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.     

Break Point Estimation and Spurious Rejections With Endogenous Unit Root Tests

This paper examines the accuracy of break point estimation using the endogenous break unit root tests of Zivot and Andrews (1992) and Perron (1997). We find that these tests tend to identify the break point incorrectly at one-period behind ( T_B -1) the true break point ( T_B ), where bias in estimating the persistence parameter and spurious rejections are the greatest. In addition, this outcome occurs under the null and alternative hypotheses, and more so as the magnitude of the break increases. Consequences of utilizing these endogenous break tests are similar to (incorrectly) omitting the break term B_t in Perron's (1989) exogenous test.     

Testing for unit roots in the possible presence of multiple trend breaks using minimum Dickey–Fuller statistics

Trend breaks appear to be prevalent in macroeconomic time series, and unit root tests therefore need to make allowance for these if they are to avoid the serious effects that unmodelled trend breaks have on power. Carrion-i-Silvestre et al. (2009) propose a pre-test-based approach which delivers near asymptotically efficient unit root inference both when breaks do not occur and where multiple breaks occur, provided the break magnitudes are fixed. Unfortunately, however, the fixed magnitude trend break asymptotic theory does not predict well the finite sample power functions of these tests, and power can be very low for the magnitudes of trend breaks typically observed in practice. In response to this problem we propose a unit root test that allows for multiple breaks in trend, obtained by taking the infimum of the sequence (across all candidate break points in a trimmed range) of local GLS detrended augmented Dickey–Fuller-type statistics. We show that this procedure has power that is robust to the magnitude of any trend breaks, thereby retaining good finite sample power in the presence of plausibly-sized breaks. We also demonstrate that, unlike the OLS detrended infimum tests of Zivot and Andrews (1992), these tests display no tendency to spuriously reject in the limit when fixed magnitude trend breaks occur under the unit root null.     

Spurious Rejections by Perron Tests in the Presence of a Break

In this paper, we concentrate on the case of an exogeneously chosen break date, but entertain the possibility that an incorrect choice is made. In fact, the Perron test statistics considered are invariant to any break in the generating process at the assumed break date. Our results therefore apply equally to the case of a generating process with two breaks, only one of which is specifically accounted for in the analysis. As in Leybourne et al . (1998), we find that a neglected relatively early break can lead to spurious rejections of the unit root null hypothesis. Moreover, for all but one of the tests analyzed, spurious rejections now also arise if a true break occurs relatively soon after the assumed break date.     

Further evidence on breaking trend functions in macroeconomic variables

This study first reexamines the findings of Perron (1989) regarding the claim that most macroeconomic time series are best construed as stationary fluctuations around a deterministic trend function if allowance is made for the possibility of a shift in the intercept of the trend function in 1929 (a crash) and a shift in slope in 1973 (a slowdown in growth). Unlike that previous study, the date of possible change is not fixed a priori but is considered as unknown. We consider various methods to select the break points and the asymptotic and finite sample distributions of the corresponding statistics. A detailed discussion about the choice of the truncation lag parameter in the autoregression and of its effect on the critical values is also included. Most of the rejections reported in Perron (1989) are confirmed using this approach. Secondly, this paper investigates an international data set of post-war quarterly real GNP (or GDP) series for the G-7 countries. Our results are compared and contrasted to those of Banerjee et al. (1992) and Zivot and Andrews (1992). In contrast to the theoretical results contained in these papers, we derive the limiting distribution of the sequential test without trimming.     

Price-wage dynamics in a transition economy: The case of Poland

In this paper we analyse the wage-price relationship of an economy in transition characterized by important structural changes. It is known (see Perron, 1989) that structural breaks in stationary time series can induce apparent unit roots. The stationarity analysis of the series employed in the present model is conducted jointly with the assumption that the breakpoint location is unknown. We follow a testing procedure recently proposed by Zivot and Andrews (1992). Cointegration analysis of wages and prices in the presence of structural breaks finds empirical evidence in favour of two cointegrating vectors involving prices and wages. Our analysis focuses on the different structural behaviour of the price-wage dynamic relationship in the short and long term; we also demonstrate the relative importance of import prices as a source of wage-price fluctuations.     

