Lectures for Part II: Time Series Models in Finance

Nonlinear Time Series Modeling 

Part II: Time Series Models in Finance 

Richard A. Davis 

Colorado State University 

(http://www.stat.colostate.edu/~rdavis/lectures) 

MaPhySto Workshop 

Copenhagen 

September 27 — 30, 2004 

MaPhySto Workshop 9/04 

1

Part II: Time Series Models in Finance 

1. Classification of white noise 

2. Examples 

3. “Stylized facts” concerning financial time series 

4. ARCH and GARCH models 

5. Forecasting with GARCH 

6. IGARCH 

7. Stochastic volatility models 

8. Regular variation and application to financial TS 

8.1 univariate case 

8.2 multivariate case 

8.3 applications of multivariate regular variation 

8.4 application of multivariate RV equivalence 

8.5 examples 

8.6 Extremes for GARCH and SV models 

8.7 Summary of results for ACF of GARCH & SV models 


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1. Classification of White Noise 

As we have already seen from financial data, such as log(returns), 

and from residuals from some ARMA model fits, one needs to 

consider time series models for white noise (uncorrelated) that 

allows for dependence. 

Classification of WN (in increasing degree of “whiteness”). 


3

1. Classification of White Noise (cont) 


4

2. Examples 

(1) All-pass processs. Satisfies W1 and not W2. 


5

2. Examples (cont) 


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Properties of ARCH(1) process: 

1. Strictly stationary solution if 0 < α 1


Likelihood function: 


13


A realization of the process 


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The sample ACF. 


15


The sample ACF of the absolute values and squares. 


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3. “Stylized Facts” of Financial Returns 

Define X t = 100*(ln (P t ) - ln (P t-1 )) (log returns) 

• heavy tailed 

P(|X 1 | > x) ~ C x -α , 0 < α < 4. 

• uncorrelated 

ˆ 

ρ X 

( h) 

near 0 for all lags h > 0 (MGD sequence?) 

• |X t | and X t2 have slowly decaying autocorrelations 

ρˆ ( ) and ˆ 

| X | 

h ρ 2 ( h) 

• process exhibits ‘stochastic volatility’. 

X 

converge to 0 slowly as h increases. 


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Log returns for IBM 1/3/62-11/3/00 (blue=1961-1981) 

100*log(returns) 

-20 -10 0 10 


1962 1967 1972 1977 1982 1987 1992 1997 

time 

19

Sample ACF IBM (a) 1962-1981, (b) 1982-2000 

(a) ACF of IBM (1st half) 

(b) ACF of IBM (2nd half) 

ACF 

0.0 0.2 0.4 0.6 0.8 1.0 

ACF 

0.0 0.2 0.4 0.6 0.8 1.0 

0 10 20 30 40 

Lag 

0 10 20 30 40 

Lag 

Remark: Both halves look like white noise. 


20

ACF of squares for IBM (a) 1961-1981, (b) 1982-2000 

(a) ACF, Squares of IBM (1st half) 

(b) ACF, Squares of IBM (2nd half) 

ACF 

0.0 0.2 0.4 0.6 0.8 1.0 

ACF 

0.0 0.2 0.4 0.6 0.8 1.0 

0 10 20 30 40 

0 10 20 30 40 

Lag 

Lag 

Remark: Series are not independent white noise? 


21

Plot of M t (4)/S t (4) for IBM 

M(4)/S(4) 

0.0 0.2 0.4 0.6 0.8 1.0 


0 2000 4000 6000 8000 10000 

Remark: For IID data, M t (k)/S t (k) → 0 as t →∞iff E|X , | k < ∞, where 

t 

k 

k 

M 

t 

= maxs= 1,..., 

t 

| X 

s 

| and St 

= ∑| 

X 

s 

| 

t 

s= 

1 

22

Hill’s estimator of tail index 

The marginal distribution F for heavy-tailed data is often modeled using 

Pareto-like tails, 

1-F(x) = x -α L(x), 

for x large, where L(x) is a slowly varying function (L(xt)/ L(x)→1, as x 

→∞). Now if 

X~ F, then P(log X > x) = P(X > exp(x))=exp(-αx)L(exp(x)), 

and hence log X has an approximate exponential distribution for large x. 

The spacings, 

log( X (j) ) − log(X (j+1) ) , j=1,2,. . . ,m, 

from a sample of size n from an exponential distr are approximately 

independent and Exp(αj) distributed. This suggests estimating α −1 by 


αˆ 

−1 

= 

= 

1 

m 

1 

m 

m 

∑( log X 

( j) 

− log X 

( j+ 

1) 

) 

j= 

1 

m 

∑( log X 

( j) 

− log X 

( m+ 

1) 

) 

j= 

1 

j 

23

Hill’s estimator of tail index 

Def: The Hill estimate of α for heavy-tailed data with distribution given 

by 

1-F(x) = x -α L(x), 

is 

αˆ 

−1 

= 

1 

m 

m 

∑( log X 

( j) 

− log X 

( j+ 

1) 

) 

j= 

1 

j 

= 

1 

m 

m 

∑( log X 

( j) 

− log X 

( m+ 

1) 

) 

j= 

1 

The asymptotic variance of this estimate for α is 

ˆ 2 

2 

α / m and estimated by α / m. 

