Range Unit‐Root (RUR) Tests: Robust against Nonlinearities, Error Distributions, Structural Breaks and Outliers

Abstract

Abstract. Since the seminal paper by Dickey and Fuller in 1979, unit-root tests have conditioned the standard approaches to analysing time series with strong serial dependence in mean behaviour, the focus being placed on the detection of eventual unit roots in an autoregressive model fitted to the series. In this paper, we propose a completely different method to test for the type of long-wave patterns observed not only in unit-root time series but also in series following more complex data-generating mechanisms. To this end, our testing device analyses the unit-root persistence exhibited by the data while imposing very few constraints on the generating mechanism. We call our device the range unit-root (RUR) test since it is constructed from the running ranges of the series from which we derive its limit distribution. These nonparametric statistics endow the test with a number of desirable properties, the invariance to monotonic transformations of the series and the robustness to the presence of important parameter shifts. Moreover, the RUR test outperforms the power of standard unit-root tests on near-unit-root stationary time series; it is invariant with respect to the innovations distribution and asymptotically immune to noise. An extension of the RUR test, called the forward backward range unit-root (FB-RUR) improves the check in the presence of additive outliers. Finally, we illustrate the performances of both range tests and their discrepancies with the Dickey Fuller unit-root test on exchange rate series.

Nonlinear Cointegration and Nonlinear Error Correction: Record Counting Cointegration Tests

“…If the two series have a common stochastic component (common trend), their ranges will tend to jump together, indicating the synchronicity of the new records. The RCC test statistic clearly outperforms the traditional DF unit root test on the cointegrating residuals (EG) as well as nonparametric tests in a similar context previously proposed by Aparicio et al (2006a). The small sample properties of RCC nonparametric test are analyzed by Monte Carlo simulations and with some empirical examples.…”

“…When there is a new record (maximum or minimum) in x 1 x t , the first difference of the ranges will be positive at time t, R x t > 0. A statistics record was proposed by Aparicio et al (2006a) for robust unit root testing. When the original series is stationary, with finite variance, the series of first differences of the ranges are positive at the beginning of the sample and the rest is almost a sequence of zeros.…”

“…Most unit root tests, like those of Dickey and Fuller (1979) or Phillips and Perron (1988), are not robust to outliers, Franses and Haldrup (1994); nor to structural breaks, Perron (1990); nor to nonlinear transformations, Granger and Hallman (1991) and Aparicio et al (2006a). Therefore, tests for noncointegration based on the augmented Dickey and Fuller (ADF) test applied to the residuals of the cointegrating relationship, Engle and Granger (1987), have size distortions and losses in power.…”

“…Therefore, tests for noncointegration based on the augmented Dickey and Fuller (ADF) test applied to the residuals of the cointegrating relationship, Engle and Granger (1987), have size distortions and losses in power. Aparicio et al (2004Aparicio et al ( , 2006a analyzed the asymptotic properties of a new range unit root (RUR) test and provide evidence of their nice behavior by Monte Carlo simulation of nonlinearities and structural breaks and by some empirical applications.…”

A New Approach to Unit Root Testing

“…Fully Nonparametric approaches to unit root testing which are robust against violations of standard assumptions have been proposed e.g. by Breitung and Gouriéroux (1997) and Aparico et al (2006).…”