带结构突变的确定趋势的单位根检验方法——兼对中国工农业产品比价的实证分析

存在未知结构突变点的DGP关于结构突变的信息是不完全的,理论上不可能得到媲美标准ADF检验或Perron检验的统计特性,且Perron检验比标准ADF检验更适合于作为其单位根统计量的极限分布。本文以Perron(2009)方法为基础,结合minSSR和FGLS详细讨论结构突变的单位根检验的三个步骤,澄清了很多模糊的认识。以我国工农业产品比价和工农业产品价格指数为例,本文的实证研究发现,中国工农业利益分配格局的重大改变是由国际经济事件造成的冲击引起的。     

考虑结构突变的单位根检验程序研究——基于“新息异常值模型”的Perron检验分析

Perron检验是一种考虑结构突变的单位根检验方法,检验统计量的分布依赖于数据生成过程中所包含的确定性趋势和所选取的检验回归式;而在实证分析中真实的数据生成过程是未知的,这使得单位根检验缺乏必要依据,因而探寻科学有效的单位根检验程序是受到广泛关注的问题。基于此,本文在"IO模型"分析框架下,依据Perron检验提出了一套考虑结构突变的单位根检验程序,并通过蒙特卡洛模拟分析了该程序在有限样本情形下的表现。本研究完善了带有结构突变的单位根检验理论,为实证分析提供了有益的建议和参考。     

International evidence on persistence in output in the presence of an episodic change

This paper uses some newly developed methods and techniques to examine the dynamic properties of international output in the presence of a structural break. We provide statistical evidence to show that the unit root test results can, in some cases, be sensitive to whether a one-time structural break in the data is modelled exogenously or endogenously. However, in most cases the unit root test results remain robust to specification of the structural break exogenously or endogenously; moreover, we find that the null hypothesis of a unit root in output can be rejected in favour of a ‘flexible’ trend alternative for a number of countries such as Canada, Denmark, France, and the United States.     

Innovational Outlier Unit Root Tests With an Endogenously Determined Break in Level

We show that a standard unit root test that permits an endogenously determined break in level can generate spurious rejections in practically interesting sample sizes when a large break occurs under the null hypothesis. This problem, which occurs for breaks of the innovational outlier type, can be corrected through a simple modification of the test procedure.     

Tests of cointegrating rank with a trend-break

The conventional testing procedure may mislead one into accepting the null of no cointegration or the null of a cointegrating rank smaller than the true rank when there is a trend-break under the alternative hypothesis. This paper proposes tests for cointegrating rank that have power against the trend-break alternative. The proposed tests are applied to the US money demand function. The results support the Campbell–Perron conjecture: money, income and interest rates are cointegrated around a broken trend.     

Evidence on purchasing power parity from univariate models: the case of smooth transition trend‐stationarity

Recent research has found that trend‐break unit root tests derived from univariate linear models do not support the hypothesis of long‐run purchasing power parity (PPP) for US dollar real exchange rates. In this paper univariate smooth transition models are utilized to develop unit root tests that allow under the alternative hypothesis for stationarity around a gradually changing deterministic trend function. These tests reveal statistically significant evidence against the null hypothesis of a unit root for the real exchange rates of a number of countries against the US dollar. However, restrictions consistent with long‐run PPP are rejected for some of the countries for which a rejection of the unit root hypothesis is obtained. Copyright © 2005 John Wiley & Sons, Ltd.     

Confidence sets for the date of a single break in linear time series regressions

This paper considers the problem of constructing confidence sets for the date of a single break in a linear time series regression. We establish analytically and by small sample simulation that the current standard method in econometrics for constructing such confidence intervals has a coverage rate far below nominal levels when breaks are of moderate magnitude. Given that breaks of moderate magnitude are a theoretically and empirically relevant phenomenon, we proceed to develop an appropriate alternative. We suggest constructing confidence sets by inverting a sequence of tests. Each of the tests maintains a specific break date under the null hypothesis, and rejects when a break occurs elsewhere. By inverting a certain variant of a locally best invariant test, we ensure that the asymptotic critical value does not depend on the maintained break date. A valid confidence set can hence be obtained by assessing which of the sequence of test statistics exceeds a single number.     

