  Power law or generalized polynomial regressions with unknown real-valued
exponents and coefficients, and weakly dependent errors, are considered for
observations over time, space or space--time. Consistency and asymptotic
normality of nonlinear least-squares estimates of the parameters are
established. The joint limit distribution is singular, but can be used as a
basis for inference on either exponents or coefficients. We discuss issues of
implementation, efficiency, potential for improved estimation and possibilities
of extension to more general or alternative trending models to allow for
irregularly spaced data or heteroscedastic errors; though it focusses on a
particular model to fix ideas, the paper can be viewed as offering machinery
useful in developing inference for a variety of models in which power law
trends are a component. Indeed, the paper also makes a contribution that is
potentially relevant to many other statistical models: Our problem is one of
many in which consistency of a vector of parameter estimates (which converge at
different rates) cannot be established by the usual techniques for coping with
implicitly-defined extremum estimates, but requires a more delicate treatment;
we present a generic consistency result.
