The Clustered Factor (CF) model is commonly used to parametrize block correlation matrices. We show that the CF model imposes additional superfluous restrictions. This can be avoided by a different parametrization, based on the logarithmic block correlation matrix

Hansen, Peter Reinhard

Tong, Chen

Carolina Digital Repository

arXiv:2308.05895v1  [econ.EM]  11 Aug 2023Characterizing Correlation Matrices that Admit a ClusteredFactor RepresentationChen Tong‡∗ and Peter Reinhard Hansen§‡Department of Finance, School of Economics, Xiamen University§University of North Carolina & Copenhagen Business SchoolAugust 14, 2023AbstractThe Clustered Factor (CF) model induces a block structure on the correlation matrixand is commonly used to parameterize correlation matrices. Our results reveal that theCF model imposes superfluous restrictions on the correlation matrix. This can beavoided by a different parametrization, involving the logarithmic transformation of theblock correlation matrix.Keywords: Block correlation matrix, copula, clustering, factor model.JEL Classification: C38∗Chen Tong acknowledges financial support from the Ministry of Education of China, Humanities andSocial Sciences Youth Fund (22YJC790117).11 IntroductionIn empirical models involving high-dimensional covariance matrices, it is often beneficialto introduce a parsimonious structure to mitigate issues stemming from overfitting. Acommonly used approach is the Clustered Factor (CF) model, which is characterized by alinear factor structure and group clustering, where factor loadings are shared within eachgroup.The clustered factor model emerged from high-dimensional copula models with a multi-factor structure, see e.g Krupskii and Joe (2013) and Creal and Tsay (2015). Clusteringbecame a popular additional structural component, because it is easy to interpret andmakes the model more parsimonious, see Krupskii and Joe (2015), Oh and Patton (2017a,2017b, 2023), Manner et al. (2019), and Opschoor et al. (2021).It is well known that the CF model induces a block structure on the correlation matrix(whenever variables are ordered by group assignments). However, the CF model is notconsistent with all block correlation matrices. It cannot generate negative within-groupcorrelations. However, this is unlikely to be problematic in empirical applications, becausevariables in the same cluster are expected to share traits and be positively correlated. Itis more problematic that the CF model imposes other restrictions that rules out a class ofblock correlation matrices, which seem perfectly reasonable and ordinary from an empiricalviewpoint. Below we fully characterize the class of block correlation matrices that are co-herent with the CF model, and express the superfluous restrictions the CF model imposesas a testable hypothesis. We then proceed to highlight an alternative parametrization ofblock correlation matrices by Archakov and Hansen (2023), which is based on the matrixlogarithm of the correlation matrix of Archakov and Hansen (2021). This parametrizationuniquely parametrizes any non-singular block correlation matrix, without imposing addi-tional structure. A simple linear factor structure is also applicable to this parametrization,if a more parsimonious model is required.The rest of this paper is organized as follows. In Section 2, we introduce the requirednotation and the CF model followed by the theoretical results that characterizes the blockcorrelation matrices that admit CF model. We express the CF structure as a testablehypothesis, and detail a testing procedure for selecting the number of factors. In Section 3,we presents the alternative parametrization by Archakov and Hansen (2023) and illustrateit with a simple example. We conclude in Section 4 with a brief summary.2 Clustered Factor Models and Block Correlation MatricesConsider an n-dimensional random variable, X ∈ Rn, with non-singular covariance matrixΣ = var(X). Then the corresponding correlation matrix is given byC = Λ−1σ ΣΛ−1σ ,2where Λσ = diag(√Σ11, . . . ,√Σnn), and the vector of standardized variables,Z = Λσ(X − µ), µ = EX,is such that C = E(ZZ ′).In the copula literature, it is common to parametrize C with the clustered factor (CF)model, which is characterized by a linear factor structure for Z and a partitioning of thevariables into clusters/groups. The factor structure is defined by,Zi = b′if + εi, i = 1, . . . , n, (1)where f ∈ Rr is a common vector with r orthogonal factors, Ef = 0 and