Limit Theory for Some Non-Linear Time Series
Models Including GARCH and SV
Richard A. Davis
Colorado State University
Thomas Mikosch
University of Copenhagen
Bojan Basrak
EURANDOM
1
Outline
Characteristics of some financial time series
IBM returns
Linear processes
Background results
Multivariate regular variation
Point process convergence
Application to sample ACVF and ACF
Applications
Stochastic recurrence equations (GARCH)
Stochastic volatility models
More on multivariate regular variation (Mikosch)
2
Characteristics of Some Financial Time Series
Define Xt = 100*(ln (Pt) - ln (Pt-1)) (log returns)
• heavy tailed
P(|X1| > x) ~ C x−α,
0 < α < 4.
• uncorrelated
ρˆ X ( h ) near 0 for all lags h > 0 (MGD sequence)
• |Xt| and Xt2 have slowly decaying autocorrelations
ρˆ |X | ( h ) and ρˆ X 2 ( h ) converge to 0 slowly as h increases.
• process exhibits ‘stochastic volatility’.
3
-10
-20
100*log(returns)
0
10
Log returns for IBM 1/3/62-11/3/00 (blue=1961-1981)
1962
1967
1972
1977
1982
time
1987
1992
1997
4
Sample ACF IBM (a) 1962-1981, (b) 1982-2000
(b) ACF of IBM (2nd half)
0.0
0.0
0.2
0.2
0.4
0.4
ACF
ACF
0.6
0.6
0.8
0.8
1.0
1.0
(a) ACF of IBM (1st half)
0
10
20
Lag
30
40
0
10
20
30
40
Lag
5
Sample ACF of abs values for IBM (a) 1961-1981, (b) 1982-2000
0.8
0.6
0.0
0.2
0.4
ACF
0.0
0.2
0.4
ACF
0.6
0.8
1.0
(b) ACF, Abs Values of IBM (2nd half)
1.0
(a) ACF, Abs Values of IBM (1st half)
0
10
20
Lag
30
40
0
10
20
30
40
Lag
6
Sample ACF of squares for IBM (a) 1961-1981, (b) 1982-2000
0.8
0.6
0.0
0.2
0.4
ACF
0.0
0.2
0.4
ACF
0.6
0.8
1.0
(b) ACF, Squares of IBM (2nd half)
1.0
(a) ACF, Squares of IBM (1st half)
0
10
20
Lag
30
40
0
10
20
30
40
Lag
7
0.0
0.0
0.2
0.2
0.4
0.4
ACF
ACF
0.6
0.6
0.8
0.8
1.0
1.0
Sample ACF of original data and squares for IBM 1962-2000
0
10
20
Lag
30
40
0
10
20
30
40
Lag
8
0.6
0.4
0.2
0.0
M(4)/S(4)
0.8
1.0
Plot of Mt(4)/St(4) for IBM
0
2000
4000
6000
8000
10000
t
9
5
4
1
2
3
Hill
3
2
1
Hill
4
5
Hill’s plot of tail index for IBM (1962-1981, 1982-2000)
0
200
400
600
m
800
1000
0
200
400
600
800
m
10
-0.04
-0.02
X(t)
0.0
0.02
500-daily log-returns of NZ/US exchange rate
0
100
200
300
400
500
t
11
0.0
0.0
0.2
0.2
ACF
0.4
ACF(|X|^2)
0.4
0.6
0.6
0.8
0.8
1.0
1.0
ACF of log-returns of NZ/US exchange rate
0
10
20
30
lag h
40
50
0
10
20
30
40
50
lag h
12
0.6
0.4
0.2
0.0
M(4)/S(4)
0.8
1.0
Plot of Mt(4)/St(4)
0
100
200
300
t
400
500
13
3
2
1
Hill
4
5
Hill’s plot of tail index
0
20
40
60
80
m
14
Models for Log(returns)
Basic model
Xt = 100*(ln (Pt) - ln (Pt-1)) (log returns)
= σt Zt ,
where
• {Zt} is IID with mean 0, variance 1 (if exists). (e.g. N(0,1) or
a t-distribution with ν df.)
• {σt}is the volatility process
• σt and Zt are independent.
