Time Series – HW 6

Please see attached pdf along with zip files.

I need question 2 through 5 to be completed (all except from question 1).

Note: part b) & c) of question 2 and 3 are not required.

HW needs to be completed with detailed answers by 2pm on November 8th NYC eastern time.

Best,

Henry

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HW Set 6 Data Sets/djao2.tsm
3621.63 1941.50 Sep 10, 93
3634.21 1938.30 DJ AO daily closing values
3615.76 1912.90
3633.65 1903.60
3630.85 1902.60
3613.25 1925.50
3575.80 1924.10
3537.24 1925.20
3547.02 1919.30
3539.75 1928.60
3543.11 1946.50
3567.70 1943.00
3566.02 1942.30
3566.30 1951.40
3555.12 1964.40
3581.11 1972.70
3577.76 1977.00
3587.26 1998.50
3598.99 2018.80
3583.63 2022.50
3584.74 2026.20
3593.41 2039.80
3593.13 2028.00
3603.19 2038.60
3621.63 2062.00
3629.73 2074.10
3642.31 2085.50
3635.32 2075.50
3645.10 2051.70
3636.16 2060.40
3649.30 2061.40
3673.61 2046.90
3672.49 2055.70
3664.66 2068.30
3687.86 2076.30
3680.59 2112.20
3692.61 2132.40
3697.64 2125.30
3661.87 2108.40
3624.98 2101.60
3643.43 2079.90
3647.90 2054.20
3640.07 2050.80
3663.55 2042.90
3662.43 2052.40
3684.51 2074.00
3677.52 2082.90
3710.77 2083.80
3704.35 2104.30
3685.34 2108.00
3694.01 2083.20
3670.25 2049.30
3674.17 2009.60
3687.58 2032.40
3687.58 2042.00
3683.95 2043.10
3677.80 2010.30
3683.95 2009.40
3697.08 2005.40
3702.11 2047.30
3704.07 2047.40
3710.21 2053.70
3718.88 2073.90
3734.53 2096.00
3729.78 2095.70
3740.67 2084.90
3764.43 2094.50
3742.63 2086.60
3716.92 2069.90
3726.14 2074.80
3751.57 2080.20
3755.21 2076.00
3745.15 2067.00
3762.19 2053.20
3757.72 2068.80
3757.72 2089.20
3792.93 2089.20
3793.77 2089.20
3794.33 2126.90
3775.88 2154.50
3754.09 2173.60
3756.60 2173.60
3783.90 2174.30
3798.82 2193.40
3803.88 2200.30
3820.77 2186.00
3865.51 2198.60
3850.31 2206.70
3848.63 2195.60
3842.43 2177.50
3867.20 2206.40
3870.29 2238.20
3870.29 2232.10
3884.37 2248.20
3891.96 2266.20
3914.48 2250.30
3912.79 2224.50
3895.34 2221.90
3908.00 2221.90
3926.30 2250.70
3945.43 2259.90
3978.36 2310.80
3964.01 2310.10
3975.54 2312.10
3967.66 2340.60
3871.42 2332.80
3906.32 2281.10
3906.03 2305.40
3931.92 2270.90
3895.34 2234.30
3894.78 2241.40
3904.06 2238.60
3928.27 2234.00
3937.27 2249.00
3922.64 2240.90
3887.46 2223.20
3887.46 2178.50
3911.66 2202.50
3891.68 2218.90
3839.90 2197.00
3838.78 2148.80
3832.02 2180.10
3809.23 2181.70
3831.74 2154.00
3824.42 2151.40
3832.30 2116.80
3856.22 2144.70
3851.72 2171.70
3853.41 2146.80
3830.62 2155.10
3862.70 2153.10
3862.98 2179.30
3849.59 2172.50
3848.15 2173.50
3865.14 2164.40
3895.65 2163.50
3864.85 2140.50
3862.55 2140.80
3869.46 2180.90
3821.09 2169.80
3774.73 2151.60
3762.35 2108.90
3699.02 2100.80
3626.75 2092.40
3635.96 2053.10
3635.96 2053.10
3593.35 2053.10
3675.41 2050.00
3679.73 2084.10
3693.26 2087.40
3674.26 2082.00
3688.83 2076.00
3681.69 2095.10
3661.47 2111.20
3663.25 2095.00
3661.47 2080.60
3620.42 2095.90
3619.82 2061.40
3598.71 2046.60
3652.54 2029.60
3648.68 2042.50
3705.78 2042.50
3699.54 2069.40
3699.54 2059.70
3668.31 2069.10
3681.69 2066.10
3701.02 2047.90
3714.41 2044.20
3697.75 2018.40
3695.97 1988.10
3669.50 2004.30
3629.04 2009.30
3656.41 2008.20
3629.04 2034.60
3652.84 2041.40
3659.68 2070.00
3671.50 2110.90
3720.61 2096.00
3732.89 2107.80
3758.98 2093.70
3766.35 2103.90
3742.41 2121.00
3745.17 2132.40
3755.30 2105.90
3753.46 2096.90
3757.14 2102.20
3757.14 2091.80
3758.37 2081.80
3760.83 2097.20
3758.99 2077.00
