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Sharp upper and lower uniform bounds are established for a general class of functionals of integrated and fractionally integrated time series. The main result is used to develop optimal uniform convergence for the Nadaraya-Watson estimator and the local linear nonparametric estimator in a nonlinear cointegrating regression model. Unlike the point-wise situation, it is shown that the performance of the local linear nonparametric estimator is superior to that of the Nadaraya-Watson estimator in uniform asymptotics.

For a class of martingales, this paper provides a framework on the uniform consistency with broad applicability. The main condition imposed is only related to the conditional variance of the martingale, which holds true for stationary mixing time series, stationary iterated random function, Harris recurrent Markov chains and I(1) processes with innovations being a linear process. Using the established results, this paper investigates the uniform convergence of the Nadaraya-Watson estimator in a non-linear cointegrating regression model. Our results not only provide sharp convergence rate, but also the optimal range for the uniform convergence to be held. This paper also considers the uniform upper and lower bound estimates for a functional of Harris recurrent Markov chain, which are of independent interests.

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