Abstract
This thesis, which consists of four chapters, focuses on three topics: discriminating between stationary and nonstationary time series, testing the constancy of the error covariance matrix of a vector model, and estimating density functions over bounded domains using kernel techniques. In Chapter 1, “Testing the unit root hypothesis against the logistic smooth transition autoregressive model”, and Chapter 2, “A nonlinear alternative to the unit root hypothesis”, the joint hypothesis of unit root and linearity allows one to distinguish between random walk processes, with or without drift, and stationary nonlinear processes of the smooth transition autoregressive type. This is important in applications because steps taken in modelling a time series are likely to be drastically different depending on whether or not the unit root hypothesis is rejected. In Chapter 1 the nonlinearity is based on the logistic function, while Chapter 2 considers the second-order logistic function. Monte Carlo simulations show that the proposed tests have about the same or higher power than the standard Dickey-Fuller unit root tests when the alternative exhibits nonlinear behavior. In Chapter 1 the tests are applied to the seasonally adjusted U.S. monthly unemployment rate, giving support to the hypothesis that the unemployment rate series follows a smooth transition autoregressive model rather than a random walk. Chapter 2 considers testing the so called purchasing power parity (PPP) hypothesis. The test results complement earlier studies, giving support to the PPP hypothesis for 44 out of 120 real exchange rates considered. Chapter 3. “Testing the constancy of the error covariance matrix in vector models”Estimating the parameters of an econometric model is necessary for any use of the model, be it forecasting or policy evaluation. Finding out thereafter whether or not the model appears to satisfy the assumptions under which it was estimated should be an integral part of a normal modelling exercise. This chapter includes the derivation of a Lagrange Multiplier test of the null hypothesis of constant variance in vector models when testing against three specific parametric alternatives. The Monte Carlo simulations show that the test has good size properties, very good power against a correctly specified alternative, but low or only up to moderate power in cases for a misspecified alternative hypothesis. Chapter 4. “ Estimating confidence regions over bounded domains”Nonparametric density estimation by kernel techniques is a standard statistical tool in the estimation of a density function in situations where its parametric form is assumed to be unknown. In many cases, the data set over which the density is to be estimated exhibits linear, or nonlinear, dependence. A solution to this problem is to apply a one-to-one transformation to the considered data set in such a way that the dependence in the data vanishes, but too often such a unique transformation does not exist. This chapter proposes a method for estimating confidence regions over bounded domains when no one-to-one transformation of the considered data exists, or if the existence of such a transformation is difficult to verify. The method, simple kernel estimation over a nonlinear grid, is illustrated by applying it to three data sets generated from the GARCH(1,1) model. The resulting confidence regions cover a reasonable area of the definition space, and are well aligned with the corresponding data sets.