Download Modeling and Forecasting the Volatility of the EUR/ROL

DISSERTATION PAPER
Modeling and Forecasting the Volatility of the
EUR/ROL Exchange Rate Using GARCH
Models.
Student :Becar Iuliana
Supervisor: Professor Moisa Altar
Table of Contents
•
The importance of forecasting exchange rate
volatility.
•
Data description.
•
Model estimates and forecasting performances.
•
Concluding remarks.
Why model and forecast volatility?

Volatility is one of the most important concepts in the whole of
finance.

ARCH models offered new tools for measuring risk, and its
impact on return.

Volatility of exchange rates is of importance because of the
uncertainty it creates for prices of exports and imports, for the
value of international reserves and for open positions in foreign
currency.
Volatility Models.

ARCH/GARCH models.
Engle(1982)
Bollerslev(1986)
Baillie, Bollerslev and Mikkelsen (1996)

ARFIMA models.
Granger (1980)
Data description



Data series: nominal daily EUR/ROL exchange rates
Time length: 04:01:1999-11:06:2004
1384 nominal percentage returns yt  100[ln( st )  ln( st 1 )]
Exchange Rate
40000
Time Series of The Exchange Rate
35000
30000
25000
20000
15000
0
100
200
300
400
500
600
700
800
900
1000
1100
1200
1300
Descriptive Statistics for the return series.
0.8
Density
Histogram of Returns together with the Normal and Return Density
0.7
Statistic t-Test
P-Value
0.6
0.5
Skewness
1.0472 15.922 4.4605e-057
0.4
Excess
Kurtosis
8.5138 64.769 0.00000
JarqueBera
4432.9
0.3
0.2
0.1
-2
-1
0
1
2
3
4
5
6
7
Heteroscedasticity
7
6
5
The Daily Return Series
4
3
2
1
0
-1
-2
0
100
200
300
400
500
600
700
800
900
1000
1100
1200
The returns are not homoskedastic.
Low serial dependence in returns.
The Ljung-Box statistic for 20 lags
equals 27.392 [0.125].
  37.6
2
20
1300
Autocorrelation and
Partial autocorrelation
of the Return Series
1.00
0.75
0.50
0.25
0.00
-0.25
-0.50
-0.75
0
5
10
15
20
Autocorrelation and Partial Autocorrelation
of Squared Returns
1.00
0.75
0.50
0.25
The Ljung-Box statistic for 20
lags equals 151.01[0.000]
0.00
ARCH 1 test: 17.955 [0.0000]**
ARCH 2 test: 18.847 [0.0000]**
-0.25
-0.50
-0.75
0
5
10
15
20
Stationarity
Unit Root Tests for EUR/ROL return series.
ADF
Test
Statistic
-35.60834
1%
Critical
Value*
-3.4380
5%
Critical
Value
10%
Critical
Value
1%
Critical
Value*
-3.4380
-2.8641
5%
Critical
Value
-2.8641
-2.5681
10%
Critical
Value
-2.5681
*MacKinnon critical values for rejection of
hypothesis of a unit root.
PP Test
Statistic
-35.57805
*MacKinnon critical values for rejection of
hypothesis of a unit root.
Model estimates and forecasting performances.

Methodology.
Ox Professional 3.30 [email protected]
4.01.1999-30.12.2002 (1018 observations) for model
estimation
06.01.2003-11.06.2004 (366 observations) for out of
sample forecast evaluation.

The Models.
Two distributions: Student, Skewed Student, QMLE.
The Mean Equations:
1. A constant mean
2. An ARFIMA(1,da,0) mean
3. An ARFIMA(0, da,1) mean
The variance equations.

GARCH(1,1) and FIGARCH(1,d,1) without the constant term
and with a non-trading day dummy variable.
The estimated twelve models.

