Determining the number of factors in a multivariate error correction–volatility factor model

1. INTRODUCTION

The concept of co-integration (Granger, 1981, Granger and Weiss, 1983, and Engle and Granger, 1987) has been successfully applied to modelling multivariate non-stationary time series. The literature on co-integration is extensive. The most frequently used representations for a cointegrated system are the ECM of Engle and Granger (1987), the common trends form of Stock and Watson (1998) and the triangular model of Phillips (1991). The error correction model has been applied in various practical problems, such as determining exchange rates, capturing the relationship between expenditure and income, modelling and forecasting inflation, etc. From the equilibrium point of view, the term ‘error correction’ reflects the correction on the long-run relationship by the short-run dynamics.

However, the ECM ignores the characteristics of time-varying volatility, which plays an important role in various financial areas such as portfolio selection, option evaluation and risk management. Kroner and Sultan (1993) argued that the neglect of either co-integration or time-varying volatility would affect the hedging performance of existing models in the literature for the futures market. Similar conclusion has been given by Ghost (1993) and Lien (1996) through empirical calculation and theoretical analysis, respectively. Therefore, the traditional ECM needs to be generalized to have conditional heteroscedasticity for capturing both co-integration and time-varying volatility.

Univariate volatility models have been extended to multivariate cases. Extensions of the generalized autoregressive heteroscedastic (GARCH) model (Bollerslev, 1986) include, e.g. vectorized GARCH (VEC-GARCH) model of Bollerslev et al. (1988), the BEKK model of Engle and Kroner (1995), a dynamic conditional correlation (DCC) model of Engle (2002) and Engle and Sheppard (2001), a generalized orthogonal GARCH model of van der Weide (2002); see a survey of multivariate GARCH models by Bauwens et al. (2006).¹ These models assume that a vector transformation of the covariance matrix can be written as a linear combination of its lagged values and the innovations. Andersen et al. (1999) showed that these models perform well relatively to competing alternatives. But the curse of dimensionality becomes a major obstacle in application. A useful approach to simplifying the dynamic structure of a multivariate volatility process is to use factor models. As is well known, factor models have been used for performance evaluation and risk measurement in finance. Moreover, it is now widely accepted that the financial volatilities move together over time across assets and markets (Anderson et al., 2006). These make it reasonable that we impose a factor structure on the residual term of a multivariate error correction model. In this sense, an error correction–volatility factor (EC–VF) model can capture the features of co-movements in both conditional mean (co-integration) and conditional variance (volatility factors) of a high dimensional time series.

The contribution of this paper is to estimate the EC–VF model. The set of parameters is divided into three subsets: structural parameter set including lag order and all autoregressive coefficient vector and matrices, co-integration parameter set including the co-integration vectors and the rank, and factor parameter set including the factor loading matrix and the number of factors. We conduct a two-step procedure to estimate relevant parameters. First, assuming that the structural and co-integration parameters are known, we give the estimation of factor loading matrix in the volatility factor model, and then give a method to determine the number of factors consistently. Our model specification and estimation approaches are general, because we impose neither i.i.d. assumption on the idiosyncratic components in the factor structure nor independence between factors and idiosyncratic errors. In contrast to the innovation expansion method in Pan and Yao (2008) and Pan et al. (2007), where they can not prove that their algorithm for the number of factors is consistent, our method in this paper is based on a penalized goodness-of-fit criterion. We prove our estimator of the number of factors is consistent. Secondly, the structural and co-integration parameters will be consistently estimated without knowing the true factor structure. The main distinction between Bai and Ng (2002) and this paper is that their factor model concerned the unconditional mean of economic variables while our factor structure is imposed on the conditional variance to reduce the dimension of volatilities.

The rest of the paper is organized as follows. Section 2 defines the EC–VF model and mentions some practical backgrounds of the model. Section 3 presents an information criterion for determining the number of factors and the consistency of our estimator. In Section 4, a simple Monte Carlo simulation is conducted to check the accuracy of the proposed estimation for the factor loading matrix and the number of factors. In Section 5, an application to financial risk management is discussed to show the advantages of the EC–VF model to other traditional alternatives. All theoretical proofs are given in the Appendix.

