3.1.1. DH Model

The idea behind the DH model is that a consumer has to overcome two hurdles before recording a positive expenditure. These two hurdles are: (1) the participation market (potential consumers), and (2) actual consumption (Angulo et al. 2001). A complete DH model consists of the participation and consumption decisions, with equations set as follows (Jones 1989; Aristei et al. 2008):

*Y<sup>i</sup>* = *D<sup>i</sup>* ∗ *Y<sup>i</sup>*

Observed consumption:

Participation decision:

$$\begin{aligned} D\_i^\* = Z\_i \mathfrak{a} + \mu\_i \; \mu\_i \sim \mathcal{N}(0, 1) \\ D\_i = 0 \; \text{else} \end{aligned} \tag{2}$$

∗∗ (1)

In Equation (2), a value of *D*∗ *i* larger than 0 and a value of *D<sup>i</sup>* of 1 indicates that consumers decide to participate in the consumption. A value of *D*∗ *i* equal to or less than 0 and a value of *D<sup>i</sup>* of 0 indicates that consumers will decide not to participate in the consumption. *Z<sup>i</sup>* is a variable influencing the participation decision.

Consumption decision:

$$\begin{aligned} Y\_i^\* &= X\_i \beta + v\_{i\prime} \ v\_i \sim N\left(0, \sigma^2\right) \\ Y\_i^{\*\*} &= Y\_i^\* \text{ if } Y\_i^\* > 0 \\ Y\_i^{\*\*} &= 0, \text{ else} \end{aligned} \tag{3}$$

In Equation (3), *Y* ∗ *i* is the latent consumption variable and *X<sup>i</sup>* is the variable influencing consumption expenditure. It can be clearly observed from Equations (2) and (3) that zero expenditure can appear in the participation decision stage when consumers choose not to participate or else choose to participate but do not have actual consumption expenditure.

Assuming that the error terms of the participation decision and consumption decision equations are mutually independent, the log-likelihood function of the independent DH model can be expressed as follows (Moffatt 2005; Aristei et al. 2008):

$$\ln \text{Ln} = \sum\_{0} \ln \left[ 1 - \Phi(Z\_i \mathfrak{a}) \Phi \left( \frac{X\_i \beta}{\sigma} \right) \right] + \sum\_{+} \ln \left[ \Phi(Z\_i \mathfrak{a}) \frac{1}{\sigma} \phi \left( \frac{Y\_i - X\_i \beta}{\sigma} \right) \right] \tag{4}$$

In Equation (4), Φ(.) is the cumulative distribution function, *φ*(.) is the standard normal density function, 0 means zero consumption, and + means that the consumption value is positive.

Assuming that the error terms of the participation and consumption decision equations are correlated and that simultaneous participation and consumption decisions are possible, the bivariate normal distribution of the error terms of the two equations of the DDH model is as follows:

$$
\begin{pmatrix} \mu\_i \\ \upsilon\_i \end{pmatrix} \sim \begin{bmatrix} \begin{pmatrix} 0 \\ 0 \end{pmatrix} / \begin{pmatrix} 1\rho\sigma \\ \rho\sigma \sigma^2 \end{pmatrix} \end{pmatrix} \tag{5}
$$

In Equation (5), *ρ* is the degree of correlation between the error terms of the participation and consumption decision equations. After adding the correlation coefficient, the log-likelihood function of the DDH model is as follows (Jones 1992):

$$\ln \ln L = \sum\_{0} \ln \left[ 1 - \Phi \left( Z\_i \alpha, \frac{X\_i \beta}{\sigma}, \rho \right) \right] + \sum\_{+} \ln \left[ \Phi \left( \frac{Z\_i \alpha + \frac{\rho}{\sigma} (Y\_i - X\_i \beta)}{\sqrt{1 - \rho^2}} \right) \frac{1}{\sigma} \phi \left( \frac{Y\_i - X\_i \beta}{\sigma} \right) \right] \tag{6}$$

The data distribution of limited dependent variables often reveals a significant positive skew, which is therefore unable to fulfill the hypothesis of a normal distribution of error terms. Therefore, if the maximum likelihood method is used to estimate the model, it is not possible to maintain parameter consistency. Through the inverse hyperbolic sine (IHS), dependent variables can generate consistent parameter estimates for model estimation (Newman et al. 2001). The IHS conversion function is as follows:

$$T(\theta Y\_i) = \log \left[ \theta Y\_i + \left( \theta^2 Y\_i^2 \right)^{1/2} \right] / \theta = \sinh^{-1} (\theta Y\_i) / \theta \tag{7}$$

After the dependent variables are converted through the IHS, the log-likelihood function of the DH model can be expressed as follows:

$$\begin{split} \ln \text{L} = \sum\_{0} \ln \left[ 1 - \Phi(Z\_i \alpha) \Phi\left(\frac{X\_i \beta}{\sigma}\right) \right] + \\ \sum\_{+} \ln \left[ \left( 1 + \theta^2 Y\_i^2 \right)^{-\frac{1}{2}} \Phi(Z\_i \alpha) \frac{1}{\sigma} \phi\left(\frac{[T(\theta\_i Y\_i) - X\_i \beta]}{\sigma}\right) \right] \end{split} \tag{8}$$

When using the DH model, different explanatory variables can be chosen for the participation and consumption decision equations (Jones and Yen 2000; Mart*ι*´nez-Espineira 2006). Early studies that applied the DH model were on cigarette and tobacco expenditures (Jones 1989, 1992; Garcia and Labeaga 1996; Aristei and Pieroni 2008) and alcoholic beverage expenditures (Angulo et al. 2001). Over the past few years, the model has been applied in a variety of fields, such as expenditure on cumulative loans (Moffatt 2005), meat products (Jones and Yen 2000; Newman et al. 2001), and nonmarket financial evaluation (Clinch and Murphy 2001; Mart*ι*´nez-Espineira 2006; Okoffo et al. 2016).
