Nested logit groups alternatives into nests of close
substitutes. Within a nest, alternatives share an unobserved component,
so substitution is stronger inside a nest than across nests. Each
non-singleton nest has a positive dissimilarity parameter λ; singleton
nests have λ fixed at 1 because no within-nest variation can identify
it. The usual random-utility interpretation of nested logit focuses on
0 < λ <= 1: λ = 1 gives the MNL limit, and smaller λ
means tighter substitution inside the nest. choicer imposes λ > 0
rather than an upper bound. Estimates above one are mathematically
computable but imply negative within-nest correlation and should be
treated as evidence against the proposed nesting structure, not as the
usual “close substitutes” interpretation.
Think of nested logit as the parsimonious middle ground between the multinomial and mixed logits. It introduces genuine within-nest correlation in unobserved utility at the cost of just one extra parameter per nest, and it stays globally well-behaved and cheap to estimate (a closed-form GEV likelihood, no simulation). The price you pay is a strong prior: you must specify the nesting tree in advance, and the model only permits correlation within the nests you draw. When the right grouping is obvious from the application (travel modes, product categories), that is a defensible restriction; when it is not, the result can be sensitive to how you nest. A good nested-logit application therefore treats the tree as an economic assumption, not as a tuning parameter chosen after looking at fit statistics. See Choosing among choice models for how this tradeoff compares with the alternatives.
Simulate a nested process
simulate_nl_data() builds two nests of inside goods plus
an outside option, with known dissimilarity parameters.
sim <- simulate_nl_data(N = 4000, seed = 1)
sim
#> <choicer_sim: nl>
#> settings:
#> N = 4000
#> J_inside = 5
#> nest_structure = (1,2), (3,4,5)
#> rows in $data: 24000
#> true_params: beta, delta, lambdasFit
Supply the nest membership column; choicer estimates the coefficients, the alternative-specific constants and the nest dissimilarity parameters jointly, using an analytical gradient and Hessian.
fit <- run_nestlogit(
data = sim$data,
id_col = "id",
alt_col = "j",
choice_col = "choice",
covariate_cols = c("X", "W"),
nest_col = "nest",
use_asc = TRUE,
include_outside_option = TRUE,
outside_opt_label = 0L
)
#> Optimization run time 0h:0m:0.11s
summary(fit)
#> Nested Logit (NL) model
#>
#> Parameter Estimate Std.Error z-value Pr(>|z|)
#> X 1.502372 0.047212 31.8220 0.00e+00 ***
#> W -0.806760 0.027163 -29.7012 0.00e+00 ***
#> Lambda_1 0.794271 0.043175 18.3964 0.00e+00 ***
#> Lambda_2 0.185597 0.013846 13.4041 0.00e+00 ***
#> ASC_1 0.547211 0.123706 4.4235 9.71e-06 ***
#> ASC_2 0.316589 0.128397 2.4657 1.37e-02 *
#> ASC_3 -0.137415 0.133417 -1.0300 3.03e-01
#> ASC_4 -0.449435 0.140065 -3.2088 1.33e-03 **
#> ASC_5 0.426713 0.122610 3.4802 5.01e-04 ***
#> ---
#> Signif. codes: '***' 0.001 '**' 0.01 '*' 0.05
#>
#> Std. Errors: Analytical Hessian
#> Log-likelihood: -3063.62
#> AIC: 6145.24 | BIC: 6201.89
#> McFadden R2: 0.573 (adj: 0.571) | Hit rate: 0.694
#> N: 4000 | Parameters: 9
#> Optimization time: 0.11 s
#> Convergence: 3 ( NLOPT_FTOL_REACHED: Optimization stopped because ftol_rel or ftol_abs (above) was reached. )Parameter recovery
The lambda rows are the nest dissimilarity parameters —
the part that is unique to nested logit.
recovery_table(fit, sim$true_params)
#> <choicer_recovery> model=choicer_nl level=0.95
#> parameter group true estimate se bias rel_bias_pct z_vs_true
#> <char> <char> <num> <num> <num> <num> <num> <num>
#> 1: X beta 1.5 1.5024 0.0472 0.0024 0.1581 0.0502
#> 2: W beta -0.8 -0.8068 0.0272 -0.0068 0.8450 -0.2489
#> 3: Lambda_1 lambda 0.8 0.7943 0.0432 -0.0057 -0.7161 -0.1327
#> 4: Lambda_2 lambda 0.2 0.1856 0.0138 -0.0144 -7.2013 -1.0402
#> 5: ASC_1 asc 0.5 0.5472 0.1237 0.0472 9.4422 0.3816
#> 6: ASC_2 asc 0.3 0.3166 0.1284 0.0166 5.5296 0.1292
#> 7: ASC_3 asc -0.2 -0.1374 0.1334 0.0626 -31.2926 0.4691
#> 8: ASC_4 asc -0.5 -0.4494 0.1401 0.0506 -10.1130 0.3610
#> 9: ASC_5 asc 0.4 0.4267 0.1226 0.0267 6.6782 0.2179
#> lower_ci upper_ci covers
#> <num> <num> <lgcl>
#> 1: 1.4098 1.5949 TRUE
#> 2: -0.8600 -0.7535 TRUE
#> 3: 0.7096 0.8789 TRUE
#> 4: 0.1585 0.2127 TRUE
#> 5: 0.3048 0.7897 TRUE
#> 6: 0.0649 0.5682 TRUE
#> 7: -0.3989 0.1241 TRUE
#> 8: -0.7240 -0.1749 TRUE
#> 9: 0.1864 0.6670 TRUEWhat identifies the dissimilarity parameters?
