Publication Type

PhD Dissertation

Publication Date



Financial and macroeconomic time series data are often nonstationary. My dissertation consists of three essays concerning time series models with nonstationarity. Chapter 1 develops a new jackknife estimator for nonstationary autoregressive model. The remaining two chapters explore the restricted maximum likelihood (REML hereafter) estimation and the restricted maximum likelihood based likelihood ratio test (RLRT hereafter) in predictive regression. Chapter 1 proposes an improved jackknife estimator of the persistence parameter that works for both the discrete time unit root model and the continuous time unit root model. Maximum likelihood estimation of the persistence parameter in the discrete time unit root model is known to suffer from a downward bias. The bias is more pronounced in the continuous time unit root model. Recently, Chambers and Kyriacou (2013) introduces a new jackknife method to remove the first order bias in the estimator of the persistence parameter in a discrete time unit root model. The proposed jackknife estimator is optimal in the sense that it minimizes the variance. Simulations highlight the performance of the proposed method in both contexts. They show that our optimal jackknife reduces the variance of the jackknife method of Chambers and Kyriacou by around 10% in both cases and the results continue to hold in near unit root cases. Chapter 2 derives the limit distribution of the RLRT statistic for univariate predictor in predictive regression and extends the result to multivariate predictors. Chen and Deo (2009a) proposes procedures based on the restricted likelihood (RL hereafter) function for estimation and inference in the context of predictive regression. Their method achieves bias reduction both in estimation and in inference which assists in overcoming size distortion in predictive hypothesis testing. This chapter provides the corresponding theoretical results for implementing RLRT when predictors are time series with unit root, local to unity and moderate deviation from unit root. Our results extend Chen, Deo and Yi (2013) by allowing for multiple predictors. Chapter 3 extends the REML approach to cases which allow for drift in predictive regressors. It is shown that, without modification, the REML approach is seriously oversized and can have unit rejection probability in the limit under the null when the drift in the regressor is dominant. A limit theory for the modified RLRT is given under a localized drift specification that accommodates predictors with varying degrees of persistence. The extension is useful in empirical work because predictors typically involve stochastic trends with drift. Simulations show that with these modifications, the good performance of RLRT is preserved and that RLRT outperforms other predictive tests in terms of size and power even when there is no drift in the regressor.


bias reduction, jackknife estimator, predictive regression, restricted likelihood ratio test, localized drift, size distortion

Degree Awarded

PhD in Economics




Phillips, Peter C.B.; Yu, Jun; Tay, Anthony S.