07350015%2E2013%2E859078

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TANJUNGPINANG, KEPULAUAN RIAU] Date: 11 January 2016, At: 20:47

Journal of Business & Economic Statistics

ISSN: 0735-0015 (Print) 1537-2707 (Online) Journal homepage: http://www.tandfonline.com/loi/ubes20

Modeling Conditional Covariances With Economic

Information Instruments

H. J. Turtle & Kainan Wang

To cite this article: H. J. Turtle & Kainan Wang (2014) Modeling Conditional Covariances With Economic Information Instruments, Journal of Business & Economic Statistics, 32:2, 217-236, DOI: 10.1080/07350015.2013.859078

To link to this article: http://dx.doi.org/10.1080/07350015.2013.859078

Accepted author version posted online: 13 Nov 2013.

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Supplementary materials for this article are available online. Please go tohttp://tandfonline.com/r/JBES

Modeling Conditional Covariances With

Economic Information Instruments

H. J. T

URTLE

Department of Finance, College of Business & Economics, West Virginia University, P.O. Box 6025, Morgantown, WV 26505 (harry.turtle@mail.wvu.edu)

Kainan W

ANG

Department of Finance, College of Business and Innovation, University of Toledo, Mail Stop 103, Toledo, OH 43606 (kainan.wang@utoledo.edu)

We propose a new model for conditional covariances based on predetermined idiosyncratic shocks as well as macroeconomic andowninformation instruments. The specification ensures positive definiteness by construction, is unique within the class of linear functions for our covariance decomposition, and yields a simple yet rich model of covariances. We introduce a property,invariance to variate order, that assures estimation is not impacted by a simple reordering of the variates in the system. Simulation results using realized covariances show smaller mean absolute errors (MAE) and root mean square errors (RMSE) for every element of the covariance matrix relative to a comparably specified BEKK model with

owninformation instruments. We also find a smaller mean absolute percentage error (MAPE) and root mean square percentage error (RMSPE) for the entire covariance matrix. Supplementary materials for practitioners as well as all Matlab code used in the article are available online.

KEY WORDS: Covariance decomposition;Owninformation variables.

1. INTRODUCTION

Modeling the second moments of financial asset returns has important implications in risk management, derivative pricing, hedging, and portfolio optimization. A vast literature related to the time-varying covariance structure of asset returns has de-veloped from the seminal work of Engle (1982) and the subse-quent GARCH extension by Bollerslev (1986). Two illustrative examples of such work are Bollerslev, Engle, and Wooldridge (1988) and Turtle, Buse, and Korkie (1994). A variety of eco-nomic models also examine multivariate relationships between first and second moments. Continued improvement in covari-ance estimation will provide important benefits in the area of financial economics.

Developments in the estimation of multivariate second mo-ments include, among others, the constant conditional correla-tion (CCC) model of Bollerslev (1990); the factor ARCH model of Engle, Ng, and Rothschild (1990); the BEKK model of Baba et al. (1991) and Engle and Kroner (1995); the dynamic condi-tional correlation (DCC) model of Engle (2002) or the related model of Tse and Tsui (2002); and the spline-GARCH model of Rangel and Engle (2012). These models provide varying degrees of success in capturing the volatility clustering, leptokurtosis, and asymmetry commonly observed in economic and financial time series.

Two important challenges exist when modeling second mo-ments. First, positive definiteness of the covariance matrix must be ensured without spuriously imposing or restricting temporal patterns in conditional covariances. Second, important known economic information instruments must be allowed to impact the resultant covariances.

To address the positive definiteness issue, numerous ap-proaches have been proposed. For instance, the popular BEKK model ensures positive definiteness by writing the covariance

matrix as the sum of multiple positive definite matrices. In the DCC framework, restrictions on correlation parameters are im-posed in a multistep procedure that facilitates large-scale es-timation. A well-known drawback with these approaches is that the model structure may inadvertently impact the under-lying economic dynamics describing the evolution of second moments. Engle and Kroner (1995) explicitly recognized this potential undesirable impact when only one lag is included. They further provided necessary conditions for a general BEKK model with additional lags to achieve full generality. One ben-efit of our model is that we do not impose implicit restrictions within the class of linear specifications that we consider for our decomposition.

With regard to information instruments, we differentiate be-tween two sources that affect covariances—aggregate informa-tion that impacts all covariances, andowninformation that may be specific to particular asset return covariances. Schwert (1989) provided an early examination of how aggregate information might be expected to impact asset return volatilities for a va-riety of sources including trading days, lagged market trading volume, lagged market leverage, recession indicators, and ag-gregate financial leverage, among others. Later work by Glosten, Jagannathan, and Runkle (1993) shows that Treasury bill rates impact equity return variances. In addition to aggregate infor-mation effects, our model allowsowninformation for a given asset return to impact both the variance of that portfolio and also the covariance of that portfolio with other portfolios.Own

effects will vary across different assets in many economic con-texts. For example, firm-specific leverage decisions give rise

© 2014American Statistical Association Journal of Business & Economic Statistics April 2014, Vol. 32, No. 2

DOI:10.1080/07350015.2013.859078

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to increases in both systematic and unsystematic risk that may not be captured in aggregate measures. Unfortunately, general specifications that allow information on any asset to impact any other asset suffer from a curse of dimensionality in large systems without further structure.

In many financial economic contexts, we are primarily in-terested in the role of owneffects with little impact expected from other assets in the system. Thus, in our study we examine a class of specifications in which the covariance between assets

iandjis impacted byowninformation related only to these two assets, and not any other assetk. Our model is parsimonious and feasible to estimate in settings withownimpacts for 10, or more assets. The proposed approach is practical for most asset allocation and performance evaluation studies as well as many empirical, portfolio-based investigations.

In this article, we consider linear specifications within a co-variance decomposition. We introduce a property, invariance to variate order, that assures estimation is not impacted by a simple reordering of the variates in the system. This simple ap-proach yields a unique result for covariances. Intuitively, we allow known information relatedonlyto asseti, say Zi,t−1, to

impact theith row of our (lower triangular) covariance decompo-sition, for all columns. This provides a large reduction in model parameters, and a rich model structure that includes informa-tion instruments forbothassetsiandjin covariances between assetsiandj. The resulting covariances from our model include a constant as well as terms inZi,t−1,Zj,t−1, andZi,t−1Zj,t−1.

Our invariance to variate orderproperty provides a valuable structure for the development of our model and produces a rich and parsimonious functional form not available in the extant literature.

Our approach will benefit many studies that examine how

owneconomic, financial, or fundamental accounting variables may impact expected returns, and potentially conditional co-variances. The literature in this area is extensive. For example, returns that are anomalous relative to the CAPM have been found for small capitalization companies (Banz1981), and value com-panies with large book-to-market equity (Rosenberg, Reid, and Lanstein1985). In addition, Fama and French (2008) provided a summary of how these types of anomalous returns may arise in models related to stock issuance, accruals, and momentum.

The extensive literature examining howownexogenous vari-ables may impact conditional variances suggests that a related investigation of covariances is worthy of pursuit. For example, Lamoureux and Lastrapes (1990) demonstrated thatownvolume has a highly significant impact on variance. Hagiwara and Herce (1999) considered howowndifferences from U.S. interest rates impact currency variances. Engle and Patton (2001, sec 2.3) discussed how non-GARCH-type variables, including macroe-conomic announcements and calendar-based regularities, may provide information to describe conditional variances. Although there is extensive research that examines how these and other fundamental portfolio characteristics may impact portfolio ex-pected returns, there is little work that studies how these same

own portfolio economic characteristics might impact covari-ances, and ultimately portfolio risks across a broad multivariate time series. In fact, beyond the highly successful BEKK ap-proach, there is little accepted practice to model covariances in multivariate settings. As demonstrated in our simulations, our

model is well behaved by standard measures of statistical per-formance and in comparison to the BEKK design. Further, the impact ofowninformation effects on conditional second mo-ments can be readily implemented without a large increase in model parameters.

