Properties of Simulated Paths

Contrary to standard time series models (ARMA), the GARCH structure allows the magnitude of the noise ε _t to be a function of its past values. Thus, periods with high‐volatility level (corresponding to large values of ) will be followed by periods where the fluctuations have a smaller amplitude. Figures 2.1–2.7 illustrate the volatility clustering for simulated GARCH models. Large absolute values are not uniformly distributed on the whole period, but tend to cluster. We will see that all these trajectories correspond to strictly stationary processes which, except for the ARCH(1) models of Figures 2.3–2.5, are also second‐order stationary. Even if the absolute values can be extremely large, these processes are not explosive, as can be seen from these figures. Higher values of α (theoretically, α > 3.56 for the distribution, as will be established below) lead to explosive paths. Figures 2.6 and 2.7, corresponding to GARCH(1, 1) models, have been obtained with the same simulated sequence (η _t ). As we will see, permuting α and β does not modify the variance of the process but has an effect on the higher‐order moments. For instance the simulated process of Figure  2.7, with α = 0.7 and β = 0.2, does not admit a fourth‐order moment, in contrast to the process of Figure  2.6. This is reflected by the presence of larger absolute values in Figure  2.7. The two processes are also different in terms of persistence of shocks: when β approaches 1, a shock on the volatility has a persistent effect. On the other hand, when α is large, sudden volatility variations can be observed in response to shocks.

2.2.1 The GARCH(1,1) Case

When p = q = 1, model ( 2.5) has the form

2.7

with ω > 0, α ≥ 0,  β ≥ 0. Let a(z) = αz ² + β .

The next result shows that non‐stationary GARCH processes are explosive.

Figure 2.8 shows the zones of strict and second‐order stationarity for the strong GARCH(1, 1) model when . Note that the distribution of η _t only matters for the strict stationarity. As noted above, the frontier of the strict stationarity zone corresponds to a random walk (for the process log(h _t  − ω)). A similar interpretation holds for the second‐order stationarity zone. If α + β = 1 we have

Thus, since the last term in this equality is centred and uncorrelated with any variable belonging to the past of h _t − 1 , the process (h _t ) is a random walk. The corresponding GARCH process is called integrated GARCH (or IGARCH(1, 1)) and will be studied later: it is strictly stationary, has an infinite variance, and a conditional variance which is a random walk (with a positive drift).

2.2.2 The General Case

In the general case of a strong GARCH(p, q) process, the following vector representation will be useful. We have

2.16

where

and

2.17

is a (p + q) × (p + q) matrix. In the ARCH( q ) case, reduces to and its q − 1 first past values, and A _t to the upper‐left block of the above matrix. Equation (2.16) defines a first‐order vector autoregressive model, with positive and iid matrix coefficients. The distribution of conditional on its infinite past coincides with its distribution conditional on z _t − 1 only, which means that is a Markov process. Model ( 2.16) is thus called the Markov representation of the GARCH(p, q) model. Iterating Eq. ( 2.16) yields

2.18

provided that the series exists a.s.. Finding conditions ensuring the existence of this series is the object of what follows. Notice that the existence of the right‐hand vector in (2.18) does not ensure that its components are positive. One sufficient condition for

2.19

in the sense that all the components of this vector are strictly positive (but possibly infinite), is that

2.20

This condition is very simple to use but may not be necessary, as we will see in Section 2.3.2.

Strict Stationarity

The main tool for studying strict stationarity is the concept of the top Lyapunov exponent. Let A be a (p + q) × (p + q) matrix. The spectral radius of A , denoted by ρ(A), is defined as the greatest modulus of its eigenvalues. Let ‖ ⋅ ‖ denote any norm on the space of the (p + q) × (p + q) matrices. We have the following algebra result:

2.21

(Exercise 2.3). This property has the following extension to random matrices.

As for ARMA models, we are mostly interested in the non‐anticipative solutions (ε _t ) to model ( 2.5), that is, those for which ε _t belongs to the σ ‐field generated by {η _t , η _t − 1, …}.

