Appendix D

Problems

Problem 1

The exercises are independent. Let (η_t) be a sequence of iid random variables satisfying E(η_t) = 0 and Var(η_t) = 1.

Exercise 1: Consider, for all t , the model

where the constants satisfy ω > 0, α_i ≥ 0,i = 1,…,q and β_j ≥ 0, j = 1,…, p. We also assume that η_t is independent of the past values of _t. Let μ = E|η_t|.

1. Give a necessary condition for the existence of E|_i|, and give the value of m = E|_t|.

2. In this question, assume that p = q = 1.

(a) Establish a sufficient condition for strict stationarity using the representation

and give a strictly stationary solution of the model. It will be assumed that this condition is also necessary.

(b) Establish a necessary and sufficient condition for the existence of a second-order stationary solution. Compute the variance of this solution.

3. Give a representation of the model which allows the coefficients to be estimated by least squares.

Exercise 2: The following parametric specifications have been introduced in the ARCH literature.

(i) Quadratic GARCH(p, q):

(ii) Qualitative ARCH of order 1:

where the α_i are real coefficients, (A_i ,i = 1,…,I) is a partition of and is equal to 1 if x A_i, and to 0 otherwise.

(iii) Autoregressive stochastic volatility:

where the u_t are iid variables with mean 0 and variance 1, and are independent of (η_t,t ).

Briefly discuss the dynamics of the solutions (in terms of trajectories and asymmetries), the constraints on the coefficients and why each of these models is of practical interest.

In the case of model (ii), determine maximum likelihood estimators of the coefficients α_i, assuming that the A_i are known and that η_t is normally distributed.

Exercise 3: Consider the model

where (η_1t, η_2t) is an iid sequence with values in ² such that E(η_1t) = E(η_2t) = 0, Var(η _1t) = Var(η_2t) = 1 and Cov(η_1t, η_2t) = 0; σ_1t and σ_2t belong to the σ-field generated by the past of _t; and ω_l > 0, α_li ≥ 0 (i = 1,…,q), and β_lj ≥ 0 (j = 1,…,p).

1. We assume in this question that there exists a second-order stationary solution (_t). Show that E_t = 0. Under a condition to be specified, compute the variance of _t.

2. Compute E(|_t−1,_t−2…) and E(|_t−1,_t−2…) Do GARCH-type solutions of the form _t = σ_tη_t exist, where σ_t belongs to the past of _t and η_t is independent of that past?

3. In the case where α_1j = β_2j for j = 1,…, p, show that (_t) is a weak GARCH process.

Solution

Exercise 1:

1. We have

and, putting r = max(p, q), m satisfies

The condition is thus .

2. (a) We obtain a(η_t−1) = α₁|η_t−1| + β₁. Arguing as in the GARCH(1, 1) case, it can be shown that a sufficient strict stationarity condition is E{log a(η_t)} < 0.

(b) First assume that there exists a second-order stationary solution (_t). We have E() = E(). Thus

and, since E() = E₋₁

Thus we necessarily have

One can then compute

Conversely, if this condition holds true, that is, if E {a²(η_t) < 1, by Jensen’s inequality we have

Thus there exists a strictly stationary solution of the form

with the convention that a(η_t−1)… a(η_t−k) = 1 if k = 0. Using Minkowski’s inequality, it follows that

3. Assume that the condition of part 1 is satisfied. We have

Let

be the innovation of |_t|. We know that (u_t) is a noise (if E < ∞). Multiplying the equation of σ_t by μ and replacing μσ_t−j by |_t−j| − u_t−j, we obtain

This is an ARMA(r, p) equation, which can be used to estimate the coefficients by least squares.

Exercise 2:

Model (i) displays some asymmetry: positive and negative past values of the noise do not have the same impact on the volatility. Its other properties are close to the standard GARCH model: in particular, it should allow for volatility clustering. Positivity constraints are not necessary on ω and the α_i but are required for the β_j. The volatility is no longer a linear function of the past squared values, which makes its study more delicate.

Model (ii) has a constant volatility on intervals. The impact of the past values depends only on the interval of the partition to which they belong. If the α_i are well chosen, the largest values will have a stronger impact than the smallest values, and the impact could be asymmetric. If I is large, then the model is more flexible, but numerous coefficients have to be estimated (and thus it is necessary to have enough observations within each interval).

Model (iii) is a stochastic volatility model. The process σ_t cannot be interpreted as a volatility because it is not positive in general. Moreover, |σ_t| is not exactly the volatility of _t in the sense that is not the conditional variance of _t. At least when the distributions of the two noises are symmetric, the model should be symmetric and the trajectories should resemble those of a GARCH.

