Unit root tests allowing for a break in the trend function at an unknown time under both the null and alternative hypotheses

Perron [Perron, P., 1989. The great crash, the oil price shock and the unit root hypothesis. Econometrica 57, 1361–1401] introduced a variety of unit root tests that are valid when a break in the trend function of a time series is present. The motivation was to devise testing procedures that were invariant to the magnitude of the shift in level and/or slope. In particular, if a change is present it is allowed under both the null and alternative hypotheses. This analysis was carried under the assumption of a known break date. The subsequent literature aimed to devise testing procedures valid in the case of an unknown break date. However, in doing so, most of the literature and, in particular the commonly used test of Zivot and Andrews [Zivot, E., Andrews, D.W.K., 1992. Further evidence on the great crash, the oil price shock and the unit root hypothesis. Journal of Business and Economic Statistics 10, 251–270], assumed that if a break occurs, it does so only under the alternative hypothesis of stationarity. This is undesirable since (a) it imposes an asymmetric treatment when allowing for a break, so that the test may reject when the noise is integrated but the trend is changing; (b) if a break is present, this information is not exploited to improve the power of the test. In this paper, we propose a testing procedure that addresses both issues. It allows a break under both the null and alternative hypotheses and, when a break is present, the limit distribution of the test is the same as in the case of a known break date, thereby allowing increased power while maintaining the correct size. Simulation experiments confirm that our procedure offers an improvement over commonly used methods in small samples.     

Testing for a break in trend when the order of integration is unknown

Harvey, Leybourne and Taylor [Harvey, D.I., Leybourne, S.J., Taylor, A.M.R. 2009. Simple, robust and powerful tests of the breaking trend hypothesis. Econometric Theory 25, 995–1029] develop a test for the presence of a broken linear trend at an unknown point in the sample whose size is asymptotically robust as to whether the (unknown) order of integration of the data is either zero or one. This test is not size controlled, however, when this order assumes fractional values; its asymptotic size can be either zero or one in such cases. In this paper we suggest a new test, based on a sup-Wald statistic, which is asymptotically size-robust across fractional values of the order of integration (including zero or one). We examine the asymptotic power of the test under a local trend break alternative. The finite sample properties of the test are also investigated.     

Unit root log periodogram regression

Log periodogram (LP) regression is shown to be consistent and to have a mixed normal limit distribution when the memory parameter d=1$d=1$. Gaussian errors are not required. The proof relies on a new result showing that asymptotically infinite collections of discrete Fourier transforms (dft's) of a short memory process at the fundamental frequencies in the vicinity of the origin can be treated as asymptotically independent normal variates, provided one does not include too many dft's in the collection.     

Confidence sets for the date of a break in level and trend when the order of integration is unknown

We propose methods for constructing confidence sets for the timing of a break in level and/or trend that have asymptotically correct coverage for both I(0) and I(1) processes. These are based on inverting a sequence of tests for the break location, evaluated across all possible break dates. We separately derive locally best invariant tests for the I(0) and I(1) cases; under their respective assumptions, the resulting confidence sets provide correct asymptotic coverage regardless of the magnitude of the break. We suggest use of a pre-test procedure to select between the I(0)- and I(1)-based confidence sets, and Monte Carlo evidence demonstrates that our recommended procedure achieves good finite sample properties in terms of coverage and length across both I(0) and I(1) environments. An application using US macroeconomic data is provided which further evinces the value of these procedures.     

Unit root testing via the stationary bootstrap

A nonparametric, residual-based stationary bootstrap procedure is proposed for unit root testing in a time series. The procedure generates a pseudoseries which mimics the original, but ensures the presence of a unit root. Unlike many others in the literature, the proposed test is valid for a wide class of weakly dependent processes and is not based on parametric assumptions on the data-generating process. Large sample theory is developed and asymptotic validity is shown via a bootstrap functional central limit theorem. The case of a least squares statistic is discussed in detail, including simulations to investigate the procedure's finite sample performance.     

