This dissertation consists of two chapters related to inference about locally ordered and common breaks in multiple regressions and one chapter pertains to modeling exchange rate volatility with random level shifts. The first chapter considers inference about locally ordered breaks in a system of equations. These apply when break dates in different equations are not separated by a positive fraction of the sample size. We extend the results of Qu and Perron (2007) in several directions allowing: a) the covariates to be any mix of trends, stationary or integrated regressors; b) breaks in the variance-covariance matrix of errors; c) an arbitrary number of breaks occurring in a subset of equations. We show that the limit distributions derived provide good approximations to the finite sample distributions and forming confidence intervals in a joint fashion allows more precision. The second chapter considers testing for common breaks and estimating locally ordered breaks in a multiple system with joined segmented trends. To test for common breaks, we consider a likelihood ratio type test. The null hypothesis is that some subsets of coefficients for slope shifts share some common breaks, while the alternative hypothesis is that the breaks dates are different and possibly locally ordered. To estimate locally ordered breaks, we use a quasi-maximum likelihood estimation method. We show consistency and derive rate of convergence and asymptotic distributions of test statistics and estimates of the break dates. Simulation results show that the asymptotic results provide useful approximations in finite samples. In the third chapter, we estimate a random level shifts model for the log-absolute returns of the Dollar-Mark and Dollar-Yen exchange rates, in order to assess whether random level shifts can explain the long memory property. The results show that there are few level shifts, but once they are taken into account, the long memory property disappears. We also provide out-of-sample forecasting comparisons, which show that the random level shifts model outperforms standard fractionally integrated models.