Examining the models page 30 to 34 the conclusions are:
• The estimated coefficients are significantly different
from zero at the 10% level.
• the ARFIMA coefficient lies between  0.5;0.5
which implies stationarity.
• all variance coefficients are positive and     1
In-sample model evaluation. Residual tests. GARCH models.
Model
SBC
Skewness
EK1
Q*
Q2**
ARCH***
Nyblom
ARMA (0,0)
GARCH(1,1)
Skewed-Student
2.210463
0.75224
3.9543
37.5958
[0.9019571]
30.3204
[0.9783154]
1.1358
[0.3395]
1.96933
ARMA (0,0)
GARCH(1,1)
Student
2.212901
0.74033
3.8319
37.5877
[0.9021277]
30.3145
[0.9783579]
1.1238
[0.3458]
1.58334
ARFIMA (1,d,0)
GARCH(1,1)
Skewed-Student
2.214579
0.76024
4.1028
36.4188
[0.9083405]
31.7529
[0.9659063]
1.2484
[0.2843]
2.24209
ARFIMA (1,d,0)
GARCH(1,1)
Student
2.216388
0.73353
3.857
36.0009
[0.9165657]
31.8411
[0.9649974]
1.1801
[0.3169]
1.89543
ARFIMA (0,d,1)
GARCH(1,1)
Skewed-Student
2.215735
0.75909
4.1153
36.1425
[0.9138359]
31.3112
[0.9701942]
1.2084
[0.3030]
2.2612
ARFIMA (0,d,1)
GARCH(1,1)
Student
2.217401
0.73390
3.8852
35.8043
[0.9202571]
31.3087
[0.9702172]
1.1360
[0.3394]
1.9047
1 EK-Excess Kurtosis;* Q-Statistics on Standardized Residuals with 50 lags; ** Q-Statistics on Squared
Standardized Residuals 50 lags; *** ARCH test with 5 lags; P-values in brackets.
In-sample model evaluation. Residual tests. FIGARCH models.
Model
SBC
Skewness
EK1
Q*
Q2**
ARCH***
Nyblom
ARMA (0,0)
FIGARCH(1,d,1)
Skewed-Student
2.222089
0.76305
3.8723
37.4681
[0.9046084]
28.4572
[0.9888560]
1.2601
[0.2790]
1.56799
ARMA (0,0)
FIGARCH(1,d,1)
Student*
2.22472
0.74698
3.7313
37.7303
[0.8991133]
28.9803
[0.9864387]
1.3297
[0.2491]
1.37719
ARFIMA (1,d,0)
FIGARCH(1,d,1)
Skewed-Student
2.226549
0.757
3.9242
36.3540
[0.9096502]
29.8994
[0.9811947]
1.3204
[0.2529]
2.05757
ARFIMA (1,d,0)
FIGARCH(1,d,1)
Student
2.228334
0.73378
3.7256
36.1801
[0.9131002]
30.4315
[0.9775013]
1.3272
[0.2501]
1.82764
ARFIMA (0,d,1)
FIGARCH(1,d,1)
Skewed-Student
2.227516
0.75901
3.96
36.2611
[0.9115043]
29.2088
[0.9852596]
1.2729
[0.2733]
2.0233
ARFIMA (0,d,1)
FIGARCH(1,d,1)
Student
2.229199
0.73799
3.7813
36.1313
[0.9140531]
29.5586
[0.9832983]
1.2630
[0.2777]
1.79097
1 EK-Excess Kurtosis;* Q-Statistics on Standardized Residuals with 50 lags; ** Q-Statistics on Squared
Standardized Residuals 50 lags; *** ARCH test with 5 lags; P-values in brackets.
Out-of-sample Forecast Evaluation

Forecast methodology
- sample window: 1018 observations
- at each step, the 1 step ahead dynamic forecast is stored
for the conditional variance and the conditional mean
-dynamic forecast is programmed in OxEdit
[email protected] package

Benchmark: ex-post volatility = squared returns.
Measuring Forecast Accuracy.


The Mincer-Zarnowitz
regression:
The Mean Absolute
Error:
2
2
ˆ
 t  alfa  beta t  ut
1 n
MAE    t2  ˆ t2
n t 1
n