For model (ii), neglecting the initial value, up to a constant the log-likelihood has the form of that of the standard GARCH models:

Thus the first-order conditions are

We have (θ) = if _t−1 A_i. Let T_i = {t; _t−1 A_i} and let |T_i| be the cardinality of this set. Thus

and finally, the maximum likelihood estimator of α_i is

Exercise 3:

1. We immediately have E(_t) = 0 by the independence between the η_it and the past. Moreover,

Thus if β_1j = β_2j := β_j for all j, we obtain

2. Let . We have

In general, there exists no constant k such that

which shows that (_t) is not a strong GARCH process.

3. Let be the innovation of . Replacing by , we have

which shows that () satisfies an ARMA{max(p, q), p} equation.

Problem 2

The three parts of this problem are independent (except part III.3).

Consider the model

where (α_t) and (ω_t) are two sequences of iid random variables with symmetric distributions such that E(α_t) = E(ω_t) = 0 and

and admitting moments of order 4 at least. Assume in addition that α_t and ω_t are independent of the past of _t (that is, independent of {_t−s, s > 0}).

Part I: Assume that the sequences (α_t) and (ω_t) are independent.

1.1. Verify that, in the sense of Definition 2.1, there exists an ARCH(1) solution (_t), with E() < ∞. Show that, in general, (_t) is not an ARCH process in the strong sense (Definition 2.2).

1.2. For k > 0, write _t as a function of _t−k and of variables from the sequences (α_t) and (ω_t). Show that

is a sufficient condition for the existence of a strictly stationary solution. Express this condition as a function of α in the case where α_t is normally distributed. (Note that if U ~ (0, 1), then E log|U| = − 0.63.)

I.3. If there exists a second-order stationary solution, determine its mean, E(_t), and its autocovariance function Cov(_t, _t−h), h ≥ 0.

I.4. Establish a necessary and sufficient condition for the existence of a second-order stationarity solution.

I.5. Compute the conditional kurtosis of _t. Is it different from that of a standard strong ARCH?

I.6. Is the model stable by time aggregation?

Part II: Now assume that α_t = λω_t, where λ is a constant.

11.1. Do ARCH solutions exist?

11.2. What is the standard property of the financial series for which this model seems more appropriate than that of part I? For that property, what should be the sign of λ?

11.3. Assume there exists a strictly stationary solution (_t) such that E() < ∞.

(a) Justify the equality , for all h > 0.

(b) Compute the autocovariance function of .

(c) Prove that (_t) admits a weak GARCH representation.

Part III: Assume that the variables α_t and ω_t are normally distributed and that they are correlated, with Corr(α_t, ω_t) = ρ [− 1, 1]. Denote the observations by (₁,…, _n) and let the parameter θ = (α, ω, ρ).

III.1. Write the log-likelihood L_n(θ) conditional on ₀.

III.2. Solve the normal equations

How can these equations be interpreted?

III.3. The model is first estimated under the constraint ? = 0, and then without constraint, on a series (r_t) of log-returns (r_t = log(p_t/r_t−1)). The series contains 2000 daily observations. The estimators are assumed to be asymptotically Gaussian. The following results are obtained (the estimated standard deviations are given in parentheses):

Comment on these results. Can we accept the model of part I? Today, the price falls by 1% compared to yesterday. What are the predicted values for the returns of the next two days? How can we obtain prediction intervals?

Solution

Part I:

I.1. The existence of E() allows us to compute the first two conditional moments of _t:

in view of the independence between α_t and ω_t on one hand, and between these two variables and _u, u < t, on the other hand. These conditional expectations characterize an ARCH(1) process. This process is not an ARCH in the strong sense because the variables are not independent (see I.5).

I.2. We have, for k > 0,

where the sum is equal to 0 if k = 1. Let us show that the series

is almost surely well defined. By the Cauchy root test, the series is almost surely absolutely convergent because

converges a.s. to exp (E log | α_t |) < 1, by the strong law of large numbers. Moreover, we have z_t = α_tz_t−1 + ω_t . Thus (z_t) is a strictly stationary solution of the model. If α_tU_t, where U_t ~ (0, 1), the condition is given by α < exp(1.26) = 3.528.

I.3. We have E(_t) = 0, Cov(_t, _t−h) = 0, for all h > 0, and Var(_t) = ω/(1 − α).

I.4. In view of the previous question, a necessary condition is α < 1. To show that this condition is sufficient, note that, by Jensen’s inequality, it implies the strict stationarity of the solution (z_t). Moreover,

I.5. Assuming E() < ∞ and using the symmetry of the distributions of α_t and ω_t, we have

The conditional kurtosis of _t is equal to the ratio E (|_u, u < t). If this coefficient were constant we would have, for a constant K and for all t,

It is easy to see that this equation has no solution.

