Singular Conditional Autoregressive Wishart Model for Realized Covariance Matrices

Abstract

Realized covariance matrices are often constructed under the assumption that richness of intra-day return data is greater than the portfolio size, resulting in nonsingular matrix measures. However, when for example the portfolio size is large, assets suffer from illiquidity issues, or market microstructure noise deters sampling on very high frequencies, this relation is not guaranteed. Under these common conditions, realized covariance matrices may obtain as singular by construction. Motivated by this situation, we introduce the Singular Conditional Autoregressive Wishart (SCAW) model to capture the temporal dynamics of time series of singular realized covariance matrices, extending the rich literature on econometric Wishart time series models to the singular case. This model is furthermore developed by covariance targeting adapted to matrices and a sector wise BEKK-specification, allowing excellent scalability to large and extremely large portfolio sizes. Finally, the model is estimated to a 20-year long time series containing 50 stocks and to a 10-year long time series containing 300 stocks, and evaluated using out-of-sample forecast accuracy. It outperforms the benchmark models with high statistical significance and the parsimonious specifications perform better than the baseline SCAW model, while using considerably less parameters.

Keywords:

• Covariance targeting
• High-dimensional data
• Realized covariance matrix
• Stock co-volatility
• Time series matrix-variate model

1 Introduction

The covariance matrix of asset returns plays a key role in several financial applications, such as portfolio allocation, risk management and option pricing. It is well-documented that this quantity changes over time, why describing and understanding its temporal dynamics is fundamental to financial decision making. A typical approach is to capture this evolution in discrete time by applying multivariate GARCH-type models, summarized in Bauwens, Laurent, and Rombouts (2006), where the conditional covariance matrix is determined by past observations of daily returns. Another classic method is to use multivariate stochastic volatility-type models, reviewed in Asai, McAleer, and Yu (2006), in where the covariance matrix process is assumed to be random.

During the last two decades, increased availability of asset price data on high frequencies has paved the way for numerous novel approaches in this area. Many of them are built upon the notion of realized covariance, where the daily return covariance matrix is estimated by a large number of intra-day returns, on for example 5 or 10 min intervals (see e.g., Andersen et al. 2003; Barndorff-Nielsen and Shephard 2004). Modeling the time series dynamics for realized covariance matrices in discrete time has become a large branch in the econometric literature. A popular approach is to assume the underlying stochastic process to be Wishart, a well-studied distribution that ensures positive-definiteness almost surely. For example, the Wishart Autoregressive (WAR) model introduced in Gouriéroux, Jasiak, and Sufana (2009), assumes realized covariances are conditionally distributed as noncentral Wishart, where the noncentrality parameter is described by historical realized covariance matrices. The High-Frequency-Based Volatility (HEAVY) model presented in Noureldin, Shephard, and Sheppard (2012) and the Conditional Autoregressive Wishart (CAW) model introduced by Golosnoy, Gribisch, and Liesenfeld (2012) rely on the centralized Wishart distribution, where the scale matrix parameters are determined by past observations. A central Wishart distribution is also considered in Jin and Maheu (2012), but here the scale matrix is decomposed into either multiplicative components or additive components determined by sample means of historical realized covariances. The General Conditional Autoregressive Wishart (GCAW) model is proposed in Yu, Li, and Ng (2017), parameterized with both a noncentral parameter as in the WAR model and a scale matrix as in CAW model. In Anatolyev and Kobotaev (2018), the CAW model is extended to the Conditional Threshold Autoregressive Wishart (CTAW) model with the aim to include the effects of price asymmetry on future realized covariances. Goodness-of-fit tests for models driven by an underlying central Wishart distribution, such as the CAW model, is presented in Alfelt, Bodnar, and Tyrcha (2020).

All of the models discussed above assume a realized covariance matrix that is positive definite. This can be ensured as long as the number, n, of intra-day returns used to compute the realized covariance matrix is larger than or equal to the number, m, of assets included into the portfolio. Regarding small and moderately sized portfolios or reasonably liquid assets, this relation can often be justified. However, in many applications it is of interest to consider portfolios of large dimensions, containing perhaps 50, 100 or even 1000 assets (see e.g., Rubio, Mestre, and Palomar 2012; Hautsch, Kyj, and Malec 2015; Ledoit and Wolf 2017; Ao, Yingying, and Zheng 2019; Cai et al. 2020; Bodnar et al. 2021; Ding, Li, and Zheng 2021; Bodnar, Bodnar, and Parolya 2022; Bodnar, Okhrin, and Parolya 2022). Furthermore, available data for the portfolio assets might be restricted, due to for example low liquidity resulting in a few price quotes per day. In addition, there might be limits to how high of an intra-day return sample frequency that is suitable, in presence of so-called market microstructure noise (see, e.g., Aït-Sahalia and Yu 2009). Any combination of these factors might result in a situation where m > n, and hence daily realized covariance matrices that are singular. Finally, Jacod and Podolskij (2013) derived an asymptotic test for inferring the rank of multivariate volatility processes.

To solve the problem with the singularity of the covariance matrix, one can consider data of higher frequency and/or can make use of the ultra high frequency data (see e.g., Engle 2000; Aït-Sahalia, Mykland, and Zhang 2011; Christensen, Oomen, and Podolskij 2014). On the other side, in spite of the broad availability of ultra high-frequency data, there are several reasons for possibly using moderate intra-daily sample sizes for the purpose of realized covariance estimation, namely market microstructure noise and zeros. Both of the issues are largely present due to illiquidity of stocks traded on financial markets.

The microstructure noise can directly be incorporated in the model for stock prices (see e.g., Bandi and Russell 2006, 2008; Bibinger et al. 2014, 2019), while the illiquidity issue related to nonsynchronous trading can be treated as a missing-value problem (see e.g., Corsi, Peluso, and Audrino 2015; Shephard and Xiu 2017; Buccheri et al. 2021), where the idea is to use all available price data and to perform the estimation of model parameters by employing the Kalman filter recursion together with the ordinary least squares estimation or quasi maximum likelihood estimation. To this end, the presence of zeros in the price process, that is, the periods where the price of a stock does not change, is another source that influences the properties of the realized covariance matrices and the way how they are constructed. The issue has recently been discussed in a number of literature studies. While Bandi, Pirino, and Renò (2017) and Bandi et al. (2020a, 2020b) consider the case of continuous-time modeling, the results in the discrete-time case can be found in Catania, Mari, and de Magistris (2020) and Sucarrat and Grønneberg (2020).

The above mentioned approaches deal with modeling the price process. Although the developed methods suggest an efficient way how to deal with the microstructure noise, zeros and nonsynchronous trading, the issues might become severe when the dimension of the holding portfolio becomes large and as such high-dimensional realized covariance matrices should be constructed from the intra-day data. To reduce the impact of nonsynchronous trading and the presence of zeros in the prices, one can reduce the sample frequency. As such the realized covariance matrices might become singular by construction and new econometric models should be developed. This is the aim of this article, namely to focus on time series models of singular realized covariance matrices directly, and to extend the family of econometric autoregressive Wishart models to the singular case by introducing the Singular Conditional Autoregressive Wishart (SCAW) model to describe such time series. It is based on the assumption that realized covariance matrices follow a conditional singular Wishart distribution, described in, for example, Srivastava (2003) and Bodnar and Okhrin (2008), where the scale matrix parameter is determined by historical observations in an autoregressive fashion similar to the BEKK-specification of Engle and Kroner (1995), alike for example the CAW model in Golosnoy, Gribisch, and Liesenfeld (2012). This specification ensures positive definiteness and allows us to directly estimate the model parameter with the maximum likelihood method. Furthermore, parameter-based conditions for weak stationarity of the model are deduced. To this end, SCAW model coincides with the multivariate GARCH process based on the BEKK specification when the intra-day data are replaced by daily asset returns, that is, when n = 1.

Since the singular case is closely related to portfolios of large dimensions, a challenge in this setting is to capture the temporal dynamics of the time series, while simultaneously maintaining a parsimonious model that can be feasibly estimated. To deal with this scaling challenge, two novel approaches are introduced. The first one regards covariance targeting (see e.g., Pedersen and Rahbek 2014), where the approach of Noureldin, Sheppard, and Shephard (2014) is adapted to the matrix case. It concerns standardizing the time series by its unconditional mean, which allows implementing straightforward conditions on the model parameters such that positive definiteness is maintained also under a covariance targeting regime. This method circumvents estimating the large number of parameters present in the constant matrix of the BEKK-specification, greatly increasing estimation feasibility. The second approach uses the similarity of assets that belong to the same market sector. This specification assumes that the parameters governing temporal dynamics of the matrix time series are homogeneous for assets of the same sector. As such, the number of parameters does not depend on the number of portfolio assets, but rather of the number of market sectors these assets stem from. Combining these approaches results in a model that is well equipped for implementation on large or extremely large portfolios. In addition, an extension using the heterogeneous autoregressive (HAR) specification, adapted from Corsi (2009), is applied to the SCAW model. This approach considers long-time memory dependence by including historical realized covariance matrices on longer horizons, such as weekly or monthly.

In the empirical part of the article the SCAW model with various specifications is estimated to a time series of 50 and 300 assets traded on the National Association of Securities Dealers Automated Quotations (NASDAQ) over 20 and 10 years, respectively. It is evaluated by several out-of-sample forecast measures and benchmarked against similarly specified Multivariate GARCH models and two DCC-type models extended to model large-dimensional dynamic covariance matrices (see Engle 2002; Engle, Ledoit, and Wolf 2019; Ledoit and Wolf 2020; De Nard et al. 2020).

The results of the empirical study reveal that the SCAW model outperforms the benchmark models with high statistical significance for the vast majority of the forecasts measures. Moreover, it suggests that the presented sectorwise parameterization, scalar parameterization and HAR-extension each can be useful, outperforming the original SCAW model in out-of-sample forecast accuracy, despite having substantially fewer parameters.

The rest of the article is organized as follows. Section 2 introduces the SCAW model and presents its stochastic properties. In Section 3, covariance targeting, the sectorwise specification, and the HAR-extension are introduced. Section 4 governs the estimation procedure for the SCAW model with its various specifications. The empirical application is presented in Section 5, while Section 6 concludes. Proofs of the obtained theoretical results and some tables with the results of the empirical study can be found in the supplementary materials.

2 Singular Conditional Autoregressive Wishart (SCAW) Model

Let ${\mathbf{R}}_t$ be an m × m realized covariance matrix, constructed using n intra-day return vectors recorded during day t. In addition, suppose that the number of intra-day return vectors used in the computation of ${\mathbf{R}}_t$ is less than the dimension of these vectors, such that n < m. As a result, ${\mathbf{R}}_t$ is a singular matrix by construction. Furthermore, let $\left\{{\mathbf{R}}_t\right\}$ be a discrete time series of such matrices, and let ${\mathcal{F}}_t$ denote the filtration associated with $\left\{{\mathbf{R}}_t\right\}$. Now, assume that conditioned on ${\mathcal{F}}_{t-1},{\mathbf{R}}_t$ follows a singular Wishart distribution of dimension m. That is,(1) $${\mathbf{R}}_t|{\mathcal{F}}_{t-1}\sim S{W}_m(n,{\mathbf{S}}_t/n),$$(1) where $S{W}_m(\nu ,\mathbf{\Sigma })$ denote the singular Wishart distribution with degrees of freedom ν and symmetric, positive-definite scale matrix $\mathbf{\Sigma }$ of dimension m × m. In addition, since $\mathbb{E}[{\mathbf{R}}_t|{\mathcal{F}}_{t-1}]={\mathbf{S}}_t$, the matrix ${\mathbf{S}}_t$ is the conditional mean matrix of $\left\{{\mathbf{R}}_t\right\}$. Note that the singularity of ${\mathbf{R}}_t$ stems from the degrees of freedom n being lower than the matrix dimension m, while the conditional mean matrix, ${\mathbf{S}}_t$, is assumed to be nonsingular.

