Appendix D

Problems

Problem 1

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

Exercise 1: Consider, for all t 2208_fmt 2124_fmt, the model

bapp04ue001_fmt

where the constants satisfy ω > 0, αi ≥ 0,i = 1,…,q and βj ≥ 0, j = 1,…, p. We also assume that ηt is independent of the past values of 2208_fmtt. Let μ = Et|.

1. Give a necessary condition for the existence of E|2208_fmti|, and give the value of m = E|2208_fmtt|.

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

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

bapp04ue002_fmt

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):

bapp04ue003_fmt

(ii) Qualitative ARCH of order 1:

bapp04ue004_fmt

where the αi are real coefficients, (Ai ,i = 1,…,I) is a partition of 211D_fmt and bapp04-ie440001_fmt is equal to 1 if x 2208_fmt Ai, and to 0 otherwise.

(iii) Autoregressive stochastic volatility:

bapp04ue005_fmt

where the ut are iid variables with mean 0 and variance 1, and are independent of (ηt,t 2208_fmt 2124_fmt).

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 αi, assuming that the Ai are known and that ηt is normally distributed.

Exercise 3: Consider the model

bapp04ue006_fmt

where (η1t, η2t) is an iid sequence with values in 211D_fmt2 such that E1t) = E2t) = 0, Var(η 1t) = Var(η2t) = 1 and Cov(η1t, η2t) = 0; σ1t and σ2t belong to the σ-field generated by the past of 2208_fmtt; and ωl > 0, αli ≥ 0 (i = 1,…,q), and βlj ≥ 0 (j = 1,…,p).

1. We assume in this question that there exists a second-order stationary solution (2208_fmtt). Show that E2208_fmtt = 0. Under a condition to be specified, compute the variance of 2208_fmtt.

2. Compute E(2208-2t_fmt|2208_fmtt−1,2208_fmtt−2…) and E(2208-4t_fmt|2208_fmtt−1,2208_fmtt−2…) Do GARCH-type solutions of the form 2208_fmtt = σtηt exist, where σt belongs to the past of 2208_fmtt and ηt is independent of that past?

3. In the case where α1j = β2j for j = 1,…, p, show that (2208_fmtt) is a weak GARCH process.

Solution

Exercise 1:

1. We have

bapp04ue007_fmt

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

bapp04ue008_fmt

The condition is thus bapp04-ie441001_fmt.

2. (a) We obtain at−1) = α1t−1| + β1. Arguing as in the GARCH(1, 1) case, it can be shown that a sufficient strict stationarity condition is E{log at)} < 0.

(b) First assume that there exists a second-order stationary solution (2208_fmtt). We have E(2208-2t_fmt) = E(sigma-2t_fmt). Thus

bapp04ue009_fmt

and, since E(2208-2t_fmt) = E2208-2t_fmt−1

bapp04ue010_fmt

Thus we necessarily have

bapp04ue011_fmt

One can then compute

bapp04ue012_fmt

Conversely, if this condition holds true, that is, if E {a2t) < 1, by Jensen’s inequality we have

bapp04ue013_fmt

Thus there exists a strictly stationary solution of the form

bapp04ue014_fmt

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

bapp04ue015_fmt

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

bapp04ue016_fmt

Let

bapp04ue017_fmt

be the innovation of |2208_fmtt|. We know that (ut) is a noise (if E2208-2t_fmt < ∞). Multiplying the equation of σt by μ and replacing μσtj by |2208_fmttj| − utj, we obtain

bapp04ue018_fmt

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 αi but are required for the βj. 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 αi 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 σt cannot be interpreted as a volatility because it is not positive in general. Moreover, |σt| is not exactly the volatility of 2208_fmtt in the sense that sigma-2t_fmt is not the conditional variance of 2208_fmtt. 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:

bapp04ue019_fmt

Thus the first-order conditions are

bapp04ue020_fmt

We have sigma-2t_fmt(θ) = alpha-2i_fmt if 2208_fmtt−1 2208_fmt Ai. Let Ti = {t; 2208_fmtt−1 2208_fmt Ai} and let |Ti| be the cardinality of this set. Thus

bapp04ue021_fmt

and finally, the maximum likelihood estimator of αi is

bapp04ue022_fmt

Exercise 3:

