Journal of Financial Econometrics

Journal of Financial Econometrics Advance Access originally published online on July 11, 2007
Journal of Financial Econometrics 2007 5(4):624-636; doi:10.1093/jjfinec/nbm010

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Positivity Conditions for a Bivariate Autoregressive Volatility Specification

C. Gourieroux
     CREST, and University of Toronto

Address correspondence to C. Gourieroux, CREST, and University of Toronto, Canada, UK, or e-mail: christian.gourieroux{at}ensae.fr

	Abstract

Top Abstract 1 Introduction 2 The Positive Definiteness... 3 Discussions Of The... 4 Concluding Remarks Appendix: A Appendix: B References Footnotes

 
We derive necessary and sufficient conditions for the positive definiteness of the predicted volatility matrix in a bivariate autoregressive volatility specification. These nonlinear inequality restrictions have strong implications in terms of causality between volatilities and covolatilities.

KEYWORDS: GARCH model, nonlinear causality, stochastic volatility, Wishart process

	1 Introduction

Top Abstract 1 Introduction 2 The Positive Definiteness... 3 Discussions Of The... 4 Concluding Remarks Appendix: A Appendix: B References Footnotes

 
The dynamics of volatility-covolatility matrices are often based on autoregressive specifications, in which the best predictions of volatilities and covolatilities are affine functions of lagged volatilities and covolatilities. Examples are multivariate ARCH models, as the so-called VEC and BEKK models [Bollerslev, Engle, and Wooldridge (1988), Engle and Kroner (1995), Bauwens, Laurent, and Rombouts (2006)], Wishart processes [Gourieroux (2006), Gourieroux, Jasiak, and Sufana (2007)], or continuous time affine factor model a la Duffie, Kan [Duffie and Kan (1996)], when the factors are the elements of a stochastic volatility matrix. The fact that the prediction of a volatility-covolatility matrix has to be symmetric, positive semi-definite for any conditioning value implies complicated restrictions on the autoregressive coefficients. The aim of this article is to explicit these restrictions in the bivariate case and for an autoregressive specification.

In Section 2, we explain why these restrictions are equivalent to the positivity of a well-chosen quartic polynomial. Then, we use a general result by Ulrich and Watson (1994) to derive the necessary and sufficient positive definiteness restrictions¹. These restrictions are discussed in Section 3 with their practical implications, especially for causality analysis. Section 4 concludes.

	2 The Positive Definiteness Restrictions

Top Abstract 1 Introduction 2 The Positive Definiteness... 3 Discussions Of The... 4 Concluding Remarks Appendix: A Appendix: B References Footnotes

 
2.1 Polynomial Restrictions
Let us consider a (2 x 2) stochastic volatility matrix

which can take any value in the set of symmetric positive definite matrices. An AR(1) model² specifies the best prediction of Y_t as:

where a_j, b_j, c_j, d_j, j = 1, 2, 3 are scalar parameters. The autoregressive parameters cannot be chosen arbitrarily, since we must have:

(1)

Since the mapping Y  E(Y_t|Y_t–1 = Y) is affine and the set is a positive convex cone, it is equivalent to write condition (1) for matrices Y belonging to the boundary of . This boundary includes the positive semi-definite matrices with diminished rank either Y = 0, or , with

We deduce that conditions (1) are equivalent to:

It is well-known that the positivity condition of a (2 x 2) symmetric matrix is equivalent to the positivity of its diagonal elements plus the positivity of its determinant. Thus, the conditions above are equivalent to:

(2)

The conditions for the positivity of a quadratic polynomial are well-known. Thus, conditions (2) are also equivalent to:

(3)

where:

(4)

It remains to write the conditions for the positivity of the quartic polynomial. This is the aim of Section 2.2.

2.2 Positivity Conditions for Quartic Polynomial
Let us first consider the degenerate cases.

2.2.1 Degenerate cases A = 0 or E = 0
If B is not equal to zero, we get a cubic polynomial with a change of sign. Thus, B = 0, and the positivity condition concerns a quadratic polynomial. The conditions are:

(5)

The case E = 0 is symmetric of the previous one by replacing z by 1/z. We get:

(6)

2.2.2 Nondegenerate case
By applying the extension of Ulrich and Watson (1994) proved in Appendix 1, we get the following Lemma.

