Volatility in Several Series The various xARCH models provide many - - PowerPoint PPT Presentation

▶

Nov 04, 2023 469 likes •648 views

ST 810-006 Statistics and Financial Risk Volatility in Several Series The various xARCH models provide many ways to model volatility dynamics in a single series. Many areas of financial risk involve the joint behavior of several variables.

SLIDE 1

ST 810-006 Statistics and Financial Risk

Volatility in Several Series The various xARCH models provide many ways to model volatility dynamics in a single series. Many areas of financial risk involve the joint behavior of several variables. Graphs show that volatility typically varies simultaneously across series. Individual xARCH models could be fitted to each variable, but would not be linked.

1 / 17

SLIDE 2

ST 810-006 Statistics and Financial Risk

GARCH Recall the GARCH model: yt = financial variable, such as log-return on some asset

r perhaps the residual of some variable from an ARIMA model

for the conditional mean structure. Assume yt = σtǫt, where {ǫt} are independent and follow some fixed distribution (standard normal, standardized t, . . . ). GARCH(1, 1): σ2

t = ω + αy 2 t−1 + βσ2 t−1.

The “standardized residuals” are zt = ˆ σ−1

t yt.

2 / 17

SLIDE 3

ST 810-006 Statistics and Financial Risk

Multivariate GARCH With several series, yt = vector of J financial variables. The analog of σ2

t = var(yt|yt−1, yt−2, . . . )

is Σt = var(yt|yt−1, yt−2, . . . ). The purpose of a multivariate xARCH model is to provide a recursive expression for Σt.

3 / 17

SLIDE 4

ST 810-006 Statistics and Financial Risk

PC-GARCH: use PCA. First stage: fit univariate GARCH models to the individual series. Put standardized residuals for time t in a vector zt = S−1

1,tyt,

where S1,t is the diagonal matrix of first stage conditional standard deviations. Set up a data matrix and use the SVD: Z = (T × J) data matrix with rows z′

t, t = 1, 2, . . . , T

=

T×J

D

J×J,diagonal

V′

J×J

4 / 17

SLIDE 5

ST 810-006 Statistics and Financial Risk

Second stage: fit univariate GARCH models to the PC scores (here columns of U; more conventionally, of UD). Write ǫt = S−1

2,tut

where S2,t is the diagonal matrix of second stage conditional standard deviations. Model ǫt as:

NJ(0, I); tJ,ν(0, I); a meta t-distribution, combining marginal t-distributions and a t-copula, all potentially with different degrees of freedom; some other non-Gaussian distribution with chosen tail length, tail dependence, and shape.

5 / 17

SLIDE 6

ST 810-006 Statistics and Financial Risk

The conditional distribution of yT+1 is represented as yT+1 = S1,T+1zT+1 = S1,T+1VDuT+1 = S1,T+1VDS2,T+1ǫT+1 where:

V and D come from the SVD; S1,T+1 and S2,T+1 come from the first and second stage GARCH recursions, respectively; ǫT+1 follows the chosen model.

Note that the distribution of ǫT+1 does not depend on the past, so it is independent of the past.

6 / 17

SLIDE 7

ST 810-006 Statistics and Financial Risk

If the chosen distribution of ǫT+1 is Gaussian or t, the conditional distribution of yT+1 is in the same family. Otherwise, the conditional distribution of yT+1 is likely to be intractable except for simulation. Either way, use it for instance to compute VaR or ES of a portfolio of the assets on which these are the returns.

7 / 17

SLIDE 8

ST 810-006 Statistics and Financial Risk

Reduced Rank PC-GARCH: recall that if d2

1 + d2 2 + · · · + d2 k ≫ d2 k+1 + · · · + d2 J then

Z ≈

U(k)

T×k

D(k)

k×k,diagonal

V(k)′

k×J

So only k principal component score series need to be modeled, and ǫt consists of only k variables whose distributions need to be modeled. If k ≪ J, modeling is much simplified.

8 / 17

SLIDE 9

ST 810-006 Statistics and Financial Risk

PC-GARCH and Reduced Rank PC-GARCH are related to Orthogonal GARCH (OGARCH) and Generalized Orthogonal GARCH (GO-GARCH). These model the conditional covariance matrix of yt directly: Σt = XDtX′ where Dt is a diagonal matrix of univariate GARCH conditional variances, and:

in OGARCH, X is (J × k) with orthogonal columns, like V(k); in GO-GARCH, X is (J × J) with no orthogonality constraints.

Then yt is modeled as Gaussian or t with covariance matrix Σt. Note: PC-GARCH makes Σt more complicated: Σt = S1,tVDS2

2,tDV′S1,t.