1. We immediately have E(2208_fmtt) = 0 by the independence between the ηit and the past. Moreover,

bapp04ue023_fmt

Thus if β1j = β2j := βj for all j, we obtain

bapp04ue024_fmt

2. Let bapp04-ie443001_fmt. We have

bapp04ue025_fmt

In general, there exists no constant k such that

bapp04ue026_fmt

which shows that (2208_fmtt) is not a strong GARCH process.

3. Let bapp04-ie443002_fmt be the innovation of 2208-2t_fmt. Replacing bapp04-ie443003_fmt by bapp04-ie443004_fmt, we have

bapp04ue027_fmt

which shows that (2208-2t_fmt) 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

bapp04ue028_fmt

where (αt) and (ωt) are two sequences of iid random variables with symmetric distributions such that Et) = Et) = 0 and

bapp04ue029_fmt

and admitting moments of order 4 at least. Assume in addition that αt and ωt are independent of the past of 2208_fmtt (that is, independent of {2208_fmtts, s > 0}).

Part I: Assume that the sequences (αt) and (ωt) are independent.

1.1. Verify that, in the sense of Definition 2.1, there exists an ARCH(1) solution (2208_fmtt), with E(2208-2t_fmt) < ∞. Show that, in general, (2208_fmtt) is not an ARCH process in the strong sense (Definition 2.2).

1.2. For k > 0, write 2208_fmtt as a function of 2208_fmttk and of variables from the sequences (αt) and (ωt). Show that

bapp04ue030_fmt

is a sufficient condition for the existence of a strictly stationary solution. Express this condition as a function of α in the case where αt is normally distributed. (Note that if U ~ 004E_fmt(0, 1), then E log|U| = − 0.63.)

I.3. If there exists a second-order stationary solution, determine its mean, E(2208_fmtt), and its autocovariance function Cov(2208_fmtt, 2208_fmtt−h), 2200_fmt 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 2208_fmtt. Is it different from that of a standard strong ARCH?

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

Part II: Now assume that αt = λωt, 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 (2208_fmtt) such that E(2208-4t_fmt) < ∞.

(a) Justify the equality bapp04-ie444001_fmt, for all h > 0.

(b) Compute the autocovariance function of 2208-2t_fmt.

(c) Prove that (2208_fmtt) admits a weak GARCH representation.

Part III: Assume that the variables αt and ωt are normally distributed and that they are correlated, with Corr(αt, ωt) = ρ 2208_fmt [− 1, 1]. Denote the observations by (2208_fmt1,…, 2208_fmtn) and let the parameter θ = (α, ω, ρ).

III.1. Write the log-likelihood Ln(θ) conditional on 2208_fmt0.

III.2. Solve the normal equations

bapp04ue031_fmt

How can these equations be interpreted?

III.3. The model is first estimated under the constraint ? = 0, and then without constraint, on a series (rt) of log-returns (rt = log(pt/rt−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):

bapp04ut001_fmt

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(2208-2t_fmt) allows us to compute the first two conditional moments of 2208_fmtt:

bapp04ue032_fmt

in view of the independence between αt and ωt on one hand, and between these two variables and 2208_fmtu, 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 bapp04-ie446001_fmt are not independent (see I.5).

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

bapp04ue033_fmt

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

bapp04ue034_fmt

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

bapp04ue035_fmt

converges a.s. to exp (E log | αt |) < 1, by the strong law of large numbers. Moreover, we have zt = αtzt−1 + ωt . Thus (zt) is a strictly stationary solution of the model. If αtroot-alpha_fmtUt, where Ut ~ 004E_fmt(0, 1), the condition is given by α < exp(1.26) = 3.528.

