3. DECOMPOSITIONS OF THE PROCESS

The process X_t is decomposed in two ways to facilitate the subsequent asymptotic analysis. The first decomposition concerns the stochastic part of the process whereas the second decomposition disentangles deterministic and stochastic parts of the process.

The first decomposition separates the eigenvalues of the companion matrix B for the stochastic part of the process. Following Herstein (1975, p. 308) there exists a regular, real matrix M that transforms B into a real block diagonal matrix with blocks U, V, and W having eigenvalues with absolute value less than one, equal to one, and larger than one, respectively. That is,

For the purpose of proving the results of Section 2 it can be assumed without loss of generality that B = diag(U,V,W) is block diagonal and S_t = (U_t′,V_t′,W_t′,D_t′)′.

The second decomposition seeks to separate the stochastic and deterministic components and is based on two arguments. The processes U_t,W_t,D_t are first separated by a similarity transformation using that the matrices U, W, and D have no common eigenvalues, whereas the processes V_t,D_t have to be discussed in more detail because the matrices V and D may in general have common eigenvalues.

The processes U_t,W_t are linear functions of the deterministic process D_t, and they are first shown to satisfy the relationships

with

. The argument is the same in both cases. Taking U_t as an example consider the companion matrix for the vector (U_t′,D_t′) and apply a similarity transformation of the form

The result (3.1) then follows by arguing that

can be chosen so

. Because the matrices U and D have no common eigenvalues a unique solution

can be found according to Gantmacher (2000, p. 225).

In the special case where B has no eigenvalues in common with D the same argument can be made for the entire process X_t as for the U_t,W_t processes. That is,

When it comes to the general situation where V and D are allowed to have common eigenvalues it is convenient first to discuss the special case where V and D have their eigenvalues at one and are both Jordan matrices

with (Λ,E) = (1,1). In that situation V_t will be shown to satisfy

with

and where

is of the form (3.4) with

. To see this write the process (V_t,D_t) in companion form as

which has the solution

The result (3.5) is therefore a consequence of the following two properties:

for some matrices

. The second property is proved as follows. Because V and D are both Jordan blocks of the form (3.4) with eigenvalues at one a real matrix M exists so that

is a block diagonal matrix with diagonal elements that are Jordan blocks of the form (3.4) with (Λ,E) = (1,1). The left-hand side of (3.7) can then be written as a sum:

for some matrices μ_j,D_j,0 and Jordan blocks D_j. Writing the vector D_j,0 as T(0,1)′ where T is an upper triangular band matrix and noting that upper triangular band matrices commute then μ_jD_j^tD_j,0 = μ_jD_j^tT(0,1)′ = μ_jTD_j^t(0,1)′. This in turn can be written as

, ensuring the desired result (3.7) with

.

In general V and D can have eigenvalues anywhere on the unit circle. Suppose these occur at l distinct complex pairs exp(iθ_j) and exp(−iθ_j) for 0 ≤ θ_j ≤ π, which of course reduce to a single eigenvalue of 1 or −1 if θ_j equals 0 or π. Following Herstein (1975, p. 308) and using Assumption 2.3 to D there exist regular, real similarity transformations M_V and M_D that block-diagonalize V and D as

where the subblocks V_j,m and D_j are real Jordan matrices of the form (3.4) and where (Λ,E) is one of the pairs

Using the same argument as before it therefore holds in general that V_t has representation (3.5) where

has subcomponents

of dimension dim V_j and dim V_j,m, respectively, and

has subcomponents

of dimension

.

Combining the results (3.1), (3.2), and (3.5) with the notation

shows that the process without loss of generality can be represented as

It is convenient also to introduce the dimensions

and also the maxima

5. THE ORDER OF MAGNITUDE OF THE PROCESS

In the following discussion the order of magnitude of the process X_t is investigated. This is a generalization of Lai and Wei (1985, Theorem 1) where the case without deterministic components is considered. Subsequently a convergence result is given for the explosive component W_t.

