Supplementary Material for "Estimation of a Multiplicative Correlation Structure in the Large Dimensional Case"

10/16/2018 ∙ by Christian M. Hafner, et al. ∙ University of Cambridge Université catholique de Louvain FUDAN University 0

Supplementary Material for "Estimation of a Multiplicative Correlation Structure in the Large Dimensional Case"

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8 Supplementary Material

This section contains supplementary materials to the main article. SM 8.1 contains additional materials related to the Kronecker product (models). SM 8.2 gives a lemma characterising a rate for , which is used in the proofs of limiting distributions of our estimators. SM 8.3, SM 8.4, and SM 8.5 provide proofs of Theorem LABEL:thm_asymptotic_normality_MD_when_D_is_unknown, Theorem LABEL:prop_Haihan_score_functions_and_second_derivatives, and Theorem LABEL:thm_one_step_estimator_asymptotic_normality, respectively. SM 8.6 gives proofs of Theorem LABEL:thm_overidentification_test_fixed_dim and Corollary LABEL:coro_diagonal_asymptotics. SM 8.7 contains miscellaneous results.

8.1 Additional Materials Related to the Kronecker Product

The following lemma proves a property of Kronecker products.

Lemma 8.1.

Suppose and that are real symmetric and positive definite matrices of sizes , respectively. Then

Proof.

We prove by mathematical induction. We first give a proof for ; that is,

Since are real symmetric, they can be orthogonally diagonalized: for , where is orthogonal, and is a diagonal matrix containing those eigenvalues of . Positive definiteness of ensures that their Kronecker product is positive definite. Then the logarithm of is:

(8.1)

where the first equality is due to the mixed product property of the Kronecker product, and the second equality is due to a property of matrix functions. Next,

(8.2)

where the third equality holds only because and have real positive eigenvalues only and commute for all (Higham (2008) p270 Theorem 11.3). Substitute (8.2) into (8.1):

We now assume that this lemma is true for . That is,

(8.3)

We prove that the lemma holds for . Let and .

where the third equality is due to (8.3). Thus the lemma holds for . By induction, the lemma is true for . ∎


Next we provide two examples to illustrate the necessity of an identification restriction in order to separately identify log parameters.

Example 8.1.

Suppose that . We have

Then we can calculate

Log parameters can be separately identified from the off-diagonal entries of because they appear separately. We now examine whether log parameters can be separately identified from diagonal entries of . The answer is no. We have the following linear system

Note that the rank of is 3. There are three effective equations and four unknowns; the linear system has infinitely many solutions for . Hence one identification restriction is needed to separately identify log parameters . We choose to set .

Example 8.2.

Suppose that . We have

Then we can calculate

Log parameters can be separately identified from off-diagonal entries of because they appear separately. We now examine whether log parameters can be separately identified from diagonal entries of . The answer is no. We have the following linear system

Note that the rank of is 4. There are four effective equations and six unknowns; the linear system has infinitely many solutions for . Hence two identification restrictions are needed to separately identify log parameters . We choose to set .

8.2 A Rate for

The following lemma characterises a rate for , which is used in the proofs of limiting distributions of our estimators.

Lemma 8.2.

Let Assumptions LABEL:assu_subgaussian_vector(i) and LABEL:assu_mixing be satisfied with . Suppose if . Then

Proof.

Let denote , similarly for , where . Let denote , similarly for where .

(8.4)
(8.5)
(8.6)
(8.7)

Display (8.5)

Assumption LABEL:assu_subgaussian_vector(i) says that for all , there exist absolute constants such that

By repeated using Lemma LABEL:lemmaexponentialtail in Appendix LABEL:secArateofconvergence, we have for all , every , absolute constants such that

where and . Use the assumption to invoke Theorem LABEL:thmbernsteininequality followed by Lemma LABEL:lemmabernsteinrate in Appendix LABEL:sec_oldappendixB to get

(8.8)

Display (8.7)

We now consider (8.7).

(8.9)
(8.10)

Consider (8.9).

where the first equality is due to Lemma LABEL:lemmaexponentialtail(ii) in Appendix LABEL:secArateofconvergence, Theorem LABEL:thmbernsteininequality and Lemma LABEL:lemmabernsteinrate in Appendix LABEL:sec_oldappendixB. Now consider (8.10).

where the equality is due to Lemma LABEL:lemmaexponentialtail(ii) in Appendix LABEL:secArateofconvergence, Theorem LABEL:thmbernsteininequality and Lemma LABEL:lemmabernsteinrate in Appendix LABEL:sec_oldappendixB. Thus

(8.11)

Display (8.4)

We first give a rate for . The index is arbitrary and could be replaced with . Invoking Lemma LABEL:lemmabernsteinrate in Appendix LABEL:sec_oldappendixB, we have

(8.12)

Then we also have

(8.13)

We now consider (8.4):

With expansion, simplification and recognition that the indices are completely symmetric, we can bound (8.4) by

(8.14)
(8.15)
(8.16)
(8.17)

We consider (8.14) first. (8.14) can be bounded by repeatedly invoking triangular inequalities (e.g., inserting terms like ) using Lemma LABEL:lemmaexponentialtail(ii) in Appendix LABEL:secArateofconvergence, (8.13) and (8.12). (8.14) is of order . (8.15) is of order by a similar argument. (8.16) and (8.17) are of the same order using a similar argument provided that both and are ; these follow from Lemma LABEL:lemmaexponentialtail(ii) in Appendix LABEL:secArateofconvergence, Theorem LABEL:thmbernsteininequality and Lemma LABEL:lemmabernsteinrate in Appendix LABEL:sec_oldappendixB. Thus

(8.18)

Display (8.6)

We now consider (8.6).

(8.19)