3772.22 2078.60
3768.52 2072.50
3755.91 2070.20
3749.45 2079.70
3753.14 2076.70
3773.45 2069.40
3783.12 2069.40
3814.83 2076.60
3790.41 2074.40
3811.34 2056.00
3776.78 2051.20
3741.90 2024.40
3707.97 1993.60
3724.77 2010.90
3699.09 2022.50
3636.94 2017.90
3685.50 1957.40
3669.64 1974.40
3667.05 1975.10
3624.96 1989.10
3646.65 1965.80
3646.65 1987.10
3652.48 2003.40
3674.50 1991.20
3688.42 1962.20
3709.14 1964.90
3702.99 1961.20
3702.66 1972.90
3704.28 1978.60
3739.25 2007.70
3753.81 2058.00
3755.43 2072.30
3748.31 2077.40
3727.27 2078.60
3732.45 2049.20
3735.04 2052.50
3741.84 2048.30
3735.68 2041.30
3720.47 2041.70
3730.83 2042.10
3764.50 2061.50
3798.17 2082.10
3796.22 2086.90
3792.16 2072.30
3765.79 2083.50
3747.02 2091.90
3753.81 2081.10
3755.76 2086.80
3766.76 2076.50
3750.90 2062.80
3768.71 2051.90
3760.29 2055.70
3784.57 2040.00
3776.48 2059.50
3755.43 2066.80
3755.11 2061.30
3751.22 2063.60
3775.83 2051.60
3846.73 2061.10
3829.89 2077.80
3881.05 2077.20 Aug 26, 94
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HW Set 6 Data Sets/IBM2012.tsm
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HW Set 6 Data Sets/m-INTC7308.txt
date rtn
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HW Set 6 Data Sets/SP500.txt
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0.0263
-0.0438
-0.0312
0.0094
0.0819
0.0112
0.0091
-0.0185
0.0115
0.0385
0.0072
0.0480
-0.0416
-0.0082
-0.0474
0.0344
0.0215
-0.0022
-0.0556
-0.0602
0.0401
-0.0250
0.0442
-0.0353
-0.0187
-0.0765
0.0527
0.0015
-0.0905
-0.0610
-0.0500
0.0733
0.0445
0.0330
-0.0114
0.0474
0.0568
0.0405
0.0091
0.0368
0.0363
-0.0416
0.0007
-0.0413
0.0361
-0.0070
-0.0418
-0.0025
0.0862
0.0181
0.0253
0.0059
0.0044
0.0173
-0.0218
0.0023
0.0345
-0.0049
0.0093
0.0456
0.0118
-0.0171
-0.0375
-0.0014
-0.0408
-0.0189
-0.0066
0.0380
-0.0367
0.0401
-0.0013
-0.1139
0.0166
-0.0100
-0.0036
-0.0233
-0.0391
-0.0336
-0.0147
-0.0778
-0.0903
-0.1193
0.1630
-0.0532
-0.0202
0.1228
0.0599
0.0217
0.0473
0.0441
0.0443
-0.0677
-0.0211
-0.0346
0.0616
0.0247
-0.0115
0.1183
-0.0114
0.0307
-0.0110
-0.0144
0.0409
-0.0081
-0.0051
0.0226
-0.0222
-0.0078
0.0525
-0.0505
-0.0217
-0.0140
0.0002
-0.0236
0.0454
-0.0162
-0.0210
-0.0025
-0.0434
0.0270
0.0028
-0.0615
-0.0248
0.0249
0.0854
0.0042
-0.0176
0.0539
0.0259
-0.0073
-0.0916
0.0166
0.0149
0.0397
-0.0365
0.0552
0.0017
-0.0263
0.0387
0.0087
0.0531
0.0000
-0.0686
0.0426
0.0168
0.0576
-0.0044
-0.1018
0.0411
0.0466
0.0270
0.0650
0.0058
0.0252
0.0160
0.1024
-0.0339
-0.0457
0.0133
0.0360
-0.0235
-0.0017
-0.0104
-0.0022
-0.0621
-0.0538
0.0491
0.0366
-0.0301
-0.0175
-0.0605
-0.0102
0.0400
-0.0392
-0.0203
-0.0230
0.1160
0.0076
0.1104
0.0361
0.0152
0.0331
0.0190
0.0331
0.0749
-0.0123
0.0352
-0.0330
0.0113
0.0102
-0.0152
0.0174
-0.0088
-0.0092
-0.0389
0.0135
0.0055
-0.0594
0.0175
-0.0165
0.1063
-0.0035
-0.0001
-0.0151
0.0224
0.0741
0.0087
-0.0029
-0.0046
0.0541
0.0121
-0.0048
-0.0120
-0.0347
0.0425
0.0651
0.0451
0.0024
0.0715
0.0528
-0.0141
0.0502
0.0141
-0.0587
0.0712
-0.0854
0.0547
0.0215
-0.0283
0.1318
0.0369
0.0264
-0.0115
0.0060
0.0479
0.0482
0.0350
-0.0242
-0.2176
-0.0853
0.0729
0.0404
0.0418
-0.0333
0.0094
0.0032
0.0433
-0.0054
-0.0386
0.0397
0.0260
-0.0189
0.0147
0.0711
-0.0289
0.0208
0.0501
0.0351
-0.0079
0.0884
0.0155
-0.0065
-0.0252
0.0165
0.0214
-0.0688
0.0085
0.0243
-0.0269
0.0920
-0.0089
-0.0052
-0.0943
-0.0512
-0.0067
0.0599
0.0248
0.0415
0.0673
0.0222
0.0003
0.0386
-0.0479
0.0449
0.0196
-0.0191
0.0119
-0.0439
0.1116