To describe the order of magnitude of X_t let λ_j denote the distinct eigenvalues of B whereas m_j is the multiplicity of λ_j. Define the multiplicity of the largest eigenvalue as

The following result then holds.

THEOREM 5.1. Suppose Assumptions 2.1 and 2.3 are satisfied. Then, for ξ < γ/(2 + γ),

Proof of Theorem 5.1. By (3.9) it holds that

. Lai and Wei (1985, Theorem 1) show the results for the purely stochastic component

, whereas the order of the deterministic component

follows from Lemma 4.2. █

When studying the process

Lai and Wei (1985) use the following generalization of the Marcinkiewicz–Zygmund theorem.

THEOREM 5.2 (Lai and Wei, 1983a, Corollaries 3 and 4). Suppose Assumptions 2.1 and 2.2 are satisfied. Then for any sequence of matrices A_t the series

converges a.s. if and only if the series

converges. If this holds, and A_t ≠ 0 for infinitely many t, then

for any variable Y that is

-measurable for some t.

This result yields a more precise statement about the order of magnitude of the explosive component.

COROLLARY 5.3. Suppose Assumptions 2.1–2.3 are satisfied. Then

Proof of Corollary 5.3.

6. CORRELATION BETWEEN STATIONARY AND DETERMINISTIC COMPONENTS

One major difference between the results presented here and the work of Lai and Wei (1985) is that deterministic terms are included in the model. Before turning to the question of how big the denominator matrix can be in Section 7 it is convenient to consider the asymptotic order of magnitude of correlations between the zero mean process with roots smaller than one,

, and the deterministic component, D_t.

As a first step toward discussing the sample correlation of

, results of Lai and Wei concerning the matrices

are stated. The results give conditions for relative compactness of sequences of such matrices. Recalling that the relative compactness of a sequence is the property that the limit points fall in a compact set, this enables a discussion of the order of magnitude of the sequence under weak assumptions. In particular, a condition is given ensuring that the limit points are bounded away from zero.

THEOREM 6.1 (Lai and Wei, 1985, Theorem 2, equation (3.7), Example 3). Suppose Assumption 2.1 is satisfied. Then, with probability one, the matrix sequences

are relatively compact with the same limit points.

If in addition Assumption 2.2 is satisfied, the limit points are positive definite.

Because e_U,t is a linear combination of ε_t the sequence

is therefore relatively compact. In addition the following results can be shown.

THEOREM 6.2 (Lai and Wei, 1985, Theorem 2, Example 3). Suppose Assumptions 2.1 and 2.2 are satisfied. Then it holds with probability one that

and also it holds that

Before turning to the sample correlation of

it is useful to cite the following univariate result by Wei (1985).

LEMMA 6.3 (Wei 1985, Lemma 2). Suppose Assumption 2.1 is satisfied. Let (x_t) be a sequence of random variables adapted to

with

. Assume

for some η > 0. Then

The result for the sample correlation of

can now be stated and proved.

THEOREM 6.4. Suppose Assumptions 2.1–2.3 are satisfied. Then, for all η > 0,

Proof of Theorem 6.4. Theorem 6.2 shows that

, so it suffices to show that

.

The main contribution arises from the sum

for an α satisfying 1 > α > 0. With this in mind and using

the object of interest can be written as

The first two terms in (6.1) are o(T^η). To see this bound their norm by

and use Theorems 4.1 and 5.1 and that ∥U∥^Tα decreases exponentially.

The third term in (6.1) is o(T^η). To see this use D_t = D^sD_t−s and the normalization N_D given in (4.1) to rewrite it as

The norm of this expression is bounded by

Because the sum

converges it suffices to show that the last two components can be approximated uniformly by a variable that is o(T^η).

The sum

is approximately equal to

, which converges to a positive definite matrix; see Theorem 4.1. The norm of the approximation error,

, is bounded by

, because of Theorem 4.1.

In a similar way

is approximately

, which is not dependent on s. Considering each element of this matrix and applying Lemma 6.3 shows that this is o(T^ηN_D⁻¹). The approximation error can be bounded by

Using Theorems 4.1 and 5.1 this is seen to be o(T^α−ξ/2), which is o(1) for a small α. █

Some immediate consequences of these results are the following examples.

