Download Ch6

Chapter 5 : Volatility Models
• Similar to linear regression analysis, many time series exhibit a
non-constant variance (heteroscedasticity). In a regression
model, suppose that yt = 0 + 1x1t + 2x2t + … + t; var(t) =
2t
then instead of using the ordinary least squares (OLS)
procedure, one should use a generalized least squares (GLS)
method to account for the heterogeneity of t.
• With financial time series, it is often observed that variations of
the time series are quite small for a number of successive
periods, then large for a while, then smaller again. It would be
desirable if these changes in volatility can be incorporated into
the model.
1
• This plot shows the weekly dollar/sterling exchange rate
from January 1980 to December 1988 (470 observations)
2
• This first difference of the series is shown here
3
• The levels exhibit wandering movement of
a random walk, and consistent with this, the
differences are stationary about zero and
show no discernable pattern, except that the
differences tend to be clustered (large
changes tend to be followed by large
changes and small changes tend to be
followed by small changes)
• An examination of the series’ ACF and
PACF reveals some of the cited
characteristics
4
The ARIMA Procedure
Name of Variable = rates
Period(s) of Differencing
1
Mean of Working Series
-0.00092
Standard Deviation
0.02754
Number of Observations
469
Observation(s) eliminated by differencing
1
Autocorrelations
Lag
0
Covariance
Correlation
-1 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 1
0.00075843
1.00000 |
1 -0.0000487
-.06416
2 6.52075E-6
0.00860
3 0.00005996
0.07906
4 0.00004290
0.05657
5 -0.0000173
-.02284
6 2.67563E-6
0.00353
7 0.00006114
0.08061
8 -9.5206E-6
-.01255
9 6.54731E-6
0.00863
10 0.00003322
0.04380
11 -0.0000507
-.06689
12 0.00001356
0.01788
13 0.00001637
0.02158
14 0.00003604
0.04752
15 1.26289E-6
0.00167
16 0.00002185
0.02881
17
3.2823E-7
0.00043
18 -0.0000340
-.04483
19 0.00005576
0.07352
20
5.5947E-6
0.00738
21 -3.8865E-6
-.00512
22 0.00001112
0.01466
23 -0.0000168
-.02212
24 0.00003914
0.05161
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Std Error
|********************|
0
.*| .
|
0.046176
.|.
|
0.046365
. |**
|
0.046369
. |*.
|
0.046655
.|.
|
0.046801
.|.
|
0.046825
. |**
|
0.046826
.|.
|
0.047121
.|.
|
0.047128
. |*.
|
0.047131
.*| .
|
0.047218
.|.
|
0.047419
.|.
|
0.047434
. |*.
|
0.047455
.|.
|
0.047556
. |*.
|
0.047556
.|.
|
0.047593
.*| .
|
0.047593
. |*.
|
0.047683
.|.
|
0.047924
.|.
|
0.047927
.|.
|
0.047928
.|.
|
0.047938
. |*.
|
0.047959
"." marks two standard errors
5
Partial Autocorrelations
Lag
Correlation
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
-0.06416
0.00450
0.08023
0.06742
-0.01626
-0.00704
0.07182
-0.00271
0.00843
0.03316
-0.07116
0.01058
0.01856
0.05192
0.01636
0.02016
-0.01202
-0.04319
0.06369
0.01375
0.00007
0.00120
-0.03788
0.05154
-1 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 1
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
.*| .
.|.
. |**
. |*.
.|.
.|.
. |*.
.|.
.|.
. |*.
.*| .
.|.
.|.
. |*.
.|.
.|.
.|.
.*| .
. |*.
.|.
.|.
.|.
.*| .
. |*.
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
6
Engle (1982, Econometrica) called this form of
heteroscedasticity, where 2t depends on 2t1, 2t2, 2t3, etc.
“autoregressive conditional heteroscedasticity (ARCH)”.
More formally, the model is
yt   0  1 x1t 
  k xkt   t ;  t  t 1 ~ N  0,  t2 
q
   0    i t2i
2
t
i 1
where  t 1  yt 1 , xt 1 , xt 2 , represents the past realized
values of the series. Alternatively we may write the error
1
process as
q
2

2 
 t  t  0   i t i  ;t ~ N 0,1
i 1


7
This equation is called an ARCH(q) model.