The essence of our approach results from two observations. First, any positive definite matrix can be written as a lower tri-angular matrix multiplied by its transpose. Second, a simple reshuffling of asset returns should not produce a different func-tional form for the resultant covariance terms. Our strategy is to consider the class of models such that eachi,jelement of the covariance matrixdecompositionis linear in the underlying

owninformation instruments,Zi,t−1andZj,t−1. Linearity in this

decomposition can be motivated with a Taylor approximation using well-chosen instruments, even for potentially complex nonlinear underlying specifications. The general principles of our development can be readily extended to include higher or-der quadratic or cubic terms, at a cost of additional parameters. We recognize that our proposed methodology will give rise to a relatively large number of parameters in its most general form. However, our approach does offer improvement over currently available options. As is well known, estimation of even sim-ple unconditional samsim-ple covariance matrices is difficult when the number of assets is large, and the problem worsens as the number of assets increases relative to the number of time series observations (Jobson and Korkie1980). As a recent example, DeMiguel, Garlappi, and Uppal (2009) claimed that over 3000 monthly observations (250 years of data) would be needed to implement optimal portfolio choices in a context with 25 un-derlying assets. A number of approaches have been proposed to mitigate some of these estimation problems. A good represen-tative Bayesian covariance example is provided by Ledoit and Wolf (2004a,2004b) who shrunk the sample covariance matrix to a target value.

When modeling covariances acrossNassets, there areN(N₂−1) unique covariance terms at each point in time. Because our in-terest is in parameterizing each unique term in a covariance matrix, the set of parameters quickly becomes large asN in-creases. Relative to existing alternatives that admitowneffects, our covariance modeling approach substantially mitigates this “curse of dimensionality.” In particular, our proposed method requires only oneowneffect regressor in each unique covariance decomposition term. This yields well-specified covariances in a parsimonious manner. For example, when examined relative to a comparably specified BEKK model, our approach requires 40% fewer parameters in a system of 10 variates. As we discuss in Section 2, our model can be further specialized to reduce parameters, at a cost of some generality.

Although the proposed model gives rise to a large number of parameters when the asset space is large, our approach is applicable in many common financial contexts. Because finan-cial decisions may often be simplified to consider a small set of well-diversified portfolios rather than the thousands of under-lying individual securities, our model is of value in most asset pricing studies and asset allocation decisions. For example, aca-demic financial literature often examines subsets of assets to replicate the important choices faced by investors. These asset subsets may be denoted asspanningorintersectingportfolios (Huberman and Kandel 1987). From a practical perspective,

Brinson, Hood, and Beebower (1986) found that a small num-ber of asset class portfolios explain a significant component of overall portfolio performance. They inferred that financial plan-ning and pension investment may be best achieved with only a small set of index funds.

Our conditional covariance model is intended to provide an economic description of second moments as a function of known economic information instruments, rather than as a time series motivated specification. This approach has the potential to be employed within an asset pricing investigation that provides nonlinear relations between economic variates and required as-set returns. Typical asas-set pricing tests are examined in the con-text of return portfolios that reduce the asset dimensionality to 2, 5, or 10. In these sorts of research designs, our estimation approach will have little difficulty for longitudinal samples of 10 years or more, as is common in application. Although our interest is largely related to economically motivated and known information instruments, our specification may also be readily adapted to a purely time series context as demonstrated in our empirical analysis.

In empirical simulations, we compare our model to a famil-iar multivariate ARCH specification for the covariance matrix suggested by the BEKK model. By construction, our linear in-formation instrument model has the potential to capture nonlin-ear economic relationships between variates that may be missed in existing specifications. Our design outperforms the BEKK alternative for all terms in the resultant conditional covariance matrix when measured by mean absolute errors (MAE) and root mean square errors (RMSE). When using heteroscedasticity-adjusted mean squared errors (HMSE), we find that for all but one covariance term, our model has greater estimation accu-racy than a comparable BEKK specification. In addition, our model contains fewer parameters than the BEKK design. Our simulation experiments also confirm that the proposed model outperforms the BEKK specification for all multivariate metrics of performance.

The research design proposed in this article may be readily nested within any well-specified model for conditional means. In particular, our approach has the potential to link an econom-ically motivated description of expected asset returns (through conditional mean specifications) within the context of a model describing evolving risks (through covariances). Although we focus on predetermined economic information instruments, a variety of parametric and nonparametric alternatives to model both means and covariances can also be considered. Fan and Yao (2003) provided a complete treatment of potential time series approaches available to examine conditional moments.

Covariance estimation may also be based on an underlying model of conditional means with a defined small set of common risk factors. These factor-based approaches effectively reduce the dimensionality of the asset portfolio covariance matrix by linking asset conditional means and covariances to underlying factor sensitivities. Examples of work in this area include Fan, Fan, and Lv (2008), who considered a defined set of risk factors for all assets, or Gagliardini, Ossola, and Scaillet (2011), and Fan, Liao, and Mincheva (2011), who employed factor model and sparse matrix techniques. Our work relates directly to this area of study. Intuitively, our focus is on how a moderate num-ber of portfolio returns behave in relation to known underlying

economic information instruments. Restricted versions of our model may extend our approach to consider larger asset sets without an excessive number of parameters, in a manner akin to factor-based models (see, e.g., the Appendix Bdevelopment). Our general model may also be used to describe the covariance dynamics within the mimicking factor portfolios from these works in order to examine how factor return moments are re-lated to known economic information instruments.

In addition, our approach can be compared to the extant lit-erature that uses Cholesky decompositions in a longitudinal context. Much of this research examines the interesting cross-correlations of disturbances over time. For example, Pinheiro and Bates (1996) provided an excellent discussion of five po-tential strategies to ensure positive definiteness in the estimation of covariance matrices that also allows estimation to proceed in an unconstrained manner. They favored Cholesky-type decom-positions to best estimate covariance behavior, with a prefer-ence for a spherical representation of decomposition elements. Pourahmadi (1999, 2000) used Cholesky decompositions to specify intertemporal covariances (or their inverses) between period tandt-jdisturbances for j₌0, 1, 2,. . .,t-1. Pourah-madi’s work is similar in spirit to the approach in Gallant (1987) who used a Cholesky decomposition to model the inverse of a re-lated covariance matrix when fitting autocovariances in whole-sale prices (chap. 2, example 1). In our study, we focus on covariances across assets at a point in time, and how these covariances may change over time. Further, we are particu-larly interested in specifications that admit the impact of known

own economic variables while ensuring positive definiteness. Thus, our research dovetails with the longitudinal literature and suggests interesting avenues for future studies as these fields converge.

The remainder of the article is organized as follows. Sec-tion2presents our model specification, discusses the resultant positive definiteness of the covariance matrix, and describes theinvariance to variate orderproperty. We also examine our specification relative to a comparable version of the BEKK de-sign. Section3presents the empirical performance of our model in a large sample simulation. To confirm that our findings are not dependent on a large time series sample, we also present supportive finite sample simulation results and robustness tests. Concluding remarks are provided in Section4.

2. MODEL DEVELOPMENT

We begin with an intuitive presentation of our approach in a simple two-asset case. Consider a system comprised of a value portfolio, r1t, and a growth portfolio, r2t, both of which are potentially influenced by the known average research and de-velopment expenditures of the firms within these portfolios. We hypothesize that firms with greater research and development expenditures are more opaque and, hence, more risky (Baker and Wurgler2006). Other examples of importantowneconomic in-formation instruments abound and include financial statement information, previous price to earnings ratios, liquidity mea-sures, leverage meamea-sures, dividend yields, or any other charac-teristics that vary in the cross-section and over time.

To focus on covariance estimation, let r_t=[r1t r2t] be a zero mean vector with conditional covariance matrix, M_t=[m11t m12t

m21t m22t]. Our primary interest is to provide an empiri-cal specification for covariances related to underlyingown port-folio research and development expenditures. Let the known economic information instruments be the average research and development expenditures for each portfolio, which we denote byZ_t−1=[Zi,t−1] fori=1 and 2. That is, we have a particular

interest in the research and development expenditures specific to each portfolio and not in the aggregate level of research and development expenditure in the overall economy. We observe that every positive definite covariance matrix can be written as M_t=L_tL′_{t, where}L_{tis a lower triangular matrix. We consider} a specification for each element in L_{t =[}lij t] that is linear in Z_{t−1, and examine a simple form for}lij t,

lij t =γij0+γij1Zi,t−1 (1)

for i_≥j, j ₌1, and 2; and model parametersγij0 andγij1.

