# Multinomial logit

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.

    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.

    sim <- simulate_mnl_data(N = 2000, J = 4, seed = 1)
    sim
    #> <choicer_sim: mnl>
    #>   settings:
    #>     N = 2000
    #>     J = 4
    #>     K_x = 2
    #>     outside_option = TRUE
    #>     vary_choice_set = TRUE
    #>   rows in $data: 7967
    #>   true_params: beta, delta

## Fit

    fit <- run_mnlogit(
      data           = sim$data,
      id_col         = "id",
      alt_col        = "alt",
      choice_col     = "choice",
      covariate_cols = c("x1", "x2")
    )
    #> Optimization run time 0h:0m:0s
    summary(fit)
    #> Multinomial Logit (MNL) model
    #>
    #> Parameter    Estimate  Std.Error  z-value  Pr(>|z|)
    #> x1           0.790385   0.057681  13.7026  0.00e+00  ***
    #> x2          -0.568065   0.056300 -10.0900  0.00e+00  ***
    #> ASC_1        0.459936   0.067736   6.7902  1.12e-11  ***
    #> ASC_2       -0.570242   0.085511  -6.6686  2.58e-11  ***
    #> ASC_3        0.488752   0.067826   7.2060  5.76e-13  ***
    #> ASC_4       -0.467326   0.081197  -5.7555  8.64e-09  ***
    #> ---
    #> Signif. codes:  '***' 0.001 '**' 0.01 '*' 0.05
    #>
    #> Std. Errors: Analytical Hessian
    #> Log-likelihood: -2437.1
    #> AIC: 4886.2  | BIC: 4919.8
    #> McFadden R2: 0.104 (adj: 0.102) | Hit rate: 0.447
    #> N: 2000  | Parameters: 6
    #> Optimization time: 0 s
    #> Convergence: 1 ( NLOPT_SUCCESS: Generic success return value. )

## 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.

    recovery_table(fit, sim$true_params)
    #> <choicer_recovery> model=choicer_mnl level=0.95
    #>    parameter  group  true estimate     se    bias rel_bias_pct z_vs_true
    #>       <char> <char> <num>    <num>  <num>   <num>        <num>     <num>
    #> 1:        x1   beta   0.8   0.7904 0.0577 -0.0096       -1.202   -0.1667
    #> 2:        x2   beta  -0.6  -0.5681 0.0563  0.0319       -5.322    0.5672
    #> 3:     ASC_1    asc   0.5   0.4599 0.0677 -0.0401       -8.013   -0.5915
    #> 4:     ASC_2    asc  -0.5  -0.5702 0.0855 -0.0702       14.048   -0.8214
    #> 5:     ASC_3    asc   0.5   0.4888 0.0678 -0.0112       -2.249   -0.1658
    #> 6:     ASC_4    asc  -0.5  -0.4673 0.0812  0.0327       -6.535    0.4024
    #>    lower_ci upper_ci covers
    #>       <num>    <num> <lgcl>
    #> 1:   0.6773   0.9034   TRUE
    #> 2:  -0.6784  -0.4577   TRUE
    #> 3:   0.3272   0.5927   TRUE
    #> 4:  -0.7378  -0.4026   TRUE
    #> 5:   0.3558   0.6217   TRUE
    #> 6:  -0.6265  -0.3082   TRUE

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:

    predict(fit, type = "shares")        # aggregate fitted shares in these data
    #>        [,1]
    #> [1,] 0.2405
    #> [2,] 0.2610
    #> [3,] 0.1080
    #> [4,] 0.2645
    #> [5,] 0.1260
    elasticities(fit, elast_var = "x2")  # own- and cross-price elasticities
    #>   0        1         2         3         4
    #> 0 0 -0.01624 -0.010341 -0.013943 -0.010855
    #> 1 0 -0.01633 -0.006815 -0.008984 -0.006413
    #> 2 0 -0.01336 -0.002613 -0.010000 -0.008505
    #> 3 0 -0.01081 -0.006343 -0.020585 -0.006275
    #> 4 0 -0.01114 -0.007366 -0.010306 -0.010266
    diversion_ratios(fit)                # where demand goes
    #>        0      1      2      3      4
    #> 0 0.0000 0.3627 0.3252 0.3652 0.3403
    #> 1 0.3227 0.0000 0.2684 0.3245 0.2587
    #> 2 0.1620 0.1503 0.0000 0.1417 0.1364
    #> 3 0.3233 0.3229 0.2519 0.0000 0.2646
    #> 4 0.1920 0.1641 0.1545 0.1686 0.0000
    wtp(fit, price_var = "x2")           # willingness to pay, with delta-method SEs
    #> Willingness to pay (WTP), price variable: 'x2' (95% CI)
    #>    Estimate Std_Error z_value CI_lower CI_upper
    #> x1    1.391    0.1643   8.467    1.069    1.713
    gof(fit)                             # McFadden R2 and hit rate
    #> Goodness of fit
    #>   Log-likelihood:       -2437.1
    #>   Null log-likelihood: -2721.21 (equal shares)
    #>   McFadden R2:          0.1044 (adj: 0.1022)
    #>   Hit rate:             0.4465
    #>   N: 2000  | Parameters: 6

## 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 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, mixed logit, or multinomial probit when the empirical question requires grouped substitution, random tastes or correlated utility shocks. The getting-started vignette 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.