Now, let ${\mathbf{R}}_t$ be partitioned as(2) $${\mathbf{R}}_t=\left[\begin{array}{cc}{\mathbf{R}}_{11,t}& {\mathbf{R}}_{12,t}\\ {\mathbf{R}}_{21,t}& {\mathbf{R}}_{22,t}\end{array}\right],$$(2) where ${\mathbf{R}}_{11,t}$ is an n × n nonsingular matrix, ${\mathbf{R}}_{12,t}$ is $n\times (m-n),{\mathbf{R}}_{21,t}={\mathbf{R}}_{12,t}^{\prime }$ and ${\mathbf{R}}_{22,t}$ is $(m-n)\times (m-n)$ with ${\mathbf{R}}_{22,t}={\mathbf{R}}_{21,t}{\mathbf{R}}_{11,t}^{-1}{\mathbf{R}}_{12,t}$. That any singular, symmetric matrix can be partitioned this way is shown by, for example, Harville (1997, Lemma 9.2.2). Consequently, in accordance with Srivastava (2003) regarding the singular Wishart distribution, the conditional density for ${\mathbf{R}}_t$ is given by(3) $$\begin{array}{c}f({\mathbf{R}}_t|{\mathcal{F}}_{t-1})=\frac{{\pi }^{n(n-m)/2}}{2^{mn/2}{\Gamma }_n(n/2)|{\mathbf{S}}_t/n{|}^{n/2}}|{\mathbf{R}}_{11,t}{|}^{(n-m-1)/2}\\ \hspace{1em}\hspace{0.17em}\text{exp}\hspace{0.17em}\left(-\frac{1}{2}\text{tr}({({\mathbf{S}}_t/n)}^{-1}{\mathbf{R}}_t)\right)\\ =\frac{{\pi }^{n(n-m)/2}{n}^{m-n/2}}{2^{mn/2}{\Gamma }_n(n/2)|{\mathbf{S}}_t{|}^{n/2}}|{\mathbf{R}}_{11,t}{|}^{(n-m-1)/2}\\ \hspace{1em}\hspace{0.17em}\text{exp}\hspace{0.17em}\left(-\frac{n}{2}\text{tr}({\mathbf{S}}_t^{-1}{\mathbf{R}}_t)\right),\end{array}$$(3) where ${\Gamma }_n(·)$ denotes the multivariate gamma function (see e.g., Gupta and Nagar 2000). In addition, the conditional covariance matrix of ${\mathbf{R}}_t$ consists of the following elements(4) $$\text{cov}[r_{ij,t},r_{kl,t}|{\mathcal{F}}_{t-1}]=\frac{1}{n}(s_{ik,t}{s}_{jl,t}+s_{il,t}{s}_{jk,t}),$$(4) for $i,j,k,l=1,\dots ,m$, where $r_{ij,t}$ and $s_{ij,t}$ denotes the element on row i and column j of ${\mathbf{R}}_t$ and ${\mathbf{S}}_t$, respectively.

The conditional mean matrix ${\mathbf{S}}_t$, which is measurable by ${\mathcal{F}}_{t-1}$, captures the time series dynamics of singular realized covariance matrices $\left\{{\mathbf{R}}_t\right\}$. In the following it is assumed that(5) $${\mathbf{S}}_t=\mathbf{CC}\prime +\sum _{i=1}^p{\mathbf{B}}_i{\mathbf{S}}_{t-i}{\mathbf{B}}_i^{\prime }+\sum _{j=1}^q{\mathbf{A}}_j{\mathbf{R}}_{t-j}{\mathbf{A}}_j^{\prime },$$(5) where we will denote the lag order of the model by (p, q) and ${\mathbf{A}}_j,{\mathbf{B}}_i,\mathbf{C}$ are m × m parameter matrices for $i=1,\dots ,p$ and $j=1,\dots ,q$ where C is lower-triangular with strictly positive diagonal elements. This form is similar to the BEKK specification of Engle and Kroner (1995) in the multivariate GARCH case, which is also adapted for the CAW model in Golosnoy, Gribisch, and Liesenfeld (2012). It ensures that ${\mathbf{S}}_t$ is symmetric and positive definite as long as the initial matrices ${\mathbf{S}}_0,{\mathbf{S}}_{-1},\dots ,{\mathbf{S}}_{-p+1}$ are symmetric and positive semidefinite. It is notable that the conditional covariance matrix in GARCH-BEKK(p, q) coincides with the expression presented in (5) with ${\mathbf{R}}_{t-j}={\mathbf{x}}_{t-j}{\mathbf{x}}_{t-j}^{\prime }$, where ${\mathbf{x}}_t$ is the one day return vector of day t. As such, the proposed SCAW model (1) and (5) is a generalization of the GARCH-BEKK(p, q) process, where the GARCH-BEKK(p, q) model is a special case corresponding to n = 1. Furthermore, the specification of the SCAW(p, q) process is similar to the CAW(p, q) model suggested in Golosnoy, Gribisch, and Liesenfeld (2012) with the difference that the SCAW(p, q) process models singular realized covariance matrices, while Golosnoy, Gribisch, and Liesenfeld (2012) consider nonsingular ones.

In the article we will further consider different structures of the parameter matrices ${\mathbf{A}}_j,{\mathbf{B}}_i,\mathbf{C},j=1,\dots ,q,i=1,\dots ,p$. Since large dimensional cases will generally be discussed, the matrices ${\mathbf{A}}_j,{\mathbf{B}}_i,\mathbf{C}$ need to be specified parsimoniously such that estimation of the model remains feasible. If for example one allows ${\mathbf{A}}_j,{\mathbf{B}}_i$ to be general m × m matrices and C lower-triangular, the model (5) will consist of $m(m+1)/2+(p+q)m^2$ parameters. With a large dimensional case, such as m = 50 and $p=q=2$, this results in 11275 parameters, a formidable estimation exercise indeed.

2.1 Stochastic Properties of the SCAW Model

In this section we will present conditions under which the matrix-variate process $\left\{{\mathbf{R}}_t\right\}$ is weakly stationary. As with the CAW(p, q) model in Golosnoy, Gribisch, and Liesenfeld (2012), the stochastic properties of the SCAW(p, q) model defined in (1) and (5) are derived using the VARMA representation of the model. The proofs of the results presented in this section can be found in the supplement.

Let $\text{vec}(·)$ be the vectorization operator and let $\text{vech}(·)$ be the half-vectorization operator. The symbol ${\mathbf{D}}_m$ denotes the duplication matrix, while ${\mathbf{L}}_m$ stands for the elimination matrix.¹ We define$${\mathbf{r}}_t=\text{vech}({\mathbf{R}}_t),\hspace{1em}{\mathbf{s}}_t=\text{vech}({\mathbf{S}}_t),\hspace{1em}\mathbf{c}=\text{vech}(\mathbf{CC}\prime ),$$such that the vector representation of (5) is(6) $$s_t=c+{\displaystyle \sum _{i=1}^p}{\mathbf{ℬ}}_i{s}_{t-i}+{\displaystyle \sum _{j=1}^q}{\mathbf{𝒜}}_j{r}_{t-j},$$(6) where ${\mathbf{𝒜}}_j$ and ${\mathbf{ℬ}}_i$ are k × k matrices, with $k=m(m+1)/2$ given by$${\mathbf{𝒜}}_j=L_m(A_j\otimes A_j)D_m,{\mathbf{ℬ}}_i=L_m(B_i\otimes B_i)D_m,$$ where the symbol ⊗ denotes the Kroenecker product. Furthermore, ${\mathbf{r}}_t$ can be written as(7) $${\mathbf{r}}_t=\mathbb{E}\left[{\mathbf{r}}_t\right|{\mathcal{F}}_{t-1}]+{\mathbf{v}}_t={\mathbf{s}}_t+{\mathbf{v}}_t,$$(7) where ${\mathbf{v}}_t$ is a martingale difference sequence such that$$\mathbb{E}\left[{\mathbf{v}}_t\right]=0\hspace{1em}\text{and}\hspace{1em}\mathbb{E}\left[{\mathbf{v}}_t{\mathbf{v}}_s^{\prime }\right]=0,\hspace{1em}\forall s\ne t.$$

Plugging in (7) into (6), the SCAW(p, q) model can be written with the following VARMA(max($p,q),p$) representation:(8) $$r_t=c+{\displaystyle \sum _{i=1}^{\max (p,q)}({\mathbf{𝒜}}_i+{\mathbf{ℬ}}_i)}{r}_{t-i}+v_t-{\displaystyle \sum _{j=1}^p}{\mathbf{ℬ}}_j{v}_{t-j},$$(8) where ${\mathbf{𝒜}}_i={\mathbf{ℬ}}_j=0$ for $i>q,j>p$. From (8) the conditions for weak stationarity of ${\mathbf{R}}_t$ can be obtained. First, we derive a condition for the existence of the unconditional expectation of the SCAW(p, q) process, given by the following proposition.

Proposition 1.

The unconditional expectation of the SCAW(p, q) model is finite if and only if all eigenvalues of the matrix(9) $${\Psi }_1={\displaystyle \sum _{i=1}^{\text{max}(\text{p},\text{q})}({\mathbf{𝒜}}_i+{\mathbf{ℬ}}_i)}$$(9) are less than 1 in modulus. In that case the unconditional expectation is given by(10) $$\mathbb{E}[r_t]=\overline{r}={\left(I_k-{\displaystyle \sum _{i=1}^{\text{max}(\text{p},\text{q})}({\mathbf{𝒜}}_i+{\mathbf{ℬ}}_i)}\right)}^{-1}c.$$(10)

Equation (8)(8) $$r_t=c+{\displaystyle \sum _{i=1}^{\max (p,q)}({\mathbf{𝒜}}_i+{\mathbf{ℬ}}_i)}{r}_{t-i}+v_t-{\displaystyle \sum _{j=1}^p}{\mathbf{ℬ}}_j{v}_{t-j},$$(8) can also be represented as an infinite vector moving average time series by (see, e.g., sec. 11.3 and 11.4 in Lütkepohl 2005)(11) $${\mathbf{r}}_t=\overline{\mathbf{r}}+\sum _{i=0}^{\infty }{\mathbf{\Phi }}_i{\mathbf{v}}_{t-i},\hspace{1em}\text{where}$$(11) (12) $${\Phi }_i=-{\mathbf{ℬ}}_i+{\displaystyle \sum _{j=1}^i({\mathbf{𝒜}}_j+{\mathbf{ℬ}}_j)}{\Phi }_{i-j},i,j=1,2,\dots ,$$(12) (13) $${\mathbf{\Phi }}_0={\mathbf{I}}_m.$$(13)

Moreover, given that they exist, the autocovariances of ${\mathbf{r}}_t$ can then be expressed as(14) $$\mathbb{E}[({\mathbf{r}}_t-\overline{\mathbf{r}})({\mathbf{r}}_{t-\tau }-\overline{\mathbf{r}})\prime ]=\sum _{i=0}^{\infty }{\mathbf{\Phi }}_{i+\tau }\mathbb{E}\left[{\mathbf{v}}_t{\mathbf{v}}_t^{\prime }\right]{\mathbf{\Phi }}_i^{\prime }.$$(14)

Let(15) $$\begin{array}{c}\mathbf{\Omega }=\frac{1}{n}({\mathbf{L}}_m\otimes {\mathbf{L}}_m)[{\mathbf{I}}_{m^2}\otimes ({\mathbf{I}}_{m^2}+{\mathbf{K}}_{mm})]\\ \hspace{1em}({\mathbf{I}}_m\otimes {\mathbf{K}}_{mm}\otimes {\mathbf{I}}_m)({\mathbf{D}}_m\otimes {\mathbf{D}}_m),\end{array}$$(15) where ${\mathbf{K}}_{mm}$ is the commutation matrix.² Then the following holds.