15
Models for Log(returns)-cont
Xt = σt Zt (observation eqn in state-space formulation)
Examples of models for volatility:
(i) GARCH(p,q) process (observation-driven specification)
σ 2t = α0 + α1X 2t -1 + m + α p X 2t -p + β1σ2t -1 + m + β q σ2t -q .
Special case: ARCH(1), X t2 = ( α0 + α1X 2t -1 ) Z 2t .
ρ X (h) = α1h , if α12 < 1 / 3.
2
(ii) stochastic volatility process (parameter-driven specification)
log σ =
2
t
∞
∑ψ jε t − j ,
j = −∞
∞
2
2
ψ
<
∞
,
{
ε
}
~
IID
N(0,
σ
)
∑ j
t
j = −∞
ρ X (h) = Cor (σ t2 , σ t2+ h ) / EZ14
2
16
Model:
Xt =
Properties:
Linear Processes
∞
−α, 0<α<2.
{Z
}~IID,
P(|
Z
|>x)
~
Cx
ψ
Z
t
∑ j t- j t
j = −∞
• P(| Xt|>x) ~ C2 x−α
• Define ρ( h ) =
∞
∑ ψ jψ j +h /
j = −∞
Case α > 2:
^ − ρ(h))
n 1/2(ρ(h)
Case 0 < α < 2:
d
→
∞
2
ψ
∑ j .
j = −∞
∞
∑ (ρ(h+j)+ρ(h-j)-2ρ(j)ρ(h)) N , {N }~IIDN
j
t
j =1
d
^ − ρ(h)) →
(n / ln n)1/α ( ρ(h)
∞
∑ (ρ(h+j)+ρ(h-j)-2ρ(j)ρ(h)) S / S ,
j =1
j
0
{St}~IID stable (α), S0 stable (α/2)
17
Background Results—multivariate regular variation
Multivariate regular variation of X=(X1, . . . , Xm): There exists a
random vector θ ∈ Sm-1 such that
P(|X|> t x, X/|X| ∈ • )/P(|X|>t) →v x−α P( θ ∈ • )
(→v vague convergence on Sm-1) .
• P( θ ∈•) is called the spectral measure
• α is the index of X.
Equivalence: There exist positive constants an and a measure µ,
nP(X/ an ∈ • ) →v µ(• )
In this case, one can choose an and µ such that
µ((x, ∞) × B) = x−α P( θ ∈ B )
18
Background Results—multivariate regular variation
Another equivalence?
MRV ⇔ all linear combinations of X are regularly varying
i.e., if and only if
P(cTX> t)/P(1TX> t) →w(c), exists for all real-valued c,
in which case,
w(tc) = t−αw(c).
(⇒) true
(⇐) established by Basrak, Davis and Mikosch (2000) for α not an
even integer—case of even integer is unknown.
19
Background Results—point process convergence
Theorem (Davis & Hsing `95, Davis & Mikosch `97). Let {Xt} be
a stationary sequence of random m-vectors. Suppose
(i) finite dimensional distributions are jointly regularly varying (let
(θ
θ−k, . . . , θk) be the vector in S(2k+1)m-1 in the definition).
(ii) mixing condition A (an) or strong mixing.
(iii) lim lim sup P ( ∨ | X t | > an y | X 0 | > an y ) = 0.
k →∞
Then
k ≤ |t |≤ rn
n →∞
γ = lim E (| θ
k →∞
exists. If γ > 0, then
n
(k ) α
0
k
| − ∨ | θ(jk ) | ) + /E | θ(0k ) |α
j =1
∞
d
N n := ∑ ε X t / an 
→
N := ∑
t =1
i =1
∞
∑ε
j =1
Pi Qij
,
20
Background Results—point process convergence(cont)
where
• (Pi) are points of a Poisson process on (0,∞) with intensity
function ν(dy)=γαy−α−1dy.
∞
•
∑ε
j =1
Qij
, i ≥ 1, are iid point process with distribution Q, and
Q is the weak limit of
lim E (| θ
k →∞
(k ) α
0
k
| − ∨ |θ
j =1
(k )
j
| ) + I • ( ∑ ε θ( k ) ) / E (| θ
|t |≤ k
t
(k ) α
0
k
| − ∨ | θ(jk ) | ) +
j =1
21
Background Results—application to ACVF & ACF
Set-up: Let {Xt} be a stationary sequence and set
Xt=Xt(m) = (Xt, . . . , Xt+m).
