Arch+and+Garch+Models
Transcript of Arch+and+Garch+Models
ARCH and GARCH MODELS
David LeblangUniversity of Colorado
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I. Motivation: Why ARCH/GARCH Models?
A. What is ARCH/GARCH?
1) Generalized—more general than ARCH2) Autoregressive—depends on its past3) Conditional—variance depends on past
info4) Heteroscedasticity—non-constant
variance.
B. Econometric—OLS assumes:
1) No Serial Correlation: -- tests and corrections are standard in the literature.
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2) Homoscedastic Errors: --errors are normally and independently distributed. Usual for papers to test for heteroscedasticity (i) in the cross-sectional context but unusual in the time-series context (t)
3) Consequences: OLS is BLUE and consistent. HOWEVER, OLS is not efficient (minimum variance) if we relax the class of estimators to include nonlinear estimators.
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C. Empirical Regularities (S&P returns).1) Volatility Clustering
10
0*[
log
(sp
(t))
-(lo
g(s
p(t
-1))
)]
Volatility Clusteringdate
22dec1999 31mar2000 09jul2000 17oct2000
-6.00451
4.65458
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2) Fat/Heavy Tails (Kurtosis) [k=3]
Fra
ctio
n
Kurtosis100*[log(sp(t))-(log(sp(t-1)))]
-6.00451 4.65458
0
.113636
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D. Theoretical—Variance is of Interest
1) What causes volatility/variance of a series? Finance literature (risk premium); economics literature (target zones). Political science—political events/information influence variability of asset prices (e.g., Leblang and Bernhard; Freeman, Hays and Stix)
2) Are some events/periods/systems conducive to more/less volatility than others?
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E. Textbook References1) Enders, Applied
Econometric Time Series
2) Patterson, An Introduction to Applied Time Series
3) Franses and van Dijk, Non-Linear Time Series Models in Empirical Finance
F. Software (others=PC-GIVE, RATS, TSP)
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Software Advantages Disadvantages
STATAwww.stata.com
My favorite in generalLots of built in modelsChoice of algorithmEasy to program
Only normal dist.Few built-in diagnostics
EVIEWSwww.eviews.com
Lots of built in modelsChoice of algorithmLots of built in diag.FAST!
Only normal dist.Difficult to program
S+ GARCHwww.insightful.com
Lots of built in modelsFIGARCHMGARCHt, ged, double exp
Difficult to programNo choice of algorithmA bit “clunky”
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dist.Terrific Graphics
II. Preliminaries: Linear Time SeriesA. Variable yt is observed for t=1,2,..,n
B. The error (t) is a white noise series if1)2) . The error is
unconditionally and conditionally homoscedastic.
3) . Note: this says that the information set does not contain information to forecast .
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C. A time series for yt can be thought of as the sum of a predictable and an unpredictable component: .
III. Relax assumption of homoscedasticity
A. Allow conditional variance of to vary over time: for some nonnegative function.
B. In general, this is expressed as: , where zt is independently and identically distributed normally with mean zero and unit variance (this can be relaxed—use student t and ged distributions to allow for fatter tails).
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C. This means that the distribution of conditional upon the history is normal with mean zero and variance ht. It also means that the unconditional variance of is constant. Using the law of iterated expectations:
.
D. We now need a model to specify how the conditional variance of evolves over time.
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IV. Autoregressive Conditional Heteroscedasticity A. Invented by Engle (1982) to explain the
volatility of inflation rates.
B. Basic ARCH (1) model: conditional variance of a shock at time t is a function of the squares of past shocks: . (Recall, h is the variance and is a “shock,” “news,” or “error”).
C. Since the conditional variance needs to be nonnegative, the conditions have to be met. If 1 = 0, then the conditional variance is constant and is conditionally homoscedastic.
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V. Generalized ARCH (GARCH)
A. Because ARCH(p) models are difficult to estimate, and because decay very slowly, Bollerslev (1986) developed the GARCH model.
