Standard Dickey-Fuller Test Research Articles

Behavior of the standard Dickey–Fuller test when there is a Fourier-form break under the null hypothesis

We derive the null asymptotic distribution of the standard Dickey–Fuller test with the existence of an unnoticed Fourier component. The so-called converse Perron phenomenon might occur, but only in the trend-case with a low-frequency Fourier component and small error variance.

UNIT ROOT TESTS IN THE PRESENCE OF MULTIPLE BREAKS IN VARIANCE

Spurious rejections of the standard Dickey–Fuller (DF) test caused by a single variance break have been reported and some solutions to correct the problem have been proposed in the literature. Kim et al. (2002) put forward a correctly-sized unit root test robust to a single variance break, called the KLN test. However, there can be more than one break in variance in time series data as documented in Zhou and Perron (2008), so allowing only one break can be too restrictive. In this paper, we show that multiple breaks in variance can generate spurious rejections not only by the standard DF test but also by the KLN test. We then propose a bootstrap-based unit root test that is correctly-sized in the presence of multiple breaks in variance. Simulation experiments demonstrate that the proposed test performs well regardless of the number of breaks and the location of the breaks in innovation variance.

Cointegration of the prices of gold and silver: RALS-based evidence

Much significant research has been done to study the cointegration link between the prices of gold and silver. Empirical results reported in earlier research suggest that the strength of the cointegration link changed over time, possibly reflecting the impact of bubble-like periods and financial crises. Against this background, a Residual Augmented Least Squares (RALS) test for noncointegration was used to study cointegration of the prices of gold and silver. Application of the RALS test yielded stronger evidence against noncointegration than was obtained based on a standard Dickey–Fuller test, but a cointegration link did not exist during extended periods of time.

Reply to Golan and Rosenblatt: Revisiting the statistical analysis of the Israeli–Palestinian conflict confirms that both sides retaliate

We thank Golan and Rosenblatt (1) for their comments on our analysis (2), which we address in turn. First, Golan and Rosenblatt suggest that the time series may be nonstationary (that is, the distribution of the variables in the data may change over time). Our original paper addressed this concern in two ways. First, we tested for stationarity using the standard Dickey Fuller test. The results (table S2 in ref. 2) supported the stationarity hypothesis. Second, we included year dummy variables to control for structural breaks. We chose years rather than political events for these dummy variables to avoid the arbitrariness inherent in the latter approach. Thus, nonstationarity is not a concern for our conclusions. Nevertheless, Golan and Rosenblatt answer an interesting additional question by showing that the patterns of retaliation vary across subperiods.

The size performance of a nonparametric unit root test under a variance shift

This paper examines the size performance of Breitung's [2002. Nonparametric tests for unit roots and cointegration. J. Econometrics 108, 343–363.] nonparametric unit root test in the presence of a variance shift. We show that the limiting distribution of the test statistic in the presence of a variance shift depends on the break point of the variance and the ratio of the prebreak to the postbreak variance, as in the case of the standard Dickey–Fuller test. However, our Monte Carlo simulations provide clear evidence that Breitung's nonparametric unit root test achieves a far superior size performance as compared with the Dickey–Fuller test.

The performance of unit root tests under level-dependent heteroskedasticity

Several financial variables exhibit level-dependent conditional heteroskedasticity. This may cause severe distortions in conventional unit root tests. Given the absence of theoretical results, we conduct Monte Carlo investigation to assess the performance of the standard Dickey–Fuller tests, a nonparametric alternative, and a heteroskedastic-robust extension of the Dickey–Fuller t-test. While these procedures have approximately correct size, we find strong distortions in the power of the standard Dickey–Fuller tests.