Root Mean Square Error RMSE 
(standard error):
n
Theil's inequality
coefficient -Theil's U:
U 
 (
t 1
2
t
 ˆ t2 ) 2
n
2
2 2
ˆ
(



 t
t )
t 1
n
2
2 2
(



 t 1 t )
t 1
One Step Ahead Forecast Evaluation Measures.
1. The Mincer-Zarnowitz regression
Model
alfa
beta
R2
Model
alfa
beta
R2
ARMA (0,0)
GARCH(1,1)
Skewed-Student
-0.104961
[0.0699]
0.624769
[0.0006]
0.0533211
ARMA (0,0)
FIGARCH(1,d,1)
Skewed-Student
-0.038611
[0.3070]
0.741465
[0.0005]
0.0822328
ARMA (0,0)
GARCH(1,1)
Student
-0.100843
[0.0766]
0.617284
[0.0007]
0.0530545
ARMA (0,0)
FIGARCH(1,d,1)
Student
-0.037921
[0.3143]
0.725906
[0.0005]
0.0793558
ARFIMA (1,d,0)
GARCH(1,1)
Skewed-Student
-0.112153
[0.0607]
0.631864
[0.0006]
0.0518779
ARFIMA (1,d,0)
FIGARCH(1,d,1)
Skewed-Student
-0.046087
[0.2517]
0.730264
[0.0006]
0.0759213
ARFIMA (1,d,0)
GARCH(1,1)
Student
-0.104983
[0.0698]
0.620363
[0.0006]
0.0522936
ARFIMA (1,d,0)
FIGARCH(1,d,1)
Student
-0.043940
[0.2681]
0.707455
[0.0006]
0.0735089
ARFIMA (0,d,1)
GARCH(1,1)
Skewed-Student
-0.112613
[0.0596]
0.634110
[0.0006]
0.052295
ARFIMA (0,d,1)
FIGARCH(1,d,1)
Skewed-Student
-0.045701
[0.254]
0.731791
[0.0006]
0.0765561
ARFIMA (0,d,1)
GARCH(1,1)
Student
-0.105667
[0.0680]
0.623092
[0.0006]
0.0527494
ARFIMA (0,d,1)
FIGARCH(1,d,1)
Student
-0.043431
[0.2715]
0.70931
[0.0006]
0.0742364
2. Forecasting the conditional mean. Loss functions.
Model
MAE
RMSE
ARMA (0,0)
GARCH(1,1)
Skewed-Student
0.2601
0.3412
ARMA (0,0)
GARCH(1,1)
Student
0.2576
ARFIMA (1,d,0)
GARCH(1,1)
Skewed-Student
TIC
Model
MAE
RMSE
TIC
0.7895
ARMA (0,0)
FIGARCH(1,d,1)
Skewed-Student
0.2606
0.3416
0.7861
0.3395
0.812
ARMA (0,0)
FIGARCH(1,d,1)
Student
0.258
0.3397
0.8086
0.2724
0.3521
0.7527
ARFIMA (1,d,0)
FIGARCH(1,d,1)
Skewed-Student
0.2726
0.3522
0.7518
ARFIMA (1,d,0)
GARCH(1,1)
Student
0.2694
0.3493
0.77
ARFIMA (1,d,0)
FIGARCH(1,d,1)
Student
0.2697
0.3496
0.7684
ARFIMA (0,d,1)
GARCH(1,1)
Skewed-Student
0.2722
0.352
0.7548
ARFIMA (0,d,1)
FIGARCH(1,d,1)
Skewed-Student
0.2724
0.3522
0.7536
ARFIMA (0,d,1)
GARCH(1,1)
Student
0.2691
0.3493
0.7729
ARFIMA (0,d,1)
FIGARCH(1,d,1)
Student
0.2694
0.3495
0.7711
3. Forecasting the conditional variance. Loss functions.
Model
MAE
RMSE
TIC
Model
MAE
RMSE
TIC
ARMA (0,0)
GARCH(1,1)
Skewed-Student
0.2844
0.3148
0.5253
ARMA (0,0)
FIGARCH(1,d,1)
Skewed-Student
0.17
0.2234
0.484
ARMA (0,0)
GARCH(1,1)
Student
0.2824
0.3131
0.5244
ARMA (0,0)
FIGARCH(1,d,1)
Student
0.1726
0.2253
0.4845
ARFIMA (1,d,0)
GARCH(1,1)
Skewed-Student
0.2907
0.3204
0.5286
ARFIMA (1,d,0)
FIGARCH(1,d,1)
Skewed-Student
0.1802
0.2299
0.4856
ARFIMA (1,d,0)
GARCH(1,1)
Student
0.2866
0.3168
0.5265
ARFIMA (1,d,0)
FIGARCH(1,d,1)
Student
0.1832
0.2322
0.4861
ARFIMA (0,d,1)
GARCH(1,1)
Skewed-Student
0.2903
0.32
0.5283
ARFIMA (0,d,1)
FIGARCH(1,d,1)
Skewed-Student
0.1794
0.2294
0.4854
ARFIMA (0,d,1)
GARCH(1,1)
Student
0.2862
0.3164
0.5263
ARFIMA (0,d,1)
FIGARCH(1,d,1)
Student
0.1822
0.2315
0.4859
Concluding remarks.

