Here (3)-(5) provide a regression with z representing a second measurement or possibly a given projection onto a set of instruments for the unobserved x $^{∗}$. Here y, z or x, y, z are observed; u is a Berkson type measurement error independent of z ; u$_{y}$, u$_{x}$ have zero conditional (on z and the other errors) expectations. Denote w$_{1}$ = E ( y | z ) , density of measurement error f. Example 3. Panel data model with two periods. Evdokimov (2010). Here let x (or z ) represent the observed variable in the first period, and z ( x ) for the second, x ∗ is the nonparametric function m ( X, α ) , where α is the idiosyncratic component and the densities are conditional on the same value X for the two periods; the same distributional assumptions as in Example 1 are used. The models lead to the same system of convolution equations. All vectors are in R $^{d}$. By independence in all cases we get For examples 1 and 3 define density of z by f$_{z}$ ; by x$_{k}$ the kth component of the vector x and consider g ∗ f = w$_{1}$. E ( f$_{z}$x$_{k}$ | z ) = E ( x ∗ $_{k}$f$_{z}$ | z ) = ∫ ( z − u )$_{k}$ g ( z − u ) f ( u ) du = x$_{k}$g ∗ f. 4