The asymptotic distribution of canonical correlations and vectors in higher-order cointegrated models

Abstract

The study of the large-sample distribution of the canonical correlations and variates in cointegrated models is extended from the first-order autoregression model to autoregression of any (finite) order. The cointegrated process considered here is nonstationary in some dimensions and stationary in some other directions, but the first difference (the “error-correction form”) is stationary. The asymptotic distribution of the canonical correlations between the first differences and the predictor variables as well as the corresponding canonical variables is obtained under the assumption that the process is Gaussian. The method of analysis is similar to that used for the first-order process.

Cointegrated stochastic processes are used in econometrics for modeling macroeconomic time series that have both stationary and nonstationary properties. The term “cointegrated” means that in a multivariate process that appears nonstationary some linear functions are stationary. Many economic time series may show inflationary tendencies or increasing volatility, but certain relationships are not affected by these tendencies. Statistical inference is involved in identifying these relationships and estimating their importance.

The family of stochastic processes studied in this paper consists of vector autoregressive processes of finite order. A vector of contemporary measures is considered to depend linearly on earlier values of these measures plus random disturbances or errors. The dependence may be evaluated by the canonical correlations between the contemporary values and the earlier values.

The nonstationarity of a process may be eliminated by treating differences or higher-order differences (over time) of the vectors. This paper treats processes in which first-order differencing accomplishes stationarity. The first-order difference is represented as a linear combination of the first lagged variable and lags of the difference variable. The stationary linear combinations are the canonical variables corresponding to the nonzero process canonical correlations between the difference variable and the first lagged variable not accounted for by the lagged differences. The number of these is defined as the degree of cointegration.

Statistical inference of the model is based on a sample of observations; that is, a vector time series over some period of time. The estimator of the parameters of the original autoregressive model is a transformation of the estimator of the (stationary) error-correction form. In the latter, one coefficient matrix is of lower rank (the degree of cointegration). It is estimated efficiently by the reduced rank regression estimator introduced by me (1). It depends on the larger canonical correlations and corresponding canonical vectors. The smaller correlations are used to determine the rank of this matrix. Inference is based on the large-sample distribution of these correlations and variables.

The asymptotic distribution of the canonical correlations and coefficients of the variates for the first-order autoregressive process was derived by me (2). The distribution for the higher-order process (that is, several lags) is obtained in this paper, using similar algebra. Hansen and Johansen (3) have independently obtained the asymptotic distribution of the canonical correlations, but by a different method and expressed in a different form.

The likelihood ratio test for the degree of cointegration that I found (1) is given in Asymptotic Distribution of the Smaller Roots; its asymptotic distribution under the null hypothesis was found by Johansen (4). To evaluate the power of such a test, one needs to know the distribution or asymptotic distribution of the sample canonical correlations corresponding to process canonical correlations different from 0. See ref. 5, for example.

For further background, the reader is referred to Johansen (6) and Reinsel and Velu (7).

The Model

The general cointegrated model is an autoregressive process {Y_t} of order m defined by

1

where Z_t is unobserved with ℰZ_t = 0, ℰZ_tZ′_t = Σ_ZZ, and ℰY_t−iZ′_t = 0, i = 1, … . Let B(λ) = λ^mI − λ^m−1B₁ − … − B_m. If the roots λ₁, … , λ_pn of |B(λ)| = 0 satisfy |λ_i| < 1, a stationary process {Y_t} can be defined by 1. If some of the roots are 1, the process will be nonstationary. In this paper, we assume that n (0 < n < p) roots of |B(λ)| = 0 are 1(λ₁ = … = λ_p = 1), and the other pm − n roots satisfy |λ_i| < 1, i = n + 1, … , pm. The first difference of the process, the “error-correction” form, is

2

Here Π = B₁ + … + B_m − I = −B(1), Π_j = −(B_j+1 + … + B_m), j = 1, … , m − 1, Π̄ = (Π₁, … , Π_m−1), and Δ̄Y_t−1 = (ΔY′_t−1, … , ΔY′_t−m+1)′.