(See also GPD=generalized Pareto distribution.) 


24

Hill’s plot of tail index for IBM (1962-1981, 1982-2000) 

Hill 

1 2 3 4 5 

Hill 

1 2 3 4 5 

0 200 400 600 800 1000 

m 

0 200 400 600 800 

m 


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4. ARCH and GARCH Models 


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4. ARCH and GARCH Models (cont) 


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i 


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SS but not WS GARCH Processes 


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Parameter Estimation for Finite-Variance GARCH Models 


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5. Forecasting with GARCH 


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5. Forecasting with GARCH (cont) 


67

6. IGARCH 


68

7. Stochastic Volatility Models 


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7. Stochastic Volatility Models (cont) 


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8. Regular variation and application to financial TS models 

8.1 Regular variation — univariate case 

Def: The random variable X is regularly varying with index α if 

P(|X|> t x)/P(|X|>t) → x −α and P(X> t)/P(|X|>t) →p, 

or, equivalently, if 

P(X> t x)/P(|X|>t) → px −α and P(X< −t x)/P(|X|>t) → qx −α , 

where 0 ≤ p ≤ 1 and p+q=1. 

Equivalence: 

X is RV(α) if and only if P(X ∈ t • ) /P(|X|>t)→ v µ(• ) 

(→ v vague convergence of measures on R\{0}). In this case, 

µ(dx) = (pα x −α−1 I(x>0) + qα (-x) -α−1 I(x

8.1 Regular variation — univariate case (cont) 

Another formulation (polar coordinates): 

Define the ± 1 valued rv θ, P(θ = 1) = p, P(θ = −1) = 1− p = q. 

Then 

X is RV(α) if and only if 

or 

P(| 

X | t x, X/ | X | ∈ S) 

→ 

P(| 

X | > t ) 

> −α 

x 

P( 

θ∈ 

S) 

P(| 

X| t x,X/| X| ∈• ) 

→ 

P(| 

X| > t ) 

> −α 

v 

x 

P( 

θ∈• 

) 

(→ v vague convergence of measures on S 0 = {-1,1}). 


84

8.2 Regular variation — multivariate case 

Multivariate regular variation of X=(X 1 , . . . , X m ): There exists a 

random vector θ ∈ S m-1 such that 

Equivalence: 

P(|X|> t x, X/|X| ∈ •)/P(|X|>t) → v x −α P( θ ∈•) 

(→ v vague convergence on S m-1 , unit sphere in R m ) . 

• P( θ ∈•) is called the spectral measure 

• α is the index of X. 

P( 

X 

∈t 

•) 

→ v 

µ ( 

• 

) 

P 

(| 

X 

| > 

t ) 

µ is a measure on R m which satisfies for x > 0 and A bounded away 

from 0, 

µ(xB) = x −α µ(xA). 


85

8.2 Regular variation — multivariate case (cont) 

Examples: 

1. If X 1 > 0 and X 2 > 0 are iid RV(α), then X= (X 1 , X 2 ) is 

multivariate regularly varying with index α and spectral distribution 

P( θ =(0,1) ) = P( θ =(1,0) ) =.5 (mass on axes). 

Interpretation: Unlikely that X 1 and X 2 are very large at the same 

time. 

Figure: plot of 

(X t1 ,X t2 ) for realization 

of 10,000. 

x_2 

0 10 20 30 40 


0 5 10 15 20 

x_1 

86

2. If X 1 = X 2 > 0, then X= (X 1 , X 2 ) is multivariate regularly varying 

with index α and spectral distribution 

P( θ = (1/√2, 1/√2) ) = 1. 

3. AR(1): X t = .9 X t-1 + Z t , {Z t }~IID symmetric stable (1.8) 

{ 

±(1,.9)/sqrt(1.81), W.P. .9898 

Distr of θ: 

±(0,1), W.P. .0102 

Figure: plot of (X t , 

X t+1 ) for realization 

of 10,000. 

x_{t+1} 

-10 0 10 20 30 


-10 0 10 20 30 

x_t 

87

8.3 Applications of multivariate regular variation 

• Domain of attraction for sums of iid random vectors 

(Rvaceva, 1962). That is, when does the partial sum 

−1 

a n 

∑ 

t= 

1 

converge for some constants a n ? 

n 

• Spectral measure of multivariate stable vectors. 