Suppose Xt satisfies the conditions of previous theorem. Then
∞
n
N n := ∑ ε X t / an 
→ N := ∑
d
t =1
i =1
∞
∑ε
j =1
Pi Qij
,
Sample ACVF and ACF:
γˆ X ( h ) = n
−1
n −h
∑X X
t =1
t
t +h
, h ≥ 0,
ρˆ X ( h ) = γˆ X ( h ) / γˆ X ( h ), h ≥ 1,
ACVF
ACF
If EX02 < ∞, then define γX(h) =EX0Xh and ρX(h)=γX(h)/ γX(0).
22
Background Results—application to ACVF & ACF
(i) If α∈(0,2), then
d
( nan−2 γˆ X (h )) h =0,l,m 
→
(Vh ) h =0,l,m
where
d
(ρˆ X (h )) h =1,l,m 
→
(Vh / V0 ) h =1,l,m ,
∞
Vh = ∑
i =1
∞
2 ( 0) ( h )
P
∑ i Qij Qij , h = 0,l, m. (jointly stable)
j =1
(ii) If α∈(2,4) + additional condition, then
(na
(na
−2
n
−2
n
d
(Vh )h =0,l,m
( γˆ X ( h ) − γ X ( h ) )h =0,l,m 
→
d
(ρˆ X ( h ) − ρ X (h )) )h =1,l,m 
→
γ −X1 (0)(Vh − ρ X ( h )V0 )h =1,l,m .
23
Applications—stochastic recurrence equations
Yt= At Yt-1+ Bt,
(At , Bt) ~ IID,
At d×d random matrices, Bt random d-vectors
Examples
ARCH(1): Xt=(α0+α1 X2t-1)1/2Zt, {Zt}~IID. Then the squares
follow an SRE with Yt = X 2t , A t = α1Z 2t , Bt = α0 Z 2t .
GARCH(2,1): X t = σ t Z t , σ 2t = α0 + α1X 2t -1 + α 2 X 2t -2 + β1σ 2t -1 .
Then Yt = (σ 2t , X 2t -1 )' follows the SRE given by
 σ 2t  α1Z 2t -1 + β1 α 2   σ 2t -1  α 0 
 2  +  
 2 =
2
0  X t -2   0 
X t -1   Z t -1
24
Stochastic Recurrence Equations (cont)
Examples (tricks)
GARCH(1,1): X t = σ t Z t , σ2t = α0 + α1X 2t -1 + β1σ2t -1 .
Although this process does not have a 1-dimensional SRE
2
σ
representation, the process t does. Iterating, we have
σ2t = α0 + α1X 2t-1 + β1σ 2t-1 = α0 + α1σ2t-1Z 2t-1 + β1σ 2t-1
= ( α1Z 2t-1 + β1 )σ2t-1 + α0 .
Bilinear(1): Xt = aXt-1+ bXt-1Zt-1 + Zt,
{Zt}~IID
= Yt-1 + Zt ,
Yt = AtYt-1 + Bt At=a+bZt, Bt= AtZt
25
Stochastic Recurrence Equations (cont)
Yt= At Yt-1+ Bt,
(At , Bt) ~ IID
Existence of stationary solution
• E ln+ || A1 || < ∞
• E ln+ || B1 || < ∞
• inf n-1 E ln || A1 … An || =: γ < 0 (γ −Lyapunov exponent)
Ex. (d=1) E ln |A1| < 0.
Strong mixing
If E ||A1||ε < ∞, E |B1|ε < ∞ for some ε > 0, then the SRE (Yt)
is geometrically ergodic ⇒ strong mixing with geometric rate
(Meyn and Tweedie `93).
26
Stochastic Recurrence Equations (cont)
Regular variation of the marginal distribution (Kesten)
Assume A and B have non-negative entries and
• E ||A1||ε < 1 for some ε > 0
• A1 has no zero rows a.s.
• W.P. 1, {ln ρ(A1… An): is dense in R for some n, A1… An >0}
• There exists a κ0 > 0 such that E A κ ln + A < ∞ and
0
κ0



E min ∑ A ij  ≥ d κ0 / 2
i =1,...,d
j =1


d
Then there exists a κ1∈ (0, κ0] such that all linear combinations of
Y are regularly varying with index κ1. (Also need E | B |κ < ∞ .)