We now give two illustrations allowing us to obtain more explicit stationarity conditions than in the theorem.

Figure 2.9, constructed from these simulations, gives a more precise idea of the strict stationarity region for an ARCH(2) process. We shall establish in Corollary 2.3 a result showing that any strictly stationary GARCH process admits small‐order moments. We begin with two lemmas which are of independent interest.

The following result, which is stated for any sequence of positive iid matrices, provides another characterisation of the strict stationarity of GARCH models.

Using Lemma 2.3 and Corollary 2.3 together, it can be seen that for s ∈ (0, 1],

2.34

The converse is generally true. For instance, we have for s ∈ (0, 1],

2.35

(Exercise 2.15).

IGARCH(p, q) Processes

When

the model is called an integrated GARCH(p, q) or IGARCH(p, q) model (see Engle and Bollerslev 1986). This name is justified by the existence of a unit root in the autoregressive part of representation ( 2.4) and is introduced by analogy with the integrated ARMA models, or ARIMA. However, this analogy can be misleading: there exists no (strict or second‐order) stationary solution of an ARIMA model, whereas an IGARCH model admits a strictly stationary solution under very general conditions. In the univariate case ( p = q = 1), the latter property is easily shown.

This property extends to the general case under slightly more restrictive conditions on the law of η _t .

Note that this strictly stationary solution has an infinite variance in view of Theorem 2.5.

2.3.1 Existence Conditions

The existence of a stationary ARCH(∞) process requires assumptions on the sequences (φ _i ) and (η _t ). The following result gives an existence condition.

Equality ( 2.42) is called a Volterra expansion (see Priestley 1988). It shows in particular that, under the conditions of the theorem, if φ ₀ = 0, the unique strictly stationary and non‐anticipative solution of the ARCH(∞) model is the identically null sequence. An application of Theorem 2.6, obtained for s = 1, is that the condition

ensures the existence of a second‐order stationary ARCH(∞) process. If

it can be shown (see Giraitis, Kokoszka, and Leipus 2000a) that , for all h and

2.45

The fact that the squares have positive autocovariances will be verified later in the GARCH case (see Section 2.5). In contrast to GARCH processes, for which they decrease at exponential rate, the autocovariances of the squares of ARCH(∞) can decrease at the rate h ^−γ with γ > 1 arbitrarily close to 1. A strictly stationary process of the form ( 2.40) such that

is called integrated ARCH(∞), or IARCH(∞). Notice that an IARCH(∞) process has infinite variance. Indeed, if , then, by ( 2.40), σ ² = φ ₀ + σ ² , which is impossible. From Theorem 2.8, the strictly stationary solutions of IGARCH models (see Corollary 2.5) admit IARCH(∞) representations. The next result provides a condition for the existence of IARCH(∞) processes.

Other integrated processes are the long‐memory ARCH, for which the rate of decrease of the φ _i is not geometric.

2.3.2 ARCH(∞) Representation of a GARCH

It is sometimes useful to consider the ARCH(∞) representation of a GARCH(p, q) process. For instance, this representation allows the conditional variance of ε _t to be written explicitly as a function of its infinite past. It also allows the positivity condition (2.20) on the coefficients to be weakened. Let us first consider the GARCH(1, 1) model. If β < 1, we have

2.46

In this case, we have

The condition A _s μ _2s  < 1 thus takes the form

For example, if α + β < 1 this condition is satisfied for s = 1. However, second‐order stationarity is not necessary for the validity of (2.46). Indeed, if (ε _t ) denotes the strictly stationary and non‐anticipative solution of the GARCH(1, 1) model, then, for any q ≥ 1,

2.47

By Corollary 2.3 there exists s ∈ (0, 1[ such that . It follows that So converges a.s. to 0 and, by letting q go to infinity in (2.47), we get ( 2.46). More generally, we have the following property.

