than any power. An important advantage of assuming the functions are in the space of generalized functions is that in that space any distribution function has a density (generalized function) that continuously depends on the distribution function, so that distributions with mass points and fractal measures have well-defined generalized densities. Any regular function majorized by a polynomial belongs to S $^{∗}$; this in- cludes polynomially growing regression functions and binary choice regres- sion as well as many conditional density functions. Another advantage is that Fourier transform is an isomorphism of this space, and thus the usual approaches in the literature that employ characteristic functions are also included. Details about the space S ∗ are in Schwartz (1966) and are sum- marized in Zinde-Walsh (2012). The model classes examined here lead to convolution equations that are similar to each other in form; the main focus of this paper is on existence, identification, partial identification and well-posedness conditions. Existence and uniqueness of solutions to some systems of convolution equations in the space S ∗ were established in Zinde-Walsh (2012). Those results are used here to state identification in each of the models. Identification requires examining support of the functions and generalized functions that enter into the models; if support excludes an open set then identification at least for some unknown functions in the model fails, however, some isolated points or lower-dimensional manifolds where the e.g. the characteristic function takes zero values (an example is the uniform distribution) does not preclude 4