Stationarity and Existence of Moments

As we have seen, specifications of the function g(·) that are different from (4.10) are possible, depending on the kind of empirical properties we are trying to mimic. The following result does not depend on the specification chosen for g(·). It is, however, assumed that Eg(η _t ) exists and is equal to 0.

The previous computation shows that in the Gaussian case, moments exist at any order. This shows that the leptokurticity property may be more difficult to capture with EGARCH than with standard GARCH models.

Assuming that , the autocorrelation structure of the process can be derived by taking advantage of the ARMA form of the dynamics of . Indeed, replacing the terms in by , we get

Let

One can easily verify that ( ) has finite variance. Since only depends on a finite number r ( r = max(p, q)) of past values of η _t , it is clear that Cov( , ) = 0 for k > r . It follows that ( ) is an MA(r) process (with intercept) and thus that is an ARMA(p, r) process. This result is analogous to that obtained for the classical GARCH models, for which an ARMA(r, p) representation was exhibited for . Apart from the inversion of the integers r and p , it is important to note that the noise of the ARMA equation of a GARCH is the strong innovation of the square, whereas the noise involved in the ARMA equation of an EGARCH is generally not the strong innovation of . Under this limitation, the ARMA representation can be used to identify the orders p and q and to estimate the parameters β _j and α _i (although the latter do not explicitly appear in the representation).

The autocorrelations of can be obtained from formula (4.13). Provided the moments exist we have, for h > 0,

the first product being replaced by 1 if h = 1. For h > 0, this leads to

Invertibility

The existence of a stationary solution (ε _t ) to a time series model is often obtained by showing that, for some white noise (η _t ) and some measurable function ψ ,

The model is said to be invertible if the reverse holds true, i.e. if

for some measurable function φ .

Under the stationarity condition given in Theorem 4.1, the model ( 4.9) expresses as a function of the unobserved innovations {η _u , u < t}. To use the EGARCH model in practice (for instance for predicting by a function of {ε _u , u < t}), it is necessary to be able to express as a function of the past observations. Take the example of the EGARCH(1,1) model, which can be rewritten as

4.15

Replacing η _t by , one can see that the EGARCH(1,1) satisfies the SRE

We will thus apply Lemma 4.1. Assuming β ∈ [0, 1) and −ς < θ < ς , which appear as reasonable restrictions for the applications (see 2. of Remark 4.3), belongs to the interval F = [ω/(1 − β), ∞). We also have θx + ς ∣ x ∣  ≥ 0 and

Lemma 4.1 then entails the following result, due to Straumann and Mikosch (2006).

To illustrate the importance of the invertibility issue, consider the EGARCH(1,1) model ( 4.15), with η _t iid and two parameter sets:

The two models are stationary because ∣β ∣  < 1. Figure 4.1 shows that, for Design A (a), the initial value required for computing recursively the approximation of the volatility has no effect asymptotically. In that case, the volatility at time t , when t is large, can be accurately approximated by a function of the available observations ε ₁, …, ε _t − 1 . The situation is dramatically different in Design B (b). The estimate varies considerably with h and may be completely different from (for visibility reasons, we have not been able to represent the more extreme values of in the graph), even for large t .

It must be emphasised that, even if an EGARCH model with parameters as in Design B is a stationary and ergodic process, it cannot be used to recover the volatility from the past observations. In this case, such a model is not very useful. In practice, when an EGARCH is fitted on a real series, it is thus important to assess if the estimated model corresponds to an invertible model (for instance by evaluating the left‐hand side of Eq. (4.16) using Monte Carlo simulations, and/or by studying how varies with t and h ). The invertibility is also crucial to be able to consistently estimate the parameter of an EGARCH (see Wintenberger 2013).

4.3.1 Stationarity of the Extended Log‐GARCH Model

First consider the case p = q = 1, which can be handled more explicitly than the general case. Write ω ₋ , α ₊ , α ₋ , and β instead of ω ₁₋ , α ₁₊ , α ₁₋ , and β ₁ . The volatility of the model can be rewritten as

Assume that

4.19

and

4.20

where π ₊ = P(η ₀ > 0), π ₋ = P(η ₀ < 0), and π ₀ = P(η ₀ = 0). Recall that, in our notations, we make the convention that 0 × ln x = 0 even if x = 0. Therefore, the case P(η _t  = 0) > 0 is not precluded by Eq. (4.19). Let

and defined by

4.21

By the Cauchy rule, the series (4.21) converges absolutely with probability one because

under (4.20), and

The last inequality comes from the fact that (see Exercise 4.12), which is implied by ( 4.19). Now note that the process ε _t  = σ _t η _t , where is defined by Eq. ( 4.21) is an extended Log‐GARCH(1,1) process of the form ( 4.18), because {ε _t  = 0} iff {η _t  = 0} and {ε _t  > 0} iff {η _t  > 0}.