I.3. We have E(2208_fmtt) = 0, Cov(2208_fmtt, 2208_fmtth) = 0, for all h > 0, and Var(2208_fmtt) = 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 (zt). Moreover,

bapp04ue036_fmt

I.5. Assuming E(2208-4t_fmt) < ∞ and using the symmetry of the distributions of αt and ωt, we have

bapp04ue037_fmt

The conditional kurtosis of 2208_fmtt is equal to the ratio E (2208-4t_fmt|2208_fmtu, u < t)sigma-2t_fmt. If this coefficient were constant we would have, for a constant K and for all t,

bapp04ue038_fmt

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

I.6. The model satisfied by the process (2208-002A-t_fmt) := (2208_fmt2t) satisfies

bapp04ue039_fmt

The independence assumptions of the initial model are not all satisfied because alpha-002A-t_fmt and omega-002A-t_fmt are not independent. The time aggregation thus holds only if a dependence between the variables ωt and αt is allowed in the model.

Part II:

II.1. We have

bapp04ue040_fmt

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 2208_fmtt−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 ωt and all the past variables.

(b) For h > 1,

bapp04ue041_fmt

using (a) (Cov 2208_fmtt−1, 2208-2t_fmtt) = 0 for h > 1). For h = 1, we obtain

bapp04ue042_fmt

We have bapp04-ie447001_fmt, because the distribution of ωt−1 is symmetric. Finally, the relation

bapp04ue043_fmt

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

bapp04ue044_fmt

which allows us to obtain Var(2208-2t_fmt) and finally the whole autocovariance function of (2208-2t_fmt).

(c) The recursive relation between the autocovariances of (2208-2t_fmt) implies that the process is an AR(1) of the form

bapp04ue045_fmt

where (ut) is a noise. The process (2208_fmtt) is thus an ARCH(1) in the weak sense.

Part III:

III.1. We have

bapp04ue046_fmt

where

bapp04ue047_fmt

III.2. The normal equations are

bapp04ue048_fmt

with

bapp04ue049_fmt

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 carret-p_fmt is what was expected. The optimal prediction is 0, regardless of the horizon. We have E(2208-2t_fmt|2208_fmtu, u < t) and E(2208-2t_fmt+1|2208_fmtu, < t). The estimated volatility for the next day is bapp04-ie448001_fmt. The 95% confidence intervals are thus:

at horizon 1, bapp04-ie448002_fmt

at horizon 2, bapp04-ie448003_fmt

Problem 3

Let (ηt) be a sequence of iid random variables such that E(ηt) = 0 and Var(ηt) = 1. Consider, for all t 2208_fmt 2124_fmt, the model

(D.1) bapp04e001_fmt

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

(D.2) bapp04e002_fmt

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|2208_fmtt|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 (2208_fmtt) of (D.1), one can define a solution (2208-002A-t_fmt) of a standard GARCH model (with r = 2).

4. Show that if E|2208_fmtt|2r < ∞, then (|2208_fmtt|r) is an ARMA process whose orders will be given.

5. Assume that we observe a series (2208_fmtt) of log-returns (2208_fmtt = log(pt/pt−1)). For different powers of | 2208_fmtt |, ARMA models are estimated by least squares. The orders of these ARMA models are identified by information criteria. We obtain the following results, υt denoting a noise.

bapp04ue050_fmt

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

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

Solution

1. (a) We have

bapp04ue051_fmt

Thus, for N > 0,

bapp04ue052_fmt

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

bapp04ue053_fmt

the process (ht), defined by

bapp04ue054_fmt

exists, takes real positive values, is strictly stationary and satisfies ht = ω + at−1)ht. A strictly stationary solution of the model is obtained by bapp04-ie450001_fmt. This solution is nonanticipative, because 2208_fmtt is function of ηt and of its past.

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

bapp04ue055_fmt

Using the Jensen inequality, it can be seen that

bapp04ue056_fmt

The conditions are thus less restrictive when r is larger.

