# Estimating Multinomial Logit Models

This vignette demonstrates an example of how to use the logitr() function to estimate multinomial logit (MNL) models with preference space and WTP space utility parameterizations.

# The data

This example uses the yogurt data set from Jain et al. (1994). The data set contains 2,412 choice observations from a series of yogurt purchases by a panel of 100 households in Springfield, Missouri, over a roughly two-year period. The data were collected by optical scanners and contain information about the price, brand, and a “feature” variable, which identifies whether a newspaper advertisement was shown to the customer. There are four brands of yogurt: Yoplait, Dannon, Weight Watchers, and Hiland, with market shares of 34%, 40%, 23% and 3%, respectively.

In the utility models described below, the data variables are represented as follows:

Symbol Variable
$$p$$ The price in US dollars.
$$x_{j}^{\mathrm{Feat}}$$ Dummy variable for whether the newspaper advertisement was shown to the customer.
$$x_{j}^{\mathrm{Hiland}}$$ Dummy variable for the “Highland” brand.
$$x_{j}^{\mathrm{Yoplait}}$$ Dummy variable for the “Yoplait” brand.
$$x_{j}^{\mathrm{Dannon}}$$ Dummy variable for the “Dannon” brand.

# Preference space model

This example will estimate the following homogeneous multinomial logit model in the preference space:

$\begin{equation} u_{j} = \alpha p_{j} + \beta_1 x_{j}^{\mathrm{Feat}} + \beta_2 x_{j}^{\mathrm{Hiland}} + \beta_3 x_{j}^{\mathrm{Yoplait}} + \beta_4 x_{j}^{\mathrm{Dannon}} + \varepsilon_{j} \label{eq:mnlPrefExample} \end{equation}$

where the parameters $$\alpha$$, $$\beta_1$$, $$\beta_2$$, $$\beta_3$$, and $$\beta_4$$ have units of utility.

To estimate the model, first load the logitr package:

library(logitr)

Estimate the model using the logitr() function:

mnl_pref <- logitr(
data       = yogurt,
choiceName = 'choice',
obsIDName  = 'obsID',
parNames   = c('price', 'feat', 'brand')
)
#> Running Model...
#> Done!

Print a summary of the results:

summary(mnl_pref)
#> =================================================
#> MODEL SUMMARY:
#>
#> Model Space:    Preference
#> Model Run:          1 of 1
#> Iterations:             21
#> Elapsed Time:  0h:0m:0.13s
#> Exit Status:             3
#> Weights Used?:       FALSE
#> robust?              FALSE
#>
#> Model Coefficients:
#>               Estimate StdError    tStat pVal signif
#> price        -0.366543 0.024365 -15.0439    0    ***
#> feat          0.491433 0.120061   4.0932    0    ***
#> brandhiland  -3.715428 0.145416 -25.5504    0    ***
#> brandweight  -0.641128 0.054498 -11.7643    0    ***
#> brandyoplait  0.734496 0.080642   9.1082    0    ***
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#>
#> Model Fit Values:
#>
#> Log-Likelihood:         -2656.8878799
#> Null Log-Likelihood:    -3343.7419990
#> AIC:                     5323.7758000
#> BIC:                     5352.7168000
#> McFadden R2:                0.2054148
#> Adj McFadden R2:            0.2039195
#> Number of Observations:  2412.0000000

View the estimated model coefficients:

coef(mnl_pref)
#>        price         feat  brandhiland  brandweight brandyoplait
#>   -0.3665429    0.4914329   -3.7154279   -0.6411280    0.7344962

Compute the WTP implied from the preference space model:

wtp_mnl_pref <- wtp(mnl_pref, priceName =  "price")
wtp_mnl_pref
#>                Estimate StdError    tStat  pVal signif
#> lambda         0.366543 0.024471  14.9786 0e+00    ***
#> feat           1.340724 0.360552   3.7185 2e-04    ***
#> brandhiland  -10.136406 0.585214 -17.3209 0e+00    ***
#> brandweight   -1.749121 0.181712  -9.6258 0e+00    ***
#> brandyoplait   2.003848 0.143493  13.9647 0e+00    ***

# WTP space model

This example will estimate the following homogeneous multinomial logit model in the WTP space:

$\begin{equation} u_{j} = \lambda ( \omega_1 x_{j}^{\mathrm{Feat}} + \omega_2 x_{j}^{\mathrm{Hiland}} + \omega_3 x_{j}^{\mathrm{Yoplait}} + \omega_4 x_{j}^{\mathrm{Dannon}} - p_{j}) + \varepsilon_{j} \label{eq:mnlWtpExample} \end{equation}$

where the parameters $$\omega_1$$, $$\omega_2$$, $$\omega_3$$, and $$\omega_4$$ have units of dollars and $$\lambda$$ is the scale parameter.