Now consider the general extended Log‐GARCH( p, q ) process. Because coefficients equal to zero can always be added, it is not restrictive to assume p > 1 and q > 1. Let the vectors

and the matrix

where I _k denote the k × k identity matrix. Model ( 4.18) is rewritten in matrix form as

4.22

Let γ be the top Lyapunov exponent of the sequence { C _t , t ∈ ℤ},

4.23

4.3.2 Existence of Moments and Log‐Moments

In Corollary 2.3, we have seen that for the GARCH models, the strict stationarity condition entails the existence of a moment of order s > 0 for ∣ε _t ∣. The following Lemma shows that this is also the case for in the Log‐GARCH model, when the condition ( 4.19) is slightly reinforced.

We now give conditions for the existence of higher‐order log‐moments, restricting ourselves to the Log‐GARCH(1,1) case. We have the Markovian representation

4.25

with

Under the conditions ( 4.19) and ( 4.20), Eq. (4.25) admits the solution

4.26

The next proposition gives conditions for the existence of and for m ≥ 1.

In the extended Log‐GARCH(1,1) model with α ₋ > max {0, α ₊} and ω ₋ ≥ 0, negative returns impact future volatilities more importantly than positive returns of the same magnitude when this magnitude is large (at least larger than 1), but the effect of the sign can be reversed for small returns in absolute value. A measure of the average leverage effect can be defined through the covariance between η _t − 1 and the current log‐volatility.

If the covariance (4.28) is negative, the leverage effect is present: past negative innovations tend to increase the log‐volatility, and hence the volatility, more than past positive innovations. The sign of the covariance depends not only on all the GARCH coefficients but also on the innovations distribution. Interestingly, the leverage effect may hold with α ₊ > α ₋ and/or ω ₋ < 0.

To illustrate the moment conditions on ε _t , as well as the computation of the autocorrelations of , let us come back to the standard Log‐GARCH(p, q) model

4.29

where (η _t ) is iid . The process thus satisfies

4.30

where , , r = max {p, q}, α _i  = 0 for i > q and β _i  = 0 for i > p . Assuming for all ∣z ∣  ≤ 1, we have

with

We thus have

Since , we have

For s ≤  − 1, the moment μ(s) is infinite. This allows to compute

and

for h ≥ 1. Noting that ln μ(s) = O(s) as s → 0, these moments exist if and only if π _i  >  − 1/4 for all i , which is somewhat a strange moment condition. Another astonishing result is that, if , then the moment E|ε _t | ^d  < ∞ if and only if . If , then E|ε _t | ^d  < ∞ for all d .

Figure 4.2 displays the theoretical autocorrelation function (ACF) of the squares of a Log‐GARCH(2,1) model (4.29) with α ₀₁ = 0.06, β ₀₁ = 1.64, and β ₀₂ =  − 0.95. Note the existence of negative autocorrelations. The Log‐GARCH is thus able to take into account lagged countercyclical heteroscedasticity, i.e. the fact that a high volatility period could entail a low volatility period at a certain horizon. According to Proposition 2.2, this is not possible with standard GARCH models.

4.3.3 Relations with the EGARCH Model

Take the example of an EGARCH(1,1) model, which can be written as

4.31

where the 's are iid with , , and with the notation x ⁺ = max {x, 0} and x ⁻ = min {x, 0}. The volatility of the EGARCH is given by

Now consider a Log‐GARCH(1,1) of the form

4.32

with the parameters α ≠ 0, , ω ₋ =  − α(c ₋ − c ₊), and the iid sequence

with constants c ₊ and c ₋ to be chosen later. The volatility of this Log‐GARCH is given by

Note that, although the processes (ε _t ) and are not the same, the volatility of the Log‐GARCH is equal to the volatility of the EGARCH. In other words, the class of the Extended Log‐GARCH volatilities includes that of the EGARCH volatilities, and it can be shown that the inclusion is strict (see Figure 4.3).

More precisely, we have the following result.

This proposition shows that the class of the Log‐GARCH models generates a richer class of volatilities than the EGARCH.

Stationarity of the TGARCH(1, 1) Model

The study of the stationarity properties of the TGARCH(1, 1) model is based on Eq. (4.36) and follows from similar arguments to the GARCH(1, 1) case. The strict stationarity condition is written as

4.37

In particular, for the TARCH(1) model ( β ₁ = 0), we have

Hence, if the distribution of (η _t ) is symmetric, the expectation of the two indicator variables is equal to 1/2 and the strict stationarity condition reduces to

Exercise 4.8 shows that the second‐order stationarity condition is

4.38

This condition can be made explicit in terms of the first two moments of and . For instance, if η _t is distributed, we get

4.39

Of course, the second‐order stationarity condition is more restrictive than the strict stationarity condition (see Figure 4.5).

