4

Temporal Aggregation and Weak GARCH Models

Most financial series are analyzed at different frequencies (daily, weekly, monthly, …). The properties of a series and, as a consequence, of the model fitted to the series, often crucially depend on the observation frequency. For instance, empirical studies generally find a stronger persistence (that is, α+β closer to 1) in GARCH(1, 1) models, when the frequency increases.

For a given asset, observed at different frequencies, a natural question is whether strong GARCH models at different frequencies are compatible. If the answer is positive, the class of GARCH models will be called stable by temporal aggregation. In this chapter, we consider, more generally, invariance properties of the class of GARCH processes with respect to time transformations frequently encountered in econometrics. It will be seen that, to obtain stability properties, a wider class of GARCH-type models, called weak GARCH and based on the L² structure of the squared returns, has to be introduced.

4.1 Temporal Aggregation of GARCH Processes

Temporal aggregation arises when the frequency of data generation is lower than that of the observations so that the underlying process is only partially observed. The time series resulting from temporal aggregation may of course have very different properties than the original time series. More formally, the temporal aggregation problem can be formulated as follows: given a process (X_t) and an integer m, what are the properties of the sampled process (X_mt) (that is, constructed from (X_t) by only keeping every mth observation)? Does the aggregated process (X_mt) belong to the same class of models as the original process (X_t)? If this holds for any (X_t) and any m, the class is said to be stable by temporal aggregation.

An elementary example of a model that is stable by temporal aggregation is obviously white noise (strong or weak): the independence (or noncorrelation) property is kept for the aggregated process, as well as the property of zero mean and fixed variance. On the other hand, ARMA models in the strong sense are generally not stable by temporal aggregation. It is only by relaxing the assumption of noise independence, that is, by considering the class of weak ARMA models, that temporal aggregation holds.

We shall see that, like many strong models (based on an iid white noise), GARCH models in the strong or semi-strong sense (that is, in the sense of Definition 2.1), are not stable by aggregation: a GARCH model at a given frequency is not compatible with a GARCH model at another frequency. As for ARMA models, temporal aggregation is obtained by enlarging the class of GARCH.

4.1.1 Nontemporal Aggregation of Strong Models

To show that temporal aggregation does not hold for GARCH models, it suffices to consider the ARCH(l) example. Let (_t) be the nonanticipative, second-order solution of the model:

The model satisfied by the even-numbered observations is easily obtained:

It follows that

because η_2t and η_2t−1 are independent of the variables involved in the conditioning. Thus, the process (_2t) is ARCH(l) in the semi-strong sense (Definition 2.1). It is a strong ARCH if the process defined by dividing _2t by its conditional standard deviation,

is iid. We have seen that and , but

If were a.s. constant, we would have α = 0 (no ARCH effect), or μ₄ = 1 (, a.s.), or, for some constant K,

the latter inequality implying that a.s., for another constant K*. By stationarity = K* a.s., for all t and would take only one value, leading us again to the case μ₄ = 1. This proves that the process (_t) is not iid, whence α > 0 (presence of ARCH), and μ₄ ≠ 1 (nondegenerate law of ). The process (_2t) is thus not strong GARCH, although (_t) is strong GARCH.

It can be shown that this property extends to any integer m (Exercise 4.1).

From this example, it might seem that strong GARCH processes aggregate in the class of semi-strong GARCH processes. We shall now see that this is not the case.

4.1.2 Nonaggregatloe In the Class of Semi-Strong GARCH Processes

Let (_t) denote the nonanticipative, second-order stationary solution of the strong ARCH(2) model:

under the same assumptions on (η_t) as before. In view of (2.4), the AR(2) representation satisfied by () is

(4.1)

where (v_t) is the strong innovation of (). Using the lag operator, this model is written as

where λ₁ and λ₂ are real positive numbers (such that λ₁ – λ₂ = α₁ and λ₁λ₂ = α₂). Multiplying this equation by (1 + λ₁L)(1 − λ₂L), we obtain

that is,

where ω* = ω(1+λ₁)(1+−₂), υ_t = v_t2 + (λ₁–λ₂)v_2t−1−λ₁λ₂v_2t−2 and Y_t = _2t. Observe that (υ_t) is an MA(1) process, such that

It follows that (υ_t) can be written as υ_t = u_t − θu_t−1, where (u_t) is a white noise and θ is a constant depending on λ₁ and λ₂. Finally, has the following ARMA(2, 1) representation:

(4.2)

The ARMA orders are compatible with a semi-strong GARCH(1, 2) model for (_2t)_t, with conditional variance:

If (_2t)_t were such a semi-strong GARCH(1, 2) process, the corresponding ARMA(2, 1) representation would then be

in view of (2.4). This equation is not compatible with (4.2), because of the sign of the coefficient of We can conclude that if (_t) is a strong ARCH(2), (_2t) is never a semi-strong GARCH.