Direct multiplication yields the resultant covariance matrix as

Mt =

m11t m12t

m21t m22t

=

γ110+γ111Z1,t−1 0

γ210+γ211Z2,t−1 γ220+γ221Z2,t−1

×

γ110+γ111Z1,t−1 γ210+γ211Z2,t−1

0 γ220+γ221Z2,t−1

. (2)

Evaluating term by term, we see that m11t is a linear func-tion of Z1,t−1 andZ12,t−1; m22t is a linear function of Z2,t−1

andZ2

2,t₋1; andm12t is a linear function ofZ1,t₋1,Z2,t₋1, and

Z1,t−1Z2,t−1. That is, the resultant variances,miit, are impacted by the own portfolio research and development expenditure,

Zi,t−1, as well as its square,Zi,t2−1, and covariances are impacted

by research and development expenditures for both portfolios in the system, throughZ1,t−1,Z2,t−1, andZ1,t−1Z2,t−1.

The model has some important features. First, the functional form of each element in M_{t with respect to the information} instruments is easily interpreted. For example, the variance of each portfolio’s return depends only on the firm-specific mation, where the covariance is driven by the economic infor-mation for both portfolios. Second, the model ensures positive definiteness ofM_{tif each diagonal element in}L_{tis nonzero (see} AppendixA). Third, the specification given by Equation (1) is theuniquelinear function ofowninformation instruments that ensures covariances are not spuriously modified by a simple reordering of the portfolios in the system. We call this final propertyinvariance to variate order.

Definition: A model is said to satisfy the condition of in-variance to variate orderif the functional form for all variance and covariance specifications includes the same independent information instruments when the variate order is perturbed.

To understand the importance of the invariance to variate orderproperty, note that Equation (1) satisfies the condition that variate order will not impact the functional form of covariances.

If we were to estimate the trivially reordered system, r∗_{t =}

r2t

r1t

,

the functional form for the two variates would not change. We view this property as desirable and only consider models that satisfy this condition. In our empirical application, this condition ensures that each pair from our five portfolio return series has the same functional form for each potential covariance specification. Surprisingly, many intuitive linear specifications forlij t do not pass the test ofinvariance to variate order. For example, consider the more exhaustive specification,

lij t =

γij0+γij1Zi,t−1, fori=j

γij0+γij1Zi,t−1+γij2Zj,t−1, fori > j

(3) for j ₌1, and 2; and model parameters γij0, γij1, and γij2.

This specification admits terms in bothZi,t−1andZj,t−1for all

lij t terms. Direct multiplication yields the resultant covariance matrix as

M_{t =}

m11t m12t

m21t m22t

=

γ110+γ111Z1,t₋1 0

γ210+γ211Z2,t−1+γ212Z1,t−1 γ220+γ221Z2,t−1

×

γ110+γ111Z1,t−1 γ210+γ211Z2,t−1+γ212Z1,t−1

0 γ220+γ221Z2,t−1

(4) Evaluating term by term, we observe that m11t is a linear function ofZ1,t−1 andZ12,t−1, where m22t is a linear function ofZ1,t−1,Z2,t−1,Z1,t−1Z2,t−1,Z21,t−1, andZ

2

2,t−1. If the order

of the two assets were changed, the functional form for the two variates would change due solelyto variate order. Therefore, specification (3) does not satisfy theinvariance to variate order

property. 2.1 Model

Our goal is to provide an economically driven specification for time varying covariances that is parsimonious, yet rich in de-scribing covariances as a function of the underlying economic state. To focus solely on covariance estimation, consider an

N_×1 zero mean random vector, r_{t, with associated} condi-tional covariance matrix, M_t≡Et₋1[rtr′t], for t =1, . . . , T. We wish to consider potential specifications forM_{t related to} underlying economic information instruments that may be ei-ther lagged endogenous variables, or any oei-ther known observ-able economic information instruments from the information set. For clarity, we consider two general types of economic in-formation instruments,owneconomic information instruments,

Zi,t−1,i=1, . . . , Nthat are specific to each series considered,

as well as aK_×1 vector of macroeconomic sources of infor-mation,Z_{agg,t−1. In equilibrium, we might consider the relation} between covariances of abnormal asset returns to both individual asset characteristics as well as observed market characteristics.

For example, stock or portfolio covariances may be dependent onownfinancial characteristics such as liquidity, opaqueness, profitability, or valuation measures like price-to-book or cash-flow-to-book. In addition, overall market forces such as general credit conditions might impact conditional covariances.

Our general framework considers the time varying condi-tional covariance matrix, M_{t =[mij t], for i, j=1,2, . . . , N,} and fort ₌1, . . . , T. For each conditional covariance matrix, M_{t, we consider the following decomposition,}

M_t=L_t(θ_|t−1)Lt(θ|t−1)′, (5)

where θ represents a vector of parameters, t−1 represents a

vector of information instruments, andL_t(θ_|t−1) is any lower

triangular matrix with nonzero diagonals.L_{tis not a Cholesky} decomposition of M_{t, as we do not require strictly positive} diagonal entries inL_t.

Our interest is in a particular form for the nonzero elements within L_t(θ_|t₋1)=[lij t(θ|t₋1)] from the general class of

linear functions in aggregate information instruments, and in

owninformation instruments defined as

lij t(θ|t₋1)=

⎧ ⎨ ⎩

γii0+γ′ii1Zagg,t−1+γii2Zi,t−1, fori=j

γij0+γ′ij1Zagg,t−1+γij2Zi,t−1 fori > j

+γij3Zj,t−1,

(6) wherei, j₌1,2, . . . , N,i_≥j; Z_{agg,t−1 is a}K_×1 vector of macroeconomic sources of information; Zi,t−1 andZj,t−1 are

known own information instruments for the ith and jth vari-ate, respectively; and whereγij0,γij1,γij2, andγij3are known

conformable parameters. For all i < j ₌1,2, . . . , N we set

lij t =0.

To simplify our notation, we write the class of linear functions as

lij t(θ|t−1)=θZ, (7)

where θ₌[γij0,γ′ij1, γij2] and Z=[1,Z′agg,t−1, Zi,t−1]′ for

i₌j; and θ₌[γij0,γ′ij1, γij2, γij3] and Z=[1,Z′agg,t₋1,

Zi,t−1, Zj,t−1]′fori > j.

Proposition 1: Within the general class of functions defined by Equation (6), for some predetermined variablesZagg,t−1and

Zi,t₋1, the functional form given by

lij t(θ|t₋1)=γij0+γ′ij1Zagg,t₋1+γij2Zi,t₋1 (8)

ensures a unique covariance specification, ensures positive def-initeness for the resultant covariance matrixM_{t provided that} the coefficients γii0 are nonzero for all i=1,2, . . . , N, and

satisfies theinvariance to variate orderproperty.

Proof: See AppendixA.

Our proposed specification is similar in spirit to a Cholesky decomposition of the conditional covariance matrix. This de-composition is often used in constant covariance applications where positive definiteness is required, and is found in the works of Pinheiro and Bates (1996) and Pourahmadi (1999, 2000) among others. Because every positive definite matrix

yields a unique Cholesky decomposition, we build our speci-fication for second moments within the lower triangular matrix context. Following Pinheiro and Bates (1996), we ignore the degenerate case of a positive semidefinite covariance matrix, and focus solely on the positive definite case. Given our con-cern that uniqueness is solely related to the resultant covariance specification and not to the decomposition, L_{t, we do not} im-pose the Cholesky condition that diagonal elements must be strictly positive. Rather, these elements must only be nonzero. An interesting alternative form provided by Pinheiro and Bates (1996) models the log of the diagonal elements ofL_{tto ensure} positivity in an unrestricted manner and uniqueness in the de-composition. Untabulated estimation results, which impose the additional condition that diagonal elements of the lower trian-gular matrixL_{t are positive for all times and variates, produce} very similar results.