Example 6.5

Suppose Assumptions 2.1 and 2.7 are satisfied. Then Theorems 6.1 and 6.2 imply

and in particular

for some matrix Ω_U, so

Example 6.6

Suppose Assumptions 2.1–2.3 are satisfied. Then Theorems 6.2 and 6.4 and equation (3.1) imply that the sequence of matrices

is relatively compact with positive definite limit points. Moreover, this series converges almost surely if

is convergent. According to Example 6.5 this is for instance the case if, in addition, Assumption 2.7 is satisfied.

7. THE LARGEST EIGENVALUE OF THE DENOMINATOR MATRIX

The order of magnitude of the largest eigenvalue of the denominator matrix

can now be described. This is followed by a convergence result for the purely explosive case and a bound for the rate of convergence of sum of powers of

.

First, the largest eigenvalue of M_T is considered in the following generalization of Lai and Wei (1985, Corollary 1).

THEOREM 7.1. Suppose Assumptions 2.1–2.3 are satisfied. Then

Proof of Theorem 7.1. If max|eigen(B)| < 1 then

by (3.1). Lai and Wei (1985, Corollary 1) show

and the result then follows from Theorems 4.1 and 6.4. If max|eigen(B)| ≥ 1 the result follows directly from Theorem 5.1. █

For the explosive part of the process the following generalization of Lai and Wei (1985, Corollary 2) can be established.

COROLLARY 7.2. Suppose Assumptions 2.1–2.3 are satisfied and min|eigen(B)| > 1 so

and recall the definition of W in Corollary 5.3. Then

where F_W is positive definite a.s.; hence

Proof of Corollary 7.2. Let R_T denote the difference between the matrices

. The decomposition (3.2) shows

which vanishes for large T because of Theorem 4.1(i) and Corollary 5.3(i). The desired result is then a direct consequence of Lai and Wei (1983b, Theorem 2). █

Whereas Theorem 7.1 gives a bound for the sum of squares of the process, the following result gives a bound for sums of higher order powers of the stationary component.

THEOREM 7.3. Suppose Assumption 2.1 is satisfied. Then, for all η > 0 and ζ < γ,

Proof of Theorem 7.3. For notational convenience define

. Using Hölder's inequality it follows that

Summation over t then gives the following bound:

Changing summation index in the double sum, this can be bounded further by

The first two sums converge, whereas the third term can be decomposed as

The latter term is of order O(T) = o(T^1+η) by Assumption 2.1. The first term is a martingale. Normalized by T^1+η it converges to zero a.s. on the set where

see Hall and Heyde (1980, Theorem 2.18). Minkowski's inequality shows that this sum is finite if the sum

is finite. Assumption 2.1 ensures that this is the case. █

8. THE SMALLEST EIGENVALUE OF THE DENOMINATOR MATRIX

Three results are given concerning the order of the inverse of the denominator matrix,

, of the least square estimator in the nonexplosive case. Using the techniques of Chan and Wei (1988) it can be proved that T⁻¹M_T is bounded from below in a weak convergence sense. The first result goes some way toward an almost sure version of this result in showing that the partial denominator matrix

is bounded from below, whereas the second result shows that the joint matrix T⁻¹M_T is bounded from below in the special case where B and D have no common eigenvalues. In combination these results can be used to establish the third result concerning the order of max_T^α≤t≤T S_t′M_T⁻¹S_t without actually establishing the order of M_T⁻¹, and this will suffice to prove the main theorems.

The first result concerns the partial denominator matrix

and is related to Lai and Wei (1985, Theorem 3).

THEOREM 8.1. Suppose Assumptions 2.1–2.3 are satisfied and |eigen(B)| ≤ 1. Then

To prove Theorem 8.1 the following lemma is needed. This lemma ensures that Lai and Wei (1982, Lemma 1) concerning the order of magnitude of normalized least squares estimators can be used.

