     f$_{x}$ ∗ ∗ f$_{u}$ = w ; (1 − ρ ) h$_{x}$ ∗ ∗ f$_{u}$ + ρzf ( z ) = w$_{x}$ ; (1 − ρ $^{2}$) h$_{x}$ ∗ ∗ f$_{u}$ + ρ $^{2}$zf ( z ) = w$_{y}$. (3) The three equations have three unknown functions, f$_{x}$ ∗ , f$_{u}$ and ρ. Assum- ing that support of ρ does not include the point 1, ρ can be expressed as a solution to a linear algebraic equation derived from the two equations in (3) that include ρ : The table below provides several regression models and the corresponding convolution equations involving density weighted conditional expectations. Table 2. Regression models: 5. Regression with classical measurement error and an additional observation; 6. Regression with Berkson error ( x, y, z are observable); 7. Regression with zero mean measurement error and Berkson instruments. 22 2.3 Regression models with classical and Berkson er- rors and the convolution equations 2.3.1 The list of models ρ = ( w$_{x}$ − zf ( z )) − $^{1}$( w$_{y}$ − w$_{x}$ ) .