The number of parameters required to estimate our proposed system offers an improvement over existing models. In general, each of the N(N₂+1) uniquelij t terms requires a specification in our context. In the case of Equation (8) with a scalarZagg,t−1,

this results in3N(N₂+1)total covariance parameters. For example, in a typical financial market application with 10 portfolios and 40 years of data, estimation of the 165 required covariance parameters is readily implementable, even with only monthly observations (480 time series observations).

In Appendix A we provide sufficient conditions to ensure the asymptotic consistency and normality of maximum like-lihood estimates and discuss some recent literature providing more general related results. Further research to estimate mod-els of this sort with semiparametric techniques (Engle and Gonzalez-Rivera 1991), estimating functions (Li and Turtle 2000), or nonparametric approaches (Wu and Pourahmadi2003 or Yao and Li2013) may also be worthwhile.

2.2 Comparisons With a BEKK-Like Covariance Model Although our interest is primarily in economically motivated changes in second moment matrices, our approach can be com-pared to the extant time series literature by specifying our in-formation instruments as time series variables. In particular, we compare the multivariate ARCH representation of the BEKK model to a restricted version of Equation (8) with no macroeco-nomic information instruments,

lij t(θ|t−1)=γij0+γij1Zi,t−1. (9)

We define the following covariance specifications for the lin-ear information instrument and BEKK models,

Mt=LtL′t =[mij t], (10)

and

M_BEKKt=C₀C′₀₊A₁r_t−1r′_t

−1A′1, (11)

respectively, where C_0and A_{1 are}N_×N parameter matrices and where C_{0 is lower triangular. Notice that to estimate the} BEKK covariance matrix, M_{BEKKt, we require} N(N+1)

2 +N 2

parameters given there are N(N₂+1) parameters in C_{0 andN}2

parameters in A_1.

To facilitate comparison with our linear information instru-ment model, we replacer_t−1withZ_{t−1for the BEKK} specifica-tion in Equaspecifica-tion (11). Therefore, in the simplest bivariate case, M_{tis given by}M_{t =[}m11t m12t

m21t m22t] with

m11t =γ1102 +2γ110γ111Z1,t−1+γ1112 Z 2 1,t−1,

m21t =m12t =γ210γ110+γ210γ111Z1,t−1+γ211γ110Z2,t−1

+γ211γ111Z1,t−1Z2,t−1, and

m22t =

γ₂₁₀2 ₊γ₂₂₀2 ₊2 (γ210γ211+γ220γ221)Z2,t−1

+γ₂₁₁2 ₊γ₂₂₁2 Z2_2,t−1. (12)

The related BEKK representation is given by M_{BEKKt =}

mBEKK11t mBEKK12t

mBEKK21t mBEKK22t

=

c11 0

c21 c22

c11 c21

0 c22

+

a11 a12

a21 a22

×

Z2_1,t−1 Z1,t−1Z2,t−1

Z1,t−1Z2,t−1 Z22,t₋1

a11 a21

a12 a22

with

mBEKK11t =c211+a 2 11Z

2

1,t−1+2a11a12Z1,t−1Z2,t−1+a122Z 2 2,t−1,

mBEKK21t =mBEKK12t=c11c21+a11a21Z12,t−1

+(a12a21+a11a22)Z1,t−1Z2,t−1

+a12a22Z22,t−1, and

mBEKK22t =c221+c 2 22+a

2 21Z

2

1,t−1+2a21a22Z1,t−1Z2,t−1

+a₂₂2Z2_2,t−1. (13)

A comparison of Equations (12) and (13) reveals four im-portant findings: (1) Both models satisfy theinvariance to vari-ate order property. (2) Our model given by Equation (9) is more parsimonious than the BEKK-like model. In particular, for anyN_×N covariance matrix, M_{t has}N(N₊1) parameters, whereM_{BEKKtrequires}N(N+1)

2 +N

2_{parameters. Thus, for any}

N _≥2, N(N₂+1)₊N2> N(N₊1). For example, in a system of 10 variates, the BEKK model will require an increase of 40% (₌(155–110)/110) in required parameters. (3) Our covariance matrix,M_{t, is the unique representation given the general class} of models described in Equation (6). (4) Our conditional covari-ance matrix has a distinctly different functional form relative to BEKK. Specifically, eachi, jth element of the BEKK covariance matrix is a function ofZ2

i,t₋1,Zj,t2 ₋1, andZi,t₋1Zj,t₋1. In

con-trast, our approach yields a covariance matrix where variances are solely functions of own instruments, and covariances be-tweeniandj are functions ofZi,t−1,Zj,t−1, andZi,t−1Zj,t−1.

Our approach, therefore, has the ability to capture interesting

own effects involving lower moments of the instruments that may be obscured within the BEKK functional form.

Much of the extant literature asserts that the BEKK and DCC models often perform similarly in forecasting conditional co-variances. For example, Caporin and McAleer (2008) found that

the scalar versions of the two models are similar in forecasting conditional covariances and value-at-risk thresholds. Further, Massimiliano and Michael (2010) suggested that the BEKK and DCC (or CCC) models produce highly comparable condi-tional covariances and correlations in both univariate and large scale contexts. Given these findings, and our primary interest inowneffects, we focus our empirical analysis on a detailed comparison of our linear information instrument model and the BEKK model.

Conceptually, however, a similar special case of our model can be compared to other models such as the factor ARCH model of Engle, Ng, and Rothschild (1990), using a restricted version of Equation (8) with only macroeconomic information instruments, such as,lij t(θ|t−1)=γij0+γ′i1Zagg,t−1. In

Ap-pendixBwe discuss the similarities between this specialized form of our model and the factor ARCH. Methodologically, this restricted approach has similarities with Fan, Fan, and Lv (2008), Gagliardini, Ossola, and Scaillet (2011), and Fan, Liao, and Mincheva (2011) who considered a small set of contempo-raneous risk factors to describe conditional asset return means. The most general representation of the multivariate ARCH model is the unrestricted vech form Bollerslev, Engle, and Wooldridge (1988), which for the multivariate ARCH(1) model yields,m_{t =}ω₊A₁η_t−1, wherem_t=vech(M_t),η_{t =}

vech(r_tr′

t) and vech is the matrix operator that stacks the lower triangular part of a symmetricN_×NmatrixM_{tinto an}N(N+1)

2

dimensional vector, and whereωandA_{1are parameter matrices} with dimensions N(N₂+1)_×1 and N(N₂+1)_×N(N₂+1), respec-tively. Although general, the vech representation is not practical to implement without additional restrictions to ensure covari-ance matrix positive definiteness.

3. EMPIRICAL ANALYSIS

We present empirical results for covariance matrix estimation using our linear information instrument model and a comparable specification of the BEKK model. Our empirical results focus on a comparison of the underlying choice parameters in Equations (10) and (11). In many empirical applications a restricted form of M_{BEKKt is considered in which the parameter matrix}A_{1 is} diagonal, orA_{1is equal to a scalar parameter times the identity} matrix. For brevity, we avoid a comparison of restricted versions of the BEKK model and similarly restricted versions of our linear information instrument model.

Our empirical analysis is comprised of three general sections. Initially, we establish that the proposed approach behaves well in a lengthy time series in comparison with the BEKK specifi-cation. Then, we demonstrate that the linear information instru-ment model has desirable finite sample properties for various time series sample sizes and asset dimensions. Finally, we offer results that demonstrate our findings are robust to alternative co-variance specifications, information instrument definitions, and asset sets.

Our primary empirical applications make use of a lengthy sample of realized conditional covariance matrices for five book-to-market (BM) portfolios that we treat as the true value to be estimated using the two competing procedures. Our methodol-ogy can also be applied to a number of other owneconomic

or financial variables of interest in asset allocation and per-formance evaluation studies, or portfolio-based asset pricing investigations.

