3.1. The Semiparametric Lower Bound

In the following discussion we characterize the SP lower bound for multivariate volatility models of the form (1). In the context of univariate GARCH models, standard references are Linton (1993), Drost and Klaassen (1997), and Gonzalez-Rivera and Drost (1999). A detailed description of general SP estimation theory and adaptivity is beyond the scope of this paper. We refer to Bickel (1982), Newey (1990), Steigerwald (1992), Bickel et al. (1993), and Drost, Klaassen, and Werker (1997) for details. We follow a standard approach to derive the SP efficiency bound. Alternative methods such as that proposed by Severini and Tripathi (2001) may be possible also.

One can estimate the SP model using a two-step procedure that uses the so-called influence function to correct an initial consistent estimator such as the quasi maximum likelihood estimator. The correction is essentially a one-step Newton–Raphson algorithm based on the score vector of the likelihood. Assume for now that the unknown density g of v_t is parameterized by some nuisance parameter η and write this density as g(v_t,η). A particular parameterization of g that contains the true parameterization as a special case is known as the parametric submodel. Let us write the log likelihood as

with

and denote by

the score vector with respect to the parameter of interest and by s_t(θ,η) = ∂l_t /∂η the score with respect to the nuisance parameter. Note that s_t(θ,η) is a vector of length dim(η), so that its dimension differs among parametric submodels. It is easily seen that E_θ [s_t(θ,η)] = 0. Also, recall that g is restricted to be density of a mean-zero, identity covariance matrix random vector, so that s_t(θ,η) is orthogonal to the vector F(v_t) where F(x) = (x′,vech(xx′ − I_N)′)′. In the following discussion we suppress the dependence of l_t and

on η for notational convenience.

To obtain efficient estimates, it is required to eliminate the variation of

that is due to the nuisance parameter η. This is achieved by projecting the score on the so-called tangent set, which is the mean square closure of κ′s_t(θ,η), where

; see, e.g., Newey (1989, 1990). Because of the properties of s_t(θ,η), the elements of the tangent set are characterized by zero mean and orthogonality to F_t. Thus, the tangent set is defined by

To do the projection on

it is often crucial to factorize the score l_t(θ) into a term that depends on θ and

and another that depends on θ and the nuisance parameter. The next proposition shows that this is possible for the model under study.

PROPOSITION 1. For the model (1), the score vector takes the form

where

and ψ_t = ψ(v_t(θ)) is given by (3).

Note that W_t is

-measurable and depends only on the specification of H_t(θ). The other term ψ_t depends on the innovation v_t, so that W_t and ψ_t are independent. Note also that the score vector is a martingale difference sequence. The reason is that

because E_θ [ψ_t] = 0 as implied by Assumptions 1 and 2.

Remark 1. If one has a model for Σ_t, the score can also be expressed in terms of derivatives of Σ_t rather than H_t. However, it turns out that the score will in general have a different form depending on whether H_t is restricted to be symmetric or not. In the first case (H_t = H_t′), W_t takes the form

and in the second case

If g has certain symmetry properties, e.g., sphericity or componentwise independence, then the score in (8) can be equivalently written as

.

does not depend on H_t being symmetric or not, because W_t(θ)D_N is the same whether (10) or (11) is used.

In the following paragraphs we discuss the conditions under which one obtains local asymptotic normality (LAN) of the SP multivariate volatility model. Consider two local parameterizations θ_n = θ₀ + O(n^−1/2) and

such that

. Suppose the log-likelihood ratio

can be written in the form

where

and where l(H_t(θ)) = l_t(θ,η) is given in (7). Let W_nt be defined such that

As we assumed continuity of H_t in θ, we see that W_nt is asymptotically equivalent to W_t. Note also that, by definition, W_nt is measurable with respect to

.

We now have the following conditions to obtain LAN.

Assumption 5. Let W_t and W_nt be defined as in (9) and (12), respectively.

(a)

(b)

(c)

, where

(d)

for all δ > 0.