B. GARCH (1,1): .
C. The variance (ht) is a function of an intercept (), a shock from the prior period () and the variance from last period ().
D. Higher order GARCH models:
.
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VI. Linear GARCH Variations.
A. Integrated GARCH (Engle and Bollerslev 1986).
1) Phenomena is similar to integrated series in regular (ARMA-type) time-series.
2) Occurs when +=1. When this is the case it means that there is a unit root in the conditional variance; past shocks do not dissipate but persist for very long periods of time.
B. Fractionally Integrated GARCH (Baillie, Bollerslev and Mikkelsen (1996)).
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C. GARCH in Mean (Engle, Lilien and Robbins (1987).
1) Idea is that there is a direct relationship between risk and return of an asset.
2) In the mean equation, include some function of the conditional variance—usually the standard deviation.
3) This allows the mean of a series to depend, at least in part, on the conditional variance of the series (more later)..
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VII. Non-Linear GARCH Variations (dozens in last 20 years). Linear GARCH models all allow prior shocks to have a symmetric affect on ht. Non-linear models allow for asymmetric shocks to volatility. I will focus on the most common: the Exponentional GARCH (1,1) (EGARCH) model developed by Nelson (1991).
A. Conditional variance: , where
and is the standardized residual. is the asymmetric component.
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B. News Impact Curve—differential impact of positive and negative shocks.
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Co
nd
itio
na
l Va
ria
nce
: GA
RC
H
News Impact Curve: dCPI w/ ARMA(1,1)error (t-1)
Co
nd
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na
l Va
ria
nce
: EG
AR
CH
Conditional Variance: GARCH Conditional Variance: EGARCH
-9.8 9.8
.587194
40.1751
.404425
83.8448
VIII. Testing for ARCH
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A. ARCH Tests (Engle 1982).1) Regress Y on X and obtain some
residuals
2) Regress on p lags of ; that is,
a. Assess joint significance of . If the coefficients are different from zero then the null of conditional homoscedasticity can be rejected.
b. T*R2 is Engle’s LM test statistic. Under the null of homoscedasticity it is asymptotically distributed
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B. Graphical Test—Ljung-Box Q Statistic
1) LB (Q) used to diagnose serial correlation in the residuals
2) LB(Q2) used to diagnose serial correlation in the squared residuals—heteroscedasticity
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IX. Example—Returns on the S&P 500. regress dlsp /returns on the S & P 500 Index
Source | SS df MS Number of obs = 220-------------+------------------------------ F( 0, 219) = 0.00 Model | 0.00 0 . Prob > F = . Residual | 391.285893 219 1.78669358 R-squared = 0.0000-------------+------------------------------ Adj R-squared = 0.0000 Total | 391.285893 219 1.78669358 Root MSE = 1.3367
------------------------------------------------------------------------------ dlsp | Coef. Std. Err. t P>|t| [95% Conf. Interval]-------------+---------------------------------------------------------------- _cons | .0096484 .0901184 0.11 0.915 -.167962 .1872588------------------------------------------------------------------------------
. predict e if e(sample), resid / obtain residuals
. gen e2=e^2 /generate squared residuals
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. reg e2 l.e2 /regress squared residuals on a lag
Source | SS df MS Number of obs = 219-------------+------------------------------ F( 1, 217) = 5.83 Model | 72.087039 1 72.087039 Prob > F = 0.0166 Residual | 2684.92529 217 12.3729276 R-squared = 0.0261-------------+------------------------------ Adj R-squared = 0.0217 Total | 2757.01233 218 12.6468456 Root MSE = 3.5175