3.1 Conditional Covariance Matrix Construction

We construct weekly conditional covariance matrices from daily returns of the value-weighted book-to-market (BM) quintile portfolios over the period July 5, 1963, to Decem-ber 30, 2009. Our sample begins on Friday, July 5, be-cause Thursday, July 4, was a market closure for the Inde-pendence Day holiday. Portfolio return data are downloaded from Ken French’s website: http://mba.tuck.dartmouth.edu/ pages/faculty/ken.french/data library.html.

To avoid spurious weekend and holiday effects, we define a week from Thursday-open to Wednesday-close. By choosing this weekly period, we mitigate the impact of market closures and provide the greatest number of possible five-day trading weeks. Because our analysis is focused at the portfolio level, missing observations reflect U.S. market closures. Our sample contains 2419 Thursday to Wednesday calendar weeks of which 2413 have either four (413) or five (2000) trading days per week. The worst period for available data was from Thursday, July 4 to Wednesday, July 10, 1968. The market was closed on July 4 (Independence Day), July 5 (the day after Independence Day), July 6 and 7 (the weekend), and July 10 (the 1968 Paperwork crisis). This produced a two-day trading week that included only Monday, July 8 and Tuesday, July 9. These closures also resulted in a related four-day trading week that followed.

Weekly realized conditional covariance matrices are con-structed as follows. For each Wednesday ending on dayt, we compute the weekly realized covariance matrices using daily re-turns for the current and previous seven weeks of daily rere-turns as

covt=5× 1

Ndays

Ndays

j=1

rd,T+1−jrd,T′ +1−j (14)

whererd,T+1−j is the vector of percentage daily returns on trad-ing dayT ₊1−j,T is the last trading day in the week under consideration, andNdaysis the number of trading days in the

cur-rent and previous seven weeks (for a total 8-week window used to construct realized covariances). Thus, similar to Andersen et al. (2001) and the empirical mixed data sampling (MIDAS) results from Ghysels, Santa-Clara, and Valkanov (2006), we cre-ate weekly realized conditional covariance matrices by scaling each daily realized covariance matrix by five to create a weekly measure.

We treat this calculated time series of realized covariance matrices as the population values to be subsequently estimated using either our linear information instrument model or the BEKK model. At each point of time,t, we draw a weekly return vector rt from a multivariate normal distribution with mean of zero and covariance matrix of covt as in Equation (14). This procedure creates a lengthy multivariate time series ofT₌2419 weekly observations for the BM quintile portfolios (N₌5) that we treat as our base case sample for later empirical simulations.

In robustness tests, we find our empirical results are compa-rable when using modified realized conditional covariance con-struction methods with different previous return windows and varied decay weights for lags of previous daily squared returns. Although the general magnitudes of the model performance met-rics change with covariance construction assumptions, the su-perior performance of the linear information instrument model relative to the BEKK model persists in all cases.

3.2 Conditional Covariance Matrix Estimation

The maximum likelihood (ML) estimator of the covariance matrix, _T1 T

t₌1(rt−r¯)(rt−r¯)′,for the 5×1 set of book-to-market simulated portfolio returns,r_{t, with sample mean ¯}r_{, is} reported for the weekly sample of 2419 simulated observations inTable 1. The simulated series is based on a normal draw with a 5 × 1 zero mean vector and a time-varying covariance matrix given by Equation (14).

To begin our empirical analysis, we report the estimated constant covariance matrix using our ML routine for the sim-ulated series. The general log-likelihood function is given by log(θ_|r_{t)= −}1

2

2419

t=1 [Nlog(2π)+log|Mt| +r′tM−

1

t rt], whereM_{tis the conditional covariance matrix at}t. For the case of a constant covariance matrix, we parameterize the lower trian-gular matrixL_=[lij] as,lij =γijfori≥j, i, j =1,2, . . . ,5; and wherelij is thei,jth element ofL. The covariance matrix is then constructed as,M₌L L′_{.Table 2reports the constant} co-variance matrix in lower triangular form,L_=[γij], along with associated standard errors in parentheses. Starting values for the mean vector are given by zeros and the initial value ofγij is the i, jth element of the Cholesky decomposition of the uncondi-tional sample covariance matrix. We use the quasi-Newton algo-rithm in Matlab with the Broyden–Fletcher–Goldfarb–Shanno (BFGS) formula for updating the Hessian.

We note that allγijterms are typically magnitudes larger than their standard errors. In addition, after multiplication, we find that the resultant estimated unconditional covariance matrix, M ₌L L′_{, is virtually identical to the ML estimate of the}

co-variance matrix inTable 1. Although we follow the conventional practice of discussing individual tests in many tables, we rec-ognize that na¨ıve usage of multiple individual tests will tend to find too many significant variables. For example, with a test size of 5%, we recognize that one in 20 tests will incorrectly reject

Table 1. The estimated unconditional maximum likelihood (ML) covariance matrix

Quintile BM

Portfolios Smallest 2nd 3rd 4th Largest

Smallest 5.3747 – – – –

2nd 4.3709 4.2613 – – –

3rd 4.1067 3.9191 4.1303 – –

4th 3.8184 3.6828 3.7135 3.9825 –

Largest 4.0617 3.8932 3.8856 3.9678 4.7037

NOTE: The maximum likelihood estimator of the covariance matrix of simulated weekly returns for five book-to-market (BM) portfolios,r_t, is given by1

T

T

t=1(rt−r¯)(rt−¯r)′

for theT=2419 observations, with sample mean vector, ¯r. The simulated series is based on a normal draw with zero mean and time-varying covariance given by Equation (14).

Table 8. Robustness results

Model Mean Std Min Max 5th Pctl. Median 95th Pctl.

Panel A: Realized covariance, past 4 weeks

MAPE Linear Inst. 0.3064 0.0653 0.1993 0.7951 0.2270 0.2958 0.4215

BEKK 0.4063 0.1726 0.2150 2.5718 0.2617 0.3607 0.7303

RMSPE Linear Inst. 0.3972 0.0853 0.2606 0.9958 0.2892 0.3840 0.5449

BEKK 0.5354 0.2405 0.2708 3.3047 0.3298 0.4713 0.9816

Panel B: Realized covariance, past 12 weeks

MAPE Linear Inst. 0.3258 0.0689 0.2110 0.6428 0.2383 0.3116 0.4695

BEKK 0.4057 0.1286 0.2265 1.1369 0.2634 0.3748 0.6652

RMSPE Linear Inst. 0.4019 0.0840 0.2633 0.7738 0.2936 0.3860 0.5717

BEKK 0.5154 0.1759 0.2754 1.4378 0.3272 0.4721 0.8626

Panel C: Instrument, past 2 weeks

MAPE Linear Inst. 0.3331 0.0702 0.2156 0.7168 0.2446 0.3183 0.4587

BEKK 0.4130 0.1329 0.2231 1.1446 0.2678 0.3795 0.6859

RMSPE Linear Inst. 0.4164 0.0873 0.2700 0.8529 0.3036 0.3990 0.5778

BEKK 0.5288 0.1846 0.2720 1.5157 0.3335 0.4793 0.9332

Panel D: Instrument, past 6 weeks

MAPE Linear Inst. 0.2002 0.0437 0.1331 0.4759 0.1473 0.1909 0.2830

BEKK 0.3115 0.1463 0.1427 1.4135 0.1782 0.2647 0.6231

RMSPE Linear Inst. 0.2487 0.0522 0.1641 0.5468 0.1840 0.2376 0.3475

BEKK 0.3977 0.2005 0.1757 1.7831 0.2225 0.3335 0.8451

Panel E: Ten momentum portfolios (N=10),T=520

MAPE Linear Inst. 0.3322 0.0591 0.1981 0.5295 0.2427 0.3282 0.4371

BEKK 0.4418 0.1350 0.2459 1.0506 0.2972 0.4059 0.7380

RMSPE Linear Inst. 0.4109 0.0714 0.2495 0.6444 0.3029 0.4065 0.5396

BEKK 0.5676 0.1920 0.3068 1.5964 0.3702 0.5140 0.9987

NOTE: The mean absolute percentage error (MAPE) and the root mean square percentage error (RMSPE) for 1,000 simulation replications for both the linear information instrument model and the BEKK model. In panels A through E, we report results forT=260 andN=3. Panels A and B compute the realized covariance matrices using Equation (14) with 4 (current and previous 3 weeks of daily returns) and 12 weeks of daily returns, respectively, whereNdaysreflects the number of days in the window of daily returns. Panels C and D compute the realized information instruments as the average of daily absolute returns over the 2- and 6-week periods beginning one week prior to the week of interest, respectively. Finally, in panel E we report simulation results for 1,000 replications with ten (N=10) momentum portfolios forT=520 observations using the research design described inTable 7.

to alternative methods of forming realized covariances, and to different lags in our information instruments.