Proposition 2.

The unconditional second moment of the SCAW(p, q) model is finite if and only if all eigenvalues of the matrix(16) $${\mathbf{\Psi }}_2=\sum _{i=1}^{\infty }({\mathbf{\Phi }}_i\otimes {\mathbf{\Phi }}_i)\mathbf{\Omega }$$(16) are less than 1 in modulus. In that case the second moment is given by(17) $$\text{vec}(\mathbb{E}[{\mathbf{r}}_t{\mathbf{r}}_t^{\prime }])=(\mathbf{\Omega }+{\mathbf{I}}_{k^2}){\left({\mathbf{I}}_{k^2}-\sum _{i=1}^{\infty }({\mathbf{\Phi }}_{\mathit{i}}\otimes {\mathbf{\Phi }}_i)\mathbf{\Omega }\right)}^{-1}\text{vec}(\overline{\mathbf{r}}\overline{\mathbf{r}}\prime ),$$(17) with $\overline{\mathbf{r}}$ given by (10).

Proposition 3.

Given that the unconditional second moments of the SCAW(p, q) model exist, the autocovariance matrix at lag τ is given by$$\begin{array}{c}\text{vec}(\mathbb{E}[({\mathbf{r}}_t-\overline{\mathbf{r}})({\mathbf{r}}_{t-\tau }-\overline{\mathbf{r}})\prime ])=\sum _{i=0}^{\infty }({\mathbf{\Phi }}_{i+\tau }\otimes {\mathbf{\Phi }}_i)\mathbf{\Omega }\\ {\left(\mathrm{}{\mathbf{I}}_{k^2}-\sum _{i=1}^{\infty }({\mathbf{\Phi }}_{\mathit{i}}\otimes {\mathbf{\Phi }}_i)\mathbf{\Omega }\mathrm{}\mathrm{}\right)}^{\mathrm{}-1}\text{vec}(\overline{\mathbf{r}}\overline{\mathbf{r}}\prime ).\end{array}$$

As such, the process $\left\{{\mathbf{R}}_t\right\}$ under the SCAW(p, q) model defined by (1)–(5) is weakly stationary if and only if the eigenvalues of the matrix (16) are less than 1 in modulus.

3 Parameterization

As mentioned in Section 2, when the dimension of $\left\{{\mathbf{R}}_t\right\}$ grows large, it is important to parameterize ${\mathbf{S}}_t$ in (5) parsimoniously, in order to maintain feasible estimation. Simultaneously, the specification must be rich enough to capture the time series dynamics observed in data. This section discusses several parameterizations that can be applied to this end.

3.1 Covariance Targeting

The constant term $\mathbf{CC}\prime$ in (5) consists of $m(m+1)/2$ parameters, rapidly increasing the estimation burden as the portfolio size grows. One approach to reduce the number of parameters is to consider covariance targeting, an extension of the idea of variance targeting (see Engle and Mezrich 1996), where the constant term $\mathbf{CC}\prime$ is consistently estimated (see Pedersen and Rahbek 2014) as follows. Let ${\mathbf{R}}_t={\mathbf{S}}_t+{\mathbf{V}}_t$, where ${\mathbf{V}}_t$ is a martingale difference, s.t. $\mathbb{E}[{\mathbf{V}}_t]=0$. Further denote the unconditional mean of $\left\{{\mathbf{R}}_t\right\}$ as $\mathbb{E}[{\mathbf{R}}_t]=\overline{\mathbf{S}}$. Then we can write (5) as$$\begin{array}{c}{\mathbf{S}}_t=\mathbf{CC}\prime +\sum _{i=1}^p{\mathbf{B}}_i{\mathbf{S}}_{t-i}{\mathbf{B}}_i^{\prime }+\sum _{j=1}^q{\mathbf{A}}_j{\mathbf{R}}_{t-j}{\mathbf{A}}_j^{\prime }\\ {\mathbf{R}}_t-{\mathbf{V}}_t=\mathbf{CC}\prime +\sum _{i=1}^p{\mathbf{B}}_i({\mathbf{R}}_{t-i}-{\mathbf{V}}_{t-i}){\mathbf{B}}_i^{\prime }+\sum _{j=1}^q{\mathbf{A}}_j{\mathbf{R}}_{t-j}{\mathbf{A}}_j^{\prime }.\end{array}$$

Taking unconditional expectations we obtain$$\begin{array}{c}\mathbb{E}\left[{\mathbf{R}}_t\right]=\mathbf{CC}\prime +\sum _{i=1}^p{\mathbf{B}}_i\mathbb{E}\left[{\mathbf{R}}_t\right]{\mathbf{B}}_i^{\prime }+\sum _{j=1}^q{\mathbf{A}}_j\mathbb{E}\left[{\mathbf{R}}_t\right]{\mathbf{A}}_j^{\prime }\\ \overline{\mathbf{S}}=\mathbf{CC}\prime +\sum _{i=1}^p{\mathbf{B}}_i\overline{\mathbf{S}}{\mathbf{B}}_i^{\prime }+\sum _{j=1}^q{\mathbf{A}}_j\overline{\mathbf{S}}{\mathbf{A}}_j^{\prime }.\end{array}$$such that(18) $$\mathbf{CC}\prime =\overline{\mathbf{S}}-\sum _{i=1}^p{\mathbf{B}}_i\overline{\mathbf{S}}{\mathbf{B}}_i^{\prime }-\sum _{j=1}^q{\mathbf{A}}_j\overline{\mathbf{S}}{\mathbf{A}}_j^{\prime }.$$(18)

The idea is then to replace $\mathbf{CC}\prime$ in (5) by the expression (18), and to estimate $\overline{\mathbf{S}}$ by the sample mean of the process. This specification determines the constant term $\mathbf{CC}\prime$ by the persistence parameters ${\mathbf{A}}_j$ and ${\mathbf{B}}_i$, such that $k=m(m+1)/2$ parameters less needs to be estimated in the model.

In order to ensure that the expression (18) is positive-definite, particular restrictions on the parameter matrices ${\mathbf{A}}_j$ and ${\mathbf{B}}_i$ must be imposed, and in general it is difficult to specify such conditions. One approach to circumvent this issue is considered in Noureldin, Sheppard, and Shephard (2014) in the case of an ARCH model, where the original series of return vectors is rotated by its unconditional mean, to create a standardized series of returns particularly suitable to model with covariance targeting. In this article we adapt this approach to singular realized covariance matrices in order to obtain a parsimonious parameterization. To this end, apply the eigenvalue decomposition to the unconditional expectation such that$$\overline{\mathbf{S}}=\mathbf{P}\mathbf{\Lambda }\mathbf{P}\prime ,$$where P is a matrix with eigenvectors of $\overline{\mathbf{S}}$ as columns and $\mathbf{\Lambda }$ is a diagonal matrix with the eigenvalues of $\overline{\mathbf{S}}$ as diagonal entries. Note that although $\left\{{\mathbf{R}}_t\right\}$ is a series of singular matrices, its unconditional mean $\overline{\mathbf{S}}$ is nonsingular and as such all the eigenvalues in $\mathbf{\Lambda }$ are positive. Further note that the symmetric square root of $\overline{\mathbf{S}}$ is ${\overline{\mathbf{S}}}^{1/2}=\mathbf{P}{\mathbf{\Lambda }}^{1/2}\mathbf{P}\prime$ and that ${\overline{\mathbf{S}}}^{-1/2}=\mathbf{P}{\mathbf{\Lambda }}^{-1/2}\mathbf{P}\prime$, since P is an orthogonal matrix.

Next we define the standardized realized covariance as(19) $${\mathbf{E}}_t={\overline{\mathbf{S}}}^{-1/2}{\mathbf{R}}_t({\overline{\mathbf{S}}}^{-1/2})\prime =\mathbf{P}{\mathbf{\Lambda }}^{-1/2}\mathbf{P}\prime {\mathbf{R}}_t\mathbf{P}{\mathbf{\Lambda }}^{-1/2}\mathbf{P}\prime .$$(19) which has expected value$$\mathbb{E}\left[{\mathbf{E}}_t\right]={\overline{\mathbf{S}}}^{-1/2}\mathbb{E}\left[{\mathbf{R}}_t\right]({\overline{\mathbf{S}}}^{-1/2})\prime ={\overline{\mathbf{S}}}^{-1/2}\overline{\mathbf{S}}({\overline{\mathbf{S}}}^{-1/2})\prime ={\mathbf{I}}_m.$$

Similarly, define ${\mathbf{G}}_t={\overline{\mathbf{S}}}^{-1/2}{\mathbf{S}}_t({\overline{\mathbf{S}}}^{-1/2})\prime$, such that(20) $${\mathbf{E}}_t\sim S{W}_m(n,{\mathbf{G}}_t/n)$$(20)

due to the affine transformation property of the singular Wishart distribution (see Theorem 2 of Bodnar, Mazur, and Okhrin 2014). As such, we will model ${\mathbf{G}}_t$ with an specification equivalent to (5) given by$${\mathbf{G}}_t=\tilde{\mathbf{C}}\tilde{\mathbf{C}}\prime +\sum _{i=1}^p{\tilde{\mathbf{B}}}_i{\mathbf{G}}_{t-i}{\tilde{\mathbf{B}}}_i^{\prime }+\sum _{j=1}^q{\tilde{\mathbf{A}}}_j{\mathbf{E}}_{t-j}{\tilde{\mathbf{A}}}_j^{\prime }.$$

Note that since ${\mathbf{E}}_t$ follows a conditional singular Wishart distribution and the specification of ${\mathbf{G}}_t$ is equivalent to that of ${\mathbf{S}}_t$, all results in Section (2.1) applies to the process $\left\{{\mathbf{E}}_t\right\}$ as well, with regards to parameters ${\tilde{\mathbf{A}}}_j$ and ${\tilde{\mathbf{B}}}_i$. Moreover, by applying the covariance targeting technique described in (18) with $\mathbb{E}[{\mathbf{E}}_t]={\mathbf{I}}_m$ we get(21) $$\begin{array}{c}{\mathbf{G}}_t=\left({\mathbf{I}}_m-\sum _{i=1}^p{\tilde{\mathbf{B}}}_i{\tilde{\mathbf{B}}}_i^{\prime }-\sum _{j=1}^q{\tilde{\mathbf{A}}}_j{\tilde{\mathbf{A}}}_j^{\prime }\right)+\sum _{i=1}^p{\tilde{\mathbf{B}}}_i{\mathbf{G}}_{t-i}{\tilde{\mathbf{B}}}_i^{\prime }\\ +\sum _{j=1}^q{\tilde{\mathbf{A}}}_j{\mathbf{E}}_{t-j}{\tilde{\mathbf{A}}}_j^{\prime }.\end{array}$$(21)

The restrictions on the persistence parameters ${\tilde{\mathbf{A}}}_j$ and ${\tilde{\mathbf{B}}}_i$ needed to ensure the positive definiteness of ${\mathbf{G}}_t$ are easily obtained for several parameterizations, as discussed below. In the following, the covariance targeting SCAW model described by (19), (20), and (21) will be referred to as SCAW_CT (p, q).