Estimation: Conditional MLE or Quasi MLE

Special Note: In this course, we estimate volatility models using

the R package fGarch with garchFit command. The program is

easy to use and allows for several types of innovational distributions:

The default is Gaussian (norm), standardized Student-t distribution

(std), generalized error distribution (ged), skew normal distribution

(snorm), skew Student-t (sstd), skew generalized error distribution

(sged), and standardized inverse normal distribution (snig). Except

for the inverse normal distribution, other distribution functions are

discussed in the textbook. Readers should check the book for details

about the density functions and their parameters.

Example: Monthly log returns of Intel stock

R demonstration: The Use fGarch package.

> library(fGarch)

> da=read.table(“m-intc

303.txt”,header=T)

> head(da)

date rtn

1 1

730

1 0.0

…..

730629 0.13333

> intc=log(da$rtn+1) <== log returns

> acf(intc)

> acf(intc^2)

> pacf(intc^2)

> Box.test(intc^2,lag=10,type=’Ljung’)

Box-Ljung test

data: intc^2

X-squared = 59.72

, df = 10, p-value = 4.091e-09

> m1=garchFit(~garch(3,0),data=intc,trace=F) <== trace=F reduces the amount of output.

> summary(m1)

Title:

GARCH Modelling

Call:

garchFit(formula = ~garch(3, 0), data = intc,

trace = F)

Mean and Variance Equation:

data ~ garch(3, 0)

[data = intc]

Conditional Distribution:

norm

Coefficient(s):

mu omega alpha1 alpha2 alpha3

0.016572 0.0

043 0.

649 0.07

37 0.049045

Std. Errors:

based on Hessian

Error Analysis:

Estimate Std. Error t value Pr(>|t|)

mu 0.016572 0.006423 2.580 0.00988 **

omega 0.012043 0.00

79 7.627 2.4e-

***

alpha1 0.208649 0.129

7 1.615 0.10626

alpha2 0.071837 0.048551 1.480 0.13897

alpha3 0.049045 0.048847 1.004 0.31536

—

Standardised Residuals Tests:

Statistic p-Value

Jarque-Bera Test R Chi^2 169.7731 0

Shapiro-Wilk Test R W 0.9606957 1.970413e-08

Ljung-Box Test R Q(10) 10.97025 0.3598405

Ljung-Box Test R Q(15) 19.59024 0.18822

Ljung-Box Test R Q(20) 20.82192 0.40768

Ljung-Box Test R^2 Q(10) 5.376602 0.864644

Ljung-Box Test R^2 Q(15) 22.73460 0.08993976

Ljung-Box Test R^2 Q(20) 23.70577 0.255481

LM Arch Test R TR^2 20.48506 0.05844884

Information Criterion Statistics:

AIC BIC SIC HQIC

-1.228111 -1.175437 -1.228466 -1.207193

> m1=garchFit(~garch(1,0),data=intc,trace=F)

> summary(m1)
Title:
GARCH Modelling
Call:

garchFit(formula = ~garch(1, 0), data = intc, trace = F)

Mean and Variance Equation:
8

data ~ garch(1, 0)

[data = intc]
Conditional Distribution:
norm
Coefficient(s):

mu omega alpha1

0.016570 0.012490 0.363447

Std. Errors:
based on Hessian
Error Analysis:
Estimate Std. Error t value Pr(>|t|)

mu 0.016570 0.006161 2.689 0.00716 **

omega 0.012490 0.001549 8.061 6.66e-16 ***

alpha1 0.363447 0.131598 2.762 0.00575 **

—

Log Likelihood:

230.2423 normalized: 0.6189309

Standardised Residuals Tests:
Statistic p-Value

Jarque-Bera Test R Chi^2 122.4040 0

Shapiro-Wilk Test R W 0.9647629 8.274158e-08

Ljung-Box Test R Q(10) 13.72604 0.1858587 <=== Meaning?

Ljung-Box Test R Q(15) 22.31714 0.09975386 <==== implication?

Ljung-Box Test R Q(20) 23.88257 0.2475594

Ljung-Box Test R^2 Q(10) 12.50025 0.2529700

Ljung-Box Test R^2 Q(15) 30.11276 0.01152131

Ljung-Box Test R^2 Q(20) 31.46404 0.04935483

LM Arch Test R TR^2 22.036 0.0371183

Information Criterion Statistics:
AIC BIC SIC HQIC

-1.221733 -1.190129 -1.221861 -1.209182

> plot(m1)

Make a plot selection (or 0 to exit):

1: Time Series

2: Conditional SD

Series with 2 Conditional SD Superimposed

4: ACF of Observations

5: ACF of Squared Observations

6: Cross Correlation

7: Residuals

8: Conditional SDs

9: Standardized Residuals

10: ACF of Standardized Residuals

11: ACF of Squared Standardized Residuals

12: Cross Correlation between r^2 and r

13: QQ-Plot of Standardized Residuals

Selection: 13

Make a plot selection (or 0 to exit):
1: Time Series
2: Conditional SD
3: Series with 2 Conditional SD Superimposed
4: ACF of Observations
5: ACF of Squared Observations
6: Cross Correlation
7: Residuals
8: Conditional SDs
9: Standardized Residuals
10: ACF of Standardized Residuals
11: ACF of Squared Standardized Residuals
12: Cross Correlation between r^2 and r
13: QQ-Plot of Standardized Residuals

Selection: 0

The fitted ARCH(1) model is

rt = 0.0176 + at, at = σt�t, �t ∼ N(0, 1)
σ2t = 0.0125 + 0.363σ

2
t−1.

Model checking statistics indicate that there are some higher order

dependence in the volatility, e.g., see Q(15) for the squared standard-

ized residuals. It turns out that a GARCH(1,1) model fares better

for the data.

