This dissertation is a collection of three essays applying modern time series techniques in the context of financial markets. There is a particular focus on disentangling persistent trend components from transitory cyclical dynamics. The information contained in these cyclical components is leveraged to garner insight into the broader macroeconomy. The first essay, Trend and Cycle in the Yield Curve: A Procedure for Forecasting Recessions, utilizes short-term (slope) dynamics present in the yield curve to predict impending economic downturns. Building on a large body of literature chronicling the relationship between the shape of the yield curve and the business cycle I employ Dynamic Nelson-Siegel modeling to define the level, slope, and curvature characteristics of the term structure through time. Given these dynamics, the trend and cycle are extracted using various decomposition techniques. I show that cycles present within the slope factor are extremely robust predictors of recessions, correctly identifying recessions as much as eighteen months in advance. Moreover, I develop a ``Predictive Power Score'' as a way to quantify my procedure's performance. This score demonstrates the superiority of my procedure over other common leading indicators including the yield spread. This first essay illustrates a common obstacle faced by researchers when attempting to measure cycles in real-time. Symmetric band-pass filters are estimated at the expense of data trimming, i. e. current estimates of the cycle must be sacrificed in order to construct the filtered series. Building on the work of Baxter and King (1999), Christiano and Fitzgerald (2003) construct a ``one-sided" filter which allows the practitioner to obtain estimates of the cycle in real-time. The second essay of this dissertation, Spurious Periodicity in Christiano-Fitzgerald Filtered Time Series, studies the cyclical properties of time series filtered by the Christiano and Fitzgerald (2003) filter. I show that in the presence of a stochastic trend the CF filter imposes spurious periodicity onto the filtered series, i. e. the filter imparts cyclicality where there is none. This is due to a common defect among band-pass filters which allows cyclical components of the error term to pass through the filter to the estimated cycle. In practice, this leads to cycle estimates of higher amplitude and longer duration. The third essay of this dissertation focuses on an emerging financial market which until recently has received little attention in the academic literature. An Analysis of Bitcoin Exchange Rates studies the relationship between bitcoin prices and the foreign exchange market in a way that has not been done before. I contend that the best way to think of bitcoins is as digital gold. Bitcoins are a purely electronic commodity traded for speculative purposes as well as in exchange for goods and services. Just like physical gold the relative price of bitcoins denominated in different currencies implies a nominal exchange rate. This is a departure from previous literature which treats bitcoin prices themselves as exchange rates. I argue that treating prices as exchange rates is inappropriate as one would not consider the price of physical gold to be an exchange rate. Therefore, I characterize the behavior of nominal exchange rates implied by relative bitcoin prices. I show that the implied nominal exchange rate is highly cointegrated with the nominal exchange rate determined in conventional foreign currency exchange markets. I also show that the direction of causality flows from the conventional markets to the bitcoin market and not vice-versa which can explain much of the volatility in bitcoin prices.