Furthermore, since ${\mathbf{S}}_t={\overline{\mathbf{S}}}^{1/2}{\mathbf{G}}_t({\overline{\mathbf{S}}}^{1/2})\prime$ and ${\mathbf{E}}_t={\overline{\mathbf{S}}}^{-1/2}{\mathbf{R}}_t({\overline{\mathbf{S}}}^{-1/2})\prime$ the model (21) for the standardized series $\left\{{\mathbf{E}}_t\right\}$ implies the following equalities(22) $${\mathbf{A}}_j={\overline{\mathbf{S}}}^{1/2}{\tilde{\mathbf{A}}}_j({\overline{\mathbf{S}}}^{-1/2})\prime$$(22) (23) $${\mathbf{B}}_i={\overline{\mathbf{S}}}^{1/2}{\tilde{\mathbf{B}}}_i({\overline{\mathbf{S}}}^{-1/2})\prime$$(23) (24) $$\mathbf{CC}\prime ={\overline{\mathbf{S}}}^{1/2}\left({\mathbf{I}}_m-\sum _{i=1}^p{\tilde{\mathbf{B}}}_i{\tilde{\mathbf{B}}}_i^{\prime }-\sum _{j=1}^q{\tilde{\mathbf{A}}}_j{\tilde{\mathbf{A}}}_j^{\prime }\right)({\overline{\mathbf{S}}}^{1/2})\prime$$(24) for the parameterization in (5), modeling the nonstandardized series $\left\{{\mathbf{R}}_t\right\}$.

Moreover, as discussed in Noureldin, Sheppard, and Shephard (2014), there are several ways to parameterize the conditional mean, in this model described by (21): scalar and diagonal specification of ${\tilde{\mathbf{A}}}_j$ and ${\tilde{\mathbf{B}}}_i$, as well as specifications with common persistence or with orthogonal parameter matrices. In this presentation we will focus on the scalar and the diagonal specification, such that ${\tilde{\mathbf{A}}}_j$ and ${\tilde{\mathbf{B}}}_i$ are all diagonal matrices, with the additional condition that the first element of each parameter matrix is positive, in order to ensure model identification. As such, the constant term in (21) will be positive-definite if and only if (see Engle and Kroner 1995)(25) $$\sum _{j=1}^q{\tilde{a}}_{j,ll}^2+\sum _{i=1}^p{\tilde{b}}_{i,ll}^2<1,\hspace{1em}l=1,\dots ,m,$$(25) where ${\tilde{a}}_{j,ll}$ is the l:th diagonal element of ${\tilde{\mathbf{A}}}_j$ and ${\tilde{b}}_{i,ll}$ is the l:th diagonal element of ${\tilde{\mathbf{B}}}_i$. Conditions for the other specifications mentioned above can be obtained correspondingly. The diagonal parameterization of (21) results in $m(p+q)$ parameters, which is substantially lower than the $m(m+1)/2+(p+q)m^2$ parameters in the original specification (5), particularly for large dimensional cases. In the example with m = 50 and $p=q=2$ above, the diagonal model suggested thus results in 200 parameters, instead of the 11,275 parameters in the original specification, making estimation much more feasible.

Hence, instead of modeling the series $\left\{{\mathbf{R}}_t\right\}$ with conditional means $\left\{{\mathbf{S}}_t\right\}$ directly, the above approach instead models the standardized realized covariances $\left\{{\mathbf{E}}_t\right\}$ with the conditional means $\left\{{\mathbf{G}}_t\right\}$. In turn, this model implies $\left\{{\mathbf{S}}_t\right\}$ to be specified as (5) with parameters obtained as (22)–(24). Note that while ${\tilde{\mathbf{A}}}_j$ and ${\tilde{\mathbf{B}}}_i$ are diagonal, the implied parameters for $\left\{{\mathbf{S}}_t\right\},{\mathbf{A}}_j$ and ${\mathbf{B}}_i$, are in general not, since the transformations (22)–(24) do not necessarily result in diagonal matrices. This does indeed suggest a rich dynamic for the original series of realized covariance matrices, as discussed in Noureldin, Sheppard, and Shephard (2014) in the equivalent ARCH case. However, it does not mean that the specification $\left\{{\mathbf{S}}_t\right\}$ results in an entirely general BEKK model, since its parameters are constrained by the unconditional mean $\overline{\mathbf{S}}$.

3.2 Sectorwise Parameterization

Prices for assets that belong to the same market sector tend to exhibit some level of similarity in price movements (see e.g., King 1966; Chan, Lakonishok, and Swaminathan 2007). To incorporate this feature, we introduce a model specification that assumes that covariance dynamics are homogeneous within market sectors. For this sectorwise parameterization, we define the diagonal elements of the parameter matrices ${\tilde{\mathbf{A}}}_j$ and ${\tilde{\mathbf{B}}}_i,j=1,\dots ,q,i=1,\dots ,p$, in (21) as ${\tilde{a}}_{ll,j}={\tilde{a}}_{kk,j}$ and ${\tilde{b}}_{ll,i}={\tilde{b}}_{kk,i}$ if asset l and asset k belong to the same market sector. The number of parameters for this specification is as such $s(p+q)$, where s denote the number of sectors that the considered assets belong to. Hence, the number of parameters for this approach is independent of process dimension m, which makes it an attractive modeling candidate when very large asset portfolios are considered.

We note that the sectorwise parameterization is applied to the model formulated for the standardized realized covariance matrices $\left\{{\mathbf{E}}_t\right\}$. Such a model specification is motivated by the observation that a high average correlation is present between the elements of ${\mathbf{E}}_t$ and ${\mathbf{R}}_t$ with the same indices (0.75 in the empirical illustration with m = 50 stocks and 0.56 in the empirical study with m = 300, both presented in Section 5), while the average correlation between the elements of two matrices with different indices as well as the average correlation between their absolute values are close to zero. As such, the sectorwise parameterization introduced in the model for $\left\{{\mathbf{E}}_t\right\}$ roughly corresponds to the sectorwise parameterization in the case of the model for $\left\{{\mathbf{R}}_t\right\}$.

3.3 HAR Extension

To account for the high persistence in volatility processes, we also adapt the SCAW model with a heterogeneous autoregressive (HAR) extension, as proposed by Corsi (2009) in the univariate case and implemented by Golosnoy, Gribisch, and Liesenfeld (2012) in a matrix-variate version. Such an approach considers the long-memory dependence in daily volatility by including lagged realized covariances observed on longer horizons, like weekly and monthly. Consequently, for this specification, we define the conditional process mean ${\mathbf{G}}_t$ as(26) $${\mathbf{G}}_t=\left({\mathbf{I}}_m-\sum _{j=1}^q{\tilde{\mathbf{A}}}_j{\tilde{\mathbf{A}}}_j^{\prime }-\tilde{\mathbf{D}}\tilde{\mathbf{D}}\prime \right)+\sum _{j=1}^q{\tilde{\mathbf{A}}}_j{\mathbf{E}}_{t-j}{\tilde{\mathbf{A}}}_j^{\prime }+\tilde{\mathbf{D}}{\mathbf{E}}_{t-1}^{(h)}\tilde{\mathbf{D}}\prime ,$$(26) where ${\mathbf{E}}_t^{(h)}={\sum }_{j=q}^h{\mathbf{E}}_{t-j}$. Further, we define $\tilde{\mathbf{D}}$ as a diagonal matrix with sectorwise parameterization as described above. As such, (26) can be specified in terms of (21), but we denote it with a separate parameter matrix $\tilde{\mathbf{D}}$ for ease of interpretation.

4 Estimation

Similar to Noureldin, Sheppard, and Shephard (2014), we apply a two-step estimation procedure in order to obtain the parameter estimates of the considered model (21). Given a sample of the realized covariance process, ${\left\{{\mathbf{R}}_t\right\}}_{1\le t\le T}$, a method of moments approach is first used to estimate the unconditional mean of the process $\overline{\mathbf{S}}$, as(27) $$\widehat{\overline{\mathbf{S}}}=\frac{1}{T}\sum _{t=1}^T{\mathbf{R}}_t.$$(27)

The estimate $\widehat{\overline{\mathbf{S}}}$ is then decomposed into estimates $\widehat{\mathbf{P}}$ and $\widehat{\mathbf{\Lambda }}$. From these estimates, a standardized series is obtained in correspondence to (19) as$${\mathbf{E}}_t=\widehat{\mathbf{P}}{\widehat{\mathbf{\Lambda }}}^{-1/2}\widehat{\mathbf{P}}\prime {\mathbf{R}}_t\widehat{\mathbf{P}}{\widehat{\mathbf{\Lambda }}}^{-1/2}\widehat{\mathbf{P}}\prime ,$$consistent with the approach in Noureldin, Sheppard, and Shephard (2014). In the second step, we estimate the diagonal parameter matrices ${\tilde{\mathbf{A}}}_j$ and ${\tilde{\mathbf{B}}}_i,i=1,\dots ,p,j=1\dots ,q$, in (21), by the maximum likelihood method. Similarly to Golosnoy, Gribisch, and Liesenfeld (2012), in order to ensure the positivity of the first diagonal element in each of the parameter matrices, the square roots of these values are estimated. To enforce the condition (25), the diagonal elements of ${\tilde{\mathbf{A}}}_j$ are specified according to the following function, for $l=1,\dots ,m$,(28) $${\tilde{a}}_{ll,j}=\{\begin{array}{cc}{a}_{ll,j}^{*}\hfill & \text{if}\hspace{1em}{s}_l<1\hfill \\ \frac{a_{ll,j}^{*}(1-ϵ)}{s_l}\hfill & \text{if}\hspace{1em}{s}_l\ge 1\hfill \end{array}$$(28) (29) $${\tilde{b}}_{ll,i}=\{\begin{array}{cc}{b}_{ll,i}^{*}\hfill & \text{if}\hspace{1em}{s}_l<1\hfill \\ \frac{b_{ll,i}^{*}(1-ϵ)}{s_l}\hfill & \text{if}\hspace{1em}{s}_l\ge 1\hfill \end{array},$$(29) where $s_l={\sum }_{j=1}^q{\tilde{a}}_{j,ll}^2+{\sum }_{i=1}^p{\tilde{b}}_{i,ll}^2$ and ϵ is positive and close to zero. As such, we define the argument vector to the log-likelihood function as $\psi \prime =(\psi {\prime }_a,\psi {\prime }_b)$ with $\psi {\prime }_a=\left(\sqrt{a_{11,1}^{*}},a_{22,1}^{*},\dots ,a_{ll,q}^{*}\right)$ and $\psi {\prime }_b=\left(\sqrt{b_{11,1}^{*}},b_{22,1}^{*},\dots ,b_{ll,q}^{*}\right)$. Furthermore, since by (20), ${\mathbf{E}}_t$ follows a singular Wishart distribution, the log-likelihood obtains directly from the density (3) as(30) $$\begin{array}{c}\mathcal{L}(\psi )=\sum _{t=1}^T[c+\frac{n}{2}\text{ln}\hspace{0.17em}\left|{\mathbf{G}}_t\right|+\frac{n-m-1}{2}\text{ln}\hspace{0.17em}\left|{\mathbf{E}}_{11,t}\right|\\ \hspace{1em}-\frac{n-m-1}{2}\text{tr}({\mathbf{G}}_t^{-1}{\mathbf{E}}_t)],\end{array}$$(30) where(31) $$c=\frac{n(n-m)}{2}\text{ln}\hspace{0.17em}(\pi )+\left(m-\frac{n}{2}\right)\hspace{0.17em}\text{ln}\hspace{0.17em}n-\hspace{0.17em}\text{ln}\hspace{0.17em}{\Gamma }_p\left(\frac{n}{2}\right).\hspace{1em}$$(31)

Finding the vector ψ that maximizes the log-likelihood function (30) can then be done by applying numerical optimization techniques.

