Fractional integration and cointegration: The representation theory suggested by S. Johansen

Transcription

1 : The representation theory suggested by S. Johansen people.ssb.no/hhu January 31, 2007

2 Univariate fractional process Consider the following univariate process d x t = ε t, (1) where d = (1 L) d is the fractional difference operator. Similarly, we define the fractional lag operator as L d = 1 (1 L) d. Therefore; d = 1 L d. From the definition of the fractional difference operator, we see that it is equal to the normal difference operator when d is an positive integer. The fractional lag operator is equal to the usual lag operator for d = 1.

3 Univariate fractional process We need to know how to understand the expression (1 L) d, which can be defined by the following expansion: (1 z) d = ( ( 1) n d n n=0 ) z n ( d)( d + 1)... ( d + n 1) = z n n! n=0 n 1 i=0 ( d + i) = 1 + z n (2) n! n=1

4 Univariate fractional process Applying this definition ((1 z) d = 1 + can write (1) ( d x t = ε t ) as n=1 n 1 i=0 ( d+i) n! z n ) we x t dx t 1 d(d 1) + x t 2 2 d(d 1)(d 2) x t d(d 1)(d 2)(d 3) + x t = ε t, and for an non-integer d we have infinite number of lags. However, for d > 0 the weight of a lag is lower than for the previous lag.

5 Univariate fractional process We can divide both sides of (1) with (1 L) d and we get x t = (1 L) d ε t. Applying the definition above yields x t = ε t + d 1 ε t 1 + d(d + 1) ε t d(d + 1)(d + 2) ε t

6 Univariate fractional process 1.0 A(L)x t = ε t 0.5 d= d= x t = B(L) ε t d= d=

7 Univariate fractional process Properties for (1) for different values of d: 0 d < 0.5; x is mean-reverting with finite variance, 0.5 d < 1; x is mean-reverting with infinite variance, 1 d < 1.5; x is mean-reverting with finite variance,...

8 Co-fractional VAR: Previous suggested formulations Granger (1986); A (L) d X t = ( 1 b) d b αβ X t 1 + d(l)ε t Lyhagen (1998) tried to find properties of the solution, but unfortunately the results and their proofs are not correct. Quote; Johansen. Dittmann (2004) attempts to derive this model from a moving average form [...]. However, the results are not correctly proved. Quote; Johansen. Problem to derive a characteristic function; I.e. univariate model with two lags and b = d: d x t = ( 1 d) γ 1 x t 1 + γ 2 d x t 1 + ε t has the characteristic function π(z) = (1 z) d γ 1 [1 (1 z) d] z γ 2 (1 z) d z, Finding the roots [...] is not a standard problem.

9 Co-fractional VAR: Formulation The formulation suggested by Johansen: k 1 d X t = αβ d b L b X t + Γ i d L i b X t + ε t, This formulation implies the following changes: ( 1 b ) X t 1 (in the co-fractional part) is changed to L b X t. The lag polynomial A (L) is changed to A(L b ), i.e. the latter is a lag polynomial in L b (and not L). The lag polynomial d(l) is ignored. Example When d = 1 and d b = 0, i.e. I (1) variables cointegrate to I (0) relationships: X t = αβ X t 1 + i=1 k Γ i X t i + ε t, i=1

10 Co-fractional VAR: Characteristic polynomial Π(z) = (1 z) d I p αβ (1 (1 z) b )(1 z) d b k Γ i (1 (1 z) b ) i (1 z) d i=1 = (1 z) d b [(1 u)i p αβ u ] k Γ i (1 u)u i i=1 where u = 1 (1 z) b [Typo in Johansen!] If I understand Johansen correct; it is (only) the expression in the square brackets above that is interesting in determining wether the system is stable or not. The roots in this polynomial should be outside the unit disc when solved for z. When sovled for u the roots should be outside the modified unit disc.

11 Co-fractional VAR: Unit disc and modified unit disc

12 Co-fractional VAR: Problems & extensions How to analyse models with polynomial cofractional processes. How to construct d X t, d b L b X t and d L i b X t (since they are sums of infinite number of lags) What are the critical values of the rank test for different values of d and b. And how to do this test when d and b are not known. How to treat deterministic variables....

13 Problems & extensions: Polynomial cofractional processes Johansen consider the model reparameterized as ( ) k 1 d X t = d 2b αβ L b X t Γ b L b X t + Ψ i d L i b X t + ε t, i=1 where we assume that α and β have rank r < p, and that α Γβ = ξη of rank s < p r, and that 0 < 2b d. This formulation implies: The process X is I (d) Some variables cointegrates down to I (d b) Some variables cointegrates down to I (d 2b)

14 Problems & extensions: Construction of data?

15 Problems & extensions: Critical values of the rank test Tables of critical values for the conitegrating rank test for I (1) (and I (2)) variables has been available for over a decade. With fractional processes new critical values must be calculated. These will probably depend on d, d b, the number of stochastic trends, the deterministic variables in the system (and perhaps also the dimension of the system). An alternative to tables of critical values is a response surface function suggested by Doornik (1998). He shows that the distribution of the rank test can be approximated with an Gamma distribution, and the Gamma distribution can therefore be used to derive p-values. This approach may be possible to extend for fractional processes. However, there is an additional problem when testing cointegration rank with fractional processes: The rank and orders of integration (d and d b) must be decided simultaneously.

16 Problems & extensions: Deterministic variables Consider the univariate process with an intercept d x t = c + ε t, (3) If 0 d < 1, the inclusion of the constant, c, implies that the process above has a non-zero level (for c 0). If 1 d < 2, the inclusion of the constant, c, implies that the process above has a non-zero trend (for c 0). For not letting the implication of the deterministic variables to change with int(d) and int(d b) (and int(d 2b)), we can analyse de-trended data: k 1 d (X t γd t ) = αβ d b L b (X t γd t )+ Γ i d L i b (X t γd t )+ε t, Examples of such formulations in the I (1)/I (0) situation are Lütkepohl and Saikkonen (2000) and Hungnes (2006). However, both these approaches needs adjustments in order to be used with fractional processes. i=1

17 References Dittmann (2004). Error correction models for fractionally cointegrated time series. Journal of Time Series Analysis 25, Doornik (1998). Approximations to the asymptotic distribution of cointegration tests. Journal of Economic Surveys 12, Granger (1986). Developments in the study of cointegrated economic variables. Oxford Bulletin of Economics and Statistics 48, Hungnes (2006). Trends and Breaks in Cointegrated VAR Models. Ph.D. Thesis, University of Oslo. Johansen (2006). A representation theory for a class of vector autoregressive models for fractional processes. Lyhagen (1998). Maximum likelihood estimation of the multivariate fractional cointegrating model. Working paper, Stockholm School of Economics. Saikkonen and Lütkepohl (2000). Testing for the cointegrating rank of a VAR process with structural shifts. Journal of Business and Economic Statistics 18,