I am trying to understand the probability formulation in this paper. This (and all other papers on this subject) refer to having applied the chage-of-variable technique to arrive at the probability formulation. I have been trying to get to the "original formulation" but cannot get very far. It reads as if this is should be simple, but I don't see it. Could someone explain how to get to the formulation before the change-of-variable technique is applied? Am I missing something very obvious? Thanks!
Here is the relevant excerpt from the paper (slightly changed for conciseness):
The likelihood of the model is derived by the distribution induced on choices by the distribution of $\boldsymbol{\epsilon}$ on the quality weights. Given the assumed joint density on $\boldsymbol{\epsilon}$, let $f_\eta\left(\eta_1, \ldots, \eta_J\right)$ denote the implied pdf of the error differences $\boldsymbol{\eta}$. We now add the index $i$ for consumers and $t$ for time. Collect all parameters that are consumer-specific in a vector $\theta_i \equiv\left(\beta_i, \boldsymbol{\psi}_i, \boldsymbol{\gamma}_i, \boldsymbol{\alpha}_i\right)$. The likelihood of a bundle $\left(x_{1, i t}^*, \ldots, x_{K, i t}^*, 0, \ldots, 0, z_{i t}^*\right)$ in which $K$ out of the $J$ goods are bought, and goods $(K+1, \ldots, J)$ are not bought, is $$ \begin{aligned} & \mathbb{L}\left(x_{1, i t}^*, \ldots, x_{K, i t}^*, 0, \ldots, 0, z_{i t}^* \mid \mathbf{w}, \mathbf{p}_{\mathbf{t}} ; \theta_i\right) \\ & =\int_{-\infty}^{\mathcal{V}_{z, i t}-\mathcal{V}_{K+1, i t}} \cdots \int_{-\infty}^{\mathcal{V}_{z, i t} \mathcal{V}_{J, i t}} f_\eta\left(\mathcal{V}_{z, i t}-\mathcal{V}_{1, i t}, \ldots, \mathcal{V}_{z, i t}\right. \\ & \left.\quad-\mathcal{V}_{K, i t}, \eta_{K+1, i t}, \ldots, \eta_{J, i t}\right)\left\|\mathscr{F}_{i t}\right\| d \eta_{K+1, i t} \cdots d \eta_{J, i t}, \end{aligned} $$ where implicitly $$ \left(\mathcal{V}_{i t}, \mathcal{V}_{z, i t}\right) \equiv\left(\mathcal{V}\left(\mathbf{w}, \mathbf{p}_{\mathbf{t}} ; \theta_i\right), \mathcal{V}_z\left(\mathbf{w}, \mathbf{p}_{\mathbf{t}} ; \theta_i\right)\right) . $$ $\mathcal{F}_{i t}$ is the $K \times K$ Jacobian matrix with cell $(l, m)$ given by $$ \mathcal{F}_{l m, i t}=\frac{\partial\left(\mathcal{V}_{z, i t}-\mathcal{V}_{l, i t}\right)}{\partial x_{m, i t}^*}, \quad l, m=(1, \ldots, K) . $$
The likelihood has two parts and can be understood as follows. First, for the chosen goods $(1, \ldots, K)$, the following defines the inverse mapping from the unobservables to demand:
$$\begin{aligned} \eta_j = \mathcal{V}_{z} - \mathcal{V}_{j}, \quad j = (1, ..., K), x^*_j > 0, \\ \eta_j \leq \mathcal{V}_{z} - \mathcal{V}_{j}, \quad j = (K+1, ..., J), x^*_j = 0, \\ \end{aligned} $$
where $\eta_j = \epsilon_j - \epsilon_z$ and
$$\begin{aligned} \mathcal{V}_{j} = \omega_j \beta_i + (\alpha_j - 1) ln\left(\frac{x^*_j}{\gamma^*_j + 1}\right) - ln(p_j). \end{aligned} $$
Thus, the first part of the likelihood involves the density of $\left(x_1^*, i t, \ldots, x_{K, \text { it }}^*\right)$ given by change-of-variable calculus. This generates the $K \times K$ Jacobian $\mathcal{g}$. The second part involves the probability of not purchasing goods $(K+1, \ldots, J)$.