In considering the problems of inference in nonlinear regression models the statistical and computational aspects of parameter estimation are discussed, and model selection procedures for both nested and nonnested hypotheses are analyzed including an optimal sequential testing procedure for ordered nested hypotheses. A distinction is made between tests of specification and tests of misspecification, and is discussed in relationship to the Wald, likelihood ratio, and Lagrange multiplier hypothesis testing principles. Degrees of freedom or small sample adjustments to the asymptotically valid test statistics are also discussed. The choice of production function from a class of CES functions, with either additive or multiplicative error specifications, for a cross section of UK industries, provides an application of the theory.