4.2 Weak GARCH

The previous example shows that the square of a process obtained by temporal aggregation of a strong or semi-strong GARCH admits an ARMA representation. This leads us to the following definition.

Definition 4.1 (Weak GARCH process) A fourth-order stationary process (_t) is said to be a weak GARCH(r, p) if:

(i) (_t) is a white noise;

(ii) () admits an ARMA representation of the form

(4.3)

where (v_t) is the linear innovation of ().

Recall that the property of linear innovation entails that

From (2.4), semi-strong GARCH(p, q) processes satisfy, under the fourth-order stationarity condition, Definition 4.1 with r = max (p, q). The linear innovation coincides in this case with the strong innovation: v_t is thus uncorrelated with any variable of the past of _t (provided this correlation exists).

Remark 4.1 (Generality of the weak GARCH class) Let (X_t) denote a strictly stationary, purely nondeterministic process, admitting moments up to order 4. By the Wold theorem, (X_t) admits an MA(∞) representation. Suppose this representation is invertible and there exists an ARMA representation of the form

where (_t) is a weak white noise with variance σ² > 0, and the polynomials Ф (z) = 1 + ø₁z + … + ø_Pz^P and Ψ(z) = 1 + ψ₁z + … + ψ_Qz^Q have all their roots outside the unit disk and have no common root. Without loss of generality, suppose that ø_P ≠ 0 and ψ_Q ≠ 0 (by convention, ø₀ = ψ₀ = 1). The process (_t) can then be Interpreted as the linear innovation of (X_t). The process ()_t is clearly second-order stationary and purely nondeterministic. It follows that it admits an MA(∞) representation by the Wold theorem. If this representation is invertible, the process (_t) is a weak GARCH process.

The class of weak GARCH processes is not limited to processes obtained by temporal aggregation. Before returning to temporal aggregation, we conclude this section with further examples of weak GARCH processes.

Example 4.1 (GARCH with measurement error) Suppose that a GARCH process (_t) is observed with a measurement error W_t. We have

(4.4)

For simplicity, it can be assumed that the sequences (Z_t) and (W_t) are mutually independent, iid and centered, with variances 1 and respectively.

It can be shown (Exercise 4.3) that (_t) is a weak GARCH process of the form

where the β_i are different from the −b_i, unless σ_w = 0. It is worth noting that the AR part of this representation Is not affected by the presence of the perturbation W_t.

Statistical inference on GARCH with measurement errors is complicated because the likelihood cannot be written In explicit form. Methods using least squares, the Kalman filter or simulations have been suggested to estimate these models.

Example 4.2 (Quadratic GARCH) Consider the modification of the semi-strong GARCH model given by

(4.5)

where the constants b_i are positive. Let u_t = − . The u_t are nonautocorrelated, uncorrelated with any variable of the future (by the martingale difference assumption) and, by definition, with any variable of the past of _t. Rewrite the equation for as

where

(4.6)

It Is not difficult to verify that (υ_t) is an MA(max{p, q}) process (Exercise 4.4). It follows that (_t) is a weak GARCH(max{p, q}, max{p, q}) process.

Example 4.3 (Markov-switching GARCH) Markov-switching models (ARMA, GARCH) allow the coefficients to depend on a Markov chain, in order to take into account the changes of regime In the dynamics of the series. The chain being unobserved, these models are also referred to as hidden Markov models.

