Trend Stationary Research Articles

Panel Data Unit Root Tests with an Application

Using several different unit root/stationarity tests on single time series Konya (2000) found the logarithm of real GDP of most OECD countries behaving as a random walk during the last four decades. This outcome, however, might be due to the generally low power of these tests. The aim of this paper is to reconsider this issue by exploiting the extra information provided by the combination of the time-series and cross-sectional data and the subsequent power advantages of panel data unit root tests. We apply the tests advocated by Levin and Lin (1993), Im, Pesaran and Shin (1997) and Maddala and Wu (1999). The joint unit root null hypothesis cannot be rejected for the whole panel, however, after having dropped the least likely stationary series from the panel, the Im, Pesaran and Shin (1997) and Maddala and Wu (1999) tests can reject the null for the remaining sub-panels. Since these tests are not valid under cross-correlated error terms, we repeat them with data specific bootstrap critical values taking contemporaneous cross-correlation into account. This way, both tests can reject the unit root null hypothesis even for the whole panel. Yet, they cannot suggest how many and which particular panel members are stationary, so these results are not really informative. For this reason finally we apply the unit root test of Breuer, McNown and Wallace (1999) with bootstrap critical values. This procedure makes use of the panel data setting and seemingly unrelated regressions, but performs separate unit-root tests on each panel member. The results are markedly different from the conventional univariate Dickey-Fuller test results, and they support trend stationarity in the case of four countries: Australia, Japan, the Netherlands and Switzerland.

Trend und Zyklus im Bruttoinlandsprodukt der Bundesrepublik Deutschland. Eine Anmerkung / Trends and Cycles in Germany’s Real Gross Domestic Product. A Note

Zusammenfassung Diese Anmerkung zeigt, dass das reale Bruttoinlandsprodukt der Bundesrepublik Deutschland einem trendstationären Prozess folgt. Dabei werden sowohl ökonometrische Tests, bei denen die Trendstationarität die Alternativhypothese ist, eingesetzt als auch solche, bei denen sie die Nullhypothese darstellt. Beide Verfahren präferieren das trendstationäre Modell. Eine explizite Berücksichtigung von Brüchen im Trend ist für dieses Ergebnis nicht notwendig. Für die praktische Trend-Zyklus-Zerlegung im Rahmen der Konjunkturforschung hat ein Modell mit Trendbrüchen gleichwohl Vorteile.

Are Net Discount Ratios Stationary? Evidence of Mean-Reversion and Persistence

This article extends the debate initiated by Haslag, Nieswiadomy, and Slottje (1991, 1994) and Gamber and Sorensen (1994) in this journal about whether the net discount ratio can be described as a stationary process. Haslag, Nieswiadomy, and Slottje found discount ratios to be stationary. Gamber and Sorensen concluded that they are nonstationary; however, they identified the source of the nonstationarity as a single shift in the mean of the series. Using the Cochrane variance ratio and Campbell-Mankiw decomposition tests, the authors find that the net discount ratio follows a trend stationary process. However, to determine the degree or level of persistence, Lo's Modified R/S Analysis (1991) is used. The authors find that the relative importance of any mean shift is a function of the duration of the discount period for expected earnings.

Testing for integration using evolving trend and seasonals models: A Bayesian approach

In this paper, we make use of state space models to investigate the presence of stochastic trends in economic time series. A model is specified where such a trend can enter either in the autoregressive representation or in a separate state equation. Tests based on the former are analogous to Dickey–Fuller tests of unit roots, while the latter are analogous to KPSS tests of trend stationarity. We use Bayesian methods to survey the properties of the likelihood function in such models and to calculate posterior odds ratios comparing models with and without stochastic trends. We extend these ideas to the problem of testing for integration at seasonal frequencies and show how our techniques can be used to carry out Bayesian variants of either the HEGY or Canova–Hansen test. Stochastic integration rules, based on Markov Chain Monte Carlo, as well as deterministic integration rules are used. Strengths and weaknesses of each approach are indicated.

The uncertain trend in U.S. GDP

Several recent papers conclude that GDP is trend stationary, implying that all shocks are transitory. We re-examine the evidence in light of test size distortion due to data-based lag selection and departures from the maintained hypothesis of temporal homogeneity, and find both effects trigger false rejections of the unit root hypothesis when it is true and signal the presence of permanent shifts in trend that did not occur. Trend stationarity is not supported by the more homogeneous post-war data, but if imposed implies cycles of implausible duration and pattern – 1997 output was 8% below trend.

