Let α_* be any basis of col(Π) and β_* be any basis of col(Π′); we note col(α_⊥) = col(α_*⊥), col(β_⊥) = col(β_*⊥) so no distinction is made between α_⊥ and α_*⊥, β_⊥ and β_*⊥ in the following discussion.

1. Condition (b) is equivalent to rank(α_⊥′ β_⊥) = s < p − r; this implies that

has dimension p₂ := p − r − s. Because

, one has that p₂ ≤ r.

2. Condition (b) can be written as α_⊥′ β_⊥ = ξη′, with ξ, η (p − r) × s full column rank matrices; α_⊥ can thus be chosen as follows:

where a is r × p₂, p₂ ≤ r, b := (b₁′ : b₂′)′ is p × s, and q is of full column rank. The matrix α_⊥′ β_⊥ becomes

so that ξ = (0 : I_p2)′ and η = b₂; thus β₁ = β_⊥ b₂, β₂ = β_⊥ b_2⊥, α₁ = α_⊥(0 : I_p2)′, α₂ = α_⊥(I_s : 0)′ = β_* a.

Using orthogonal projections, one can represent α_* as α_* = (β_*,β_⊥)c; from α_⊥′α = 0 one deduces that c = q_⊥, where c = q_⊥ can be chosen as follows:

where v is a nonsingular square matrix and ω := −(b₂′(β_⊥′ β_⊥)⁻¹b₂)⁻¹b₁′ a_⊥. Note that the second block of columns is equal to β₂ = β_⊥κ, where we have named κ := b_2⊥.

Now let α := (β_* a_⊥ + β₁ω : β₂) and define β′ := α′Π; because β and β_* are two bases of col(Π′), they are related by the relation βu = β_* for a nonsingular matrix u. Substituting within α one finds α := (βφ + β₁ω : β₂) with φ := ua_⊥ of full column rank. Summarizing, the coefficients (α,β,Ω) are constrained as follows:

where β₁ = β_⊥κ_⊥, β₂ = β_⊥κ. The I(2) model is thus indexed by the parameters τ := (β, κ, φ, ω, Ω), which are freely varying.

3. From the I(2) representation theorem of Johansen (1992), one finds that δ := α′β₂. By (1) one sees that δ = (0,I_p2)′, which is full rank p₂. Let β¹ := β(I_r−p2 : 0)′ and β² := β(0,I_p2)′; one sees that β¹′X_t are the CI(2,2) relations and β²′X_t + (β₂′ β₂)⁻¹β₂′ΔX_t are the multicointegrating relations.

4. By the I(2) representation theorem of Johansen (1992), C₂ = β₂(α₂′θβ₂)⁻¹α₂′, where θ := (I + βα′). The choice β is by construction conformable with α := (βφ + β₁ω : β₂). Going back to the representation of α₂ one finds β_* = β(u′)⁻¹ and thus α₂ = β_* a = βφ_⊥, where we note that (u′)⁻¹a = φ_⊥. Thus α₂′θβ₂ = φ_⊥′ β′(I + βα′)β₂ = φ_⊥′ β′βα′β₂ = φ_⊥′ β′βδ, given the structure (1) of α, and hence

Note that C₂ depends on β, κ, φ and it does not depend on ω, Ω.

5. The likelihood function depends on τ through the product αβ′ and Ω. Specifically αβ′ = (βφ + β₁ω)β¹′ + β_⊥κβ²′, where β = (β¹ : β²) is partitioned conformably with α. It is simple to see that τ is not identified because one can insert two square invertible matrices φ_i, i = 1,2, and their inverses within the two factors in αβ′; i.e., τ := (β, κ, φ, ω, Ω) and τ* := (βφ′, κφ₂⁻¹, (φ′)⁻¹ φφ₁⁻¹, ωφ₁⁻¹, Ω) give the same likelihood, where φ := diag(φ₁,φ₂). Moreover because β_⊥, β₁ are any bases of col(β_⊥) and col(β₁), one also needs to identify them.

For any matrix γ, let a be of the same dimensions such that a′γ is nonsingular; define γ_a := γ(a′γ)⁻¹ (see, e.g., Johansen, 1996, ch. 5). To achieve identification one can identify β¹, β², β_⊥, and β₁ by defining β_a1¹, β_a2², β_⊥a3, and β_1a4 for appropriate matrices a_i, i = 1,2,3,4. This solves the identification problem for τ.