The dissertation consists of three chapters on econometric methods related to parameter instability, forecasting and long memory. The first chapter introduces a new frequentist-based approach to forecast time series in the presence of in and out-of-sample breaks in the parameters. We model the parameters as random level shift (RLS) processes and introduce two features to make the changes in parameters forecastable. The first models the probability of shifts according to some covariates. The second incorporates a built-in mean reversion mechanism to the time path of the parameters. Our model can be cast into a non-linear non-Gaussian state-space framework. We use particle filtering and Monte Carlo expectation maximization algorithms to construct the estimates. We compare the forecasting performance with several alternative methods for different series. In all cases, our method allows substantial gains in forecasting accuracy. The second chapter extends the RLS model of Lu and Perron (2010) for the volatility of asset prices. The extensions are in two directions: a) we specify a time-varying probability of shifts as a function of large negative lagged returns; b) we incorporate a mean reverting mechanism so that the sign and magnitude of the jump component change according to the deviations of past jumps from their long run mean. We estimate the model using daily data on four major stock market indices. Compared to competing models, the modified RLS model yields the smallest mean square forecast errors overall. The third chapter proposes a method of inference about the mean or slope of a time trend that is robust to the unknown order of fractional integration of the errors. Our tests have the standard asymptotic normal distribution irrespective of the value of the long-memory parameter. Our procedure is based on using quasi-differences of the data and regressors based on a consistent estimate of the long-memory parameter obtained from the residuals of a least-squares regression. We use the exact local-Whittle estimator proposed by Shimotsu (2010). Simulation results show that our procedure delivers tests with good finite sample size and power, including cases with strong short-term correlations.