In-sample analysis:
Residual tests:
-all models may be appropriate.
-the Student distribution is better than the Skewed Student.

Out-of-sample analysis:
-the FIGARCH models are superior.
-for the conditional mean the Student distribution is
superior.
-the two ARFIMA mean equations don't provide a better
forecast of the conditional mean.
- for the conditional variance the Skewed Student
distribution is superior.
Concluding remarks.

Model construction problems;

Further research:
-option prices, which reflect the market’s expectation
of volatility over the remaining life span of the option.
-daily realized volatility can be computed as the sum of
squared intraday returns
Bibliography










Alexander, Carol (2001) – Market Models - A Guide to Financial Data Analysis, John Wiley
&Sons, Ltd.;
Andersen, T. G. and T. Bollerslev (1997) - Answering the Skeptics: Yes, Standard Volatility
Models Do Provide Accurate Forecasts, International Economic Review;
Andersen, T. G., T. Bollerslev, Francis X. Diebold and Paul Labys (2000)- Modeling and
Forecasting Realized Volatility, the June 2000 Meeting of the Western Finance Association.
Andersen, T. G., T. Bollerslev and Francis X. Diebold (2002)- Parametric and
Nonparametric Volatility Measurement, Prepared for Yacine Aït-Sahalia and Lars Peter
Hansen (eds.), Handbook of Financial Econometrics, North Holland.
Andersen, T. G., T. Bollerslev and Peter Christoffersen (2004)-Volatility Forecasting, Rady
School of Management at UCSD
Baillie, R.T., Bollerslev T., Mikkelsen H.O. (1996)- Fractionally Integrated Generalized
Autoregressive Conditional Heteroskedasticity, Journal of Econometrics, Vol. 74, No.1, pp. 330.
Bollerslev, Tim, Robert F. Engle and Daniel B. Nelson (1994)– ARCH Models, Handbook of
Econometrics, Volume 4, Chapter 49, North Holland;
Diebold, Francis and Marc Nerlove (1989)-The Dynamics of Exchange Rate Volatility: A
Multivariate Latent factor Arch Model, Journal of Applied Econometrics, Vol. 4, No.1.
Diebold, Francis and Jose A. Lopez (1995)- Forecast Evaluation and Combination, Prepared
for G.S. Maddala and C.R. Rao (eds.), Handbook of Statistics, North Holland.
Enders W. (1995)- Applied Econometric Time Series, 1st Edition, New York: Wiley.
Bibliography
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