We require that 0 > 0 and i ≥ 0 to ensure
that the conditional variance is positive.
Stationarity of the series requires that
q

i 1
i
 1.
8
Typical stylized facts about the ARCH(q) process
include:
1. {t} is heavy tailed, much more so than the
Gaussian White noise process.
2. Although not much structure is revealed in the
correlation function of {t}, the series {t2} is
highly correlated.
3. Changes in {t} tends to be clustered.
9
As far as testing is concerned, there are many
methods. Three simple approaches are as follows:
1. Time series test. Since an ARCH(p) process
implies that {t2} follows an AR(p), one can
use the Box-Jenkins approach to study the
correlation structure of t2 to identify the AR
properties
2. Ljung-Box-Pierce test
10
3. Lagrange multipler test
H0: 1 = 2 = … q = 0
H1: 1 ≥ 0, i = 1, …, q (with at least one inequality)
To conduct the test,
i)
Regress et2 on its lags depends on the assumed
order of the ARCH process. For an ARCH(q)
process, we regress et2 on e2t1 … e2tq.
a
ii) The LM statistic is LM   n  q  R ~ q2
under
H0, where R2 is the coefficient of determination
from the auxiliary regression.
2
11
•
The following SAS program estimates an ARCH model for the monthly
stock returns of Intel Corporation from January 1973 to December 1977
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
data intel;
infile 'd:\teaching\ms6217\m-intc.txt';
input r t;
r2=r*r;
lr2=lag(r2);
proc reg;
model r2=lr2;
proc arima;
identify var=r nlag=10;
run;
proc arima;
identify var=r2 nlag=10;
run;
proc autoreg;
model r= /garch =(q=4);
run;
proc autoreg;
model r= /garch =(q=1);
output out=out1 r=e;
run;
proc print data=out1;
var e;
run;
12
•
•
•
The REG Procedure
Model: MODEL1
Dependent Variable: r2
•
Analysis of Variance
•
•
Source
•
•
•
Model
Error
Corrected Total
•
•
•
DF
1
297
298
Sum of
Squares
0.01577
0.49180
0.50757
Root MSE
0.04069
Dependent Mean 0.01766
Coeff Var
230.46618
•
Mean
Square
F Value
0.01577
0.00166
R-Square
Adj R-Sq
9.53
Pr > F
0.0022
0.0311
0.0278
Parameter Estimates
•
•
Variable
DF
Parameter
Estimate
•
•
Intercept
1
lr2
1
0.01455
0.17624
Standard
Error
0.00256
0.05710
t Value
Pr > |t|
5.68
<.0001
3.09
0.0022
13
H0: 1 = 0
H1: otherwise
LM = 299(0.0311)
= 9.2989 > 21, 0.05 = 3.84
Therefore, we reject H0
14
•
The ARIMA Procedure
•
Name of Variable = r
•
•
•
Mean of Working Series 0.028556
Standard Deviation
0.129548
Number of Observations
300
•
Autocorrelations
•
Lag
Covariance
•
•
•
•
•
•
•
•
•
•
•
0
1
2
3
4
5
6
7
8
9
10
0.016783
0.00095235
-0.0000497
0.00098544
-0.0005629
-0.0007545
0.00038362
-0.0002817
-0.0006309
-0.0009289
0.00097606
•
Correlation
1.00000
0.05675
-.00296
0.0587
-.03354
-.04496
0.0228
-.00678
-.03759
-.05535
0.05816
-1 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 1
|
|
|
|
|
|
|
|
|
|
|********************|
. |*.
|
.|.
|
. |*.
|
.*| .
|
.*| .
|
.|.
|
.*| .
|
.*| .
|
.*| .
|
. |*.
|
Std Error
0
0.057735
0.057921
0.057921
0.058119
0.058184
0.058299
0.058329
0.059918
0.059996
0.060166
"." marks two standard errors
15
Partial Autocorrelations
Lag
Correlation
1
2
3
4
5
6
7
8
9
10
0.05675
-0.00620
0.05943
-0.04059
-0.04022
0.02403
-0.16854
-0.01354
-0.06333
0.08700
-1 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 1
|
|
|
|
|
|
|
|
|
|
. |*.
.|.
. |*.
