Asymptotic Properties of GARCH Models in the Presence of

Faculty of Economics and Business Administration

Asymptotic Properties of GARCH Modelsin the Presence of Time VaryingUnconditional Variance:A Simulation Study

Oliver Old

Applied Statistics and Methods of Empirical Social ResearchOliver Old, M.Sc.


Outline

1. Model Specification

2. Simulation Study

3. Conclusion

4. References


Slide 2 Statistische Woche Trier 2019 - 12th September


Model Specification: Short Term Volatility

yt = 100 · ln (pt/pt−1) (1)

E[yt|Ψt−1] = µt = φ1yt−1 + φ2yt−2 (2)

Ψt−1 = {yt−1, yt−2, yt−3, ...}

yt − µt = εt =√htzt (3)

zti.i.d.∼ N (0, 1)

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 100000

500

1000

1500

2000

2500

3000

Figure: S&P500 1980-2018 prices

0 2000 4000 6000 8000 10000-25

-20

-15

-10

-5

0

5

10

15

Figure: S&P500 1980-2018 log-returns




Model Specification: Short Term Volatility

symmetric: GARCH(1,1) Bollerslev (1986)

E[ε2t |Ψt−1] = ht = α0 + α1ε2t−1 + β1ht−1 (4a)

E[ht] = σ2 = α0/(1− α1 − β1) (4b)

asymmetric: GJR-GARCH(1,1) Glosten et al. (1993)

E[ε2t |Ψt−1] = ht = α0 +(α1 + δ11εt−1<0

)ε2t−1 + β1ht−1 (5a)

E[ht] = σ2 = α0/(1− α1 −1

2δ1 − β1) (5b)




I Example: AR(2)-GARCH(1,1): S&P500 01/1980 - 12/2018ht = 0.0154 + 0.0847ε2t−1 + 0.9032ht−1

I α1 + β1 = 0.9879 ≈ 1 Engle and Bollerslev (1986)

σ2 causes non-stationarity Hillebrand (2005); Mikosch and Starica (2004)

0 2000 4000 6000 8000 100000

5

10

15

20

25

30

35

40

45

50

h t

1.24

1.26

1.28

1.3

1.32

1.34

1.36

2

ht

2whole sample

21980-1999

21999-2018

Figure: S&P500 1980-2018 ht with σ2




I Example: AR(2)-GARCH(1,1): S&P500 01/1980 - 12/2018ht = 0.0154 + 0.0847ε2t−1 + 0.9032ht−1

I α1 + β1 = 0.9879 ≈ 1 Engle and Bollerslev (1986)

σ2 causes non-stationarity Hillebrand (2005); Mikosch and Starica (2004)

0 2000 4000 6000 8000 100000

5

10

15

20

25

30

35

40

45

50

h t

1.24

1.26

1.28

1.3

1.32

1.34

1.36

2

ht

2whole sample

21980-1999

21999-2018

Figure: S&P500 1980-2018 ht with σ2




Model Specification: Long Term Volatility

Spline-GARCH: original framework Engle and Rangel (2008)

εt = σtzt (6)

σt =√ht√τt

τt = c · exp

(w0t+

K∑i=1

wi((t− ti−1)+)2 + γxt

)(7)

(t− ti)+ =

{(t− ti) if t > ti

0 otherwise

t0 = 0; t1, t2, ...., tK−1




Spline Basis Functions with K = 9 knots and T = 9835 data points

0 2000 4000 6000 8000 10000

time

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Figure: (a) Basis Functions

0 2000 4000 6000 8000 10000

time

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Figure: (c) Sum of Scaled Basis Functions

0 2000 4000 6000 8000 10000

time

-150

-100

-50

0

50

100

150

200

Figure: (b) Scaled Basis Functions

(a)t

T,

((t− t0)+

T

)2

,...,

((t− t8)+

T

)2

(b) c′, w0

t

T, w1

((t− t0)+

T

)2

,...,w9

((t− t8)+

T

)2

(c)

c′ + w0t

T+

K∑i=1

wi

((t− ti−1)+

T

)2

⇒ weights are estimated within σ2t framework





Spline-GARCH: original framework Engle and Rangel (2008)