A sample consists of T observations: Y₁, … , Y_T. Because the rank of Π is k, it is to be estimated by the reduced rank regression estimator introduced by me (1) as the maximum likelihood estimator when Z₁, … , Z_T are normally distributed and Y₀, Y₋₁, … , Y_−m+1 are nonstochastic and known. The matrices Π₁, … , Π_m−1 are unrestricted except for the condition |λ_i| < 1, i = n + 1, … , pm. The estimator depends on the canonical correlations and vectors of ΔY_t and Y_t−1 conditioned on ΔY_t−1, … , ΔY_t−m+1.

Define

where S_Δ̄Y,Δ̄Y = T⁻¹ ∑Δ̄Y_t−1Δ̄Y′_t−1, S_ΔY,Δ̄Y = T⁻¹ ∑ ΔY_tΔ̄Y′_t−1, and S_Ȳ,Δ̄Y = T⁻¹ ∑ Y_t−1Δ̄Y′_t−1. The vectors ΔŶ and Ŷ are the sample residuals of ΔY_t−1 and Y_t−1 regressed on Δ̄Y_t−1. Define Ŝ = T⁻¹ ∑ ΔŶΔŶ = S_ΔY,ΔY − S_ΔY,Δ̄YSS_Δ̄Y,ΔY, Ŝ = T⁻¹ ∑ ΔŶŶ = S_ΔY,Ȳ − S_ΔY,Δ̄YSS_Δ̄Y, and Ŝ = T⁻¹ ∑ ŶŶ = S_ȲȲ − S_Ȳ,Δ̄YSS_Δ̄Y,Ȳ, where S_ΔY,ΔY = T⁻¹ ∑ ΔY_tΔY′_t, S_ΔY,Ȳ = T⁻¹ ∑ ΔY_tY′_t−1, and S_ȲȲ = T⁻¹ ∑ Y_t−1Y′_t−1. The sample canonical correlations between ΔŶ and Ŷ and variates are defined by

3

4

More information on canonical analysis is covered in chapter 12 of ref. 8. One form of the reduced rank regression estimator is Π̂_(k) = ŜΓ̂₂Γ̂′₂, where Γ̂₂ = (γ̂_n+1, … , γ̂_p) and r < … < r.

We shall assume that there are exactly n linearly independent solutions to ω′B(1) = 0; that is, ω′Π = 0. Then the rank of Π is p − n = k and there exists a p × n matrix Ω₁ of rank n such that Ω′₁Π = 0. See Anderson (9). There is also a p × k matrix Ω₂ of rank k such that Ω′₂Π = Υ₂Ω′₂, where Υ₂ (k × k) is nonsingular, and Ω = (Ω₁, Ω₂) is nonsingular.

To distinguish between the stationary and nonstationary coordinates, we make a transformation of coordinates. Define

Ψ_j = Ω′B_j(Ω′)⁻¹, j = 1, … , m. Then the process 1 is transformed to

5

If we define Υ = Ψ₁ + … + Ψ_m − I = Ω′Π(Ω′)⁻¹, Υ_j = −∑ Ψ_j = Ω′Π_j(Ω′)⁻¹, Ῡ = (Υ₁, … , Υ_m−1), and Δ̄X_t−1 = (ΔX′_t−1, … , ΔX′_t−m+1)′, the form 2 is transformed to

6

Note that Υ = diag(0, Υ²²).

Define ΔX̂, X̂, S_Δ̄X,Δ̄X, S_ΔX,Δ̄X, S_X̄,Δ̄X, Ŝ, Ŝ, and Ŝ in a manner analogous to the definitions in the Y-coordinates. The reduced rank regression estimator of Υ is based on the canonical correlations and canonical variates between ΔX̂ and X̂ defined by

7

8

The estimator of Υ of rank k is Υ̂_(k) = ŜG₂G′₂, where G₂ = (g_n+1, … , g_p) and g_i is the solution for g in 8 when r = r_i, the solution to 7 and r₁ < … < r_p. The rest of this paper is devoted to finding the asymptotic distribution of {g_i, r_i}. Note that Υ̂_(k) = Ω′Π̂_(k)(Ω′)⁻¹.

The vectors ΔX̂ = ΔX_t − S_ΔX,Δ̄XSΔ̄X_t−1 and X̂ = X_t−1 − S_X̄,Δ̄XSΔ̄X_t−1 are the residuals of ΔX_t and X_t−1 regressed on Δ̄X_t−1, and r₁ is the maximum correlation between ΔX̂ and X̂, which is the correlation between ΔX_t and X_t−1 after taking account of the dependence “explained” by Δ̄X_t−1. The canonical correlations are the canonical correlations between (ΔX′_t, Δ̄X_t−1) and (X_t−1, Δ̄X_t−1) other than ±1.