Because our research design is focused entirely on covari-ance evolution, we also examine the impact that a poor mean specification might have on our model’s ability with respect to MAPE and RMSPE. To do this we extend our research design to consider the actual demeaned return series along with the re-alized covariances described earlier. Returns are demeaned with a simple (and intentionally inadequate) constant that differs for each variate considered. These results provide robustness re-sults in the context of a misspecified conditional mean, where the misspecification is given by the actual sampled multivariate returns. We then consider the behavior of the estimation results for 12 sequential draws ofT₌200 multivariate returns forN₌

3 series. Unreported results show that the linear information in-strument model continues to show marked improvement relative to the BEKK model in terms of both MAPE and RSMPE. Inter-estingly the (unreported) differences in mean RMSPE values is nearly identical to the values reported inTable 7, although both values are larger due to unspecified underlying mean dynam-ics. Of course, both models would also clearly benefit from an improved conditional mean specification.

Our simulation results show that the linear information in-strument model outperforms the BEKK model in both three and five portfolio examples based on book to market portfolios. To examine the behavior of our model in larger systems and with

different underlying asset returns, we replicate our finite sample simulations for ten momentum portfolios. The portfolio return data are downloaded from French’s website covering the period from November 4, 1926, to December 31, 2012. We follow the methodology of Section 3.5to construct the weekly realized conditional covariance matrices and the underlyingown infor-mation instruments. For each replication we choose a multivari-ate time series ofT₌520, which corresponds to approximately 10 years of return data. We perform 1000 simulation runs and document the results in panel E ofTable 8.

Not surprisingly, we again find that the linear information instrument model provides improved performance relative to the BEKK model for both the MAPE and RMSPE. In particular, the linear information instrument model improves the mean MAPE by 11% (₌ 0.4418–0.3322) and the mean RMSPE by 15% (₌0.5676–0.4109).

4. CONCLUDING REMARKS

The second moments of financial asset returns play an impor-tant role in risk management, derivative pricing, hedging, and portfolio optimization. We propose a new model for conditional covariance estimation based on available information instru-ments. Our approach is based on a Cholesky-like decomposition and ensures positive definiteness andinvariance to variate order

by construction. Compared to existing time-series approaches,

our linear instrument model provides a parsimonious and ac-curate description of second moments. The proposed approach is expected to be especially valuable whenowneffects are im-portant. Simulation results suggest that our linear information instrument model has better estimation accuracy than the BEKK model. Our particular empirical application with well diversi-fied portfolio returns may be limited in revealing the benefits from our covariance specification; however, the rapid growth in highly nonlinear hedge fund payoffs may provide an interesting application for our model in future research.

The linear information instrument model should provide a natural application in an asset pricing context where return mo-ments evolve with predetermined economic information instru-ments. Our approach can be used to estimate both time varying risk premiums and asset pricing risks that arise as a function of asset moments and economic fundamentals. In sum, we add to the burgeoning conditional covariance literature with a simple, yet parsimonious model to incorporate economic information. Future work may be helpful to consider different functional forms and instrument choices within the decomposition suggested.

APPENDIX A A.1 Proof of Proposition 1

To see that Equation (8) results in a positive definite covari-ance matrix, M_{t, note that because} L_{t is a lower triangular} matrix, the determinant of L_{t is given as Det(}L_t)=N

i=1liit,

where liit represents theith diagonal element in Lt. For any

nonzero coefficientsγii0, we have thatliit=0 with probability

one. Therefore, Det(L_{t)=0, hence}L_{tis invertible. According} to the equivalent conditions of matrix positive definiteness, for any lower triangular invertible matrixL_{t, we have that} L_tL′

t is

positive definite. Therefore, M_{tis positive definite.}

To show that Equation (8) satisfies theinvariance to variate orderproperty, we first show sufficiency followed by necessity within the class of linear functions defined by Equation (6).

(1) Sufficiency

Consider the subgroup of linear functions given by Equation (8). GivenM_{t =}L_tL′

t, theith diagonal element inMt,miit, is

miit = i

k=1

l_ikt2 ₌ϕii0+ϕ′ii1Zagg,t−1+ϕii2Zi,t−1

+ϕ′_ii3Z_agg,t−1⊗Z_agg,t−1 +ϕ′_ii4Z_agg,t−1Zi,t−1+ϕii5Z2i,t−1

where ϕii0=i_k=1γ_ik2₀,ϕii1= i

k=12γik0γik1, ϕii2=i_k=1 2γik0γik2, ϕii3=

i

k=1γik1⊗γik1, ϕii4= i

k=12γik2γik1,

ϕii5= i k=1γ

2

ik2, and ⊗ denotes the Kronecker product operator.

Themij t element is mij t =

j

k=1liktlj kt =φij0+φ ′

ij1Zagg,t₋1 +φij2Zi,t−1+φij3Zj,t−1

+φ_ij′₄Z_agg,t−1⊗Z_agg,t−1+φ′_ij5Z_agg,t−1Zi,t−1 +φ_ij′₆Z_agg,t−1Zj,t−1+φij7Zi,t−1Zj,t−1, for any i > j, and where φij0 =

j

k=1γik0γj k0, φij1= j

k=1(γik0γj k1+γik1γj k0), φij2= j

k=1γik2γj k0, φij3= j k=1

γik0γj k2, φij4= j

k=1γik1⊗γj k1, φij5= j

k=1γik2γj k1,

φ_ij6=j

k=1γik1γj k2, andφij7= j

k=1γik2γj k2.

Note that miit is a linear function of Zagg,t−1, Zi,t−1, Z_agg,t−1⊗Z_agg,t−1, Z_agg,t−1Zi,t−1, and Z2_i,t−1; where mij t is

a function of Z_agg,t−1, Zi,t₋1, Zj,t₋1, Zagg,t₋1⊗Zagg,t₋1, Z_agg,t−1Zi,t₋1,Zagg,t₋1Zj,t₋1, andZi,t₋1Zj,t₋1. BecauseMtis

symmetric, it follows thatmij t =mj it, for anyi > j.Therefore,

L_{tsatisfies the}invariance to variate orderproperty. (2) Necessity

By way of contradiction, we will show that any subgroup of the defined class of linear functions other than Equation (8) violates the invariance to variate order property. In the case of the general specification given by Equation (6), we need only show that theinvariance to variate orderis violated in a simple 2_×2 case whenZj,t−1are admitted into the empirical specification. Consider a lower triangular matrixL_t=[lij t] with lij t =γij0+γ′ij1Zagg,t−1+γij2Zi,t−1+γij3Zj,t−1. (A.1) GivenM_{t =}L_tL′_{t, it is straightforward to confirm that}

m11t =γ1102 +(2γ110γ111)′Zagg,t−1 +(2γ110γ112+2γ110γ113)Z1,t−1 +(γ_111⊗γ₁₁₁)′(Z_agg,t−1⊗Z_agg,t−1) +(2γ₁₁₁γ112+2γ111γ113)′Zagg,t−1Z1,t−1 +γ₁₁₂2 ₊2γ112γ113+γ1132

Z2_1,t−1.

and

m22t =

γ₂₁₀2 ₊γ₂₂₀2 ₊(2γ210γ211+2γ220γ221)′Zagg,t−1 +2γ210γ213Z1,t−1

+(2γ210γ212+2γ220γ222+2γ220γ223)Z2,t−1 +(γ_211⊗γ₂₁₁₊γ_221⊗γ₂₂₁)′(Z_agg,t−1⊗Z_agg,t−1) +(2γ₂₁₁γ213)′Zagg,t−1Z1,t−1

+(2γ₂₁₁γ212+2γ221γ222+2γ221γ223)′Zagg,t₋1Z2,t₋1 +2γ212γ213Z1,t−1Z2,t−1+γ2132 Z

2 1,t−1 +

γ₂₁₂2 ₊γ₂₂₂2 ₊γ₂₂₃2 ₊2γ222γ223Z22,t−1

Observe that m11t is a function of Zagg,t−1, Z1,t−1, Z_agg,t−1⊗Z_agg,t−1,Z_{agg,t−1Z1,t−1,andZ}2

1,t−1. In contrast,m22t

is a function of Z_agg,t−1, Z1,t−1, Z2,t−1, Zagg,t−1⊗Zagg,t−1, Z_agg,t−1Z1,t−1,Zagg,t−1Z2,t−1, Z1,t−1Z2,t−1, Z12,t−1, andZ

2 2,t−1.