PROPOSITION 2. Under Assumption 5, it follows that R_n = o_Pθn(1) and

Furthermore, the probability measures

and P_θn are contiguous.

Denote M_ψF(θ) = E_θ [ψ_t F_t′], M_Fψ(θ) = M_ψF′(θ), and M_FF(θ) = E[F_t F_t′], where ψ_t = ψ(v_t(θ)) and F_t = F(v_t(θ)). We can now derive the projection of

on the tangent set. The orthogonal complement of this projection is the so-called efficient score function, which can be used to do a one-step Newton–Raphson improvement of the quasi maximum likelihood estimator.

PROPOSITION 3. The projection of

on

is given by

Define by

the efficient score function. The SP lower bound is then given by

where J*(θ) = E_θ [l_t*(θ)l_t*′(θ)] is the SP information matrix. By definition, adaptive estimation is possible if and only if P_t(θ₀) = 0, which means that the SP efficient score, l_t*, is equal to the parametric score

and, hence, V_sp is equal to the parametric lower bound, V_ml = J(θ₀)⁻¹. In the following proposition we characterize the asymptotic covariance matrices of the three estimation methods in terms of W_t and ψ_t.

PROPOSITION 4. Under Assumptions 1–5, the information matrices can be expressed as

where dependence of W_t, M_ψψ, M_ψF, and M_FF on θ is suppressed and where

As a corollary, we obtain that the difference of the information between the MLE and the SP estimator is given by the positive semidefinite matrix

The matrix Q determines the inefficiency of the SP estimator with respect to maximum likelihood estimation. Adaptive estimation is possible if and only if Q = 0. This would clearly be a special case, and we show later that Q = 0 happens only if g is Gaussian. The conclusion is that adaptive estimation of the model in (1) without reparameterization of H_t is not possible. Similar to Gonzalez-Rivera and Drost (1999) it can also be verified that V_sp⁻¹ − V_qml⁻¹ is positive semidefinite, meaning that the SP estimator is at least as efficient as the quasi maximum likelihood estimator.

To characterize the distance of the SP estimator from adaptivity, it will be necessary to evaluate the matrices M_ψψ and M_FF by numerical integration. However, there are special cases where they take simple forms. In the following discussion we consider two subclasses of

for which calculation turns out to be particularly simple. The first one is the class of spherical distributions:

and the second one is the class where the components are independent with identical and symmetric marginal densities:

where h(·) is the marginal density of any component of v_t.

The following discussion analyzes the properties of the SP estimator with respect to maximum likelihood estimation in the case that g happens to be in

or in

without having made this assumption to construct the estimator. The situation where one has superior knowledge about g and, e.g., knows a priori that g is in

, is different and is treated in Section 3.4.

The matrix M_FF depends on the structure of fourth moments of v_t. Lemma 3 in the Appendix implies that, for spherical distributions, the marginal kurtosis κ = E [v_ti⁴] is linked to any co-kurtosis c = E [v_ti²v_tj²],j ≠ i by κ = 3c, and M_FF depends on only one parameter.

PROPOSITION 5.

(1) If

, then

(2) If

, then

(3) If

, then

Note that Assumption 2 implies that τ in (23) and τ₂ in (26) are finite. The scalar τ₂ can be interpreted as the Fisher information for location that for distributions in

is the same for all components.

Clearly, the parametric and SP lower bounds coincide in the Gaussian case because then c = 1, τ = 1, and therefore Q = 0. In a univariate framework, Gonzalez-Rivera (1997) has shown that a class of symmetric bimodal distributions allows one to attain the parametric lower bound. The following proposition states that, for higher dimensions, the Gaussian distribution is the only one for which parametric efficiency can be attained.

PROPOSITION 6. If N ≥ 2, then the multinormal distribution is the only one for which the parametric and SP lower bounds for the estimation of model (1) coincide.

See however Section 3.3 for examples where reparameterizations may yield adaptive estimates of particular linear combinations of θ.

