Kuhn-Tucker and Multiple Discrete-Continuous Extreme Value (MDCEV) Model Estimation and Simulation in R: The rmdcev Package
The rmdcev R package estimate and simulates Kuhn-Tucker demand models with individual heterogeneity. The models supported by rmdcev are the multiple-discrete continuous extreme value (MDCEV) model and Kuhn-Tucker specification common in the environmental economics literature on recreation demand. Latent class and random parameters specifications can be implemented and the models are fit using maximum likelihood estimation or Bayesian estimation. All models are implemented in Stan, which is a C++ package for performing full Bayesian inference (see https://mc-stan.org/). The rmdcev package also implements demand forecasting and welfare calculation for policy simulation.
Current Status
Development is in progress. Currently users can estimate the following models:
- Bhat (2008) MDCEV model specifications
- Kuhn-Tucker model specification in environmental economics (von Haefen and Phaneuf, 2005)
Models can be estimated using
- Fixed parameter models (maximum likelihood or Bayesian estimation)
- Latent class models (maximum likelihood estimation)
- Random parameters models (Bayesian estimation)
Installation
I recommend you first install rstan by following these steps:
https://github.com/stan-dev/rstan/wiki/RStan-Getting-Started
Once rstan is installed, you can install rmdcev from CRAN using
install.packages("rmdcev")Or install the latest version of rmdcev from GitHub using devtools
if (!require(devtools)) {
install.packages("devtools")
library(devtools)
}
install_github("plloydsmith/rmdcev", dependencies = TRUE, INSTALL_opts="--no-multiarch")If you have any issues with installation or use of the package, please let me know by filing an issue.
References
Background on the models, estimation, and simulation as well as a walk through of the package is provided in
Lloyd-Smith, P (2021). “Kuhn-Tucker and Multiple Discrete-Continuous Extreme Value Model Estimation and Simulation in R: The rmdcev Package” The R Journal, 12(2): 251-265.
For more details on the model specification and estimation:
Bhat, C.R. (2008) “The Multiple Discrete-Continuous Extreme Value (MDCEV) Model: Role of Utility Function Parameters, Identification Considerations, and Model Extensions” Transportation Research Part B, 42(3): 274-303.
von Haefen, R. and Phaneuf D. (2005) “Kuhn-Tucker Demand System Approaches to Non-Market Valuation” In: Scarpa R., Alberini A. (eds) Applications of Simulation Methods in Environmental and Resource Economics. The Economics of Non-Market Goods and Resources, vol 6. Springer, Dordrecht.
For more details on the demand and welfare simulation:
Pinjari, A.R. and Bhat , C.R. (2011) “Computationally Efficient Forecasting Procedures for Kuhn-Tucker Consumer Demand Model Systems: Application to Residential Energy Consumption Analysis.” Technical paper, Department of Civil & Environmental Engineering, University of South Florida.
Lloyd-Smith, P (2018). “A New Approach to Calculating Welfare Measures in Kuhn-Tucker Demand Models.” Journal of Choice Modeling, 26: 19-27
Estimation
As an example, we can simulate some data using Bhat (2008)‘s ’Gamma’ specification. In this example, we are simulating data for 2,000 individuals and 10 non-numeraire alternatives. We will randomly generate the parameter values to simulate the data and then check these values to our estimation results.
library(pacman)
p_load(tidyverse, rmdcev)
set.seed(12345)
model <- "gamma"
nobs <- 2000
nalts <- 10
sim.data <- GenerateMDCEVData(model = model, nobs = nobs, nalts = nalts)
#> Sorting data by id.var then alt...
#> Checking data...
#> Data is goodEstimate model using MLE (note that we set “psi_ascs = 0” to omit any alternative-specific constants)
mdcev_est <- mdcev(~ b1 + b2 + b3 + b4 + b5 + b6,
data = sim.data$data,
psi_ascs = 0,
model = model,
algorithm = "MLE")
#> Using MLE to estimate KT model
#> Chain 1: Initial log joint probability = -100801
#> Chain 1: Iter log prob ||dx|| ||grad|| alpha alpha0 # evals Notes
#> Chain 1: Error evaluating model log probability: Non-finite gradient.
#> Error evaluating model log probability: Non-finite gradient.