GARCH(1,1) ht = α0 + α1

(ε2t−1

τt−1

)+ β1ht−1 (8a)

GJR-GARCH(1,1) ht = α0 +(α1 + δ11εt−1<0

)( ε2t−1

τt−1

)+ β1ht−1 (8b)

variance targeting α0 = (1− α1 − β1) (9a)

α0 = (1− α1 −1

2δ1 − β1) (9b)

“unit-GARCH process“ ⇒ σ2 = 1





σ2t = α0τt−1 + α1ε

2t−1 + β1σ

2t−1 (10a)

σ2t = α0τt−1 +

(α1 + δ11εt−1<0

)ε2t−1 + β1σ

2t−1 (10b)

until now:

I E[ε2t |Ψt−1] = ht

I E[E[ε2t |Ψt−1]] = E[ε2t ] = E[ht] = σ2 (constant)

now:

I E[ε2t ] = E[htτtz2t ] = E[ht]τt = σ2τt = τt (time varying)





spline-GARCH (modified)

τt = exp

(c′ + w0

t

T+

K∑i=1

wi

((t− ti−1)+

T

)2)

(11)

⇒ t is rescaled⇒ to ensure τt > 0 ∀t, c′ = exp(c)⇒ xt not considered




Example: S&P500 1980-2018 AR(2)-SPLINE(9)-GARCH(1,1)

0 2000 4000 6000 8000 100000

5

10

15

20

25

30

35

40

45

50

h t

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

2

Figure: (a) ht with GARCH(1,1) and σ2 = 1

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 100000

0.5

1

1.5

2

2.5

3

Figure: τt and σ2 = 1

0 2000 4000 6000 8000 100000

10

20

30

40

50

60

2 t

0

0.5

1

1.5

2

2.5

3

t

Figure: (b) σ2t and τt with SPLINE(9)-GARCH(1,1)

(a) ht = 0.016 + 0.0804ε2t−1 + 0.9036ht−1

(b) σ2t = 0.0319τt−1 + 0.0881ε

2t−1 + 0.88σ

2t−1

(c) τt = exp

c′ + w0t

T+

K∑i=1

wi

((t− ti−1)+

T

)2

θτt = [0.28′,−4.075, 15.39, 16.41,−152.92, 345.81

− 455.28, 369.37,−185.18, 7.98, 148.69]




Simulation Setup

1.Sampling of real data

I S&P500 equity index (1980-2018) with 9835 data points

2.Estimation of innovations serie εt

I εt = yt − φ1yt−1 − φ2yt−2 AR(2)

3.Estimation of conditional variance

I LSE starting values own algorithm

I BFGS with line searching MATLAB fminunc and own algorithm




Simulation Setup

4.Data Generating Processes (DGP)

I ht = α0 + α1ε2t−1 + β1ht−1 GARCH(1,1)

I ht = α0 +(α1 + δ11εt−1<0

)ε2t−1 + β1ht−1 GJR-GARCH(1,1)

I σ2t = α0τt−1 + α1ε

2t−1 + β1σ

2t−1 SPLINE-GARCH(1,1)

I σ2t = α0τt−1 +

(α1 + δ11εt−1<0

)ε2t−1 + β1σ

2t−1 SPLINE-GJR-GARCH(1,1)

I SPLINE-GARCH(1,1) and SPLINE-GJR-GARCH(1,1) each with 1,4,9 and14 knots




Simulation Setup

5. Monte Carlo Simulation

I εt =√τt√htzt zt

i.i.d.∼ N (0, 1)

I 6 different time series lengthT ∈ {100, 500, 1000, 5000, 10000, 25000}each for every DGP

I resulting in N = 60 different model setups {j = 1, ..., N}each with M = 1000 replications {i = 1, ...,M}

I θji = (αji0, αji1, βji1, δji1, cji′, wji0, ..., wjiK)