(c) If (2208_fmtt) is nonanticipative and stationary and admits a moment of order 2r, we have bapp04-ie450002_fmt and, since ηt−1 and σt−1 are independent,

bapp04ue057_fmt

Thus, by stationarity,

bapp04ue058_fmt

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

bapp04ue059_fmt

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

bapp04ue060_fmt

(d) If r < 0, at) is not defined when ηt = 0. This is the same for the strictly stationary solution, when one of the variables ηtj is null. However, the probability of such an event is null if Pt = 0) = 0.

2. The specification

bapp04ue061_fmt

with αi,+ > 0 and αt,− > 0, induces a different impact on the volatility at time t for the positive and negative past values of 2208_fmtt. For αi,+ = αi,− we retrieve model (1).

3. Let bapp04-ie451001_fmt, where (υt) is an iid process, independent of (2208_fmtt) and taking the values −1 and 1 with probability 1/2. We have bapp04-ie451002_fmt with

bapp04ue062_fmt

and (small-eta-002A-t_fmt) = (|ηt|r/2vt) is an iid process with mean 0 and variance 1. The process (2208-002A-t_fmt) is thus a standard GARCH.

4. The innovation of |2208_fmtt|r is defined by

bapp04ue063_fmt

with Et|r = μr. It follows that

bapp04ue064_fmt

and (|2208_fmtt|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

bapp04ue065_fmt

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 (ηt) be a sequence of iid random variables such that Et) = 0, Var(ηt) = 1, E(small-eta-4t_fmt) = μ4. Let (at) be another sequence of iid random variables, independent of the sequence (ηt), taking the values 0 and 1 and such that

bapp04ue066_fmt

Consider, for all t 2208_fmt 2124_fmt, the model

(D.3) bapp04e003_fmt

(D.4) bapp04e004_fmt

where

bapp04ue067_fmt

A solution such that 2208_fmtt is independent of the future variables ηt+h and at+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) bapp04e005_fmt

where At−1 is a matrix depending on ηt−1, at−1, a1, a2.

3. Deduce a strict stationarity condition for the process (Zt), and then for the process (2208_fmtt), as function of the Lyapunov coefficient

bapp04ue068_fmt

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

bapp04ue069_fmt

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 2208_fmtt.

5. We now consider predictions of future values of 2208_fmtt and of its square. Give an expression for E(2208_fmtt+h|2208_fmtt−1, 2208_fmtt−2,…) and bapp04-ie452001_fmt for h > 0, as a function of 2208_fmtt−1.

6. What is the conditional kurtosis of 2208_fmtt? Is there a standard ARCH solution to the model?

7. Assuming that the distribution of ηt 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, Ln(·) denotes the likelihood):

bapp04ut002_fmt

  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, 2200_fmtp. It is also obtained for p = 0, 2200_fmtα1, α2 and for p = 1, 2200_fmtα1, 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

bapp04ue070_fmt

3. The existence of γ requires E log+ 2016_fmt At 2016_fmt < ∞. This condition is satisfied because E2016_fmt At 2016_fmt < ∞, for example with the norm defined by 2016_fmt A2016_fmt = Σ2016_fmtaij. 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

bapp04ue071_fmt

Note that

bapp04ue072_fmt

Thus

bapp04ue073_fmt

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

bapp04ue074_fmt

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

bapp04ue075_fmt

It follows that

bapp04ue076_fmt

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 (2208_fmtt) is second-order stationary, we have

bapp04ue077_fmt

The necessary condition is thus

bapp04ue078_fmt

and we have

bapp04ue079_fmt

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

bapp04ue080_fmt

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

5. We clearly have E(2208_fmtt+h|2208_fmtt−1, 2208_fmtt−2,…) = 0. Since at(1 − at) = 0 we have

bapp04ue081_fmt

Letting omega-bar_fmt = Etilde-omega_fmtt and since alpha-bar_fmt = Etilde-alpha_fmtt, it follows that

bapp04ue082_fmt

for h > 0.

