In this dissertation, I provide methods to robustly estimate the parameters of a wide variety of time series models in the potential presence of additive low-frequency contamination. The types of contamination covered include level shifts (changes in mean) and monotone or smooth time trends, both of which have been shown to bias parameter estimates towards regions of persistence in a variety of contexts.

The first chapter of this dissertation deals with the fully parametric estimation of strictly stationary short-memory time series models. The estimators employed in this chapter minimize trimmed frequency domain quasi-maximum likelihood (FDQML) and Whittle objective functions without requiring specification of the low-frequency contaminating component. When proper sample size-dependent trimmings are used, the FDQML estimators are consistent and asymptotically normal, asymptotically eliminating the presence of any spurious persistence. These asymptotic results also hold in the absence of additive low-frequency contamination. In the presence of contamination, the trimmed estimators entail substantial gains over standard estimators.

The second chapter addresses semiparametric estimation of the memory parameter of a time series that exhibits possible long-memory. The semiparametric estimators used in this context are simple trimmed versions of the popular log-periodogram regression estimator that employ certain sample size-dependent, and in some cases, data-dependent trimmings which discard low-frequency components. Regardless of whether the underlying long/short-memory process is indeed contaminated, the estimators are shown to be consistent and asymptotically normal with the same limiting variance as the standard log-periodogram estimator. The tradeoffs involved with their use when such components are not present but the underlying process exhibits strong short-memory dynamics, or is contaminated by noise, are also assessed and a particular version of the estimators is recommended to balance these tradeoffs. Empirical estimation results of the first two chapters suggest that a large portion of the apparent persistence in certain economic and financial time series may indeed be spurious.

The final chapter provides the conditions under which the fully parametric FDQML estimators introduced in the first chapter can be used to consistently estimate the long-memory stochastic volatility model parameters in the presence of additive low-frequency contamination in log-squared returns.