2. MODEL

2.1. Definition

Suppose that {Y_t} is a d × 1 time series. The EC–VF model is of the form

((2.1))

where ΔY_t=Y_t−Y_t−1, μ is a d × 1 vector, Γ_i, i = 1, …, k are d ×d matrices. The rank of Γ₀, denoted by m, is called the co-integration rank. {Z_t} is strictly stationary with and , where is a r × 1 time series, r < d is unknown, A is a d ×r unknown constant matrix. F_t and e_t are assumed to satisfy

((2.2))

where Σ_e is a positive definite matrix independent on t. The components of F_t are called ‘factors’, and r is the number of factors. Note that F_t and e_t are conditional uncorrelated. There is no loss of generality in assuming that E(F_tF′_t) is a r ×r positive definite matrix (otherwise, the above model may be expressed equivalently in terms of a smaller number of factors).

Remark 2.1. The error term {Z_t} in an EC–VF model is conditionally heteroscedastic and follows a factor structure, while the error term in the traditional ECM developed by Engle and Granger (1987) is covariance stationary with mean 0. Here, the factor structure is not the classical one because we assume neither that the idiosyncratic components e_t are i.i.d. with a diagonal covariance matrix nor that the factor components F_t is independent of e_t.

Model (2.1) assumes that the volatility dynamics of ΔY_t is determined by a lower dimensional volatility dynamics of F_t and the static variation of e_t, as

((2.3))

where and . Without loss of generality, we assume rank (A) =r. The lower dimensional volatility dynamics Σ_f(t) can be fitted by, e.g. the dynamic conditional correlation model of Engle (2002) or the conditionally uncorrelated components model of Fan et al. (2008).

3. ESTIMATION OF THE NUMBER OF FACTORS

The parameter set of the EC–VF model (2.1) is {Θ; Γ₀; A}, in which Θ={μ, Γ₁, …, Γ_k−1} is called the structural parameter, Γ₀ the co-integration parameter and A the factor parameter. In first two subsections, {Θ, Γ₀} is assumed known and its determination will be discussed later in subsection 3.3.

3.1. Determining A

Note that the factor loading matrix A and the vector of factors F_t in equation (2.1) are not separately identifiable. Our goal is to determine the rank of A and the space spanned by the columns of A. Without loss of generality, we may assume A′A =I_r, where I_r denotes the r ×r identity matrix. Let be the linear subspace of R^d spanned by the columns of A, which is called the factor loading space. Then, we need to estimate or its orthogonal complement , where B is a d × (d −r) matrix for which (A, B) forms a d ×d orthogonal matrix, i.e. B′A = 0 and B′B =I_d−r. Now it follows from equation (2.1) that

((3.1))

From equation (3.1) and the assumption that {e_t} is a conditional homoscedastic sequence of martingale differences (see equation (2.2)), we have

where Σ_z=E(Z_tZ_t′). This implies that

((3.2))

or equivalently

((3.3))

where consists of some subsets in R^d, and ∥M∥=[tr(M′M)]^1/2 denotes the norm of matrix M. Hence, we may estimate B by minimizing

((3.4))

subject to the condition B′B =I_d−r, where τ₀ is a prescribed positive integer and . This is a high-dimensional optimization problem, but it does not explicitly address the issue how to determine the number of factors r consistently. We first assume r is known and introduce some properties of the estimator of B derived by Pan et al. (2007) before we present a consistent estimator of r.

Let be the set of all d × (d −r) (d ≥r) matrix B satisfying B′B =I_d−r. We partition into equivalent classes such that belong to the same class if and only if , which is equivalent to

((3.5))

Define

The equivalent classes can be regarded as the elements of the quotient space defined by D-distance. It can be shown that D is a well-defined metric distance on the space , and thus , which is our parametric space, is a metric space; see Pan and Yao (2008).

