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 non-stationary. 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 Chi-squared-test. NONSTATIONARY COINTEGRATION IN THE FCVAR MODEL 521 2. ASSUMPTIONS AND MAIN RESULTSIn Johansen and Nielsen (2012a), asymptotic properties of maximum likelihood estimators and test statistics were derived for model (1) with the parameter space ≤ b ≤ d ≤ d 1 for some d 1 > 0, which can be arbitrarily large, and some such that 0 < ≤ 1∕2. The parameter space was extended by Johansen and Nielsen (2018) toagain for an arbitrarily large d 1 > 0 and an arbitrarily small such that 0 < ≤ 1∕2. While is exactly the same as in Johansen and Nielsen (2012a), we have in (3) introduced the new constant 1 > 0, which is zero in Johansen and Nielsen (2012a). We note that the parameter space  explicitly includes the linein the interior precisely because 1 > 0. Although > 0 can be arbitrarily small, a smaller implies a stronger moment condition for consistency in both Johansen and Nielsen (2012a) and Johansen and Nielsen (2018). This moment condition is relaxed below. We will assume that the data for t ≥ 1 are generated by model (1 ). A standard approach for autoregressive models, which we follow, is to conduct inference using the conditional likelihood function of X 1 , … , X T given initial values {X −n } n≥0 . That is, we interpret (1) as a model for X t , t = 1, … , T, given the past, and use the conditional density to build a conditional likelihood function. Thus, since our entire approach is conditional on the initial values {X −n } n≥0 , we consider these non-random, as is standard for (especially non-stationary) autoregressive models.However, it is difficult to imagine a situation where {X s } T s=−∞ are available, or perhaps even exist, so we assume that the data are only observed for t = −N +1, … , T. Johansen and Nielsen (2016) argue in favor of the assumption that series was initialized in the finite past using two leading examples, political opinion poll data and financial volatility data, but we maintain the more general assumption from J...

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