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““Data analysis of multiData analysis of multi--wavelength wavelength magnitudes the SDSSmagnitudes the SDSS--DR3 using a DR3 using a

AutoRegressive Conditional AutoRegressive Conditional Heteroskedasticity (ARCH) processHeteroskedasticity (ARCH) process””The Sloan Digital Sky Survey (SDSS):The Sloan Digital Sky Survey (SDSS):FROM ASTEROIDS TO COSMOLOGY FROM ASTEROIDS TO COSMOLOGY

International Symposium International Symposium Friday, August 15 Friday, August 15 –– Saturday, August 16, 2008 Saturday, August 16, 2008

Chicago, Illinois Chicago, Illinois by by

Byron E. BellByron E. Bell

KennedyKennedy--King CollegeKing College

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AbstractAbstractThe view of multiThe view of multi--wavelength magnitudes of the wavelength magnitudes of the quasar datasetquasar dataset optical bands of the Sloan Digital optical bands of the Sloan Digital Sky Survey Data Release 3 (SDSSSky Survey Data Release 3 (SDSS--DR3) of DR3) of Penn State University is more unique by using a Penn State University is more unique by using a Autoregressive Conditional Heteroskedasticity Autoregressive Conditional Heteroskedasticity (ARCH) method from econometrics. Viewing (ARCH) method from econometrics. Viewing square error terms of SDSSsquare error terms of SDSS--DR3, sig_u, sig_g, DR3, sig_u, sig_g, sig_r, sig_i as independent variables. And sig_r, sig_i as independent variables. And variance of the sig_z (variance of the sig_z (sig_zsig_z, squared ) as a , squared ) as a forecast term is utilized. forecast term is utilized. KEYTERMS: Quasar, MBH, ARCH, Variability.KEYTERMS: Quasar, MBH, ARCH, Variability.

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IntroductionIntroductionThe model of the ARCH is a nonlinear function and in The model of the ARCH is a nonlinear function and in this study the redshift is the component that determines this study the redshift is the component that determines the apparent magnitudes of light [ultraviolet (u), green the apparent magnitudes of light [ultraviolet (u), green (g), red (r), more red (i), even more red (z)], ugriz of (g), red (r), more red (i), even more red (z)], ugriz of quasars. In the ARCH theory of (Engel 1982) it takes quasars. In the ARCH theory of (Engel 1982) it takes past data squared error terms of stock prices, to past data squared error terms of stock prices, to predictions of future variance of the stock prices. In the predictions of future variance of the stock prices. In the ARCH method (Bell 2008) of this paper makes future ARCH method (Bell 2008) of this paper makes future terms the factors (sig_ugri) squared and variance of terms the factors (sig_ugri) squared and variance of sig_z, even more red (z), apparent magnitude of light, as sig_z, even more red (z), apparent magnitude of light, as the forecast variable. In this process, the model will use the forecast variable. In this process, the model will use variables in a mathematical analysis to try to draw a variables in a mathematical analysis to try to draw a clearer view of quasar phenomenon from out of a black clearer view of quasar phenomenon from out of a black hole.hole.

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Basic conceptsBasic concepts

Conditional variance is not constant Conditional variance is not constant over timeover timeConditional variance may affect the Conditional variance may affect the conditional meanconditional meanA regression model for the mean should A regression model for the mean should include some functions of the include some functions of the conditional variance.conditional variance.

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ARCH (ARCH (Engle, 1982)Engle, 1982) and three and three stylized factsstylized facts

Conditional variance change over time, Conditional variance change over time, sometimes quite substantiallysometimes quite substantiallyThere is volatility clustering There is volatility clustering –– large (small) large (small) changes in unpredictable returns tend to be changes in unpredictable returns tend to be followed by large (small) changes followed by large (small) changes The unconditional distribution of returns has The unconditional distribution of returns has ‘‘fatfat’’tails giving a relatively large probability of tails giving a relatively large probability of ‘‘outliersoutliers’’ relative to the normal distribution.relative to the normal distribution.

Engle, Robert F. (1982) “Autoregressive Conditional Heteroskedasticity with Estimates of the Variance of U.K. Inflation,” Econometrica, 50, 987?008.

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The Graph: The Graph: Apparent Magnitudes of ultraviolet (u) Apparent Magnitudes of ultraviolet (u) light shows clusteringlight shows clustering

Z

6543210

U_M

AG

28

26

24

22

20

18

16

14 Rsq = 0.5372

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The Graph: The Graph: Apparent Magnitudes of green (g) lightApparent Magnitudes of green (g) lightshows clusteringshows clustering

Z

6543210

G_M

AG

28

26

24

22

20

18

16

14 Rsq = 0.2894

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The Graph: The Graph: Apparent Magnitudes of red (r) lightApparent Magnitudes of red (r) lightshows clusteringshows clustering

Z

6543210

R_M

AG

24

22

20

18

16

14 Rsq = 0.1179

99

Z

6543210

I_M

AG

22

21

20

19

18

17

16

15 Rsq = 0.0879

The Graph: Apparent Magnitudes of more red (i) light shows clustering

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RegressionRegressionVariables Entered/Removedb

(sig_i)**2,(sig_g)**2,(sig_u)**2,(sig_r)**2

a. Enter

Model1

VariablesEntered

VariablesRemoved Method

All requested variables entered.a.

Dependent Variable: Var(sig_z)b.

Model Summary

.988a .976 .976 **********Model1

R R SquareAdjustedR Square

Std. Error ofthe Estimate

Predictors: (Constant), (sig_i)**2, (sig_g)**2,(sig_u)**2, (sig_r)**2

a.

ANOVAb

.000 4 .000 467322.6 .000a

.000 46415 .000

.000 46419

RegressionResidualTotal

Model1

Sum ofSquares df Mean Square F Sig.

Predictors: (Constant), (sig_i)**2, (sig_g)**2, (sig_u)**2, (sig_r)**2a.

Dependent Variable: Var(sig_z)b.

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The Model The Model σσzz22 == aaoo + + aauuεεii

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Using Einstein notation of the Using Einstein notation of the above model for each term from above model for each term from

sig_usig_u, , sig_gsig_g, , sig_rsig_r, , sig_isig_iThe null hypothesis is HThe null hypothesis is Hoo: : aaugriugri=0=0

The alternative The alternative hypothesis is hypothesis is HH11: : aaugriugri≠≠00

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SUMMARYSUMMARYF=467322.6, Alpha=1%=.01F=467322.6, Alpha=1%=.01

PP--value of value of εεugriugri^2^2 =.000=.000Alpha > PAlpha > P--value value Reject NReject Nullull HypothesisHypothesis (H0)(H0)

Alpha < PAlpha < P--value value Accept H0Accept H0.01>.000.01>.000 RejectReject H0H0

ConclusionConclusionTheThe NNull Hypothesisull Hypothesis (H0) can be rejected at (H0) can be rejected at

Alpha equal 0.01 levelAlpha equal 0.01 level

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ReferencesReferences

Engle, Robert F. (1982) “Autoregressive Conditional Heteroskedasticity with Estimates of the Variance of U.K. Inflation,” Econometrica, 50, 987?008.