I.6. The model satisfied by the process () := (2t) satisfies

The independence assumptions of the initial model are not all satisfied because and are not independent. The time aggregation thus holds only if a dependence between the variables ω_t and α_t is allowed in the model.

Part II:

II.1. We have

which is incompatible with the conditional variance of an ARCH process.

II.2. The model is asymmetric because the volatility depends on the sign of _t−1. We should have λ < 0, so that the volatility increases more when the stock price falls than when it rises by the same magnitude.

II.3. (a) The equality follows from the independence between ω_t and all the past variables.

(b) For h > 1,

using (a) (Cov _t−1, _−t) = 0 for h > 1). For h = 1, we obtain

We have , because the distribution of ω_t−1 is symmetric. Finally, the relation

is true for h > 0. For h =0 we have

which allows us to obtain Var() and finally the whole autocovariance function of ().

(c) The recursive relation between the autocovariances of () implies that the process is an AR(1) of the form

where (u_t) is a noise. The process (_t) is thus an ARCH(1) in the weak sense.

Part III:

III.1. We have

where

III.2. The normal equations are

with

These equations can be interpreted, for n large, as orthogonality relations.

III.3. Note that the estimated coefficients are significant at the 5% level. The constrained model (and thus that of part I) is rejected by the likelihood ratio test. The sign of is what was expected. The optimal prediction is 0, regardless of the horizon. We have E(|_u, u < t) and E(₊₁|_u, < t). The estimated volatility for the next day is . The 95% confidence intervals are thus:

at horizon 1,

at horizon 2,

Problem 3

Let (η_t) be a sequence of iid random variables such that E(η_t) = 0 and Var(η_t) = 1. Consider, for all t , the model

(D.1)

where r is a positive real. The other constants satisfy:

(D.2)

1. Assume in this question that p = q = 1.

(a) Establish a sufficient condition for strict stationarity and give a strictly stationary solution of the model.

(b) Give this condition in the case where β₁ = 0 and compare the conditions corresponding to the different values of r.

(c) Establish a necessary and sufficient condition for the existence of a nonanticipative strictly stationary solution such that E|_t|^2r < ∞. Compute this expectation.

(d) Assume in this question that r < 0. What might be the problem with that specification?

2. Propose an extension of the model that could take into account the typical asymmetric property of the financial series.

3. Show that from a solution (_t) of (D.1), one can define a solution () of a standard GARCH model (with r = 2).

4. Show that if E|_t|^2r < ∞, then (|_t|^r) is an ARMA process whose orders will be given.

5. Assume that we observe a series (_t) of log-returns (_t = log(p_t/p_t−1)). For different powers of | _t |, ARMA models are estimated by least squares. The orders of these ARMA models are identified by information criteria. We obtain the following results, υ_t denoting a noise.

  What values of r are compatible with model (D.1)–(D.2)? Assuming that the distribution of η_t is known, what are the corresponding parameters α_t and β_j ?

6. In view of the previous questions discuss the interest of the models defined by (D.1).

Solution

1. (a) We have

Thus, for N > 0,

Proceeding as for the standard GARCH(1, 1) model, using the Cauchy root test, it is shown that if

the process (h_t), defined by

exists, takes real positive values, is strictly stationary and satisfies h_t = ω + a(η_t−1)h_t. A strictly stationary solution of the model is obtained by . This solution is nonanticipative, because _t is function of η_t and of its past.

(b) If β₁ = 0, the condition is given by

Using the Jensen inequality, it can be seen that

The conditions are thus less restrictive when r is larger.

(c) If (_t) is nonanticipative and stationary and admits a moment of order 2r, we have and, since η_t−1 and σ_t−1 are independent,

Thus, by stationarity,

A necessary condition is thus E {a(η_t)²} < 1. Conversely, if this condition is satisfied we have

The stationary solution that was previously given thus admits a moment of order 2r. Using the previous calculation, we obtain

(d) If r < 0, a(η_t) is not defined when η_t = 0. This is the same for the strictly stationary solution, when one of the variables η_t−j is null. However, the probability of such an event is null if P(η_t = 0) = 0.

2. The specification

with α_i,+ > 0 and α_t,− > 0, induces a different impact on the volatility at time t for the positive and negative past values of _t. For α_i,+ = α_i,− we retrieve model (1).

3. Let , where (υ_t) is an iid process, independent of (_t) and taking the values −1 and 1 with probability 1/2. We have with

and () = (|η_t|^r/2v_t) is an iid process with mean 0 and variance 1. The process () is thus a standard GARCH.