6. The conditional kurtosis is equal to

bapp04ue083_fmt

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 2208_fmtt is written as

bapp04ue084_fmt

and the log-likelihood of the sample is the product of the lt 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 xstyle2(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 (ηt) be a sequence of iid random variables, such that Et) = 0. When Et|m < ∞, denote bapp04-ie456001_fmt. Consider the model

(D.6) bapp04e006_fmt

1. Strict stationarity

(a) Let

bapp04ue085_fmt

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

bapp04ue086_fmt

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

(c) We have

bapp04ue087_fmt

Give the strict stationarity condition when ηt ~ 004E_fmt(0, μ2).

2. Second-order stationarity

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

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

(c) Show that if b2μ2 < 1 then (2208_fmtt) = (Zt) 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 (2208_fmtt) is the second-order stationary solution of (D.6).

(a) Show that (2208_fmtt) is a weak GARCH process whose orders will be specified.

(b) Compare the conditional variance of (2208_fmtt) with that of a strong ARCH(1). Does the sign of 2208_fmtt−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 bapp04-ie456002_fmt and that 2208_fmtt,…, 2208_fmtn is a second-order stationary realization of model (D.6). Let

bapp04ue088_fmt

(a) What is the interpretation of vt = 2208-2t_fmth0t = (small-eta-2t_fmt − μ02)(1 + b02208_fmtt−1)2?

(b) Assume that E2208-4t_fmt < ∞. Show that, almost surely,

bapp04ue089_fmt

(c) Assume that the distribution of 2208_fmtt is not concentrated on one or two points (in particular, μ2 ≠ 0). Show that

bapp04ue090_fmt

(d) Under the previous assumptions, consider the criterion

bapp04ue091_fmt

Show that, almost surely,

bapp04ue092_fmt

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

bapp04ue093_fmt

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

(a) Compute the autocovariance function of 2208-2t_fmt. Prove that 2208-2t_fmt 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

bapp04ue094_fmt

and

bapp04ue095_fmt

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 carret-theta_fmt of the parameter θ0 = (b0, μ02)′ satisfies

bapp04ue096_fmt

where E0) = J−1 IJ−1.

(e) Let

bapp04ue097_fmt

When it exists, compute

bapp04ue098_fmt

and show that

bapp04ue099_fmt

where

bapp04ue100_fmt

What moment condition is necessary for the existence of JQML?

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

bapp04ue101_fmt

(g) What is the particular form of ΣQml0) at θ0 = (0, μ02)′? What are the consequences on the asymptotic properties of the estimator?

(h) Compare Σ(θ0) and ΣQml0) at θ0 = (0, μ 02)′.

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

bapp04ue102_fmt

Solution

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

bapp04ue103_fmt

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

bapp04ue104_fmt

which shows the result.

(b) For all n, we have

bapp04ue105_fmt

Taking the limit, Zt = ηt +bηt Zt−1, which shows that (2208_fmtt) = (Zt) is a nonanticipativ/sub solution of (D.6). Since Zt = ft, ηt−1,−) (where f : 211D_fmt211D_fmt is measurable) and (ηt) is ergodic and stationary, (Zt) is also stationary and ergodic.

(c) We have bapp04-ie459001_fmt. The stationarity condition is thus written as

bapp04ue106_fmt

or equivalently

bapp04ue107_fmt

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

bapp04ue108_fmt

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

bapp04ue109_fmt

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

bapp04ue110_fmt

(c) When b2μ2 < 1, we have seen that (Zt,n)n is a Cauchy sequence. Thus it converges in L2 to some limit Zt. It also converges almost surely to Zt. Thus Zt = tilde-Z_fmtt almost surely, and bapp04-ie459002_fmt. To show the uniqueness of the solution, assume the existence of two second-order stationary solutions (Zt) and (Z-002A-t_fmt). Then, for any n ≥ 1,

bapp04ue111_fmt

By the Cauchy-Schwarz and triangular inequalities,

bapp04ue112_fmt

Since this is true for all n, the condition b2μ2 < 1 entails E|ZtZ-002A-t_fmt, which implies ZtZ-002A-t_fmt almost surely.