9 / 17

SLIDE 10

ST 810-006 Statistics and Financial Risk

Non-PCA Approaches MGARCH is a very general extension of univariate GARCH. MGARCH(1, 1): vech(Σt) = Avech(yt−1y′

t−1) + Bvech(Σt−1) + c

where vech(·) vectorizes the lower triangle of a symmetric matrix: vech(S) =              s1,1 s2,1 . . . sJ,1 s2,2 s3,2 . . . sJ,J             

10 / 17

SLIDE 11

ST 810-006 Statistics and Financial Risk

MGARCH is over-parametrized: vech(S) is (J(J + 1)/2 × 1). So A and B are ((J(J + 1)/2) × (J(J + 1)/2)), with ∼ J4/4 entries each, and c is ((J(J + 1)/2) × 1). Constraining Σt to be non-negative definite is a challenge.

11 / 17

SLIDE 12

ST 810-006 Statistics and Financial Risk

In diag-MGARCH, A and B are constrained to be diagonal–some improvement. Note: diagonal multiplication of vech(·) is equivalent to entrywise multiplication: Σt = A ◦ (yt−1y′

t−1) + B ◦ Σt−1 + C

where A, B, and C are now (J × J), and “◦” denotes entrywise (Hadamard, or Schur) product. Constraining Σt to be non-negative definite is still a challenge: requiring A, B, and C to be non-negative definite is sufficient, but not necessary. No “cross-talk” in diag-MGARCH.

12 / 17

SLIDE 13

ST 810-006 Statistics and Financial Risk

BEKK is a different simplified form of MGARCH: Σt = A′(yt−1y′

t−1)A + B′Σt−1B + C

Here A and B are unrestricted, and C is positive definite symmetric. Off-diagonal entries in A and B introduce cross-talk: volatility in

ne variable can flow into another from either the yt−1y′

t−1 term

r the Σt−1 term.

13 / 17

SLIDE 14

ST 810-006 Statistics and Financial Risk

Constant Conditional Correlation (CCC) We can always decompose Σt: Σt = DtRtDt, where Dt is the diagonal matrix of conditional standard deviations, and Rt is the conditional correlation matrix. In CCC, assume that Rt = R, constant. Use separate GARCH models to build Dt, and estimate R from the standardized residuals zt.

14 / 17

SLIDE 15

ST 810-006 Statistics and Financial Risk

Dynamic Conditional Correlation (DCC) As in CCC, use separate GARCH models to build Dt. Then Rt = (diag(Qt))−1/2Qt(diag(Qt))−1/2 where Qt satisfies the recursion Qt = θ1zt−1z′

t−1 + θ2Qt−1 + (1 − θ1 − θ2) ¯

Q. That is, Qt follows an even simpler MGARCH model with scalar multipliers, but driven by the standardized residuals zt instead of the original returns yt. Rt simply extracts the correlation structure from Qt.

15 / 17

SLIDE 16

ST 810-006 Statistics and Financial Risk

Role of the Copula In each case, we construct Σt = var(yt|yt−1, yt−2, . . . ). Write ǫt = Σ−1/2

t

yt so that var(ǫt) = IJ. Here Σ−1/2

t

could be any inverse square root of Σt; e.g. triangular (Choleski), symmetric positive definite (spectral decomposition). In PC-GARCH, a specific version was implied.

16 / 17

SLIDE 17

ST 810-006 Statistics and Financial Risk

The distribution of ǫt might be assumed to be independent normal (with no tail dependence), or multivariate t (with positive tail dependence). It could alternatively be constructed to have appropriate tail lengths and appropriate tail dependences by separately estimating:

1 / 17

GARCH Recall the GARCH model: yt = financial variable, such as log-return on some asset

for the conditional mean structure. Assume yt = σtǫt, where {ǫt} are independent and follow some fixed distribution (standard normal, standardized t, . . . ). GARCH(1, 1): σ2

t = ω + αy 2 t−1 + βσ2 t−1.

The “standardized residuals” are zt = ˆ σ−1

t yt.

2 / 17

Multivariate GARCH With several series, yt = vector of J financial variables. The analog of σ2

t = var(yt|yt−1, yt−2, . . . )

is Σt = var(yt|yt−1, yt−2, . . . ). The purpose of a multivariate xARCH model is to provide a recursive expression for Σt.

3 / 17

PC-GARCH: use PCA. First stage: fit univariate GARCH models to the individual series. Put standardized residuals for time t in a vector zt = S−1

1,tyt,

where S1,t is the diagonal matrix of first stage conditional standard deviations. Set up a data matrix and use the SVD: Z = (T × J) data matrix with rows z′

t, t = 1, 2, . . . , T

=

T×J

D

J×J,diagonal

V′

J×J

Second stage: fit univariate GARCH models to the PC scores (here columns of U; more conventionally, of UD). Write ǫt = S−1

2,tut

where S2,t is the diagonal matrix of second stage conditional standard deviations. Model ǫt as:

NJ(0, I); tJ,ν(0, I); a meta t-distribution, combining marginal t-distributions and a t-copula, all potentially with different degrees of freedom; some other non-Gaussian distribution with chosen tail length, tail dependence, and shape.