Next, consider Student-t innovations.
R demonstration
> m2=garchFit(~garch(1,0),data=intc,cond.dist=”std”,trace=F)

�3 �2 �1 0 1 2 3

�4
�2

0
2

qnorm � QQ Plot

Theoretical Quantiles

Sa
m

ple
Q

ua
nt

ile
s

Figure 3: QQ-plot for standardized residuals of an ARCH(1) model with Gaussian innova-
tions for monthly log returns of INTC stock: 1973 to 2003.

> summary(m2)

Title:
GARCH Modelling
Call:

garchFit(formula = ~garch(1, 0), data = intc, cond.dist = “std”,

trace = F)
Mean and Variance Equation:
data ~ garch(1, 0)
[data = intc]

Conditional Distribution: <====== Standardized Student-t.

std

Coefficient(s):

mu omega alpha1 shape

0.021571 0.013424 0.259867 5.985979

Std. Errors:
based on Hessian
11

Error Analysis:
Estimate Std. Error t value Pr(>|t|)

mu 0.021571 0.006054 3.563 0.000366 ***

omega 0.013424 0.001968 6.820 9.09e-12 ***

alpha1 0.259867 0.119901 2.167 0.030209 *

shape 5.985979 1.660030 3.606 0.000311 *** <== Estimate of degrees of freedom

—
Log Likelihood:

242.9678 normalized: 0.6531391

Standardised Residuals Tests:
Statistic p-Value

Jarque-Bera Test R Chi^2 130.8931 0

Shapiro-Wilk Test R W 0.9637529 5.744026e-08

Ljung-Box Test R Q(10) 14.31288 0.1591926

Ljung-Box Test R Q(15) 23.34043 0.07717449

Ljung-Box Test R Q(20) 24.87286 0.2063387

Ljung-Box Test R^2 Q(10) 15.35917 0.1195054

Ljung-Box Test R^2 Q(15) 33.96318 0.003446127

Ljung-Box Test R^2 Q(20) 35.46828 0.01774746

LM Arch Test R TR^2 24.11517 0.01961957

Information Criterion Statistics:
AIC BIC SIC HQIC

-1.284773 -1.242634 -1.285001 -1.268039

> plot(m2)

Selection: 13 <== The plot shows that the model needs further improvements.

> predict(m2,5) <===== Prediction

meanForecast meanError standardDeviation

1 0.02157100 0.1207911 0.1207911

2 0.02157100 0.1312069 0.1312069

3 0.02157100 0.1337810 0.1337810

4 0.02157100 0.1344418 0.1344418

5 0.02157100 0.1346130 0.1346130

The fitted model with Student-t innovations is

rt = 0.0216 + at, at = σt�t, � ∼ t5.99
σ2t = 0.0134 + 0.260a

2
t−1.

We use t5.99 to denote the standardized Student-t distribution with

5.99 d.f.

Comparison with normal innovations:

• Using a heavy-tailed dist for �t reduces the ARCH effect.
• The difference between the models is small for this particular
instance.

You may try other distributions for �t.

GARCH Model

at = σt�t,

σ2t = α0 +
m∑

i=1
αia

2
t−i +

s∑

j=1
βjσ

2
t−j

where {�t} is defined as before, α0 > 0, αi ≥ 0, βj ≥ 0, and
∑max(m,s)
i=1 (αi + βi) < 1.

Re-parameterization:

Let ηt = a
2
t − σ2t . {ηt} un-correlated series.

The GARCH model becomes

a2t = α0 +
max(m,s)∑

i=1
(αi + βi)a

2
t−i + ηt −

s∑

j=1
βjηt−j.

This is an ARMA form for the squared series a2t .

Use it to understand properties of GARCH models, e.g. moment

equations, forecasting, etc.

Focus on a GARCH(1,1) model

σ2t = α0 + α1a
2
t−1 + β1σ

2
t−1,

• Weak stationarity: 0 ≤ α1, β1 ≤ 1, (α1 + β1) < 1. • Volatility clusters • Heavy tails: if 1− 2α21 − (α1 + β1)2 > 0, then

E(a4t )

[E(a2t )]
2
=

3[1− (α1 + β1)2]
1− (α1 + β1)2 − 2α21

> 3.

• For 1-step ahead forecast,
σ2h(1) = α0 + α1a

2
h + β1σ

2
h.

For multi-step ahead forecasts, use a2t = σ
2
t �

2
t and rewrite the

model as

σ2t+1 = α0 + (α1 + β1)σ
2
t + α1σ

2
t (�

2
t − 1).

2-step ahead volatility forecast

σ2h(2) = α0 + (α1 + β1)σ
2
h(1).

In general, we have

σ2h( ) = α0 + (α1 + β1)σ
2
h( − 1), > 1.

This result is exactly the same as that of an ARMA(1,1) model

with AR polynomial 1− (α1 + β1)B.

Example: Monthly excess returns of S&P 500 index starting from

1926 for 792 observations.

The fitted of a Gaussian AR(3) model

r̃t = rt − 0.0062
r̃t = .089r̃t−1 − .024r̃t−2 − .123r̃t−3 + .007 + at,

σ̂2a = 0.00333.

For the GARCH effects, use a GARCH(1,1) model, we have

A joint estimation:

rt = 0.032rt−1 − 0.030rt−2 − 0.011rt−3 + 0.0077 + at
σ2t = 7.98× 10−5 + .853σ2t−1 + 0.124a2t−1.

Implied unconditional variance of at is

0.0000798

1− 0.853− 0.1243 = 0.00352
close to the expected value. All AR coefficients are statistically in-

significant.

A simplified model:

rt = 0.00745 + at, σ
2
t = 8.06× 10−5 + .854σ2t−1 + .122a2t−1.

Model checking:

For ãt: Q(10) = 11.22(0.34) and Q(20) = 24.30(0.23).