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.     

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.     

Power of Tests for Unit Roots in the Presence of a Linear Trend*

Dickey and Fuller [Econometrica (1981) Vol. 49, pp. 1057–1072] suggested unit‐root tests for an autoregressive model with a linear trend conditional on an initial observation. TPower of tests for unit roots in the presence of a linear trendightly different model with a random initial value in which nuisance parameters can easily be eliminated by an invariant reduction of the model. We show that invariance arguments can also be used when comparing power within a conditional model. In the context of the conditional model, the Dickey–Fuller test is shown to be more stringent than a number of unit‐root tests motivated by models with random initial value. The power of the Dickey–Fuller test can be improved by making assumptions to the initial value. The practitioner therefore has to trade‐off robustness and power, as assumptions about initial values are hard to test, but can give more power.     

Asymptotic and bootstrap tests for linearity in a TAR-GARCH(1,1) model with a unit root

This paper derives the limiting distribution of the Lagrange Multiplier (LM) test for threshold nonlinearity in a TAR model with GARCH errors when one of the regimes contains a unit root. It is shown that the asymptotic distribution is nonstandard and depends on nuisance parameters that capture the degree of conditional heteroskedasticity and non-Gaussian nature of the process. We propose a bootstrap procedure for approximating the exact finite-sample distribution of the test for linearity and establish its asymptotic validity.     

Estimating a common deterministic time trend break in large panels with cross sectional dependence

This paper develops an estimation procedure for a common deterministic time trend break in large panels. The dependent variable in each equation consists of a deterministic trend and an error term. The deterministic trend is subject to a change in the intercept, slope or both, and the break date is common for all equations. The estimation method is simply minimizing the sum of squared residuals for all possible break dates. Both serial and cross sectional correlations are important factors that decide the rate of convergence and the limiting distribution of the break date estimate. The rate of convergence is faster when the errors are stationary than when they have a unit root. When there is no cross sectional dependence among the errors, the rate of convergence depends on the number of equations and thus is faster than the univariate case. When the errors have a common factor structure with factor loadings correlated with the intercept and slope change parameters, the rate of convergence does not depend on the number of equations and thus reduces to the univariate case. The limiting distribution of the break date estimate is also provided. Some Monte Carlo experiments are performed to assess the adequacy of the asymptotic results. A brief empirical example using the US GDP price index is offered.     

Panel unit root tests in the presence of a multifactor error structure

This paper extends the cross-sectionally augmented panel unit root test (CIPS$CIPS$) proposed by Pesaran (2007) to the case of a multifactor error structure, and proposes a new panel unit root test based on a simple average of cross-sectionally augmented Sargan–Bhargava statistics (CSB$CSB$). The basic idea is to exploit information regarding the m$m$ unobserved factors that are shared by k$k$ observed time series in addition to the series under consideration. Initially, we develop the tests assuming that m⁰$m^0$, the true number of factors, is known and show that the limit distribution of the tests does not depend on any nuisance parameters, so long as k≥m⁰−1$k\ge m^0-1$. Small sample properties of the tests are investigated by Monte Carlo experiments and are shown to be satisfactory. Particularly, the proposed CIPS$CIPS$ and CSB$CSB$ tests have the correct size for all   combinations of the cross section (N$N$) and time series (T$T$) dimensions considered. The power of both tests rises with N$N$ and T$T$, although the CSB$CSB$ test performs better than the CIPS$CIPS$ test for smaller sample sizes. The various testing procedures are illustrated with empirical applications to real interest rates and real equity prices across countries.     

A simple,robust and powerful test of the trend hypothesis

In this paper we develop a simple test procedure for a linear trend which does not require knowledge of the form of serial correlation in the data, is robust to strong serial correlation, and has a standard normal limiting null distribution under either I(0)$I(0)$ or I(1)$I(1)$ shocks. In contrast to other available robust linear trend tests, our proposed test achieves the Gaussian asymptotic local power envelope in both the I(0)$I(0)$ and I(1)$I(1)$ cases. For near-I(1)$I(1)$ errors our proposed procedure is conservative and a modification for this situation is suggested. An estimator of the trend parameter, together with an associated confidence interval, which is asymptotically efficient, again regardless of whether the shocks are I(0)$I(0)$ or I(1)$I(1)$, is also provided.     