4. The innovation of |_t|^r is defined by

with E|η_t|^r = μ_r. It follows that

and (|_t|^r) is an ARMA{max(p, q), p} process.

5. Noting that the order and coefficients of the AR part in the ARMA representation are greater (in absolute value) than those of the MA part, we see that only the model for r = 1 is compatible with the class of model that is considered here. We have r = p = 1 and q = 2, and the estimated coefficients are

6. The proposed class can take into account the same empirical properties as the standard GARCH models, but is more flexible because of the extra parameter r.

Problem 4

Let (η_t) be a sequence of iid random variables such that E(η_t) = 0, Var(η_t) = 1, E() = μ₄. Let (a_t) be another sequence of iid random variables, independent of the sequence (η_t), taking the values 0 and 1 and such that

Consider, for all t , the model

(D.3)

(D.4)

where

A solution such that _t is independent of the future variables η_t+h and a_t+h, h > 0, is called nonanticipative.

1. What are the values of the parameters corresponding to the standard ARCH(1)? What kind of trajectories can be obtained with that specification?

2. In order to obtain a strict stationarity condition, write (D.4) in the form

(D.5)

where A_t−1 is a matrix depending on η_t−1, a_t−1, a₁, a₂.

3. Deduce a strict stationarity condition for the process (Z_t), and then for the process (_t), as function of the Lyapunov coefficient

(justify the existence of γ and briefly outline the steps of the proof). Note that A_t is the product of a column vector by a row vector. Deduce the following simple expression for the strict stationarity condition:

for a constant c that will be specified. How can the condition be interpreted?

4. Give a necessary condition for the existence of a second-order and nonanticipative stationary solution. It can be assumed that this condition implies strict stationarity. Deduce that the necessary second-order stationarity condition is also sufficient. Compute the variance of _t.

5. We now consider predictions of future values of _t and of its square. Give an expression for E(_t+h|_t−1, _t−2,…) and for h > 0, as a function of _t−1.

6. What is the conditional kurtosis of _t? Is there a standard ARCH solution to the model?

7. Assuming that the distribution of η_t is standard normal, write down the likelihood of the model. For a given series of 2000 observations, a standard ARCH(1) model is estimated, and then model (D.3)–(D.4) is estimated. The estimators are assumed to be asymptotically Gaussian.

The results are presented in the following table (the estimated standard deviations are given in parentheses, L_n(·) denotes the likelihood):

  Comment on these results. Can we accept the general model? (we have P {?²(3) > 7.81} = 0.05).

8. Discuss the estimation of the model by OLS.

Solution

1. The standard ARCH(1) is obtained for α₁ = α₂, p. It is also obtained for p = 0, α₁, α₂ and for p = 1, α₁, α₂. The trajectories may display abrupt changes of volatility (for instance, if ω₁ and ω₂ are very different).

2. We obtain equation (D.5) with

3. The existence of γ requires E log⁺ A_t < ∞. This condition is satisfied because E A_t < ∞, for example with the norm defined by A = Σa_ij. The strict stationarity condition γ < 0 is shown as for the standard GARCH. Under this condition the strictly stationary solution of (D.5) is given by

Note that

Thus

because α₁ and α₂ are positive. It follows that, by the strong law of large numbers,

almost surely as t → ∞. The second expectation is equal to

It follows that

We note that the condition is satisfied even if one of the coefficients, for instance α₁, is large, provided that the corresponding probability, p, is not too large.

4. If (_t) is second-order stationary, we have

The necessary condition is thus

and we have

Conversely, suppose that this condition is satisfied and that it implies that γ < 0. We have

This matrix has rank 1, and thus admits a zero eigenvalue and a nonzero eigenvalue that is equal to its trace, . This coefficient, less than 1 by assumption, is also the spectral radius of E(A_t). It follows that the expectation of the stationary solution Z_t defined above is finite because EA_t A_t−1 … A_t−i = {E(A_t)}ⁱ⁺¹.

5. We clearly have E(_t+h|_t−1, _t−2,…) = 0. Since a_t(1 − a_t) = 0 we have

Letting = E_t and since = E_t, it follows that

for h > 0.

6. The conditional kurtosis is equal to

This coefficient depends on t in general, which shows that there is no standard GARCH solution to this model (except in the cases mentioned in part 1).

7. The conditional density of _t is written as

and the log-likelihood of the sample is the product of the l_t for t = 1,…, n. The estimation results display very different estimated coefficients α₁ and α₂. Moreover, the likelihood ratio test is equivalent to comparing the difference of the log-likelihoods and the quantile of order 1 − α of a ²(3). Since we have 2 × (1275.2 − 1268.2) > 7.81, the standard ARCH(1) is rejected in favor of the general model at the 5% level.