The simplest Markov-switching GARCH model is obtained when a single parameter ω is allowed to depend on the Markov chain. More precisely, let (Δ_t) be a Markov chain with state space 0, 1,…, K − 1. Suppose that this chain Is homogenous, stationary, Irreducible and aperiodic, and let p_ij = P[Δ_t = j|Δ_t−1 = i], for i, j = 0, 1,…, K − 1, be its transition probabilities. The model is given by

(4.7)

with

(4.8)

where (η_t) Is an iid (0, 1) process with finite fourth-order moment, the sequence (η_t) being independent of the sequence (Δ_t). Tedious computations show that (_t) is a weak GARCH(max{p, q] + K − 1, p+ K− 1) process of the form

(4.9)

where λ₁,…, λ_K−1 are the eigenvalues different from 1 of the matrix = (p_ji). The β_i generally do not have simple expressions as functions of the initial parameters, but can be numerically obtained from the first autocorrelations of the process () (Exercise 4.7).

Example 4.4 (Stochastic volatility model) An example of stochastic volatility model is given by

(4.10)

where (η_t and (υ_t) are iid (0, 1) sequences, with η_t independent of the υ_t−j, j ≥ 0. Note that the GARCH(1, 1) process Is obtained by taking and a = 0. Under the assumption d² + b² < 1, It can be shown (Exercise 4.5) that the autocovariance structure of () is characterized by

It follows that (_t) is a weak GARCH(1, 1) process with

(4.11)

where (u_t) is a weak white noise and β can be explicitly computed.

Example 4.5 (Contemporaneous aggregation of GARCH processes) It Is standard in finance to consider linear combinations of several series (for instance, to define portfolios). If these series are GARCH processes, is their linear combination also a GARCH process? To simplify the presentation, consider the sum of two GARCH(1, 1) processes, defined as the second-order stationary and nonanticipative solutions of

and suppose that the sequences (η_1t) and (η_2t) are Independent. Let _t = _1t + _2t. It is easy to see that (_t) is a white noise. We have, for for i ≠ j, because the processes (_1t) and (_2t) are Independent. Moreover, for h > 0,

because η_1t is independent of the other variables. It follows that, for h>0,

(4.12)

By formula (2.61), we deduce that

(4.13)

If f is a function defined on the integers, denote by Lf the function such that Lf(h) = f(h − 1), h > 0. We have (1 − βL)β^h = 0 for h > 0. Relation (4.13) shows that

It follows that () is a weak GARCH(2, 2) process of the form

where (u_t) is a noise. Since , we obtain ω = {1 − (α₁ + α₂)}ω₂ + {1 − (α₂ + β₂)}ω₁- Note that the orders obtained for the ARMA representation of are not necessarily the minimum ones. Indeed, if α₁ + β₁ = α₂ + β₂, then , from (4.13). Therefore, {1 − (α₁ + β₁)L}(h) = 0 if h > 1. Thus () is a weak GARCH(1, 1) process. This example can be generalized to higher-order GARCH models (Exercise 4.9).

Example 4.6 (β-ARCH process) Consider the conditionally heteroscedastic AR(1) model defined by

where (η_t is an lid (0, 1) symmetrically distributed sequence. A difference between this model, called β-ARCH, and the standard ARCH is that the conditional variance of X_t is specified as a function of X_t−1, not as a function of the noise.

Suppose β = 1 and let

We have

where . By expanding the squared term we obtain the representation

where . Note that the process (υ_t − ø²υ_t−1) is MA(1). Consequently, () is an ARMA(1, 1) process. Finally, the process (X_t) admits a weak AR(1)-GARCH(1, 1) representation.

4.3 Aggregation of Strong GARCH Processes in the Weak GARCH Class

We have seen that the class of semi-strong GARCH models (defined in terms of conditional moments) is not large enough to include all processes obtained by temporal aggregation of strong GARCH. In this section we show that the weak GARCH class of models is stable by temporal aggregation. Before dealing with the general case, we consider the GARCH(1, 1) model, for which the solution is more explicit.

Theorem 4.1 (Aggregation of the GARCH(1, 1) process) Let (_t) be a weak GARCH(1, 1) process. Then, for any integer m ≥ 1, the process (_mt) is also a weak GARCH(1, 1) process. The parameters of the ARMA representations

are related by the relations

Proof. First note that, () being stationary by assumption, and (v_t) being its linear innovation, a is strictly less than 1 in absolute value. Now, if (_t) is a white noise, (_mt) is also a white noise. By successive substitutions we obtain

(4.14)

where υ_t = v_t + (a − b[v_t−1 + av_t−2 + … + a^m−2v_t−m−1] −a^m−1bv_t−m. Because (v_t) is a noise, we have

Hence, (υ_mt)_t is an MA(1) process, from which it follows that (_mt) is an ARMA(1, 1) process. The constant term and the AR coefficient of this representation appear directly in (4.14), whereas the MA coefficient is obtained as the solution, of absolute value less than 1, of

which, after simplification, gives the claimed formula.                  

