Nonstationary Cointegration in the Fractionally Cointegrated VAR Model

Abstract

We consider the fractional cointegrated vector autoregressive (CVAR) model of Johansen and Nielsen (2012a) and make two distinct contributions. First, in their consistency proof, Johansen and Nielsen (2012a) imposed moment conditions on the errors that depend on the parameter space, such that when the parameter space is larger, stronger moment conditions are required. We show that these moment conditions can be relaxed, and for consistency we require just eight moments regardless of the parameter space. Second, Johansen and Nielsen (2012a) assumed that the cointegrating vectors are stationary, and we extend the analysis to include the possibility that the cointegrating vectors are nonstationary. Both contributions require new analysis and results for the asymptotic properties of the likelihood function of the fractional CVAR model, which we provide. Finally, our analysis follows recent research and applies a parameter space large enough that the usual (non-fractional) CVAR model constitutes an interior point and hence can be tested against the fractional model using a χ 2 -test.

Weak convergence to derivatives of fractional Brownian motion

“…One motivation for the results on the weak convergence of derivatives of the fractional process comes from the analysis of the multifractional cointegrated vector autoregressive (MFCVAR) model; see Johansen and Nielsen (2021). The model with parameters λ = (Ω, d, b, α, β) and no lags is given by Λ…”

“…In Section 5 we presented an application of our results in (11) and Theorem 2 to the asymptotic distribution theory for the maximum likelihood estimators of the fractional parameters in the so-called "multifractional" vector autoregressive model of Johansen and Nielsen (2021). In this setting, the derivative D d Z ⌊T r⌋ (d) and its weak limit D d W (r; d) play an important role because they allow avoiding linear dependence in the limit and because the asymptotic distribution is expressed in terms of both W (r, d) and D d W (r, d).…”

“…One interpretation of this model is a generalization of Hualde's (2014) bivariate unbalanced cointegrated regression model to a multivariate system framework. Johansen and Nielsen (2021) show that, in this setting, the derivative D d Z ⌊T r⌋ (d) and its weak limit D d W (r; d) play an important role in the asymptotic distribution theory for the maximum likelihood estimators of the fractional parameters. We present some details of this analysis in Section 5…”