The ARCH(∞) representation can be used to weaken the condition ( 2.20) imposed to ensure the positivity of . Consider the GARCH(p, q) model with ω > 0, without any a priori positivity constraint on the coefficients α _i and β _j , and assuming that the roots of the polynomial ℬ(z) = 1 − β ₁ z − ⋯ − β _p z ^p have moduli strictly greater than 1. The coefficients φ _i introduced in ( 2.48) are then well defined, and under the assumption

2.49

we have

Indeed, φ ₀ > 0 because otherwise B(1) ≤ 0, which would imply the existence of a root inside the unit circle since ℬ(0) = 1. Moreover, the proofs of the sufficient parts of Theorem 2.4, Lemma 2.3, and Corollary 2.3 do not use the positivity of the coefficients of the matrices A _t . It follows that if the top Lyapunov exponent of the sequence (A _t ) is such that γ < 0, the variable is a.s. finite‐valued. To summarise, the conditions

imply that there exists a strictly stationary and non‐anticipative solution to model ( 2.5). The condition (2.49) is not generally simple to use, however, because it implies an infinity of constraints on the coefficients α _i and β _j . In the ARCH( q ) case, it reduces to the condition ( 2.20), that is, α _i  ≥ 0, i = 1, …, q . Similarly, in the GARCH(1, 1) case, it is necessary to have α ₁ ≥ 0 and β ₁ ≥ 0. However, for p ≥ 1 and q > 1, the condition ( 2.20) can be weakened (Exercise 2.16).

2.3.3 Long‐Memory ARCH

The introduction of long memory into the volatility can be motivated by the observation that the empirical autocorrelations of the squares, or of the absolute values, of financial series decay very slowly in general (see for example Table  2.1). We shall see that it is possible to reproduce this property by introducing ARCH(∞) processes, with a sufficiently slow decay of the modulus of the coefficients φ _i . ⁸ A process (X _t ) is said to have long memory if it is second‐order stationary and satisfies, for h → ∞,

2.50

and K is a non‐zero constant. An alternative definition relies on distinguishing ‘intermediate‐memory’ processes for which d < 0 and thus , and ‘long‐memory’ processes for which d ∈ (0, 1/2[ and thus (see Brockwell and Davis 1991, p. 520). The autocorrelations of an ARMA process decrease at exponential rate when the lag increases. The need for processes with a slower autocovariance decay leads to the introduction of the fractionary ARIMA models. These models are defined through the fractionary difference operator

Denoting by π _j the coefficient of B ^j in this sum, it can be shown that π _j ∼Kj ^{−d − 1} when j → ∞, where K is a constant depending on d . An ARIMA(p, d, q) process with d ∈ ( − 0.5, 0.5) is defined as a stationary solution of

where ε _t is a white noise, and ψ and θ are polynomials of degrees p and q , respectively. If the roots of these polynomials are all outside the unit disk, the unique stationary and purely deterministic solution is causal, invertible and its covariances satisfy condition (2.50) (see Brockwell and Davis 1991, Theorem 13.2.2.). By analogy with the ARIMA models, the class of FIGARCH(p, d, q) processes is defined by the equations

2.51

where ψ and θ are polynomials of degrees p and q , respectively, such that ψ(0) = θ(0) = 1, the roots of ψ have moduli strictly greater than 1 and φ _i  ≥ 0, where the φ _i are defined by

We have φ _i ∼Ki ^{−d − 1} , where K is a positive constant, when i → ∞ and . The process introduced in (2.51) is thus IARCH(∞), provided it exists. Note that existence of this process cannot be obtained by Theorem 2.7 because, for the FIGARCH model, the φ _i decrease more slowly than the geometric rate. The following result, which is a consequence of Theorem 2.6, and whose proof is the subject of Exercise 2.22, provides another sufficient condition for the existence of IARCH(∞) processes.

This result can be used to prove the existence of FIGARCH(p,d,q) processes for d ∈ (0,1) sufficiently close to 1, if the distribution of is assumed to be non‐degenerate (hence, ); see Douc, Roueff, and Soulier (2008). The FIGARCH process of Corollary 2.6 does not admit a finite second‐order moment. Its square is thus not a long‐memory process in the sense of definition ( 2.50). More generally, it can be shown that the squares of a ARCH(∞) processes do not have the long‐memory property. This motivated the introduction of an alternative class, called linear ARCH (LARCH) and defined by

2.53

Under appropriate conditions, this model is compatible with the long‐memory property for .