LEMMA 8.2. Suppose Assumptions 2.1–2.3 are satisfied. Then

Proof of Lemma 8.2. It suffices to show that u′M_Tu > 0 for all u ∈ R^{dim S} so u ≠ 0 and some T. Because

it is equivalent that (S₀,…,S_T−1)R spans R^{dim S} for some invertible matrix R.

The decomposition (3.9) shows that

where

. The Cayley–Hamilton theorem (see Herstein, 1975, p. 334) implies that if

with d₀ = 1 is the characteristic polynomial of

then

and in particular

, say. Define

and partition R as a (2 × 2)-block matrix so the upper right block is a dim

-dimensional square matrix. The preceding properties then show that (S₁,…,S_T)R is an upper triangular (2 × 2)-block matrix. The lower right block is

, which spans R^{dim D} by Assumption 2.3. It is left to prove that the upper left block

spans R^{dim X}. The process z_t is a linear combination of the process

that satisfies a first-order autoregression without deterministic terms. The desired result then follows from Lai and Wei (1985, Theorem 3) using Assumptions 2.1 and 2.2. █

Proof of Theorem 8.1. Let m = dim X. Using the model equation (2.3) and that D_t−1−j = D^−1−jD_t the process X_t can be rewritten as

where e_X,0 = X₀, e_X,−t = −μD_t−1, and X_−t = 0 for t > 0. It follows that

It is now argued that the lower bound in (8.1) satisfies

The norm of the difference between the left-hand side and the first term on the right is bounded by

The first sum is finite, so when normalized by the denominator term it is seen to be O(1) as a result of Lemma 8.2. The normalized second term is O[(log T)^1/2] as a result of Lai and Wei (1982, Lemma 1), which can be used because of Lemma 8.2.

Using Lai and Wei (1982, Lemma 1) once again in combination with Theorem 6.2 shows that, for j < k,

This in turn implies that

The proof is completed by combining (8.2), (8.3), and Theorem 6.2. █

When B and D have no common eigenvalues Theorem 8.1 can be extended.

THEOREM 8.3. Suppose Assumptions 2.1–2.3 are satisfied and B and D have no common eigenvalues. Then lim inf_T→∞ λ_min(T⁻¹M_T) > 0 a.s.

Proof of Theorem 8.3. Because of the representation

given in (3.3) it suffices to show the result for sums of squares of

If

is the characteristic polynomial of B the Cayley–Hamilton theorem (see Herstein, 1975, p. 334) implies

, and hence

Because B and D have no common eigenvalues then

. It follows that

using Theorems 4.1, 6.2, and 6.4 and Lai and Wei (1985, equation (3.19)). The argument is finished as in the proof of Lai and Wei (1985, Theorem 3). █

The final and more technical result addresses the order of S_t′M_T⁻¹S_t. Lai and Wei (1985, Lemma 4) show that max_t≤T S_t′M_T⁻¹S_t vanishes when |eigen(B)| ≤ 1 and dim D = 0. For the subsequent analysis it suffices to take a maximum over just T^α ≤ t ≤ T for some 0 < α < 1 requiring that 0 < |eigen(B)| but allowing dim D > 0.

THEOREM 8.4. Suppose Assumptions 2.1–2.3 are satisfied and that 0 < |eigen(B)| ≤ 1. Then

for all α,ζ so 0 < α < 1 and ζ < min[γ/(2 + γ),½].

Theorem 8.4 will be proved in a few steps following Lai and Wei (1983b). The first step is to strengthen their Lemma 3.

LEMMA 8.5. Let (a_t) be a sequence of nonnegative numbers satisfying the following conditions.

Then a_T = o(T^−ρ) for all ρ < min(1,κ/2).

Proof of Lemma 8.5. Condition (i) implies that for every 0 < ρ < 1 it holds that min_{T>t≥T−T^ρ} a_t ≥ a_T − 2CT^ρ−κ for all large T. In particular, choosing ρ to satisfy 0 < ρ < min(1,κ/2), it is seen that

Combining this with (ii) it follows that

. Because δ can be chosen arbitrarily small this proves the desired result. █

The second step is to generalize Lai and Wei (1983b, Lemma 6ii).