Problem 5

Let (η_t) be a sequence of iid random variables, such that E(η_t) = 0. When E|η_t|^m < ∞, denote . Consider the model

(D.6)

1. Strict stationarity

(a) Let

Show that if E log |bη_t| < 0 then the sequence (|Z_t,n|)n ≥ 1 converges almost surely. Under this condition, let

(b) Show that if E log |bη_t | < 0 then equation (D.6) admits a nonanticipative and ergodic strictly stationary solution.

(c) We have

Give the strict stationarity condition when η_t ~ (0, μ2).

2. Second-order stationarity

(a) Under what condition is (Z_t,n)_n a Cauchy sequence in L² ?

(b) Show that b²μ₂ < 1 entails E log |bη_t| < 0.

(c) Show that if b²μ₂ < 1 then (_t) = (Z_t) is the second-order stationary solution of (D.6).

(d) Assume that μ₂ ≠ 0. Show that the condition b²μ₂ < 1 is also necessary for the existence of a nonanticipative second-order stationary solution.

3. Properties of the marginal distribution and conditional moments Assume that b²μ₂ < 1 and that (_t) is the second-order stationary solution of (D.6).

(a) Show that (_t) is a weak GARCH process whose orders will be specified.

(b) Compare the conditional variance of (_t) with that of a strong ARCH(1). Does the sign of _t−1 have an impact on the volatility at time t? Is this property of interest for financial series?

4. Estimation Denote by b₀ and μ02 the true value of the parameters b and μ₂. Assume that and that _t,…, _n is a second-order stationary realization of model (D.6). Let

(a) What is the interpretation of v_t = − h_0t = ( − μ₀₂)(1 + b_0t−1)²?

(b) Assume that E < ∞. Show that, almost surely,

(c) Assume that the distribution of _t is not concentrated on one or two points (in particular, μ₂ ≠ 0). Show that

(d) Under the previous assumptions, consider the criterion

Show that, almost surely,

with equality if and only if b = b₀ and μ₂ = μ₀₂. Give an estimation method for the parameters.

(e) Describe the quasi-maximum likelihood method.

5. Extension Without giving detailed proofs, extend the stationarity and estimation results to the model

6. Further extensions Assume that μ₂ ≠ 0, μ₃ = 0 and μ₄b⁴ < 1.

(a) Compute the autocovariance function of . Prove that follows a weak ARMA model whose orders will be given.

(b) Propose a moment estimator for μ₂ and b². Show that this estimator is consistent.

(c) When they exist, compute the matrices

and

What moment condition is necessary for the existence of I and J ?

(d) Give the scheme of proof which would establish that, under some assumptions, the least-squares estimator of the parameter θ₀ = (b₀, μ₀₂)′ satisfies

where E(θ₀) = J⁻¹ IJ⁻¹.

(e) Let

When it exists, compute

and show that

where

What moment condition is necessary for the existence of J^QML?

(f) Give the scheme of proof which would establish that, under some assumptions, the quasimaximum likelihood estimator ^QML satisfies

(g) What is the particular form of Σ^Qml(θ₀) at θ₀ = (0, μ₀₂)′? What are the consequences on the asymptotic properties of the estimator?

(h) Compare Σ(θ₀) and Σ^Qml(θ₀) at θ₀ = (0, μ ₀₂)′.

(i) Without giving detailed proofs, extend the stationarity and QML estimation results to the model

Solution

1. (a) In view of the Cauchy root test, it suffices to show that almost surely

By the law of large numbers, this limit is equal to

which shows the result.

(b) For all n, we have

Taking the limit, Z_t = η_t +bη_t Z_t−1, which shows that (_t) = (Z_t) is a nonanticipativ/sub solution of (D.6). Since Z_t = f(η_t, η_t−1,−) (where f : ^∞ → is measurable) and (η_t) is ergodic and stationary, (Z_t) is also stationary and ergodic.

(c) We have . The stationarity condition is thus written as

or equivalently

2. (a) For n < m, we have

as n, m → ∞ (that is, the sequence is a Cauchy sequence) if and only if

(b) If b²μ₂ < 1 then, using the Jensen inequality, we have

(c) When b²μ₂ < 1, we have seen that (Z_t,n)n is a Cauchy sequence. Thus it converges in L² to some limit Z_t. It also converges almost surely to Z_t. Thus Z_t = _t almost surely, and . To show the uniqueness of the solution, assume the existence of two second-order stationary solutions (Z_t) and (). Then, for any n ≥ 1,

By the Cauchy-Schwarz and triangular inequalities,

Since this is true for all n, the condition b²μ₂ < 1 entails E|Z_t − , which implies Z_t almost surely.