Alternative bootstrap procedures for testing cointegration in fractionally integrated processes

This paper considers alternative methods of testing cointegration in fractionally integrated processes, using the bootstrap. The investigation focuses on (a) choice of statistic, (b) use of bias correction techniques, and (c) designing the simulation of the null hypothesis. Three residual-based tests are considered, two of the null hypothesis of non-cointegration, the third of the null hypothesis that cointegration exists. The tests are compared in Monte Carlo experiments to throw light on the relative roles of issues (a)–(c) in test performance.     

Testing for unit roots in the presence of uncertainty over both the trend and initial condition

In this paper we provide a joint treatment of two major problems that surround testing for a unit root in practice: uncertainty as to whether or not a linear deterministic trend is present in the data, and uncertainty as to whether the initial condition of the process is (asymptotically) negligible or not. We suggest decision rules based on the union of rejections of four standard unit root tests (OLS and quasi-differenced demeaned and detrended ADF unit root tests), along with information regarding the magnitude of the trend and initial condition, to allow simultaneously for both trend and initial condition uncertainty.     

Exact local Whittle estimation of fractionally cointegrated systems

Semiparametric estimation of a bivariate fractionally cointegrated system is considered. We propose a two-step procedure that accommodates both (asymptotically) stationary (δ<1/2)$(\delta <1/2)$ and nonstationary (δ≥1/2)$(\delta \ge 1/2)$ stochastic trend and/or equilibrium error. A tapered version of the local Whittle estimator of Robinson (2008) is used as the first-stage estimator, and the second-stage estimator employs the exact local Whittle approach of Shimotsu and Phillips (2005). The consistency and asymptotic distribution of the two-step estimator are derived. The estimator of the memory parameters has the same Gaussian asymptotic distribution in both the stationary and the nonstationary case. The convergence rate and the asymptotic distribution of the estimator of the cointegrating vector are affected by the difference between the memory parameters. Further, the estimator has a Gaussian asymptotic distribution when the difference between the memory parameters is less than 1/2.     

A class of simple distribution-free rank-based unit root tests

We propose a class of distribution-free rank-based tests for the null hypothesis of a unit root. This class is indexed by the choice of a reference densityg, which need not coincide with the unknown actual innovation density f. The validity of these tests, in terms of exact finite-sample size, is guaranteed, irrespective of the actual underlying density, by distribution-freeness. Those tests are locally and asymptotically optimal under a particular asymptotic scheme, for which we provide a complete analysis of asymptotic relative efficiencies. Rather than stressing asymptotic optimality, however, we emphasize finite-sample performances, which also depend, quite heavily, on initial values. It appears that our rank-based tests significantly outperform the traditional Dickey-Fuller tests, as well as the more recent procedures proposed by Elliott et al. (1996), Ng and Perron (2001), and Elliott and Müller (2006), for a broad range of initial values and for heavy-tailed innovation densities. Thus, they provide a useful complement to existing techniques.     

Estimation and testing of Euler equation models with time-varying reduced-form coefficients

This paper studies the estimation and testing of Euler equation models in the framework of the classical two-step minimum-distance method. The time-varying reduced-form model in the first step reflects the adaptation of private agents’ beliefs to the changing economic environment. The presumed ability of Euler conditions to deliver stable parameters indexing tastes and technology is interpreted as a time-invariant second-step model. This paper shows that, complementary to and independent of one another, both standard specification test and stability test are required for the evaluation of an Euler equation. As an empirical application, a widely used investment Euler equation is submitted to examination. The empirical outcomes appear to suggest that the standard investment model has not been a success for aggregate investment.