The Process

The process {X_t} defined by 5 can be put in the form of the Markov model

9

(section 5.4, ref. 10). Multiplication of 9 on the left by

yields a form that includes the error-correction form 6

10

The first n components of 10 constitute

11

Here Υ_j has been partitioned into n and k rows and columns. Assume X₁₀ = X_1,−1 = … = 0 and W₁₀ = W_1,−1 = … = 0. The sum of 11 for t = −∞ to t = s is X_1s = ∑ [ΥX_1,s−j + ΥX_2,s−j] + ∑ W_1t, or

12

Write 12 as

13

where Γ = (I − ∑ Υ)⁻¹, Γ⁻¹H is a linear combination of Υ₁, … , Υ_m−1, and X̃_s = (X′_2s, Δ̄X′_s)′. [The matrix on the left-hand side of 12 is nonsingular because otherwise there would be a linear combination of the right-hand side identically 0.] The right-hand side of 13 is the sum of a stationary process and a random walk (∑ W_1t).

The last pm − n = k + p(m − 1) components of 10 constitute a stationary process satisfying

14

where X̃′_t = (X′_2t, Δ̄X′_t), W̃′_t = (W′_2t, W′_t, 0), and Υ̃ consists of the last pm − n rows and columns of the coefficient matrix in 10. Note that the first n columns and last pm − n rows of that matrix consist of 0s. Because the eigenvalues of Υ̃ are less than 1 in absolute value (9), X̃_t = ∑ Υ̃^sW̃_t−s, ℰX̃_tX̃′_t = Σ̃ = ∑ Υ̃^sΣ̃_WWΥ̃′^s, ℰX̃_tX̃′_t−h = Υ̃^hΣ̃. The covariance Σ̃ satisfies

15

Given Υ̃ and Σ̃_WW, 15 can be solved for Σ̃ [Anderson (10), section 5.5]. Further we write 13 as X_1t = Γ ∑ W_1,t−s + H ∑ Υ̃^sW̃_t−s. Then

since I − Υ̃^t → I. Here ℰW_1tW̃′_t = Σ is the second set of rows in Σ̃_WW. Then T⁻¹ℰS = T⁻² ∑ ℰX_1tX′_1t → 2⁻¹ΓΣΓ′ because ∑ t = T(T + 1)/2. Further

Define

16

where Σ_ΔX,Δ̄X = ℰΔX_tΔ̄X′_t−1, Σ_X̄,Δ̄X = ℰX_t−1Δ̄X′_t−1 depends on t, and Σ_Δ̄X,Δ̄X = ℰΔ̄X_t−1Δ̄X′_t−1 does not depend on t. Note that ΔX and X correspond to ΔX̂ and X̂ with S_ΔX,Δ̄X, S_X̄,Δ̄X and S_Δ̄X,Δ̄X replaced by Σ_ΔX,Δ̄X, Σ_X̄,Δ̄X and Σ_Δ̄X,Δ̄X, respectively. Then 6 can be written as a regression model

17

with ℰXW′_t = 0. Note that this model has the form of 2.10 in Anderson (2).

From 16 and 17 we calculate

The process analogs of 7 and 8 are

18

19

These define the process canonical correlations and variates in the X-coordinates.

Sample Statistics

The canonical correlations and vectors depend on Ŝ, Ŝ, and Ŝ, which in turn depend on the submatrices of S_X̄X̄, S_X̄,Δ̄X, and S_Δ̄X,Δ̄X (equivalently S, S̃, S̃). The vector X̃_t satisfies the first-order stationary autoregressive model 14. The sample covariance matrices S̃_XX, S̃_WX, and S_WW are consistent estimators of Σ̃, 0, and Σ_WW, and S̃ = (S̃_XX − Σ̃), S̃ = S̃_WX, S = (S_WW − Σ_WW) have a limiting normal distribution with means 0 and covariances that have been given in refs. 2 and 11.

Let W(u) be the Brownian motion process defined by T⁻ ∑ W_t →^wW(u). Define I₁₁ by

See Anderson (2) and theorem B.12 of Johansen (6). Define J_j1 by

Then T⁻¹S →^dΓI₁₁Γ′ by 13, T⁻¹S̃_XX →^p0, and the Cauchy–Schwarz inequality.