Under the second‐order stationarity condition, it is easily seen that the property of symmetry ( 4.1) is generally violated. For instance if the distribution of η _t is symmetric, we have, for the TARCH(1) model:

whenever α _{1, +} ≠ α _{1, −} .

Strict Stationarity of the TGARCH(p, q) Model

The study of the general case relies on a representation analogous to Eq. (2.16), obtained by replacing, in the vector , the variables by , the by σ _t − i , and by an adequate modification of and A _t . Specifically, using (4.35), we get

4.40

where

and

4.41

is a matrix of size (p + 2q) × (p + 2q),

The following result is analogous to that obtained for the strict stationarity of the GARCH(p, q).

Numerical evaluation, by means of simulation, of the Lyapunov coefficient γ can be time‐consuming because of the large size of the matrices A _t . A condition involving matrices of smaller dimensions can sometimes be obtained. Suppose that the asymmetric effects have a factorisation of the form α _i− = θα _i+ for all lags i = 1, …, q . In this constrained model, the asymmetry is summarised by only one parameter θ ≠ 1, the case θ > 1 giving more importance to the negative returns.

mth‐Order Stationarity of the TGARCH(p, q) Model

Contrary to the standard GARCH model, the odd‐order moments are not more difficult to obtain than the even‐order ones for a TGARCH model. The existence condition for such moments is provided by the following theorem.

The proof of this theorem is identical to that of Theorem 2.9.

Kurtosis of the TGARCH(1, 1) Model

For the TGARCH(1, 1) model with positive coefficients, the condition for the existence of E|ε _t | ^m can be obtained directly. Using the representation

we find that exists and satisfies

if and only if

4.44

If this condition is satisfied for m = 4, then the kurtosis coefficient exists. Moreover, if , we get

and, using the notation a _i  = Ea ⁱ (η _t ), the moments can be computed successively as

Many moments of the TGARCH(1, 1) can be obtained similarly, such as the autocorrelations of the absolute values (Exercise 4.9) and squares, but the calculations can be tedious.

Stationarity of the APARCH(1, 1) Model

Relation (4.47) is an extension of (2.6) which allows us to obtain the stationarity conditions, as in the classical GARCH(1, 1) case. The necessary and sufficient strict stationarity condition is thus

4.48

For the APARCH(1, 0) model, we have

showing that, if the distribution of (η _t ) symmetric, the strict stationarity condition reduces to

Note that in the limit case where ∣ς ₁ ∣  = 1, the model is strictly stationary for any value of α ₁ , as might be expected. Under condition (4.48), the strictly stationary solution is given by

Assuming E|η _t | ^δ  < ∞, the condition for the existence of (and of ) is

4.49

which reduces to

when the distribution of (η _t ) symmetric, with

when η _t is Gaussian (Γ denoting the Euler gamma function). Figure 4.6 shows the strict and second‐order stationarity regions of the APARCH(1, 0) model when η _t is Gaussian.

Obviously, if δ ≥ 2, condition (4.49) is sufficient (but not necessary) for the existence of a strictly stationary and second‐order stationary solution to the APARCH(1, 1) model. If δ ≤ 2, condition ( 4.49) is necessary (but not sufficient) for the existence of a second‐order stationary solution.

Descriptive Statistics

Figure 4.7 displays the first 500 values of the series. The volatility clustering phenomenon is clearly evident. The correlograms in Figure 4.8 indicate absence of autocorrelation. However, squared returns present significant autocorrelations, which is another sign that the returns are not independent. Ljung–Box portmanteau tests, such as those available in SAS (see Table 4.3; Chapter 5 gives more details on these tests), confirm the visual analysis provided by the correlograms. Figure 4.9a, compared to Figure  4.8b, seems to indicate that the absolute returns are slightly more strongly correlated than the squares. Figure  4.9b displays empirical correlations between the series ∣r _t ∣ and r _t − h . It can be seen that these correlations are negative, which implies the presence of leverage effects (more accentuated, apparently, for lags 2 and 3 than for lag 1).

Fit by Symmetric and Asymmetric GARCH Models

We will consider the classical GARCH(1, 1) model and the simplest asymmetric models (which are the most widely used). Using the AUTOREG and MODEL procedures of SAS, the estimated models are

• GARCH(1, 1) model
  4.57
• EGARCH(1, 1) model
  4.58
• QGARCH(1, 1) model
  4.59
• GJR‐GARCH(1, 1) model
  4.60
• TGARCH(1, 1) model
  4.61

Interpretation of the Estimated Coefficients

Note that all the estimated models are stationary. The standard GARCH(1, 1) admits a fourth‐order moment since, in view of the computation on p. 45, we have 3α ² + β ² + 2αβ < 1. It is thus possible to compute the variance and kurtosis in this estimated model (which are, respectively, equal to 1.3 × 10⁻⁴ and 3.49 for the standard GARCH(1,1)). Given the ARMA(1, 1) representation for , we have for any h > 1. Since is close to 1, the decay of to zero will be slow when h → ∞, which can be interpreted as a sign of strong persistence of shocks. ⁴

Note that in the EGARCH model the parameter θ =  − 0.53 is negative, implying the presence of the leverage effect. A similar interpretation can be given to the negative sign of the coefficient of ε _t − 1 in the QGARCH model, and to that of in the GJR‐GARCH model. In the TGARCH model, the leverage effect is present since α _{1, −} > α _{1, +} > 0.