5 Empirical Application

5.1 Data and Estimation

The SCAW model presented in Section 2 is applied to analyze the daily realized covariance matrices of 50 assets traded at National Association of Securities Dealers Automated Quotations (NASDAQ) from mid 1997 to mid 2017, and 300 assets traded in the same market between mid 2007 to mid 2017. The assets have been classified with an associated market sector following the NASDAQ sector classification (as e.g., Litimi, BenSaïda, and Bouraoui 2016; BenSaïda 2017). Furthermore, the assets have been selected such that the sample sector distribution is proportional to the sector distribution of assets traded at NASDAQ for the considered time period. The realized covariance matrix of these assets, for trading day t, is constructed as ${\mathbf{R}}_t={\sum }_i^n{\mathbf{x}}_{t,i}{\mathbf{x}}_{t,i}^{\prime }$, where ${\mathbf{x}}_{t,i}$ is the $m\times 1$ return vector obtained for the i:th 10-minute interval of day t between 09:30 and 16:00. In turn, this results in n = 39 return vectors, such that the rank of the m × m matrix ${\mathbf{R}}_t$ is 39, making it a singular matrix. The sample period for the time series with 50 assets starts 2nd of June 1997 and ends 15th of June 2017, resulting in about 20 years of data, and 4994 trading days. The time series with 300 assets starts 27th of June 2007 and ends 15th of June 2017, resulting in about 10 years of data, and 2498 trading days. As such, the considered series covers two exceptionally volatile time periods: the so-called Dot-com bubble, which had its peak around the year 2000, and the global financial crisis of 2007–2008.

Tables 1 and 2 summarize statistics for the realized variance in each of the 12 market sectors in the two samples. According to these statistics, the energy sector experiences the largest average variance, while assets in the financial sector are the most right skewed and leptokurtic.

Table 1 Summary statistics for the realized variance (multiplied by 10⁴) of 50 assets in each of the 12 market sectors considered.

Sector	Nr.assets	Mean	Median	Std.dev.	Skewness	Kurtosis
Basic industries	1	17.23	8.47	24.45	4.34	33.21
Capital goods	4	17.68	7.61	59.31	82.02	9377.41
Consumer durables	1	10.37	2.61	75.15	54.69	3459.49
Consumer nondurables	2	15.90	2.96	130.01	83.19	7769.35
Consumer services	6	7.61	2.98	19.80	18.38	629.25
Energy	1	20.62	12.33	33.13	10.30	206.96
Finance	11	7.02	2.43	39.47	159.67	32358.94
Health care	11	11.75	4.48	36.99	63.56	8048.78
Miscellaneous	2	11.41	3.44	106.37	49.79	2823.32
Public utilities	1	4.11	1.42	8.59	8.30	125.28
Technology	9	10.06	4.23	48.09	146.27	26943.34
Transportation	1	16.24	3.73	63.71	18.54	476.35

Table 2 Summary statistics for the realized variance (multiplied by 10⁴) of 300 assets in each of the 12 market sectors considered.

Sector	Nr.assets	Mean	Median	Std.dev.	Skewness	Kurtosis
Basic industries	8	14.78	5.73	38.07	10.68	184.96
Capital goods	23	11.41	4.91	30.58	17.85	561.30
Consumer durables	8	8.37	3.61	70.77	63.84	4333.05
Consumer nondurables	12	6.98	2.91	88.50	107.42	13240.57
Consumer services	37	8.48	3.22	31.29	57.64	7008.07
Energy	6	21.40	10.64	40.97	8.74	130.46
Finance	63	8.64	3.18	80.44	209.24	51151.97
Health care	67	19.61	7.03	318.11	194.25	43606.03
Miscellaneous	10	11.27	4.96	59.19	79.04	8548.82
Public utilities	6	7.16	3.49	14.59	11.85	256.04
Technology	53	7.46	3.20	80.25	190.35	41194.67
Transportation	7	13.96	4.18	81.76	71.08	6526.02

In the medium dimension case of m = 50, the models discussed in Section 5.2 have been estimated with a rolling window approach. Every 20 trading days (corresponding to roughly one month) in the time series sample, the models are re-estimated using the last 500 trading days (roughly two years). These estimates are then used to make 1-step ahead forecasts for the coming 20 trading days.

Regarding the large dimension case of m = 300, the models are instead estimated with a fixed window approach. The parameters of the models are estimated on the first 80% (1998 trading days) of the time series sample, while the forecast accuracy of the models are evaluated on the last 20% (500 trading days) of the sample.

The parameters of the considered models are estimated as described in Section 4, where $ϵ={10}^{-7}$ is used in Equations (28)(28) $${\tilde{a}}_{ll,j}=\{\begin{array}{cc}{a}_{ll,j}^{*}\hfill & \text{if}\hspace{1em}{s}_l<1\hfill \\ \frac{a_{ll,j}^{*}(1-ϵ)}{s_l}\hfill & \text{if}\hspace{1em}{s}_l\ge 1\hfill \end{array}$$(28) and (29)(29) $${\tilde{b}}_{ll,i}=\{\begin{array}{cc}{b}_{ll,i}^{*}\hfill & \text{if}\hspace{1em}{s}_l<1\hfill \\ \frac{b_{ll,i}^{*}(1-ϵ)}{s_l}\hfill & \text{if}\hspace{1em}{s}_l\ge 1\hfill \end{array},$$(29) . Such a value of ϵ has only a minor impact on the resulting estimators, since only 0.8% of s_l values across all models are larger than 1.

5.2 Models

To study these data using the suggested SCAW model, the estimation and forecasting are performed using the various model specifications discussed in Section 3. The following SCAW-parameterizations are considered:

• SCAW_CT (p, q): Parameter matrices ${\tilde{\mathbf{A}}}_j$ and ${\tilde{\mathbf{B}}}_i,j=1,\dots ,q,i=1,\dots ,p$ of (21) are diagonal.

• SCAW-SCALAR(p, q): Parameter matrices ${\tilde{\mathbf{A}}}_j$ and ${\tilde{\mathbf{B}}}_i,j=1,\dots ,q,i=1,\dots ,p$ of (21) are proportional to the identity matrix.

• SCAW-SS_CT (p, q): Parameter matrices ${\tilde{\mathbf{A}}}_j$ and ${\tilde{\mathbf{B}}}_i,j=1,\dots ,q,i=1,\dots ,p$ of (21) are diagonal. Further, ${\tilde{a}}_{ll,j}={\tilde{a}}_{kk,j}$ and ${\tilde{b}}_{ll,i}={\tilde{b}}_{kk,i}$ if asset l and asset k belongs to the same sector.

• SCAW-SS-HAR_CT (q, h): Parameter matrices ${\tilde{\mathbf{A}}}_j$ and $\tilde{\mathbf{D}}$ of (26) are diagonal. Further, ${\tilde{a}}_{ll,j}={\tilde{a}}_{kk,j}$ and ${\tilde{d}}_{ll}={\tilde{d}}_{kk}$ if asset l and asset k belongs to the same sector.

• SCAW-SCALAR-HAR(q, h): Parameter matrices ${\tilde{\mathbf{A}}}_j$ and $\tilde{\mathbf{D}}$ of (26) are proportional to the identity matrix.

Similarly to Golosnoy, Gribisch, and Liesenfeld (2012), Multivariate GARCH models fitted to daily return data are used as forecast accuracy benchmarks to the parameterizations described above. To have comparable results, the MGARCH benchmark models follow equivalent specifications and are consequently denoted MGARCH_CT (p, q), MGARCH-SCALAR(p, q), MGARCH-SS_CT (p, q), MGARCH-SS-HAR_CT (q, h), and MGARCH-SCALAR-HAR(q, h). As discussed in Section 2, the Multivariate GARCH model with BEKK-specification can be thought of as a special case of the SCAW model, with the number of intra-day returns (and matrix rank) n = 1. Apart from MGARCH, two additional models applicable to singular realized covariance matrices, the DCC-S and DCC-NL models discussed in Engle, Ledoit, and Wolf (2019), are included as benchmarks.

5.3 Forecasting

The models discussed in Section 5.2 are evaluated by out-of-sample forecast accuracy. As described in Section 5.1, for m = 50 the considered models are evaluated with a rolling window approach throughout the sample, while for the m = 300 sample, a fixed window technique is applied. For each of the models, the $\mathcal{l}$-step-ahead forecast is computed recursively as(32) $$\mathbb{E}\left[{\mathbf{R}}_{t+\mathcal{l}}\right|{\mathcal{F}}_t]=\mathbb{E}[{\mathbf{S}}_{t+\mathcal{l}}\left|{\mathcal{F}}_t\right]=\mathbf{P}{\mathbf{\Lambda }}^{1/2}\mathbf{P}\prime \mathbb{E}\left[{\mathbf{G}}_{t+\mathcal{l}}\right|{\mathcal{F}}_t]\mathbf{P}{\mathbf{\Lambda }}^{1/2}\mathbf{P}\prime ,\hspace{1em}\text{with}$$(32) (33) $$\begin{array}{c}\mathbb{E}\left[{\mathbf{G}}_{t+\mathcal{l}}\right|{\mathcal{F}}_t]=\left({\mathbf{I}}_m-\sum _{i=1}^p{\tilde{\mathbf{B}}}_i{\tilde{\mathbf{B}}}_i^{\prime }-\sum _{j=1}^q{\tilde{\mathbf{A}}}_j{\tilde{\mathbf{A}}}_j^{\prime }\right)+\\ +\sum _{i=1}^p{\tilde{\mathbf{B}}}_i\mathbb{E}\left[{\mathbf{G}}_{t+\mathcal{l}-i}\right|{\mathcal{F}}_t]{\tilde{\mathbf{B}}}_i^{\prime }\\ +\sum _{j=1}^q{\tilde{\mathbf{A}}}_j\mathbb{E}\left[{\mathbf{E}}_{t+\mathcal{l}-j}\right|{\mathcal{F}}_t]{\tilde{\mathbf{A}}}_j^{\prime },\end{array}$$(33) (34) $$\mathbb{E}\left[{\mathbf{E}}_{t+\mathcal{l}-j}\right|{\mathcal{F}}_t]=\mathbb{E}[{\mathbf{G}}_{t+\mathcal{l}-j}\left|{\mathcal{F}}_t\right],$$(34) where the parameter matrices are estimated as described in Section 4. The specifications SCAW-SS-HAR_CT (q, h) and SCAW-SCALAR-HAR(q, h) are computed similarly, employing that they can be represented in the form of (21).