Lemma 1.
The quartic polynomial P(z)  0 for any z, if and only if,

and

where:  = BA^–3/4E^–1/4, ß = CA^–1/2E^–1/2,

2.3 The Positivity Conditions
We immediately deduce the proposition below.

Proposition 1.
The conditions for the positivity of the prediction of the volatility-covolatility matrix E(Y_t|Y_t–1) are:

where A, B, C, D, E are given in (4).

Note that the positivity restrictions are union of equality and inequality restrictions on parameters.

	3 Discussions Of The Restrictions

Top Abstract 1 Introduction 2 The Positive Definiteness... 3 Discussions Of The... 4 Concluding Remarks Appendix: A Appendix: B References Footnotes

 
3.1 Causal Implications
The positivity restrictions have strong implications for linear causality analysis between volatility and covolatility processes. The basic reason is the Cauchy–Schwarz inequality: , which implies that a shock on Y_{12, t–1} (resp. on Y_{11, t–1}) can imply a movement on Y_{11, t–1}( resp. on Y_{12, t–1}), if the Cauchy–Schwarz inequality is almost binding. We provide below several illustrations.

3.1.1 The lagged volatilities contain all useful information
This hypothesis underlies the literature on volatility transmission across national markets, which considers the dynamics of the market volatilities and usually disregards the covolatility between markets.

This hypothesis is satisfied when b₁ = b₂ = b₃ = 0. Under these restrictions we deduce that

Moreover, by applying the inequalities of Proposition 1, we see that the transformed parameter ß takes the limiting value ß = –2. Therefore, we have:

that is a deterministic nonlinear relationship between the a_j and c_j coefficients.

Finally, note that:

To summarize, under the additional conditions b₁ = b₂ = b₃ = 0, the a_j, c_j parameters are constrained by:

(7)

3.1.2 The lagged covolatility contains all useful information
This situation arises when a_j = c_j = 0, j = 1, 2, 3. These additional conditions imply b₁ = b₃ = 0 (see (3)), A = B = D = C = 0 (see 4), then , that is b₂ = 0. To summarize it is not compatible with positivity restrictions to assume that the lagged covolatility contains all useful information.

3.1.3 The lagged volatility of asset 2 has no influence on the volatility of asset 1
This arises when c₁ = 0. This additional condition implies b₁ = 0 (from (3)). Then from (4), is nonnegative, if and only, if C₂ = 0. Thus, when the lagged volatility of asset 2 is not introduced in the autoregression for Y_{11, t}, the lagged covolatility has also not to be introduced, and the lagged volatility of asset 2 has not to be introduced in the autoregression for Y_{12, t}.

3.1.4 Both volatilities are unpredictable
This arises when a₁ = b₁ = c₁ = a₃ = b₃ = c₃ = 0. It is easily checked that this implies: a₂ = b₂ = c₂ = 0. Thus, when the volatility processes are unpredictable, the covolatility process is also unpredictable.

3.1.5 The VEC-diagonal specification
Sufficient conditions for positive definiteness have been proposed for the VEC-diagonal model by Ding and Engle (1994), Ledoit, Santa-Clara, and Wolf (2003). The VEC-diagonal specification is such that:

They imply: Therefore, the conditions of Proposition 1 become:

that is, the matrices and have to be positive semi-definite. Thus, in the bivariate VEC – diagonal specification, the sufficient condition by Ding and Engle (1994) is also a necessary condition.

3.2 Constrained Specification
The general specification considered in this article depends on 12 parameters, which can be fixed freely up to the complicated inequality restrictions of Proposition 1. In the literature some constrained specifications, depending on a smaller number of "independent" parameters, have been introduced to satisfy easily the positive definiteness restriction. We first recall the main constrained autoregressive specification³, which underlies the BEKK model [Engle and Kroner (1995), Engle and Mezrich (1996)], and the Wishart process [Gourieroux and Jasiak (2006), Gourieroux, Jasiak, and Sufana (2007)].