Thus, Equation (A.1) violates the invariance to variate order

property.

The only other alternative linear specification from Equation (6) is given byL_t=[lij t] with

lij t =γij0+γ′ij1Zagg,t−1+γij3Zj,t−1. (A.2) We can now constructM_t=L_tL′_{t, to demonstrate that}

m11t =γ1102 +(2γ110γ111)′Zagg,t−1+2γ110γ113Z1,t−1 +(γ_111⊗γ₁₁₁)′₍Z_agg,t−1⊗Z_agg,t−1) +(2γ₁₁₁γ113)′Zagg,t−1Z1,t−1+γ1132 Z 2 1,t−1 and

m22t =

γ₂₁₀2 ₊γ₂₂₀2 ₊(2γ210γ211+2γ220γ221)′Zagg,t−1 +2γ210γ213Z1,t−1+2γ220γ223Z2,t−1

+(γ_211⊗γ₂₁₁₊γ_221⊗γ₂₂₁)′(Z_agg,t−1⊗Z_agg,t−1) +(2γ₂₁₁γ213)′Zagg,t−1Z1,t−1+(2γ221γ223)′Zagg,t−1 ×Z2,t₋1+γ2132 Z

2 1,t−1+γ

2 223Z

2 2,t−1.

We observe thatm11tis a function of onlyZagg,t−1andZ1,t−1, wherem22t is a function of Zagg,t−1,Z1,t−1, andZ2,t−1.Thus, Equation (A.2) also violates the invariance to variate order

property.

Therefore, all subgroups of the defined class of linear func-tions except Equation (8) violate theinvariance to variate order

property. Thus, within the class of linear functions having both aggregate andowninformation instruments, any model that sat-isfies the invariance to variate order property must take the form of Equation (8).

A.2 Sufficient Conditions for Consistency and Asymptotic Normality

We provide sufficient conditions for consistency and asymp-totic normality following Newey and MacFadden (1994). Comte and Lieberman (2003) building on the work of Jeantheau (1998) provided very general results for comparable multivari-ate GARCH(p,q) models, and especially BEKK-like models. Consider the normalized log-likelihood, or objective function,

QT(θ)= −

1 2T

T

t=1

Nlog (2π)+log|M₍θ)t| +r′tM(θ)− 1 t rt,

where for the purposes of this appendix we simplify notation and let all parameters of the covariance matrix be given in the vectorθ, for given zero mean vector,r_{t, and covariance matrix,} M₍θ)t. Assumert is square integrable, strictly stationary, and

ergodic.

Asymptotic consistency of the MLE estimator is assured un-der Theorem 2.5 of Newey and MacFadden (1994). In particular, the required conditions are:

(i) Ifθ₌θ0thenQT(θ)=QT(θ0) (ii) θ0 ∈ℵ, which is compact

(iii) QT(θ) is continuous at eachθ∈ℵwith probability one

(iv) E[supθ∈ℵ|QT(θ)|]<∞

(i) Because M₍θ)t has a unique specification in θ from

Proposition I, given ∀θεℵ, _∀θ0εℵ, we must have,

QT(θ)=QT(θ0). Soθ0is identified. (ii) We assumeℵis compact.

(iii) QT(θ) is continuous at eachθ∈ℵwith probability one

by inspection.

(iv) Following Comte and Lieberman’s (p. 67,2003) condi-tions to ensure strong consistency, letx_i(θ) be the eigen-values ofM₍θ)t for a fixedt, we have,

log_|M₍θ)t| = n

i₌1

log (x_i(θ))_≤

n

i₌1

x_i(θ)₌Tr (M₍θ)t).

Taking expectations with respect to θ on both sides of the inequality, we have,

Eθ(log_|M₍θ)t|)≤Eθ(tr (M(θ)t))= n

i=1

Eθ([M₍θ)t]ii).

It is straightforward to show that, r′

tM(θ)− 1 t rt=

n

i₌1

r_i,t2[M₍θ)t]ii+2 n

i₌1,j <i

ri,trj,t[M(θ)t]ij.

Taking expectations with respect toθ, we find,

Eθr′

tM(θ)− 1 t rt

=

n

i₌1

r_i,t2Eθ([M₍θ)t]ii)+2 n

i₌1,j <i

ri,trj,tEθ([M(θ)t]ij).

Becauser_{tis square integrable, each element in}M₍θ)texists

and is finite. Thus, we must have,

Eθ([M₍θ)t]ii)<∞ fori=1, . . . , nandEθ([M(θ)t]ij) <_∞fori₌1, . . . , nandj < i

Therefore,

Eθ(QT(θ))<∞.

Conditions (i) to (v) in Theorem 3.1 of Newey and MacFadden (p. 2143,1994) ensure that the maximum likelihood estimator forθ, say ˆθ, is asymptotically normal:

(i) θis in the interior ofℵ

(ii) QT(θ) is twice continuously differentiable in a

neighbor-hoodℵofθ0 (iii) √T∂QT(θ)

∂θi d

→N(0_{,) and}

(iv) The expected Hessian, H₍θ), is continuous at θ0 and supθ_∈ℵ|

∂2_Q

T(θ)

∂θ∂θ′ −H(θ0) p

→0 (v) H₌ H₍θ0) is nonsingular.

We consider these conditions in turn.

(i) Assumeθis in the interior of the parameter spaceℵ. (ii) Following Comte and Lieberman (2003, Lemma A.1 and

the associated proof) and Jeantheau (1998), the second-order derivative ofQT(θ) with respect to theith andjth

element inθis given by

∂2_Q T(θ) ∂θiθj = −

1 2T

T

t=1 tr

_∂2_M(θ) t ∂θiθj

M₍θ)−t1− ∂M₍θ)t

∂θi

M₍θ)−t1

×∂M_∂θ(θ)t

j

M₍θ)−t1+rtr′tM(θ)− 1 t

∂M₍θ)t ∂θj

M₍θ)−t1

×∂M(θ)t

∂θi

M₍θ)−t1−rtr′tM(θ)− 1 t

∂2M₍θ)t ∂θiθj

M₍θ)−t1

+r_tr′_tM₍θ)−t1 ∂M₍θ)t

∂θi

M₍θ)−t1 ∂M₍θ)t

∂θj

M₍θ)−t 1

Because each element in M₍θ)t is a linear function of the

instruments, it is twice differentiable. SoQT(θ) is twice

differ-entiable.

(i) The first-order derivative ofQT(θ) with respect to theith

element inθis (also see Comte and Lieberman (2003) for a similar development),

∂QT(θ)

∂θi = −

1 2T

T

t=1

tr

M₍θ)−t1

∂M₍θ)t ∂θi

−

r′_tM₍θ)−t1 ∂M₍θ)t

∂θi

M₍θ)−t1rt

= − 1 2T

T

t=1

tr

M₍θ)−t1

∂M₍θ)t ∂θi

−

r′_tM₍θ)−t1

∂M₍θ)t ∂θi

M₍θ)−t1rt

= − 1 2T

T

t=1

tr

M₍θ)−t1

∂M₍θ)t ∂θi

−

r_tr′_tM₍θ)−t1

∂M₍θ)t ∂θi

M₍θ)−t1

UsingEt₋1(rtrt′)=M(θ0)t, and taking expectations, yields, E

_∂Q

T(θ0)

∂θi

= − 1 2T

T

t=1

tr

M₍θ0)−t1

∂M₍θ)t

∂θi −

_∂M (θ0)t ∂θi

M₍θ0)−t1

= − 1 2T

T

t=1

tr

M₍θ0)−t1

∂M₍θ0)t

∂θi −

M₍θ0)−t1

∂M₍θ0)t ∂θi

= 0

Thus,E(∂QT(θ0)

∂θ )=[0].

