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

itio

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

ns

of e

2

CorrelogramLag

0 10 20 30 40

-1.00

-0.75

-0.50

-0.25

0.00

0.25

0.50

0.75

1.00

-1.00

-0.75

-0.50

-0.25

0.00

0.25

0.50

0.75

1.00

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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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