Residuals‐based tests for cointegration with generalized least‐squares detrended data

Abstract

We provide GLS-detrended versions of single-equation static regression or residualsbased tests for testing whether or not non-stationary time series are cointegrated. Our approach is to consider nearly optimal tests for unit roots and apply them in the cointegration context. We derive the local asymptotic power functions of all tests considered for a triangular DGP imposing a directional restriction such that the regressors are pure integrated processes. Our GLS versions of the tests do indeed provide substantial power improvements over their OLS counterparts. Simulations show that the gains in power are important and stable across various con…gurations.JEL Classi…cation Number: C22, C32, C52.

Tests for segmented cointegration: an application to US governments budgets

“…, T (2.4) with m = 0 corresponding to the no change case. Furthermore, regarding the deterministic kernels μ yi in (2.1) and μ xi in (2.2), three empirically relevant cases (i = 0, 1, 2) are considered: a) μ y0 = 0 and μ x0 = 0; b) μ y1 = φ y1 and μ x1 = φ x1 ; and c) μ y2 = φ y1 + φ y2 t and μ x2 = φ x1 + φ x2 t, where φ xi , i = 1, 2 are K × 1 vectors of parameters; see also Perron and Rodriguez (2016). Following Kim (2003) and Davidson and Monticini (2010), we assume that μ yi , μ xi , i = 0, 1, 2, and β are not subject to shifts so that ε t can be estimated using the entire sample by standard methods in a single-equation cointegration framework.…”

“…r ) , K is the number of exogenous regressors considered in (2.1), and R 2 = with = −1/2 xx ω xy ω −1/2 yy and 0 ≤ R 2 ≤ 1; see e.g.,Perron and Rodriguez (2016).Under H 0 : c j = 0 and γ j = 0, for all j, considering b := 1, − β , where β is the OLS estimate of β in (2.1) and β x t = e t , it follows that b → b, where b := 1, −β . Furthermore, )B(r ) dr b, where B (r ) := B 1 (r ) 1×1 , B 2 (r ) K ×1 is a (K + 1) × 1 vector Brownian motion with covariance matrix ¨= 0 + 1 + 1 .To remove the nuisance parameters present in the distributions of the test statistics,consider ¨:= LL , where L := l 11 0 l 21 L 22 with l 11 := ω yy − ω xy ¨)W(r ) dr = f yy f xy f xy F xx ,…”

Cointegration in high frequency data

“…where σ X and σ Z are càdlàg adapted processes, and [50], or the more recent work of [46]. On the other hand, σ M encompasses possibly non ergodic trends in volatility and is assumed to be a common factor in X and Y .…”

“…We are well aware that linear drift is rather a strong assumption, but overall there is a lack of literature on drift. Some exceptions include [46], which deals with optimal methods for unit-root and cointegration tests when a trend is present (but no time-varying volatility), and [7] (Section 4.3), which gives a general detrending method for time-linear trends in the presence of timevarying variance for the unit-root test. Theorem 4.3 from the latter work gives the related theoretical asymptotic results.…”

Long run dynamic volatilities between OPEC and non-OPEC crude oil prices

“…Notes: For EG test: the one sided (lower tail) test of the null hypothesis is that the variables are not cointegrated; at the 1%, 5%, and 10% significance levels, the critical values are -4.02, -3.40 and -3.09, respectively (Rapach and Weber 2004). For PR test: the one sided (lower-tail) test of the null hypothesis is that the variables are not cointegrated; at the 1%, 5%, and 10% significance levels, the critical values are equal to -3.33, -2.76 and -2.47, respectively (Perron and Rodriguez 2001).…”