semiBRM: Semiparametric Binary Response Models in R

This R package semiBRM offers an implementation of single-index semiparametric binary response models, theorized in a seminal work in the semiparametric econometrics literature Klein and Spady (1993).

This is built upon Rcpp along with OpenMP, which parallelizes computation of nonparametric conditional expectations over given data points. This improves computation efficiency enormously in parameter estimation, which is one of the main features of this package.

The package is still in development, with the goal of making it as easy to employ Klein and Spady (1993) as to run Probit or Logistic regressions in R.

Identifiable Set of Parameters

In a single-index semiparametric approach, not all parameters are identifiable. First of all, intercept cannot be estimated. Secondly, the coefficients can be estimated as ratios to a “basis” coefficient. This package sets the coefficient of the first explanatory variable as the basis so that its coefficient is normalized to 1, and those of the rest of the variables are estimated conformably to it.

In the context of binary response models, or nonlinear models more broadly, this is not problematic at all, as the coefficients themselves have little interpretation. As the matter of fact, the identifiable set of parameters in this approach can correctly estimate the conditional probability of Pr(Y=1|X), which would be the quantity of interest in many cases.

Installation

The development version can be installed from GitHub:

library(devtools)
install_github(repo="henrykye/semiBRM")

This version was built with:

• R version 4.0.5 (2021-03-31)
• maxLik 1.4-8
• Rcpp 1.0.6
• RcppArmadillo 0.10.4.0.0

Example

Setup

library(semiBRM)
set.seed(20190815) # for reproduction of results

Data Generating Process

## data generating process
N <- 1500L
X1 <- rnorm(N)
X2 <- (X1 + 2*rnorm(N))/sqrt(5) + 1
X3 <- rnorm(N)^2/sqrt(2)
X <- cbind(X1, X2, X3)
beta <- c(2, 2, -1, -1) # this is the original set of coefficients
V <- as.vector(cbind(X, 1)%*%beta)
Y <- ifelse(V >= rnorm(N), 1L, 0L)

Parameter Estimation

## estimands: the rescaled coefficients by the first coefficient excluding intercept
coefs_true <- c(1, -.5)
data <- data.frame(Y, X1, X2, X3)

## Klein and Spady (1993): semiparametric approach
semi <- semiBRM(Y ~ X1 + X2 + X3, data = data, control = list(iterlim=50))
coefs_semi <- coef(semi)

## Probit: parametric approach
probit <- glm(Y ~ X1 + X2 + X3, family = binomial(link = "probit"), data = data)
coefs_probit <- probit$coefficients[-1L][-1L]/probit$coefficients[-1L][1L]

## formatted print
{
    cat(sprintf("        %7s %7s %7s\n", "True", "Probit", "Semi"))
    cat(sprintf("parm 1: %7.4f %7.4f %7.4f\n", coefs_true[1L], coefs_probit[1L], coefs_semi[1L]))
    cat(sprintf("parm 2: %7.4f %7.4f %7.4f\n", coefs_true[2L], coefs_probit[2L], coefs_semi[2L]))
}
#>            True  Probit    Semi
#> parm 1:  1.0000  0.9296  0.9430
#> parm 2: -0.5000 -0.4069 -0.4103

In-Sample Prediction

## in-sample conditional probability
in_prob_true <- pnorm(V)
in_prob_semi <- predict(semi)
in_prob_probit <- fitted(probit)

## formatted print
target <- sample.int(N, size = 10L)
{
    cat(sprintf("%7s %8s %8s %8s %12s\n", "Obs.", "True", "Probit", "Semi", "non.endpoint") )
    for (i in target){
        cat(sprintf("[%04d]: %8.6f %8.6f %8.6f %12s\n",
                    i, in_prob_true[i], in_prob_probit[i], in_prob_semi$prob[i], in_prob_semi$non.endpoint[i]))
    }
}
#>    Obs.     True   Probit     Semi non.endpoint
#> [0034]: 0.999400 0.999262 0.991678         TRUE
#> [0514]: 0.898871 0.859331 0.774815         TRUE
#> [0480]: 0.995355 0.994697 0.970159         TRUE
#> [1285]: 0.004911 0.006525 0.034459         TRUE
#> [0065]: 0.990712 0.993818 0.966249         TRUE
#> [0966]: 0.896823 0.862113 0.780052         TRUE
#> [0558]: 0.997327 0.997560 0.981306         TRUE
#> [1001]: 0.000001 0.000000 0.000653         TRUE
#> [0843]: 0.998430 0.998133 0.984302         TRUE
#> [0381]: 0.781482 0.773341 0.697235         TRUE