(d) If _t is a second-order stationary solution then

that is,

If b²μ₂ were greater than 1, the left-hand side of the previous inequality would be negative, which is impossible because the right-hand side is strictly positive.

3. (a) Such a solution is nonanticipative and satisfies

It is thus a white noise. Let us show that () is an ARMA process. Using the independence between η_t and _t−k and E() = μ₂, we have, for k > 0,

For k > 1,

It follows that, for k > 1,

which shows that () admits an ARMA(1, 1) representation. Finally, (_t) admits a weak GARCH(1, 1) representation.

(b) The volatility of the model is

whereas it is of the form

for an ARCH(1). The sign of _t−1 is thus important. If b < 0, a negative return _t−1 increases the volatility more than it does the return −_t−1 > 0. Such an asymmetry in the shocks is observed in real series, but is not taken into account by standard GARCH models.

4. (a) Since h_0t is the conditional expectation of , υ_t is the strong innovation of .

(b) The process {υ_t(h_0t − h_t)}_t is ergodic and stationary, by arguments already used. The ergodic theorem entails that

because h_0t (h_0t − h_t) is independent of ( − μ₀₂), as measurable function of {η_u, u ≤ t − 1}.

(c) We have E(h_0t − h_t)² = 0 if and only if

This second-order equation in _t−1 (or in _t by stationarity) admits a solution if and only if the coefficients are null, that is, if and only if b = b₀ and μ₂ = μ₀₂.

(d) Using the two last questions, almost surely,

with equality if and only if b = b₀ and μ₂ = μ₀₂. This suggests looking for a value of (b, μ₂) minimizing the criterion Q_n(b, μ₂). This is the least-squares method.

(e) If η_t is (0,μ₀₂) distributed then the distribution of _t given {_u, u < t} is (0,h_0t). Given the initial value ₁, the quasi-log likelihood of ₂,…, _n is thus

A quasi-maximum likelihood estimator satisfies

where Θ × ]0, ∞[ is the parameter space. If Θ is a compact set, since the criterion is continuous, there always exists at least one QMLE.

Problem 6

Consider the model

(D.7)

where (η_t) is an iid sequence of random variables with mean 0 and variance 1 and finite moments of order 4 at least, and where ω(·) is a function with strictly positive values. Let = E{ω(η_t)}.

1. What is the difference between this model and the standard GARCH(1, 1) model and why is it of interest for the modeling of financial series? An example of simulated trajectory of the model is given in Figure D.1.

2. Strict stationarity

(a) Show that under the assumption

where a is a function that will be specified, the model admits a unique nonanticipative strictly stationary solution.

(b) Show that if E log a(η_t) > 0, the model does not admit a strictly stationary solution.

3. Second-order stationarity, kurtosis

(a) Establish the necessary and sufficient condition for the existence of a second-order stationary solution, and compute E(). Prove that the process has the same second-order properties as a standard GARCH(1, 1) (that is, with ω(·) constant).

(b) Assuming that the fourth-order moments exist, compare the kurtosis coefficients of these processes.

4. Asymmetries Give an example of a specification of ω that can take into account the usual asymmetry property of the financial series.

5. ARMA representations

(a) Denote by v_t = − E ( | _u, u < t) the innovation of . Show that, under assumptions to be specified, we have

(b) Show that (u_t) is an MA(1) process. Prove that admits an ARMA(1, 1) representation. Is this representation different from that obtained for the standard GARCH(1, 1) such that ω(η_t−1) = ? (The case β = 0 can be considered.)

6. Estimation and tests

(a) With the aid of part 5, note that the autocorrelation function ρ(h) of the process () satisfies ρ(h) = αρ(h − 1), for h > 1. Give a simple estimator of α that does not depend on the specification of ω. The following values were obtained for the first empirical autocorrelations of ():

Give an estimate of α. Is a standard ARCH(1) model (β = 0 and ω constant) plausible for these data?

(b) Assume that the function ω(·) is parameterized by some parameter γ: for example, ω (η_t−1) = 1 + γ −1 with γ > 0. Consider the estimation of θ = (γ,α, β)′ using the observations _t,…, _n. Write down the quasi-maximum likelihood criterion, given initial values for _u, u > 1.

7. Extension Outline how the previous results are modified if ω(η_t−1) is replaced by ?(η_t−k) in (D.7), with k > 1.

Problem 7

Consider the ARCH models

(I) _t = σ_tη_t, = ω + α₋₁, (II) _t = σ_tη_t, = ω + α₋₂,

where ω > 0, α ≥ 0, and (η_t) denotes an iid sequence of random variables, such that E(η_t) = 0 and Var(η_t) = 1.

