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. In a standard Tobit model, represented as y = ( x β + u ) 1 [ x β + u > 0 ] {\displaystyle y=(x\beta +u)1[x\beta +u>0]} , where u | x ∼ N ( 0 , σ 2 ) {\displaystyle u|x\sim N(0,\sigma ^{2})} This model construction implicitly imposes two first order assumptions: Since: ∂ P [ y > 0 ] / ∂ x j = φ ( x β / σ ) β j / σ {\displaystyle \partial P[y>0]/\partial x_{j}=\varphi (x\beta /\sigma )\beta _{j}/\sigma } and ∂ E ⁡ [ y ∣ x , y > 0 ] / ∂ x j = β j { 1 − θ ( x β / σ } {\displaystyle \partial \operatorname {E} [y\mid x,y>0]/\partial x_{j}=\beta _{j}\{1-\theta (x\beta /\sigma \}} , the partial effect of x j {\displaystyle x_{j}} on the probability P [ y > 0 ] {\displaystyle P[y>0]} and the conditional expectation: E ⁡ [ y ∣ x , y > 0 ] {\displaystyle \operatorname {E} [y\mid x,y>0]} has the same sign: The relative effects of x h {\displaystyle x_{h}} and x j {\displaystyle x_{j}} on P [ y > 0 ] {\displaystyle P[y>0]} and E ⁡ [ y ∣ x , y > 0 ] {\displaystyle \operatorname {E} [y\mid x,y>0]} are identical, i.e.: ∂ P [ y > 0 ] / ∂ x h ∂ P [ y > 0 ] / ∂ x j = ∂ E ⁡ [ y ∣ x , y > 0 ] / ∂ x h ∂ E ⁡ [ y ∣ x , y > 0 ] / ∂ x j = β h β j | {\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.

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