Adaptive Testing for Cointegration With Nonstationary Volatility

Abstract

This article develops a class of adaptive cointegration tests for multivariate time series with nonstationary volatility. Persistent changes in the innovation variance matrix of a vector autoregressive model lead to size distortions in conventional cointegration tests, which may be resolved using the wild bootstrap, as shown in recent work by Cavaliere, Rahbek, and Taylor. We show that it also leads to the possibility of constructing tests with higher power, by taking the time-varying volatilities and correlations into account in the formulation of the likelihood function and the resulting likelihood ratio test statistic. We find that under suitable conditions, adaptation with respect to the volatility process is possible, in the sense that nonparametric volatility matrix estimation does not lead to a loss of asymptotic local power relative to the case where the volatilities are observed. The asymptotic null distribution of the test is nonstandard and depends on the volatility process; we show that various bootstrap implementations may be used to conduct asymptotically valid inference. Monte Carlo simulations show that the resulting test has good size properties, and higher power than existing tests. Empirical analyses of the U.S. term structure of interest rates and purchasing power parity illustrate the applicability of the tests.

Adaptive information-based methods for determining the co-integration rank in heteroskedastic VAR models

“…In what follows, σ t will be referred to as the volatility matrix of ε t . Elements of Assumption 1 have previously been employed by, inter alia, Cavaliere et al (2010), Boswijk, Cavaliere, Rahbek and Taylor (2016), Cavaliere et al (2018) and Boswijk and Zu (2022). In particular, Assumption 1 allows for a countable number of discontinuities in σ (•) therefore allowing for a wide class of potential models for the timevarying behaviour of the unconditional variance matrix of ε t .…”

“…Under the assumption of a known autoregressive lag length, Boswijk and Zu (2022) develop an procedure based on adaptive PLR tests for determining the co-integration rank in possibly heteroskedastic VAR models. Specifically, they propose a procedure where the volatility process is estimated using a non-parametric kernel estimator, with this estimate then used in the adaptive PLR test procedure.…”

“…In practice the autoregressive lag length will need to be estimated along with the co-integration rank. To that end, the practitioner can use either a sequential procedure, where the lag length is consistently estimated in a first step and then subsequently employed in the second step in a procedure such as either the adaptive PLR test approach of Boswijk and Zu (2022) or an information criterion for determining the co-integration rank, or a joint information criteria-based approach can be used whereby the lag length and co-integration rank are determined simultaneously. Cavaliere et al (2018) show that both joint and sequential procedures based on standard information criteria consistently determine both the lag length and the co-integration rank in the presence of nonstationary unconditional volatility, provided standard conditions hold on the penalty term.…”

“…We also establish the asymptotic validity of a sequential procedure selecting the autoregressive lag length by an adaptive information criterion [ALS-IC] in the first step and then determining the co-integration rank using again an ALS-IC in the second step based on the first step estimate of the lag length. Because the co-integration rank is determined by minimising an adaptive information criterion over all possible values of the co-integration rank from zero up to the dimension of the system, the practitioner does not therefore need to obtain p-values by bootstrap methods, making the procedure considerably less time consuming than the Boswijk and Zu (2022) procedure based on adaptive PLR tests.…”

“…We also establish the asymptotic validity of a sequential procedure selecting the autoregressive lag length by an ALS-IC in the first step and then using the adaptive PLR test-based approach of Boswijk and Zu (2022) in the second step based on the first step estimate of the lag length.…”