------------------------------------------------------------------------------e2 | Coef. Std. Err. t P>|t| [95% Conf. Interval]-------------+----------------------------------------------------------------e2 | L1 | .1616863 .0669855 2.41 0.017 .0296608 .2937118_cons | 1.49788 .2661034 5.63 0.000 .9734018 2.022358------------------------------------------------------------------------------
. test l1.e2 /test H0: homoscedastic residuals
( 1) L.e2 = 0.0
F( 1, 217) = 5.83 Prob > F = 0.0166
. display 219*.02615.7159
. display chiprob(1, 5.7159) /the value is the p-value to reject H0 of Homoscedasticity
.01681195
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Autocorrelation Function. ac e2 /autocorrelation function of the squared residuals
Bartlett's formula for MA(q) 95% confidence bands
Au
toco
rre
latio
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of e
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CorrelogramLag
0 10 20 30 40
-1.00
-0.75
-0.50
-0.25
0.00
0.25
0.50
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1.00
-1.00
-0.75
-0.50
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0.00
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. corrgram e2 /correlegram gives the ac and pacs
-1 0 1 -1 0 1 LAG AC PAC Q Prob>Q [Autocorrelation] [Partial Autocor]-------------------------------------------------------------------------------1 0.1615 0.1617 5.8191 0.0159 |- |- 2 0.1511 0.1282 10.937 0.0042 |- |- 3 -0.0107 -0.0555 10.963 0.0119 | | 4 0.0577 0.0505 11.715 0.0196 | | 5 0.0724 0.0695 12.906 0.0243 | | 6 0.1087 0.0765 15.603 0.0161 | | 7 -0.0132 -0.0594 15.643 0.0286 | | 8 0.0007 -0.0123 15.643 0.0478 | | 9 -0.0317 -0.0189 15.876 0.0695 | | 10 0.0070 0.0027 15.887 0.1029 | |
. wntestq e2, lags(1)Portmanteau test for white noise--------------------------------------- Portmanteau (Q) statistic = 5.8191 Prob > chi2(1) = 0.0159
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Remedy: GARCH (1,1) Model
. arch dlsp, arch(1) garch(1) nolog
ARCH family regression
Sample: 4 to 223 Number of obs = 220 Wald chi2(.) = .Log likelihood = -366.1473 Prob > chi2 = .
------------------------------------------------------------------------------ | OPGdlsp | Coef. Std. Err. z P>|z| [95% Conf. Interval]-------------+----------------------------------------------------------------dlsp |_cons | .0232815 .0826522 0.28 0.778 -.1387138 .1852768-------------+----------------------------------------------------------------ARCH |arch | L1 | .1652834 .045527 3.63 0.000 .0760521 .2545146garch | L1 | .7815966 .0783583 9.97 0.000 .6280172 .935176_cons | .1121176 .0913255 1.23 0.220 -.066877 .2911122------------------------------------------------------------------------------
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RESIDUAL TESTS
. predict e, resid
. predict v, variance
. gen s=sqrt(v)
. gen se=e/s
. gen se2=se^2
. wntestq se2
Portmanteau test for white noise--------------------------------------- Portmanteau (Q) statistic = 30.0623 Prob > chi2(40) = 0.8735
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. corrgram se2
-1 0 1 -1 0 1 LAG AC PAC Q Prob>Q [Autocorrelation] [Partial Autocor]-------------------------------------------------------------------------------1 -0.0100 -0.0100 .02243 0.8810 | | 2 0.0873 0.0875 1.7295 0.4212 | | 3 -0.0914 -0.0911 3.6084 0.3070 | | 4 0.0091 0.0009 3.6269 0.4588 | | 5 -0.0114 0.0044 3.6562 0.5999 | | 6 0.0214 0.0127 3.7612 0.7090 | | 7 -0.0549 -0.0547 4.4529 0.7264 | | 8 -0.0243 -0.0290 4.5894 0.8004 | | 9 -0.0238 -0.0117 4.7205 0.8580 | | 10 0.0067 0.0017 4.7311 0.9084 | |
No Remaining ARCH…BUT, what about normality??
Recall: Normal distribution has skewness of 0 and kurtosis of 3 and we know that financial series tend to be fat tailed.
