A mixture autoregressive model based on Gaussian and Student’s t-distributions

C.1 Proof of Theorem 1

Suppose { y t } t = 1 ∞ is a G-StMAR process. Then the process y _t = (y _t , …, y_t−p+1) is clearly a Markov chain on R p . Let y ₀ = (y₀, …, y_−p+1) be a random vector whose distribution is characterized by the density function f ( y 0 ; θ ) = ∑ m = 1 M 1 α m n p ( y 0 ; μ m 1 p , Γ m , p ) + ∑ m = M 1 + 1 M α m × t p ( y 0 ; μ m 1 p , Γ m , p , ν m ) . According to Eqs. (2.3)–(2.5), (2.8)–(2.10), (2.13), and (2.15), the density of the conditional distribution of y₁ given y ₀ is

The random vector ( y ₁, y ₀) therefore has the density function

Using properties of marginal densities of multivariate normal and t-distributions, by integrating y_−p+1 out we obtain the density of y ₁ as f ( y 1 ; θ ) = ∑ m = 1 M 1 α m n p ( y 1 ; μ m 1 p , Γ m , p ) + ∑ m = M 1 + 1 M α m × t p ( y 1 ; μ m 1 p , Γ m , p , ν m ) .^[11] Thus, the random vectors y ₀ and y ₁ are identically distributed. As the process { y t } t = 1 ∞ is a (time homogeneous) Markov chain, it follows that { y t } t = 1 ∞ has a stationary distribution π _y (⋅) characterized by the density f ( ⋅ ; θ ) = ∑ m = 1 M 1 α m n p ( ⋅ ; μ m 1 p , Γ m , p ) + ∑ m = M 1 + 1 M α m t p ( ⋅ ; μ m 1 p , Γ m , p , ν m ) (Meyn and Tweedie 2009, pp. 230–231).

For ergodicity, let P y p ( y , ⋅ ) = P ( y p ∈ ⋅ | y 0 = y ) signify the p-step transition probability measure of the process y _t . Using the pth order Markov property of y _t , it is easy to check that P y p ( y , ⋅ ) has the density

Clearly f( y _p | y ₀; θ ) > 0 for all y p ∈ R p and all y 0 ∈ R p , so we can conclude that y _t is ergodic in the sense of Meyn and Tweedie (2009, Ch. 13) by using arguments identical to those used in the proof of Theorem 1 in Kalliovirta, Meitz, and Saikkonen (2015). □

C.2 Proof of Theorem 2

First note that L T ( c ) ( θ ) is continuous, and that together with Assumption 1 of the main paper it implies existence of a measurable maximizer θ ̂ T . In order to conclude strong consistency of θ ̂ T , it needs to be shown that (see, e.g., Newey and McFadden 1994, Theorem 2.1 and the discussion on page 2122)

1. the uniform strong law of large numbers holds for the log-likelihood function; that is, sup θ ∈ Θ L T ( c ) ( θ ) − E L T ( c ) ( θ ) → 0 almost surely as T → ∞,

2. and that the limit of L T ( c ) ( θ ) is uniquely maximized at θ = θ ₀.

Proof of (i). Because the initial values are assumed to be from the stationary distribution, the process y _t = (y _t , …, y_t−p+1), and hence also y _t , is stationary and ergodic, and E L T ( c ) ( θ ) = E l t ( θ ) . To conclude (i), it thus suffices to show that E sup θ ∈ Θ l t ( θ ) < ∞ (see Rao 1962). This is done by using compactness of the parameter space to derive finite lower and upper bounds for l _t ( θ ) which is given by

We know from the structure of the parameter space that c 1 ≤ σ m 2 ≤ c 2 and c₁ ≤ α _m ≤ 1 − c₁ for all m = 1, …, M, and c₃ ≤ ν _m ≤ c₂ for all m = M₁ + 1, …, M, for some 0 < c₁ < 1, c₂ < ∞ and c₃ > 2. Because the exponential function is bounded from above by one on the non-positive real axis, and in addition c 1 ≤ σ m 2 , there exists a constant U₁ < ∞ such that

for all m = 1, …, M₁.

We also have c₃ ≤ ν _m + p ≤ c₂ + p for all m = M₁ + 1, …, M. Combined with the fact that the Gamma function is continuous on the positive real axis, this implies that there exist constants c₄ > 0 and c₅ < ∞ such that

for all m = M₁ + 1, …, M. Because Γ _m and hence Γ m − 1 is positive definite, σ m 2 ≥ c 1 and c₃ ≤ ν _m ≤ c₂, we can find some c₆ > 0 such that

for all m = M₁ + 1, …, M. Combined with (C.7) and (C.8), the inequality −(1 + ν _m + p)/2 < 0 implies that there exists a constant U₂ < ∞ for which

for all m = M₁ + 1, …, M. According to (C.6), (C.9) and the restriction 0 ≤ α_m,t ≤ 1, there exists a constant U₃ < ∞ such that

We know from compactness of the parameter space that

implying

for all m = 1, …, M₁, and for some finite constant c₇. By σ m 2 ≤ c 2 it also holds that ( 2 π σ m 2 ) − 1 / 2 ≥ ( 2 π c 2 ) − 1 / 2 , so

for all m = 1, …, M₁.