Our estimator of B is the minimizer of Φ_n(·) in , i.e.

Under the assumptions listed below, the estimator is consistent with a convergence rate .

Assumption 3.1 {Z_t} is a strictly stationary d-dimensional time series with E∥Z_t∥^2p < ∞ for some p > 2. The β-mixing coefficients

satisfy β_n=O(n^−b) for some , where is the σ-algebra generated by {Z_t, i ≤t ≤j}.

Assumption 3.2 Denote . There exists a matrix which minimizes Φ(B), and Φ(B) reaches its minimum value at a matrix if and only if D(B, B₀) = 0.

Assumption 3.3 There exists a positive constant a such that Φ(B) −Φ(B₀) ≥aD(B, B₀) for any matrix .

By the similar way to that in proof of Theorem 2 in Pan et al. (2007), we can prove the following result, which is useful in deriving a consistent estimator for the number of factors in next subsection.

Theorem 3.1. If the collection of subsets in R^d is a VC-class, and Assumptions 3.1 and 3.2 hold, then

((3.6))

If, in addition, Assumption 3.3 also holds,

((3.7))

Remark 3.1 The definition of Vapnik-C̆ervonenkis (VC) class can be found in van der Vaart and Wellner (1996).

3.2. Determining r

Let r₀ be the true number of factors and A₀ the true factor loading matrix with rank r₀. We discuss how to estimate r₀ based on the estimated factor loading matrix (or its counterpart ) derived in the previous subsection. The basic idea is to treat the number of factors as the ‘order’ of model (2.1) and to determine the order in terms of an appropriate information criterion.

In the following, we always assume that Assumptions 3.1–3.3 hold. Let M^l denote a matrix with rank d −l. In particular, B₀ and ) denote the matrices B₀ and with ranks d −r₀ and d −r, respectively.

Let

((3.8))

where

Our penalized goodness-of-fit criterion is defined as

((3.9))

where g(n) is a penalty for ‘overfitting’. We may estimate r₀ by minimizing PC(r), i.e.

We call equation (3.9) a penalized goodness-of-fit criterion because of Lemma A.1.

Remark 3.2 Φ_n(·) can be regarded as fitting error, because a model with r + 1 factors can fit no worse than a model with r factors, while Lemma A.1 shows that Φ_n(·) is a non-increasing function of r. But the efficiency is lost as more factors are estimated. For example, there is neither error nor efficiency in the extreme case when with .

The following theorem shows that is a consistent estimator of r₀ provided that the penalty function g(n) satisfies some mild conditions.

Theorem 3.2 Under Assumptions 3.1–3.3, as provided that g(n) → 0 and .

3.3. Determining {Θ, Γ₀}

In this subsection, we give an estimation of the structural and co-integration parameter sets without knowledge of the true factor structure for Z_t. By the Grange representation theorem, if there are exactly m co-integration relations among the components of Y_t, and Γ₀ admits the decomposition Γ₀=γα′, then α is a d ×m matrix with linearly independent columns and α′Y_t is stationary. In this sense, α consists of m co-integration vectors. As α and γ are not separately identifiable, our goal is to determine the rank of α, i.e. the dimension of the space spanned by the columns of α. Besides Assumptions 3.1–3.3 on {Z_t}, we need an additional assumption on {Y_t} as follows.

Assumption 3.4 The process Y_t satisfies the basic assumptions of the Granger representation theorem given by Engle and Granger (1987), and E∥α′Y_t−1∥⁴ < ∞.

Our estimation of co-integration vectors is the solution to the following optimization problem

((3.10))

where . The solution of equation (3.10) is , where are the m generalized eigenvectors of S₁₀S₀₁ with respect to S₁₁ corresponding to the m largest generalized eigenvalues.