The TGARCH model seems easier to interpret than the other asymmetric models. The volatility (that is, the conditional standard deviation) is the sum of four terms. The first is the intercept ω = 8 × 10⁻⁴ . The term ω/(1 − β ₁) = 0.006 can be interpreted as a ‘minimal volatility’, obtained by assuming that all the innovations are equal to zero. The next two terms represent the impact of the last observation, distinguishing the sign of this observation, on the current volatility. In the estimated model, the impact of a positive value is 3.5 times less than that of a negative one. The last coefficient measures the importance of the last volatility. Even in absence of news, the decay of the volatility is slow because the coefficient β ₁ = 0.87 is rather close to 1.

Likelihood Comparisons

Table 4.4 gives the log‐likelihood, ln L _n , of the observations for the different models. One cannot directly compare the log‐likelihood of the standard GARCH(1, 1) model, which has one parameter less, with that of the other models, but the log‐likelihoods of the asymmetric models, which all have five parameters, can be compared. The largest likelihood is observed for the GJR threshold model, but, the difference being very slight, it is not clear that this model is really superior to the others.

Resemblances between the Estimated Volatilities

Figure 4.10 shows that the estimated volatilities for the five models are very similar. It follows that the different specifications produce very similar prediction intervals (see Figure 4.11).

Distances between Estimated Models

Differences can, however, be discerned between the various specifications. Table 4.5 gives an insight into the distances between the estimated volatilities for the different models. From this point of view, the TGARCH and EGARCH models are very close and are also the most distant from the standard GARCH. The QGARCH model is the closest to the standard GARCH. Rather surprisingly, the TGARCH and GJR‐GARCH models appear quite different. Indeed, the GJR‐GARCH is a threshold model for the conditional variance and the TGARCH is a similar model for the conditional standard deviation.

Figure 4.12 confirms the results of Table  4.5. The left‐hand scatterplot shows

and the right‐hand one

The left‐hand graph shows that the difference between the estimated volatilities of the TGARCH and the standard GARCH, denoted by , is always very close to the difference between the estimated volatilities of the EGARCH and the standard GARCH, denoted by (the difference from the standard GARCH is introduced to make the graphs more readable). The right‐hand graph shows much more important differences between the TGARCH and GJR‐GARCH specifications.

Comparison between Implied and Sample Values of the Persistence and of the Leverage Effect

We now wish to compare, for the different models, the theoretical autocorrelations ρ(∣r _t  ∣ , ∣ r _t − h ∣) and ρ(∣r _t  ∣ , r _t − h ) to the empirical ones. The theoretical autocorrelations being difficult – if not impossible – to obtain analytically, we used simulations of the estimated model to approximate these theoretical autocorrelations by their empirical counterparts. The length of the simulations, 50 000, seemed sufficient to obtain good accuracy (this was confirmed by comparing the empirical and theoretical values when the latter were available).

Figure 4.13 shows satisfactory results for the standard GARCH model, as far as the autocorrelations of absolute values are concerned. Of course, this model is not able to reproduce the correlations induced by the leverage effect. Such autocorrelations are adequately reproduced by the TGARCH model, as can be seen from the top and bottom right panels. The autocorrelations for the other asymmetric models are not reproduced here but are very similar to those of the TGARCH. The negative correlations between r _t and the r _t − h appear similar to the empirical ones.

Implied and Empirical Kurtosis

Table 4.6 shows that the theoretical variances obtained from the estimated models are close to the observed variance of the CAC 40 index. In contrast, the estimated kurtosis values are all much below the observed value.

In all these five models, the conditional distribution of the returns is assumed to be . This choice may be inadequate, which could explain the discrepancy between the estimated theoretical and the empirical kurtosis. Moreover, the normality assumption is clearly rejected by statistical tests, such as the Kolmogorov–Smirnov test, applied to the standardised returns. A leptokurtic distribution is observed for those standardised returns.

Table 4.7 reveals a large number of returns outside the interval , whatever the specification used for . If the conditional law were Gaussian and if the conditional variance were correctly specified, the probability of one return falling outside the interval would be 2{1 − Φ(3)} = 0.0027, which would correspond to an average of 6 values out of 2385.