For ã2t : Q(10) = 9.92(0.45) and Q(20) = 16.75(0.67).

Forecast: 1-step ahead forecast:

σ2h(1) = 0.00008 + 0.854σ
2
h + 0.122a

2
h

Horizon 1 2 3 4 5 ∞
Return .0074 .0074 .0074 .0074 .0074 .0074

Volatility .054 .054 .054 .054 .054 .059

R demonstration:

> sp5=scan(“sp500.txt”)

Read 792 items

> pacf(sp5)

> m1=arima(sp5,order=c(3,0,0))

> m1

Call:

arima(x = sp5, order = c(3, 0, 0))

Coefficients:

ar1 ar2 ar3 intercept

0.0890 -0.0238 -0.1229 0.0062

s.e. 0.0353 0.0355 0.0353 0.0019

sigma^2 estimated as 0.00333: log likelihood = 1135.25, aic=-2260.5

> m2=garchFit(~arma(3,0)+garch(1,1),data=sp5,trace=F)

> summary(m2)
Title:
GARCH Modelling
Call:

garchFit(formula = ~arma(3,0)+garch(1, 1), data = sp5, trace = F)

Mean and Variance Equation:

data ~ arma(3, 0) + garch(1, 1)

[data = sp5]

Conditional Distribution: norm

Error Analysis:
Estimate Std. Error t value Pr(>|t|)

mu 7.708e-03 1.607e-03 4.798 1.61e-06 ***

ar1 3.197e-02 3.837e-02 0.833 0.40473

ar2 -3.026e-02 3.841e-02 -0.788 0.43076

ar3 -1.065e-02 3.756e-02 -0.284 0.77677

omega 7.975e-05 2.810e-05 2.838 0.00454 **

alpha1 1.242e-01 2.247e-02 5.529 3.22e-08 ***

beta1 8.530e-01 2.183e-02 39.075 < 2e-16 ***

—
Log Likelihood:

1272.179 normalized: 1.606287

Standardised Residuals Tests:
Statistic p-Value

Jarque-Bera Test R Chi^2 73.04842 1.110223e-16

Shapiro-Wilk Test R W 0.985797 5.961994e-07

Ljung-Box Test R Q(10) 11.56744 0.315048

Ljung-Box Test R Q(15) 17.78747 0.2740039

Ljung-Box Test R Q(20) 24.11916 0.2372256

Ljung-Box Test R^2 Q(10) 10.31614 0.4132089

Ljung-Box Test R^2 Q(15) 14.22819 0.5082978

Ljung-Box Test R^2 Q(20) 16.79404 0.6663038

LM Arch Test R TR^2 13.34305 0.3446075

Information Criterion Statistics:
AIC BIC SIC HQIC

-3.194897 -3.153581 -3.195051 -3.179018

> m2=garchFit(~garch(1,1),data=sp5,trace=F)

> summary(m2)

Title: GARCH Modelling

Call:

garchFit(formula = ~garch(1, 1), data = sp5, trace = F)

Mean and Variance Equation:

data ~ garch(1, 1)

[data = sp5]
Conditional Distribution: norm
Error Analysis:
Estimate Std. Error t value Pr(>|t|)

mu 7.450e-03 1.538e-03 4.845 1.27e-06 ***

omega 8.061e-05 2.833e-05 2.845 0.00444 **

alpha1 1.220e-01 2.202e-02 5.540 3.02e-08 ***

beta1 8.544e-01 2.175e-02 39.276 < 2e-16 ***

—
Log Likelihood:

1269.455 normalized: 1.602848

Standardised Residuals Tests:
Statistic p-Value

Jarque-Bera Test R Chi^2 80.32111 0

Shapiro-Wilk Test R W 0.9850517 3.141228e-07

Ljung-Box Test R Q(10) 11.22050 0.340599

Ljung-Box Test R Q(15) 17.99703 0.262822

Ljung-Box Test R Q(20) 24.29896 0.2295768

Ljung-Box Test R^2 Q(10) 9.920157 0.4475259

Ljung-Box Test R^2 Q(15) 14.21124 0.509572

Ljung-Box Test R^2 Q(20) 16.75081 0.6690903

LM Arch Test R TR^2 13.04872 0.3655092

Information Criterion Statistics:
17

AIC BIC SIC HQIC

-3.195594 -3.171985 -3.195645 -3.186520

> plot(m2)
Make a plot selection (or 0 to exit):
1: Time Series
2: Conditional SD
3: Series with 2 Conditional SD Superimposed
4: ACF of Observations
5: ACF of Squared Observations
6: Cross Correlation
7: Residuals
8: Conditional SDs
9: Standardized Residuals
10: ACF of Standardized Residuals
11: ACF of Squared Standardized Residuals
12: Cross Correlation between r^2 and r
13: QQ-Plot of Standardized Residuals

Selection: 3

> predict(m2,6)

meanForecast meanError standardDeviation

1 0.007449721 0.05377242 0.05377242

2 0.007449721 0.05388567 0.05388567

3 0.007449721 0.05399601 0.05399601

4 0.007449721 0.05410353 0.05410353

5 0.007449721 0.05420829 0.05420829

6 0.007449721 0.05431038 0.05431038

Turn to Student-t innovation. (R output omitted.)

Estimation of degrees of freedom:

rt = 0.0085 + at, at = σt�t, �t ∼ t7
σ2t = .000125 + .113a

2
t−1 + .842σ

2
t−1,

where the estimated degrees of freedom is 7.00.

Forecasting evaluation

Not easy to do; see Andersen and Bollerslev (1998).