.*| .
.*| .
.|.
***| .
.|.
.*| .
. |**
|
|
|
|
|
|
|
|
|
|
16
The ARIMA Procedure
Name of Variable = r2
Mean of Working Series
Standard Deviation
Number of Observations
0.017598
0.041145
300
Autocorrelations
Lag
Covariance
Correlation
0
1
2
3
4
5
6
7
8
9
10
0.0016929
0.00029832
0.00018962
0.00037532
0.00033045
0.00019604
0.00016872
0.00016590
0.00005835
0.00011312
0.00008283
1.00000
0.17622
0.11201
0.22169
0.19519
0.11580
0.09966
0.09799
0.03447
0.06682
0.04893
-1 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 1
|
|
|
|
|
|
|
|
|
|
|
.
.
.
.
.
.
.
.
.
.
|********************|
|****
|
|**
|
|****
|
|****
|
|**.
|
|**.
|
|**.
|
|* .
|
|* .
|
|* .
|
"." marks two standard errors
Std Error
0
0.057735
0.059501
0.060200
0.062862
0.064851
0.065537
0.066040
0.066523
0.066582
0.066805
17
Partial Autocorrelations
Lag
Correlation
1
2
3
4
5
6
7
8
9
10
0.17622
0.08355
0.19631
0.13255
0.04335
0.02042
0.01435
-0.04129
0.02083
-0.00074
-1 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 1
|
|
|
|
|
|
|
|
|
|
. |****
. |**
. |****
. |***
. |*.
.|.
.|.
.*| .
.|.
.|.
|
|
|
|
|
|
|
|
|
|
18
The AUTOREG Procedure
Dependent Variable
r
Ordinary Least Squares Estimates
SSE
MSE
SBC
Regress R-Square
Durbin-Watson
Variable
Intercept
5.03481522
0.01684
-369.15479
0.0000
1.8834
DFE
Root MSE
AIC
Total R-Square
299
0.12976
-372.85857
0.0000
DF
Estimate
Standard
Error
t Value
Approx
Pr > |t|
1
0.0286
0.007492
3.81
0.0002
Algorithm converged.
GARCH Estimates
SSE
MSE
Log Likelihood
SBC
Normality Test
5.03595361
0.01679
205.372347
-376.522
83.9101
Observations
Uncond Var
Total R-Square
AIC
Pr > ChiSq
300
0.01661204
.
-398.74469
<.0001
The AUTOREG Procedure
Variable
Intercept
ARCH0
ARCH1
ARCH2
ARCH3
ARCH4
DF
Estimate
Standard
Error
t Value
Approx
Pr > |t|
1
1
1
1
1
1
0.0266
0.009459
0.2474
0.0737
0.0421
0.0673
0.006782
0.001418
0.1107
0.0686
0.0714
0.0622
3.92
6.67
2.24
1.08
0.59
1.08
<.0001
<.0001
0.0254
0.2822
0.5552
0.2792
19
The AUTOREG Procedure
Dependent Variable
r
Ordinary Least Squares Estimates
SSE
MSE
SBC
Regress R-Square
Durbin-Watson
Variable
Intercept
5.03481522
0.01684
-369.15479
0.0000
1.8834
DFE
Root MSE
AIC
Total R-Square
299
0.12976
-372.85857
0.0000
DF
Estimate
Standard
Error
t Value
Approx
Pr > |t|
1
0.0286
0.007492
3.81
0.0002
Algorithm converged.
GARCH Estimates
SSE
MSE
Log Likelihood
SBC
Normality Test
5.0357148
0.01679
202.39693
-387.68251
54.6463
Observations
Uncond Var
Total R-Square
AIC
Pr > ChiSq
300
0.01855167
.
