Notes. Notation: k = 1 , 2 ...d ; in model 5. w = E ( yf$_{z}$ ( z ) | z ); w$_{k}$ = E ( x$_{k}$f$_{z}$ ( z ) | z ); in model 6. w = E ( y | z ); in model 7. w = E ( y | z ); w$_{k}$ = E ( x$_{k}$y | z ) . Theorem 2. Under Assumption 1 for each of the models 5-7 the corre- sponding convolution equations hold. The proof is in the derivations of the next subsection. 2.3.2 Discussion of the regression models and derivation of the 23 Model Distributional assumptions Convolution equations Known functions Unknown functions 5. y = g ( x $^{∗}$) + v z = x ∗ + u ; x = x ∗ + u$_{x}$ x $^{∗}$⊥ u ; E ( u ) = 0; E ( u$_{x}$ | x $^{∗}$, u ) = 0; E ( v | x $^{∗}$, u, u$_{x}$ ) = 0 . f$_{x}$ ∗ ∗ f$_{u}$ = f$_{z}$ ; ( gf$_{x}$ ∗ ) ∗ f$_{u}$ = w, h$_{k}$ ∗ f$_{u}$ = w$_{k}$ ; with h$_{k}$ ( x ) ≡ x$_{k}$g ( x ) f$_{x}$ ∗ ( x ); k = 1 , 2 ...d f$_{z}$ ; w ; w$_{k}$ f$_{x}$ ∗ ; f$_{u}$ ; g. 6. y = g ( x ) + v z = x + u ; E ( v | z ) = 0; z ⊥ u ; E ( u ) = 0 . f$_{x}$ = f$_{−}$$_{u}$ ∗ f$_{z}$ ; g ∗ f$_{−}$$_{u}$ = w f$_{z}$ ; f$_{x}$, w f$_{u}$ ; g. 7. y = g ( x $^{∗}$) + v ; x = x ∗ + u$_{x}$ ; z = x ∗ + u ; z ⊥ u ; E ( v | z, u, u$_{x}$ ) = 0; E ( u$_{x}$ | z, v ) = 0 . g ∗ f$_{u}$ = w ; h$_{k}$ ∗ f$_{u}$ = w$_{k}$, with h$_{k}$ ( x ) ≡ x$_{k}$g ( x ); k = 1 , 2 ...d w, w$_{k}$ f$_{u}$ ; g.