Engle, R.F. (1982) – Autoregressive conditional heteroskedasticity with estimates of the
variance of UK inflation, Econometrica, 50, pp. 987-1007;
Engle, R.F. and Victor K. Ng (1993) – Measuring and Testing the Impact of News on
Volatility, The Journal of Finance, Vol. XLVIII, No. 5;
Engle, R. (2001) – Garch 101: The Use of ARCH/GARCH Models in Applied
Econometrics, Journal of Economic Perspectives – Volume 15, Number 4 – Fall 2001 –
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Engle, R. and A. J. Patton (2001) – What good is a volatility model?, Research Paper,
Quantitative Finance, Volume 1, 237-245;
Engle, R. (2001) – New Frontiers for ARCH Models, prepared for Conference on
Volatility Modelling and Forecasting, Perth, Australia, September 2001;
Hamilton, J.D. (1994) – Time Series Analysis, Princeton University Press;
Lopez, J.A.(1999) – Evaluating the Predictive Accuracy of Volatility Models,
Economic Research Deparment, Federal Reserve Bank of San Francisco;
Peters, J. and S. Laurent (2001) – A Tutorial for G@RCH 2.3, a Complete Ox Package
for Estimating and Forecasting ARCH Models;
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for Estimating and Forecasting ARCH Models;
West, Kenneth and Dongchul Cho (1994)-The Predictive Ability of Several Models of
Exchange Rate Volatility, NBER Technical Working Paper #152.
Appendix 1.
The ARMA (0, 0), GARCH (1, 1) Skewed Student model.
Robust Standard Errors (Sandwich formula)
Coefficient
Std.Error
t-value
Probability
Constant(Mean)
0.091930
0.021613
4.253
0.0000
dummyFriday (V)
0.048977
0.019781
2.476
0.0134
ARCH(Alpha1)
0.036076
0.011561
3.121
0.0019
GARCH(Beta1)
0.924490
0.018052
51.21
0.0000
Asymmetry
0.145722
0.047250
3.084
0.0021
Tail
9.872213
3.3488
2.948
0.0033
For more details see Appendix 1, page 45.
Appendix 2
The ARMA (0, 0), GARCH (1, 1) Student model.
Robust Standard Errors (Sandwich formula)
Coefficient
Std.Error
t-alue
Probability
Constant(Mean)
0.077795
0.021673
3.589
0.0003
dummyFriday (V)
0.049240
0.020163
2.442
0.0148
ARCH(Alpha1)
0.037186
0.011975
3.105
0.0020
GARCH(Beta1)
0.923353
0.018479
49.97
0.0000
Student(DF)
8.921340
2.8119
3.173
0.0016
For more details, see Appendix 2, page 47.
Appendix 3
The ARFIMA (1, da, 0),GARCH (1, 1) Skewed Student model.
Robust Standard Errors (Sandwich formula)
Coefficient
Std.Error
t-value
Probability
0.089939
0.010527
8.544
0.0000
-0.128224
0.045067
-2.845
0.0045
AR(1)
0.123269
0.054553
2.260
0.0241
dummyFriday (V)
0.048860
0.019703
2.480
0.0133
ARCH(Alpha1)
0.033897
0.011677
2.903
0.0038
GARCH(Beta1)
0.926283
0.018096
51.19
0.0000
Asymmetry
0.139771
0.047194
2.962
0.0031
Tail
9.189523
2.9091
3.159
0.0016
Constant(Mean)
d-Arfima
For more details, see Appendix 3, page 49.
Appendix 4
The ARFIMA (1, da, 0),GARCH (1, 1) Student model.
Robust Standard Errors (Sandwich formula)
Coefficient
Std.Error
t-value
Probabilty
0.082711
0.010237
8.080
0.0000
-0.136317
0.045875
-2.971
0.0030
AR(1)
0.140455
0.055832
2.516
0.0120
dummyFriday (V)
0.049635
0.020117
2.467
0.0138
ARCH(Alpha1)
0.036517
0.012510
2.919
0.0036
GARCH(Beta1)
0.923503
0.018602
49.64
0.0000
Student(DF)
8.436809
2.5257
3.340
0.0009
Constant(Mean)
d-Arfima
For more details, see Appendix 4, page 52.
Appendix 5
The ARFIMA (0, da,1),GARCH (1, 1) Skewed Student model.
