Number Of Unit Roots Research Articles

Modeling high-dimensional unit-root time series

This paper proposes a new procedure to build factor models for high-dimensional unit-root time series by postulating that a p-dimensional unit-root process is a nonsingular linear transformation of a set of unit-root processes, a set of stationary common factors that are dynamically dependent, and some idiosyncratic white noise components. For the stationary components, we assume that the factor process captures the temporal dependence, and that the idiosyncratic white noise series explains, jointly with the factors, the cross-sectional dependence. The estimation of nonsingular linear loading spaces is carried out in two steps. First, we use an eigenanalysis of a nonnegative definite matrix of the data to separate the unit-root processes from the stationary ones, and a modified method to specify the number of unit roots. We then employ another eigenanalysis and a projected principal component analysis to identify the stationary common factors and the white noise series. We propose a new procedure to specify the number of white noise series and, hence, the number of stationary common factors. We establish asymptotic properties of the proposed method for both fixed and diverging p as the sample size n increases, and use a simulation and a real example to demonstrate the performance of the proposed method in finite samples. We also compare our method with some commonly used ones in the literature regarding the forecast ability of the extracted factors, and find that the proposed method performs well in out-of-sample forecasting of a 508-dimensional PM2.5 series in Taiwan.

Evaluating trends in time series of distributions: A spatial fingerprint of human effects on climate

We analyze a time series of global temperature anomaly distributions to identify and estimate persistent features in climate change. We employ a formal test for the existence of functional unit roots in the time series of these densities, and we develop a new test to distinguish functional unit roots from functional deterministic trends or explosive behavior. Results suggest that temperature anomalies contain stochastic trends (as opposed to deterministic trends or explosive roots), two trends are present in the Northern Hemisphere while one stochastic trend is present in the Southern Hemisphere, and the probabilities of observing moderately positive anomalies have increased. We postulate that differences in the pattern and number of unit roots in each hemisphere may be due to a natural experiment which causes human emissions of greenhouse gases and sulfur to be greater in the Northern Hemisphere, decreasing the mean temperature anomaly but increasing the spatial variance relative to the Southern Hemisphere. Together, these results are consistent with the theory of anthropogenic climate change.

A Strongly Consistent Criterion to Decide Between I(1) and I(0) Processes Based on Different Convergence Rates

The usual procedure to determine whether a univariate time series is stationary or first-difference stationary is to perform some unit root test. In this article, an alternative methodology is presented that leads to a strongly consistent two-step criterion to estimate the number of unit roots. The criterion is based on estimating some autoregressive polynomials using regression procedures and exploiting the fact that the nonstationary roots converge at a faster rate than the stationary ones. The proposed procedure requires at most four regressions and is easy to implement. A simulation study demonstrates that it can perform significantly better in practice than the Dickey–Fuller and the generalized least squares (GLS)-detrended Dickey–Fuller tests.

On seasonal error correction when the processes include different numbers of unit roots

AbstractWe propose a seasonal cointegration model (SECM) for quarterly data which includes variables with different numbers of unit roots and thus needs to be transformed in different ways in order to yield stationarity. A Monte Carlo simulation is carried out to investigate the consequences of specifying a SECM with all variables in annual diffrerences in this situation. The SECM in annual differences is compared to the correctly specified model. Pre‐testing for unit roots using two different approaches, and where the models are specified according to the unit root test results, is also considered. The forecast mean squared error criterion and certain parameter estimation results indicate that, in practice, a cointegration model where all variables are transformed with the annual difference filter is more robust than one obtained by pre‐testing for a smaller number of unit roots. The second‐best choice when the true model is not known and when the aim is to forecast, is an ordinary VAR model also in annual differences. Copyright © 2003 John Wiley & Sons, Ltd.

Finding Cointegration Rank in High Dimensional Systems Using the Johansen Test: An Illustration Using Data Based Monte Carlo Simulations

The authors examine the ability of the Johansen (1991) test to estimate the number of unit roots in high dimensional systems. They use data based Monte Carlo methods as a simple means of evaluating the validity of inference using asymptotic critical values. These simulations for a typical annual post-World War II dataset illustrate how the estimated number of unit roots change in a nonmonotone fashion with the dimension of the system, and with the number of lags in the VAR representation. The authors find that overparametrization in high dimensions is as bad as underparametrization. The Bayes information criteria outperforms the Akaike information criteria in their setup. Copyright 1996 by MIT Press.

MODEL SELECTION IN PERIODIC AUTOREGRESSIONS†

textabstractThis paper focuses on the issue of period autoagressive time series models (PAR) selection in practice. One aspect of model selection is the choice for the appropriate PAR order. This can be of interest for the valuation of economic models. Further, the appropriate PAR order is important for an adequate empirical application of tests for unit roots since too many parameters affect the performance of such tests. In fact, another aspect of PAR model selection is the decision on the number of unit roots. Finally, in case of unit roots, model choice involves a decision on the most suitable differencing filter to ensure stationarity of the transformed series.

Semiparametric Tests for Double Unit Roots

In this article, I argue that if an explosive root is considered a serious alternative to an 1(2) process, then joint testing for the number of unit roots along the lines of Hasza and Fuller is preferable instead of prior differencing as suggested by Dickey and Pantula. A semiparametric equivalent of the Hasza–Fuller test that permits rather general assumptions about the innovation process is developed. The test is a straightforward generalization of the Phillips Z test for a single unit root. Through Monte Carlo simulations, the different tests are compared, and the article is completed by an empirical application examining the purchasing power parity for several Latin American countries for which both explosive and double unit roots may be a possibility.

A Two-Step Procedure for Estimating Linear Simultaneous Equations with Unit Roots

A two-step procedure for estimating linear simultaneous structural equations with unit roots is presented. It generalizes the procedure of differencing for univariate time series via G. Box and J. M. Jenkins (1970). First, one finds the number of unit roots and the canonical variables that are stationary. Second, one retains only the stationary canonical variables and estimates a stationary model by standard methods. The procedure is easy to use. It is robus t against the difficult testing problem of finding the correct number of unit roots. A multiplier-accelerator model is estimated with interesting conclusions. Copyright 1993 by MIT Press.