1
27
Application to GARCH
Proposition: Let (Yt) be the soln to the SRE based on the squares of
a GARCH model. Assume
• Lyapunov exponent γ < 0. (See Bougerol and Picard`92)
• Z has a positive density on (−∞, ∞) with all moments finite or
E|Z|h = ∞, for all h ≥ h0 and E|Z|h < ∞ for all h < h0 .
• Not all the GARCH parameters vanish.
Then (Yt) is strongly mixing with geometric rate (Boussama `98)
and all finite dimensional distributions are multivariate regularly
varying with index κ1.
Corollary: The corresponding GARCH process is strongly mixing
and has all finite dimensional distributions that are MRV with
index κ = 2κ1.
28
Application to GARCH (cont)
Remarks:
~
~
1. Kesten’s result applied to an iterate of Yt , i.e., Ytm = A t Y(t-1)m + B t
2. Determination of κ is difficult. Explicit expressions only known
in two(?) cases.
• ARCH(1): E|α1 Z2|κ/2 = 1.
α1
κ
.312
8.00
.577
4.00
1.00
2.00
1.57
1.00
• GARCH(1,1): E|α1 Z2+ β1|κ/2 = 1 (Mikosch and Stărică)
• For IGARCH (α1 + β1 = 1), then κ = 2 ⇒infinite variance.
• Can estimate κ empirically by replacing expectations with
sample moments.
29
Summary for GARCH(p,q)
κ∈(0,2):
κ∈(2,4):
d
(ρˆ X ( h )) h =1,,m 
→
(Vh / V0 ) h =1,,m ,
(n
1− 2 / κ
d
ρˆ X ( h ) )h =1,,m 
→
γ −X1 (0)(Vh )h =1,,m .
κ∈(4,∞):
(n
d
ρˆ X ( h ) )h =1,,m 
→
γ −X1 (0)(Gh )h =1,,m .
1/ 2
Remark: Similar results hold for the sample ACF based on |Xt| and
Xt2.
30
Results for IBM Data
Fitted GARCH(1,1) model IBM data with t-noise:
1st half:
X t = σ t Z t , σ 2t = .0342 + .0726 2t -1 + .9107 σ 2t -1 , df = 7.185, κ = 4.56
2nd half:
X t = σ t Z t , σ 2t = .0513 + .0472 2t -1 + .9364σ 2t -1 , df = 5.398, κ = 4.16
both:
X t = σ t Z t , σ 2t = .0320 + .0557 X 2t -1 + .9319σ 2t -1 , df = 6.089, κ = 4.96
31
Realization of GARCH Process
Fitted GARCH(1,1) model for NZ-USA exchange:
X t = σ t Z t , σ2t = (6.70)10 −7 + .1519 X 2t -1 + .772σ 2t -1
(Zt) ~ IID t-distr with 5 df. κ is approximately 3.8
-20
-10
X(t)
0
10
Realization of fitted GARCH
0
100
200
300
400
500
t
32
ACF of Fitted GARCH(1,1) Process
0.8
0.6
0.0
0.2
0.4
ACF
0.0
0.2
0.4
ACF
0.6
0.8
1.0
A C F o f s q u a re s o f re a liz a tio n 2
1.0
A C F o f s q u a re s o f re a liz a tio n 1
0
10
20
L ag
30
40
50
0
10
20
L ag
30
40
50
33
1.0
0.8
ACF(|X|^2)
0.4
0.6
0.2
0.0
0.0
0.2
ACF(|X|^2)
0.4
0.6
0.8
1.0
ACF of 2 realizations of an (ARCH)2: Xt=(.001+.7 Xt-1)1/2 Zt
0
10
20
30
lag h
40
50
0
10
20
30
40
50
lag h
34
0.0
0.0
0.2
0.2
ACF(|X|^2)
0.4
0.6
ACF(|X|^2)
0.4
0.6
0.8
0.8
1.0
1.0
ACF of 2 realizations of an |ARCH|: Xt=(.001+ Xt-1)1/2 Zt
0
10
20
30
lag h
40
50
0
10
20
30
lag h
40
50
35
Stochastic Volatility Models
SVM: Xt= σt Zt
• (Zt) ~ IID with mean 0 (if it exists)
• (σt) is a stationary process (2 log σt is a linear process) given
by
log σt2 =
∞
∑ ψ j εt − j ,
j = −∞
∞
2
2
ψ
<
∞
,
(
ε
)
~
IID
N(0,
σ
)
∑ j
t
j = −∞
Heavy tails: Assume Zt has Pareto tails with index α, i.e.,
P(| Zt| > z) ~ C z−α ⇒ P(| Xt| > z) ~ C Eσα z−α.
