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 g ∗ f = w$_{1}$, (10) h ∗ f = w$_{2}$, with w$_{1}$ = E ( y | z ) , w$_{2}$ = E ( xy | z ) known. 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 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 ∫ g (( z − u ) | x$_{c}$ ) f ( u | x$_{c}$ ) du = w ( z | x$_{c}$ ) . 7