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.graph se, norm bin(50)F
ract
ion
se-4.20146 2.73526
0
.1
. sktest se Skewness/Kurtosis tests for Normality ------- joint ------ Variable | Pr(Skewness) Pr(Kurtosis) adj chi2(2) Prob>chi2-------------+------------------------------------------------------- se | 0.067 0.012 8.77 0.0125
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Solution: Use Robust Standard Errors—robust to departures from normality (Bollerslev & Wooldridge 1982)
. arch dlsp, arch(1) garch(1) nolog robust
ARCH family regression
Sample: 4 to 223 Number of obs = 220 Wald chi2(.) = .Log likelihood = -366.1473 Prob > chi2 = .
------------------------------------------------------------------------------ | Semi-robustdlsp | Coef. Std. Err. z P>|z| [95% Conf. Interval]-------------+----------------------------------------------------------------dlsp |_cons | .0232815 .0786518 0.30 0.767 -.1308732 .1774362-------------+----------------------------------------------------------------ARCH |arch | L1 | .1652834 .2083251 0.79 0.428 -.2430264 .5735931garch | L1 | .7815966 .3140995 2.49 0.013 .165973 1.39722_cons | .1121176 .2578869 0.43 0.664 -.3933314 .6175666------------------------------------------------------------------------------
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Inclusion of Exogenous Variables
. arch dlsp, arch(1) garch(1) nolog robust het(gore) bhhh
ARCH family regression -- multiplicative heteroskedasticity
Sample: 4 to 223 Number of obs = 220 Wald chi2(.) = .Log likelihood = -365.4092 Prob > chi2 = .
------------------------------------------------------------------------------ | Semi-robustdlsp | Coef. Std. Err. z P>|z| [95% Conf. Interval]-------------+----------------------------------------------------------------dlsp |_cons | .0135455 .080307 0.17 0.866 -.1438533 .1709443-------------+----------------------------------------------------------------HET |gore | -.1355259 .0615727 -2.20 0.028 -.2562061 -.0148457_cons | 5.006925 3.286556 1.52 0.128 -1.434607 11.44846-------------+----------------------------------------------------------------ARCH |arch | L1 | .1945511 .0973455 2.00 0.046 .0037575 .3853447garch | L1 | .6837859 .1219819 5.61 0.000 .4447057 .922866------------------------------------------------------------------------------
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. wntestq se2
Portmanteau test for white noise--------------------------------------- Portmanteau (Q) statistic = 29.1930 Prob > chi2(40) = 0.8965
. sktest se2
Skewness/Kurtosis tests for Normality ------- joint ------ Variable | Pr(Skewness) Pr(Kurtosis) adj chi2(2) Prob>chi2-------------+------------------------------------------------------- se2 | 0.000 0.000 . 0.0000
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ARCH IN MEAN
. arch dlsp, arch(1) garch(1) nolog robust het(gore) archm archmexp(sqrt(X))
ARCH family regression -- multiplicative heteroskedasticity
Sample: 4 to 223 Number of obs = 220 Wald chi2(1) = 4.84Log likelihood = -362.6718 Prob > chi2 = 0.0278
------------------------------------------------------------------------------ | Semi-robustdlsp | Coef. Std. Err. z P>|z| [95% Conf. Interval]-------------+----------------------------------------------------------------dlsp |_cons | -.80408 .3986817 -2.02 0.044 -1.585482 -.0226782-------------+----------------------------------------------------------------ARCHM |sigma2ex | .7068768 .3213948 2.20 0.028 .0769545 1.336799-------------+----------------------------------------------------------------HET |gore | -.1067959 .016462 -6.49 0.000 -.1390609 -.0745308_cons | 3.790934 .9837613 3.85 0.000 1.862797 5.71907-------------+----------------------------------------------------------------ARCH |arch | L1 | .1835399 .0981365 1.87 0.061 -.0088041 .3758838garch | L1 | .6634369 .1133509 5.85 0.000 .4412733 .8856006------------------------------------------------------------------------------
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