IGARCH model

0 200 400 600 800

�0
.2

0.
0

0.
2

0.
4

Series with 2 Conditional SD Superimposed

Inde

Figure 4: Monthly S&P 500 excess returns and fitted volatility

An IGARCH(1,1) model:

at = σt�t, σ
2
t = α0 + β1σ

2
t−1 + (1− β1)a2t−1.

For the monthly excess returns of the S&P 500 index, we have

rt = .007 + at, σ
2
t = .0001 + .806σ

2
t−1 + .194a

2
t−1

For an IGARCH(1,1) model,

σ2h( ) = σ
2
h(1) + ( − 1)α0, ≥ 1,

where h is the forecast origin.

Effect of σ2h(1) on future volatilities is persistent, and the volatility

forecasts form a straight line with slope α0. See Nelson (1990) for

more info.

Special case: α0 = 0.

used in RiskMetrics to VaR calculation.

Example: An IGARCH(1,1) model for the monthly excess returns

of S&P500 index from 1926 to 1991 is given below via R.

rt = 0.0074 + at, at = σt�t

σ2t = 5.11× 10−5 + .143a2t−1 + .857σ2t−1.
R demonstration: Using R script Igarch.R.

> source(“Igarch.R”)

> m4=Igarch(sp5)

Maximized log-likehood: -1268.205

Coefficient(s):
Estimate Std. Error t value Pr(>|t|)

mu 7.41587e-03 1.52545e-03 4.86144 1.1653e-06 ***

omega 5.10855e-05 1.74923e-05 2.92046 0.0034952 **

beta 8.57124e-01 2.14420e-02 39.97404 < 2.22e-16 ***

—
20

Lecture Note of Bus

02, Spring 20

More Volatility Models. Mr. Ruey Tsay

The GARCH-M model

rt = μ + cσ
2
t + at, at = σt�t, σ

2
t = α0 + α1a

2
t−1 + β1σ

2
t−1

where c is referred to as risk premium, which is expected to be posi-

tive.

Example: A GARCH(1,1)-M model for the monthly excess returns

of S&P

00 index from January 1

to December 1991. For numer-

ical stability, I use percentage returns.

The fitted model is

rt = 0.

+ 0.04

σ
2
t + at, σ

2
t = 0.812 + 0.123a

2
t−1 + .854σ

2
t−1.

Std err of risk premium is 0.141 so that the estimate is not statistically

significant at the usual 5% level.

R demonstration

> source(“garchM.R”)

> mn=garchM(sp5*

[1] 2399.822

0: 2380.0229: 0.422452 0.00561297 0.806149 0.121976 0.854361

3: 2379.9525: 0.427596 0.00652832 0.806107 0.121466 0.855756

6: 2378.4838: 0.606807 0.0375055 0.801889 0.126193 0.846833

9: 2377.9587: 0.673997 0.00166937 0.798191 0.125517 0.851915

12: 2377.8474: 0.692209 0.0444464 0.802574 0.122278 0.854141

15: 2377.7922: 0.742796 0.0480959 0.812187 0.122530 0.853507

Maximized log-likehood: 2377.792

Coefficient(s):

Estimate Std. Error t value Pr(>|t|)

mu 0.7427956 0.1540336 4.82229 1.4192e-06 ***

gamma 0.0480959 0.1408765 0.34140 0.7327991

omega 0.8121873 0.2858128 2.84168 0.0044877 **

alpha 0.1225297 0.0220596 5.55449 2.7843e-08 ***

beta 0.8535072 0.0219079 38.95885 < 2.22e-16 ***

—

Remarks: This R script is relatively slow. It takes longer time due

to its use of recursive loop in evaluating likelihood function.

The EGARCH model

Asymmetry in responses to past positive and negative returns:

g(�t) = θ�t + γ[|�t|− E(|�t|)],

with E[g(�t)] = 0.

To see asymmetry of g(�t), rewrite it as

g(�t) =

⎧⎪⎪⎨
⎪⎪⎩
(θ + γ)�t − γE(|�t|) if �t ≥ 0,
(θ − γ)�t − γE(|�t|) if �t < 0.

An EGARCH(m, s) model:

at = σt�t, ln(σ
2
t ) = α0 +

1 + β1B + · · · + βs−1Bs−1
1 − α1B −···− αmBm

g(�t−1).

Some features of EGARCH models:

• uses log trans. to relax the positiveness constraint
• asymmetric responses

Consider an EGARCH(1,1) model

at = σt�t, (1 − αB) ln(σ2t ) = (1 − α)α0 + g(�t−1),

Under normality, E(|�t|) =
√
2/π and the model becomes

(1 − αB) ln(σ2t ) =
⎧⎪⎪⎨
⎪⎪⎩
α∗ + (θ + γ)�t−1 if �t−1 ≥ 0,
α∗ + (θ − γ)�t−1 if �t−1 < 0

where α∗ = (1 − α)α0 −
√
2
π
γ.

This is a nonlinear fun. similar to that of the threshold AR model of

Tong (1978, 1990).

Specifically, we have

σ2t = σ
2α
t−1 exp(α∗)

⎧⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎩

exp[(θ + γ)
at−1√
σ2t−1

] if at−1 ≥ 0,
exp[(θ − γ) at−1√

σ2t−1
] if at−1 < 0.

The coefs (θ + γ) & (θ − γ) show the asymmetry in response to
positive and negative at−1. The model is, therefore, nonlinear if

θ �= 0. Thus, θ is referred to as the leverage parameter.
Focus on the function g(�t−1). The leverage parameter θ shows the

effect of the sign of at−1 whereas γ denotes the magnitude effect.

