Simulation WTPs with categorical parameters that follow different distribution or do not follow it at all

[3G FL]

Dear Professor Bierlaire,

 

I need your assistance. I am trying to simulate WTP. I ran a DCE where respondents got to decide which option they would choose among 3 products to purchase and no purchase option. Those products are characterized by categorical variables and price, which is a continuous variable. My aim was to see how adding some extra specifications (changing attribute values) to the products would affect customers' willingness to pay for them. I estimated parameters using the Mixed Logit Model with Monte Carlo Draws. I also analytically calculated WTP and obtained good results. The problem is now that I get WTPs to be around 0 with the simulation. I think the problem is with the fact that my model specification includes ASCs that are not following any specific distribution (e.g. ASC_Product1 = Beta(' ASC_Product1', logit_results.loc[' ASC_Product1   ','Value'], None, None, 0)), attributes are following normal distribution (e.g. att1_rnd = att1   + att1_s  * bioDraws('att1_rnd', 'NORMAL')), while the price coefficient is defined as log normal (e.g. price_rnd = -exp(price   + price_s  * bioDraws('price_rnd', 'NORMAL'))). 

To calculate WTP analytically, I transformed sigma and mean price from lognormal to normal distributions as described in the article (page 395), and then divided each beta with it to obtain good values.

I wanted to calculate the standard errors and CI of WTP. I simulated the results. I calculated them using the following principle: ASC_Product1_WTP  = Derive(V1,' ASC_Product1')/Derive(V1,'b_price') and use MonteCarlo draws. I read somewhere that this is a good approach as long as the dominator is a continuous random variable. However, my WTPs are around zero.

The next approach I used only for the ASC since they don't follow any specific distribution is ASC_Product1_WTP  = ASC_Product1/Derive(V1,'b_price') . The results are still meaningless.

Then I used the article's formula to convert the mean price into the normal distribution: ASC_Product1_WTP = ASC_Product1/-exp (Derive (V1, 'b_compens')+0.5*Derive (V1, 'b_compens_sigma')**2.I also tried it without the derive function. However, I couldn't obtain similar results that I get when I calculated WTPs analitically.

What should be the right way to proceed? Can CI and standard errors somehow be calculated analitically? And how this should be done within the simulation?

 

Thank you very much for your time and help in advance!

Best Regards

Kate