The estimated co-integration vectors are consistent with the standard root-n convergence rate. The corresponding estimator of the co-integration loading matrix and the estimator of the structural parameter are also consistent. These conclusions are obtained by Li et al. (2006), who also give a joint estimation for the co-integration rank and the lag order of the error correction model by a penalized goodness-of-fit measure

((3.11))

where

((3.12))

g₁(n) is the penalty for ‘overfitting’ and n_m,k is the number of free parameters. Note that n_m,k=d +d²(k − 1) + 2dm −m² for model (2.1). We may estimate m₀ by minimizing , i.e.

where K is a prescribed positive integer. Let k₀ be the true lag order. The theorem below ensures that is a consistent estimator for (m₀, k₀).

Theorem 3.3 Under Assumptions 3.1–3.4, as provided that g₁(n) → 0 and ng₁(n) →∞.

In practice, the choice of penalty function g(·) is flexible, e.g. or .

4. MONTE CARLO SIMULATION

We present a simple Monte Carlo experiment to illustrate the proposed approach in this section. Particularly, we check the accuracy of our estimation for the factor loading matrix A and the number of factors r.

Consider a simple EC–VF model with d = 6, m = 1, r = 1,

((4.1))

where σ²_t=β₀+β₁F²_t−1+β₂σ²_t−1, e_t is independent of F_t, and the values of parameters are given as follows: and β= (β₀, β₁, β₂)′= (0.02, 0.10, 0.76)′.

Note that A′A = 1. We conduct 2000 replications, and for each replication, the sample sizes are n = 500 and 1000, respectively. We estimate the transformation matrix B by minimizing Φ_n(B) defined by equation (3.4), and measure the estimation error of the factor loading space by

The coefficients β_i, i = 0, 1, 2, are estimated by quasi-maximum likelihood estimation (MLE) based on a Gaussian likelihood. The resulting estimates are summarized in Table 1.

Table 1. Simulation results: summary statistics of estimation errors.

n = 500	Mean	0.0563	0.0179	0.0894	0.7414
	Median	0.0438	0.0183	0.0827	0.7521
	STD	0.0601	0.0022	0.0403	0.0935
	Bias	–	−0.0021	−0.0106	−0.0186
	RMSE	–	0.0029	0.0454	0.0958
n = 1000	Mean	0.0477	0.0193	0.0922	0.7481
	Median	0.0390	0.0199	0.0897	0.7543
	STD	0.0426	0.0010	0.0276	0.0724
	Bias	–	−0.0007	−0.0078	−0.0119
	RMSE	–	0.0013	0.0295	0.0766

The mean of estimation errors is less than 0.06, while it decreases over 15% as the sample size increases from 500 to 1000. The negative biases indicate a slight underestimation for the heteroscedastic coefficients. The relative frequencies for taking different values are listed in Table 2. It shows that when the sample size n increases, the estimation of r becomes more accurate.

Table 2. Relative frequencies for taking different values, when r = 1.

	0	1	2	3	4	5	6
n = 500	0.0120	0.8425	0.1310	0.0105	0.0040	0	0
n = 1000	0.0090	0.9765	0.0100	0.0045	0	0	0

5. APPLICATION TO REAL DATA

The VaR is widely adopted by banks and other financial institutions to measure and manage market risk, as it reflects downside risk of a given portfolio or investment. Specifically, at a given confidence level 1 −a, the VaR of a portfolio with weight ω_t is defined as the solution to

((5.1))

where ΔY_t is a vector of log returns of assets in the portfolio. In the case when the conditional density is normal, equation (5.1) reduces to the well-known formula

((5.2))

where z_a is the ath quantile of the univariate standard normal distribution.

In this section, we attempt to compare the VaR forecasting results by assuming three different models: AR-DCC, EC-DCC, EC-VF-DCC for the asset price series {Y_t}. The DCC refers to dynamic conditional correlation, a volatility model proposed by Engle (2002). Focusing on the methodology, we only consider the case when the conditional multivariate density is normal, while the impact of other distributions (like Student-t and some non-parametric densities) on VaR computation is beyond our scope here.