(d) If 2208_fmtt is a second-order stationary solution then

bapp04ue113_fmt

that is,

bapp04ue114_fmt

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

bapp04ue115_fmt

It is thus a white noise. Let us show that (2208-2t_fmt) is an ARMA process. Using the independence between ηt and 2208_fmttk and E(small-eta-2t_fmt) = μ2, we have, for k > 0,

bapp04ue116_fmt

For k > 1,

bapp04ue117_fmt

It follows that, for k > 1,

bapp04ue118_fmt

which shows that (2208-2t_fmt) admits an ARMA(1, 1) representation. Finally, (2208_fmtt) admits a weak GARCH(1, 1) representation.

(b) The volatility of the model is

bapp04ue119_fmt

whereas it is of the form

bapp04ue120_fmt

for an ARCH(1). The sign of 2208_fmtt−1 is thus important. If b < 0, a negative return 2208_fmtt−1 increases the volatility more than it does the return −2208_fmtt−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 h0t is the conditional expectation of 2208-2t_fmt, υt is the strong innovation of 2208-2t_fmt.

(b) The process {υt(h0tht)}t is ergodic and stationary, by arguments already used. The ergodic theorem entails that

bapp04ue121_fmt

because h0t (h0tht) is independent of (small-eta-2t_fmt − μ02), as measurable function of {ηu, ut − 1}.

(c) We have E(h0tht)2 = 0 if and only if

bapp04ue122_fmt

This second-order equation in 2208_fmtt−1 (or in 2208_fmtt 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,

bapp04ue123_fmt

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

(e) If ηt is 004E_fmt(0,μ02) distributed then the distribution of 2208_fmtt given {2208_fmtu, u < t} is 004E_fmt(0,h0t). Given the initial value 2208_fmt1, the quasi-log likelihood of 2208_fmt2,…, 2208_fmtn is thus

bapp04ue124_fmt

A quasi-maximum likelihood estimator satisfies

bapp04ue125_fmt

where Θ 2282_fmt 211D_fmt × ]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) bapp04e007_fmt

where (ηt) 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 omega-bar_fmt = E{ω(ηt)}.

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

bapp04ue126_fmt

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

(b) Show that if E log at) > 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(2208-2t_fmt). 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 vt = 2208-2t_fmtE (2208-2t_fmt | 2208_fmtu, u < t) the innovation of 2208-2t_fmt. Show that, under assumptions to be specified, we have

bapp04ue127_fmt

Figure D.1 Simulated trajectory of model (D.7) with α = 0.2, β = 0.5 and ω(ηt−1) = 4 (left) ω(ηt−1) = 1 + small-eta-4t_fmt−1 (right), for the same sequence of variables ηt ~ 004E_fmt(0, 1).

bapp04f001_fmt

(b) Show that (ut) is an MA(1) process. Prove that 2208-2t_fmt admits an ARMA(1, 1) representation. Is this representation different from that obtained for the standard GARCH(1, 1) such that ω(ηt−1) = omega-bar_fmt? (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 (2208-2t_fmt) 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 (2208-2t_fmt):

bapp04ue128_fmt

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 (ηt−1) = 1 + γ small-eta-2t_fmt−1 with γ > 0. Consider the estimation of θ = (γ,α, β)′ using the observations 2208_fmtt,…, 2208_fmtn. Write down the quasi-maximum likelihood criterion, given initial values for 2208_fmtu, u > 1.

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

Problem 7

Consider the ARCH models

(I) 2208_fmtt = σtηt, sigma-2t_fmt = ω + α2208-2t_fmt−1, (II) 2208_fmtt = σtηt, sigma-2t_fmt = ω + α2208-2t_fmt−2,

where ω > 0, α ≥ 0, and (ηt) denotes an iid sequence of random variables, such that Et) = 0 and Var(ηt) = 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(2208_fmtt) of any given function f of 2208_fmtt is the same in the two models.