• Domain of attraction for componentwise maxima of iid 

random vectors (Resnick, 1987). Limit behavior of 

1 

a − 

n 

n 

∨ 

t= 

1 

• Weak convergence of point processes with iid points. 

• Solution to stochastic recurrence equations, Y t = A t Y t-1 + B t 

• Weak convergence of sample autocovariances. 

X 

t 

X 

t 


88

8.3 Applications of multivariate regular variation (cont) 

Linear combinations: 

X ~RV(α) ⇒ all linear combinations of X are regularly varying 

i.e., there exist α and slowly varying fcn L(.), s.t. 

P(c T X> t)/(t -α L(t)) →w(c), exists for all real-valued c, 

where 

w(tc) = t −α w(c). 

Use vague convergence with A c ={y: c T y > 1}, i.e., 

T 

P( 

X 

∈ 

tA 

c 

) 

P 

( c 

X 

> t) 

= 

→µ (A ) 

: 

w( 

), 

c 

= c 

−α 

t L 

( 

t) 

P(| 

X 

| 

> 

t ) 

A c 

where t -α L(t) = P(|X| > t). 


89

8.3 Applications of multivariate regular variation (cont) 

Converse? 

X ~RV(α) ⇐ all linear combinations of X are regularly varying? 

There exist α and slowly varying fcn L(.), s.t. 

(LC) P(c T X> t)/(t -α L(t)) →w(c), exists for all real-valued c. 

Theorem (Basrak, Davis, Mikosch, `02). Let X be a random vector. 

1. If X satisfies (LC) with α non-integer, then X is RV(α). 

2. If X > 0 satisfies (LC) for non-negative c and α is non-integer, 

then X is RV(α). 

3. If X > 0 satisfies (LC) with α an odd integer, then X is RV(α). 


90

8.4 Applications of theorem 

1. Kesten (1973). Under general conditions, (LC) holds with L(t)=1 

for stochastic recurrence equations of the form 

Y t = A t Y t-1 + B t , 

(A t , B t ) ~ IID, 

A t d×d random matrices, B t random d-vectors. 

It follows that the distributions of Y t , and in fact all of the finite dim’l 

distrs of Y t are regularly varying (if α is non-even). 

2. GARCH processes. Since squares of a GARCH process can be 

embedded in a SRE, the finite dimensional distributions of a 

GARCH are regularly varying. 


91

8.5 Examples 

Example of ARCH(1): X t =(α 0 +α 1 X 2 t-1) 1/2 Z t , 

{Z t }~IID. 

α found by solving E|α 1 Z 2 | α/2 = 1. 

α 1 .312 .577 1.00 1.57 

α 8.00 4.00 2.00 1.00 

Distr of θ: 

P(θ ∈•) = E{||(B,Z)|| α I(arg((B,Z)) ∈ •)}/ E||(B,Z)|| α 

where 

P(B = 1) = P(B = -1) =.5 


92

8.4 Examples (cont) 

Example of ARCH(1): α 0 =1, α 1 =1, α=2, X t =(α 0 +α 1 X 2 t-1) 1/2 Z t , 

{Z t }~IID 

Figures: plots of (X t , X t+1 ) and estimated distribution of θ for 

realization of 10,000. 

x_{t+1} 

-20 -10 0 10 20 

-20 -10 0 10 20 

x_t 

0.06 0.08 0.10 0.12 0.14 0.16 0.18 

-3 -2 -1 0 1 2 3 

theta 


93

8.4 Applications of theorem (cont) 

Example: SV model X t = σ t Z t 

Suppose Z t ~ RV(α) and 

logσ 

2 

t 

= 

∞ 

∑ 

j=−∞ 

ψ ε 

j 

t− 

j 

, 

∞ 

∑ 

j=−∞ 

ψ 

2 

j 

< ∞,{ 

ε 

t 

}~ IIDN(0, σ 

2 

). 

Then Z n =(Z 1 ,…,Z n )’ is regulary varying with index α and so is 

X n = (X 1 ,…,X n )’ = diag(σ 1 ,…, σ n ) Z n 

with spectral distribution concentrated on (±1,0), (0, ±1). 

Figure: plot of 

(X t ,X t+1 ) for 

realization of 10,000. 

x_2 

-5000 0 5000 10000 


-5000 0 5000 10000 

x_1 

94

8.6 Extremes for GARCH and SV processes 

Setup 

• X t = σ t Z t , {Z t } ~ IID (0,1) 

• X t is RV (α) 

• Choose {b n } s.t. nP(X t > b n ) →1 

Then 

P 

n − 

( 1 − 

b X1 

≤ x) 

→exp{ 

−x 

α 

n 

}. 

Then, with M n = max{X 1 , . . . , X n }, 

(i) GARCH: 

P( 

b 


γ is extremal index ( 0 < γ < 1). 

