Many non/semi-parametric time series estimates may be regarded as different forms of sieve extremum estimates. For stationary $\beta$-mixing observations, we obtain convergence rates of sieve extremum estimates and root-n asymptotic normality of "plug-in" sieve extremum estimates of smooth functionals. As applications to time series models, we give convergence rates for nonparametric ARX(p, q) regression via neural networks, splines, and wavelets; root-n asymptotic normality for partial linear additive AR(p) models, and monotone transformation AR(1) models.