Out-of-Sample Prediction

## conditional probability at the means
Xbar <- colMeans(X)
newdata <- as.data.frame(as.list(Xbar))

## predictions
out_prob_true <- pnorm(as.vector(c(Xbar, 1)%*%beta))
out_prob_semi <- predict(semi, newdata, boot.se = TRUE)
out_prob_probit <- pnorm(as.vector(coef(probit)%*%c(1, Xbar)))

## standard errors of Probit
grad <- dnorm(as.vector(coef(probit)%*%c(1, Xbar))) * c(1, Xbar)
out_stde_probit <- sqrt(as.vector(crossprod(grad, vcov(probit))%*%grad))

## formatted print
{
    cat(sprintf("            %7s %7s %7s\n", "True", "Probit", "Semi"))
    cat(sprintf("Prob. Est.: %7.4f %7.4f %7.4f\n", out_prob_true, out_prob_probit, out_prob_semi$prob))
    cat(sprintf(" Std. Err.: %7s %7.4f %7.4f\n", "", out_stde_probit, out_prob_semi$boot.se))
}
#>                True  Probit    Semi
#> Prob. Est.:  0.5833  0.5792  0.5543
#>  Std. Err.:          0.0230  0.0221

Average Marginal Effects

## marginal effects as difference between conditional probabilities with and without perturbation
delta <- sd(X1) # size of perturbation

me_true <- pnorm(as.vector(cbind(X1+delta, X2, X3, 1)%*%beta)) - pnorm(as.vector(cbind(X, 1)%*%beta))

## average marginal effects
av_me_true <- mean(me_true)
av_me_semi <- MarginalEffects(semi, variable = "X1", delta = delta)

## formatted print
{
    cat(sprintf("           %7s %7s \n", "True", "Semi"))
    cat(sprintf("Marg.Eff.: %7.4f %7.4f\n", av_me_true, av_me_semi[1L]))
    cat(sprintf("Std. Err.: %7s %7.4f\n", "", av_me_semi[2L]))
}
#>               True    Semi 
#> Marg.Eff.:  0.2021  0.2029
#> Std. Err.:          0.0050

Quantile Marginal Effects

## percentile cutoffs
p.cutoffs <- c(1/4, 2/4, 3/4)

## group indicators
q_vals <- quantile(X1, p.cutoffs)
q1 <- ifelse(X1 <= q_vals[1L], 1L, 0L)
q2 <- ifelse(q_vals[1L] < X1 & X1 <= q_vals[2L], 1L, 0L)
q3 <- ifelse(q_vals[2L] < X1 & X1 <= q_vals[3L], 1L, 0L)
q4 <- ifelse(q_vals[3L] < X1, 1L, 0L)

## quantile marginal effects
q_me_true <- c("G1" = mean(me_true[q1==1]), "G2" = mean(me_true[q2==1]),
               "G3" = mean(me_true[q3==1]), "G4" = mean(me_true[q4==1]))

q_me_semi <- MarginalEffects(semi, variable = "X1", delta, p.cutoffs)

## formatted print
{
    cat("         ", sprintf(" %5s  ", paste0("G", 1:4)), fill = TRUE)
    cat("True    :", sprintf(" %.4f ", q_me_true), fill = TRUE)
    cat("Semi    :", sprintf(" %.4f ", q_me_semi[,1L]), fill = TRUE)
    cat("Std.Err.:", sprintf("(%.4f)", q_me_semi[,2L]), fill = TRUE)
}
#>               G1       G2       G3       G4  
#> True    :  0.2232   0.3175   0.2196   0.0481 
#> Semi    :  0.2186   0.3130   0.2248   0.0552 
#> Std.Err.: (0.0100) (0.0088) (0.0096) (0.0061)

References

Eddelbuettel, D., & François, R. (2011). Rcpp: Seamless R and C++ integration. Journal of Statistical Software, 40(8), 1-18. https://dirk.eddelbuettel.com/code/rcpp/Rcpp-introduction.pdf

Klein, R. W., & Spady, R. H. (1993). An Efficient Semiparametric Estimator for Binary Response Models. Econometrica, 61(2), 387-421. https://doi.org/10.2307/2951556.

Klein, R., & Vella, F. (2009). A Semiparametric Model for Binary Response and Continuous Outcomes Under Index Heteroscedasticity. Journal of Applied Econometrics, 24(5), 735-762. https://doi.org/10.1002/jae.1064