Mixed logit model in WTP space - WTP estimates and Delta method

[nimslkg]

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:

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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

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?

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deltaMethod_settings <- list(operation="lognormal", parName1="mu_logcost", parName2="sigma_logcost")
apollo_deltaMethod(model, deltaMethod_settings)

Here is the output:

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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

(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:

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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

Any suggestion would be greatly appreciated.

Many thanks,
Manori

[stephanehess]

Hi

if you're happy with the use of Normal distributions for WTP (which would imply that some people like and some dislike the attribute) then:

1) the mu will give you the mean wtp, the sigma the standard devaition
2) the sign of the mu has no relation to the use of the negative lognormal, but is simply to do with the MRS. So for a desirable attribute, the mu will be negative as it's the relative impact on utility of an increase in that attribute vs an increase in cost
3) high sds are often the case for lognormals

Stephane