Truncated normal hurdle model

In econometrics, the truncated normal hurdle model is a variant of the Tobit model and was first proposed by Cragg in 1971.[1]

In a standard Tobit model, represented as ${\displaystyle y=(x\beta +u)1[x\beta +u>0]}$, where ${\displaystyle u|x\sim N(0,\sigma ^{2})}$This model construction implicitly imposes two first order assumptions:[2]

1. Since: ${\displaystyle \partial P[y>0]/\partial x_{j}=\varphi (x\beta /\sigma )\beta _{j}/\sigma }$ and ${\displaystyle \partial \operatorname {E} [y\mid x,y>0]/\partial x_{j}=\beta _{j}\{1-\theta (x\beta /\sigma \}}$, the partial effect of ${\displaystyle x_{j}}$ on the probability ${\displaystyle P[y>0]}$ and the conditional expectation: ${\displaystyle \operatorname {E} [y\mid x,y>0]}$ has the same sign:[3]
2. The relative effects of ${\displaystyle x_{h}}$ and ${\displaystyle x_{j}}$ on ${\displaystyle P[y>0]}$ and ${\displaystyle \operatorname {E} [y\mid x,y>0]}$ are identical, i.e.:

${\displaystyle {\frac {\partial P[y>0]/\partial x_{h}}{\partial P[y>0]/\partial x_{j}}}={\frac {\partial \operatorname {E} [y\mid x,y>0]/\partial x_{h}}{\partial \operatorname {E} [y\mid x,y>0]/\partial x_{j}}}={\frac {\beta _{h}}{\beta _{j}}}|}$

However, these two implicit assumptions are too strong and inconsistent with many contexts in economics. For instance, when we need to decide whether to invest and build a factory, the construction cost might be more influential than the product price; but once we have already built the factory, the product price is definitely more influential to the revenue. Hence, the implicit assumption (2) doesn't match this context.[4] The essence of this issue is that the standard Tobit implicitly models a very strong link between the participation decision ${\displaystyle (y=0}$ or ${\displaystyle y>0)}$ and the amount decision (the magnitude of ${\displaystyle y}$ when ${\displaystyle y>0}$). If a corner solution model is represented in a general form: ${\displaystyle y=s\centerdot w,}$ , where ${\displaystyle s}$ is the participate decision and ${\displaystyle w}$ is the amount decision, standard Tobit model assumes:

${\displaystyle s=1[x\beta +u>0];}$

${\displaystyle w=x\beta +u.}$

To make the model compatible with more contexts, a natural improvement is to assume:

${\displaystyle s=1[x\gamma +u>0],{\text{ where }}u\sim N(0,1);}$

${\displaystyle w=x\beta +e,}$ where the error term (${\displaystyle e}$) is distributed as a truncated normal distribution with a density as ${\displaystyle \varphi (\cdot )/\Phi \left({\frac {x\beta }{\sigma }}\right)/\sigma$ ;}

${\displaystyle s}$ and ${\displaystyle w}$ are independent conditional on ${\displaystyle x}$.

This is called Truncated Normal Hurdle Model, which is proposed in Cragg (1971).[1] By adding one more parameter and detach the amount decision with the participation decision, the model can fit more contexts. Under this model setup, the density of the ${\displaystyle y}$ given ${\displaystyle x}$ can be written as:

${\displaystyle f(y\mid x)=[1-\Phi (\chi \gamma )]^{1[y=0]}\cdot \left[{\frac {\Phi \ (\chi \gamma )}{\Phi (\chi \beta /\sigma )}}\left.\varphi \left({\frac {y-\chi \beta }{\sigma }}\right)\right/\sigma \right]^{1[y>0]}}$

From this density representation, it is obvious that it will degenerate to the standard Tobit model when ${\displaystyle \gamma =\beta /\sigma .}$ This also shows that Truncated Normal Hurdle Model is more general than the standard Tobit model.

The Truncated Normal Hurdle Model is usually estimated through MLE. The log-likelihood function can be written as:

${\displaystyle {\begin{aligned}\ell (\beta ,\gamma ,\sigma )={}&\sum _{i=1}^{N}1[y_{i}=0]\log[1-\Phi (x_{i}\gamma )]+1[y_{i}>0]\log[\Phi (x_{i}\gamma )]\\[5pt]&{}+1[y_{i}>0]\left[-\log \left[\Phi \left({\frac {x_{i}\beta }{\sigma }}\right)\right]+\log \left(\varphi \left({\frac {y_{i}-x_{i}\beta }{\sigma }}\right)\right)-\log(\sigma )\right]\end{aligned}}}$

From the log-likelihood function, ${\displaystyle \gamma }$ can be estimated by a probit model and ${\displaystyle (\beta ,\sigma )}$ can be estimated by a truncated normal regression model.[5] Based on the estimates, consistent estimates for the Average Partial Effect can be estimated correspondingly.