The forecast accuracy of ${\widehat{\mathbf{R}}}_{t+\mathcal{l}}=\mathbb{E}[{\mathbf{R}}_{t+\mathcal{l}}|{\mathcal{F}}_t]$ is evaluated using several measures. First, the average Frobenius norm of the $\mathcal{l}$-step-ahead forecast error is computed as(35) $${\text{FN}}_{\mathcal{l}}=\frac{1}{T_{\mathcal{l}}}\sum _t\left|\right|{\widehat{\mathbf{R}}}_{t+\mathcal{l}}-{\mathbf{R}}_{t+\mathcal{l}}\left|\right|,$$(35) where $T_{\mathcal{l}}$ is the sample-size for $\mathcal{l}$-step-ahead forecasts and $\left|\right|\mathbf{M}\left|\right|$ denote the Frobenius norm of the matrix M. Further, in practice, one is often interested in applying covariance matrix forecasts in a portfolio setting. As such, we also compute mean squared error of the standard deviation of an equally weighted (EW) portfolio, a popular portfolio in financial literature (see DeMiguel, Garlappi, and Uppal 2009), using the obtained covariance forecast and the realized covariance:$${\text{SD}}_{\text{EW},\mathcal{l}}=\frac{1}{T_{\mathcal{l}}}\sum _t\frac{1}{m^2}{\left(\sqrt{\mathbf{1}\prime {\widehat{\mathbf{R}}}_{t+\mathcal{l}}\mathbf{1}}-\sqrt{\mathbf{1}\prime {\mathbf{R}}_{t+\mathcal{l}}\mathbf{1}}\right)}^2.$$

Another important portfolio is the global minimum variance (GMV) portfolio (see Frahm and Memmel 2010; Glombeck 2014; Bodnar, Parolya, and Schmid 2018; Bodnar et al. 2019; Ding, Li, and Zheng 2021). This portfolio has the lowest risk of all possible portfolios of risky assets, and its weight vector is solely determined by the covariance matrix of asset returns. We employ the l-step ahead forecast of the covariance matrix ${\widehat{\mathbf{R}}}_{t+\mathcal{l}}$ produced by each model for the computation of the weights of the GMV portfolio expressed as(36) $${\widehat{\mathbf{w}}}_{t+\mathcal{l}}={\widehat{\mathbf{R}}}_{t+\mathcal{l}}^{-1}\mathbf{1}/(\mathbf{1}\prime {\widehat{\mathbf{R}}}_{t+\mathcal{l}}^{-1}\mathbf{1}).$$(36)

As a performance measure of the constructed portfolio for different models, we use the standard deviation of the GMV portfolio of future time periods given by$${\text{SD}}_{\text{GMV},\mathcal{l}}=\frac{1}{T_{\mathcal{l}}}\sum _t\sqrt{{\widehat{\mathbf{w}}}_{t+\mathcal{l}}^{\prime }{\mathbf{R}}_{t+\mathcal{l}}{\widehat{\mathbf{w}}}_{t+\mathcal{l}}}.$$

The quantity ${\text{SD}}_{\text{GMV},\mathcal{l}}$ measures the average forecasted standard deviation of the GMV portfolios in the out-of-sample period. Hence, ${\text{SD}}_{\text{EW},\mathcal{l}}$ and ${\text{SD}}_{\text{GMV},\mathcal{l}}$ illustrate the ability of each model to forecast different things. The former is a measure of the squared difference between the predicted standard deviation of the equally weighted portfolio, and the observed standard deviation of the equally weighted portfolio. The latter measures the predicted standard deviation of the GMV portfolio. For this measure, more accurate predictions will result in lower values, and the minimum value is obtained when inserting ${\widehat{\mathbf{R}}}_{t+\mathcal{l}}={\mathbf{R}}_{t+\mathcal{l}}$ into (36). An accurate prediction of this quantity has considerable economic value, since the standard deviation of the GMV portfolio is a key input value in many financial applications.

In addition, to further asses the properties of the portfolios constructed using the GMV weights predicted by the models, ${\widehat{\mathbf{w}}}_{t+\mathcal{l}}$, we measure the empirical variance, turnover and leverage they produce. Hence, we introduce the following measures$$\begin{array}{c}{\text{EGMV}}_{\text{var},\mathcal{l}}=\frac{1}{T_{\mathcal{l}}}\sum _t{\left({\widehat{\mathbf{w}}}_{t+\mathcal{l}}{\mathbf{r}}_{t+\mathcal{l}}-\left(\sum _t{\widehat{\mathbf{w}}}_{t+\mathcal{l}}{\mathbf{r}}_{t+\mathcal{l}}\right)/T_{\mathcal{l}}\right)}^2,\\ {\text{EGMV}}_{\text{tu},\mathcal{l}}=\frac{1}{T_{\mathcal{l}}}\sum _t\Vert {\widehat{\mathbf{w}}}_{t+\mathcal{l}}-{\widehat{\mathbf{w}}}_{t+\mathcal{l}-1}\Vert ,\\ {\text{EGMV}}_{\text{lev},\mathcal{l}}=\frac{1}{T_{\mathcal{l}}}\sum _t\sum _{i=1}^m|{\widehat{w}}_{i,t+\mathcal{l}}|,\end{array}$$where ${\mathbf{r}}_t$ is the observed return vector at time t and ${\widehat{w}}_{i,t}$ is the ith element of ${\widehat{\mathbf{w}}}_t$.

Finally, it is relevant to see if the difference in a forecast accuracy measure between the SCAW model and its benchmarks is statistically significant. To this end, a two-sided paired t-test is applied to the sample of terms in ${\text{FN}}_{\mathcal{l}},{\text{SD}}_{\text{EW},\mathcal{l}},{\text{SD}}_{\text{GMV},\mathcal{l}},{\text{EGMV}}_{\text{tu},\mathcal{l}}$ or ${\text{EGMV}}_{\text{lev},\mathcal{l}}$ for the SCAW model, and to the sample of terms for the same measure in the equivalent MGARCH specification, the DCC-S model, and the DCC-NL model, respectively. Similarly, the measure ${\text{EGMV}}_{\text{var},\mathcal{l}}$ is evaluated using the ${\text{HAC}}_{\text{PW}}$ method described in Ledoit and Wolf (2011). Significance level 0.05 is used for all the applied tests. Beside the two-sided paired t-test and ${\text{HAC}}_{\text{PW}}$ method used to compare the SCAW models and its benchmark, we also construct the 90% Hansen’s model confidence set (Hansen, Lunde, and Nason 2011) with respect to the several forecast measures, namely, ${\text{FN}}_{\mathcal{l}},{\text{SD}}_{\text{EW},\mathcal{l}}$, and ${\text{SD}}_{\text{GMV},\mathcal{l}}$.

5.4 Results

Table 3 and Tables S.1–S.3 in the supplementary materials summarize the forecasts performance of the models discussed in Section 5.2, with rolling window for forecast horizon $\mathcal{l}=1$ and with fixed window for forecast horizons $\mathcal{l}=1,5,10$. For each measure, the most favorable value in each column is highlighted bold.

Table 3 Summary of the rolling window forecasts for each respective model for 20 years of data (mid 1997 to mid 2017) on 50 assets proportionally distributed among the 12 sectors in NASDAQ classification, evaluated on a monthly basis.

Model	Parameters	${\text{FN}}_{\mathcal{l}}$	${\text{SD}}_{\text{EW},\mathcal{l}}$	${\text{SD}}_{\text{GMV},\mathcal{l}}$	${\text{EGMV}}_{\text{var},\mathcal{l}}$	${\text{EGMV}}_{\text{tu},\mathcal{l}}$	${\text{EGMV}}_{\text{lev},\mathcal{l}}$	Sharpe Ratio
SCAWCT (0, 1)	50	${0.0189}^{(\text{MSN})}$	$7.44e^{-3^{(\mathrm{M})}}$	${0.105}^{(\text{MSN})}$	$1.87e^{-2^{(\text{SN})}}$	${0.31}^{(\text{SN})}$	$\mathbf{1}.\mathbf{5}{\mathbf{8}}^{(\text{MSN})}$	0.014
SCAWCT (1, 1)	100	${0.0189}^{(\text{MSN})}$	$5.53e^{-3^{(\text{MSN})}}$	$1.92e^{-2^{(\text{MSN})}}$	$1.21e^{-3^{(\text{MSN})}}$	${0.70}^{(\mathrm{M})}$	${1.72}^{(\text{MS})}$	0.021
SCAW-SSCT (0, 1)	12	${0.0187}^{(\text{MSN})}$	$6.99e^{-3^{(\mathrm{M})}}$	${0.105}^{(\text{MSN})}$	$1.85e^{-2^{(\text{SN})}}$	${0.34}^{(\text{SN})}$	${1.59}^{(\text{MSN})}$	0.016
SCAW-SSCT (1, 1)	24	${0.0185}^{(\text{MSN})}$	$4.95e^{-3^{(\text{MSN})}}$	${0.107}^{(\text{MSN})}$	$1.76e^{-2^{(\text{MSN})}}$	0.65	${1.71}^{(\text{MS})}$	$\mathbf{0}.\mathbf{0}\mathbf{3}\mathbf{5}$
SCAW-SSCT (1, 2)	36	$\mathbf{0}.\mathbf{0}\mathbf{1}\mathbf{8}{\mathbf{3}}^{(\text{MSN})}$	$4.84e^{-3^{(\text{MSN})}}$	${0.106}^{(\text{MSN})}$	$1.83e^{-2^{(\text{MSN})}}$	0.54	${1.69}^{(\text{MS})}$	0.026
SCAW-SSCT (2, 2)	48	${0.0184}^{(\text{MSN})}$	$4.61e^{-3^{(\text{MSN})}}$	${0.105}^{(\text{MSN})}$	$1.81e^{-2^{(\text{MSN})}}$	${0.55}^{(\mathrm{M})}$	${1.69}^{(\text{MS})}$	0.026
SCAW-SS-HARCT (1, 5)	24	${0.0197}^{(\text{MSN})}$	$8.73e^{-3^{(\mathrm{M})}}$	${0.107}^{(\text{MSN})}$	$1.96e^{-2^{(\text{SN})}}$	${0.40}^{(\text{SN})}$	${1.61}^{(\text{MSN})}$	0.019
SCAW-SS-HARCT (1, 20)	24	${0.0201}^{(\text{MSN})}$	$1.00e^{-2^{(\mathrm{M})}}$	${0.109}^{(\text{MSN})}$	$1.98e^{-2^{(\text{SN})}}$	${0.33}^{(\text{SN})}$	${1.58}^{(\text{MSN})}$	0.020
SCAW-SCALAR(0, 1)	1	${0.0187}^{(\text{MSN})}$	$6.88e^{-3^{(\mathrm{M})}}$	${0.104}^{(\text{MSN})}$	$1.85e^{-2^{(\text{SN})}}$	${0.36}^{(\text{SN})}$	${1.59}^{(\text{MSN})}$	0.018
SCAW-SCALAR(1, 1)	2	${0.0184}^{(\text{MSN})}$	$4.71e^{-3^{(\text{MSN})}}$	${0.103}^{(\text{MSN})}$	$1.79e^{-2^{(\text{SN})}}$	${0.31}^{(\text{SN})}$	${1.65}^{(\text{MSN})}$	$\mathbf{0}.\mathbf{0}\mathbf{3}\mathbf{5}$
SCAW-SCALAR(1, 2)	3	${0.0184}^{(\text{MSN})}$	$4.69e^{-3^{(\text{MSN})}}$	${0.103}^{(\text{MSN})}$	$1.78e^{-2^{(\text{SN})}}$	${0.29}^{(\text{SN})}$	${1.65}^{(\text{MSN})}$	$\mathbf{0}.\mathbf{0}\mathbf{3}\mathbf{5}$
SCAW-SCALAR(2, 2)	4	$\mathbf{0}.\mathbf{0}\mathbf{1}\mathbf{8}{\mathbf{3}}^{(\text{MSN})}$	$\mathbf{4}.\mathbf{5}\mathbf{3}{\mathbf{e}}^{-{\mathbf{3}}^{(\text{MSN})}}$	$\mathbf{0}.\mathbf{1}\mathbf{0}{\mathbf{2}}^{(\text{MSN})}$	$\mathbf{1}.\mathbf{7}\mathbf{1}{\mathbf{e}}^{-{\mathbf{2}}^{(\text{SN})}}$	${0.32}^{(\text{SN})}$	${1.66}^{(\text{MSN})}$	0.031
SCAW-SCALAR-HAR(1, 5)	2	${0.0196}^{(\text{MSN})}$	$8.51e^{-3^{(\mathrm{M})}}$	${0.106}^{(\text{MSN})}$	$1.88e^{-2^{(\text{SN})}}$	${0.40}^{(\text{SN})}$	${1.61}^{(\text{MSN})}$	0.020
SCAW-SCALAR-HAR(1, 20)	2	${0.0199}^{(\text{MSN})}$	$9.86e^{-3^{(\mathrm{M})}}$	${1.07}^{(\text{MSN})}$	$1.95e^{-2^{(\text{SN})}}$	${0.35}^{(\text{SN})}$	${1.59}^{(\text{MSN})}$	0.020
MGARCHCT (0, 1)	50	0.0207	$1.18e^{-2}$	0.121	$1.79e^{-2}$	0.19	2.07	0.017
MGARCHCT (1, 1)	100	0.0219	$1.11e^{-2}$	0.148	$3.66e^{-2}$	0.84	2.34	0.011
MGARCH-SSCT (0, 1)	12	0.0207	$1.20e^{-2}$	0.121	$1.77e^{-2}$	0.13	2.06	0.017
MGARCH-SSCT (1, 1)	24	0.0215	$1.13e^{-2}$	0.131	$2.19e^{-2}$	0.36	2.15	0.007
MGARCH-SSCT (1, 2)	36	0.0215	$1.05e^{-2}$	0.128	$2.05e^{-2}$	0.42	2.14	0.010
MGARCH-SSCT (2, 2)	48	0.0212	$1.00e^{-2}$	0.136	$2.34e^{-2}$	0.59	2.20	0.019
MGARCH-SS-HARCT (1, 5)	24	0.0207	$1.20e^{-2}$	0.121	$1.79e^{-2}$	0.14	2.01	0.017
MGARCH-SS-HARCT (1, 20)	24	0.0207	$1.20e^{-2}$	0.121	$1.79e^{-3}$	0.14	2.06	0.017
MGARCH-SCALAR(0, 1)	1	0.0207	$1.19e^{-2}$	0.121	$1.79e^{-2}$	0.14	2.06	0.017
MGARCH-SCALAR(1, 1)	2	0.0207	$1.15e^{-2}$	0.121	$1.79e^{-2}$	0.14	2.06	0.015
MGARCH-SCALAR(1, 2)	3	0.0206	$1.19e^{-2}$	0.121	$1.78e^{-2}$	$\mathbf{0}.\mathbf{1}\mathbf{2}$	2.06	0.017
MGARCH-SCALAR(2, 2)	4	0.0206	$1.19e^{-2}$	0.121	$1.79e^{-2}$	$\mathbf{0}.\mathbf{1}\mathbf{2}$	2.06	0.017
MGARCH-SCALAR-HAR(1, 5)	2	0.0207	$1.20e^{-2}$	0.121	$1.79e^{-2}$	0.14	2.06	0.017
MGARCH-SCALAR-HAR(1, 20)	2	0.0207	$1.20e^{-2}$	0.121	$1.79e^{-2}$	0.14	2.06	0.017
DCC-S	153	0.0234	$6.04e^{-3}$	0.148	$3.03e^{-2}$	0.52	1.79	0.019
DCC-NL	153	0.0229	$5.79e^{-3}$	0.144	$2.94e^{-2}$	0.47	1.67	0.023