On stationarity and cointegration of international health expenditure and GDP

This paper examines stationarity and cointegration of health expenditure and GDP, for a sample of 21 OECD countries using data for the period 1960–1997, by applying a test battery that allows robust inference to be made on the stationarity and cointegration issue. Trend stationarity and no-cointegration are tested using new country-by-country and panel tests, not previously applied in this setting. New results for country-by-country and panel tests of non-stationarity and cointegration are presented. Our unit root and trend stationarity results indicate that both health expenditure and GDP are non-stationary. The no-cointegration and cointegration results indicate that health expenditure and GDP are cointegrated.

OECD unemployment: structural breaks and stationarity

The paper is concerned with testing the unemployment rate of twenty two OECD countries for stationarity. A sequential testing procedure was applied where the break data is endogenized. Three different models were tested for unit roots. It was found that the ‘crash’ model, which allows for a shift in the level of the unemployment rate, was most relevant. Furthermore, most breaks were associated with the first oil price shock. Results suggest that in nine countries the unit root can be rejected, in ten countries the null hypothesis cannot be rejected and in three cases the results suggest possible trend stationarity.

LM tests for unit roots in the presence of missing observations: small sample evidence

The purpose of this paper is to derive the asymptotic distributions of some Lagrange Multiplier (LM) tests for unit roots in time series models in the presence of missing observations, and to provide evidence on the small sample properties of these tests. LM tests for a unit root in a first-order autoregressive process for two types of null and alternative hypotheses are considered: a unit root without drift versus level stationarity, and a unit root with drift versus trend stationarity. Modifications of the tests to account for serially correlated errors are suggested. The small sample size and power properties of the tests are investigated using a Monte Carlo simulation.

THE U.S. PRODUCTIVITY SLOWDOWN: A PEAK THROUGH THE STRUCTURAL BREAK WINDOW

This paper provides a formal test of the null hypothesis of a unit root in the log‐level of labor productivity against the alternative of linear trend stationarity with a one‐time structural break in the level and slope of the trend at an a priori unknown date. Using some newly developed time series tests, we show that the log‐level of productivity is more accurately modeled as following a deterministic trend with a regime shift rather than as a unit root process. Some implications of the results for detrending and for testing cointegration relationships between productivity and other variables are discussed. (JEL C22. O40)

Trend stationarity in the I(2) cointegration model

A vector autoregressive model for I(2) processes which allows for trend-stationary components and restricts the deterministic part of the process to be at most linear is defined. A two-step statistical analysis of the model is derived. The joint test of I(1) and I(2) cointegrating ranks is shown to be asymptotically similar with respect to the drift terms and the asymptotic distribution is tabulated. The cointegrating parameters are shown to be mixed Gaussian and an application for UK monetary data illustrates the proposed analysis.

Monte Carlo inference in econometric models with symmetric stable disturbances

The paper develops Markov Chain Monte Carlo methods to perform exact posterior analysis in models with symmetric stable Paretian disturbances. It is shown how posterior moments and marginal densities of functions of parameters can be computed methodically by combining a Gibbs sampler with Metropolis independence chains. The new method is shown to perform satisfactorily in constructed examples. The method is also applied to a set of monthly real exchange rates and the question of difference versus trend stationarity is taken up jointly with the problem of inference about the parameter of the stable distribution.

On trends and constants in periodic autoregressions

Periodic autoregressions are characterised by autoregressive structures that vary with the season. If a time series is periodically integrated, one needs a seasonally varying differencing filter to remove the stochastic trend. When the periodic regression model contains constants and trends with unrestricted parameters, the data can show diverging seasonal deterministic trends. In this paper we derive explicit expressions for parameter restrictions that result in common deterministic trends under periodic trend stationarity and periodic integration.

A Primer on Unit Root Testing

The immense literature and diversity of unit root tests can at times be confusing even to the specialist and presents a truly daunting prospect to the uninitiated. In consequence, much empirical work still makes use of the simplest testing procedures because it is unclear from the literature and from recent reviews which tests if any are superior. This paper presents a survey of unit root theory with an emphasis on testing principles and recent developments. The general framework adopted makes it possible to consider tests of stochastic trends against trend stationarity and trend breaks of a general type. The main tests are listed, and asymptotic distributions are given in a simple form that emphasizes commonalities in the theory. Some simulation results are reported, and an extensive list of references and all annotated bibliography are provided.