1. Show that the strict stationarity condition is the same for the two models, and show that this condition implies the existence of a unique strictly stationary nonanticipative solution. For each model, give the unique strictly stationary nonanticipative solution. Prove that, when it exists, the expectation Ef(_t) of any given function f of _t is the same in the two models.

2. From the observation of empirical autocovariances of , how can we determine the data generating process between model (I) and model (II)?

3. For model (II), write down the likelihood and the equations that allow ω and α to be estimated (without trying to solve these equations). Recall that the asymptotic variance of the quasimaximum likelihood estimator is (E − 1)J⁻¹, where

Compare the asymptotic variances of the estimators of θ = (ω, α)′ in the two models. In each model, how is the hypothesis α = 0 tested?

Problem 8

Exercise 1: Recall that the autocorrelation function ρ(·) of a second-order stationary ARMA(p, q) process satisfies

where the φ_i are the AR coefficients. Denote by B the lag operator: BX_t = X_t−1. Let (_t) be a strictly stationary solution of a GARCH(p, q) model such that E() < ∞.

1. Show that () admits an ARMA representation. What is the relation satisfied by the autocorrelation function ρ² of this process?

2. The aim of this question is to check that the function ρ² has positive values.

(a) Show the property when p = q = 1.

(b) Using the ARMA representation of , show that there exist constants c_i and a noise υ_t such that

(c) Let P and Q be two polynomials with positive coefficients such that P(0) = Q(0) = 0. Assume that 1 − Q(z) = 0 implies that |z| > 1. Show by induction that {1 − Q(B)}⁻¹ P(B) is a series in B with positive terms.

(d) Prove that the coefficients c_i are positive.

(e) Deduce the property.

3. The aim of this question is to show that the function ρ² is not always decreasing.

(a) Verify that for a GARCH(1, 1) model the function ρ² is decreasing.

(b) Assume that q =2 and p = 0, that is, .

(i) What is the relationship between ρ²(h), ρ²(h − 1) and ρ²(h − 2) for h > 0?

(ii) Deduce an expression for {ρ²(1)}⁻ ρ²(2) as a function of α₁ and α₂.

(iii) Prove that for some set of values of α₁ and α₂, the function ρ² is not decreasing.

Exercise 2: Consider the ARCH(q) model

where ω₀ > 0, α_0i ≥ 0, i = 1,…,q, and (η_t) is an iid sequence such that E(η_t) = 0, and Var(η_t) = 1. Let ₁,…,_n be n observations of the process (_t) and let _o,…., _1−q be initial values. Introduce the vector Z_{t−1 q} defined by Z′_t−1 = (1, ₋₁, …, _−q), and the n × q matrix X and the n × 1 vector Y given by

Denote by θ₀ = (ω₀, α₀₁,…, α_0q)′ the true value of the parameter.

1. Show that the OLS estimator = (, ₁,…, _q)′ of θ₀ is given by

We use the notation and _t = {σ_t()}⁻¹_t.

2. Give conditions ensuring the following almost sure convergences as n → ∞ :

3. Prove that converges almost surely to θ₀.

4. In order to take into account the conditional heteroscedasticity, define the weighted least-squares estimator

where

(a) Without going into all the mathematical details, justify the introduction of such an estimator.

(b) Show that

(D.8)

(c) Let . Justify the following results:

(d) Assume that the asymptotic distribution of the right-hand side of (D.8) does not change when () is replaced by . Deduce the asymptotic distribution of ( − θ₀).

Problem 9

For any random variable X, denote

Thus X⁺ ≥ 0 and X⁻ ≤ 0 almost surely. Consider the threshold GARCH(1, 1) model, or TGARCH(1, 1), defined by

(D.9)

where (η_t) is a centred iid sequence with variance 1, and where ω is strictly positive, and α₊, α⁻ and β are nonnegative numbers. In model (D.9), the parameter β is called the shock persistence parameter. In order to introduce some asymmetry for this persistence parameter, that is, a different value when _t−1 < cσ_t−1 and when _t−1 > cσ_t−1 for some constant c (that is, depending on whether the price fell abnormally or not), consider the model

(D.10)

1. Explain briefly the difference between the TGARCH model defined by (D.9) and the standard GARCH(1, 1) model, and why the TGARCH model might be of interest for financial series modeling.

2. Rewrite (D.10) to introduce an asymmetric persistence parameter. Why might such a model be of interest?

3. Study of the TGARCH model defined by (D.9)

(a) Give expressions for and as functions of σ_t, and .

(b) Show that σ_t = ω + a(η_t−1)σ_t−1, where a is a function that will be specified.

(c) Give a sufficient condition for the existence of a nonanticipative and ergodic strictly stationary solution.

