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.
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.
fit <- run_mnlogit(
data = sim$data,
id_col = "id",
alt_col = "alt",
choice_col = "choice",
covariate_cols = c("x1", "x2")
)
#> Optimization run time 0h:0m:0.01s
summary(fit)
#> Multinomial Logit (MNL) model
#>
#> Parameter Estimate Std.Error z-value Pr(>|z|)
#> x1 0.790385 0.057681 13.7026 0.00e+00 ***
#> x2 -0.568065 0.056300 -10.0900 0.00e+00 ***
#> ASC_1 0.459936 0.067736 6.7902 1.12e-11 ***
#> ASC_2 -0.570242 0.085511 -6.6686 2.58e-11 ***
#> ASC_3 0.488752 0.067826 7.2060 5.76e-13 ***
#> ASC_4 -0.467326 0.081197 -5.7555 8.64e-09 ***
#> ---
#> Signif. codes: '***' 0.001 '**' 0.01 '*' 0.05
#>
#> Std. Errors: Analytical Hessian
#> Log-likelihood: -2437.1
#> AIC: 4886.2 | BIC: 4919.8
#> McFadden R2: 0.104 (adj: 0.102) | Hit rate: 0.447
#> N: 2000 | Parameters: 6
#> Optimization time: 0.01 s
#> Convergence: 1 ( NLOPT_SUCCESS: Generic success return value. )recovery_table() lines up each estimate against the
value that generated the data, with the bias, a z-score, and whether the
95% confidence interval covers the truth.
recovery_table(fit, sim$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 TRUEIn 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.
The full demand and welfare toolkit is available on the fitted
object. Treating x2 as price:
predict(fit, type = "shares") # predicted market shares
#> [,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: 6The 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.