Unit Root Testing under a Local Break in Trend using Partial Information on the Break Date*

We consider unit root testing allowing for a break in trend when partial information is available regarding the location of the break date. This takes the form of knowledge of a relatively narrow window of data within which the break takes place, should it occur at all. For such circumstances, we suggest employing a union of rejections strategy, which combines a unit root test that allows for a trend break somewhere within the window with a unit root test that makes no allowance for a trend break. Asymptotic and finite sample evidence shows that our suggested strategy works well, provided that, when a break does occur, the partial information is correct. An empirical application to UK interest rate data containing the 1973 ‘oil shock’ is also considered.     

Tests for Stationarity in Series with Endogenously Determined Structural Change*

We consider tests of the null hypothesis of stationarity against a unit root alternative, when the series is subject to structural change at an unknown point in time. Three extant tests are reviewed which allow for an endogenously determined instantaneous structural break, and a related fourth procedure is introduced. We further propose tests which permit the structural change to be gradual rather than instantaneous, allowing the null hypothesis to be stationarity about a smooth transition in linear trend. The size and power properties of the tests are investigated, and the tests are applied to four economic time series.     

Testing parameter constancy in linear models against stochastic stationary parameters

This paper considers testing parameter constancy in a linear model when the alternative is that a subset of the parameters follows a stationary vector autoregressive process of known finite order. This kind of a linear model is only identified under the alternative, which usually precludes finding a test statistic with an analytic null distribution. In the present situation, however, it is still possible to derive a test statistic with an asymptotic chi-squared distribution under the null hypothesis and this is done in the paper. The small-sample properties of the test statistic are investigated by simulation and found statisfactory. The test retains its power when the alternative to parameter constancy is a random walk parameter process.     

Armax model specification testing,with an application to unemployment in the Netherlands

In this paper we present a consistent standard normal model specification test for ARMAX models. The null hypothesis is that the ARMAX model represents the conditional expectation of the dependent variable relative to the entire past of the economic vector time series process under review. This null is tested against the general alternative hypothesis that the null is false. The test is applied to testing the rational expectations-natural rate (RE-NR) hypothesis for the Netherlands according to the approach of Sargent (1976). It appears that RE-NR hypothesis has to be rejected.     

Dickey–Fuller Type of Tests against Nonlinear Dynamic Models*

In this paper, we introduce several test statistics testing the null hypothesis of a random walk (with or without drift) against models that accommodate a smooth nonlinear shift in the level, the dynamic structure and the trend. We derive analytical limiting distributions for all the tests. The power performance of the tests is compared with that of the unit‐root tests by Phillips and Perron [Biometrika (1988), Vol. 75, pp. 335–346], and Leybourne, Newbold and Vougas [Journal of Time Series Analysis (1998), Vol. 19, pp. 83–97]. In the presence of a gradual change in the deterministics and in the dynamics, our tests are superior in terms of power.     

Detection of Structural Change in the Long‐run Persistence in a Univariate Time Series*

In this paper, we investigate a test for structural change in the long‐run persistence in a univariate time series. Our model has a unit root with no structural change under the null hypothesis, while under the alternative it changes from a unit‐root process to a stationary one or vice versa. We propose a Lagrange multiplier‐type test, a test with the quasi‐differencing method, and ‘demeaned versions’ of these tests. We find that the demeaned versions of these tests have better finite‐sample properties, although they are not necessarily superior in asymptotics to the other tests.     

Income convergence within the MENA countries: A panel unit root approach

This article aims at testing the convergence hypothesis in MENA region using new tests of a unit root in panel data. Evans and Karras [Evans P., & Karras G. (1996). Convergence revisited. Journal of Monetary Economics, 37, 249–265] and Bernard and Jones [Bernard A., & Jones C. I. (1996). Productivity across industries and countries: Time series theory and evidence. The Review of Economics and Statistics, 135–146] recommend this technique to evaluate the income convergence hypothesis. According to them it avoids econometric problems of the cross-countries growth regressions testing convergence and sample bias of the multivariate cointegration techniques. We test for both absolute and the conditional convergence with panel unit root tests using the Summers and Heston's data 5.6 and 6.1 on the periods of 1960 to 1990 and from 1960 to 2000. The absolute convergence hypothesis use panel unit roots test with no fixed individual effects. The catching-up hypothesis is not rejected for most groups of countries of the region during both periods. If we allow a break in the unit root tests, the hypothesis is not rejected for more groups. The conditional convergence requires panel unit root tests with fixed individual effects. Again, during the whole periods, the conditional convergence is not rejected for the major part of the remaining groups of MENA countries.