Granger Causality Between Money and Income for the Japanese Economy in the Presence of a Structural Change

This paper examines Granger causality between money and income in the Japanese economy based upon a bivariate VAR model with a structural change in the trend function. We employ a stratified testing strategy incorporating preliminary tests for a unit root and for the order of cointegration rank. Our study reveals that the choice of either trend stationarity or difference stationarity, as well as the order of cointegration rank, crucially affect the test results for Granger causality. It is found that the causality from money to income was strong before 1980 but weakened or virtually disappeared after 1980; the opposite causality existed weakly before 1980 but not after 1980. The result confirms the claim by the Bank of Japan (1992) and Honda et al. (1995) among others that the role of money as a leading indicator for predicting movements in income has weakened or even disappeared in the 1980s. JEL Classification Numbers: C32, E40

Economic growth and stationarity of real exchange rates: Evidence from some fast-growing Asian countries

The possible implications of economic growth for the trending behavior of real exchange rates are explored. Given that prior studies consider mainly industrial countries, an issue concerns whether parity reversion holds for countries with sharply different growth experiences. For countries undergoing dramatic income growth from a low level, substantial changes in the relative price structure between tradables and nontradables can often occur. An implication is that real exchange rates for these countries are likely afflicted by trend shifts, which should bear on empirical testing for unit-root nonstationarity. This study analyzes the trend stationarity property of dollar-based real exchange rates for several fast-growing Asian countries. Considerable evidence is found to support the trend-shift hypothesis and reject the unit-root hypothesis.

Dickey–Fuller, Lagrange Multiplier and Combined Tests for a Unit Root in Autoregressive Time Series

In this paper we investigate (augmented) Dickey–Fuller (DF) and Lagrange multiplier (LM) type unit root tests for autoregressive time series through comprehensive Monte Carlo simulations. We consider two sorts of null and alternative hypotheses: a unit root without drift versus level stationarity and a unit root with drift versus trend stationarity. The DF‐type coef ficient tests are found to show the best overall performance in both cases, at least if the sample size is sufficiently large. How ever, it is also found that the DF and LM tests are roughly complementary with regard to their finite‐sample power. We therefore consider combining these two types of unit root tests to obtain (ad hoc‘but’) ‘robust’ test procedures. Critical values for the proposed tests are provided

Transitory shock or structural shift? The impact of the early 1980s recession on British regional unemployment

This paper examines the impact of the early 1980s recession on regional unemployment in Britain. More specifically, it seeks to evaluate the hypothesis that this recessionary shock was so severe that it caused an upward structural shift in the underlying mean unemployment rates of the regions. This proposition is analysed using Dickey — Fuller tests for difference versus trend stationarity, and the augmented structural shift and trend break time-series models developed by Perron. The results suggest that the apparent stochastic non-stationarity in regional unemployment over the period 1965 – 95 is in fact due to an upward structural shift in regional mean unemployment rates in 1980. Furthermore, the nature of this shift appears to have differed as between the northern and southern regions of the country.

F-tests for seasonal differencing with a break-point

The classification between stochastic trend stationarity and deterministic broken trend stationarity is important because incorrect inferences can follow if a stationary series with a broken trend is incorrectly classified as integrated. In this paper, we consider joint tests for regular and seasonal unit roots null hypothesis against broken trend stationarity alternatives where the location of the break is known or unknown. Based on the F-test proposed by Hasza and Fuller (1982, Ann. Statist. 10, 1209–1216), we develop testing procedures for distinguishing these two types of process. The asymptotic distributions of test statistics are derived as functions of Wiener processes. A response surface regression analysis directed to relating the finite sample distributions and the breaking position is studied. Simulation experiments suggest that the power of the test is reasonable. The testing procedure is illustrated by the Canadian consumer price index series.

Testing the unit root with drift hypothesis against nonlinear trend stationarity, with an application to the US price level and interest rate

This paper is concerned with testing the unit root with drift hypothesis against a very general trend stationarity hypothesis, namely the alternative that the time series is stationary about an almost arbitrary deterministic function of time. Our approach employs the fact that any function of time can be approximated arbitrarily close by a linear function of Chebishev polynomials. We propose various tests on the basis of an Augmented Dickey-Fuller auxiliary regression with linear and nonlinear deterministic trends, where the nonlinear deterministic trend is approximated by detrended Chebishev time polynomials. Also, we propose a model-free test. We apply our tests to the GNP deflator, the consumer price index, and the interest rate for the USA, taken from the extended Nelson-Plosser data set. The results indicate that these series are nonlinear trend stationary.

Testing for Unit Roots with Breaks: Evidence on the Great Crash and the Unit Root Hypothesis Reconsidered

Nelson and Plosser (1982), in a classic paper, failed to find strong evidence against the null hypothesis of a generating process with a unit autoregressive root for thirteen US macroeconomic time series. Perron (1989) claimed that such evidence was available for a majority of these series if the alternative hypothesis was of trend stationarity with a break in 1929. Zivot and Andrews (1992) treated the break date as endogenous, then finding strong evidence agcainst the null for a minority of these series. Our own analysis extends theirs by permitting a break under the null as well as the alternative hypothesis, and allowing for the sequential nature of the testing. Our empirical findings complete the circle. We find no strong evidence against the unit root hypothesis for any of the thirteen Nelson–Plosser series.