The Rescaled VAR Model with an Application to Mixed-Frequency Macroeconomic Forecasting

A.3.1 Preliminary Lemmas

The next two lemmas are well-known but are included here to make the paper self contained.

A.3.2 Theorem 3

Let θ = [A′, B′, Γ′]′. Process (3) is linear and Gaussian and therefore there exists a unique family of probability measures {P_θ,s,x: s ≥ 0, x ∈ ℝ^K} induced by the solutions of (3) for t ≥ s, given θ and X_s = x – see Pedersen (1995). Define the transition probability function implied by this family of probabilities as

for a Borel set D ∈ ℝ^K. Define Δ = t − s and let p(s, x, t, y; θ) denote the density of P with respect to the Lebesgue measure in ℝ^K. The Euler-Maruyama approximation of order N = 1, 2, … under P_θ,s,x to the continuous time stochastic process, then, is

where Wtθ,s=∫st(Γ′Γ)−1/2d[Xu−x−∫suAXv+Bdv], for t ≥ 0, is a K-dimensional Wiener process starting at time s, under P_θ,s,x. Then it can be shown that X_nN → X_t in L¹ norm (with measure P_θ,s,x) as N diverges (Pedersen, 1995). The transition probabilities of the Euler-Maruyama approximation are such that

where the expectation is taken with respect to P_θ,s,x. Intuitively, p^(N) is equal to the probability that the Euler-Maruyama process will transition to y at time t, from the position XnN−1(N) that it is expected to take at t − Δ/N. Then p^(N) → p as N → ∞ (Pedersen, 1995, Theorem 1). Backward substitution in the Euler-Maruyama N-approximation yields

where εnj=Wnjθ,s−Wnj−1θ,s. It follows that XnN−1(N) is normal with

Together with (12), these imply that the probability that XnN(N)=y at n_N = t can be found by taking the expectation of the following expression

Moreover, because XnN−1(N) is normal, we have that p^(N)(s, x, t, y; θ) also is normal with mean (I+ΔNA)Nx+∑i=0N−1(I+ΔNA)iΔNB and covariance matrix ΔN∑i=0N−1(I+ΔNA)iΓ′Γ(I+ΔNA)i′.

Define now S=∑i=0∞(I+ΔNA)iΓ′Γ(I+ΔNA)i′. Under Assumption 2 and Assumption 3, none of the eigenvalues of I+ΔNA can be the reciprocal of another one and therefore S is the unique solution to the Lyapunov equation (9), and it satisfies equation (10). Now, from the definition of S it follows that

which shows that p^(N)(s, x, s + Δ, y; θ) is normal with mean

and covariance

From Lemma 1 the limiting distribution of p^(N) as N → ∞ is normal with mean

which is the desired result for the mean. To derive the covariance matrix, use Lemma 1 and Lemma 2 and since Assumption 2 ensures that A ⊕ A is invertible, we have

Finally,

This completes the proof.

A.4 Proof of Theorem 4

Let ℓ({Xi}i=1T;C1,Φ1,Σ1) denote the likelihood function for the observed data set, given X₀; we have

where m = [I − Φ₁]⁻¹C₁. The maximum likelihood estimates for the LF-VAR are (C~1,Φ~1,Σ~1)=arg⁡maxlog⁡ℓ({Xi}i=1T;C1,Φ1,Σ1). Under the assumption that (3) is the DGP we can also express the likelihood of these data as follows

where p is the transition probability function defined in the proof of Theorem 3: given X(t) at t the distribution of X(t + 1) at t + 1 is normal with mean e^AX(t) + [e^A − I]A⁻¹B and (vectorized) variance vec(Σ) = (I − e^A⊕A)(−A ⊕ A)⁻¹vec(Γ^′Γ). Define now X⁰(t) ≡ X(t) + A⁻¹B, so that p(t, X⁰(t), t + 1, X⁰(t + 1)) is normal with mean e^AX⁰(t) and variance Σ. It follows that this likelihood satifies

Under the mapping defined by the rescaling algorithm, we have that (i) e^A = Φ₁; (ii) C₁ = (e^A − I)⁻¹A⁻¹B so that m = −A⁻¹B; and (iii) vec(Σ) = (e^A⊕A − I)(A ⊕ A)⁻¹vec(Γ^′Γ) = vec(Σ₁); and therefore the rescaling algorithm maps one-to-one the maximizers of (13) into the maximizers of (14). It is now immediate from Theorem 3 that the rescaling algorithm also maps one-to-one the maximizers of (14) into the maximizers of the likelihood of the (unobserved) data set in which all the variables are measured at the higher frequency

and therefore the matrices C̃_τ, Φ~τ, and Σ~τ are, asymptotically, ML estimates of the R-VAR’s parameters. The distributions of the estimators follow from standard results; see, for example, (Lütkepohl, 2007, Proposition 3.1).