Unit root, outliers and cointegration analysis with macroeconomic applications

Abstract

In this thesis, we deal with three particular issues in the literature on nonstationary time series. The first essay deals with various unit root tests in the context of structural change. The second paper studies some residual based tests in order to identify cointegration. Finally, in the third essay, we analyze several tests in order to identify additive outliers in nonstationary time series. The first paper analyzes the hypothesis that some time series can be characterized as stationary with a broken trend. We extend the class of M-tests and ADF test for a unit root to the case where a change in the trend function is allowed to occur at an unknown time. These tests (MGLS, ADFGLS) adopt the Generalized Least Squares (GLS) detrending approach to eliminate the set of deterministic components present in the model. We consider two models in the context of the structural change literature. The first model allows for a change in slope and the other for a change in slope as well as intercept. We derive the asymptotic distribution of the tests as well as that of the feasible point optimal test (PF-Ls) which allows us to find the power envelope. The asymptotic critical values of the tests are tabulated and we compute the non-centrality parameter used for the local GLS detrending that permits the tests to have 50% asymptotic power at that value. Two methods to select the break point are analyzed. A first method estimates the break point that yields the minimal value of the statistic. In the second method, the break point is selected such that the absolute value of the t-statistic on the change in slope is maximized. We show that the MGLS and PTGLS tests have an asymptotic power function close to the power envelope. An extensive simulation study analyzes the size and power of the tests in finite samples under various methods to select the truncation lag for the autoregressive spectral density estimator. In an empirical application, we consider two U.S. macroeconomic annual series widely used in the unit root literature: real wages and common stock prices. Our results suggest a rejection of the unit root hypothesis. In other words, we find that these series can be considered as trend stationary with a broken trend. Given the fact that using the GLS detrending approach allows us to attain gains in the power of the unit root tests, a natural extension is to propose this approach to the context of tests based on residuals to identify cointegration. This is the objective of the second paper in the thesis. In fact, we propose residual based tests for cointegration using local GLS detrending to eliminate separately the deterministic components in the series. We consider two cases, one where only a constant is included and one where a constant and a time trend are included. The limiting distributions of various residuals based tests are derived for a general quasi-differencing parameter and critical values are tabulated for values of c = 0 irrespective of the nature of the deterministic components and also for other values as proposed in the unit root literature. Simulations show that GLS detrending yields tests with higher power. Furthermore, using c = -7.0 or c = -13.5 as the quasi-differencing parameter, based on the two cases analyzed, is preferable. The third paper is an extension of a recently proposed method to detect outliers which explicitly imposes the null hypothesis of a unit root. it works in an iterative fashion to select multiple outliers in a given series. We show, via simulation, that under the null hypothesis of no outliers, it has the right size in finite samples to detect a single outlier but when applied in an iterative fashion to select multiple outliers, it exhibits severe size distortions towards finding an excessive number of outliers. We show that this iterative method is incorrect and derive the appropriate limiting distribution of the test at each step of the search. Whether corrected or not, we also show that the outliers need to be very large for the method to have any decent power. We propose an alternative method based on first-differenced data that has considerably more power. The issues are illustrated using two US/Finland real exchange rate series.