Note, in particular, that the aggregate of an ARCH(l) process is another ARCH(l): b = 0 ⇒ b(_m) = 0.

It is also worth noting that a^m tends to 0 when m tends to infinity, thus a_(m) and b_(m) also tend to 0. In other words, the conditional heteroscedasticity tends to vanish by temporal aggregation of GARCH processes. This conforms to the empirical observation that low-frequency series (weekly, monthly) display less ARCH effect than daily series, for instance.

The previous result can be straightforwardly extended to the GARCH(1, p) case. Denote by [x] the integer part of x.

Theorem 4.2 (Aggregation of the GARCH(1, p) process) Let (_t) be a weak GARCH(1, p) process. Then, for any integer m > 1, the process (_mt) is a weak GARCH process.

Proof. In the proof of Theorem 4.1, equation (4.14) remains valid subject to a modification of the definition of υ_t. Introduce the lag polynomial Q(L) = 1 − b₁L − … − b_pL^p. We have υ_t = Q(L))[1 + aL + … + a^m−1L^m−1]v_t. Thus, because (v_t) is a noise,

Hence, (υ_mt) is an MA process, and the conclusion follows.                  

It can be seen from (4.14) that the constant term and the AR coefficient of the ARMA representation of (_mt) are the same as in Theorem 4.1. The coefficients of the MA part can be determined through the first autocovariances of the process (υ_mt).

Note that temporal aggregation always entails a reduction of the MA order in the ARMA representation (except when p = 1, for which it remains equal to 1) - all the more so as m increases. Let us now turn to the general case.

Theorem 4.3 (Aggregation of the GARCH(r, p) process) Let (_t) be a weak GARCH(r, p) process. Then for any integer m > 1, the process (_mt) is a weak GARCH process.

Proof. Denote by λ_i (1 ≤ i ≤ r) the inverses of the complex roots of the AR polynomial of the ARMA representation for (). Write model (4.3) in the form

where μ = E and Applying the operator to this equation we get

Consider now the process defined by . We have the model

with the convention that Lv_mt = v_mt−1. Observe that v_t = f(v_mt, v_mt−1,… v_mt−r(m−1)−p). This suffices to show that the process (υ_t) is a moving average. The largest index k for which υ_t and υ_t−k have a common term v_i is such that r(m − 1) + p − m < mk ≤ r(m − 1) + p. Thus , which gives the order of the moving average part, and subsequently the orders of the ARMA representation for .                  

This proof suggests the following scheme for deriving the exact form of the ARMA representation:

(i) The AR coefficients are deduced from the roots of the AR polynomial; but in the previous proof we saw that these roots, for the aggregate process, are the mth powers of the roots of the initial AR polynomial.

(ii) The constant term immediately follows from the AR coefficients and the expectation of the process: .

(iii) The derivation of the MA part is more tedious and requires computing the first r + autocovariances of the process ; these autocovariances follow from the ARMA representation for .

An alternative method involves multiplying the ARMA representation of (), written in polynomial form, by a well-chosen lag polynomial so as to directly obtain the AR part of the ARMA representation of . Let denote the AR polynomial of the representation of (). The AR polynomial of the representation of is given by

Example 4.7 (Computation of a weak GARCH representation) Consider the GARCH (2, 1) process defined by

and let us derive the weak GARCH representation of the process (_2t).

The ARMA representation of () is written as

that is,

Multiplying this equation by (1 + 0.5L)(1 − 0.2L), we obtain

Set υ_t = (1 + 0.5L)(1 − 0.2L)²v_t. The process (υ_2t) is MA(1), υ_2t = u_t − θu_t−1, where θ = 0.156 is the solution, with absolute value less than 1, of the equation

The weak GARCH(2, 1) representation of the process (_2t) is then

Observe that the sign of the coefficient of is not compatible with a strong or semi-strong GARCH.