LEMMA 8.6. Suppose Assumptions 2.1–2.3 are satisfied and max|eigen(B)| ≤ 1. Define T₀ as in Lemma 8.2. Then

Proof of Lemma 8.6. The proof is the same as that of Lai and Wei (1983b, Lemma 6ii) using the generalizations of their Theorem 3 and Lemma 6i presented previously in Theorem 7.1 and Lemma 8.2. █

The third step is to generalize Lai and Wei (1983b, Lemma 7).

LEMMA 8.7. Suppose Assumptions 2.1–2.3 are satisfied and that 0 < |eigen(B)| ≤ 1. Then

Proof of Lemma 8.7. (i) Using Lai and Wei (1983b, Lemma 5i) in the same way as in the proof of Lai and Wei (1983b, Lemma 7i) it holds that

where

This expression can be rewritten using the identities

for ι = (I_{dim X},0). The desired result follows by noting that

according to Lai and Wei (1982, Lemma 1), which can be used because of Lemma 8.2, whereas the term

by Theorem 8.1.

(ii) Noting that S_T = SS_T−1 + e_S,T it holds that

The norm of the first term is less than

by (i). This is in turn bounded by

because of Lemma 8.6(i). The second term equals

according to the identities (8.4) and is seen to be o(T^−ξ/2) by Theorems 5.1 and 8.1. █

Theorem 8.4 can now be proved.

Proof of Theorem 8.4. Lemmas 8.6(ii) and 8.7(ii) show that the conditions of Lemma 8.5 are satisfied for the sequence S_t′M_t⁻¹S_t with κ = ζ/2 and therefore t^ζ/4S_t′M_t+1⁻¹S_t = o(1) for large t. For t > T₀ then M_t⁻¹ > M_t+1⁻¹ so that S_t′M_T⁻¹S_t ≤ S_t′M_t+1⁻¹S_t, and thus for all ε > 0 and almost every outcome a T₁ exists so that for all t,T so that T > t ≥ T₁ it holds that S_t′M_T⁻¹S_t ≤ S_t′M_t+1⁻¹S_t < ε. This in turn implies that for all ε > 0 and almost every outcome a T₁ exists so that for all T so that T > T₁ it holds that max_T^α≤t<T S_t′M_T⁻¹S_t ≤ S_t′M_t+1⁻¹S_t < ε as desired. █

9. SAMPLE CORRELATIONS

It has already been established in Section 6 that the sample correlation of

and D_t vanishes asymptotically. In the following discussion the remaining sample correlations of pairs of the processes

are studied. A first result concerns the sample correlation of W_t and U_t,D_t.

THEOREM 9.1. Suppose Assumptions 2.1–2.3 are satisfied. Then

The bound for the sample correlation of

should be viewed in light of the results of Anderson (1959). He found that

is convergent when the innovations ε_t are independent, identically distributed, but in general divergent. The stated result combined with Theorem 6.1 shows that the order of Y_T is at most o[T^(1−ξ)/2] for martingale difference sequence innovations.

Proof of Theorem 9.1. The norms of the two expression are bounded by

where m is either of

The last two terms of (9.1) are convergent according to Corollaries 5.3 and 7.2. It holds that m_D = O(T⁻¹) by Theorem 4.1, whereas m_U = o(T^−ξ) because Theorem 5.1 shows

, whereas the denominator term is O(T⁻¹) by Example 6.6. █

For the sample correlation between W_t and S_t=(V_t′,D_t′)′ a different type of proof is needed using the results of Section 8. This is because the order of the smallest eigenvalue of

is unknown.

