# To obtain the distribution/variance of two random coefficients

I want to estimate the distribution/variance of the ratio $WTP=\frac{\gamma}{\alpha}$. The problem is, both $\gamma$ and $\alpha$ are random coefficients. In other words, both are random to begin with (please note that here their true values are random; this is different from the usual regression models where the true values are fixed constants but the estimators are random). For those interested, the model in mind is a random coefficients model (BLP).

Assume $\gamma \sim F(\mu_{\gamma},\sigma_{\gamma})$ and $\alpha \sim G(\mu_{\alpha},\sigma_{\alpha})$. Assume that $\alpha$ is bounded away from zero so that we do not worry about 0 in the denominator.

When we estimate the model, the estimation package produces $\hat{\mu}_{\gamma}$, $\hat{\sigma}_{\gamma}$, $\hat{\mu}_{\alpha}$, and $\hat{ \sigma}_{\alpha}$, plus their standard errors $\mbox{se}(\hat{\mu}_{\gamma})$, $\mbox{se}(\hat{\sigma}_{\gamma})$, $\mbox{se}(\hat{\mu}_{\alpha})$, and $\mbox{se}(\hat{\sigma}_{\alpha})$. For simplicity, assume all estimators follow normal distributions.

If $\gamma$ and $\alpha$ are not random coefficients, once I estimate the parameters I can simply use the Delta method or the Krinsky-Robb method to obtain the variance of WTP. When they are random coefficients, things get complicated.

To obtain the variance or distribution of WTP, Daly, Stephane Hess, and Train (2011) and Rischatsch (2009) seem to suggest taking random draws $\gamma \sim F(\hat{\mu}_{\gamma},\hat{\sigma}_{\gamma})$ and $\alpha \sim G(\hat{\mu}_{\alpha},\hat{\sigma}_{\alpha})$. But I think this is wrong, because this completely ignores the sampling error in the estimated parameters of the distributions. In other words, this procedure ignores $\mbox{se}(\hat{\mu}_{\gamma})$, $\mbox{se}(\hat{\sigma}_{\gamma})$, $\mbox{se}(\hat{\mu}_{\alpha})$, and $\mbox{se}(\hat{\sigma}_{\alpha})$ (by assuming they are all zeros).

To me, a "correct" simulation approach would require some sort of double randomization:

1. First randomly draw $\mu_{\gamma}^m$ from $N(\hat{\mu}_{\gamma},\mbox{se}(\hat{\mu}_{\gamma}))$ and ${\sigma}_{\gamma}^m$ from $N(\hat{\sigma}_{\gamma},\mbox{se}(\hat{\sigma}_{\gamma}))$.
2. Then randomly draw $\gamma_i$ from $F(\mu_{\gamma}^m,\sigma_{\gamma}^m)$.
3. Do the same for $\alpha_i$. Obtain the ratio.
4. Repeat this process to get a distribution of WTP.

The "typical" simulation approach to find the distribution of WTP seems to ignore step 1 and thus underestimate the variance of WTP by treating the estimated quantities as true ones.

Anybody can shed light on whether my approach is right or wrong? And are the approaches in the two references right or wrong?

Thanks in advance.

• The answers to this question on Cross Validated SE may help. – Adam Bailey Mar 22 '18 at 20:49
• Thank you Adam for pointing me to a related question. If in my question $\hat{\mu}_{\gamma}$, $\hat{\sigma}_{\gamma}$, $\hat{\mu}_{\alpha}$, and $\hat{ \sigma}_{\alpha}$ are not estimated, instead they are known quantities, then the question is almost identical to the one you linked. However, to estimate the standard error of this ratio (Willingness-to-Pay), one should account for (1) the numerator and the denominator are not fixed parameters but are random coefficients following two random distributions; (2) the parameters of the distributions are estimated which introduces sampling errors. – John Neat Mar 25 '18 at 14:35