This thesis is comprised of three independent essays. One essay is in the field of macroeconomics and the other two are in time-series econometrics. The first essay, "Productivity and Business Investment over the Business Cycle", is co-authored with my co-supervisor Hashmat Khan. This essay documents a new stylized fact: the correlation between labour productivity and real business investment in the U.S. data switching from 0.54 to -0.1 in 1990. With the assistance of a bivariate VAR, we find that the response of investment to identified technology shocks has changed signs from positive to negative across two sub-periods: ranging from the time of the post-WWII era to the end of 1980s and from 1990 onwards, whereas the response to non-technology shocks has remained relatively unchanged. Also, the volatility of technology shocks declined less relative to the non-technology shocks. This raises the question of whether relatively more volatile technology shocks and the negative response of investment can together account for the decreased correlation. To answer this question, we consider a canonical DSGE model and simulate data under a variety of assumptions about the parameters representing structural features and volatility of shocks. The second and third essays are in time series econometrics and solely authored by myself. The second essay, however, focuses on the impact of ignoring structural breaks in the conditional volatility parameters on time-varying volatility parameters. The focal point of the third essay is on empirical relevance of structural breaks in time-varying volatility models and the forecasting gains of accommodating structural breaks in the unconditional variance. There are several ways in modeling time-varying volatility. One way is to use the autoregressive conditional heteroskedasticity (ARCH)/generalized ARCH (GARCH) class first introduced by Engle (1982) and Bollerslev (1986). One prominent model is Bollerslev (1986) GARCH model in which the conditional volatility is updated by its own residuals and its lags. This class of models is popular amongst practitioners in finance because they are able to capture stylized facts about asset returns such as fat tails and volatility clustering (Engle and Patton, 2001; Zivot, 2009) and require maximum likelihood methods for estimation. They also perform well in forecasting volatility. For example, Hansen and Lunde (2005) find that it is difficult to beat a simple GARCH(1,1) model in forecasting exchange rate volatility. Another way of modeling time-varying volatility is to use the class of stochastic volatility (SV) models including Taylor's (1986) autoregressive stochastic volatility (ARSV) model. With SV models, the conditional volatility is updated only by its own lags and increasingly used in macroeconomic modeling (i.e.Justiniano and Primiceri (2010)). Fernandez-Villaverde and Rubio-Ramirez (2010) claim that the stochastic volatility model fits better than the GARCH model and is easier to incorporate into DSGE models. However, Creal et al. (2013) recently introduced a new class of models called the generalized autoregressive score (GAS) models. With the GAS volatility framework, the conditional variance is updated by the scaled score of the model's density function instead of the squared residuals. According to Creal et al. (2013), GAS models are advantageous to use because updating the conditional variance using the score of the log-density instead of the second moments can improve a model's fit to data. They are also found to be less sensitive to other forms of misspecification such as outliers. As mentioned by Maddala and Kim (1998), structural breaks are considered to be one form of outliers. This raises the question about whether GAS volatility models are less sensitive to parameter non-constancy. This issue of ignoring structural breaks in the volatility parameters is important because neglecting breaks can cause the conditional variance to exhibit unit root behaviour in which the unconditional variance is undefined, implying that any shock to the variance will not gradually decline (Lamoureux and Lastrapes, 1990). The impact of ignoring parameter non-constancy is found in GARCH literature (see Lamoureux and Lastrapes, 1990; Hillebrand, 2005) and in SV literature (Psaradakis and Tzavalis, 1999; Kramer and Messow, 2012) in which the estimated persistence parameter overestimates its true value and approaches one. However, it has never been addressed in GAS literature until now. The second essay uses a simple Monte-Carlo simulation study to examine the impact of neglecting parameter non-constancy on the estimated persistence parameter of several GAS and non-GAS models of volatility. Five different volatility models are examined. Of these models, three --the GARCH(1,1), t-GAS(1,1), and Beta-t-EGARCH(1,1) models -- are GAS models, while the other two -- the t-GARCH(1,1) and EGARCH(1,1) models -- are not. Following Hillebrand (2005) who studied only the GARCH model, this essay examines the extent of how biased the estimated persistence parameter are by assessing impact of ignoring breaks on the mean value of the estimated persistence parameter. The impact of neglecting parameter non-constancy on the empirical sampling distributions and coverage probabilities for the estimated persistence parameters are also studied in this essay. For the latter, studying the effect on the coverage probabilities is important because a decrease in coverage probabilities is associated with an increase in Type I error. This study has implications for forecasting. If the size of an ignored break in parameters is small, then there may not be any gains in using forecast methods that accommodate breaks. Empirical evidence suggests that structural breaks are present in data on macro-financial variables such as oil prices and exchange rates. The potentially serious consequences of ignoring a break in GARCH parameters motivated Rapach and Strauss (2008) and Arouri et al. (2012) to study the empirical relevance of structural breaks in the context of GARCH models. However, the literature does not address the empirical relevance of structural breaks in the context of GAS models. The third and final essay contributes to this literature by extending Rapach and Strauss (2008) to include the t-GAS model and by comparing its performance to that of two non-GAS models, the t-GARCH and SV models. The empirical relevance of structural breaks in the models of volatility is assessed using a formal test by Dufour and Torres (1998) to determine how much the estimated parameters change over sub-periods. The in-sample performance of all the models is analyzed using both the weekly USD trade-weighted index between January 1973 and October 2016 and spot oil prices based on West Texas Intermediate between January 1986 and October 2016. The full sample is split into smaller subsamples by break dates chosen based on historical events and policy changes rather than formal tests. This is because commonly-used tests such as CUSUM suffer from low power (Smith, 2008; Xu, 2013). For each sub-period, all models are estimated using either oil or USD returns. The confidence intervals are constructed for the constant of the conditional parameter and the score parameter (or ARCH parameter in GARCH and t-GARCH models). Then Dufour and Torres's union-intersection test is applied to these confidence intervals to determine how much the estimated parameter change over sub-periods. If there is a set of values that intersects the confidence intervals of all sub-periods, then one can conclude that the parameters do not change that much. The out-of-sample performance of all time-varying volatility models are also assessed in the ability to forecast the mean and variance of oil and USD returns. Through this analysis, this essay also addresses whether using models that accommodate structural breaks in the unconditional variance of both GAS and non-GAS models will improve forecasts.