In the first chapter, we consider the logistic regression model with an integrated regressor. In particular, we derive the limit distributions of the nonlinear least squares (NLS) estimators and their t-ratios. It is shown that the NLS estimators are generally not efficient. Moreover, the t-ratios for the level parameters have limit distributions that are non-normal and dependent upon nuisance parameters, due to the asymptotic correlation between the innovations of the regressor and the regression errors. We propose an efficient NLS procedure to deal with the inefficiency of the estimators and inferential difficulty. The new NLS procedure yields estimators that are efficient and have asymptotically normal t-ratios.The second chapter considers a state space model with integrated latent variables. The model provides an effective framework to specify, test and extract common stochastic trends in a set of integrated time series. The standard Kalman filter is used to estimate the model, and the asymptotic theory of the Kalman filter is derived. In particular, we establish the consistency and asymptotic mixed normality of the maximum likelihood estimator, and validate the conventional inference for this class of models. Moreover, we derive a trace statistic to test the number of common stochastic trends in a system of integrated time series. The asymptotic distribution of the trace statistic is derived as normal. Chapter 3 provides an empirical illustration by investigating common stochastic trends of interest rates with different maturities.
