nl_bhhh_parallel: BHHH/OPG information matrix for the Nested Logit model

In choicer: Discrete Choice Models for Economic Applications

BHHH/OPG information matrix for the Nested Logit model

Description

Computes the weighted outer product of per-individual scores \sum_i w_i\, s_i s_i^\top for the Nested Logit model. The per-individual score s_i (over the beta, lambda and delta/ASC blocks) is the (positive) gradient of individual i's log-likelihood contribution and is weight-free; the supplied weights enter only as the leading multiplier. Passing weights = w yields the ordinary weighted BHHH/OPG information; passing weights = w^2 yields the sandwich meat B = \sum_i w_i^2 s_i s_i^\top for robust (WESML) inference. Singleton-nest lambdas are fixed to 1 and contribute no score (mirroring the gradient kernel).

Usage

nl_bhhh_parallel(
  theta,
  X,
  alt_idx,
  choice_idx,
  nest_idx,
  M,
  weights,
  use_asc = TRUE,
  include_outside_option = FALSE
)

Arguments

theta	(K + n_non_singleton_nests + n_delta) vector with model parameters. Order: ⁠[beta (K), lambda (n_non_singleton_nests), delta (n_delta)]⁠
X	sum(M) x K design matrix with covariates.
alt_idx	sum(M) x 1 vector with indices of alternatives; 1-based indexing.
choice_idx	N x 1 vector with indices of chosen alternatives; 0 for outside option, 1-based index relative to rows in X_i otherwise.
nest_idx	J x 1 vector with indices of nests for each alternative; 1-based indexing (1 to n_nests).
M	N x 1 vector with number of alternatives for each individual.
weights	N x 1 vector with weights for each observation.
use_asc	whether to use alternative-specific constants.
include_outside_option	whether to include outside option normalized to V=0, lambda=1.

Value

A symmetric positive-semidefinite information matrix \sum_i w_i\, s_i s_i^\top (same sign convention as the negated Hessian).

Examples

library(data.table)
set.seed(42)
N <- 50; J <- 4
dt <- data.table(id = rep(1:N, each = J), alt = rep(1:J, N))
dt[, `:=`(x1 = rnorm(.N), x2 = rnorm(.N))]
dt[, nest := ifelse(alt <= 2, "A", "B")]
dt[, choice := 0L]
dt[, choice := sample(c(1L, rep(0L, J - 1))), by = id]
d <- prepare_nl_data(dt, "id", "alt", "choice", c("x1", "x2"), "nest")
K_x <- ncol(d$X); K_l <- length(unique(d$nest_idx))
theta <- c(rep(0, K_x), rep(0.5, K_l), rep(0, J - 1))
B <- choicer:::nl_bhhh_parallel(theta, d$X, d$alt_idx, d$choice_idx,
  d$nest_idx, d$M, d$weights)
dim(B)

choicer documentation built on Sept. 5, 2026, 1:07 a.m.