See Nelson (1991) for an exmaple of EGARCH model.

Another example: Monthly log returns of IBM stock from Jan-

uary 1926 to December 1997 for 864 observations.

For textbook, an AR(1)-EGARCH(1,1) is obtained (RATS program):

rt = 0.0105 + 0.092rt−1 + at, at = σt�t

ln(σ2t ) = −5.496 +
g(�t−1)

1 − .856B,
g(�t−1) = −.0795�t−1 + .2647[|�t−1|−

√
2/π],

Model checking:

For ãt: Q(10) = 6.31(0.71) and Q(20) = 21.4(0.32)

For ã2t: Q(10) = 4.13(0.90) and Q(20) = 15.93(0.66)

Discussion:

Using
√
2/π ≈ 0.7979 ≈ 0.8, we obtain

ln(σ2t ) = −1.0 + 0.856 ln(σ2t−1) +
⎧⎪⎪⎨
⎪⎪⎩
0.1852�t−1 if �t−1 ≥ 0
−0.3442�t−1 if �t−1 < 0.

Taking anti-log transformation, we have

σ2t = σ
2×0.856
t−1 e

−1.001 ×
⎧⎪⎪⎨
⎪⎪⎩
e0.1852�t−1 if �t−1 ≥ 0
e−0.3442�t−1 if �t−1 < 0.

For a standardized shock with magnitude 2, (i.e. two standard devi-

ations), we have

σ2t (�t−1 = −2)
σ2t (�t−1 = 2)

=
exp[−0.3442 × (−2)]

exp(0.1852 × 2) = e
0.318 = 1.374.

Therefore, the impact of a negative shock of size two-standard de-

viations is about 37.4% higher than that of a positive shock of the

same size.

Forecasting: some recursive formula available

Another parameterization of EGARCH models

ln(σ2t ) = α0 + α1
|at−1| + γ1at−1

σt−1
+ β1 ln(σ

2
t−1),

where γ1 denotes the leverage effect.

Below, I re-analyze the IBM log returns by extending the data to

December 2009. The sample size is 1008.

The fitted model is

rt = 0.012 + at, at = σt�t

ln(σ2t ) = −0.6

+
0.231|at−1|− 0.250at−1

σt−1
+ 0.92 ln(σ2t−1).

Since EGARCH and TGARCH (below) share similar objective and

the latter is easier to estimate. We shall use TGARCH model.

The Threshold GARCH (TGARCH) or GJR Model A

TGARCH(s, m) or GJR(s, m) model is defined as

rt = μt + at, at = σt�tσ
2
t = α0 +

s∑
i=1

(αi + γiNt−i)a
2
t−i +

m∑
j=1

βjσ
2
t−j,

where Nt−i is an indicator variable such that

Nt−i =

⎧⎪⎪⎨
⎪⎪⎩
1 if at−i < 0,

0 otherwise.

One expects γi to be positive so that prior negative returns have

higher impact on the volatility.

The Asymmetric Power ARCH (APARCH) Model.

This model was introduced by Ding, Engle and Granger (1993) as a

general class of volatility models. The basic form is

rt = μt + at, at = σt�t, �t ∼ D(0, 1)
σδt = ω +

s∑
i=1

αi(|at−i|− γiat−i)δ +
m∑
j=1

βjσ
δ
t−j

where δ is a non-negative real number. In particular, δ = 2 gives rise

to the TGARCH model and δ = 0 corresponds to using log(σt).

Theoretically, one can use any power δ to obtain a model. In practice,

two things deserve further consideration. First, δ will also affect the

specification of the mean equation, i.e., model for μt. Second, it is

hard to interpret δ, except for some special values such as 0, 1, 2.

In R, one can fix the value of δ a priori using the subcommand

include.delta=F, delta = 2.

Here I pre-fix δ = 2. Thus, we can use APARCH model to estimate

TGARCH model. Consider the percentage log returns of monthly

IBM stock from 1926 to 2009.

R demonstration

> da=read.table(“m-ibm2609.txt”,header=T)

> head(da)

date ibm

1 19260130 -0.010381

…..

6 19260630 0.068493

> ibm=log(da$ibm+1)*100

> m1=garchFit(~aparch(1,1),data=ibm,trace=F,delta=2,include.delta=F)

> summary(m1)

Title:

GARCH Modelling

Call:

garchFit(formula = ~aparch(1, 1), data = ibm, delta = 2, include.delta = F,

trace = F)

Mean and Variance Equation:

data ~ aparch(1, 1)

[data = ibm]

Conditional Distribution: norm

Coefficient(s):

mu omega alpha1 gamma1 beta1

1.18659 4.33663 0.10767 0.22732 0.79468

Std. Errors: based on Hessian

Error Analysis:

Estimate Std. Error t value Pr(>|t|)

mu 1.18659 0.20019 5.927 3.08e-09 ***

omega 4.33663 1.34161 3.232 0.00123 **

alpha1 0.10767 0.02548 4.225 2.39e-05 ***

gamma1 0.22732 0.10018 2.269 0.02326 *

beta1 0.79468 0.04554 17.449 < 2e-16 ***

—

Log Likelihood:

-3329.177 normalized: -3.302755

Standardised Residuals Tests:

Statistic p-Value

Jarque-Bera Test R Chi^2 67.07416 2.775558e-15

Shapiro-Wilk Test R W 0.9870142 8.597234e-08

Ljung-Box Test R Q(10) 16.90603 0.07646942

Ljung-Box Test R Q(15) 24.19033 0.06193099

Ljung-Box Test R Q(20) 31.89097 0.04447407

Ljung-Box Test R^2 Q(10) 4.591691 0.9167342

Ljung-Box Test R^2 Q(15) 11.98464 0.6801912

Ljung-Box Test R^2 Q(20) 14.79531 0.7879979

LM Arch Test R TR^2 7.162971 0.8466584

Information Criterion Statistics:

AIC BIC SIC HQIC

6.615430 6.639814 6.615381 6.624694

> plot(m1) <= shows normal distribution is not a good fit.