Then if α∈(0,2) ,
(n / ln n )
1/ α
ρˆ X ( h ) 
→
d
σ1σ h +1
σ1
2
α
α
Sh
.
S0
36
Stochastic Volatility Models (cont)
Other powers:
1. Absolute values: α∈(1,2),
Ε|Xt | = Ε|σt |Ε|Zt |, Ε|Xt Xt+h | = (Ε|σt σt+h |)(Ε|Zt |Ε|Zt+h |)
Cov(|Xt| , |Xt+h|) = Cov(σt , σt+h)(Ε|Z |)2
Cor(|Xt| , |Xt+h|) = Cor(σt , σt+h)(Ε|Z |)2/ EZ2
= 0 (?).
We obtain
−1 / α
d
n (n ln n )
γˆ |X | ( h ) − γ |X | ( h ) 
→
σ1σ h +1 α Sh
and
σ1σ h +1 α Sh
1/ α
d
(n / ln n ) ρˆ |X | (h ) →
.
2
σ1 α S0
(
)
37
Stochastic Volatility Models (cont)
2. Higher order: α∈(0,2)
The squares are again a SV process and the results of the previous
proposition apply. Namely,
(n / ln n )
2/α
ρˆ X 2 ( h ) 
→
d
In particular,
σ12σ2h +1
α/2
2 2
1 α/2
σ
Sh
.
S0
P
ρˆ X 2 ( h ) 
→
0.
38
Stochastic Volatility Models (cont)
0
-2
-4
-6
-8
log X^2(t)
2
4
6
(log X2 ) - mean for NZ-USA exchange rates
0
100
200
300
400
500
39
Stochastic Volatility Models (cont)
ACF/PACF for (log X2 ) suggests ARMA (1,1) model:
µ= -11.5403, Yt = .9646Yt-1 + εt −.8709 εt-1 , (εt)∼WN(0,4.6653)
ACF
1.00
model
sample
.80
.60
.80
.40
.20
.20
.00
.00
-.20
-.20
-.40
-.40
-.60
-.60
-.80
-.80
0
5
10
15
20
25
30
35
model
sample
.60
.40
-1.00
PACF
1.00
40
-1.00
0
5
10
15
20
25
30
35
40
40
Stochastic Volatility Models (cont)
The ARMA (1,1) model for log X2 leads to the SV model
with
Xt= σt Zt
2 ln σt = -11.5403 + vt + εt
vt = .9646 vt-1 + γt ,
(γt)∼WN(0,.07253)
(εt) ∼WN(0,4.2432).
41
Stochastic Volatility Models (cont)
Simulation of SVM model.
Took Zt to be distributed as a t with 3 df (suitable normalized).
0.0
0.2
ACF
0.4
0.6
0.8
1.0
ACF: abs(realization)
0
10
20
Lag
30
40
50
42
Stochastic Volatility Models (cont)
0.8
0.6
0.0
0.2
0.4
ACF
0.0
0.2
0.4
ACF
0.6
0.8
1.0
A C F : ( r e a liz a t io n ) 4
1.0
A C F : ( r e a liz a t io n ) 2
0
10
20
Lag
30
40
50
0
10
20
Lag
30
40
50
43
Sample ACF for GARCH and SV Models (1000 reps)
-0.3
-0.1
0.1
0.3
(a) GARCH(1,1) Model, n=10000
-0.06
-0.02
0.02
(b) SV Model, n=10000
44
Sample ACF for Squares of GARCH and SV (1000 reps)
0.0
0.2
0.4
0.6
(a) GARCH(1,1) Model, n=10000
0.0
0.05
0.10
0.15
(b) SV Model, n=10000
45
Sample ACF for Squares of GARCH and SV (1000 reps)
0.0
0.2
0.4
0.6
(c) GARCH(1,1) Model, n=100000
0.0
0.01
0.02
0.03
0.04
(d) SV Model, n=100000
46