A strategy for testing the unit root in AR(1) model with intercept: a Monte Carlo experiment

In this paper, we introduce a strategy for testing the unit root hypothesis in a first-order autoregressive process with an unknown intercept where the initial value of the variable is a known constant. In the context of this model the standard Dickey–Fuller test is non-similar, the intercept being the nuisance parameter. The testing strategy we propose takes into account this non-similarity. It is an unusual two-sided test of the random walk hypothesis since it involves two distributions where the acceptance region is constructed by taking away equal areas for the lower tail of the Student's t distribution and the upper tail of the distribution tabulated by Dickey and Fuller under the null hypothesis of unit root. In some cases, this strategy does not allow a direct decision to be taking regarding the existence of a unit root. To deal with these situations we suggest testing for the significance of the intercept, and if doubt persists, we use the Φ 1 test proposed by Dickey and Fuller (Econometrica 49 (1981) 1057). Finally, Monte Carlo simulations are used to demonstrate the relevance of non-similarity and to show that the testing strategy is more powerful at most stable alternatives and has less size distortions than the two-sided tests considered by Dickey and Fuller.

Further empirical analysis of the time series properties of financial ratios based on a panel data approach

A new panel unit root by Chang (Journal of Econometrics, 110, 261–92, 2002) is employed on a set of financial ratios with a view to improving the power of unit root tests when applied to a relatively small number of observations (in the present case 38 annual observations). The test is innovative in that it allows for cross-sectional dependencies and the asymptotic distribution of the test is standard. Although standard Dickey–Fuller tests suggest that individual financial ratio series are nonstationary, panel unit root tests strongly reject the null hypothesis of a joint unit root in the ratios. Taken together the evidence from the proposed new analysis implies strong persistence in the ratios but that their characterization as I(1) processes may be misleading. These findings have important implications for accounting and finance researchers who employ financial ratios as explanatory variables.

Modified unit root tests and momentum threshold autoregressive processes

The properties of modified Dickey–Fuller tests are examined when applied to momentum threshold autoregressive processes. The tests are found to possess high power, outperforming the standard Dickey–Fuller test and even the asymmetric momentum threshold autoregressive unit root test. The results suggest modifications made to increase the power of symmetric unit root tests could potentially be employed to increase the power of asymmetric unit root tests.

Nonlinear mean reversion in real exchange rates

This paper modifies a unit-root test procedure in the nonlinear STAR framework recently advanced by Kapetanios et al. [Journal of Econometrics (2001) in press]. Using a detrending methodology suggested by Schmidt and Phillips [Oxford Bulletin of Economics and Statistics 54 (1992) 257], we derive an alternative unit-root test and apply it to the bilateral real exchange rates for the G7 countries. We find that the use of our test is able to uncover evidence of nonlinear mean-reversion for most cases whereas the standard Dickey–Fuller test based on the linear model cannot.

Testing for a unit root in the nonlinear STAR framework

In this paper, we propose a simple testingprocedure to detect the presence of nonstationarity against nonlinear but globally stationary exponential smooth transition autoregressive processes. We provide an advance over the existingliterature in three senses. First, we derive the limiting nonstandard distribution of the proposed tests. Second, we 0nd via Monte Carlo simulation exercises that under the alternative of a globally stationary ESTAR process, our proposed test has better power than the standard Dickey–Fuller test, in the region of the null, where the processes are highly persistent. Third, we provide an application to ex post real interest rates and bilateral real exchange rates with the US Dollar from the 11 major OECD countries, and 0nd our test is able to reject a unit root in many cases, whereas the linear DF tests fail, providing some evidence of nonlinear mean-reversion in both real interest and exchange rates. c 2002 Elsevier Science B.V. All rights reserved. JEL classi)cation: C12; C32; E43; F31

Size and power properties of powerful unit root tests in the presence of variance breaks

The standard Dickey–Fuller (DF) test is routinely employed to analyse the integrated nature of economic and financial time series. However, recent research has shown the test to suffer severe size distortion in the presence of breaks in innovation variance under the unit root null hypothesis. In this paper, the properties of alternative, more powerful unit root tests are examined in such circumstances. Simulation experiments are undertaken to permit a comparison of the behaviour of the alternative tests and a recently proposed ‘variance break robust’ unit root test. The results derived across a range of breakpoints and break sizes show a marked difference in the size and power properties of the rival tests. It is found that over a range of breaks and sample sizes, weighted symmetric or recursively mean-adjusted unit root tests are to be preferred to both the standard DF test and the robust test.