3.2. Efficient Estimation

To construct an estimator that attains the lower bound J*⁻¹, we rely on a discretization of the first-stage estimator and on sample splitting as in Linton (1993) and Drost et al. (1997). This is convenient for the proofs but may not be necessary in practice. Let

be the point in

that is closest to the quasi maximum likelihood estimator. The vector

can then be considered deterministic with

. Furthermore, denote

.

Let α_n be an integer sequence such that α_n /n → α ∈ (0,1). Defining

, consistent estimators of M_FF are given by

and

. For the matrix M_ψF an explicit expression is available for all

, given by (21).

To estimate the unknown score function ψ, we use a kernel estimator of the multivariate density of

; see, e.g., the monograph of Scott (1992). Because the scale of the components of v_t is the same by construction, it is reasonable to use a scalar bandwidth, h_n > 0, and a product kernel with univariate kernel function K. Our density estimators for the two parts of the sample are

and

where K(u) = (2π)^−1/2 exp(−u²/2).

Using

, we can estimate ψ by

i = 1,2, where d_n → 0, c_n → ∞, and e_n → ∞. Further, let

and

. Finally, we can estimate

in (13) by

and

and the efficient score function by

, i = 1,2.

Assumption 6. There exists a square integrable matrix W₀(θ₀) such that

.

Our estimator for the SP information matrix is

We propose the following SP estimator for θ:

For the estimator

in (28) we now have the following result.

PROPOSITION 7. If Assumptions 1–6 hold, c_n h_n → 0, and c_n²e_n²n⁻¹ ×h_n^−(2+N) → 0, then

and

where J*(θ₀)⁻¹ is the SP lower bound defined in (14).

3.3. Adaptive Estimation

As we have seen in Proposition 6, adaptive estimation of the euclidean parameter, θ, is not possible. For a given model it may be possible, however, to find linear combinations of θ, possibly after reparameterization, such that adaptive estimation of these linear combinations is possible. Technically, this amounts to an additional orthogonality condition so that the efficient influence function is equal to the heuristic influence function. This condition is stated in the following assumption.

Assumption 7. There exists a matrix

, 1 ≤ r ≤ K, of rank r, such that

Define the estimator of the information matrix by

and the estimator of ν = A′θ by

PROPOSITION 8. Under Assumptions 1–7,

is an adaptive estimator of ν, i.e.,

By Corollary 3.1 of Drost et al. (1997), a matrix A that satisfies Assumption 7 can be found if B′W_t(θ) is invertible and does not depend on t, where B is a K × N² matrix such that A′B = 0. We consider two examples.

Example 1.

Suppose a model for H_t(θ) can be parameterized such that H_t(θ) = Z_t(ν)Ω, where θ = (ν′,vec(Ω)′)′ and

symmetric and invertible. Define A = (I_K−N²,0_K−N²×N²)′ and B = (0_N²×K−N²,I_N²)′. Then, using (9),

and therefore A′θ = ν is adaptively estimable. Most of the well-known multivariate GARCH-type models can be parameterized in this form, but unfortunately they are all for Σ_t and not for H_t. For example, if the BEKK model was formulated for H_t, then the “dynamic part” of the model could be estimated adaptively after a similar reparametrization as in the univariate case.

The second example is for a model for Σ_t.

Example 2.

Let Z_t, Ω, θ, ν, A, and B be defined as in Example 1. Suppose a model for Σ_t(θ) can be parameterized such that Σ_t(θ) = Z_t(ν)Ω²Z_t(ν). Let the square root be H_t(θ) = Z_t(ν)Ω, which is not restricted to be symmetric. Thus, we can use (11) to obtain

and therefore A′θ = ν is adaptively estimable. For example, the well-known CCC model of Bollerslev (1990) falls into this category. Assuming univariate GARCH(1,1) models for the conditional variances, define Z_t = diag(1 + α_iε_i,t−1² + β_i Z_ii,t−1)_i=1^N, D = diag(ω₁,…, ω_N), and R the CCC matrix. Then the CCC model can be written in the form Σ_t(θ) = Z_t(ν)Ω²Z_t(ν) with Ω = (DRD)^1/2. The parameter ν = (α₁,…, α_N,β₁,…, β_N)′ is adaptively estimable.

