In the first chapter; we consider nonlinear transformations of random walks driven by thick-tailed innovations with infinite means or variances. In particular, we show how nonlinearity, nonstationarity, and thick tails interact to generate persistency in memory, and we clearly demonstrate that this triad may generate a broad spectrum of persistency patterns. Time series generated by nonlinear transformations of random walks with thick-tailed innovations have asymptotic autocorrelations that decay very slowly as the number of lags increases or do not even decay at all and remain constant at all lags. We also discuss nonlinear regression asymptotics when the regressor is observable and an alternative regression technique when it is unobservable. We use our model to analyze two empirical applications: exchange rates governed by a target zone and electricity price spikes driven by capacity shortfalls. In the second chapter of the dissertation, we show that typical tests for purchasing power parity using target zone exchange rates are inherently misspecified. The real exchange rate cannot be mean-reverting by construction, since the nominal exchange rate is generated by a nonlinear transformation of a nonstationary economic fundamental. As an alternative, we propose basing the real exchange rate (and thus the test) on conditional expectations of the fundamental itself. Real exchange rates based on the true fundamental may in fact exhibit mean reversion, if the long-run purchasing power parity hypothesis holds. The final chapter investigates the statistical properties of the Kalman filter for state space models including integrated time series. In particular, we derive the full asymptotics of maximum likelihood estimation for a prototypical class of such models: those with a single latent common stochastic trend. We demonstrate the utility of the state space model by extracting a common stochastic trend in three empirical analyses: interest rates, stock return volatility and trading volume, and a stock price index.