Accordingly, since σ m , t 2 ≥ c 6 and ν _m ≥ c₃, it holds for some c₈ < ∞ that

Thus, because ν _m ≤ c₂ and the inner functions below take values larger than one, we have

As Meitz, Preve, and Saikkonen (2021) state in the proof of Theorem 3, the quadratic form on the right-hand-side of (C.8) satisfies

for all m = M₁ + 1, …, M, and for some c₉ < ∞. Since also 0 < ν _m − 2 ≤ c₂ and σ m 2 ≤ c 2 , we have σ m , t 2 ≤ c 10 ( 1 + y t − 1 ′ y t − 1 ) for some finite constant c₁₀. Combining the former inequality with (C.7) and (C.15) yields a lower bound

Finally, the restriction ∑ m = 1 M α m , t = 1 together with (C.13) and (C.17) implies

As E y t 2 + y t − 1 ′ y t − 1 < ∞ (because y _t is stationary and has finite second moments), it follows from Jensen’s inequality that

The upper bound (C.10) together with (C.18) and finiteness of the aforementioned expectations shows that E sup ( θ , ν ) ∈ Θ l t ( θ ) < ∞ . □

Proof of (ii). Given that condition (3.3) of the main paper sets a unique order for the mixture components, proving that this identification condition is satisfied amounts to showing that E l t ( θ ) ≤ E l t ( θ 0 ) , and that the equality E l t ( θ ) = E l t ( θ 0 ) implies

for some permutations {τ₁(1), …, τ₁(M₁)} and {τ₂(M₁ + 1), …, τ₂(M)}. For notational clarity, we omit the subscripts from y _t and y _t−1, and write μ_m,t = μ( y ; ϑ _m ), σ m 2 = σ m 2 ( ϑ m ) , σ m , t 2 = σ m , t 2 ( y ; ϑ m , ν m ) for the expressions in (C.5) making clear their dependence on the parameter value. We leave the dependence of α_m,t on θ and y unmarked and denote by α_m,0,t mixing weights based on the true parameter value.

Making use of the fact that the density function of (y _t , y _t−1) has the form f ( ( y t , y t − 1 ) ; θ ) = ∑ m = 1 M 1 α m n p + 1 ( ( y t , y t − 1 ) ) ; μ m 1 p + 1 , Γ m , p + 1 + ∑ m = M 1 + 1 M α m t p + 1 ( ( y t , y t − 1 ) ) ; μ m 1 p + 1 , Γ m , p + 1 , ν m (see proof of Theorem 1) and reasoning based on Kullback–Leibler divergence, one can use arguments analogous to those in Kalliovirta, Meitz, and Saikkonen (2015, p. 265) to conclude E l t ( θ ) − E l t ( θ 0 ) ≤ 0 with equality if and only if for almost all ( y , y ) ∈ R p + 1

For each fixed y at a time, the mixing weights, conditional means and variances in (C.21) are constants, so we may apply the result on identification of finite mixtures of normal and t-distributions in Holzmann, Munk, and Gneiting (2006, Example 1) (their parametrization of the t-distribution slightly differs from ours, but identification with their parametrization implies identification with our parametrization). For each fixed y , there thus exists a permutation {τ₁(1), …, τ₁(M₁)} (that may depend on y ) of the index set {1, …, M₁} such that

for almost all y ∈ R (m = 1, …, M₁). Analogously, for each fixed y there exists a permutation {τ₂(M₁ + 1), …, τ₂(M)} (that may depend on y ) of the index set {M₁ + 1, …, M} such that

for almost all y ∈ R (m = M₁ + 1, …, M).

As argued by Kalliovirta, Meitz, and Saikkonen (2015, pp. 265–266) for the GMAR type components, it follows from (C.22) that ϑ m = ϑ τ 1 ( m ) , 0 and α m = α τ 1 ( m ) , 0 for m = 1, …, M₁. Accordingly, Meitz et al. (2021) showed that (C.23) implies ϑ m = ϑ τ 2 ( m ) , 0 , ν m = ν τ 2 ( m ) , 0 and α m = α τ 2 ( m ) , 0 for m = M₁ + 1, …, M, completing the proof of strong consistency.

Given consistency and assumptions of the theorem, asymptotic normality of the ML estimator can now be concluded using standard arguments. The required steps can be found, for example, in Kalliovirta, Meitz, and Saikkonen (2016, proof of Theorem 3). We omit the details for brevity.□