3.4. Efficient Estimation in

In practice one may want to impose more structure on the multivariate density g to avoid the curse of dimensionality, e.g., the data sparseness problem described by Silverman (1986). One way to do so is to assume that g belongs to the class of spherical distributions, which we do in Assumption 8.

Assumption 8. For the density g, assume

, where

is defined by (19).

The density g(v_t) is said to be spherical if there exists a positive function f such that g(v_t) = f (w_t) with w_t = v_t′v_t. Examples of spherical distributions are the multivariate versions of the normal, the t and Laplace distributions. The class of spherical distributions (or elliptical, respectively, if the conditional distribution of ε_t is considered) is the most general one that is consistent with the conditional capital asset pricing model, as shown by Berk (1997); see also Hodgson and Vorkink (2003) and Brown and Hodgson (2004). In our framework, the main advantage of Assumption 8 is that it allows for univariate convergence rates of nonparametric estimators of g.

SP estimators of model (1) that are based on Assumption 8 have a lower bound that is larger than the bound in the general case. The score in the nonparametric direction, s(θ,η), is now a function of v_t only through w_t, and therefore, the tangent set contains functions that all depend on w_t only. But still, these functions have to be orthogonal to F_t because g is restricted to have mean zero and identity covariance matrix. Thus, the tangent set under Assumption 8 is defined by

Note that

. We can now derive the projection of

on

.

PROPOSITION 9. Under Assumption 8, the projection of

on

_s is given by

where ψ_t^s = ψ^s(w_t(θ)) = E_θ [ψ(v_t(θ))|w_t], F_t^s = F^s(w_t(θ)) = E_θ [F(v_t)|w_t], M_ψ^sF = E_θ [ψ_t^sF_t′], and M_F^sF = E_θ [F_t^sF_t′].

Note that F_t^s contains the conditional expectations E[v_t|w_t] and E[v_t v_t′|w_t]. The first one is zero because the distribution of v_t conditional on w_t is symmetric in v_t. The second one, E[v_t v_t′|w_t], can be estimated with arbitrary precision as described by Hodgson and Vorkink (2003). We can therefore consider it known and denote it by m(w_t).

In the following discussion we propose an estimator of θ in the spherical case. To simplify the presentation we do not use sample splitting here. Because y has a positive support, one faces the problem of estimating its density near the boundary. To solve this problem, Hodgson, Linton, and Vorkink (2002) and Hodgson and Vorkink (2003) apply a Box–Cox transformation to y and then use the standard kernel density estimator to the transformed variable. We use an alternative method by applying a gamma kernel estimator; see the paper by Chen (2000), who shows that gamma kernels are particularly suited for the estimation of density functions with bounded support. As before we use a first-stage consistent estimator,

, say, and let

and

. First, the gamma kernel estimator of g(x) is given by

where G_p,q is the density function of a gamma(p,q) random variable. As

depends only on y = x′x, we can write

. Using

, we can estimate ψ by

where d_n → 0, c_n → ∞, e_n → ∞, and f_n → 0. Further, let

. Finally, we can estimate

in (29) by

and the efficient score function by

. Analogous to the general case define

and the estimator

PROPOSITION 10. If Assumptions 1–8 hold, c_n b_n → 0, and either c_n²n⁻¹{(b_n f_n)^−1/2 + (b_n e_n)^1/2} → 0 for N = 1 or c_n²n⁻¹b_n^−1/2 f_n^1/2−N → 0 for N ≥ 2, then

, where

and

Note that this result is similar to that of Proposition 7, but because the gamma kernel is unbounded near zero, we have to take care that

is bounded by using appropriate trimming conditions.