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conditional (on z and the other errors) expectations. For this model Newey (2001) proposed to consider an additional equation assuming that the condi- tional moment E ( g ( x ) x | z ) exists. In the univariate case this gives a system of two equations with two unknown functions (as discussed in Schennach, 2007, Zinde-Walsh, 2009). Define h ( x ) = xg ( x ) , then 

with w$_{1}$ = E ( y | z ) , w$_{2}$ = E ( xy | z ) known. 

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In the multivariate case for x = ( x$_{1}$, ..., x$_{d}$ ) ∈ R $^{d}$, w$_{2}$$_{k}$ = E ( x$_{k}$y | z ) , k = 1 , ..., d , h$_{k}$ ( x ) = x$_{k}$g ( x ) this can be generalized to equations (3) . If the density f$_{x}$ ∗ of x ∗ were of interest, equation f$_{x}$ ∗ = f$_{z}$ ∗ f$_{−}$$_{u}$ could be added. 

If the independence between the mismeasured variable and measurement error holds conditionally then the equation (2) can be written for densities conditional on some x$_{c}$ , where if such densities exist this equation is 

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Then equation (2) is defined for functions in spaces where the dimension or the argument is augmented by the dimension of the conditioning variable, x$_{c}$. 

A common way of providing solutions to (2) and other equations is to 