THEOREM 9.2. Suppose Assumptions 2.1–2.3 are satisfied. Then, for all ζ < min[γ/(2 + γ),½], it holds that

Proof of Theorem 9.2. For convenience define

Because

is convergent according to Corollary 7.2 it suffices to show that

. Follow Anderson (1959, Theorem 2.2) in writing

Because

it follows, for any 0 < α < 1, that

The first two terms in (9.3) vanish exponentially fast. Their norm is less than

where ∥W∥^Tα−T vanishes exponentially, M_T⁻¹ = O(1) by Lemma 8.2, and the remaining terms are of polynomial order according to Theorem 5.1 and Corollary 5.3.

The final term in (9.3) is o(T^−ζ/8). Its norm is less than

where the first two components converge (see Corollary 5.3) and the last component is o(T^−ζ/8) by Theorem 8.4. █

Remark 9.3. The bottleneck in the proof of Theorem 9.2 is the order of magnitude of max_T^α<t≤T S_t′M_T⁻¹S_t. By extending the weak convergence results of Chan and Wei (1988) it can be proved that this term is

when |eigen(B)| ≤ 1, implying that the sample correlation between W_t and (V_t′,D_t′) is

.

Wei (1992, Theorem A.1) considers the sample correlation between U_t and V_t in the univariate case dim X = 1 when D_t is absent and ε_t is a martingale difference sequence satisfying Assumptions 2.1 and 2.7. That result can be generalized and strengthened by a proof resembling that of Theorem 6.4.

THEOREM 9.4. Suppose Assumptions 2.1–2.3 are satisfied. Then, for all ξ < γ/(2 + γ),

Proof of Theorem 9.4. Define S_t, S, e_S,t, and M_T as in (9.2). Theorem 6.2 shows that

, so it suffices to show that

. Inspired by the proof of Theorem 6.4 and Wei (1992, Theorem A.1) write

for some 0 < α < 1 so that

The first term in (9.4) is o[T^(1−ξ)/2]. The norm of the sum

is bounded by

, which is of the desired order when α is chosen small enough and using Theorem 5.1, whereas M_T^−1/2 is bounded as a result of the positive definiteness of M_T stated in Lemma 8.2.

The second term in (9.4) is o(1). By Cauchy–Schwarz inequality its norm is bounded by

where ∥U∥^Tα vanishes exponentially and the other terms are O[(T log T)^1/2] as a result of Theorem 6.2 and Lemma 8.6(ii).

The third term of (9.4) is o[T^(1−ξ)/2]. Its norm is bounded by

according to the triangle inequality. Hölder's inequality implies

for 2 < p < 2 + γ and p⁻¹ + q⁻¹ = 1. Because q/2 < 1 and T^α > T₀ for large T, then (S_t′M_T⁻¹S_t)^q/2 ≤ S_t′M_T⁻¹S_t ≤ S_t′M_t⁻¹S_t, so the last term is o(T^η) for all η > 0 according to Lemma 8.6(ii). Because of Theorem 7.3 then

uniformly in s. Overall the sum in t is therefore o[T^(1−ξ)/2] uniformly in s. The desired result then follows because

converges. █

Tables 1 and 2 give an overview of the sample correlation results of Theorems 6.4, 9.1, 9.2, and 9.4. All pairs of

have been considered except for V_t,D_t, which has nonnegligible sample correlation when V and D have common characteristic roots. To produce these tables it is used that the marginal sample correlation, C(x,y), of processes x_t,y_t relates to joint correlations by

according to the formula for partitioned inversion, and also by

Order of pairwise sample correlations, with η > 0 and ξ < γ/(2 + γ)

Order of pairwise sample correlations, with η > 0 and ξ < γ/(2 + γ)

As a consequence of the results summarized in Table 2 the condition |eigen(B)| ≤ 1 can be eliminated in Theorem 8.1 concerning the lower bound for

.

COROLLARY 9.5. Suppose Assumptions 2.1–2.3 are satisfied. Then

Proof of Corollary 9.5. Let R_t = (U_t′,V_t′)′. Using a similarity transformation M as described in Section 3 and the results in Table 2 shows that

equals

a.s. Apply Theorem 8.1 to the upper left block and Corollary 7.2 and Theorem 9.1 to the lower left block. █