DescriptionEconometric models usually relate a known function of a dependent variable, Y , with some observable covariates X. A misspeci ed transformation function, however, can cause inconsistency. The estimated parameters will be biased and inconsistent. The marginal e ects and the elasticities calculated using these estimated parameters can be far from the truth and any inference based on them will be misleading. The problem of misspeci cation can be resolved simply by estimating the transformation function. There are di erent methods of estimating the model based on the parametric assumptions on the transformation function, or the distribution of the error term, or both. In a completely parametric setup, both the transformation function and the distribution of the error term are known up to a vector of parameters. The Box-Cox transformation is a good example of using parametric transformation functions. It is also possible, however, to estimate the transformation function without parametric assumptions. In this dissertation two Hausman tests for transformation functions are proposed where validity does not depend on distributional assumptions. These tests compare estimators that remain consistent regardless of the transformation function to an estimator whose consistency depends on the transformation function. The properties of these test statistics are studied in nite sample and under di erent designs. The behavior of these test statistics is studied both when the adopted transformation function is correct and when the true transformation function deviates from the hypothesized one. This dissertation applies the semiparametric transformation function test to study reported crimes in the U.S. metropolitan areas. Most studies in this literature adopt a logarithmic transformation of reported crimes. There is no theoretical justi cation for this speci c function. In addition, it is likely that city level crime is misreported (underreported). Therefore, testing is particularly relevant. I show that, although for particular types of crimes the log function is appropriate, the same log function cannot be used for broadly de ned categories of crimes.