content_image/300768.jpg content_image/300769.jpg
## 3.2.2 Solutions to the system of equations 

For models 3,4,5 and 7 a system of equations of the form 

<img src='content_image/300768.jpg'>

(with β continuously differentiable) arises. Theorem 3 in Zinde-Walsh (2012) provides the solution and uniqueness conditions for this system of equations. It is first established that a set of continuous functions κ $_{k}$, k = 1 , ..., d, that solves the equation 

<img src='content_image/300769.jpg'>

in the space S ∗ exists and is unique on W = supp ( γ ) as long as supp ( β ) ⊃ W. Then β ′ $_{k}$β − 1 = $^{κ}$k and substitution into (9) leads to a system of first-order differential equations in β. 

Case 1. Continuous functions; W is an open set. 

For the models 3 and 4 the system (8) involves continuous characteristic functions thus there W is an open set. In some cases W can be an open set under conditions of models 5 and 7, e.g. if the regression function is integrable in model 7. 

For this case represent the open set W as a union of (maximal) connected components ∪$_{v}$ W$_{v}$. 

Then by the same arguments as in the proof of Theorem 3 in Zinde- 