#>
#> Chain 1: 19 -35837.5 0.834357 710.653 1 1 38
#> Chain 1: Iter log prob ||dx|| ||grad|| alpha alpha0 # evals Notes
#> Chain 1: 39 -35733.9 0.0404306 61.8681 0.7313 0.7313 58
#> Chain 1: Iter log prob ||dx|| ||grad|| alpha alpha0 # evals Notes
#> Chain 1: 59 -35725.9 0.0298134 12.9001 0.994 0.994 80
#> Chain 1: Iter log prob ||dx|| ||grad|| alpha alpha0 # evals Notes
#> Chain 1: 79 -35723.7 0.0276683 16.9662 1 1 102
#> Chain 1: Iter log prob ||dx|| ||grad|| alpha alpha0 # evals Notes
#> Chain 1: 99 -35723.4 0.00404058 2.37686 1 1 123
#> Chain 1: Iter log prob ||dx|| ||grad|| alpha alpha0 # evals Notes
#> Chain 1: 108 -35723.4 0.000840364 0.431756 1 1 133
#> Chain 1: Optimization terminated normally:
#> Chain 1: Convergence detected: relative gradient magnitude is below toleranceSummarize results
summary(mdcev_est)
#> Model run using rmdcev for R, version 1.2.5
#> Estimation method : MLE
#> Model type : gamma specification
#> Number of classes : 1
#> Number of individuals : 2000
#> Number of non-numeraire alts : 10
#> Estimated parameters : 18
#> LL : -35723.35
#> AIC : 71482.7
#> BIC : 71583.52
#> Standard errors calculated using : Delta method
#> Exit of MLE : successful convergence
#> Time taken (hh:mm:ss) : 00:00:3.16
#>
#> Average consumption of non-numeraire alternatives:
#> 1 2 3 4 5 6 7 8 9 10
#> 59.90 10.43 0.84 71.64 4.89 1.46 10.40 13.49 21.86 0.37
#>
#> Parameter estimates --------------------------------
#> Estimate Std.err z.stat
#> psi_b1 -4.897 0.115 -42.58
#> psi_b2 0.556 0.091 6.11
#> psi_b3 2.010 0.062 32.42
#> psi_b4 -1.501 0.057 -26.33
#> psi_b5 2.079 0.046 45.20
#> psi_b6 -1.089 0.055 -19.79
#> gamma_1 6.971 0.411 16.96
#> gamma_2 8.437 0.740 11.40
#> gamma_3 7.373 1.526 4.83
#> gamma_4 8.724 0.534 16.34
#> gamma_5 4.876 0.425 11.47
#> gamma_6 2.142 0.234 9.15
#> gamma_7 3.445 0.232 14.85
#> gamma_8 5.589 0.385 14.52
#> gamma_9 7.669 0.509 15.07
#> gamma_10 7.822 2.758 2.84
#> alpha_num 0.503 0.008 62.86
#> scale 1.000 0.015 66.70
#> Note: All non-numeraire alpha's fixed to 0.Compare estimates to true values
coefs <- as_tibble(sim.data$parms_true) %>%
mutate(true = as.numeric(true)) %>%
cbind(summary(mdcev_est)[["CoefTable"]]) %>%
mutate(cl_lo = Estimate - 1.96 * Std.err,
cl_hi = Estimate + 1.96 * Std.err)
head(coefs, 200)
#> parms true Estimate Std.err z.stat cl_lo cl_hi
#> psi_b1 psi_b1 -5.000000 -4.897 0.115 -42.58 -5.12240 -4.67160
#> psi_b2 psi_b2 0.500000 0.556 0.091 6.11 0.37764 0.73436
#> psi_b3 psi_b3 2.000000 2.010 0.062 32.42 1.88848 2.13152
#> psi_b4 psi_b4 -1.500000 -1.501 0.057 -26.33 -1.61272 -1.38928
#> psi_b5 psi_b5 2.000000 2.079 0.046 45.20 1.98884 2.16916
#> psi_b6 psi_b6 -1.000000 -1.089 0.055 -19.79 -1.19680 -0.98120
#> gamma_1 gamma1 7.488135 6.971 0.411 16.96 6.16544 7.77656
#> gamma_2 gamma2 8.881959 8.437 0.740 11.40 6.98660 9.88740
#> gamma_3 gamma3 7.848841 7.373 1.526 4.83 4.38204 10.36396
#> gamma_4 gamma4 8.975121 8.724 0.534 16.34 7.67736 9.77064
#> gamma_5 gamma5 5.108329 4.876 0.425 11.47 4.04300 5.70900
#> gamma_6 gamma6 2.497346 2.142 0.234 9.15 1.68336 2.60064
#> gamma_7 gamma7 3.925858 3.445 0.232 14.85 2.99028 3.89972
#> gamma_8 gamma8 5.583019 5.589 0.385 14.52 4.83440 6.34360
#> gamma_9 gamma9 7.549347 7.669 0.509 15.07 6.67136 8.66664
#> gamma_10 gamma10 9.907632 7.822 2.758 2.84 2.41632 13.22768
#> alpha_num alpha1 0.500000 0.503 0.008 62.86 0.48732 0.51868
#> scale scale 1.000000 1.000 0.015 66.70 0.97060 1.02940Compare outputs using a figure
coefs %>%
ggplot(aes(y = Estimate, x = true)) +
geom_point(size=2) +
geom_text(label=coefs$parms,position=position_jitter(width=.5,height=1)) +
geom_abline(slope = 1) +
geom_errorbar(aes(ymin=cl_lo,ymax=cl_hi,width=0.2))Welfare simulation
Create policy simulations (these are ‘no change’ policies with no effects)
npols <- 2 # Choose number of policies
policies <- CreateBlankPolicies(npols, mdcev_est)
df_sim <- PrepareSimulationData(mdcev_est, policies, nsims = 1) Simulate welfare changes
wtp <- mdcev.sim(df_sim$df_indiv,
df_common = df_sim$df_common,
sim_options = df_sim$sim_options,
cond_err = 1,
nerrs = 15,
sim_type = "welfare")
#> Using general approach in simulation...
#>
#> 6.00e+04simulations finished in0.42minutes.(2398per second)
summary(wtp)
#> # A tibble: 2 x 5
#> policy mean std.dev `ci_lo2.5%` `ci_hi97.5%`
#> <chr> <dbl> <dbl> <dbl> <dbl>
#> 1 policy1 -3.63e-11 NA -3.63e-11 -3.63e-11
#> 2 policy2 -3.63e-11 NA -3.63e-11 -3.63e-11Thanks
This package was not developed in isolation and I gratefully acknowledge Joshua Abbott, Allen Klaiber, Lusi Xie, the apollo team, and the Stan team, whose codes or suggestions were helpful in putting this package together.