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## 4.2 Implications of partial identification. 

Consider the case of Model 1. Essentially lack of identification, say in the case when the error distribution has characteristic function supported on a convex domain W$_{u}$ around zero results in the solution for φ$_{x}$ ∗ = φ$_{1}$φ$_{2}$, with φ$_{1}$ non-zero and unique on W$_{u}$, and thus captures the lower-frequency components of x $^{∗}$, and with φ$_{2}$ is a characteristic function of a distribution with arbitrary high frequency components. Transforming back to densities provides a corresponding model with independent components 

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where x ∗ 1 uniquely extracts the lower frequency part of observed z. The more important the contribution of x ∗ 1 to x ∗ the less important is lack of identifi- cation. 

If the feature of interest as discussed e.g. by Matzkin (2007) involves only low frequency components of x $^{∗}$, it may still be fully identified even when the distribution for x ∗ itself is not. An example of that is a deconvolution applied to an image of a car captured by a traffic camera; although even after deconvolution the image may still appear blurry the licence plate number may be clearly visible. In nonparametric regression the polynomial growth of the regression or the expectation of the response function may be identifiable even if the regression function is not fully identified. 

Features that are identified include any functional, Φ , linear or non-linear 