We shall find the limit in distribution of S from the limit of S by using equation B.20 of theorem B.13 of Johansen (6). A specialization to the model here is

where W̃(u) = [W′₂(u), W′(u), 0]′. [In theorem B.13, let θ_i = (I, 0), ψ_i = (0, Υ̃ⁱ), ɛ′_t = (W′_1t, W̃′_t), and Ω = ℰɛ_tɛ′_t, V_t = X̃_t.] Then

Because {X̃_t} is stationary, T⁻¹ ∑ W_tX̃′_t−1 →^p0 and

20

Now we wish to show that ΔX and X lead to the same asymptotic results as ΔX̂ and X̂. First note that T⁻¹S →^dΓI₁₁Γ′ and T⁻¹ times any other sample covariance converges in probability to 0. Hence T⁻¹S →^dΓI₁₁Γ′ and T⁻¹Ŝ →^dΓI₁₁Γ′. Because {X̃_t} is stationary, {X} is stationary, and S →^pΣ, Ŝ →^pΣ. Moreover S̃ = (S̃_X̄X̄ − Σ̃) has a limiting normal distribution. Expansion of S and Ŝ in terms of the submatrices of S̃ shows that S and Ŝ have the same limiting normal distribution. (See Asymptotic Distribution of the Larger Roots.) Finally, T⁻S →^p0 and T⁻Ŝ →^p0 because S, S, S_Δ̄X,Δ̄X, and S and hence S have finite limits in distribution.

From 17 we find that plim_T→∞S = plim_T→∞Ŝ = Σ and

where S = S_WX̄ − S_W,Δ̄XΣΣ_Δ̄X,X̄, which converges in distribution to the right-hand side of 20.

As noted above, S →^dS, which consists of the first k rows of the weak limit of S. Then

Asymptotic Distribution of the Smaller Roots

Let Q⁺ = S S S. The n smaller roots of |Q⁺ − r²S| = 0 converge in probability to 0 and the k larger roots converge to the roots of

by the analysis of ref. 2. Let d = Tr². The n smaller roots of |Q − T⁻¹ dS| = 0 converge in distribution to the roots of

by the algebra used in ref. 2, section 5, resulting in the same distribution as in ref. 2. The likelihood ratio criterion for testing that the rank of Υ is k (2) is −2 log λ = −T ∑(1 − r) = ∑ d_i + o_p(1), the limiting distribution of which was found by Johansen (4, 6) and was given in equation 5.4 in ref. 2.

Asymptotic Distribution of the Larger Roots

We now turn to deriving the asymptotic distribution of the k larger roots of |Q⁺ − r²S| = 0 and the associated vectors solving Q⁺g = r²Sg. First we show that the asymptotic distribution of r, … , r is the same as the asymptotic distribution of the zeros of |Q − r²S|. Then we transform from the X-coordinates to the coordinates of the process canonical correlations and vectors.

Let R̂ = diag(r, … , r) and G′₂ = (G′₁₂, G′₂₂)′ consist of the corresponding solutions to Q⁺G₂ = SG₂R̂. The normalization of the columns of G₂ is G′₂SG₂ = I, that is,

21

The probability limit of 21 shows that G₁₂ = O_p(1) and G₂₂ = O_p(1). The submatrix equations in Q⁺G₂ = SG₂R̂ can be written as

22

23

Because T⁻¹Q →^p0, T⁻Q →^p0, T⁻S →^p0, T⁻¹S →^dΓI₁₁Γ′ and R̂ →^pR = diag(ρ, … , ρ), the probability limit of the left-hand side of 22 is 0; this shows that G₁₂ →^p0. Then the asymptotic distribution of G₂₂ is the asymptotic distribution of G₂₂ defined by

24

where the elements of R̂ are defined by |Q − r²S| = 0. Note that when G₁₂ →^p0 is combined with 23, we obtain QG₂₂ = SG₂₂R̂ + o_p (T⁻).