Asymmetric GARCH Models with Non‐Gaussian Innovations

To take into account the leptokurtic shape of the residuals distribution, we re‐estimated the five GARCH models with a Student t distribution – whose parameter is estimated – for η _t .

For instance, the new estimated TGARCH model is

4.62

It can be seen that the estimated volatility is quite different from that obtained with the normal distribution (see Table 4.8).

Model with Interventions

Analysis of the residuals show that the values observed at times t = 228, 682, and 845 are scarcely compatible with the selected model. There are two ways to address this issue: one could either research a new specification that makes those values compatible with the model, or treat these three values as outliers for the selected model.

In the first case, one could replace the distribution of the noise η _t with a more appropriate (leptokurtic) one. The first difficulty with this is that no distribution is evident for these data (it is clear that distributions of Student t or generalised error type would not provide good approximations of the distribution of the standardised residuals). The second difficulty is that changing the distribution might considerably enlarge the confidence intervals. Take the example of a 99% confidence interval at horizon 1. The initial interval simply becomes the dilated interval with t _0.995 > 2.57, provided that the estimates are not much affected by the change of conditional distribution. Even if the new interval does contain 99% of returns, there is a good chance that it will be excessively large for most of the data.

So for this first case, we should ideally change the prediction formula for σ _t so that the estimated volatility is larger for the three special data (the resulting smaller standardized residuals would become consistent with the distribution), without much changing volatilities estimated for other data. Finding a reasonable model that achieves this change seems quite difficult.

We have, therefore, opted for the second approach, treating these three values as outliers. Conceptually, this amounts to assuming that the model is not appropriate in certain circumstances. One can imagine that exceptional events occurred shortly before the three dates t = 228, 682, and 845. Other special events may occur in the future, and our model will be unable to anticipate the changes in volatility induced by these extraordinary events. The ideal would be to know the values that returns would have had if these exceptional event had not occurred, and to work with these corrected values. This is, of course, not possible, and we must also estimate the adjusted values. We will use an intervention model, assuming that only the returns of the three dates would have changed in the absence of the above‐mentioned exceptional events. Other types of interventions can, of course, be envisaged. To estimate what would have been the returns of the three dates in the absence of exceptional events, we can add these three values to the parameters of the likelihood. This can easily be done using a SAS program (see Table 4.9).

GARCH‐X Models

The usual GARCH‐type models predict the squared returns by means of the past returns only {ε _u , u < t}. It is, however, often the case that some extra information is available, under the form of a vector x _t − 1 of exogenous covariates, such as the daily volume of transactions, or high frequency intraday data, or even series of other returns. To take advantage of the extra information – in order to improve the prediction of the squares – GARCH models that are augmented with additional explanatory variables (GARCH‐X models in short), can be considered. A simple version of GARCH‐X is of the form

4.63

where x _t  = (x _{1, t} , …, x _{r, t} )^′ is a vector of r exogenous covariates. The term ‘exogenous’ does not refer to the concepts of weak or strong exogeneity introduced in the econometric literature, but is employed because the dynamics if the vector x _t is not specified by the GARCH‐X model (4.63).

To ensure with probability one, assume that the covariates are a.s. positive and that the coefficients satisfy α ≥ 0, β ≥ 0, ω > 0, and π  = (π ₁, …, π _r )^′ ≥ 0 componentwise. In the GARCH‐type models, the sequence (η _t ) is traditionally assumed to be iid(0, 1), but the assumption is not necessary for the following result.

It is interesting to note that, under Assumption A, the stationarity condition of (ε _t ) is not modified by the presence of the exogenous variables. Note that if

where ℱ _t − 1 is the information set generated by {ε _u , x _u  : u < t}, then is the conditional variance of ε _t given ℱ _t − 1 .

HEAVY Models

Intraday high‐frequency data, and thus realised volatility measures RM _t (such as the realized variance), are increasingly available. This extra information can be included as exogenous variable x _t  = RM _t in model ( 4.63), which generally greatly improves the prediction of . The drawback of the GARCH‐X model is however that the dynamics of the covariate is not specified. This forces the practitioner to put in the volatility equation of only variables that will be available (or predictable) at time t + h , which is restrictive if the horizon of prediction h is larger than 1. To specify the dynamics of (RM _t ), Shephard and Sheppard (2010) proposed the high‐frequency‐based volatility (HEAVY) model, in which Model ( 4.63)

4.64

is completed by the Multiplicative Error Model (MEM)

4.65

where (z _t ) is a stationary sequence of positive variables such that E(z _t |ℱ _t − 1) = 1, ω ^* > 0, α ^* ≥ 0 and β ^* ≥ 0.