5 / 17

The conditional distribution of yT+1 is represented as yT+1 = S1,T+1zT+1 = S1,T+1VDuT+1 = S1,T+1VDS2,T+1ǫT+1 where:

V and D come from the SVD; S1,T+1 and S2,T+1 come from the first and second stage GARCH recursions, respectively; ǫT+1 follows the chosen model.

Note that the distribution of ǫT+1 does not depend on the past, so it is independent of the past.

6 / 17

7 / 17

Reduced Rank PC-GARCH: recall that if d2

1 + d2 2 + · · · + d2 k ≫ d2 k+1 + · · · + d2 J then

Z ≈

T×k

D(k)

k×k,diagonal

V(k)′

k×J

So only k principal component score series need to be modeled, and ǫt consists of only k variables whose distributions need to be modeled. If k ≪ J, modeling is much simplified.

8 / 17

PC-GARCH and Reduced Rank PC-GARCH are related to Orthogonal GARCH (OGARCH) and Generalized Orthogonal GARCH (GO-GARCH). These model the conditional covariance matrix of yt directly: Σt = XDtX′ where Dt is a diagonal matrix of univariate GARCH conditional variances, and:

in OGARCH, X is (J × k) with orthogonal columns, like V(k); in GO-GARCH, X is (J × J) with no orthogonality constraints.

Then yt is modeled as Gaussian or t with covariance matrix Σt. Note: PC-GARCH makes Σt more complicated: Σt = S1,tVDS2

2,tDV′S1,t.

9 / 17

Non-PCA Approaches MGARCH is a very general extension of univariate GARCH. MGARCH(1, 1): vech(Σt) = Avech(yt−1y′

t−1) + Bvech(Σt−1) + c

where vech(·) vectorizes the lower triangle of a symmetric matrix: vech(S) =              s1,1 s2,1 . . . sJ,1 s2,2 s3,2 . . . sJ,J             

10 / 17

MGARCH is over-parametrized: vech(S) is (J(J + 1)/2 × 1). So A and B are ((J(J + 1)/2) × (J(J + 1)/2)), with ∼ J4/4 entries each, and c is ((J(J + 1)/2) × 1). Constraining Σt to be non-negative definite is a challenge.

11 / 17

In diag-MGARCH, A and B are constrained to be diagonal–some improvement. Note: diagonal multiplication of vech(·) is equivalent to entrywise multiplication: Σt = A ◦ (yt−1y′

t−1) + B ◦ Σt−1 + C

where A, B, and C are now (J × J), and “◦” denotes entrywise (Hadamard, or Schur) product. Constraining Σt to be non-negative definite is still a challenge: requiring A, B, and C to be non-negative definite is sufficient, but not necessary. No “cross-talk” in diag-MGARCH.

12 / 17

BEKK is a different simplified form of MGARCH: Σt = A′(yt−1y′

t−1)A + B′Σt−1B + C

Here A and B are unrestricted, and C is positive definite symmetric. Off-diagonal entries in A and B introduce cross-talk: volatility in

t−1 term

13 / 17

14 / 17

Dynamic Conditional Correlation (DCC) As in CCC, use separate GARCH models to build Dt. Then Rt = (diag(Qt))−1/2Qt(diag(Qt))−1/2 where Qt satisfies the recursion Qt = θ1zt−1z′

t−1 + θ2Qt−1 + (1 − θ1 − θ2) ¯

Q. That is, Qt follows an even simpler MGARCH model with scalar multipliers, but driven by the standardized residuals zt instead of the original returns yt. Rt simply extracts the correlation structure from Qt.

15 / 17

Role of the Copula In each case, we construct Σt = var(yt|yt−1, yt−2, . . . ). Write ǫt = Σ−1/2

t

yt so that var(ǫt) = IJ. Here Σ−1/2

t

could be any inverse square root of Σt; e.g. triangular (Choleski), symmetric positive definite (spectral decomposition). In PC-GARCH, a specific version was implied.

16 / 17

The distribution of ǫt might be assumed to be independent normal (with no tail dependence), or multivariate t (with positive tail dependence). It could alternatively be constructed to have appropriate tail lengths and appropriate tail dependences by separately estimating:

the marginal distribution of each component of ǫt; a copula to introduce nonlinear dependence.

Note: if Σ−1/2

t

is non-sparse, the tail properties of components

The symmetric positive definite version relates each component

17 / 17