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 (Xt) and an integer m, what are the properties of the sampled process (Xmt) (that is, constructed from (Xt) by only keeping every mth observation)? Does the aggregated process (Xmt) belong to the same class of models as the original process (Xt)? If this holds for any (Xt) 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 (2208_fmtt) be the nonanticipative, second-order solution of the model:

c04ue001_fmt

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

c04ue002_fmt

It follows that

c04ue003_fmt

because η2t and η2t−1 are independent of the variables involved in the conditioning. Thus, the process (2208_fmt2t) is ARCH(l) in the semi-strong sense (Definition 2.1). It is a strong ARCH if the process defined by dividing 2208_fmt2t by its conditional standard deviation,

c04ue004_fmt

is iid. We have seen that c04-ie80006_fmt and c04-ie80001_fmt, but

c04ue005_fmt

If c04-ie80002_fmt were a.s. constant, we would have α = 0 (no ARCH effect), or μ4 = 1 (c04-ie80003_fmt, a.s.), or, for some constant K,

c04ue006_fmt

the latter inequality implying that c04-ie80004_fmt a.s., for another constant K*. By stationarity 2208-2t_fmt = K* a.s., for all t and c04-ie80005_fmt would take only one value, leading us again to the case μ4 = 1. This proves that the process (small-eta-tilde_fmtt) is not iid, whence α > 0 (presence of ARCH), and μ4 ≠ 1 (nondegenerate law of small-eta-2t_fmt). The process (2208_fmt2t) is thus not strong GARCH, although (2208_fmtt) 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 (2208_fmtt) denote the nonanticipative, second-order stationary solution of the strong ARCH(2) model:

c04ue007_fmt

under the same assumptions on (ηt) as before. In view of (2.4), the AR(2) representation satisfied by (2208-2t_fmt) is

(4.1) c04e001_fmt

where (vt) is the strong innovation of (2208-2t_fmt). Using the lag operator, this model is written as

c04ue008_fmt

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

c04ue009_fmt

that is,

c04ue010_fmt

where ω* = ω(1+λ1)(1+−2), υt = vt2 + (λ1–λ2)v2t−1−λ1λ2v2t−2 and Yt = 2208_fmt2t. Observe that (υt) is an MA(1) process, such that

c04ue011_fmt

It follows that (υt) can be written as υt = ut − θut−1, where (ut) is a white noise and θ is a constant depending on λ1 and λ2. Finally, c04-ie81001_fmt has the following ARMA(2, 1) representation:

(4.2) c04e002_fmt

The ARMA orders are compatible with a semi-strong GARCH(1, 2) model for (2208_fmt2t)t, with conditional variance:

c04ue012_fmt

If (2208_fmt2t)t were such a semi-strong GARCH(1, 2) process, the corresponding ARMA(2, 1) representation would then be

c04ue013_fmt

in view of (2.4). This equation is not compatible with (4.2), because of the sign of the coefficient of c04-ie81002_fmt We can conclude that if (2208_fmtt) is a strong ARCH(2), (2208_fmt2t) 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 (2208_fmtt) is said to be a weak GARCH(r, p) if:

(i) (2208_fmtt) is a white noise;

(ii) (2208-2t_fmt) admits an ARMA representation of the form

(4.3) c04e003_fmt

where (vt) is the linear innovation of (2208-2t_fmt).

Recall that the property of linear innovation entails that

c04ue014_fmt

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: vt is thus uncorrelated with any variable of the past of 2208_fmtt (provided this correlation exists).

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

c04ue015_fmt

where (2208_fmtt) is a weak white noise with variance σ2 > 0, and the polynomials Ф (z) = 1 + ø1z + … + øPzP and Ψ(z) = 1 + ψ1z + … + ψQzQ have all their roots outside the unit disk and have no common root. Without loss of generality, suppose that øP ≠ 0 and ψQ ≠ 0 (by convention, ø0 = ψ0 = 1). The process (2208_fmtt) can then be Interpreted as the linear innovation of (Xt). The process (2208-2t_fmt)t2208_fmt2124_fmt 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 (2208_fmtt) 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 (2208_fmtt) is observed with a measurement error Wt. We have

(4.4) c04e004_fmt

For simplicity, it can be assumed that the sequences (Zt) and (Wt) are mutually independent, iid and centered, with variances 1 and c04-ie83001_fmt respectively.

It can be shown (Exercise 4.3) that (2208_fmtt) is a weak GARCH process of the form

c04ue016_fmt

where the βi are different from the −bi, unless σw = 0. It is worth noting that the AR part of this representation Is not affected by the presence of the perturbation Wt.