Realised‐GARCH Models

An alternative model has been proposed by Hansen, Huang, and Shek (2012) (see also Hansen and Huang, 2016). In this model, called realised‐GARCH, a GARCH‐X equation similar to Eq. ( 4.64), or alternatively a Log‐GARCH‐X equation of the form

4.66

is completed by a measurement equation for the realised volatility measure, that can take the form

4.67

where (u _t ) is a strong white noise independent of (η _t ). Note that the dynamics are written on the logarithm of the volatility and of the realised volatility measure, in order to avoid positivity constraints. The term τ(η _t ) is assumed to be centred, for instance of the form τ(z) = τ ₁ z + τ ₂(z ² − 1), and can thus take into account the leverage effect. Defining a strong white noise ( ) by  = τ(η _t − 1) + u _t − 1, one can see that satisfies the AR(1) model

from which stationarity conditions follow. Note, however, that the realised‐GARCH is not a stochastic volatility model (see Chapter 12) because is measurable with respect to the sigma‐field ℱ _t − 1 generated by {RM _u  : u < t}.

Intraday Scale Models

Visser (2011) proposed a model that directly links the daily returns and the intraday price movements. Let p _t (u) be the price of an asset at the intraday time u ∈ [0, 1] of the day t ∈ ℕ. Consider the continuous‐time log‐return process R _t (u) = ln p _t (u)/p _t (0). Note that the open‐to‐close daily return is ε _t  = R _t (1). If, for example, R _t (u) = σ _t W _t (u) where {W _t (·)} _t is an iid sequence of standard Brownian motions, then we have the ARCH‐type equation ε _t  = σ _t η _t where (η _t ) is iid . In this model, the realised volatility – usually defined as the sum of the squared five‐minute returns – should be a good proxy of the volatility since

4.68

in mean square as h → 0 (see Exercise 4.13). More generally, denoting by D[0, 1] the set of the real‐valued function on [0,1] which are right continuous with left limits (càdlàg), if one assumes

and

that is, H(cR) = cH(R) for all c ≥ 0 and R ∈ D[0, 1], then the realized measure RM _t  = H(R _t (·)) satisfies the ARCH, or more precisely the MEM‐type equation

Noting that , Visser (2011) showed that, up to the unknown scaling constant , the parameters giving the dynamics of the daily volatility σ _t can also be estimated from a MEM model on the sequence (RM _t ). The resulting estimators are generally more accurate than those based on (ε _t ), intuitively because the variance of RM _t is often much smaller than that of .

MIDAS Models

The previous models aim at explaining a daily variable (the squared returns) by an intraday variable. More generally, MIxed DAta Sampling (MIDAS) regression is a technique that can be used when independent variables appear at higher frequencies than the dependent variable (see Wang and Ghysels (2015) and the references therein). Engle, Ghysels, and Sohn (2013) proposed a GARCH‐MIDAS model with a short‐run volatility component g _t and a long‐run volatility component τ _t . The short‐run component g _t is the volatility of a unit‐variance GARCH(1,1) process, defined by

4.69

with α > 0, β ≥ 0, and α + β < 1. As usual, (η _t ) denotes an iid (0,1) sequence. The observed sequence of returns (r _t ) is then assumed to follow an ARCH‐type model with the error term ε _t and a volatility τ _t driven by past realised volatilities :

4.70

The parameters m and θ are positive. The weights ϕ _k are positive, and their sum is equal to 1. For instance, one can take

Note that

with ω = m and q = K + N − 1. The first ARCH coefficient is α ₁ = θϕ ₁ and the last one is α _q  = θϕ _K . Compared to a standard ARCH(q), the error term ε _t is not iid(0,1), but is a GARCH(1,1). The stationarity condition remains, however, the same (see e.g. Remark 4.1, or Theorem 4.9 showing that the iid assumption can be replaced by an assumption of the form A).

Duration Models

Many financial data – in particular intra‐daily high‐frequency data – are irregularly spaced in time. Several approaches have been proposed to tackle this problem. Engle and Russell (1998) introduced the so‐called Autoregressive Conditional Duration (ACD) model for durations between events (such as trades, quotes, price changes).

Let t _i be the time at which the i th event of interest occurs, and denote by x _i  = t _i  − t _i − 1 the i th duration between two consecutive events. The ACD model is specified as

4.76

where p and q are non‐negative integers. Since the duration x _i is necessarily non‐negative, it is common to assume that (z _i ) is a sequence of non‐negative iid random variables with E(z _i ) = 1, and to impose α ₀ > 0, α _j  ≥ 0, β _ℓ ≥ 0.

Let ℱ _i − 1 be the information set consisting of all information up to and including time t _i − 1 . It is clear that, if the roots of the polynomial are outside the unit circle, and if z _i is independent of ℱ _i − 1 , the variable ψ _i can be interpreted as the conditional mean of the i ‐th duration,

The existence of a strictly stationary non‐anticipative solution to model (4.76) follows straightforwardly from the study of GARCH models. For instance, when p = q = 1, the condition E ln(α ₁ z ₁ + β ₁) < 0 ensures the existence of a strictly stationary and non‐anticipative solution. Higher‐order models can be studied by rewriting the conditional duration as , and by deducing a Markov vector representation.