Robust Standard Errors (Sandwich formula)
Coefficient
Std.Error
t-value
Probability
Constant(Mean)
0.090415
0.011041
8.189
0.0000
d-Arfima
-0.117757
0.037429
-3.146
0.0017
MA(1)
0.114844
0.046060
2.493
0.0128
dummyFriday (V)
0.048681
0.019787
2.460
0.0140
ARCH(Alpha1)
0.033847
0.011641
2.908
0.0037
GARCH(Beta1)
0.926414
0.018172
50.98
0.0000
Asymmetry
0.138631
0.047049
2.947
0.0033
Tail
9.279306
2.9613
3.134
0.0018
For more details, see Appendix 5, page 54.
Appendix 6
The ARFIMA (0, da,1),GARCH (1, 1) Student model.
Robust Standard Errors (Sandwich formula)
Coefficient
Std.Error
t-value
Probability
0.082822
0.010833
7.645
0.0000
-0.122519
0.036843
-3.325
0.0009
MA(1)
0.128311
0.045146
2.842
0.0046
dummyFriday (V)
0.049380
0.020207
2.444
0.0147
ARCH(Alpha1)
0.036344
0.012449
2.919
0.0036
GARCH(Beta1)
0.923788
0.018703
49.39
0.0000
Student(DF)
8.516429
2.5689
3.315
0.0009
Constant(Mean)
d-Arfima
For more details, see Appendix 6, page 56.
Appendix 7
The ARMA (0, 0), FIGARCH-BBM (1,d,1) Skewed Student model.
Robust Standard Errors (Sandwich formula)
Coefficient
Std.Error
t-value
Probability
Constant(Mean)
0.094259
0.021931
4.298
0.0000
dummyFriday (V)
0.047278
0.025975
1.820
0.0690
d-Figarch
0.358622
0.098899
3.626
0.0003
ARCH(Alpha1)
0.288896
0.094598
3.054
0.0023
GARCH(Beta1)
0.635309
0.058513
10.86
0.0000
Asymmetry
0.147588
0.046529
3.172
0.0016
Tail
9.545031
3.0964
3.083
0.0021
For more details, see Appendix 7, page 59.
Appendix 8
The ARMA (0, 0), FIGARCH-BBM (1,d,1) Student model.
Robust Standard Errors (Sandwich formula)
Coefficient
Std.Error
t-value
Probability
Constant(Mean)
0.079807
0.021915
3.642
0.0003
dummyFriday (V)
0.049310
0.027926
1.766
0.0777
d-Figarch
0.351448
0.10506
3.345
0.0009
ARCH(Alpha1)
0.312018
0.11026
2.830
0.0047
GARCH(Beta1)
0.644842
0.057580
11.20
0.0000
Student(DF)
8.596805
2.6044
3.301
0.0010
For more details, see Appendix 8, page 61.
Appendix 9
The ARFIMA (1,da,0), FIGARCH-BBM (1,d,1) Skewed Student model.
Robust Standard Errors (Sandwich formula)
Coefficient
Std.Error
t-value
Probability
0.090400
0.010719
8.434
0.0000
-0.126724
0.046241
-2.741
0.0062
AR(1)
0.119364
0.054745
2.180
0.0295
dummyFriday
(V)
0.052164
0.030787
1.694
0.0905
d-Figarch
0.332074
0.10662
3.115
0.0019
ARCH(Alpha1)
0.339292
0.13642
2.487
0.0130
GARCH(Beta1)
0.649620
0.053779
12.08
0.0000
Asymmetry
0.139501
0.046638
2.991
0.0028
Tail
8.871259
2.6840
3.305
0.0010
Constant(Mean)
d-Arfima
For more details, see Appendix 9, page 63.
Appendix 10
The ARFIMA (1,da,0), FIGARCH-BBM (1,d,1) Student model.
Robust Standard Errors (Sandwich formula)
Coefficient
Std.Error
t-value
Probability
0.083221
0.010263
8.109
0.0000
-0.136270
0.047181
-2.888
0.0040
AR(1)
0.138494
0.056208
2.464
0.0139
dummyFriday (V)
0.054562
0.034015
1.604
0.1090
d-Figarch
0.328545
0.12291
2.673
0.0076
ARCH(Alpha1)
0.360347
0.17155
2.101
0.0359
GARCH(Beta1)
0.659966
0.057997
11.38
0.0000
Student(DF)
8.093551
2.3226
3.485
0.0005
Constant(Mean)
d-Arfima
For more details, see Appendix 10, page 66.
Appendix 11
The ARFIMA (0,da,1), FIGARCH-BBM (1,d,1) Skewed Student model.
Robust Standard Errors (Sandwich formula)
Coefficient
Std.Error
t-value
Probability
Constant(Mean)
0.090938
0.011202
8.118
0.0000
d-Arfima
-0.117093
0.039118
-2.993
0.0028
MA(1)
0.112312
0.047184
2.380
0.0175