It is worth being explicit about where each parameter block gets its
information. The coefficients on X and W are
identified, as in any logit, by how utilities respond to covariate
variation. The dissimilarity parameters are identified by
reallocation: when an observed utility shifter moves, does the
displaced demand stay inside the nest or spill across nests?
Alternative-specific covariates that vary within nests are therefore the
variation that disciplines λ. If the covariates move whole nests
together — or the specification is mostly alternative-specific constants
— λ is pinned down mainly by functional form, and a wide confidence
interval on a lambda row is the model saying the data
cannot see inside that nest.
The estimate’s position in (0, 1] is itself informative. A λ near 1
says the data see no extra within-nest correlation: the MNL was
adequate, and the nest costs a parameter without buying substitution
structure. A λ near 0 says the nest’s alternatives are nearly perfect
substitutes at the nest margin, which carries a sharp welfare
implication: the nested logsum discounts within-nest variety by λ, so
adding an alternative to a tight nest adds almost no expected consumer
surplus, while adding one to a loose nest adds a lot. Welfare
counterfactuals computed with consumer_surplus() inherit
exactly this structure — one more reason to treat the tree as an
economic assumption rather than a fit device.
Nest-consistent elasticities
This is the payoff of nesting: cross-elasticities are larger for
alternatives in the same nest than for alternatives in
different nests. choicer’s elasticities() respects the nest
structure automatically.
elasticities(fit, elast_var = "W")
#> 0 1 2 3 4 5
#> 0 0 -0.1118 -0.1097 -0.07679 -0.07381 -0.1138
#> 1 0 -0.1529 -0.1444 -0.07679 -0.07381 -0.1138
#> 2 0 -0.1423 -0.1271 -0.07679 -0.07381 -0.1138
#> 3 0 -0.1118 -0.1097 -0.79069 -0.65893 -0.7997
#> 4 0 -0.1118 -0.1097 -0.63239 -0.66527 -0.7997
#> 5 0 -0.1118 -0.1097 -0.63239 -0.65893 -0.8606
diversion_ratios(fit)
#> 0 1 2 3 4 5
#> 0 0.0000 0.0553 0.05027 0.03971 0.04039 0.04317
#> 1 0.2725 0.0000 0.36377 0.23303 0.22071 0.27854
#> 2 0.2300 0.3379 0.00000 0.20647 0.21384 0.24121
#> 3 0.1533 0.1826 0.17416 0.00000 0.24129 0.24856
#> 4 0.1319 0.1463 0.15263 0.20418 0.00000 0.18852
#> 5 0.2123 0.2780 0.25917 0.31661 0.28378 0.00000Those substitution patterns are credible only to the extent that the nesting tree is credible. If plausible alternative trees imply materially different diversion or welfare conclusions, that sensitivity is part of the empirical result rather than a nuisance to hide.
Share inversion with BLP
blp() runs the Berry-Levinsohn-Pakes contraction to
recover the mean utilities that reproduce a set of target shares —
useful for calibration and demand estimation from aggregate data. For
strongly-nested models a damping factor below 1 stabilizes the
iteration.
target_shares <- predict(fit, type = "shares")
head(blp(fit, target_shares, damping = 0.5))
#> [,1]
#> [1,] 0.5472
#> [2,] 0.3166
#> [3,] -0.1374
#> [4,] -0.4494
#> [5,] 0.4267As always, predict(), wtp() and
consumer_surplus() are available on the fitted object with
the same syntax used throughout choicer.
Empirical checklist
A reportable nested-logit specification should show the proposed tree
and its economic rationale; report every estimated lambda and
uncertainty interval; identify singleton nests, whose lambda is fixed at
1 by choicer because there is no within-nest variation to learn from;
and flag estimates outside the conventional random-utility range
0 < lambda <= 1. Refit plausible alternative trees
and compare fit, diversion, elasticities, WTP, and welfare—not just the
coefficient table.
When using blp(), verify that shares reconstructed from
the returned mean utilities match the target within the stated
tolerance, and report sensitivity to damping. (The current return value
is the utility vector, not a convergence diagnostics object.) Damping is
a numerical aid, not evidence for the tree or the economic validity of
lambda. NL does not search over trees or estimate cross-nested
substitution, so uncertainty about the grouping remains specification
uncertainty outside the reported standard errors.
The full derivations — the GEV likelihood, analytical gradient and Hessian, nest-consistent elasticities and diversion, and how choicer’s utility-maximization-consistent parameterization relates to the non-normalized nested logit found in some other software — are in the math companion.
References
Heiss, F. (2002). Structural choice analysis with nested logit models. The Stata Journal, 2(3), 227-252.
McFadden, D. (1978). Modelling the choice of residential location. In A. Karlqvist et al. (Eds.), Spatial Interaction Theory and Planning Models. North-Holland.
Train, K. E. (2009). Discrete Choice Methods with Simulation (2nd ed.). Cambridge University Press, Chapter 4.