We consider a semiparametric model whereby the response variable follow-
ing a transformation can be expressed by means of a regression model. In this model,
the form of the transformation is specified analytically (up to an unknown transfor-
mation parameter), while the regression function is completely unknown. We develop
testing procedures for the null hypothesis that this semiparametric model adequately
describes the data at hand. In doing so, the test statistic is formulated on the basis of
Fourier-type conditional expectations, an idea first put forward by Bierens (J Econom
20:105–134, 1982). The asymptotic distribution of the test statistic is obtained under
the null as well as under alternative hypotheses. Since the limit null distribution is
nonstandard, a bootstrap version is utilized in order to actually carry out the test pro-

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• Testing the adequacy of semiparametric transformation models