---
title: "Multinomial logit"
output: rmarkdown::html_vignette
vignette: >
  %\VignetteIndexEntry{Multinomial logit}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---

```{r, include = FALSE}
knitr::opts_chunk$set(collapse = TRUE, comment = "#>")
options(digits = 4)
```

The multinomial logit (MNL) is the workhorse of discrete choice. choicer fits it
by maximum likelihood with a C++ core, analytical gradients and an analytical
Hessian, so estimation and standard errors are fast even with many
alternative-specific constants.

This vignette does two things: it shows the full MNL workflow, and — because the
data are simulated from a known process — it checks that choicer **recovers the
true parameters**.

```{r setup}
library(choicer)
set_num_threads(2)
```

## Simulate a known data-generating process

`simulate_mnl_data()` draws choices from a logit model with i.i.d. Gumbel
errors. The returned object carries both the data and the true parameters.

```{r sim}
sim <- simulate_mnl_data(N = 2000, J = 4, seed = 1)
sim
```

## Fit

```{r fit}
fit <- run_mnlogit(
  data           = sim$data,
  id_col         = "id",
  alt_col        = "alt",
  choice_col     = "choice",
  covariate_cols = c("x1", "x2")
)
summary(fit)
```

## Did we recover the truth?

`recovery_table()` lines up each estimate against the value that generated the
data, with the bias, a z-score, and whether the 95% confidence interval covers
the truth.

```{r recovery}
recovery_table(fit, sim$true_params)
```

In this run the intervals cover their true values, which is indicative of
correct behavior. What matters formally, though, is coverage over repeated
simulations; a full Monte Carlo exercise is left outside this vignette for
brevity.

## Post-estimation

The full demand and welfare toolkit is available on the fitted object. Treating
`x2` as price:

```{r post}
predict(fit, type = "shares")        # aggregate fitted shares in these data
elasticities(fit, elast_var = "x2")  # own- and cross-price elasticities
diversion_ratios(fit)                # where demand goes
wtp(fit, price_var = "x2")           # willingness to pay, with delta-method SEs
gof(fit)                             # McFadden R2 and hit rate
```

## Substitution restrictions

The MNL is useful partly because its substitution structure is transparent.
Conditional on the included covariates, the odds ratio
$P_{ij}/P_{ik} = \exp(V_{ij} - V_{ik})$ depends only on alternatives $j$ and
$k$. This is the familiar individual-level IIA implication, but the more useful
empirical statement is about counterfactual diversion: for a given decision
maker, demand leaving one alternative is reallocated across the remaining
alternatives in proportion to their fitted probabilities.

Aggregate substitution can be less mechanical than that statement suggests.
The elasticities and diversion ratios reported above are averages over choice
situations. When covariates, demographics or choice sets vary across situations,
the aggregate diversion matrix need not equal the simple market-share formula
$DR(j\to k) = s_k / (1 - s_j)$. The
[math companion](https://fpcordeiro.github.io/choicer/articles/multinomial_logit_math.html) gives the derivation.

The remaining restriction is substantive: all heterogeneity that matters for
substitution must be observed and included in the utility index. If closeness is
driven by unobserved tastes, product groupings, networks, peer groups or other
latent features, the MNL will not recover that margin. It is then not enough to
fit shares well; the model may still give the wrong diversion matrix for the
counterfactual of interest.

That makes the MNL a disciplined baseline, not a straw man. Use it when the
included variables carry the relevant substitution margin, or when the target is
an object that does not require richer unobserved structure. Move to
[nested logit](nl.html), [mixed logit](mxl.html), or
[multinomial probit](mnp.html) when the empirical question requires grouped
substitution, random tastes or correlated utility shocks. The
[getting-started vignette](choicer.html#choosing-among-choice-models) compares
those choices directly.

## Empirical checklist

Before reporting an MNL result:

- state the estimand and the within-choice-set variation that identifies each
  coefficient; inspect signs, units, scale, and collinearity;
- compare inverse-Hessian and robust covariance estimates, cluster repeated
  decision makers or markets, and use WESML plus the robust sandwich when the
  sample was drawn by the chosen alternative;
- distinguish sample shares from population shares and document the source of
  any external population totals;
- inspect elasticities and diversion—not fit alone—and compare them with at least
  one economically plausible richer substitution model when the counterfactual
  depends on closeness among alternatives;
- audit WTP and welfare for the sign and magnitude of the price coefficient,
  units, functional-form restrictions, and support of the counterfactual; and
- treat MNL as the transparent benchmark against which added heterogeneity,
  nesting, or covariance must earn its empirical conclusions.

MNL does not recover unobserved product closeness, persistent random tastes, or
endogenous-price corrections by itself. Robust or clustered standard errors can
repair inference under their assumptions; they do not change that estimand.

## References

McFadden, D. (1974). Conditional logit analysis of qualitative choice
behavior. In P. Zarembka (Ed.), *Frontiers in Econometrics*. Academic Press.

Train, K. E. (2009). *Discrete Choice Methods with Simulation* (2nd ed.).
Cambridge University Press.