var(f) = Ir, andthe idiosyncratic variables (ε1, . . . , εn) are mutually uncorrelated and uncorrelated with thefactors, cov(f, εi) = 0 and cov(εi, εj) = 0 for all i 6= j. The vectors of factor loadings aregiven by bi = corr(f, Zi) ∈ Rr, for i = 1, . . . , n, and it follows that var(εi) = 1 − b′ibi.A partitioning of the n elements into K < n groups adds additional structure. It isassumed that the factor loadings are common within each group, such thatbi = βk for all i ∈ Gk, (2)where (G1, . . . , GK) is a partition of {1, . . . , n}, with Gk containing nk elements, k =1, . . . , K, such that n =∑k nk.The CF model implies that the correlation between two variables is solely determinedby their group classification. For i ∈ Gk and j ∈ Gl with i 6= j we haveCij = corr(zi, zj) = ρkl ≡ β′kβl.By rearranging X1, . . . , Xn according to their group assignments, we obtain a block corre-lation matrix that can be expresses asC =C[1,1] C[1,2] · · · C[1,K]C[2,1] C[2,2].... . .C[K,1] C[K,K], (3)where C[k,l] is an nk × nl matrix given byC[k,k] =1 ρkk · · · ρkkρkk 1. . ..... . .. . .ρkk 1and C[k,l] =ρkl · · · ρkl.... . .ρkl ρklif k 6= l.3The advantage of a block correlation matrix, is that its number of unique correlations is(at most) d = K (K + 1) /2, whereas the number of distinct correlations in an unrestrictedn × n correlation matrix is n (n − 1) /2. If K is fixed, then d does not increase with n,which makes it possible to model high-dimensional correlation matrices with relatively fewparameters.2.1 When is the Clustered Factor Structure ApplicableAn interesting question is whether a give block correlation matrix can be expressed as a CFmodel. Any block correlation matrix is clearly coherent with a group partitioning (2), butadditional structure is needed for it to have the representation in (1). In some applications,it will be relevant to know if the factor structure rules out empirically relevant correlationmatrices. It is therefore interesting to characterize the set of block correlation matrices thatare compatible with the CF model.For later use, we define the matrix B = (β1, β2, . . . , βK)′ ∈ RK×r, where βk ∈ Rr×1 isthe vector of group-specific factor loadings, as defined in (1) and (2).From a block correlation matrix, C, we define the following K × K matrixA∗ =ρ11 ρ12 · · · ρ1Kρ21 ρ22.... . .ρK1 ρKK. (4)In this paper we focus, without loss of generality, on the case where C is nonsingular, whichimplies that ρkk < 1 for all within-group correlations, k = 1, . . . , K.Remark. If nk = 1 for some k, then A∗ is not fully defined by C, because there is no within-group correlation in the k-th group, leaving ρkk undefined. This is the reason that we treatthe case with singleton groups separately in the following Theorem.Theorem 1. Let C be a nonsingular block correlation matrix. If mink nk ≥ 2, then C hasa factor representation if and only if A∗ is positive semi-definite. If, instead, nk1 = · · · =nkm =1 (and nk ≥ 2 for all other), then C has a factor representation if and only if thereexist ρk1k1 , . . . , ρkmkm < 1 such that A∗ is positive semi-definite.Proof. Suppose that A∗ is positive semi-definite (psd) with ρkk < 1 for all k, then it canbe expressed as A∗ = BB′, where β′kβk = ρkk < 1. This shows that C admits a factorrepresentation. (The matrix of factor loadings, B, is not unique, but one valid choiceis B = QΛ1/2, where A∗ = QΛQ′ is the eigendecomposition of A∗). That the resultingcorrelation matrix, C, is nonsingular follows from Lemma 1.Now suppose C is a nonsingular block correlation matrix that is given from CF modelwith factor loadings B = (β1, . . . , βK)′. Then it follows that ρkl = β′kβl, such that A∗ = BB′is psd with ρkk = β′kβk < 1 for all k. This completes the proof.4Lemma 1. Suppose that the matrix A∗ is positive semi-definite with ρkk < 1, for allk = 1, . . . , K. Then the corresponding block correlation matrix, C, is positive definite.Proof. From Archakov and Hansen (2023) it follows that a block correlation matrix is pos-itive definite if and only ifA =1 + (n1 − 1) ρ11 ρ12√n1n2 · · · ρ1K√n1nKρ21√n1n2 1 + (n2 − 1) ρ22.... . .