-398.79386
<.0001
The AUTOREG Procedure
Variable
Intercept
ARCH0
ARCH1
DF
Estimate
Standard
Error
t Value
Approx
Pr > |t|
1
1
1
0.0268
0.0105
0.4355
0.006102
0.001351
0.1127
4.40
7.75
3.86
<.0001
<.0001
0.0001
20
Obs
e
292
293
294
295
296
297
298
299
300
0.07416
-0.03744
-0.09077
0.26844
-0.02342
-0.02479
-0.19238
-0.01871
-0.12183
21
ˆ
2
301
 0.0105  0.4355  0.12183 
2
 0.01696
ˆ
2
302
 0.0105  0.4355  0.06196 
2
 0.01789
22
ˆ t21  ˆ 0  ˆ 1et2  ˆ 2et21    ˆ q et21q
2
2
2
2
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
 t  2   0  1 t 1   2et     q et  2q
In general, the l-step ahead forecast is
q
2
2
ˆ
ˆ
ˆ
ˆ
 t l   0   i t li
i 1
23
Generalized
Autoregressive
Heteroscedasticity (GARCH)
Conditional
The first empirical application of ARCH models was
done by Engle (1982, Econometrica) to investigate the
relationship between the level and volatility of
inflation. It was found that a large number of lags was
required in the variance functions. This would
necessitate the estimation of a large number of
parameters subject to inequality constraints. Using the
concept of an ARMA process. Bollerslev (1986,
Journal of Econometrics) generalized Engle’s ARCH
model and introduced the GARCH model.
24
Specifically, a GARCH model is defined as
yt   0  1 x1t     t ;  t  t 1
q
p
i 1
j 1
~ N 0, 
2
t
 t2   0    i t2i    j t2 j
with 0 > 0, i ≥ 0, i =1, … q, j ≥ 0, j = 1,
… p imposed to ensure that the conditional
variances are positive.
25
Usually, we only consider lower order GARCH
processes such as GARCH (1, 1), GARCH (1, 2),
GARCH (2, 1) and GARCH (2, 2) processes
For a GARCH (1, 1) process, for example the
forecasts are
2
2
2
ˆ
ˆ
ˆ
ˆ
ˆ
 t 1   0  1et   1 t
2
2
ˆ
ˆ
ˆ
ˆ
ˆ
 t l   0  1   1  t l 1
for l  1
26
Other diagnostic checks:
• AIC, SBC
• Note that t = tt. So we should consider
“standardized” residuals
ˆ t  et ˆ t
and conduct Ljung-Box-Pierce test for ˆ t and ˆ t2 .
27
Consider the monthly excess return of the S&P500 index
from 1926 for 792 observations:
data sp500;
infile 'd:\teaching\ms4221\sp500.txt';
input r;
proc autoreg;
model r=/garch = (q=1);
run;
proc autoreg;
model r=/garch = (q=2);
run;
proc autoreg;
model r=/garch = (q=4);
run;
proc autoreg;
model r=/garch =(p=1, q=1);
run;
proc autoreg;
model r=/garch =(p=1, q=2);
run;
28
The AUTOREG Procedure
Dependent Variable
r
Ordinary Least Squares Estimates
SSE
MSE
SBC
Regress R-Square
Durbin-Watson
Variable
Intercept
2.70318776
0.00342
-2244.3895
0.0000
1.8161
DFE
Root MSE
AIC
Total R-Square
791
0.05846
-2249.0641
0.0000
DF
Estimate
Standard
Error
t Value
Approx
Pr > |t|
1
0.006143
0.002077
2.96
0.0032
Algorithm converged.
GARCH Estimates
SSE
MSE
Log Likelihood
SBC
Normality Test
2.70465846
0.00341
1156.49078
-2292.9579
3455.8233
Observations
Uncond Var
Total R-Square
AIC
Pr > ChiSq
792
0.00332216
.
-2306.9816
<.0001
The AUTOREG Procedure
Variable
Intercept
ARCH0
ARCH1
DF
Estimate
Standard
Error
t Value
Approx
Pr > |t|
1
1
1
0.007506
0.002742
0.1748
0.001928
0.0000651
0.0397
3.89
42.10
4.40
<.0001
<.0001
<.0001
29
GARCH Estimates
SSE
MSE
Log Likelihood
SBC
Normality Test
2.70325396
0.00341
1216.10915
-2405.52
653.3418
Observations
Uncond Var
Total R-Square
AIC
Pr > ChiSq
792
0.00322879
.