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) c04e005_fmt

where the constants bi are positive. Let ut = 2208-2t_fmtsigma-2t_fmt. The ut are nonautocorrelated, uncorrelated with any variable of the future (by the martingale difference assumption) and, by definition, with any variable of the past of 2208_fmtt. Rewrite the equation for sigma-2t_fmt as

c04ue017_fmt

where

(4.6) c04e006_fmt

It Is not difficult to verify that (υt) is an MA(max{p, q}) process (Exercise 4.4). It follows that (2208_fmtt) 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 (Δt) be a Markov chain with state space 0, 1,…, K − 1. Suppose that this chain Is homogenous, stationary, Irreducible and aperiodic, and let pij = Pt = jt−1 = i], for i, j = 0, 1,…, K − 1, be its transition probabilities. The model is given by

(4.7) c04e007_fmt

with

(4.8) c04e008_fmt

where (ηt) Is an iid (0, 1) process with finite fourth-order moment, the sequence (ηt) being independent of the sequence (Δt). Tedious computations show that (2208_fmtt) is a weak GARCH(max{p, q] + K − 1, p+ K− 1) process of the form

(4.9) c04e009_fmt

where λ1,…, λK−1 are the eigenvalues different from 1 of the matrix 2119_fmt = (pji). The βi generally do not have simple expressions as functions of the initial parameters, but can be numerically obtained from the first autocorrelations of the process (2208-2t_fmt) (Exercise 4.7).

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

(4.10) c04e010_fmt

where (ηt and (υt) are iid (0, 1) sequences, with ηt independent of the υt−j, j ≥ 0. Note that the GARCH(1, 1) process Is obtained by taking c04-ie84001_fmt and a = 0. Under the assumption d2 + b2 < 1, It can be shown (Exercise 4.5) that the autocovariance structure of (2208-2t_fmt) is characterized by

c04ue018_fmt

It follows that (2208_fmtt) is a weak GARCH(1, 1) process with

(4.11) c04e011_fmt

where (ut) 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

c04ue019_fmt

and suppose that the sequences (η1t) and (η2t) are Independent. Let 2208_fmtt = 2208_fmt1t + 2208_fmt2t. It is easy to see that (2208_fmtt) is a white noise. We have, for c04-ie84002_fmt for i ≠ j, because the processes (2208_fmt1t) and (2208_fmt2t) are Independent. Moreover, for h > 0,

c04ue020_fmt

because η1t is independent of the other variables. It follows that, for h>0,

(4.12) c04e012_fmt

By formula (2.61), we deduce that

(4.13) c04e013_fmt

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 − βLh = 0 for h > 0. Relation (4.13) shows that

c04ue021_fmt

It follows that (2208-2t_fmt) is a weak GARCH(2, 2) process of the form

c04ue022_fmt

where (ut) is a noise. Since c04-ie85001_fmt, we obtain ω = {1 − (α1 + α2)}ω2 + {1 − (α2 + β2)}ω1- Note that the orders obtained for the ARMA representation of 2208-2t_fmt are not necessarily the minimum ones. Indeed, if α1 + β1 = α2 + β2, then c04-ie85002_fmt, from (4.13). Therefore, {1 − (α1 + β1)L}Y_e-2_fmt(h) = 0 if h > 1. Thus (2208-2t_fmt) 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

c04ue023_fmt

where (ηt 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 Xt is specified as a function of Xt−1, not as a function of the noise.

Suppose β = 1 and let

c04ue024_fmt

We have

c04ue025_fmt

where c04-ie85003_fmt. By expanding the squared term we obtain the representation

c04ue026_fmt

where c04-ie85004_fmt. Note that the process (υt − ø2υt−1) is MA(1). Consequently, (2208-2t_fmt) is an ARMA(1, 1) process. Finally, the process (Xt) 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 (2208_fmtt) be a weak GARCH(1, 1) process. Then, for any integer m ≥ 1, the process (2208_fmtmt) is also a weak GARCH(1, 1) process. The parameters of the ARMA representations

c04ue027_fmt

are related by the relations

c04ue028_fmt

Proof. First note that, (2208-2t_fmt) being stationary by assumption, and (vt) being its linear innovation, a is strictly less than 1 in absolute value. Now, if (2208_fmtt) is a white noise, (2208_fmtmt) is also a white noise. By successive substitutions we obtain

(4.14) c04e014_fmt

where υt = vt + (a − b[vt−1 + avt−2 + … + am−2vtm−1] −am−1bvtm. Because (vt) is a noise, we have

c04ue029_fmt

Hence, (υmt)t2208_fmt2124_fmt is an MA(1) process, from which it follows that (2208_fmtmt) 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

c04ue030_fmt

which, after simplification, gives the claimed formula.                  Box_fmt

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 am 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 (2208_fmtt) be a weak GARCH(1, p) process. Then, for any integer m > 1, the process (2208_fmtmt) is a weak GARCH c04-ie86001_fmt process.