(d) Specify this stationarity condition when β = 0 and ηt is (0, 1) distributed.

(e) Give a necessary condition for the existence of a nonanticipative stationary solution such that Eσ_t < ∞. Give Eσ_t, when it exists.

(f) Give a necessary condition for the existence of a nonanticipative stationary solution such that E < ∞. Give E, when it exists.

(g) Assume that η_t has a symmetric distribution. When they exist, give the almost sure limits of

as n → ∞ . Give a simple empirical method for checking if α₊ < α₋ (do not go into the details of the test).

4. Study of the model defined by (D.10)

(a) Give a sufficient condition for the existence of a nonanticipative and ergodic strictly stationary solution.

(b) Give a necessary and sufficient condition for the existence of a nonanticipative stationary solution such that Eσ_t < ∞.

(c) Assume that η_t has a strictly positive density on . Show that, except in the degenerate case where α₊ = α₋ = 0, the model is identifiable, that is, denoting the ‘true’ value of the parameter by σ_t = σ_t(θ), where θ = (ω, α₊, α₋, c, β), we have

(d) Give a method for estimating the parameter θ.

Solution

1. In the standard GARCH formulation, the conditional variance does not depend on the sign of _t−1. In the TGARCH formulation with α₋ > α₊, a negative return _t−1 entails a greater increase in the volatility than a positive return of the same magnitude. Empirical studies have shown the existence of such asymmetries in most financial series.

2. We have

In this model the persistence coefficient is equal to (β − ca₋)σ_t−1 when η_t-1 ≥ c (that is, when _t−1 ≥ cσ_t−1), and (β − cα₊)σ_t−1 when η_t−1 ≤ c. A motivation for considering this model is that a negative shock should increase volatility more than a positive one of the same amplitude, and also that this increase should have a longer effect. By testing the assumption that c = 0, one can test whether the persistence of the negative shocks is the same as that of the positive shocks.

3. (a) The positivity of the coefficients guarantees that σ_t ≥ 0 (starting from an initial value σ₀ ≥ 0). We thus have and .

(b) In view of (a), σ_t = ω + a(η_t−1)σ_t−1, where a(η) = α₊η⁺ − α₋ η₋ + β.

(c) For all n ≥ 1, let

We have s_t(n) = ω + a(η_t−1)s_t−1(n − 1) for all n. If s_t = lim _{n → ∞} s_t(n) exists, then the previous equality still holds true at the limit, and the solution is given by σ_t = s_t (stationarity and ergodicity follow from the fact that s_t = f (η_t−1, η_t−2,…), where f is a measurable function and (η_t) is ergodic and stationary). Since all the terms involved in s_t(n) are positive, st is an increasing limit which exists in ⁺ {+ ∞}. By the Cauchy root criterion, the limit exists in ⁺ if

By the law of large numbers, logλ = E log a(η₁). The condition E loga(η₁) > 0 thus guarantees the existence of the solution.

(d) When β = 0 and the distribution of η₁ is symmetric, we have

The condition is then given by α₊ α₋ > exp (− 2E log |η₁|).

(e) If there exists a nonanticipative stationary solution such that Eσ_t < ∞, then

but this is possible only if Ea(η_t) > 1, that is, .

(f) If there exists a nonanticipative stationary solution such that E < ∞, then

which is possible only if Ea²(η₁) < 1.

(g) By the ergodic theorem, and noting that , under the assumption that the moments exist we have

and

Since the distribution of η_t is symmetric, we have

and For testing α₊ < α₋ one can thus use the statistic , which should converge to .

4. (a) The arguments of part 3(c) show that a condition for the existence of such a solution is E log b(η₁) < 0 with

(b) The arguments of part 3(e) show that the conditionEb(η₁) < 1 is necessary. By Jensen’s inequality, E log b(η₁) ≤ log Eb(η₁). The condition Eb(η₁) < 1 is thus sufficient for the existence of a strictly stationary solution of the form

By Beppo Levi’s theorem, this solution satisfies

(c) Let and consider the case c* ≤ c. The case c* ≥ c is handled similarly. Denote by R_t any measurable function with respect to σ{_u, u ≤ t}. We have

and, writing ,

Assume that σ_t = almost surely (for any t). Then we have

This implies that , , and if c* < c. Since , we have . and α₋ = . We then have ω − ω* + (β − β*)η_t−1 R_t−2 = 0 a.s., which entails ω = ω* and β = β*.

(d) One could estimate θ by the quasi-maximum likelihood method, that is, with the aid of the estimator

where Θ is a compact parameter space which constrains the parameters to be positive, and is defined recursively by

with for instance = ω as initial value.