I θji = arg maxθ lnL(θj)




Simulation Results

Consistency

θj =

∑Mi=1 θji

Mj(12)

volatility persistence (VP): αj1 + βj1 = θjVP (13)

0 1 4 9 14

knots

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

T = 100

T = 500

T = 1000

True Values

Figure: VP/knots for T ∈ {100, 500, 1000}

0 1 4 9 14

knots

0.94

0.945

0.95

0.955

0.96

0.965

0.97

0.975

0.98

0.985

0.99

T = 5000

T = 10000

T = 25000

True Values

Figure: VP/knots for T ∈ {5000, 10000, 25000}




Consistency

θjp→ θj

bias: ‖θjVP − θjVP‖2 (14)

0 2 4 6 8 10 12 14

knots

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

BIA

S

T=100T=500

T=1000

Figure: BIAS/knots for T ∈ {100, 500, 1000}

0 2 4 6 8 10 12 14

knots

0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

BIA

S

T=5000T=10000

T=25000

Figure: BIAS/knots for T ∈ {5000, 10000, 25000}




Consistency

Standard Deviation (STD): sθj =

(∑Mji=1(θji − θj)2

Mj − 1

) 12

(15)

‖sαj1 + sβj1‖2 = ‖sθjVP‖2 (16)

0 2 4 6 8 10 12 14

knots

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

ST

D

T=100T=500

T=1000

Figure: STD/knots for T ∈ {100, 500, 1000}

0 2 4 6 8 10 12 14

knots

0.005

0.01

0.015

0.02

ST

D

T=5000T=10000

T=25000

Figure: STD/knots for T ∈ {5000, 10000, 25000}




Asymptotic Normality of (β1 − β1)

-4 -3 -2 -1 0 1 2 30

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Coun

ts

Spline(14)-Garch(1,1) T = 100

KernelGaussian

Figure: Jarque-Bera = 220 (p-Value= 0.000)

-0.08 -0.06 -0.04 -0.02 0 0.02 0.040

5

10

15

20

25

Coun

ts


KernelGaussian


-2 -1.5 -1 -0.5 0 0.5 10

0.5

1

1.5

2

2.5

3

Coun

ts


KernelGaussian


-0.03 -0.02 -0.01 0 0.01 0.02 0.030

10

20

30

40

50

60

Coun

ts

Spline(14)-Garch(1,1) T = 25000

KernelGaussian

Figure: Jarque-Bera = 1.8 (p-Value= 0.4)





-5 -4 -3 -2 -1 0 1 2 30

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Coun

ts


KernelGaussian


-0.04 -0.03 -0.02 -0.01 0 0.01 0.02 0.030

5

10

15

20

25

30

35

40

Coun

ts


KernelGaussian


-0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.20

2

4

6

8

10

12

14

Coun

ts


KernelGaussian


-0.015 -0.01 -0.005 0 0.005 0.01 0.0150

10

20

30

40

50

60

70

80

90

100

Coun

ts


KernelGaussian






-3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.50

0.2

0.4

0.6

0.8

1

1.2

Coun

ts


KernelGaussian


-0.04 -0.03 -0.02 -0.01 0 0.01 0.02 0.030

5

10

15

20

25

30

35

40

45

Coun

ts


KernelGaussian


-0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.20

5

10

15

Coun

ts


KernelGaussian


-0.015 -0.01 -0.005 0 0.005 0.01 0.0150

10

20

30

40

50

60

70

80

90

100

Coun

ts


KernelGaussian






-2 -1.5 -1 -0.5 0 0.5 10

0.5

1

1.5

2

2.5

Coun

ts

Garch(1,1) T = 100

KernelGaussian


-0.04 -0.03 -0.02 -0.01 0 0.01 0.02 0.030

5

10

15

20

25

30

35

40

45

Coun

ts

Garch(1,1) T = 5000

KernelGaussian


-0.15 -0.1 -0.05 0 0.05 0.10

2

4

6

8

10

12

14

16

18

Coun

ts

Garch(1,1) T = 1000

KernelGaussian


-0.015 -0.01 -0.005 0 0.005 0.01 0.0150

10

20

30

40

50

60

70

80

90

100

Coun

ts

Garch(1,1) T = 25000

KernelGaussian





Coverage Probability of β1 Lumsdaine (1995)

standard error: se(θji) =[diag

(H−1(θji) · J(θji) · H−1(θji)