Estimate the model using the logitr() function:

mnl_wtp <- logitr(
data       = yogurt,
choiceName = 'choice',
obsIDName  = 'obsID',
parNames   = c('feat', 'brand'),
priceName  = 'price',
modelSpace = 'wtp',
options = list(
# Since WTP space models are non-convex, run a multistart:
numMultiStarts = 10,
# If you want to view the results from each multistart run,
# set keepAllRuns=TRUE:
keepAllRuns = TRUE,
# Use the computed WTP from the preference space model as the starting
# values for the first run:
startVals = wtp_mnl_pref\$Estimate)
)
#> Running Multistart 1 of 10...
#> NOTE: Using user-provided starting values for this run
#> Running Multistart 2 of 10...
#> Running Multistart 3 of 10...
#> Running Multistart 4 of 10...
#> Running Multistart 5 of 10...
#> Running Multistart 6 of 10...
#> Running Multistart 7 of 10...
#> Running Multistart 8 of 10...
#> Running Multistart 9 of 10...
#> Running Multistart 10 of 10...
#> Done!

Print a summary of the results:

summary(mnl_wtp)
#> =================================================
#> SUMMARY OF ALL MULTISTART RUNS:
#>
#>    run    logLik iterations status
#> 1    1 -2656.888         48      3
#> 2    2 -2656.888         45      3
#> 3    3 -2803.817         71      3
#> 4    4 -2656.888         36      3
#> 5    5 -2656.888         37      3
#> 6    6 -2803.795         89      3
#> 7    7 -2656.888         39      3
#> 8    8 -2804.379         89      3
#> 9    9 -2803.827         94      3
#> 10  10 -2656.888         43      3
#> ---
#> Use statusCodes() to view the meaning of the status codes
#>
#> Below is the summary of run 10 of 10 multistart runs
#> (the run with the largest log-likelihood value)
#> =================================================
#> MODEL SUMMARY:
#>
#> Model Space:   Willingness-to-Pay
#> Model Run:               10 of 10
#> Iterations:                    43
#> Elapsed Time:         0h:0m:0.23s
#> Exit Status:                    3
#> Weights Used?:              FALSE
#> robust?                     FALSE
#>
#> Model Coefficients:
#>                Estimate StdError    tStat  pVal signif
#> lambda         0.366585 0.024366  15.0449 0e+00    ***
#> feat           1.340571 0.355864   3.7671 2e-04    ***
#> brandhiland  -10.135735 0.576086 -17.5941 0e+00    ***
#> brandweight   -1.749063 0.179896  -9.7227 0e+00    ***
#> brandyoplait   2.003818 0.142377  14.0740 0e+00    ***
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#>
#> Model Fit Values:
#>
#> Log-Likelihood:         -2656.8878779
#> Null Log-Likelihood:    -3343.7419990
#> AIC:                     5323.7758000
#> BIC:                     5352.7168000
#> McFadden R2:                0.2054148
#> Adj McFadden R2:            0.2039195
#> Number of Observations:  2412.0000000

View the estimated model coefficients:

coef(mnl_wtp)
#>       lambda         feat  brandhiland  brandweight brandyoplait
#>    0.3665854    1.3405708  -10.1357346   -1.7490629    2.0038178

# Compare WTP from both models

Since WTP space models are non-convex, you cannot be certain that the model reached a global solution, even when using a multi-start. However, homogeneous models in the preference space are convex, so you are guaranteed to find the global solution in that space. Therefore, it can be useful to compute the WTP from the preference space model and compare it against the WTP from the WTP space model. If the WTP values and log-likelihood values from the two model spaces are equal, then the WTP space model is likely at a global solution.

To compare the WTP and log-likelihood values between the preference space and WTP space models, use the wtpCompare() function:

wtp_mnl_comparison <- wtpCompare(mnl_pref, mnl_wtp, priceName = 'price')
wtp_mnl_comparison
#>                      pref           wtp  difference
#> lambda           0.366543     0.3665854  0.00004238
#> feat             1.340724     1.3405708 -0.00015315
#> brandhiland    -10.136406   -10.1357346  0.00067139
#> brandweight     -1.749121    -1.7490629  0.00005807
#> brandyoplait     2.003848     2.0038178 -0.00003018
#> logLik       -2656.887880 -2656.8878779  0.00000194