Proof. In the proof of Theorem 4.1, equation (4.14) remains valid subject to a modification of the definition of υt. Introduce the lag polynomial Q(L) = 1 − b1L − … − bpLp. We have υt = Q(L))[1 + aL + … + am−1Lm−1]vt. Thus, because (vt) is a noise,

c04ue031_fmt

Hence, (υmt) is an MAc04-ie86002_fmt process, and the conclusion follows.                  Box_fmt

It can be seen from (4.14) that the constant term and the AR coefficient of the ARMA representation of (2208_fmtmt) are the same as in Theorem 4.1. The coefficients of the MA part can be determined through the first c04-ie86003_fmt autocovariances of the process (υmt).

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 (2208_fmtt) be a weak GARCH(r, p) process. Then for any integer m > 1, the process (2208_fmtmt) is a weak GARCHc04-ie86001_fmt process.

Proof. Denote by λi (1 ≤ ir) the inverses of the complex roots of the AR polynomial of the ARMA representation for (2208-2t_fmt). Write model (4.3) in the form

c04ue032_fmt

where μ = E 2208-2t_fmt and c04-ie87002_fmt Applying the operator c04-ie87003_fmt to this equation we get

c04ue033_fmt

Consider now the process c04-ie87004_fmt defined by c04-ie87005_fmt. We have the model

c04ue034_fmt

with the convention that Lvmt = vmt−1. Observe that vt = f(vmt, vmt−1,… vmt−r(m−1)−p). This suffices to show that the process (υt) is a moving average. The largest index k for which υt and υt−k have a common term vi is such that r(m − 1) + pm < mkr(m − 1) + p. Thus c04-ie87006_fmt, which gives the order of the moving average part, and subsequently the orders of the ARMA representation for c04-ie87007_fmt.                  Box_fmt

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: c04-ie87008_fmt.

(iii) The derivation of the MA part is more tedious and requires computing the first r + c04-ie87009_fmt autocovariances of the process c04-ie87010_fmt; these autocovariances follow from the ARMA representation for 2208-2t_fmt.

An alternative method involves multiplying the ARMA representation of (2208-2t_fmt), written in polynomial form, by a well-chosen lag polynomial so as to directly obtain the AR part of the ARMA representation of c04-ie87011_fmt. Let c04-ie87012_fmt denote the AR polynomial of the representation of (2208-2t_fmt). The AR polynomial of the representation of c04-ie87013_fmt is given by

c04ue035_fmt

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

c04ue036_fmt

and let us derive the weak GARCH representation of the process (2208_fmt2t).

The ARMA representation of (2208-2t_fmt) is written as

c04ue037_fmt

that is,

c04ue038_fmt

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

c04ue039_fmt

Set υt = (1 + 0.5L)(1 − 0.2L)2vt. The process (υ2t) is MA(1), υ2t = ut − θut−1, where θ = 0.156 is the solution, with absolute value less than 1, of the equation

c04ue040_fmt

The weak GARCH(2, 1) representation of the process (2208_fmt2t) is then

c04ue041_fmt

Observe that the sign of the coefficient of c04-ie88001_fmt 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 (vt) is not the strong innovation of (2208-2t_fmt), 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 2208-2t_fmt (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 (2208_fmtmt) obtained by temporal aggregation of a strong ARCH(l) process

(2208_fmtt) 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 c04-ie89001_fmt, for all h > 0. Use this result

to deduce the weak GARCH representation of (2208_fmtt).

4.4 (Quadratic ARCH)

Verify that the process (υt) 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

c04ue042_fmt

where At) = c + aυt, Bt) = d + bυt. Suppose that d2 + b2 < 1.

1. Show that

c04ue043_fmt

2. Express c04-ie89002_fmt as a function of c04-ie89003_fmt and of the process (υt) and deduce that, for all h > 0,

c04ue044_fmt

3. Using the second-order stationarity of (sigma-2t_fmt), compute E(sigma-2t_fmt) and Var(sigma-2t_fmt) and determine c04-ie89004_fmt 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, sigma-2t_fmt = ω(Δt)) and take for (Δt) a two-state Markov chain with 0 < p01 < 1, 0 < p10 < 1. Let π(i) = Pt = i). Denote by p(k)(i, j) the k-step transition probabilities, that is, the entries of 2119_fmtk. Set λ = p(1, 1) + p(2,2)−1.

1. Compute E2208-2t_fmt.

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

(4.15) c04e015_fmt

3. Deduce that, for k > 0,

c04ue045_fmt

4. Compute Var(2208-2t_fmt).

5. Deduce that 2208-2t_fmt 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:

c04ue046_fmt

where ηt ~ N(0,1).

4.8 (Bilinear model)

Let 2208_fmtt = ηtηt−1, where (ηt) is a strong white noise with unit variance such that E(small-eta--8t_fmt) < ∞.

1. Show that the process (2208_fmtt) is a weak GARCH.

2. Show that the process (2208-2t_fmt - 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.

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