NOTE: The one-step ahead forecasts ($\mathcal{l}=1$) are then evaluated toward the observed realized covariance with the measures: Frobenius norm (${\text{FN}}_{\mathcal{l}}$), discrepancy in the standard deviation of the equally weighted portfolio (${\text{SD}}_{\text{EW},\mathcal{l}}$), forecasted standard deviation of the GMV portfolio (${\text{SD}}_{\text{GMV},\mathcal{l}}$), variance of the empirical return computed from the forecasted weights of the GMV portfolio (${\text{EGMV}}_{\text{var},\mathcal{l}}$), the turnover (${\text{EGMV}}_{\text{tu},\mathcal{l}}$), and the leverage (${\text{EGMV}}_{\text{lev},\mathcal{l}}$). Each SCAW model’s forecast is compared with the forecast of the equivalent MGARCH model, the DCC-S model and the DCC-NL model. The performance of the SCAW models is also compared with equivalent MGARCH model, the DCC-S model and the DCC-NL model through the test based on prewhitened ${\text{HAC}}_{\text{PW}}$ method described in Ledoit and Wolf (2011) in the case of ${\text{EGMV}}_{\text{var},\mathcal{l}}$ criteria and the pairwise forecast test for other measures. Letters M, S, and N indicate statistical significance at 5% or less in favor of the SCAW model when compared to MGARCH, DCC-S, and DCC-NL models, respectively. In the last column, the reward to risk through the Sharpe ratio is presented. The strategy with the best performance with respect to each model is highlighted bold in each row.

To have a robust forecast evaluation we run a rolling window estimation (see e.g., de Brito, Medeiros, and Ribeiro 2018; Archakov, Hansen, and Lunde 2020; De Nard et al. 2020), available in Table 3, for each respective model evaluated on a monthly basis based on 50 assets. The forecasts are then evaluated toward the observed realized covariances with the following measures: Frobenius norm (${\text{FN}}_{\mathcal{l}}$), discrepancy in the standard deviation of the equally weighted portfolio (${\text{SD}}_{\text{EW},\mathcal{l}}$), forecasted standard deviation of the GMV portfolio (${\text{SD}}_{\text{GMV},\mathcal{l}}$), variance of the empirical return computed from the forecasted weights of the GMV portfolio (${\text{EGMV}}_{\text{var},\mathcal{l}}$), the turnover (${\text{EGMV}}_{\text{tu},\mathcal{l}}$), and the leverage (${\text{EGMV}}_{\text{lev},\mathcal{l}}$). For ${\text{SD}}_{\text{EW},\mathcal{l}},{\text{SD}}_{\text{GMV},\mathcal{l}}$, and ${\text{EGMV}}_{\text{var},\mathcal{l}}$ the annualized values are provided. Each SCAW model’s forecast is compared with the forecast of the equivalent MGARCH model, the DCC-S model and the DCC-NL model. As can be seen, with respect to each measure, proposed models, such as, SCAW-SCALAR(2,2) and SCAW-SS_CT (1,2) outperform the others, where the most favorable value in each column is emphasized in bold. Overall, the MGARCH models tend to be superior in terms of ${\text{EGMV}}_{\text{tu},\mathcal{l}}$. These models tend to have smaller ${\mathbf{A}}_j$ parameters and, hence, their predictions tend to vary less with data. Thus, it is not surprising the turnover is also lower. For this particular case, the SCAW models tend to have a lower ${\text{EGMV}}_{\text{lev},\mathcal{l}}$ and outperformed the other benchmark models with respect to this criteria.

Models in the upper panel are also compared with the equivalent MGARCH model, the DCC-S model and the DCC-NL model through a pairwise forecast test. For brevity, we report these result in the same table with the labels M, S, and N, where each of the letters indicates statistical significance at 5% or less in favor of the SCAW model when compared with MGARCH, DCC-S and DCC-NL, respectively. The table also presents reward to risk through the Sharpe ratio, presented in the last column. Here again, our proposed models appear to outperform the benchmarks

Next, we evaluate the suggested models forecast ability on high-dimensional data and compare it with the results obtained for the benchmark models. Tables S.1–S.3 in the supplementary materials depict fixed window forecasts for each respective model for 10 years of data (mid 2007 to mid 2017) on 300 assets proportionally distributed among the 12 sectors in NASDAQ classification and evaluated on a monthly basis. In this application, models with parameterizations adapted to large portfolio sizes are included. For the evaluation, the model parameters are estimated on first 80% of the observations, and evaluated on the last 20% observations, at different forecasting horizons, namely $\mathcal{l}=1,5$, and 10, toward the observed realized covariance with the following performance measures: Frobenius norm (${\text{FN}}_{\mathcal{l}}$), discrepancy in the standard deviation of the equally weighted portfolio (${\text{SD}}_{\text{EW},\mathcal{l}}$), forecasted variance of the GMV portfolio (${\text{SD}}_{\text{GMV},\mathcal{l}}$), variance of the empirical return computed from the forecasted weights of the GMV portfolio (${\text{EGMV}}_{\text{var},\mathcal{l}}$), the turnover (${\text{EGMV}}_{\text{tu},\mathcal{l}}$), and the leverage (${\text{EGMV}}_{\text{lev},\mathcal{l}}$). For ${\text{SD}}_{\text{EW},\mathcal{l}},{\text{SD}}_{\text{GMV},\mathcal{l}}$, and ${\text{EGMV}}_{\text{var},\mathcal{l}}$ the annualized values are presented. Each SCAW model’s forecast is compared to the forecast of the equivalent MGARCH model, the DCC-S model and the DCC-NL model. As can be seen, with respect to each performance measure, results at different forecast horizons suggest that models from the group SCAW-SS_CT and SCAW-SCALAR outperform the other competitors, where the most favorable value in each column is emphasized bold.

In the case of a high-dimensional portfolio, the MGARCH models tend again to be superior in terms of the turnover. The DCC-NL model possesses the lowest ${\text{EGMV}}_{\text{lev},\mathcal{l}}$. Finally, the SCAW models are also compared to the equivalent MGARCH model, the DCC-S model and the DCC-NL model by using a pairwise forecast test. The labels M, S, and N have the same meaning as in Table 3. We observe that the proposed SCAW models statistically outperform the benchmark approaches in almost all of the considered cases. To this end, the table also present reward to risk defined by the Sharpe ratio in the last column. Here, the proposed models outperform the benchmarks at the shorter horizon, while the DCC-NL is found to beat the other models at the longer forecast horizons $(\mathcal{l}=5,10)$.

To further assess the strength of proposed models, we consider model confidence set of Hansen, Lunde, and Nason (2011) among the different methodologies for the rolling and the fixed forecasts at different forecast horizons ($\mathcal{l}=1,5,10)$. Table 4 provides ranking among models for the MCS approach based on three different loss functions, namely ${\text{FN}}_{\mathcal{l}},{\text{SD}}_{\text{EW},\mathcal{l}}$, and ${\text{SD}}_{\text{GMV},\mathcal{l}}$, in the case of data consisting of 50 stocks. The notation ‘-’ indicates the absence of the model in the corresponding confidence set with respect to the considered loss function, while the numbers are the ranking of the models, which are included in the 90% confidence set. We observe that that none of the benchmark models managed to outperform the suggested SCAW models.

Table 4 Ranking among models computed by using the MCS test (see Hansen, Lunde, and Nason 2011) with three loss functions (${\text{FN}}_{\mathcal{l}},{\text{SD}}_{\text{EW},\mathcal{l}}$, and ${\text{SD}}_{\text{GMV},\mathcal{l}}$, respectively) computed for rolling window forecasting in the case of data consisting of 50 stocks.