> m1=garchFit(~aparch(1,1),data=ibm,trace=F,delta=2,include.delta=F,cond.dist=”std”)

> summary(m1)
Title:
GARCH Modelling
Call:

garchFit(formula = ~aparch(1, 1), data = ibm, delta = 2, cond.dist = “std”,

include.delta = F, trace = F)

Mean and Variance Equation:
data ~ aparch(1, 1)
[data = ibm]

Conditional Distribution: std

Coefficient(s):

mu omega alpha1 gamma1 beta1 shape

1.20476 3.98975 0.10468 0.22366 0.80711 6.67329

Std. Errors: based on Hessian
Error Analysis:
Estimate Std. Error t value Pr(>|t|)
7

mu 1.20476 0.18715 6.437 1.22e-10 ***

omega 3.98975 1.45331 2.745 0.006046 **

alpha1 0.10468 0.02793 3.747 0.000179 ***

gamma1 0.22366 0.11595 1.929 0.053738 .

beta1 0.80711 0.04825 16.727 < 2e-16 ***

shape 6.67329 1.32779 5.026 5.01e-07 ***

—
Log Likelihood:

-3310.21 normalized: -3.283938

Standardised Residuals Tests:
Statistic p-Value

Jarque-Bera Test R Chi^2 67.82336 1.887379e-15

Shapiro-Wilk Test R W 0.9869698 8.212564e-08

Ljung-Box Test R Q(10) 16.91352 0.07629962

Ljung-Box Test R Q(15) 24.08691 0.06363224

Ljung-Box Test R Q(20) 31.75305 0.04600187

Ljung-Box Test R^2 Q(10) 4.553248 0.9189583

Ljung-Box Test R^2 Q(15) 11.66891 0.7038973

Ljung-Box Test R^2 Q(20) 14.18533 0.8209764

LM Arch Test R TR^2 6.771675 0.872326

Information Criterion Statistics:
AIC BIC SIC HQIC

6.579782 6.609042 6.579711 6.590898

> plot(m1)

Make a plot selection (or 0 to exit):

1: Time Series

2: Conditional SD

3: Series with 2 Conditional SD Superimposed

4: ACF of Observations

5: ACF of Squared Observations

6: Cross Correlation

7: Residuals

8: Conditional SDs

9: Standardized Residuals

10: ACF of Standardized Residuals

11: ACF of Squared Standardized Residuals

12: Cross Correlation between r^2 and r

13: QQ-Plot of Standardized Residuals

Selection: 13

For the percentage log returns of IBM stock from 1926 to 2009, the

fitted GJR model is

rt = 1.20 + at, at = σt�t, �t ∼ t∗6.67
σ2t = 3.99 + 0.105(|at−1|− 0.224at−1)2 + .807σ2t−1,

where all estimates are significant, and model checking indicates that

the fitted model is adequate.

Note that, we can obtain the model for the log returns as

rt = 0.012 + at, at = σt�t

σ2t = 3.99 × 10−4 + 0.105(|at−1|− 0.224at−1)2 + .807σ2t−1.

The sample variance of the IBM log returns is about 0.005 and the

empirical 2.5% percentile of the data is about −0.130. If we use these
two quantities for σ2t−1 and at−1, respectively, then we have

σ2t (−)
σ2t (+)

=
0.0004 + 0.105(0.130 + 0.224 × 0.130)2 + 0.807 × 0.005
0.0004 + 0.105(0.130 − 0.224 × 0.130)2 + 0.807 × 0.005

= 1.849.

In this particular case, the negative prior return has about 85% higher

impact on the conditional variance.

Stochastic volatility model

A (simple) SV model is

at = σt�t, (1 − α1B −···− αmBm) ln(σ2t ) = α0 + vt
where �t’s are iid N(0, 1), vt’s are iid N(0, σ

2
v), {�t} and {vt} are

independent.

�3 �2 �1 0 1 2 3

�
2

0
2

qnorm � QQ Plot

Theoretical Quantiles

S
a
m

p
le

Q
u
a
n
til

e
s

Figure 1: Normal probability plot for TGARCH(1,1) model fitted to monthly percentage log
returns of IBM stock from 1926 to 2009

�4 �2 0 2 4

�
2
0
2
4

qstd � QQ Plot

Theoretical Quantiles
S
a
m
p
le
Q
u
a
n
til
e
s

Figure 2: QQ plot for TGARCH(1,1) model fitted to monthly percentage log returns of IBM
stock from 1926 to 2009.

Long-memory SV model

A simple LMSV is

at = σt�t, σt = σ exp(ut/2), (1 − B)dut = ηt
where σ > 0, �t’s are iid N(0, 1), ηt’s are iid N(0, σ

2
η) and indepen-

dent of �t, and 0 < d < 0.5.

The model says

ln(a2t) = ln(σ
2) + ut + ln(�

2
t)

= [ln(σ2) + E(ln �2t)] + ut + [ln(�
2
t) − E(ln �2t)]

≡ μ + ut + et.

Thus, the ln(a2t) series is a Gaussian long-memory signal plus a non-

Gaussian white noise; see Breidt, Crato and de Lima (1998).

Application

see Examples 3.4 & 3.5

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