4.4 Bibliographical Notes

The main results concerning the temporal aggregation of GARCH models were established by Drost and Nijman (1993). It should be noted that our definition of weak GARCH models is not exactly the same as theirs: in Definition 4.1, the noise (v_t) is not the strong innovation of (), but only the linear one. Drost and Werker (1996) introduced the notion of the continuous-time GARCH process and deduced the corresponding weak GARCH models at the different frequencies (see also Drost, Nijman and Werker, 1998). The problem of the contemporaneous aggregation of independent GARCH processes was studied by Nijman and Sentana (1996).

Model (4.5) belongs to the class of quadratic ARCH models introduced by Sentana (1995). GARCH models observed with measurement errors are dealt with by Harvey, Ruiz and Sentana (1992), Gouriéroux, Monfort and Renault (1993) and King, Sentana and Wadhwani (1994). Example 4.4 belongs to the class of stochastic autoregressive volatility (SARV) models introduced by Andersen (1994). The β-ARCH model was introduced by Diebolt and Guégan (1991).

Markov-switching ARMA(p,q) models were introduced by Hamilton (1989). Pagan and Schwert (1990) considered a variant of such models for modeling the conditional variance of financial series. Model (4.7) was studied by Cai (1994) and Dueker (1997); see also Hamilton and Susmel (1994). The probabilistic properties of the Markov-switching GARCH models were studied by Francq, Roussignol and Zakoïan (2001). The existence of ARMA representations for powers of (as in (4.9)) was established by Francq and Zakoïan (2005), and econometric applications of this property were studied by Francq and Zakoïan (2008).

The examples of weak GARCH models discussed in this chapter were analyzed by Francq and Zakoïan (2000), where a two-step least-squares method of estimation of weak ARMA-GARCH was also proposed.

4.5 Exercises

4.1 (Aggregate strong ARCH(1) process)

Show that the process (_mt) obtained by temporal aggregation of a strong ARCH(l) process

(_t) is a semi-strong ARCH.

4.2 (Aggregate weak GARCH (1, 2) process)

State the equivalent of Theorem 4.1 for a GARCH(1, 2) process.

4.3 (GARCH with measurement error)

Show that in Example 4.1 we have , for all h > 0. Use this result

to deduce the weak GARCH representation of (_t).

4.4 (Quadratic ARCH)

Verify that the process (υ_t) defined in (4.6) is an MA(max{p, q}) process.

4.5 (Stochastic volatility model)

In model (4.10), the volatility equation can be written as

where A(υ_t) = c + aυ_t, B(υ_t) = d + bυ_t. Suppose that d² + b² < 1.

1. Show that

2. Express as a function of and of the process (υ_t) and deduce that, for all h > 0,

3. Using the second-order stationarity of (), compute E() and Var() and determine for h > 0.

4. Conclude that (4.11) holds and explain how to obtain β.

4.6 (Independent-switching model)

Consider model (4.7)–(4.8) in the particular case where the chain (Δ) is an iid process (that is, when p(i, j) does not depend on i, for any (i, j)). Give a more explicit form for the weak GARCH representation (4.9).

4.7 (A two-regime Markov-switching model without ARCH coefficients)

In model (4.7)–(4.8), suppose that p = q = 0 (that is, = ω(Δ_t)) and take for (Δ_t) a two-state Markov chain with 0 < p01 < 1, 0 < p10 < 1. Let π(i) = P(Δ_t = i). Denote by p^(k)(i, j) the k-step transition probabilities, that is, the entries of ^k. Set λ = p(1, 1) + p(2,2)−1.

1. Compute E.

2. Show that, for i, j = 1,2,

(4.15)

3. Deduce that, for k > 0,

4. Compute Var().

5. Deduce that has an ARMA(1, 1) representation and determine the AR coefficient.

6. Simplify this representation in the case p01 + P10 = 1.

7. Determine numerically the ARMA(1, 1) representation for the model:

where η_t ~ N(0,1).

4.8 (Bilinear model)

Let _t = η_tη_t−1, where (η_t) is a strong white noise with unit variance such that E() < ∞.

1. Show that the process (_t) is a weak GARCH.

2. Show that the process ( - 1) is a weak ARMA-GARCH.

4.9 (Contemporaneous aggregation)

Using the method of the proof of Theorem 4.3, generalize Example 4.5 by considering the contemporaneous aggregation of independent strong GARCH processes of any orders.