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04.3.1 An I(2) Model for VAR(1) Processes

Published online by Cambridge University Press:  08 June 2004

Paolo Paruolo
Paolo Paruolo
Affiliation:
University of Insubria, Varese, Italy
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This problem discusses an I(2) model in the VAR(1) case. The I(2) representation theorem of Johansen (1992) (JRT) holds also for VAR(1) processes. The I(2) model for VAR(k) processes has been discussed for k ≥ 2 in Johansen (1996, Ch. 9; 1997). We here discuss a parametrization for the I(2) case of VAR(1) that differs from the VAR(k) model.

Information

Type
PROBLEMS AND SOLUTIONS
Information
Econometric Theory , Volume 20 , Issue 3 , June 2004 , pp. 639 - 640
Copyright
© 2004 Cambridge University Press

This problem discusses an I(2) model in the VAR(1) case. The I(2) representation theorem of Johansen (1992) (JRT) holds also for VAR(1) processes. The I(2) model for VAR(k) processes has been discussed for k ≥ 2 in Johansen (1996, Ch. 9; 1997). We here discuss a parametrization for the I(2) case of VAR(1) that differs from the VAR(k) model.

Consider the VAR(1) process with no deterministics,

with εt i.i.d. N(0,Ω) of dimension p, Ω positive definite. It is assumed that |I − (I + Π)z| = 0 has solutions either at z = 1 or |z| > 1 (Assumption A). Under Assumption A, the JRT states that Xt is I(2) iff the following three conditions hold.

(a) Π = αβ′, with α, β, full rank p × r matrices, r < p,

(b) Pα⊥Pβ⊥ = α1β1′

for α1, β1 p × s full column rank matrices, s < p − r, and

(c) α2′θβ2 is square and invertible, where θ := I + βα′, α2 := (α : α1)⊥, β2 := (β : β1)⊥, a := a(a′a)−1, Pa := aa′. In the following p2 := p − r − s.

1. Let a VAR(1) process be I(2), i.e., let Assumption A and conditions (a)–(c) hold. Prove that p2 ≤ r.

2. Define the I(1) model as the collection of VAR(1) processes indexed by the parameters τ := (α, β, Ω). Find the restrictions on τ that ensure that (b) holds. Also find an explicit parametrization. This model is called the I(2) model.

3. Let Yt := β′Xt + δ(β2′ β2)−1β2′ΔXt describe the (possibly multi-) cointegrating relations. Show that δ is of full column rank p2 and find δ in terms of the parameters in part 2. Indicate also within Yt the r − p2 polynomially cointegrating relations and the p2 CI(2,2) relations.

4. Xt has representation Δ2Xt = C2(L)εt. Find the expression for C2 := C2(1) in terms of the parameters in part 2.

5. Discuss the identification of τ.

References

REFERENCES

Johansen, S. (1992) A representation of vector autoregressive processes integrated of order 2. Econometric Theory 8, 188–202.Google Scholar
Johansen, S. (1996) Likelihood-Based Inference in Cointegrated Vector Auto-regressive Models. Oxford University Press.
Johansen, S. (1997) A likelihood analysis of the I(2) model. Scandinavian Journal of Statistics 24, 433–462.Google Scholar