rnorm(n)
    v <- rnorm(n)
    s <- sample(0:1, n, replace = TRUE)
    XB0 <- cbind(1,z,v) %*% t(B0)
    u <- s * XB0[,1] + (1-s) * XB0[,2]
    y <- (u > 0) - 0
    D <- data.frame(z = z, v = v, y = y)
    f <- rcbr(y ~ z + v, mode = "KW", data = D)
    plot(f)
    # Simple Test Problem for rcbr
    set.seed(15)
    n <- 100
    B0 = rbind(c(0.7,-0.7,1),c(-0.7,0.7,1))
    z <- rnorm(n)
    v <- rnorm(n)
    s <- sample(0:1, n, replace = TRUE)
    XB0 <- cbind(1,z,v) %*% t(B0)
    u <- s * XB0[,1] + (1-s) * XB0[,2]
    y <- (u > 0) - 0
    D <- data.frame(z = z, v = v, y = y)
    f <- rcbr(y ~ z + v, mode = "GK", data = D)
    contour(f$u, f$v, matrix(f$w, length(f$u)))
    points(x = 0.7, y = -0.7, col = 2)
    points(x = -0.7, y = 0.7, col = 2)
    f <- rcbr(y ~ z + v, mode = "GK", data = D, T = 7)
    contour(f$u, f$v, matrix(f$w, length(f$u)))
    points(x = 0.7, y = -0.7, col = 2)
    points(x = -0.7, y = 0.7, col = 2)
    }
}