  We study the problem of parameter estimation for a univariate discretely
observed ergodic diffusion process given as a solution to a stochastic
differential equation. The estimation procedure we propose consists of two
steps. In the first step, which is referred to as a smoothing step, we smooth
the data and construct a nonparametric estimator of the invariant density of
the process. In the second step, which is referred to as a matching step, we
exploit a characterisation of the invariant density as a solution of a certain
ordinary differential equation, replace the invariant density in this equation
by its nonparametric estimator from the smoothing step in order to arrive at an
intuitively appealing criterion function, and next define our estimator of the
parameter of interest as a minimiser of this criterion function. Our main
results show that under suitable conditions our estimator is
$\sqrt{n}$-consistent, and even asymptotically normal. We also discuss a way of
improving its asymptotic performance through a one-step Newton-Raphson type
procedure and present results of a small scale simulation study.