(ii) SV model: 

P( 

b 

M 

≤ x) 

→exp{ 

−γx 

−1 −α 

n n 

M 

≤ x) 

→ exp{ −x 

}, 

−1 −α 

n n 

}, 

extremal index γ = 1 no clustering. 

95

8.6 Extremes for GARCH and SV processes (cont) 

(i) GARCH: 

P( 

b 

(ii) SV model: P( 

b 

M 

≤ x) 

→ exp{ −γx 

−1 −α 

n n 

M 

≤ x) 

→exp{ 

−x 

−1 −α 

n n 

} 

} 

Remarks about extremal index. 

(i) 

(ii) 

γ < 1 implies clustering of exceedances 

Numerical example. Suppose c is a threshold such that 

Then, if γ = .5, P( 

b 

(iii) 1/γ is the mean cluster size of exceedances. 

(iv) Use γ to discriminate between GARCH and SV models. 

(v) 

P 

n −1 

( bn 

X1 

−1 

n 

M 

n 

≤ c) 

~ .95 

≤ c) 

~ (.95) 

= .975 

Even for the light-tailed SV model (i.e., {Z t } ~IID N(0,1), the 

extremal index is 1 (see Breidt and Davis `98 ) 

.5 


96

8.6 Extremes for GARCH and SV processes (cont) 

0 10 20 30 

** * * *** *** 

0 20 40 60 

time 


97

8.7 Summary of results for ACF of GARCH(p,q) and SV models 

GARCH(p,q) 

α∈(0,2): 

α∈(2,4): 

α∈(4,∞): 

d 

( ρˆ 

X 

( h)) 

h= , K , m 

⎯⎯→( 

Vh 

/ V0 

) 

h= 

1, K, 

1 m 

1−2/ 

α 

d −1 

( n ρ ( h) 

) ⎯⎯→ γ (0)( V ) . 

ˆ 

X h= 

1, K , m 

X h h= 

1, K, 

m 

1/2 

d −1 

( n ρ ( h) 

) ⎯⎯→ 

(0)( G ) . 

ˆ 

X 

γ 

h= 1, K , m 

X h h= 

1, K, 

m 

, 

Remark: Similar results hold for the sample ACF based on |X t | and 

X t2 . 


98

8.7 Summary of results for ACF of GARCH(p,q) and SV models (cont) 

SV Model 

α∈(0,2): 

1/ α 

d 1 h+ 

1 α h 

( n / ln n) ρˆ 

( h) 

⎯⎯→ 

. 

X 

σ σ 

σ 

1 

2 

α 

S 

S 

0 

α∈(2, ∞): 

1/2 

d −1 

( n ρ ( h) 

) ⎯⎯→ 

(0)( G ) . 

ˆ 

X 

γ 

h= 1, K , m 

X h h= 

1, K, 

m 


99

Sample ACF for GARCH and SV Models (1000 reps) 

(a) GARCH(1,1) Model, n=10000 

-0.3 -0.1 0.1 0.3 

(b) SV Model, n=10000 

-0.06 -0.02 0.02 


100

Sample ACF for Squares of GARCH (1000 reps) 

(a) GARCH(1,1) Model, n=10000 

0.0 0.2 0.4 0.6 

b) GARCH(1,1) Model, n=100000 

0.0 0.2 0.4 0.6 


101

Sample ACF for Squares of SV (1000 reps) 

(c) SV Model, n=10000 

0.0 0.01 0.02 0.03 0.04 

0.0 0.05 0.10 0.15 

(d) SV Model, n=100000 


102

Example: Amazon returns May 16, 1997 to June 16, 2004. 

.40 

Series 

.20 

.00 

-.20 

-.40 

-.60 

-.80 

-1.00 

1.00 

0 400 800 1200 1600 

Residual ACF: Abs values 

Residual ACF: Squares 

1.00 

.80 

.80 

.60 

.60 

.40 

.40 

.20 

.20 

.00 

.00 

-.20 

-.20 

-.40 

-.40 

-.60 

-.60 

-.80 

-.80 

-1.00 


0 5 10 15 20 25 30 35 40 

-1.00 

0 5 10 15 20 25 30 35 40 

103

Wrap-up 

• Regular variation is a flexible tool for modeling both dependence 

and tail heaviness. 

• Useful for establishing point process convergence of heavy-tailed 

time series. 

• Extremal index γ < 1 for GARCH and γ =1for SV. 

Unresolved issues related to RV⇔ (LC) 

• α = 2n? 

• there is an example for which X 1 , X 2 > 0, and (c, X 1 ) and (c, X 2 ) 

have the same limits for all c > 0. 

• α = 2n−1 and X > 0 (not true in general). 


104

Lectures for Part II: Time Series Models in Finance

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