We proceed to find the asymptotic distribution of G₂₂ and R̂ defined by 24 in the manner of ref. 2. Let

where W_2⋅1,t = W_2t − Σ(Σ)⁻¹W_1t and ℰW_2⋅1,tW′_2⋅1,t = Σ = Σ − Σ (Σ)⁻¹Σ. We expand {Q − [Σ(Σ)⁻¹Σ]₂₂} to obtain

25

where Λ = Υ²²ΣΥ^22′ + Σ. See equation 6.5 of ref. 2.

To express the covariances of the sample matrices, we use the “vec” notation. For A = (a₁, … a_n), we define vec A = (a′₁, … , a′_n)′. The Kronecker product of two matrices A = (a_ij) and B is A ⊗ B = (a_ijB). A basic relation is vec ABC = (C′ ⊗ A) vec B, which implies vec xy′ = vec x1y′ = (y ⊗ x) vec 1 = y ⊗ x. Define the commutator matrix K as the (square) permutation matrix such that vec A′ = K vec A for every square matrix of the same order as K.

Define C = (I, −Σ_2,Δ̄XΣ) and D = [I, −Σ(Σ)⁻¹, 0]. Then X = CX̃_t, W_2⋅1,t = DW̃_t, DΣ̃_WW = ΣJ′_(k), CΣ̃ = ΣI′_(k), J′_(k) = (I, 0, I, 0), I′_(k) = (I, 0), Σ = CΣC′, and Σ = DΣ̃_WWD′.

Theorem 1. If the W_t are independently normally distributed, S, S, and S have a limiting normal distribution with means 0, 0, and 0 and covariances

26

27

28

29

30

31

Lemma 1. If X is normally distributed with ℰX = 0 and ℰXX′ = Σ, then ℰ vec XX′(vec XX′)′ = (I + K)(Σ ⊗ Σ) + vec Σ(vec Σ)′. If X and Y are independent, ℰ vec XX′(vec YY′)′ = vec ℰXX′ ⊗ (vec ℰ′YY′)′, ℰ vec XY′(vec XY′)′ = ℰYY′ ⊗ ℰXX′, and ℰ vec XY′(vec(YX′)′ = KℰXX′ ⊗ ℰYY′.

Proof of Theorem 1: First 26 is equivalent to the first expression in Lemma 1. Next vec S = T⁻ ∑(X ⊗ W_2⋅1,t) implies 27 because X and W_2⋅1,s are independent for t − 1 ≤ s. Similarly 28 follows. To prove 29, 30, and 31, we use the following lemma.

Lemma 2.

Proof of Lemma 2: We have from X̃_t = Υ̃X̃_t−1 + W̃_t

32

Because S̃_XX − S̃_X̄X̄ = (1/T)(X̃_TX̃′_T − X̃₀X̃′₀) and {X̃_t} is a stationary process, S̃_X̄X̄ in 32 can be replaced by S̃_XX + o_p(1). Then Lemma 2 results from 32 and vec Υ̃S̃_X̄W = K vec S̃_WX̄Υ̃′.▪

Lemma 3.

Proof of Lemma 3: Write W_2t = W_2⋅1,t + Σ(Σ)⁻¹W_1t. Then

from which the lemma follows.▪

Proof of Theorem 1 Continued: Then 29 follows from Lemma 2, 26, and 28, and 30 follows from Lemma 2, 27, and 28 for X̃_t. To prove 31, use Lemma 2, 26, 27, and 10 to obtain

33

Then substitution of Σ̃_WW = Σ̃ − Υ̃Σ̃Υ̃′ in 33 yields 31.▪

Let Ξ be a k × k matrix such that Ξ′(Υ²²ΣΥ^22′)Ξ = Θ and Ξ′ΣΞ = I, where Θ = diag(θ_n+1, … , θ_p) = R(I − R)⁻¹, R = diag(ρ, … , ρ), and ρ is a root of 18 with 0 < ρ < … < ρ. Let U = Ξ′X, V_2t = Ξ′W_2t, V_1t = W_1t, Δ₂ = Ξ′(Υ²² + I)(Ξ′)⁻¹, M₂ = Ξ′Υ²²(Ξ′)⁻¹, Ξ̃ = diag[Ξ, I_m−1 ⊗ diag(I_n, Ξ)], Δ̃ = Ξ̃′Υ̃(Ξ̃′)⁻¹, Ũ_t = Ξ̃′X̃_t, C_U = Ξ′C(Ξ̃′)⁻¹. Then {Ũ_t} is generated by Ũ_t = Δ̃Ũ_t−1 + Ṽ_t, where Ṽ_t = Ξ̃′W̃_t and U satisfies U = Δ₂U + V_2t, ΔU = M₂U + V_2t. Multiplication of 25 on the left by Ξ′ and right by Ξ yields

