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The third expression is a double integral which always exists if E ‖ x $^{∗}$‖ < ∞ ; this is a consequence of boundedness of the expectations of z and u. The fourth is a result of change of variables ( x $^{∗}$, z ) into ( x $^{∗}$, u ) , the fifth uses independence of x $^{∗}$and u, and the sixth expression follows from the corre- sponding expression for the convolution of generalized functions (Schwartz, 1967, p.246). The conditions of model 3 are not sufficient to identify the distribution of u$_{x}$ ; this is treated as a nuisance part in model 3. 

The model in 4 with all the errors and mis-measured variable independent of each other was investigated by Kotlyarski (1967) who worked with the joint characteristic function. In 4 consider in addition to the equations written for model 3 another that uses the independence between x ∗ and u$_{x}$ and involves f$_{u}$$_{x}$. 

In representation 4a the convolution equations involving the density f$_{u}$$_{x}$ are obtained by applying the derivations that were used here for the model in 3.: 

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