2. From the observation of empirical autocovariances of 2208-2t_fmt, 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 (Esmall-eta-4t_fmt − 1)J−1, where

bapp04ue129_fmt

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

bapp04ue130_fmt

where the φi are the AR coefficients. Denote by B the lag operator: BXt = Xt−1. Let (2208_fmtt) be a strictly stationary solution of a GARCH(p, q) model such that E(2208-4t_fmt) < ∞.

1. Show that (2208-2t_fmt) admits an ARMA representation. What is the relation satisfied by the autocorrelation function ρ2208_fmt2 of this process?

2. The aim of this question is to check that the function ρ2208_fmt2 has positive values.

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

(b) Using the ARMA representation of 2208-2t_fmt, show that there exist constants ci and a noise υt such that

bapp04ue131_fmt

(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 ci are positive.

(e) Deduce the property.

3. The aim of this question is to show that the function ρ2208_fmt2 is not always decreasing.

(a) Verify that for a GARCH(1, 1) model the function ρ2208_fmt2 is decreasing.

(b) Assume that q =2 and p = 0, that is, bapp04-ie465001_fmt.

(i) What is the relationship between ρ2208_fmt2(h), ρ2208_fmt2(h − 1) and ρ2208_fmt2(h − 2) for h > 0?

(ii) Deduce an expression for {ρ2208_fmt2(1)} ρ2208_fmt2(2) as a function of α1 and α2.

(iii) Prove that for some set of values of α1 and α2, the function ρ2208_fmt2 is not decreasing.

Exercise 2: Consider the ARCH(q) model

bapp04ue132_fmt

where ω0 > 0, α0i ≥ 0, i = 1,…,q, and (ηt) is an iid sequence such that Et) = 0, and Var(ηt) = 1. Let 2208_fmt1,…,2208_fmtn be n observations of the process (2208_fmtt) and let 2208_fmto,…., 2208_fmt1−q be initial values. Introduce the vector Zt−1 2208_fmt 211D_fmtq defined by Zt−1 = (1, 2208-2t_fmt−1, …, 2208-2t_fmtq), and the n × q matrix X and the n × 1 vector Y given by

bapp04ue133_fmt

Denote by θ0 = (ω0, α01,…, α0q)′ the true value of the parameter.

1. Show that the OLS estimator carret-theta_fmt = (carret-omega_fmt, carret-alpha_fmt1,…, carret-alpha_fmtq)′ of θ0 is given by

bapp04ue134_fmt

We use the notation bapp04-ie466001_fmt and 2208-tilde_fmtt = {σt(carret-theta_fmt)}−12208_fmtt.

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

bapp04ue135_fmt

3. Prove that carret-theta_fmt converges almost surely to θ0.

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

bapp04ue136_fmt

where

bapp04ue137_fmt

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

(b) Show that

(D.8) bapp04e008_fmt

(c) Let bapp04-ie466002_fmt. Justify the following results:

bapp04ue138_fmt

(d) Assume that the asymptotic distribution of the right-hand side of (D.8) does not change when sigma-2t_fmt(carret-theta_fmt) is replaced by sigma-2t_fmt. Deduce the asymptotic distribution of root-n_fmt( tilde-theta_fmt − θ0).

Problem 9

For any random variable X, denote

bapp04ue139_fmt

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

(D.9) bapp04e009_fmt

where (ηt) 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 2208_fmtt−1 < cσt−1 and when 2208_fmtt−1 > cσt−1 for some constant c (that is, depending on whether the price fell abnormally or not), consider the model

(D.10) bapp04e010_fmt

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 bapp04-ie467001_fmt and bapp04-ie467002_fmt as functions of σt, bapp04-ie467003_fmt and bapp04-ie467004_fmt.

(b) Show that σt = ω + at−1t−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 004E_fmt(0, 1) distributed.

