Mixed logit model in WTP space - WTP estimates and Delta method
Posted: 16 Jun 2024, 23:35
Dear Prof. Hess & Palma,
I estimated a Mixed logit model with normally distributed non-cost attributes and the cost with a negative log-normal distribution in WTP space. I first estimated an MNL model in preference space and used MNL parameters as starting values for the mixed logit model.
Here is the output:
My questions are:
(1) - Can I just consider the mu_ coefficients as WTPs for respective attributes or do I need to draw distributions using mu_ & sigma_ parameters to get the expected values?
(2) - The signs of mu_ coefficients are the opposite of what I expected. Is it because of the negative log-normal distribution of the cost? So, when interpreting results, can I just consider the opposite signs of the mu_ coefficients produced in the output above?
(3) - I used the code below to get the mean and standard deviation of the cost and I am getting very high values for standard deviation and standard error. Are these high values indicating errors in my model?
Here is the output:
(4) Can I use the following codes to get the mean and standard deviation of cost instead of Delta method or should both ways produce the same results? I am getting very different results:
Any suggestion would be greatly appreciated.
Many thanks,
Manori
I estimated a Mixed logit model with normally distributed non-cost attributes and the cost with a negative log-normal distribution in WTP space. I first estimated an MNL model in preference space and used MNL parameters as starting values for the mixed logit model.
Here is the output:
Code: Select all
Model run by nimsl using Apollo 0.3.3 on R 4.3.1 for Windows.
Please acknowledge the use of Apollo by citing Hess & Palma (2019)
DOI 10.1016/j.jocm.2019.100170
www.ApolloChoiceModelling.com
Model name : MMNL_WTPspace
Model description : MMNL model1
Model run at : 2024-06-16 01:40:41.03767
Estimation method : bgw
Model diagnosis : Relative function convergence
Optimisation diagnosis : Maximum found
hessian properties : Negative definite
maximum eigenvalue : -0.337944
reciprocal of condition number : 3.19278e-05
Number of individuals : 457
Number of rows in database : 4570
Number of modelled outcomes : 4570
Number of cores used : 10
Number of inter-individual draws : 1000 (sobol)
LL(start) : -2962.06
LL at equal shares, LL(0) : -3167.68
LL at observed shares, LL(C) : -3071.15
LL(final) : -2498.55
Rho-squared vs equal shares : 0.2112
Adj.Rho-squared vs equal shares : 0.204
Rho-squared vs observed shares : 0.1864
Adj.Rho-squared vs observed shares : 0.1793
AIC : 5043.11
BIC : 5190.94
Estimated parameters : 23
Time taken (hh:mm:ss) : 01:11:42.35
pre-estimation : 00:01:58.86
estimation : 00:22:30.47
initial estimation : 00:19:43.26
estimation after rescaling : 00:02:47.21
post-estimation : 00:47:13.01
Iterations : 117
initial estimation : 99
estimation after rescaling : 18
Unconstrained optimisation.
Estimates:
Estimate s.e. t.rat.(0) Rob.s.e. Rob.t.rat.(0)
mu_c -0.40536 0.03816 -10.6232 0.009486 -42.7347
mu_dm -14.09705 0.86117 -16.3697 0.280432 -50.2690
mu_dam -25.21634 0.46471 -54.2625 0.106997 -235.6739
mu_rs -3.01536 0.64694 -4.6610 0.139426 -21.6270
mu_fs -19.81948 0.44811 -44.2293 0.124119 -159.6818
mu_ws -7.30486 0.73289 -9.9673 0.177994 -41.0399
mu_ms -17.59294 0.79972 -21.9988 0.178754 -98.4200
mu_md -10.40502 0.95543 -10.8904 0.218507 -47.6188
mu_st -1.55766 0.97695 -1.5944 0.291004 -5.3527
mu_vs 3.61041 0.89405 4.0383 0.290001 12.4497
mu_logcost -4.10555 0.22190 -18.5022 0.226635 -18.1153
sigma_c 1.26807 0.02223 57.0547 0.006328 200.3752
sigma_dm 0.29919 0.64612 0.4631 0.138257 2.1640
sigma_dam -25.45291 0.35423 -71.8534 0.093830 -271.2656
sigma_rs -1.83377 0.45603 -4.0211 0.119856 -15.2998
sigma_fs 25.76983 0.37546 68.6354 0.091523 281.5682
sigma_ws -3.43261 0.59285 -5.7900 0.117395 -29.2399
sigma_ms 4.71622 0.62605 7.5333 0.099400 47.4469
sigma_md 0.26842 0.36892 0.7276 0.048431 5.5424
sigma_st -13.11152 0.69208 -18.9451 0.179352 -73.1051
sigma_vs 11.03608 0.47333 23.3156 0.127723 86.4063
sigma_logcost 3.69903 0.47294 7.8213 0.496675 7.4476
asc_1 0.04245 0.04569 0.9290 0.062197 0.6825
asc_2 0.00000 NA NA NA NA
(1) - Can I just consider the mu_ coefficients as WTPs for respective attributes or do I need to draw distributions using mu_ & sigma_ parameters to get the expected values?
(2) - The signs of mu_ coefficients are the opposite of what I expected. Is it because of the negative log-normal distribution of the cost? So, when interpreting results, can I just consider the opposite signs of the mu_ coefficients produced in the output above?
(3) - I used the code below to get the mean and standard deviation of the cost and I am getting very high values for standard deviation and standard error. Are these high values indicating errors in my model?
Code: Select all
deltaMethod_settings <- list(operation="lognormal", parName1="mu_logcost", parName2="sigma_logcost")
apollo_deltaMethod(model, deltaMethod_settings)
Code: Select all
Running Delta method computations
Value Robust s.e.
Mean for exp(N( mu_logcost , sigma_logcost ) 15.42 28.59
SD for exp(N( mu_logcost , sigma_logcost ) 14433.12 53167.44
Rob t-ratio (0)
Mean for exp(N( mu_logcost , sigma_logcost ) 0.5395
SD for exp(N( mu_logcost , sigma_logcost ) 0.2715
Code: Select all
b_cost = -exp(rnorm(N, mean=model$estimate["mu_logcost"], sd=model$estimate["sigma_logcost"]))
> mean(b_cost)
[1] -7.275067
> sd(b_cost)
[1] 93.44634
Many thanks,
Manori