dummyFriday (V)
0.051724
0.030327
1.706
0.0884
d-Figarch
0.332759
0.10397
3.200
0.0014
ARCH(Alpha1)
0.334340
0.12765
2.619
0.0089
GARCH(Beta1)
0.647135
0.052822
12.25
0.0000
Asymmetry
0.138659
0.046925
2.955
0.0032
Tail
8.973744
2.7438
3.270
0.0011
For more details, see Appendix 11, page 68.
Appendix 12
The ARFIMA (0,da,1), FIGARCH-BBM (1,d,1) Student model.
Robust Standard Errors (Sandwich formula)
Coefficient
Std.Error
t-value
Probability
0.083434
0.010870
7.675
0.0000
-0.122620
0.038276
-3.204
0.0014
MA(1)
0.126887
0.045925
2.763
0.0058
dummyFriday (V)
0.054060
0.033155
1.631
0.1033
d-Figarch
0.329579
0.11765
2.801
0.0052
ARCH(Alpha1)
0.353442
0.15661
2.257
0.0242
GARCH(Beta1)
0.656630
0.055867
11.75
0.0000
Student(DF)
8.182206
2.3695
3.453
0.0006
Cst(M)
d-Arfima
For more details, see Appendix 12, page 70.
Stationarity tests. Appendix 13.
1. Dickey-Fuller Test.
Augmented Dickey-Fuller Test Equation
Dependent Variable: D(RETURNS)
Method: Least Squares
Date: 06/26/04 Time: 07:50
Sample(adjusted): 3 1384
Included observations: 1382 after adjusting endpoints
Variable
Coefficient
Std. Error
t-Statistic
Prob.
RETURNS(-1)
-0.957262
0.026883
-35.60834
0.0000
C
0.078392
0.018264
4.292148
0.0000
R-squared
0.478843
Mean dependent var
-0.000589
Adjusted R-squared
0.478465
S.D. dependent var
0.933223
S.E. of regression
0.673949
Akaike info criterion
2.050121
Sum squared resid
626.8057
Schwarz criterion
2.057692
F-statistic
1267.954
Prob(F-statistic)
0.000000
Log likelihood
Durbin-Watson stat
-1414.634
1.994863
ADF
Test
-17.25675
1%
Critical
Value*
-3.4380
Dependent Variable: D(RETURNS)
Method: Least Squares
Sample(adjusted): 7 1384
Included observations: 1378 after adjusting endpoints
5%
Critical
Value
10%
Critical
Value
-2.8641
-2.5681
*MacKinnon critical values for
rejection of hypothesis of a unit root.
Variable
Coefficient
Std. Error
t-Statistic
Prob.
RETURNS(-1)
-1.047183
0.060682
-17.25675
0.0000
D(RETURNS(-1))
0.091319
0.053927
1.693396
0.0906
D(RETURNS(-2))
0.039379
0.046166
0.852989
0.3938
D(RETURNS(-3))
0.009635
0.037319
0.258186
0.7963
D(RETURNS(-4))
0.015333
0.026967
0.568585
0.5697
C
0.086684
0.018835
4.602399
0.0000
R-squared
0.480683
Mean dependent var
0.000495
Adjusted R-squared
0.478791
S.D. dependent var
0.933787
S.E. of regression
0.674146
Akaike info criterion
2.053604
Sum squared resid
623.5364
Schwarz criterion
2.076369
F-statistic
253.9867
Prob(F-statistic)
0.000000
Log likelihood
Appendix 14.
ADF Test.
Durbin-Watson stat
-1408.933
1.998880
Appendix 15.Phillips-Perron Test.
Lag truncation for Bartlett kernel: 7
( Newey-West suggests: 7 )
Residual variance with no correction
0.453550
Residual variance with correction
0.407637
Phillips-Perron Test Equation
Dependent Variable: D(RETURNS)
Method: Least Squares
Sample(adjusted): 3 1384
Included observations: 1382 after adjusting endpoints
Variable
Coefficient
Std. Error
t-Statistic
Prob.
RETURNS(-1)
-0.957262
0.026883
-35.60834
0.0000
C
0.078392
0.018264
4.292148
0.0000
R-squared
0.478843
Mean dependent var
-0.000589
Adjusted R-squared
0.478465
S.D. dependent var
0.933223
S.E. of regression
0.673949
Akaike info criterion
2.050121
Sum squared resid
626.8057
Schwarz criterion
2.057692
F-statistic
1267.954
Prob(F-statistic)
0.000000
Log likelihood
Durbin-Watson stat
-1414.634
1.994863