# Estimation of Random Coefficient Binary Response Models 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. ```r 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. ## Returns of object of class `GK`, `KW1`, with components described in further detail in the respective fitting functions. ## 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. ## Examples ```r { 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) } } ``` ## 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](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. ## Author(s) Jiaying Gu and Roger Koenker

rcbr function