34

Theorem 2. If the V_t are independently normally distributed, S, S, and S have a limiting normal distribution with means 0, 0, and 0 and covariances

35

Let L_2,t−1 = M₂U_2,t−1(= Ξ′Υ²²X_2,t−1). Then 34 becomes

The covariances of the limiting normal distribution of vec S, vec S = (M₂ ⊗ I) vec S, and vec S = (M₂ ⊗ M₂) vec S are found from Theorem 2. We write the transform of 35 as

36

where

37

Let H₂₂ = (M′₂)⁻¹Ξ⁻¹G₂₂ [= Ξ⁻¹(Υ²²′)⁻¹G₂₂]. Then QG₂₂ = SG₂₂R̂ and G′₂₂SG₂₂ = I transform to

38

Because (SSS)₂₂ →^pΘR and S →^pΘ, the probability limits of 38 and h_ii > 0 imply H₂₂ →^pΘ⁻.

Define H = (H₂₂ − Θ⁻) and R̂ = (R̂ − R). Then we can write 38 as

39

40

where

Lemma 4.

41

Lemma 5.

42

Proof of Lemma 5: We use the facts that M₂ = Δ₂ − I, J_(k)M₂ = Δ̃I_(k) − I_(k) = (Δ̃ − I)I_(k), and (I + K)K = I + K. Then the left-hand side of 42 is

which is the right-hand side of 42.▪

Theorem 3. If Z_t are normally distributed and the roots of 18 are distinct,

43

Proof:  Theorem 4 follows from Theorem 2, 37, 41, 42, and the transpose of 42 and the fact that K(R ⊗ R) = [(I − R) ⊗ R]K(Θ ⊗ Θ)[(I − R) ⊗ R].▪

Note that 43 is equation 6.14 of ref. 2 with Φ⁺ replacing Φ.

Let Ẽ = ∑ ɛ_i(ɛ′_i ⊗ ɛ′_i), where ɛ_i is the k-vector with 1 in the ith position and 0s elsewhere. The matrix Ẽ has 1 in the ith row and i,ith column and 0s elsewhere. Define r^2∗ = (r, … , r)′. Then

The matrix Ẽ has the effect of selecting the i,ith element of Θ⁻PΘ⁻ and placing it in the ith position of r^2∗.

Theorem 4. If the Z_t vectors are independently normally distributed and the roots of 18 are distinct, the limiting distribution of r^2∗ is normal with mean 0 and covariance matrix

44

In terms of the components of r^2∗ the asymptotic covariance of r and r is 2[(1 − ρ)²φρ + ρ(φ(1 − ρ)²]. Here φ denotes the element in the ith row of the ith block of rows and the jth column of the jth block of columns in Φ⁺.

We now derive the limiting distribution of H = H + H, where H = diag(h, … , h). From vec HR = (R ⊗ I) vec H, and vec RH = (I ⊗ R) vec H we obtain vec(HR − RH) = NH = NH, where

The Moore–Penrose generalized inverse of N (denoted N⁺) has a 0 where N has a 0 and has (ρ − ρ)⁻¹ where N has (ρ − ρ), i ≠ j. Note that NN⁺ = (I ⊗ I) − E, where E = ∑ (ɛ_i ⊗ ɛ_i)(ɛ′_i ⊗ ɛ′_i). The k² × k² matrix E is idempotent of rank k; the k² × k² matrix NN⁺ is idempotent of rank k² − k; and E is orthogonal to N and N⁺.

From 39 we obtain vec H = N⁺(Θ⁻ ⊗ Θ⁻¹)vec P. From 40 we find H = −½Θ⁻ diagS and vec H = −½ EΘ⁻ vec S.

Theorem 5. If the Z_t vectors are independently normally distributed and the roots of 18 are distinct, vec H and vec H have a limiting normal distribution with means 0 and 0 and covariances

and

respectively.

From G₂₂ = Υ²²′ΞH₂₂ we can transform Theorem 5 into the asymptotic covariances of vec G₂₂ = (I ⊗ Υ²²′Ξ) vec H₂₂.