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

(f) Give a necessary condition for the existence of a nonanticipative stationary solution such that Esigma-2t_fmt < ∞. Give Esigma-2t_fmt, when it exists.

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

bapp04ue140_fmt

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 Eσt < ∞.

(c) Assume that ηt has a strictly positive density on 211D_fmt. 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 σt = σt(θ), where θ = (ω, α+, α, c, β), we have

bapp04ue141_fmt

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

Solution

1. In the standard GARCH formulation, the conditional variance bapp04-ie469001_fmt does not depend on the sign of 2208_fmtt−1. In the TGARCH formulation with α > α+, a negative return 2208_fmtt−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

bapp04ue142_fmt

In this model the persistence coefficient is equal to (β − cat−1 when ηt-1 ≥ c (that is, when 2208_fmtt−1cσt−1), and (β − +t−1 when ηt−1c. 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 σt ≥ 0 (starting from an initial value σ0 ≥ 0). We thus have bapp04-ie469002_fmt and bapp04-ie469003_fmt.

(b) In view of (a), σt = ω + at−1t−1, where a(η) = α+η+ − α η + β.

(c) For all n ≥ 1, let

bapp04ue143_fmt

We have st(n) = ω + at−1)st−1(n − 1) for all n. If st = lim n → ∞ st(n) exists, then the previous equality still holds true at the limit, and the solution is given by σt = st (stationarity and ergodicity follow from the fact that st = ft−1, ηt−2,…), where f is a measurable function and (ηt) is ergodic and stationary). Since all the terms involved in st(n) are positive, st is an increasing limit which exists in 211D_fmt+ 22C3_fmt {+ ∞}. By the Cauchy root criterion, the limit exists in 211D_fmt+ if

bapp04ue144_fmt

By the law of large numbers, logλ = E log a1). 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

bapp04ue145_fmt

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

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

bapp04ue146_fmt

but this is possible only if Eat) > 1, that is, bapp04-ie469004_fmt.

(f) If there exists a nonanticipative stationary solution such that E sigma-2t_fmt < ∞, then

bapp04ue147_fmt

which is possible only if Ea21) < 1.

(g) By the ergodic theorem, and noting that bapp04-ie470001_fmt, under the assumption that the moments exist we have

bapp04ue148_fmt

and

bapp04ue149_fmt

Since the distribution of ηt is symmetric, we have

bapp04ue150_fmt

and bapp04-ie470002_fmt For testing α+ < α one can thus use the statistic bapp04-ie470003_fmt, which should converge to bapp04-ie470004_fmt.

4. (a) The arguments of part 3(c) show that a condition for the existence of such a solution is E log b1) < 0 with

bapp04ue151_fmt

(b) The arguments of part 3(e) show that the conditionEb1) < 1 is necessary. By Jensen’s inequality, E log b1) ≤ log Eb1). The condition Eb1) < 1 is thus sufficient for the existence of a strictly stationary solution of the form

bapp04ue151_fmt

By Beppo Levi’s theorem, this solution satisfies

bapp04ue151_fmt

(c) Let bapp04-ie470005_fmt and consider the case c* ≤ c. The case c* ≥ c is handled similarly. Denote by Rt any measurable function with respect to σ{2208_fmtu, ut}. We have

bapp04ue154_fmt

and, writing bapp04-ie470006_fmt,

bapp04ue155_fmt

Assume that σt = bapp04-ie471001_fmt almost surely (for any t). Then we have

bapp04ue156_fmt

This implies that bapp04-ie471002_fmt, , and bapp04-ie471003_fmt if c* < c. Since bapp04-ie471004_fmt, we have bapp04-ie471005_fmt. and α = bapp04-ie471001c. We then have ω − ω* + (β − β*)ηt−1 Rt−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

bapp04ue157_fmt

where Θ is a compact parameter space which constrains the parameters to be positive, and bapp04-ie471002c is defined recursively by

bapp04ue158_fmt

with for instance bapp04-ie471003c = ω as initial value.

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