)] 12

95% Confidence Interval (CI): βji1 ± 1.96 · se(βji1) (17)

Coverage Probability:#{β1 ∈ CI}

Mj(18)

0 1 4 9 14

Number of knots

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Fra

ctio

n 1 in

side

CI

coverage probability small samples

T = 100T = 500T = 1000

Figure: Fraction/knots for T ∈ {100, 500, 1000}

0 1 4 9 14

Number of knots

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Fra

ctio

n 1 in

side

CI

coverage probability large samples

T = 5000T = 10000T = 25000

Figure: Fraction/knots for T ∈ {5000, 10000, 25000∗}




Conclusion

I Number of knots per oberservation lowers volatility persistence.

I In finite samples the volatility persistence is under estimated.

I Simulation results are basicly in line with asymptotic QML theory.

I BIAS and variance decline with time series length, but increase with numberof knots.

I With increasing number of knots, bias distribution converges slower tonormality.

I The coverage probability understates the advertised significance level.




References

Bollerslev, T. (1986), ‘Generalized autoregressive conditional heteroskedasticity’, Journal ofEconometrics 31(3), 307–327.

Engle, R. F. and Bollerslev, T. (1986), ‘Modelling the persistence of conditional variances’,Econometric Reviews 5(1), 1–50.

Engle, R. F. and Rangel, J. G. (2008), ‘The spline-garch model for low-frequency volatility andits global macroeconomic causes’, Review of Financial Studies 21(3), 1187–1222.

Glosten, L. R., Jagannathan, R. and Runkle, D. E. (1993), ‘On the relation between theexpected value and the volatility of the nominal excess return on stocks’, The Journal ofFinance 48(5), 1779–1801.

Hillebrand, E. (2005), ‘Neglecting parameter changes in garch models’, Journal ofEconometrics 129(1-2), 121–138.

Lumsdaine, R. L. (1995), ‘Finite-sample properties of the maximum likelihood estimator ingarch(1,1) and igarch(1,1) models: A monte carlo investigation’, Journal of Business &Economic Statistics 13(1), 1.

Mikosch, T. and Starica, C. (2004), ‘Nonstationarities in financial time series, the long-rangedependence, and the igarch effects’, Review of Economics and Statistics 86(1), 378–390.





Estimation Procedure: 1.Starting Values

1.1. Maximum Likelihood estimation of GARCH parameters

I ht ≈ h0t (MLE)

1.2. Least Square Estimation of spline parameters

I τt ≈ τ0t (LSE)εt =

√h0t τ

0t zt ⇒ ε2t = h0

t τ0t z

2t ⇒ τ0t = ε2t/h

0t z

2t

I lnτ0t = c0 + w00tT

+∑Ki=1 w

0i

((t−ti−1)+

T

)2





Estimation Procedure: 1.Starting Values

1.2. Least Square Estimation of spline parameterslnτ0

1

lnτ02...

lnτ0T

︸︷︷︸

Y

=

1 1T

((1−t1)+

T

)2. .

((1−tK )+

T

)2

1 2T

((2−t1)+

T

)2. .

((2−tK )+

T

)2

. . . . . .

. . . . . .

1 TT

((T−t1)+

T

)2. .

((T−tK )+

T

)2

︸︷︷︸

X

c0

w00

w01..w0K

︸︷︷︸θ0

+

u1

u2

.

.

.uT

︸︷︷︸U

θ0= (X

′X)−1X′Y

Estimation Procedure: 2.QML-Optimization

I No parameter constraints imposed

I BFGS with line search algorithm



Documents

Asymptotic Properties of GARCH Models in the Presence of