ρK1√nKn1 1 + (nK − 1) ρKK,is positive definite with |ρkk| < 1, for all k. This matrix can be expressed as A =Λ1/2n A∗Λ1/2n + Ψ1−ρ, whereΨ1−ρ =1 − ρ11 · · · 0.... . ....0 · · · 1 − ρKK, and Λn =n1 0. . .0 nK.If matrix A∗ is psd with ρkk < 1, then Ψ1−ρ is positive definite and Λ1/2n A∗Λ1/2n is psd. Itnow follows that A is positive definite because it is the sum of a positive definite matrix anda positive semidefinite matrix, and from Archakov and Hansen (2023, theorem 1) it followsthat the corresponding block correlation matrix is also positive definite.Although Lemma 1 shows that a psd A∗ with ρkk < 1 implies that A is positive definite,the converse is not true. We have that A∗ = Λ−1/2n (A − Ψ1−ρ) Λ−1/2n , which is psd ifand only if A − Ψ1−ρ is psd. The latter is not guaranteed. The simplest example is if awithin-group correlation is negative, since the diagonal elements of A − Ψ1−ρ equal nkρkk,k = 1, . . . , K. Another, less obvious, example is the following block correlation matrix:C =1 • • • • • • • •0.70 1 • • • • • • •0.70 0.70 1 • • • • • •0.58 0.58 0.58 1 • • • • •0.58 0.58 0.58 0.63 1 • • • •0.58 0.58 0.58 0.63 0.63 1 • • •0.54 0.54 0.54 0.19 0.19 0.19 1 • •0.54 0.54 0.54 0.19 0.19 0.19 0.71 1 •0.54 0.54 0.54 0.19 0.19 0.19 0.71 0.71 1, (5)5which is a positive definite, it’s smallest eigenvalue is λmin(C) = 0.26. However,A∗ =0.70 0.58 0.540.58 0.63 0.190.54 0.19 0.71,has a negative eigenvalue, λmin(A∗) = −0.02, and C can therefore not be expresses asa CF model. It seems unjustified that the Clustered Factor (CF) model precludes thecorrelation matrix in (5) (and similar matrices) in advance, because there does not appearto be anything bizarre or unusual about this particular block correlation matrix.Another question is how the number of factors, r, are needed to to generate a givencorrelation matrix. We address this in the following Theorem.Theorem 2 (Minimal factor model). If C admits a CF model, then the minimal numberof factors is r = rank (A∗).Proof. If C is generated by a factor model with r̃ factors, then A∗ = B̃B̃′ where B̃ ∈ RK×r̃.We have r = rank(A∗) ≤ min(K, r̃), and we can express A∗ = BB′ where rank (B) = r,which shows that C admits a factor structure with r orthogonal factors.Theorem 2 shows that increasing the number of factors beyond K does not broaden therange of attainable correlation matrices. The (maximum) number of free parameters in theCF model is K(K + 1)/2, because we can always rotate the latent factors, f , such that Bis upper (or lower) triangular.2.2 Testing if C Admits a Factor RepresentationTheorem 1 shows that the block correlation matrices that admits a CF representation, arecharacterized the smallest eigenvalue of A∗ being nonnegative, i.e. λmin(A∗) ≥ 0. Here A∗is defined by (4) where we set ρkk = 1 whenever nk = 1.In practice, it is therefore relatively straight forward to test the CF representation usingthe null hypothesis:HF : λmin(A∗) ≥ 0.2.3 Determining the Minimal Number of FactorsGiven a block correlation matrix with a CF structure, we can proceed to estimate theminimum number of factors, r, as defined by the rank of A∗. This could be done with thesequential procedure propose in Pantula (1989), which is commonly used to determine thecointegration rank in VAR models, see Johansen (1988, 1991).11The same testing principle is used for lag-length selection in time series model, see Ng and Perron (2001),and in multiple comparisons to determine the model confidence set, see Hansen et al. (2011).6Let λ1 ≥ λ2 ≥ · · · ≥ λK denote the eigenvalues of A∗, and consider the hypothesisHF,q : λr+1 = · · · = λK = 0, implying that A∗ is psd with rank(A∗) ≤ r. The sequentialprocedure begins by testing HF,0 against HF,K. In the event of a rejection, we proceedto test HF,1 against HF,K, and so forth until a null hypothesis is not rejected. We canset r̂ = r∗, where r∗ is the first instance (the smallest r) where HF,r is not rejected. Thedegrees of freedom in B for a model with r ≤ K factors is r[K + (K − r) + 1]/2.3 A Better Parametrization Block Correlation MatricesA new parametrization of correlation matrices was proposed in Archakov and Hansen (2021),and is defined byγ = vecl (log C) ∈ Rd, d = n(n − 1)/2, (6)where vecl(·) extracts and vectorizes the elements below the diagonal and log C is the matrixlogarithm of the correlation matrix.2 The identity (6) defines a one-to-one mapping betweenRd and the set of non-singular n × n correlation matrices.Because the matrix logarithm preserves the block structure in C, see Archakov and Hansen(2023), γ will contain many “duplicates”. Thus, we can parametrize block correlation ma-trices with η ∈ Rq, where η is a subvector of γ and q ≤ K(K + 1)/2. The follow examplewill serve as an illustration,1.0 0.4 0.4 0.3 0.3 0.30.4 1.0 0.4 0.3 0.3 0.30.4 0.4 1.0 0.3 0.3 0.30.3 0.3 0.3 1.0 0.6 0.60.3 0.3 0.3 0.6 1.0 0.60.3 0.3 0.3 0.6 0.6 1.0︸ ︷︷ ︸=C−.19 .326 .326 .162 .162 .162.326 −.19 .326 .162 .162 .162.326 .326 −.19 .162 .162 .162.162 .162 .162 −.38 .533 .533.162 .162 .162 .533 −.38 .533.162 .162 .162 .533 .533 −.38︸ ︷︷ ︸=log C.Here we can use η = ( 0.326 0.162 0.533 )′ as the condensed vector parametrization ofthe block correlation matrix. For a given block partitioning, (n1, . . . , nK), any non-singularblock correlation matrix will map to a unique vector η, and any vector η ∈ Rq will map toa unique non-singular block correlation matrix, see Archakov and Hansen (2023).It is straightforward to add additional structure onto the η-parametrization, for instanceby restricting η to be in a subspace of lower dimension than q. For a multivariate GARCHmodel, these ideas are explored in Archakov et al. (2020).Creal and Kim (2022) recently adopted this parametrization for Bayesian modeling ofblock correlation matrices. An attractive feature of this parametrization, it that it facilitatespriors with full support on the entire set of non-singular block correlation matrices.2For a nonsingular correlation matrix, we have log C = Q log ΛQ′, where C = QΛQ′ is the spectraldecomposition of C, so that Λ is a diagonal matrix with the eigenvalues of C.74 SummaryWe have characterized the class of block correlation matrices that can be expressed asa clustered factor model. While the clustered factor model serves as a valuable tool forgenerating clustered correlation structures, it does introduce unnecessary constraints onthe correlation matrix, which may limit its practical relevance for some empirical problems.The alternative parametrization, which is based on the matrix logarithm of the correlationmatrix, seems better suited for the modeling of block correlation matrices. It avoids theimposition of superfluous constraints on the correlation matrix and, if needed, it providesa flexible framework for further reduction of the degrees of freedom.ReferencesArchakov, I. and Hansen, P. R. (2021). A new parametrization of correlation matrices.Econometrica, 89:1699–1715.Archakov, I. and Hansen, P. R. (2023). A canonical representation of block matrices withapplications to covariance and correlation matrices. Forthcoming in Review of Economicsand Statistics.Archakov, I., Hansen, P. R., and Lunde, A. (2020). A multivariate Realized GARCH model.arXiv:2012.02708, [econ.EM].Creal, D. and Kim, J. (2022). Bayesian estimation of block covariance matrices. Workingpaper.Creal, D. and Tsay, R. S. (2015). High dimensional dynamic stochastic copula models.Journal of Econometrics, 189:335–345.Hansen, P. R., Lunde, A., and Nason, J. M. (2011). The model confidence set. Econometrica,79:456–497.Johansen, S. (1988). Statistical analysis of cointegration vectors. Journal of EconomicDynamics and Control, 12:231–254.Johansen, S. (1991). Estimation and hypothesis testing of cointegration vectors in gaussianvector autoregressive models. Econometrica, 59:1551–1580.Krupskii, P. and Joe, H. (2013). Factor copula models for multivariate data. Journal ofMultivariate Analysis, 120:85–101.Krupskii, P. and Joe, H. (2015). Structured factor copula models: Theory, inference andcomputation. Journal of Multivariate Analysis, 138:53–73.8Manner, H., Stark, F., and Wied, D. (2019). Testing for structural breaks in factor copulamodels. Journal of Econometrics, 208:324–345.Ng, S. and Perron, P. (2001). Lag length selection and the construction of unit root testswith good size an power. Econometrica, 69:1519–1554.Oh, D. H. and Patton, A. J. (2017a). Modeling dependence in high dimensions with factorcopulas. Journal of Business & Economic Statistics, 35:139–154.Oh, D. H. and Patton, A. J. (2017b). Time-varying systemic risk: Evidence from a dynamiccopula model of cds spreads. Journal of Business & Economic Statistics, 36:181–195.Oh, D. H. and Patton, A. J. (2023). Dynamic factor copula models with estimated clusterassignments. Journal of Econometrics, Forthcoming.Opschoor, A., Lucas, A., Barra, I., and van Dijk, D. (2021). Closed-form multi-factorcopula models with observation-driven dynamic factor loadings. Journal of Business andEconomic Statistics, 39:1066–1079.Pantula, S. G. (1989). Testing for unit roots in time series data. Econometric Theory,5:256–271.9

Characterizing correlation matrices that admit a clustered factor representation

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Characterizing correlation matrices that admit a clustered factor representation

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