3.2.1 A basic constrained specification
Let us consider the specification:

(8)

where D is a (2 x 2) symmetric semi-definite positive matrix and M is a (2 x 2) matrix, not necessarily symmetric:

By construction, the prediction is a symmetric semi-definite positive matrix for any Y_t–1 which belongs to . This specification involves seven parameters only, and is very constrained compared to the general specification considered in this article. The additional constraints are:

or equivalently:

where m¹, m² are the columns of matrix M and A* [resp. B*, C*] denotes the matrix

3.2.2 Interpretation of the basic specification
Let us consider the set of linear mappings (A*, B*, C*), which transform into . The set is clearly a positive convex cone. This convex cone admits finite boundary elements due to the inequality restrictions derived in Proposition 1. A mapping is a boundary element of ,if and only if there exists a boundary element of , which is transformed into a boundary element of . We will focus on the mappings which transform any boundary element of , that is a symmetric positive matrix with reduced rank, into a boundary element of . Let us assume:

which is a matrix of rank 1. The associated linear component of the prediction is:

The transformed matrix has a reduced rank for any , if and only, if

(9)

This is equivalent to the five following constraints:

(10)

It is proved in Appendix 2 that the mappings of satisfying restrictions (10) are of the type: Y  MYM' for some (2 x 2) matrix M. We deduce the proposition below.

Proposition 2.
The mappings belonging to , which transform any boundary element of into a boundary element of can be written as Y  MYM'. They are boundary elements of .

It is also interesting to note that a mapping of which is a one-to-one mapping from onto has to transform the boundary elements of into boundary elements of .

Corollary 1
The elements of , which are one-to-one mappings of onto , can be written as: Y  MYM', where M is invertible.

The corollary above shows that the basic constrained specification corresponds to an extreme case. However, it is easy to understand how to pass from the basic constrained specification to the general case. Indeed, it is known that any element of a convex cone can be written as a combination with positive coefficients of n + 1 boundary elements, where n is the dimension of the space. Thus, any specification A*Y₁₁ + B*Y₁₂ + C*Y₂₂ can be written as , where the choice of the boundary elements M_jYM'_j depend on A*, B*, C*, and is not unique in general. In some sense a BEKK specification with several terms of the type M_jYM'_j can be seen as a way to approximate the general specification by means of boundary elements.

3.2.3 Causal implications of the basic constrained specification
The basic constrained specification has stronger implications in terms of linear causality.

1. Noncausality from Y₁₂ to (Y₁₁, Y₂₂).
   This arises when b₁ = b₃ = 0, that is, if one of the following set of conditions is satisfied:

   case 1: m₁₁ = m₂₁ = 0
   In this case the predicted volatility depends on the past by means of Y₂₂ only.
   

   case 2: m₁₁ = m₂₂ = 0
   The covolatility depends on the lagged covolatility only, whereas there is a feedback between the two volatilities.
   

   case 3: m₁₂ = m₂₁ = 0
   We get a diagonal VEC model.
   

   case 4: m₁₂ = m₂₂ = 0
   The predicted volatility depends on the past by means of Y₁₁ only.
   

   
   
2. Noncausality from (Y₁₁, Y₂₂) to Y₁₂.
   The noncausality restrictions are m₁₁m₂₁ = m₁₂m₂₂ = 0. We get 4 cases including case 2 and case 3, above. The other cases are:

   case 5: m₁₁ = m₁₂ = 0
   We see that both Y₁₁ and Y₁₂ do not depend on lagged values.
   

   case 6: m₂₂ = m₂₁ = 0
   Similarly, both Y₂₂ and Y₁₂ do not depend on lagged values.
   

   
   

	4 Concluding Remarks

Top Abstract 1 Introduction 2 The Positive Definiteness... 3 Discussions Of The... 4 Concluding Remarks Appendix: A Appendix: B References Footnotes

 
In this article, we have considered an autoregressive specification for volatilities and covolatilities, and derived the restrictions on autoregressive parameters to ensure that the predicted volatility matrix is positive semi-definite. We have seen that these restrictions have important consequences in terms of causality between volatilities and covolatilities, and discussed constrained autoregressive specifications.

These are also important for statistical inference. Usually, autoregressive specifications are estimated by unconstrained estimation approaches, for instance by unconstrained quasi-maximum likelihood approach if the observations concern a sequence of realized volatilities-covolatilities or outer-products of lagged return innovations⁴. It is important to check ex post if the positivity restrictions are satisfied by the estimated parameters to ensure that the predictions can really be considered as volatilities-covolatilities. This can be the basis for a specification test.

Finally, the extension of the positivity conditions to volatility matrices with dimension larger than 3 is an open question. Indeed, it is equivalent to the positivity condition for a polynomial with degree larger than 6. In the mathematical literature it takes almost 10 years to pass from degree 3 to degree 4, and the approach developed by Ulrich and Watson (1944) for degree 4 does not seem to extend easily to degree 5,6...

