**A New Approach to Forecasting in the Presence of In and Out-of-Sample Breaks*** (Job Market Paper by Jiawen Xu and Pierre Perron)*

We present a novel frequentist-based approach to forecast time series in the presence of in-sample of out-of-sample breaks in the parameters of the forecasting model. We first model the parameters as following a random level shift process, with the occurrence of a shift governed by a Bernoulli process. In order to have a structure so that changes in the parameters are forecastable, we introduce two modifications. The first models the probability of shifts according to some covariates that can be forecasted. The second incorporates a built-in mean reversion mechanism to the time path of the parameters. Similar modifications can also be made to model changes in the variance of the error process. Our full model can be cast into a non-linear non-Gaussian state space framework. To estimate it, we use particle filtering and a Monte Carlo expectation maximization algorithm. Simulation results show that the algorithm delivers accurate in-sample estimates, in particular the filtered estimates of the time path of the parameters follow closely to their true variations. We provide a number of empirical applications and compare the forecasting performance of our model with a variety of alternative methods. These show that substantial gains in forecasting accuracy are obtained.

**Forecasting Return Volatility: Level Shifts with Varying Jump Probability and Mean Reversion*** (Jiawen Xu, Pierre Perron) *

We extend the random level shift (RLS) model of Lu and Perron (2010) for the volatility of asset prices, which consists of a short memory process and a random level shift component. Motivated by empirical features, we extend it 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. This allows the possibility of forecasting the sign and magnitude of the jumps. We estimate the model using daily data on four major stock market indices. We compare its forecasting performance with competing models. A striking feature is that the modified RLS model is the only one that belongs to the 10% model confidence set of Hansen et al. (2011) using all comparisons, for all series and all forecasting horizons. The modified RLS model also yields the smallest mean square forecast errors in 23 out of 24 cases. This is strong evidence that our modified RLS model offers important gains in forecasting performance.

**Robust Testing of Time Trend and Mean with Unknown Integration Order Errors** *(Jiawen Xu, Pierre Perron)*

This paper 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. In particular, 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 using the residuals from a least-squares regression. To that effect, we use the exact local-Whittle estimator proposed by Shimotsu and Phillips (2005). Simulation results show that our procedure delivers tests with good finite sample size and power, including for cases with strong short-term correlations. Our procedure thereby extends that of Perron and Yabu (2009) who considered only the case with errors being I(1) or I(0).