The Integration Order of Vector Autoregressive Processes



These two equivalences allow us to understand the standard I(1) and
I(2) conditions as imposing a particular shape to the adjoint matrix
polynomial, which in turn ensures that the pole at the unit root has
order one or two; in these cases the principal part of the Laurent expan-
sion of (3.2) around
z = 1 consists of one or two terms and translates
into a moving average representation which involves the cumulation
(or the double cumulation) of the stationary process that generates the
stochastic trends.

5. Polynomial cointegration

The results of the previous section can thus be interpreted as al-
ternative proofs of the Granger Representation Theorem in the
I(1)
and
I(2) cases: the order of integration is established by Theorem
3.3, the explicit expression for
H(1) indicates the directions in which
cointegration is to be found, and the restrictions implied by (4.1) and
its derivatives define the (polynomial) cointegration properties of the
process. In the
I(2) case, for example, we write the inverse of Π(z) as
π(
z ) 1 =( z -1)22g ( z ) ,z = ʃ1, ■■■ ,znr}
where H(1) 6= 0p and g(1) 6= 0, and H(z) as

H ( z ) = H (1) + TH(1)( z — 1) + ( z — 1)2 H2( z ).

From the calculations in the proof of Proposition 4.2, we have that

(5.1)               H(1) = βηψξα, ∣ψ∣=0,

(5.2)             β 0H(1) = α 0 Π(1) H (1).

Thus the polynomial cointegration properties of the process are the
following:

Proposition 5.1. Let Xt be I(2) and β 1 = βη; then β'Xt and β1Xt
are I(1), and β0Xt + α0∏(1)∆Xt is I(0).

Proof. From (5.1) we have that β0H(1) = 0r×p and β10 H(1) = 0s×p;
thus the functions
β0Π(z)-1 and β10 Π(z)-1 have a pole of order one at
z = 1 and correspond to β0Xt and β10 Xt being I (1). Using (5.2) it is
easy to see that the function ∣
β0 + (1 — z)ex0∏(1) ∣ ∏(z) 1 has no pole
at
z = 1 and corresponds to β0Xt + α0∏(1)∆Xt being I(0). ■



More intriguing information

1. The Shepherd Sinfonia
2. The name is absent
3. An Intertemporal Benchmark Model for Turkey’s Current Account
4. Food Prices and Overweight Patterns in Italy
5. Detecting Multiple Breaks in Financial Market Volatility Dynamics
6. Regional specialisation in a transition country - Hungary
7. Willingness-to-Pay for Energy Conservation and Free-Ridership on Subsidization – Evidence from Germany
8. A Multimodal Framework for Computer Mediated Learning: The Reshaping of Curriculum Knowledge and Learning
9. The quick and the dead: when reaction beats intention
10. The name is absent
11. Handling the measurement error problem by means of panel data: Moment methods applied on firm data
12. The name is absent
13. The Modified- Classroom ObservationScheduletoMeasureIntenticnaCommunication( M-COSMIC): EvaluationofReliabilityandValidity
14. Financial Market Volatility and Primary Placements
15. Getting the practical teaching element right: A guide for literacy, numeracy and ESOL teacher educators
16. Modeling industrial location decisions in U.S. counties
17. Incorporating global skills within UK higher education of engineers
18. Before and After the Hartz Reforms: The Performance of Active Labour Market Policy in Germany
19. Output Effects of Agri-environmental Programs of the EU
20. The name is absent