-2424.2183
<.0001
The AUTOREG Procedure
Variable
DF
Estimate
Standard
Error
Intercept
ARCH0
ARCH1
ARCH2
1
1
1
1
0.006432
0.001793
0.1286
0.3160
0.001747
0.0000827
0.0297
0.0335
t Value
Approx
Pr > |t|
3.68
21.69
4.34
9.42
0.0002
<.0001
<.0001
<.0001
30
GARCH Estimates
SSE
MSE
Log Likelihood
SBC
Normality Test
2.70432935
0.00341
1247.06474
-2454.0821
387.4606
Observations
Uncond Var
Total R-Square
AIC
Pr > ChiSq
792
0.0037328
.
-2482.1295
<.0001
The AUTOREG Procedure
Variable
DF
Estimate
Standard
Error
Intercept
ARCH0
ARCH1
ARCH2
ARCH3
ARCH4
1
1
1
1
1
1
0.007344
0.001095
0.0586
0.1925
0.2217
0.2339
0.001452
0.0000950
0.0321
0.0231
0.0482
0.0432
t Value
Approx
Pr > |t|
5.06
11.53
1.83
8.31
4.60
5.42
<.0001
<.0001
0.0678
<.0001
<.0001
<.0001
31
GARCH Estimates
SSE
MSE
Log Likelihood
SBC
Normality Test
2.70454696
0.00341
1269.46195
-2512.2257
95.0051
Observations
Uncond Var
Total R-Square
AIC
Pr > ChiSq
792
0.00324989
.
-2530.9239
<.0001
The AUTOREG Procedure
Variable
DF
Estimate
Standard
Error
Intercept
ARCH0
ARCH1
GARCH1
1
1
1
1
0.007453
0.0000818
0.1203
0.8545
0.001547
0.0000238
0.0197
0.0189
t Value
Approx
Pr > |t|
4.82
3.44
6.12
45.15
<.0001
0.0006
<.0001
<.0001
32
GARCH Estimates
SSE
MSE
Log Likelihood
SBC
Normality Test
2.70373438
0.00341
1271.02525
-2508.6777
81.4625
Observations
Uncond Var
Total R-Square
AIC
Pr > ChiSq
792
0.00355913
.
-2532.0505
<.0001
The AUTOREG Procedure
Variable
DF
Estimate
Standard
Error
Intercept
ARCH0
ARCH1
ARCH2
GARCH1
1
1
1
1
1
0.006974
0.0000913
0.0578
0.0883
0.8282
0.001510
0.0000284
0.0374
0.0440
0.0228
t Value
Approx
Pr > |t|
4.62
3.21
1.54
2.01
36.26
<.0001
0.0013
0.1226
0.0446
<.0001
33
•
•
•
•
•
•
•
•
•
•
•
•
proc autoreg;
model r=/garch =(p=1, q=2);
output out=out1 r=e cev=vhat;
run;
data out1;
set out1;
shat=sqrt(vhat);
s=e/shat;
ss=s*s;
proc arima;
identify var=ss nlag=10;
run;
34
The ARIMA Procedure
Name of Variable = ss
Mean of Working Series
Standard Deviation
Number of Observations
1.000064
1.80578
792
Autocorrelations
Lag
Covariance
Correlation
-1 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 1
0
1
2
3
4
5
6
7
8
9
10
3.260843
0.030967
-0.115510
0.055883
-0.084938
-0.109266
-0.029573
-0.150360
0.165407
0.177608
0.015212
1.00000
0.00950
-.03542
0.01714
-.02605
-.03351
-.00907
-.04611
0.05073
0.05447
0.00466
|
|
|
|
|
|
|
|
|
|
|
Std
|********************|
.|.
|
*|.
|
.|.
|
*|.
|
*|.
|
.|.
|
*|.
|
.|*
|
.|*
|
.|.
|
"." marks two standard errors
35
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
Partial Autocorrelations
Lag
Correlation
1
2
3
4
5
6
7
8
9
10
0.00950
-0.03552
0.01785
-0.02771
-0.03177
-0.01067
-0.04751
0.05160
0.04893
0.00743
-1 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 1
|
|
|
|
|
|
|
|
|
|
.|.
*|.
.|.
*|.
*|.
.|.
*|.
.|*
.|*
.|.
|
|
|
|
|
|
|
|
|
|
Autocorrelation Check for White Noise
To
Lag
ChiSquare
6
2.81
DF
6
Pr >
ChiSq
0.8324
--------------------Autocorrelations-------------------0.009
-0.035
0.017
-0.026
-0.03436 -0.