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
�
HW Set 6 Data Sets/IBM2012.tsm
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0.40241503
2.036730282
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HW Set 6 Data Sets/m-INTC7308.txt
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0.0051
-0.0110
0.0353
-0.0418
-0.0326
0.0196
0.0370
0.0369
-0.0013
0.0114
-0.0561
-0.0619
-0.0321
0.0161
-0.0415
0.0428
-0.0206
0.0309
0.0318
0.0150
0.0261
0.0431
0.0119
0.0484
0.0254
0.0224
0.0520
0.0038
-0.0002
0.0005
0.0388
0.0189
-0.0036
0.0349
-0.0150
-0.0456
0.0113
0.0132
0.0276
-0.0715
0.0092
-0.0139
-0.0175
0.0269
0.0195
-0.0248
0.0261
-0.0604
-0.0024
0.0403
0.0463
0.0632
0.0269
0.0255
0.0038
0.0191
-0.0288
0.0328
0.0196
-0.0197
0.0283
0.0393
0.0032
-0.0379
0.0163
-0.0059
-0.0620
-0.0860
-0.0818
0.0636
0.0153
-0.0482
0.0044
0.1016
0.0135
0.0491
-0.0289
0.0355
0.0485
0.0143
-0.0202
-0.0035
0.0487
-0.0110
0.0322
-0.0105
0.0244
0.0269
0.0099
0.0152
0.0061
0.0115
0.0164
0.0182
-0.0162
0.0287
0.0081
-0.0052
0.0039
0.0332
-0.0015
-0.0145
0.0342
-0.0077
-0.0486
0.0134
0.0225
0.0320
0.0273
-0.0088
0.0090
0.0049
-0.0179
-0.0218
0.0205
-0.0541
-0.0161
-0.0135
-0.0778
-0.0070
0.0475
0.0031
-0.0015
0.0782
0.0020
0.0394
0.0422
-0.0524
0.0175
0.0453
-0.0117
0.0328
-0.0291
0.0011
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
7
303.txt”,header=T)
> head(da)
date rtn
1 1
9
730
13
1 0.0
10
05
…..
6
19
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
16
, 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:
7
data ~ garch(3, 0)
[data = intc]
Conditional Distribution:
norm
Coefficient(s):
mu omega alpha1 alpha2 alpha3
0.016572 0.0
12
043 0.
20
8
649 0.07
18
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
15
79 7.627 2.4e-
14
***
alpha1 0.208649 0.129
17
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
11
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
3:
Series with 2 Conditional SD Superimposed
4: ACF of Observations
5: ACF of Squared Observations
9
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)
10
�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)
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 <== The plot shows that the model needs further improvements.
12
> 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.
13
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.
14
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
15
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
16
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
18
0 200 400 600 800
�0
.2
0.
0
0.
2
0.
4
Series with 2 Conditional SD Superimposed
Inde
x
x
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.
19
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
4
1
2
02, Spring 20
12
:
>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
5
00 index from January 1
9
2
6
to December 1991. For numer-
ical stability, I use percentage returns.
The fitted model is
rt = 0.
7
4
3
+ 0.04
8
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*
10
0)
[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 ***
—
1
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
2
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)
3
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
4
ln(σ2t ) = −0.6
11
+
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
5
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 **
6
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
8
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.
9
�3 �2 �1 0 1 2 3
�
2
0
2
4
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
10
�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.
11
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
12