Let₌E(∂QT(θ0)

∂θ

∂QT(θ0)

∂θ′ ). Then, by the central limit

theo-rem, we have, √

T

_∂Q

T(θ)

∂θ −[0]

d

→N([0],)

Therefore,√T∂QT(θ) ∂θ

d

→N([0],).

(i) BecauseQT(θ) is twice differentiable,H(θ)=E(∂

2_Q

T(θ) ∂θ∂θ′ )

is continuous at θ0. Given θ p

→θ0, supθ∈ℵ

∂2_Q

T(θ) ∂θ∂θ′ −

H₍θ0) p

→0.

(ii) We assume H₍θ0) is nonsingular soH(θ0)−1exists.

A.3 Development of the Covariance Matrix forh( ˆγ) The standard error for each element in the estimated covari-ance matrix can be obtained using the Delta method. Letγ be a vector of parameters for the decomposition matrixL_{t where} each element is given by Equation (9). Theith diagonal element inM_{t, denoted}miit, is

miit = i

k₌1

γ_ik2₀₊2

i

k₌1

γik1γik0Zi,t−1+ i

k₌1

γ_ik2₁Z_i,t2₋₁,

and thei, jth off-diagonal elementmij t is mij t =

j

k=1

γik0γj k0+ j

k=1

γik1γj k0Zi,t₋1+ j

k=1

γik0γj k1Zj,t₋1

+

j

k=1

γik1γj k1Zi,t−1Zj,t−1,

It is straightforward to show that the first order derivative of

miitis given by ∂miit ∂γ = ⎧ ⎪ ⎨ ⎪ ⎩

2γik0+2γik1Zi,t₋1, for allγik0givenk≤i 2γik0Zi,t−1+2γik1Z2i,t−1, for allγik1givenk≤i 0, for all other elements inγ

and the first-order derivative ofmij tis ∂mij t

∂γ = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

γik0+γik1Zi,t−1, for allγj k0givenk≤j

γj k0+γj k1Zj,t−1, for allγik0givenk≤j

γik0Zj,t₋1+γik1Zi,t₋1Zj,t₋1, for allγj k1givenk≤j

γj k0Zi,t−1+γj k1Zj,t−1Zj,t−1, for allγik1givenk≤j 0, for all other elements

inγ ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ From the Delta method, the standard errors for each element in the estimated covariance matrix at each point of time are then computed as (∂[mij t]

∂γ )′·ˆ ·

∂[mij t]

∂γ , where ˆis the estimated

covariance matrix of ˆγ and can be estimated using the inverse of the Fisher information matrix, which is readily available in most software applications (the “hessian” option in Matlab).

APPENDIX B: COMPARISON WITH THE FACTOR-ARCH MODEL

Factor models provide another popular form of covariance specification motivated by economic theory (Engle, Ng, and Rothschild 1990; van der Weide 2002; Vrontos et al. 2003;

Lanne and Saikkonen2007). For example, Engle, Ng, and Roth-schild (1990) proposed a factor structure for the conditional covariance matrix as

M_{F t=+}

K

k=1

βkβ′kfk,t, (B.1)

where is an N_×N positive semidefinite matrix; βk, k=

1, . . . K,is a linearly independentN_×1 weight vector for factor

k; andfk,t is thekth factor that is derived from the stochastic

process ofr_t.

Assumingfk,t follows a first-order ARCH structure, and

as-suming a mimicking portfolio approach can be used to create factors we have,

fk,t =λk+φk(ω′krt−1)2, (B.2)

whereλkandφkare scalar parameters andωkis a givenN×1

vector of weights. The model can be estimated using a two-step ML method as described in Engle, Ng, and Rothschild (1990). In particular, consistent estimates offk,tare typically first obtained

by ML. The estimates ofβ_k are then obtained in a second step using first stage consistent estimates offk,t.

To compare our specification with the factor ARCH model, we consider a simple bivariate case ignoring the typical multi-step estimation procedure. Also, to facilitate comparison with the single factor ARCH model, we consider a single macroe-conomic information instrument Z_agg,t−1=ω′r_{t−1 and a} par-ticular model specification given bylij t =γij0+γi1f1,t. This

specification can be readily generalized to aK factor ARCH model by extending the definition of the aggregate information instrument as Z_agg,t−1=ω′r_{t−1, where} ω is now an N_×K

matrix of weights andr_{t−1is an}N_×1 vector of lagged model disturbances. Note that this is a restricted form of our general model where the impact of the factor on covariances is limited through a further restriction to provide comparability with the factor ARCH specification. We do not consider restrictions to ensure portfolio weights sum to one, or other potentially in-teresting restrictions on the weight vector that are commonly used in practice. Instead, we focus only on differences in these specifications for a given weight vector.

In a simple case with two assets and one macroeconomic information instrument, the conditional covariance matrix of our linear information instrument model is given by

Mt =

m11t m12t m21t m22t

=

γ110+γ11(ω′rt−1) 0

γ210+γ21(ω′rt−1) γ220+γ21(ω′rt−1)

,

×

γ110+γ11(ω′rt₋1) γ210+γ21(ω′rt₋1) 0 γ220+γ21(ω′rt−1)

, (B.3)

whereωis a 2×1 vector of weights andr_{t−1is a 2×1 vector of} lagged model disturbances. Evaluating term by term, we have,

m11t =γ1102 +2γ110γ11(ω′rt₋1)+γ112(ω′rt₋1)2,

m21t =m12t=γ110γ210+(γ110γ21+γ210γ11)(ω′rt−1) +γ11γ21(ω′rt−1)2, and

m22t =γ2102 +γ 2

220+2γ21(γ210+γ220)

ω′r_t−1 +2γ₂₁2(ω′r_t−1)2

(B.4) For the factor ARCH model with a single factor, the condi-tional covariance matrix is given by

M_{F t =}

mF11t mF12t mF21t mF22t

=

γ11 γ21

γ21 γ22

+

β1

β2

β1 β2

[λ₊φ(ω′r_t−1)2_].

(B.5) Evaluating term by term, we get,

mF11t =γ11+β12[λ+φ(ω′rt−1)2],

mF21t =mF12t=γ21+β1β2[λ+φ(ω′rt−1)2], and

mF22t =γ22+β22[λ+φ(ω′rt−1)2]. (B.6) Comparing Equations (B.4) and (B.6), we observe that our lin-ear information instrument model differs from the factor ARCH model in a number of manners. First, our specialized model has a richer specification than the factor ARCH model. In particular, our model involves terms in bothri,t₋1andri,t₋1rj,t₋1, whereas the factor ARCH model only contains terms inri,t₋1rj,t₋1. Sec-ond, our model has fewer parameters. For anyN_×Ncovariance matrix withKfactors, our model requiresN(N₂+1)+N K param-eters, where the factor ARCH model has N(N₂+1)₊N K₊2K

parameters. Third, our model requires fewer restrictive assump-tions to determine the number of factors.

SUPPLEMENTARY MATERIALS

Matlab code that allows practitioners to estimate the condi-tional covariance matrix using the linear information instrument model is available online in the file entitled practitioner.zip. This file includes sample data, Matlab code, and a published Matlab user guide. We also provide all Matlab code and data used in the article in the online file entitled research.zip.

ACKNOWLEDGMENTS

The authors appreciate the helpful comments of Sung Ahn, Ranjini Jha, Bob Korkie, Ron Mittelhammer, Adam Nowak, Rick Sias, and especially Feng Yao, as well as seminar partic-ipants at Washington State University. The helpful comments of the editor, associate editor, and two referees that greatly im-proved the article are also gratefully acknowledged.

[Received November 2012. Revised October 2013.]

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