Behaviour of the standard and symmetric Dickey-Fuller-type tests when there is a break under the null hypothesis

We examine the behaviour of tests of the unit root null hypothesis when the true generating model is I (1) with a break. Asymptotic distributional results are derived predicting that the standard Dickey–Fuller test will in that case yield frequent rejections of the null for relatively early breaks. This phenomenon is most severe for lowest values of the break fraction under a break in level, and for a break in slope occurring about 15% of the way through the series. However, asymptotics predict that the phenomenon will not be present for a modified test based on a symmetric weighted estimator. The theoretical predictions are confirmed by simulation evidence.

Spurious rejections by Dickey–Fuller tests in the presence of a break under the null

It is well known that if a series is generated by a process that is stationary around a broken trend, conventional Dickey–Fuller tests can have very low power. In this paper, the converse phenomenon is studied and illustrated. Suppose that the true generating process is integrated of order one, but with a break. Then it is shown that, if the break occurs early in the series, routine application of standard Dickey–Fuller tests can lead to a very serious problem of spurious rejection of the unit root null hypothesis.

Unit-Root Tests and Asymmetric Adjustment with an Example Using the Term Structure of Interest Rates

This article develops critical values to test the null hypothesis of a unit root against the alternative of stationarity with asymmetric adjustment. Specific attention is paid to threshold and momentum threshold autoregressive processes. The standard Dickey–Fuller tests emerge as a special case. Within a reasonable range of adjustment parameters, the power of the new tests is shown to be greater than that of the corresponding Dickey–Fuller test. The use of the tests is illustrated using the term structure of interest rates. It is shown that the movements toward the long-run equilibrium relationship are best estimated as an asymmetric process.

Unit-Root Tests and Asymmetric Adjustment With an Example Using the Term Structure of Interest Rates

This article develops critical values to test the null hypothesis of a unit root against the alternative of stationarity with asymmetric adjustment. Specific attention is paid to threshold and momentum threshold autoregressive processes. The standard Dickey–Fuller tests emerge as a special case. Within a reasonable range of adjustment parameters, the power of the new tests is shown to be greater than that of the corresponding Dickey–Fuller test. The use of the tests is illustrated using the term structure of interest rates. It is shown that the movements toward the long-run equilibrium relationship are best estimated as an asymmetric process.

TESTING FOR UNIT ROOTS USING FORWARD AND REVERSE DICKEY‐FULLER REGRESSIONS

This article suggests a simple test for the null hypothesis that a process contains a unit root against a (trend) stationary alternative. It is based on the maximum value of the standard Dickey-Fuller tests when applied to both the forward and reverse data realizations. Thus, it requires no special computation and can easily be calculated from most existing econometric software packages. The null distribution of the maximum test is tabulated using Monte Carlo simulation and it is demonstrated to have considerably more power to reject the false null of a unit root than the standard Dickey-Fuller test. The tests are also shown to share very similar robustness properties. When applied to a number of well known U.S. macroeconomics time series, the new test rejects the unit root hypothesis more frequently than does the usual Dickey-Fuller test. Copyright 1995 by Blackwell Publishing Ltd

Mean reversion in real exchange rates

A modified Dickey–Fuller test, which is approximately uniformly most power invariant, is employed to examine mean reversion in real exchange rates. Results from this test provide a wider and more significant support for mean reversion than the standard Dickey-Fuller test.

An interpolation test for a unit root in the presence of drift

This paper considers tests of the null hypothesis of a unit root in an observed time series against the alternative that the series is level stationary. Interpolation tests whose critical values depend upon the value of the drift parameter are compared to the standard Dickey-Fuller tests in Monte Carlo simulations. The interpolation test performs quite well especially against one-sided alternatives.