2.4.1 Even‐Order Moments

We are interested in finding existence conditions for the moments of order 2m , where m is any positive integer. ⁹ Let ⊗ denote the tensor product, or Kronecker product, and recall that it is defined as follows: for any matrices A = (a _ij ) and B , we have A ⊗ B = (a _ij B). For any matrix A , let A ^⊗m  = A ⊗ ⋯ ⊗ A . We have the following result.

This example shows that for a non‐trivial GARCH process, that is, when the α _i and β _j are not all equal to zero, the moments cannot exist for any order.

2.4.2 Kurtosis

An easy way to measure the size of distribution tails is to use the kurtosis coefficient. This coefficient is defined, for a centred (zero‐mean) distribution, as the ratio of the fourth‐order moment, which is assumed to exist, to the squared second‐order moment. This coefficient is equal to 3 for a normal distribution, this value serving as a gauge for the other distributions. In the case of GARCH processes, it is interesting to note the difference between the tails of the marginal and conditional distributions. For a strictly stationary solution, (ε _t ) of the GARCH(p, q) model defined by ( 2.5), the conditional moments of order k are proportional to :

The kurtosis coefficient of this conditional distribution is thus constant and equal to the kurtosis coefficient of η _t . For a general process of the form

where σ _t is a measurable function of the past of ε _t , η _t is independent of this past and (η _t ) is iid centred, the kurtosis coefficient of the stationary marginal distribution is equal, provided that it exists, to

where denotes the kurtosis coefficient of (η _t ). It can thus be seen that the tails of the marginal distribution of (ε _t ) are fatter when the variance of is large relative to the squared expectation. The minimum (corresponding to the absence of ARCH effects) is given by the kurtosis coefficient of (η _t ),

with equality if and only if is a.s. constant. In the GARCH(1, 1) case we thus have, from the previous calculations,

2.60

The excess kurtosis coefficients of ε _t and η _t , relative to the normal distribution, are related by

The excess kurtosis of ε _t increases with that of η _t and when the GARCH coefficients approach the zone of non‐existence of the fourth‐order moment. Notice the asymmetry between the GARCH coefficients in the excess kurtosis formula. For the general GARCH(p, q), we have the following result.

It will be seen in Chapter 7 that the Gaussian quasi‐maximum likelihood estimator of the coefficients of a GARCH model is consistent and asymptotically normal even if the distribution of the variables η _t is not Gaussian. Since the autocorrelation function of the squares of the GARCH process does not depend on the law of η _t , the autocorrelations obtained by replacing the unknown coefficients by their estimates are generally very close to the empirical autocorrelations. In contrast, the kurtosis coefficients obtained from the theoretical formula, by replacing the coefficients by their estimates and the kurtosis of η _t by 3, can be very far from the coefficients obtained empirically. This is not very surprising since the preceding result shows that the difference between the kurtosis coefficients of ε _t computed with a Gaussian and a non‐Gaussian distribution for η _t ,

is not bounded as a approaches .

2.5.1 Positivity of the Autocovariances

For a GARCH(1, 1) model such that , the autocorrelations of the squares take the form

2.61

where

2.62

(Exercise 2.8). It follows immediately that these autocorrelations are non‐negative. The next property generalises this result.

Note that the property of positive autocorrelations for the squares, or for the absolute values, is typically observed on real financial series (see, for instance, the second row of Table  2.1).

2.5.2 The Autocovariances Do Not Always Decrease

Formulas (2.61) and (2.62) show that for a GARCH(1,1) process, the autocorrelations of the squares decrease. An illustration is provided in Figure 2.11. A natural question is whether this property remains true for more general GARCH(p, q) processes. The following computation shows that this is not the case. Consider an ARCH(2) process admitting moments of order 4 (the existence condition and the computation of the fourth‐order moment are the subject of Exercise 2.8):

We know that is an AR(2) process, whose autocorrelation function satisfies

It readily follows that

and hence that

The latter equality is, of course, true for the ARCH(1) process ( α ₂ = 0) but is not true for any (α ₁, α ₂). Figure 2.12 gives an illustration of this non‐decreasing feature of the first autocorrelations (and partial autocorrelations). The sequence of autocorrelations is, however, decreasing after a certain lag (Exercise 2.18).