	Appendix: A

Top Abstract 1 Introduction 2 The Positive Definiteness... 3 Discussions Of The... 4 Concluding Remarks Appendix: A Appendix: B References Footnotes

 
The following property is proved in Ulrich and Watson (1994), Theorem 2.

Proposition A.1
[Ulrich and Watson (1994)]: Let us consider the quartic polynomial:

then P(z)  0 for any z > 0, if and only, if one of the following conditions is satisfied:

1. ß < –2 and   0 and  +  > 0,
   
2. 
   
3. 
   

where:  = BA^–3/4E^–1/4, ß = CA^–1/2E^–1/2,  = DA^–1/4E^–3/4,

The proposition above is written for the positivity of polynomial when z > 0. We need a similar condition valid for any z.

To impose the positivity for any z, the conditions above have to be written for both polynomials P(z) and P*(z) = P(–z). We get:

The corresponding transformed parameters are:

Since ß* = ß, we get the same type of conditions as in Proposition A.1.

1. Case 1: ß = ß* < –2
   We must have  = * < 0 and  +  < 0 and * + * = –( + ) < 0.
   This is impossible.
   
2. Case 2: –2  ß  6
   The same argument shows that the only relevant condition is:   0 and ₁  0 and Note that the conditions ₁  0 and are equivalent to
   

   
   

3. Case 3: 6  ß
   The only relevant condition is   0 and ₂  0 and .
   The conditions ₂  0 and are equivalent to:
   

   
   

Note finally that:

We deduce that (ß + 2)²/(ß – 2)  16, when ß > 2, or equivalently:

To summarize we get the following proposition.

Proposition A.2
The quartic polynomial P(z)  0, for any z, if and only if,

	Appendix: B

Top Abstract 1 Introduction 2 The Positive Definiteness... 3 Discussions Of The... 4 Concluding Remarks Appendix: A Appendix: B References Footnotes

 
Characterization of the basic constrained specification

1. Expressions of A* and C*

The first restriction in (10) implies that matrix A has a diminished rank and can be written as:

since a₁ is positive⁵ from Proposition 1. Thus, we can write

where m = (m₁, m₂)'.

The same argument can be applied to matrix C to get: C* = µµ', say, where µ = (µ₁, µ₂)'.

2. Expression of matrix B*.

From the second and fourth equations of (10), we deduce:

(B1)

By substituting in the third equation of (10), we get:

By taking into account (a.1), we have:

(B2)

Finally, can always be fixed to  = +1; otherwise we have just to replace m by –m.

	Footnotes

Top Abstract 1 Introduction 2 The Positive Definiteness... 3 Discussions Of The... 4 Concluding Remarks Appendix: A Appendix: B References Footnotes

 
We thanks F. Trojani for helpful discussion. The author gratefully acknowledges financial support of NSERC Canada and of the Chair AXA: "Large Risk in Insurance".

¹ Sufficient conditions have recently been derived by Silberberg and Pafka (2001).

² The extension to an AR(p) model is straightforward. If the best prediction E(Y_t|Y_t–1) = D + A₁(Y_t–1) +  + A_p(Y_t–p), where A_j(Y_t–j) is a linear function of Y_t–j, the positive semi-definiteness conditions have to be written on D, A₁, ..., A_p, separately.

³ Other restricted or modified multivariate ARCH models have also been introduced to satisfy easily the positive definiteness condition. Among others are the VEC-diagonal formulation [Bollerslev, Engle, and Wooldridge (1988), the Generalized Orthogonal GARCH [Weide (2002)], the DCC Model [Engle (2002), Tse and Tsui (2002)],], the Cholesky Factor GARCH [Kawakatsu (2003)].

⁴ When a quasi-maximum likelihood approach is directly applied to the return themselves, such as in ARCH models, the softwares generally include some ad-hoc transformation to ensure that the determinant of the ARCH volatility is positive for each observation and to allow the computation of the quasi log-likelihood. These conditions det(A*y_11t + B*y_12t + C*y_22t + D*) > 0, t = 1, ..., T, are much weaker than the positivity condition of the autoregressive volatility specification.

⁵ The limiting case a₁ = 0 is easily treated directly.

Received July 26, 2006; revised April 17, 2007; accepted April 18, 2007

	References

Top Abstract 1 Introduction 2 The Positive Definiteness... 3 Discussions Of The... 4 Concluding Remarks Appendix: A Appendix: B References Footnotes

 

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