Tests are developed for inference on a parameter vector whose dimension grows slowly with sample size. The statistics are based on the Lagrange Multiplier, Wald and (pseudo) Likelihood Ratio principles, admit standard normal asymptotic distributions under the null and are straightforward to compute. They are shown to be consistent and possessing non-trivial power against local alternatives. The settings considered include multiple linear regression, panel data models with fixed effects and spatial autoregressions. When a nonparametric regression function is estimated by series we use our statistics to propose specification tests, and in semiparametric adaptive estimation we provide a test for correct error distribution specification. These tests are nonparametric but handled in practice with parametric techniques. A Monte Carlo study suggests that our tests perform well in finite samples. Two empirical examples use them to test for correct shape of an electricity distribution cost function and linearity and equality of Engel curves.
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