2.5.3 Explicit Computation of the Autocovariances of the Squares

The autocorrelation function of will play an important role in identifying the orders of the model. This function is easily obtained from the ARMA(max(p, q), p) representation

The autocovariance function is more difficult to obtain (Exercise 2.9) because one has to compute

One can use the method of Section 2.4.1. Consider the vector representation

defined in ( 2.16) and ( 2.17). Using the independence between and , together with elementary properties of the Kronecker product ⊗, we get

Thus

2.63

where

To compute A ^(m) , we can use the decomposition , where B and C are deterministic matrices. We then have, letting ,

We obtain and similarly. All the components of are equal to ω/(1 −  ∑ α _i  −  ∑ β _i ). Note that for h > 0, we have

2.64

Let . The following algorithm can be used:

• Define the vectors , , , and the matrices , , A ⁽¹⁾ , A ⁽²⁾ as a function of α _i , β _i , and ω , μ ₄ .
• Compute from Eq. (2.63).
• For h = 1, 2, …, compute from Eq. (2.64).
• For h = 0, 1, …, compute .

This algorithm is not very efficient in terms of computation time and memory space, but it is easy to implement.

2.8 Exercises

1. 2.2 (Strict stationarity of GARCH(1, 1) for two laws of ηt) For the GARCH(1, 1) give an explicit strict stationarity condition in the two following cases: (i) the only possible values of η _t are −1 and 1; (ii) η _t follows a uniform distribution.
2. 2.3 (Lyapunov coefficient of a constant sequence of matrices) Prove equality ( 2.21) for a diagonalisable matrix. Use the Jordan representation to extend this result to any square matrix.
3. 2.4 (Lyapunov coefficient of a sequence of matrices) Consider the sequence (A _t ) defined by A _t  = z _t A , where (z _t ) is an ergodic sequence of real random variables such that E log⁺ ∣ z _t  ∣  < ∞, and A is a square non‐random matrix. Find the Lyapunov coefficient γ of the sequence (A _t ) and give an explicit expression for the condition γ < 0.
4. 2.5 (Alternative definition of the top Lyapunov exponent)
   1. Show that, everywhere in Theorem 2.3, the matrix product A _t A _t−1…A ₁ can be replaced by A ₋₁ A ₋₂…A _−t .
   2. Justify the first convergence after (2.25).
5. 2.6 (Multiplicative norms) Show the results of footnote 6 on page 28.
6. 2.7 (Another vector representation of the GARCH(p, q) model) Verify that the vector allows us to define, for p ≥ 1 and q ≥ 2, a vector representation which is equivalent to those used in this chapter, of the form .
7. 2.8 (Fourth‐order moment of an ARCH(2) process) Show that for an ARCH(2) model, the condition for the existence of the moment of order 4, with , is written as
   

   Compute this moment.

8. 2.9 (Direct computation of the autocorrelations and autocovariances of the square of a GARCH(1, 1) process) Find the autocorrelation and autocovariance functions of when (ε _t ) is solution of the GARCH(1, 1) model
   

   where and 1 − 3α ² − β ² − 2αβ > 0.

