2. AN EXAMPLE OF HYPOTHESES ALLOWING FOR ASYMPTOTIC χ² TESTS

Denote by m_t the log nominal money stock, by p_t the log price level, by y_t log real income, and by R_t a long-term interest rate and define X_t = (m_t,p_t,y_t,R_t)′. Suppose m_t ∼ I(2), p_t ∼ I(2), y_t ∼ I(1), and R_t ∼ I(1). Moreover consider the following cointegration relations:

• m_t − p_t ∼ I(1) (i.e., log real money is I(1))
• m_t − p_t − y_t + β_R R_t ∼ I(0) (i.e., there exists a stationary money demand relation with unit income elasticity)
• R_t − Δp_t ∼ I(0) (i.e., the “Fisher effect” holds, meaning that the real interest rate is stationary)

The unit root and cointegration properties of the variables are in line with those found in Lütkepohl and Wolters (2003) for a system of German quarterly data except for some of the values assumed for the cointegration parameters. Therefore one may wish to formulate these hypotheses in the cointegrated model for I(2) variables and develop the likelihood ratio test to check whether the structure is compatible with the data.

We want to show that the hypotheses discussed earlier can be formulated as hypotheses on the parameters ρ, τ, and ψ that have the property that likelihood ratio tests of the restrictions are asymptotically χ².

Under the assumption that the process X_t is I(2) it holds that ρ′τ′X_t−1 + δτ_⊥′ΔX_t−1 and τ′ΔX_t are I(0). Therefore we want to express the preceding relations in terms of the parameters τ, β = τρ, ψ, and δ = ψ′τ_⊥(τ_⊥′τ_⊥)⁻¹. For our system p = 4, and from the cointegrating relations

we find that τ is 4 × 3, so that r + s = 3. We see that there are two relations that involve levels, so that r = 2, and β has dimension 4 × 2 and is given by

This shows that the hypotheses formulated previously imply that

Hence the relations fully specify the matrices τ, ρ, and β = τρ in this case. We denote the specific τ, ρ, and β matrices by τ₀, ρ₀, and β₀, respectively. With this notation the model reduces to

and we next find the implications of the assumptions for the 4 × 2 parameter ψ.

Because τ₀′ΔX_t−1 is stationary we decompose ψ′ΔX_t−1 as

where we used that δ is 2 × 1 and that τ₀(τ₀′τ₀)⁻¹τ₀′ + τ_0⊥(τ_0⊥′τ_0⊥)⁻¹τ_0⊥′ is equal to the identity matrix. Hence we can rewrite the equations in (1) as

Notice that the coefficients δ are identified by the choice of β = β₀ and τ_⊥ = τ_0⊥ and that

Now consider the stationary (multicointegrating) relation

By a linear transformation of the rows, which can be absorbed in α, we can eliminate δ₁(Δm_t + Δp_t) from the first equation and find that the model implies stationarity of the linear combinations

Because Δm_t + Δp_t = 2Δp_t + Δm_t − Δp_t and Δm_t − Δp_t is stationary, we have that the model, with τ₀ and β₀ as given before, allows the stationary relations

Hence we see that we can define β_R = −δ₁ /δ₂, and the only extra restriction we need to test is the hypothesis δ₂ = −0.5. Thus the restrictions formulated previously can be tested successively as the hypotheses

The first hypothesis is a test on cointegrating ranks, and the asymptotic distribution is nonstandard and tabulated by simulation (see Johansen, 1997). It follows from the results in the same paper (see also Boswijk, 2000; Johansen, 2005) that

are asymptotically distributed as χ²(4) and χ²(1), respectively. In general one cannot expect hypotheses on the coefficient δ to give asymptotic χ² tests (see Paruolo, 2000), but

specifies τ_⊥ completely, and when that is the case, one can in fact show that a test on δ, and hence

, is asymptotically distributed as χ²(1).

This can be seen by the “nominal-to-real” transformation (see Kongsted, 2005),

which satisfies a model of the form

Premultiplying with (τ₀,τ_0⊥)′ gives the I(1) cointegration model

with parameters

and

. Thus the transformed model is an I(1) model with linear restrictions on

partly known. A hypothesis on δ is therefore a hypothesis on

in an I(1) model, which is known to give asymptotic χ² tests (see Johansen, 1991).