5.1. Data set and estimation of the EC-VF-DCC model

Our data set consists of 2263 daily log prices of CSCO, DELL, INTC, MSFT and ORCL, the five most active stocks in US market, from 19 June 1997 to 16 June 2006. The plots of log returns (in percentage) are presented in Figure 1 which shows significant time-varying volatilities. Descriptive statistics are listed in Table 3. All unconditional distributions of these series exhibit excessive kurtosis and non-zero skewness, indicating significant departure from the normal distribution.

Table 3. Summary statistics of the log-returns.

n = 2263	CSCO	DELL	INTC	MSFT	ORCL
Mean	0.000423	0.000523	1.95 × 10−5	0.000200	0.000418
Stdev	0.031847	0.030270	0.030313	0.023074	0.036400
Min	−0.145000	−0.20984	−0.248680	−0.169760	−0.346150
Max	0.218239	0.163532	0.183319	0.178983	0.270416
Skewness	0.149215	−0.118260	−0.391560	−0.173470	−0.226370
Kurtosis	4.558020	3.690575	5.631860	5.955046	8.519630

The estimation procedure for the EC-VF-DCC model is given step by step as follows.

• Step 1. Fit an ECM for Y_t to determine the structural and co-integration parameters. Compute the estimate of conditional mean vector .

• Step 2. Conduct a multivariate portmanteau test for the squared residuals obtained from the previous step to detect conditional heteroscedasticity. If there exists serial dependence, fit a volatility factor model for the residual series {Z_t} to determine the factor loading matrix , otherwise switch to Step 3 with and r =d.

  Denote B = (b₁, b₂, …, b_d−r), the objective function (3.4) can be modified to

  
  where w(C) ≥ 0 are weights which ensure that the sum over converges. In numerical implementation, we simply take as the collection of all the balls centred at the origin in R^d and .

  An algorithm for estimating B and r is given as follows. Put

  
  Compute by minimizing Ψ (b) subject to the constraint b′b = 1. For l = 2, …, d, compute which minimizes Ψ_l(b) subject to the constraint for i = 1, 2, …, l − 1.

  Let with , where PC(r) is defined by equation (3.9). Note that . Let consist of the (orthogonal) unit eigenvectors, corresponding to the common eigenvalue 1, of matrix .

• Step 3. Fit a DCC volatility model (Engle, 2002) for and compute its conditional covariance .

  To this end, we first fit each element of D_t with a univariate GARCH(1,1) model using the ith component of only, and then model the conditional correlation matrix R_t by

  
  where ɛ_t is a vector of the standardized residuals obtained from the separate GARCH(1,1) fittings for the components of , and S is the sample correlation matrix of .

  If , the estimate of conditional covariance matrix of ΔY_t is equal to and terminate the algorithm. Otherwise, proceed to Step 4.

• (5.3)

  Step 4. The factor structure in equation (2.1) and the facts B′A = 0, B′e_t=B′Z_t, AA′+BB′=I_d lead to a dynamics for Σ_y(t) ≡Σ_z(t) as follows

  ((5.3))
  where .

We determine the co-integration rank by minimizing defined by equation (3.11). The surface of is plotted against m and k in Figure 2. The minimum point of the surface is attained at (m, k) = (1, 1), leading to an error correction model for this data set with lag order 1 and co-integration rank 1. Applying the Ljung-Box statistics to the squared residuals, we have Q₅(1) = 63.2724, Q₅(5) = 305.7613 and Q₅(10) = 633.7103. Based on asymptotic χ₂ distributions with degrees of freedom 11, 111 and 236, the p-values of these Q statistics are all close to zero.² Consequently, the portmanteau test confirms the existence of conditional heteroscedasticity. The algorithm stated in Step 2 leads to an estimator for the number of factors, and PC(r) is plotted against r in Figure 3. Clearly, a two-factor structure (i.e. ) is determined for the residual series {Z_t}.