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

Abstract

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®cient tests are found to show the best overall performance in both cases, at least if the sample size is suf®ciently large. However, it is also found that the DF and LM tests are roughly complementary with regard to their ®nite-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.

On LM type tests for seasonal unit roots in quarterly data

“…Several test procedures have been suggested based on this principle. In the context of testing for conventional unit roots, see, for example, Ahn (1993), Oya and Toda (1998), Schmidt and Lee (1991), Schmidt and Phillips (1992) and Solo (1984), and when testing for seasonal unit roots, see, inter alia , Ahn and Cho (1993a), Ahn and Cho (1993b), Breitung and Franses (1998), Li (1991) and Park and Cho (1994).…”

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

“…As in Oya and Toda (1998), we can show that the above test statistic is asymptotically equivalent to the sum of the t ‐statistics for ρ 1 and ρ 2 in the regression where D 1 t = 1 for and zero otherwise, D 2 t = 1 − D 1 t , φ =[ φ 1 ,…, φ p ]′ and . Then, we define the test statistic for as where and are t ‐statistics for ρ 1 and ρ 2 .…”

“…As x t = y t − μ 0 − μ 1 t from , we can substitute y t − μ 0 − μ 1 t for x t in and then the log‐likelihood can be written as Note that we replaced μ 0 by y 0 , as did Ahn (1993), Oya and Toda (1998) and Schmidt and Phillips (1992), because it is not identified under H 0 . As where φ j = ψ j for all j under H 0 , we have the (approximate) maximum likelihood estimator (MLE) of μ 1 as using the relation of (see Oya and Toda, 1998). Similarly, as the MLE of φ j under H 0 is given by the following regression: where .…”

“…Both cases of a known and unknown break point are investigated. We propose four tests for the null of no structural change in the long‐run persistence: a Lagrange multiplier (LM)‐type test, tests based on the quasi‐differencing method used by Elliott, Rothenberg and Stock (1996) and Xiao and Phillips (1999), and ‘demeaned versions’ of these tests as used by Oya and Toda (1998) and Toda and Oya (1993). We derive the limiting distributions of these test statistics under local alternatives and compare the power functions.…”

Similarity Issues in Cointegration Analysis

“…As an example Phillips (1991) has suggested an analysis based on regression rather than canonical correlations adopted in Johansen's approach. Similarity with respect to deterministic trend parameters can often be obtained using de-trending strategies as discussed by for instance Dickey and Fuller (1981), Kiviet andPhillips (1992), Lu Ètkepohl andSaikkonen (1997) and Oya and Toda (1998). In this paper such strategies are implicitly used, but deliberately in such a way that the tests remain likelihood ratio tests.…”