Model	$\text{MCS}\mathrm{ }-\mathrm{ }\text{Ranking}$
	${\text{FN}}_{\mathcal{l}}$	${\text{SD}}_{\text{EW},\mathcal{l}}$	${\text{SD}}_{\text{GMV},\mathcal{l}}$
SCAWCT (0, 1)	10	–	–
SCAWCT (1, 1)	9	–	–
SCAW-SSCT (0, 1)	8	–	–
SCAW-SSCT (1, 1)	6	–
SCAW-SSCT (1, 2)	1	5	–
SCAW-SSCT (2, 2)	5	2	–
SCAW-SS-HARCT (1, 5)	–	–	–
SCAW-SS-HARCT (1, 20)	–	–	–
SCAW-SCALAR(0, 1)	7	–	–
SCAW-SCALAR(1, 1)	3	4	–
SCAW-SCALAR(1, 2)	2	3	–
SCAW-SCALAR(2, 2)	4	1	1
SCAW-SCALAR-HAR(1, 5)	–	–	–
SCAW-SCALAR-HAR(1, 20)	–	–	–
MGARCHCT (0, 1)	–	–	–
MGARCHCT (1, 1)	–	–	–
MGARCH-SSCT (0, 1)	–	–	–
MGARCH-SSCT (1, 1)	–	–	–
MGARCH-SSCT (1, 2)	–	–	–
MGARCH-SSCT (2, 2)	–	–	–
MGARCH-SS-HARCT (1, 5)	–	–	–
MGARCH-SS-HARCT (1, 20)	–	–	–
MGARCH-SCALAR(0, 1)	–	–	–
MGARCH-SCALAR(1, 1)	–	–	–
MGARCH-SCALAR(1, 2)	–	–	–
MGARCH-SCALAR(2, 2)	–	–	–
MGARCH-SCALAR-HAR(1, 5)	–	–	–
MGARCH-SCALAR-HAR(1, 20)	–	–	–
DCC-S	–	–	–
DCC-NL	–	–	–

NOTE: The symbol ‘-’ indicates the absence of the model in the confidence set with respect to the considered loss function. The numbers are the ranking of the models, which are included in the 90% confidence set.

Table 5 summarizes ranking among models obtained by the fixed window forecasting at horizons $\mathcal{l}=1,5,10$ in the case of the high-dimensional data consisting of 300 stocks. For each horizon the ranking obtained via the same three loss functions ${\text{FN}}_{\mathcal{l}},{\text{SD}}_{\text{EW},\mathcal{l}}$, and ${\text{SD}}_{\text{GMV},\mathcal{l}}$ as presented and separated by the slash symbol in respective order. Also, the notation ‘-’ states that the model does not belong to the confidence set, while the numbers specifies the ranks of the model within the confidence set. For a shorter horizon, quite a few benchmark models lie in the confidence set, which are outperformed by the SCAW models showing higher ranks. For longer forecasting horizons $\mathcal{l}=5,10$, none of the benchmark models appeared in the confidence set and the proposed SCAW-type models clearly show better performance for all considered loss functions.

Table 5 Ranking among models computed by using the MCS test (see Hansen, Lunde, and Nason 2011) with three loss functions (${\text{FN}}_{\mathcal{l}}$/${\text{SD}}_{\text{EW},\mathcal{l}}$/${\text{SD}}_{\text{GMV},\mathcal{l}}$) computed for fixed window estimation with forecasting horizon $\mathcal{l}=1,5,10$ in the case of data consisting of 300 stocks.

Model	$\text{MCS}\mathrm{ }-\mathrm{ }\text{Ranking}$
	$\mathcal{l}=1$	$\mathcal{l}=5$	$\mathcal{l}=10$
SCAW-SSCT (0, 1)	2∕.-∕.-	5∕.-∕.-	6∕.-∕.-
SCAW-SSCT (1, 1)	3∕.-∕.-	1∕.-∕.-	1∕.-∕.-
SCAW-SSCT (1, 2)	8∕.-∕.-	11∕.-∕.-	10∕.-∕.-
SCAW-SSCT (2, 2)	10∕.-∕.-	14∕.-∕.	12∕.-∕.-
SCAW-SS-HARCT (1, 5)	19∕.-∕.-	3∕.-∕.-	4∕.-∕.-
SCAW-SS-HARCT (1, 20)	-∕.-∕.-	2∕.-∕.3	2∕.3∕.-
SCAW-SCALAR(0, 1)	1∕.-∕.-	6∕.-∕.-	7∕.-∕.-
SCAW-SCALAR(1, 1)	6∕.-∕.-	7∕.-∕.-	3∕.-∕.-
SCAW-SCALAR(1, 2)	9∕.-∕.-	9∕.-∕.2	9∕.1∕.2
SCAW-SCALAR(2, 2)	7∕.1∕.1	8∕.1∕.1	8∕.1∕.2
SCAW-SCALAR-HAR(1, 5)	11∕.-∕.-	4∕.-∕.-	5∕.-∕.-
SCAW-SCALAR-HAR(1, 20)	-∕.-∕.-	10∕.-∕.-	11∕.-∕.-
MGARCH-SSCT (0, 1)	20∕.-∕.-	-∕.-∕.-	-∕.-∕.-
MGARCH-SSCT (1, 1)	17∕.-∕.-	-∕.-∕.-	-∕.-∕.-
MGARCH-SSCT (1, 2)	12∕.-∕.-	-∕.-∕.-	-∕.-∕.-
MGARCH-SSCT (2, 2)	13∕.-∕.-	-∕.-∕.-	-∕.-∕.-
MGARCH-SS-HARCT (1, 5)	23∕.-∕.-	-∕.-∕.-	-∕.-∕.-
MGARCH-SS-HARCT (1, 20)	-∕.-∕.-	-∕.-∕.-	-∕.-∕.-
MGARCH-SCALAR(0, 1)	18∕.-∕.-	-∕.-∕.-	-∕.-∕.-
MGARCH-SCALAR(1, 1)	15∕.-∕.-	-∕.-∕.-	-∕.-∕.-
MGARCH-SCALAR(1, 2)	16∕.-∕.-	-∕.-∕.-	-∕.-∕.-
MGARCH-SCALAR(2, 2)	14∕.-∕.-	-∕.-∕.-	-∕.-∕.-
MGARCH-SCALAR-HAR(1, 5)	21∕.-∕.-	-∕.-∕.-	-∕.-∕.-
MGARCH-SCALAR-HAR(1, 20)	22∕.-∕.-	-∕.-∕.-	-∕.-∕.-
DCC-S	4∕.-∕.-	12∕.-∕.-	13∕.-∕.-
DCC-NL	5∕.-∕.-	13∕.-∕.-	14∕.-∕.-

NOTE: The symbol ‘-’ indicates the absence of the model in the confidence set with respect to the considered loss function. The numbers are the ranking of the models, which are included in the 90% confidence set.

In general, it is noteworthy that any specification of SCAW-SCALAR model turns out to be best with respect to ${\text{SD}}_{\text{EW},\mathcal{l}},{\text{SD}}_{\text{GMV},\mathcal{l}}$, and ${\text{EGMV}}_{\text{var},\mathcal{l}}$ measures at different forecasting setup (rolling or fixed), while the SCAW-SS models are found to be best w.r.t. to the ${\text{FN}}_{\mathcal{l}}$ measure. This implies that different SCAW models may be suitable, depending on the application at hand. Finally, it is notable that the specifications SCAW-SS_CT (1, 2) and $\text{SCAW}-\text{SCALAR}(2,2)$ most of the times outperform both SCAW_CT (0, 1) and SCAW_CT (1, 1) in terms of ${\text{FN}}_{\mathcal{l}},{\text{SD}}_{\text{EW},\mathcal{l}},{\text{SD}}_{\text{GMV},\mathcal{l}}$, and ${\text{EGMV}}_{\text{var},\mathcal{l}}$ for portfolio sizes m = 50 and m = 300, despite using a lower number of parameters (36 and 4 vs. 50 and 100, respectively). This suggests that the sectorwise and scalar approaches introduced in Section 3.2 indeed can be useful, especially in the high-dimensional case: they reduce the number of model parameters and outperform the diagonally parameterized model in most of the cases.

To summarize, the SCAW approach performs very well comparing to the considered benchmark approaches in terms of out-of-sample forecast accuracy for the time period studied. Among the various SCAW-specifications, the sectorwise parameterization, SCAW-SS_CT (p, q), the sectorwise parameterization with an HAR-extension, SCAW-SS-HAR_CT (q, h) and the scalar specification, SCAW-SCALAR(p, q), display the most favorable results. The performance of these specifications are important, since the number of parameters in these approaches are independent of the number of assets m. As such, these parameterizations are likely to be a feasible when very large asset portfolios are considered.

6 Conclusion

In this article, we present the Singular Conditional Autoregressive Wishart (SCAW) model to capture the temporal dynamics for time series of singular realized covariance matrices. The model employs a BEKK-type specification, thus, ensuring positive definitiveness, and allowing for straight forward estimation through the maximum likelihood method. Since the case of singular realized covariance is closely related to large portfolio dimensions, we also introduce methods to maintain parsimony in large dimensions. First, a covariance targeting approach adapted to the matrix case is presented. Second, we propose a sectorwise specification, using asset homogeneity within market sectors. As an additional extension, the well-established HAR-approach is adapted to our model. These approaches results in a model well adapted for large or extremely large portfolio sizes.

The SCAW model is estimated to 50 stocks in a time series covering 20 years and to 300 stocks covering 10 years, and evaluated out-of-sample with Multivariate GARCH models of similar specifications and the DCC-type model recently suggested in the literature to model large-dimensional dynamic covariance matrices. This study reveals that the SCAW models outperform the benchmark models in the vast majority of the forecast accuracy measures, with high statistical significance. Furthermore, it suggests that the SCAW-SCALAR(p, q) specifications, where parameter matrices are proportional to the identity matrix, the sectorwise specification and its HAR-extension show great promise, greatly improving the out-of-sample performances in relation to the baseline fully parameterized SCAW model, while using only a fraction of parameters. This empirical finding becomes very important by noting that the number of parameters in the parsimonious specifications does not depend on the portfolio size and thus can provide a very useful alternative to the general specification of the SCAW model, especially in the high-dimensional case where the portfolio size m is large.

Future venues of research include extension of the SCAW model by for example the MIDAS-extension employed in Golosnoy, Gribisch, and Liesenfeld (2012), or an adaptation including the leverage effect in the spirit of Anatolyev and Kobotaev (2018).

Acknowledgments

The authors thank Professor Christian Hansen, the Associate Editor, and two anonymous Reviewers for their comments and suggestions which have improved the presentation of the article. We gratefully acknowledge the comments and the discussion from the participants at the Workshop on Financial Econometrics 2019 (Örebro University School of Business) and at the International Conference on Computational and Financial Econometrics 2019 (University of London).

Supplementary Materials

The supplementary materials contain proofs of the propositions in Section 2.1, as well as additional Tables with forecast results to complement the results in Section 5.4

Additional information

Funding

Taras Bodnar was partially supported by the Swedish Research Council (VR) via the project Bayesian Analysis of Optimal Portfolios and Their Risk Measures. The computations and data storage were enabled by resources provided by the Swedish National Infrastructure for Computing (SNIC) at HPC2N partially funded by the Swedish Research Council through grant agreement no. 2018-05973. Farrukh Javed acknowledges financial support from the project “Models for macro and financial economics after the financial crisis” (Dnr: P18-0201) funded by the Jan Wallander and Tom Hedelius. Foundation.

Notes

1 The matrices ${\mathbf{D}}_m$ and ${\mathbf{L}}_m$ are defined as the matrices which satisfy the following equalities $\text{vec}(\mathbf{A})={\mathbf{D}}_m\text{vech}(\mathbf{A})$ and $\text{vech}(\mathbf{A})={\mathbf{L}}_m\text{vec}(\mathbf{A})$ for a symmetric matrix A, respectively (see, e.g., Harville 1997).

2 It is defined by the following equality ${\mathbf{K}}_{mm}\text{vec}(\mathbf{A})=\text{vec}(\mathbf{A}\prime )$ for any m × m matrix A (see Harville 1997)