Estimation of the parameters, θ ₀ = (α ₀, …, α _p , β ₁, …, β _q )^′ , can be performed using the QML approach. Gouriéroux, Monfort, and Trognon (1984) showed that, in order to estimate the parameters of a correctly specified conditional mean model, the QML estimator is consistent regardless of the true innovations distribution if and only if the QML is based on a distribution belonging to the linear exponential family. In particular, the QML based on the standard exponential distribution will provide consistent estimation (under regularity conditions that we will not discuss here) of θ ₀ . The exponential QML is obtained by minimising on some compact set of θ 's the criterion

where the ψ _i (θ)s are computed recursively using initial values for the x _i s. Note that this criterion coincides with the Gaussian QML criterion of a GARCH( p, q ) model, provided the x _i s are replaced by the squared returns . This is not surprising since an ACD process is nothing else than the square of a GARCH process: with

where η _i  = z _i s _i , (s _i ) being an iid sequence, independent of the sequence (z _i ), and uniformly distributed on {+1, − 1}. It follows that standard packages for estimating GARCH models can be directly used to estimate ACD models. Other distributions with a positive support, like the Weibull distribution considered by Engle and Russell (1998), or the Burr distribution used by Grammig and Maurer (2000), will not produce consistent QML estimators in general.

In practical applications, ACD models encounter some weaknesses. For instance, zero durations are often ignored. Another difficulty is that intraday transactions of a stock often exhibit diurnal patterns. In other words, the strict stationarity assumption in not realistic. Different approaches have been proposed to remove the diurnal patterns in particular using deterministic functions of the time of the day (see e.g. Engle and Russell 1998; Tsay 2005).

Note that the ACD model can also be used to model non‐negative time series. An example is the range of an asset during a trading day, which can be seen as a measure of its volatility (see Parkinson 1980). The Conditional Autoregressive Range (CARR) model introduced by Chou (2005) is an ACD model for such data. These models can also be seen as particular instances of the multiplicative error model (MEM) introduced by Engle (200b).

INGARCH Models

Time series of counts are often observed in the real world, in particular in finance (see e.g. Rydberg and Shephard 2003). Heinen (2003) considered a model called Autoregressive Conditional Poisson (ACP) and applied it to the daily number of price change durations of $0.75 on the IBM stock. Ferland, Latour, and Oraichi (2006) further studied the ACP model, under the name INteger‐valued GARCH (INGARCH). A process (X _t ), valued in ℕ, satisfies an INGARCH( p, q ) model if the conditional distribution of X _t given its past values is Poisson with intensity parameter

4.77

where ω > 0, α _i  ≥ 0, and β _j  ≥ 0 for i = 1, …, q and j = 1, …, p . The name INGARCH comes from the fact that the conditional variance (4.77) resembles the volatility of a GARCH. Note, however, that, contrary to a GARCH, the conditional mean of X _t is not zero (but is equal to λ _t ). Other count time series models can be obtained by assuming alternative discrete conditional distributions, with other time‐varying parameters. Note, however, that studying the probabilistic structure of count time series models, in particular giving conditions for ergodicity, may be a difficult issue (see Fokianos, Rahbek, and Tjøstheim 2009; Tjøstheim 2012; Davis et al. 2016).

Functional GARCH Models

Given the increasing amount of financial data at a high, or ultra‐high frequency, it makes sense to describe such high‐resolution tick data as functions. A functional version of the ARCH model was proposed in Hörmann, Horváth, and Reeder (2013) and recently extended by Aue et al. (2017). Their approach relies on a daily segmentation of the data, which is more realistic than only one continuous time process. More precisely, it is assumed that we have a functional time series {ε _k (t), 1 ≤ k ≤ T, 0 ≤ t ≤ 1}, where ε _k (t) are intraday log‐returns on day k at time t . For some lag h (for instance 1 or 5 minutes), we thus have ε _k (t) = ln p _k (t) − ln p _k (t − h), where {p _k (t)} is the underlying price process. For convenience, the price p _k (t) is supposed to be defined on the set [−h, 1], so that ε _k (t) is defined for t belonging to [0, 1].

To define sequences of real‐valued functions with domain [0, 1], we use the following notation. Let ℋ = L ²[0, 1] denote the Hilbert space of square integrable functions with norm , which is generated by the inner product , for x, y ∈ ℋ. If x, y ∈ ℋ, xy stands for point‐wise multiplication, xy(s) = x(s)y(s) for s ∈ [0, 1]. By ℋ⁺ we denote the set of non‐negative functions in ℋ. A non‐negative integral operator α , mapping ℋ⁺ on ℋ⁺ is defined, for some integrable kernel function K _α  ≥ 0 defined on [0, 1]² , by

A functional ARCH(1) process {ε _k (t), k ∈ ℤ, 0 ≤ t ≤ 1} can be defined as the solution of the model

4.83

where ω ∈ ℋ⁺ , α is a non‐negative integral operator with kernel K _α , and {η _k } is an iid sequence of random elements of ℋ.

