Common trends and cycles in I(2) VAR systems

A Reduced Rank Regression Approach to Coincident and Leading Indexes Building*

“…Linear restrictions on and may be tested by a linear switching algorithm similar to those proposed by Johansen (1995) and Paruolo (2003). In particular, the following system of hypotheses is considered where H * is a matrix of known elements and ϕ * is a g × 1 parameter matrix.…”

Common Feature Analysis of Economic Time Series: An Overview and Recent Developments

“…The CCF analysis was extended even to the case of series having different forms of stationarity than I(1)-ness. In particular, Cubadda (1999Cubadda ( , 2001) explored the presence of common cycles in seasonal time series that are also integrated at (a subset of) the seasonal frequencies, whereas Paruolo (2006) focused on the case of I(2) systems. Franchi and Paruolo (2011) offered a comprehensive theoretical analysis of the conditions of existence of the various form of CCF's and of the characterization of the CCF relations in I(0), I(1) and I(2) systems.…”

“…the conditional vector but there exist a linear combination of them whose conditional expectation is linear w.r.t. the conditional vector; common seasonal cycles (Cubadda, 1999), when there exists a linear combination of seasonally differenced series which follows an MA process of low order; common panel structures (Hecq et al, 2000), when there is a linear combination of the variables in a panel data which is white noise for all individuals of the panel; nonlinear cotrending (Bierens, 2000), when a linear combination of the components of a set of stationary time series around nonlinear deterministic time trends is stationary around a linear trend or a constant; polynomial common serial correlation (Cubadda and Hecq, 2001), when there exists a polynomial combination of serially correlated time series that is an innovation; long-run pure variance common feature (Engle and Marcucci, 2006), when the conditional variances of a collection of assets all depend upon a small number of variance factors; unpredictable polynomial combinations (Paruolo, 2006), when a polynomial linear combination of series integrated with different orders is an innovation; weak form of common serial correlation (Hecq et al, 2006), when a linear combination of serially correlated series adjusted for the equilibrium errors is an innovation; common periodic correlation (Haldrup et al, 2007), which extends the notion of common serial correlation to periodic autoregressive models. From a statistical point of view, common features imply a reduction to more parsimonious structures such as common factor representations (see, i.a., Cubadda, 2007), which can often be estimated by reduced-rank regression techniques (Anderson, 1984(Anderson, , 1999.…”

Simulation-based tests of forward-looking models under VAR learning dynamics

“…Suppose that β takes the form (23) so that the term o t = β′ X̃ t = ỹ t − ϕ w̃ t measures deviations from equilibrium. Define the p × 1 process W t = ( X̃ ′ t β, Δ X̃ ′ t τ)′, where τ is a p × ( p − r ) selection matrix such that det(τ′β ⟂ )≠ 0; Paruolo (2003, Theorem 2) shows that the VEqC (22) can be written in terms of a VAR process for W t without loss of information, i.e. where $\varepsilon_{t}^{0} = (\varepsilon'_{t} \beta, \varepsilon'_{t}\tau)'$ is a MDS with respect to ℋ︁ t , and with the coefficients in B i , i = 1, … k which depend on the VEqC coefficients α, Γ 1 , …, Γ k −1 (see also Mellader et al , 1992).…”