9. 2.10 (Computation of the autocovariance of the square of a GARCH(1,1) process by the general method) Use the method of Section 2.5.3 to find the autocovariance function of when (ε _t ) is solution of a GARCH(1,1) model. Compare with the method used in Exercise 2.9.
10. 2.11 (Fekete's lemma for sub‐additive sequences) A sequence (a _n ) _n≥1 is called sub‐additive if a _n+m  ≤ a _n  + a _m for all n and m. Show that we then have lim _n‐>∞ a _n /n = inf _n≥1 a _n /n.
11. 2.12 (Characteristic polynomial of EAt) Let A = EA _t , where {A _t , t ∈ ℤ} is the sequence defined in Eq. ( 2.17).
    1. Prove equality ( 2.38).
    2. If , show that ρ(A) = 1.
12. 2.13 (A condition for a sequence Xn to be o(n)) Let (X _n ) be a sequence of identically distributed random variables, admitting a finite expectation. Show that
    

    with probability 1. Prove that the convergence may fail if the expectation of X _n does not exist (an iid sequence with density f(x) = x ⁻² _x ≥ 1 may be considered).

13. 2.14 (Expectation of a product of dependent random variables equal to the product of their expectations) Prove equality (2.31).
14. 2.15 (Necessary condition for the existence of the moment of order 2s) Suppose that (ε _t ) is the strictly stationary solution of model ( 2.5) with for s ∈ (0, 1]. Let
    2.66
    1. Show that when K → ∞,
       
    2. Use this result to prove that as k →  ∞ .
    3. Let (X _n ) a sequence of ℓ × m matrices and Y = (Y ₁, …, Y _m )^′ a vector which is independent of (X _n ) and such that for all i , 0 < E|Y _i | ^s  < ∞. Show that, when n → ∞,
       
    4. Let A = EA _t , and suppose there exists an integer N such that (in the sense that all elements of this vector are strictly positive). Show that there exists k ₀ ≥ 1 such that
       
    5. Deduce condition (2.35) from the preceding question.
    6. Is the condition α ₁ + β ₁ > 0 necessary?
15. 2.16 (Positivity conditions) In the GARCH(1, q) case give a more explicit form for the conditions in ( 2.49). Show, by taking q = 2, that these conditions are less restrictive than ( 2.20).
16. 2.17 (A minoration for the first autocorrelations of the square of an ARCH) Let (ε _t ) be an ARCH( q ) process admitting moments of order 4. Show that, for i = 1, …, q ,
    
17. 2.18 (Asymptotic decrease of the autocorrelations of the square of an ARCH(2) process) Figure  2.12 shows that the first autocorrelations of the square of an ARCH(2) process, admitting moments of order 4, can be non‐decreasing. Show that this sequence decreases after a certain lag.
18. 2.19 (Convergence in probability to −∞) If (X _n ) and (Y _n ) are two independent sequences of random variables such that X _n  + Y _n  →  − ∞ and in probability, then Y _n  →  − ∞ in probability.
19. 2.20 (GARCH model with a random coefficient) Proceeding as in the proof of Theorem 2.1, study the stationarity of the GARCH(1, 1) model with random coefficient ω = ω(η _t − 1),
    2.67

    under the usual assumptions and ω(η _t − 1) > 0 a.s. Use the result of Exercise 2.19 to deal with the case γ ≔ E log a(η _t ) = 0.

20. 2.21 (RiskMetrics model) The RiskMetrics model used to compute the value at risk (see Chapter 11) relies on the following equations:
    

    where 0 < λ < 1. Show that this model has no stationary and non trivial solution.

21. 2.22 (IARCH(∞) models: proof of Corollary 2.6)
    1. In model ( 2.40), under the assumption A ₁ = 1, show that
       
    2. Suppose that condition ( 2.41) holds. Show that the function f : [p,1] ↦ ℝ defined by f(q) = log(A _q μ _2q ) is convex. Compute its derivative at 1 and deduce that condition (2.52) holds.
    3. Establish the reciprocal and show that E|ε _t | ^q  < ∞ for any q ∈ [0, 2).
22. 2.23 (On the predictive power of GARCH) In order to evaluate the quality of the prediction of obtained by using the volatility of a GARCH(1, 1) model, econometricians have considered the linear regression
    

    where is replaced by the volatility estimated from the model. They generally obtained a very small determination coefficient, meaning that the quality of the regression was bad. It this surprising? In order to answer that question compute, under the assumption that exists, the theoretical R ² defined by

    

    Show, in particular, that R ² < 1/κ _η .