Different choices are possible for the kernel K _α , leading to different interpretations. For a constant kernel, K _α (t, s) = a for some positive constant a , we have It follows that the volatility at time t of day k has the form

in which the effects of t and k are separate. In this simple specification, the volatility of day k depends on time t through a deterministic function. Now suppose that K _α (t, s) = aφ(s − t), where φ denotes the Gaussian density, a a positive constant. It follows that

Hence, the volatility at time t of day k depends on the returns of day k − 1, but with maximal weight around time t .

As in usual GARCH models, it is necessary to constrain the parameters to obtain a strictly stationary solution. Hörmann, Horváth, and Reeder (2013) showed that if (i) η _k  ∈ L ⁴[0, 1], (ii) the operator α is bounded and

where , then the functional ARCH(1) model admits a unique strictly stationary solution. Moreover, this solution is non‐anticipative, with

for some measurable functional g : ℋ^ℕ ↦ ℋ. Consequently, the strictly stationary solution is also ergodic.

The intra‐day correlations of the stationary solutions can be characterised. Assuming that and for all t ∈ [0, 1], we find that Eε _k (t) = E(ε _k (t) ∣ ℱ _k − 1) = 0 and for s, t ∈ [0, 1]

where ℱ _k − 1 is the σ ‐field generated by the sequence {η _i , i < t}, and C _η (s, t) = Cov(η _k (t), η _k (s).

To estimate the model, an identifiability assumption is required. It can be assumed that and . Similar to standard GARCH models, the functional ARCH(1) admits a ‘weak’ functional AR(1) representation. Noting that the expectation of is the zero function, we get

which can serve as a basis for estimating α and ω (see Hörmann, Horváth, and Reeder 2013 for details).

4.13 Exercises

1. 4.2 (The expectation of a product of independent variables is not always the product of the expectations) Find a sequence X _i of independent real random variables such that exists a.s., EY and EX _i exist for all i , and exists, but such that .
2. 4.3 (Convergence of an infinite product entails convergence of the infinite sum of logarithms) Prove that, under the assumptions of Theorem 4.1, condition ( 4.12) entails the absolute convergence of the series of general term ln g _η (λ _i ).
3. 4.4 (Variance of an EGARCH) Complete the proof of Theorem 4.1 by showing in detail that ( 4.13) entails the desired result on .
4. 4.5 (A Gaussian EGARCH admits a variance) Show that, for an EGARCH with Gaussian innovations, condition (4.14) for the existence of a second‐order moment is satisfied.
5. 4.6 (ARMA representation for the logarithm of the square of an EGARCH) Compute the ARMA representation of when ε _t is an EGARCH(1, 1) process with η _t Gaussian. Provide an explicit expression by giving numerical values for the EGARCH coefficients.
6. 4.7 (β‐mixing of an EGARCH) Using Exercise 3.5, give simple conditions for an EGARCH(1, 1) process to be geometrically β ‐mixing.
7. 4.8 (Stationarity of a TGARCH) Establish the second‐order stationarity condition (4.39) of a TGARCH(1, 1) process.
8. 4.9 (Autocorrelation of the absolute value of a TGARCH) Compute the autocorrelation function of the absolute value of a TGARCH(1, 1) process when the noise η _t is Gaussian. Would this computation be feasible for a standard GARCH process?
9. 4.10 (A TGARCH is an APARCH) Check that the results obtained for the APARCH(1, 1) model can be used to retrieve those obtained for the TGARCH(1, 1) model.
10. 4.11 (Study of a threshold model) Consider the model
    

    To which class does this model belong? Which constraints is it natural to impose on the coefficients? What are the strict and second‐order stationarity conditions? Compute Cov(σ _t , ε _t − 1) in the case where , and verify that the model can capture the leverage effect.

11. 4.12 (A condition for lim sup Xn/n ≤ 0) Show that if (X _n ) is an identically distributed sequence of real variables such that , then a.s.
12. 4.13 (Quadratic variations of the Brownian motion) Show Equality (4.68).
13. 4.14 (Beta‐t‐GARCH) The density of a Student distribution with ν > 2 degrees of freedom and variance σ ² is
    

    where

    

    Compute ∂ ln f _σ (x)/∂σ ² and deduce ( 4.75). Hint: if X ∼ f , then follows a Beta distribution with mean 1/(ν + 1) and variance 2ν/{(ν + 1)²(ν + 3)}.