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On the volatility On the volatility of ENSOof ENSO

CMOS 2012 Congress / AMS 21st NWP and 25th WAF ConferenceThe Changing Environment and its Impact on Climate, Ocean and Weather Services

Reza Modarres1, Taha B.M.J Ouarda1,2

1 Canada Research Chair on the Estimation of HydrometeorologicalVariables, INRS-ETE, 490 de la Couronne, Quebec, Qc, Canada, G1K

9A9,

2 Masdar Institute of Science and Technology, P.O. Box 54224, Abu Dhabi, UAE

reza.modarres@ete.inrs.ca, taha_ouarda@ete.inrs.ca

Plan Plan of of PresentationPresentation

IntroductionIntroductionProblemProblem definitiondefinition and Objectivesand ObjectivesData and Data and MethodologyMethodologyResultsResultsConclusionsConclusions

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IntroductionIntroduction

El Nino/Southern oscillation (ENSO) is a phenomenon of climatic

interannual variability which is found to be associated with global andy g

regional climate variations throughout the world.

The ENSO cycle refers to the year-to-year variations in

sea- surface temperatures,

convective rainfall,

surface air pressure

atmospheric circulation

that occur across the equatorial Pacific Ocean.

ENSO includes three phases, El Nino, La Nina and Neutral phases according to Sea Surface Temperature (SST).

Normal Pacific pattern. Equatorial winds gather warm water pool toward west. Cold water upwells along South American coast

Normal

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El NinoLa Nina

El Niño refers to the above-average sea-surface temperatures that periodically develop across the east-central equatorial Pacific (between approximately the date line

La Niña refers to the periodic cooling of sea-surface temperatures across the east-central equatorial Pacific. It represents the cold phase

http://en.wikipedia.org/wiki/El_Ni%C3%B1o‐Southern_Oscillation

Pacific (between approximately the date line and 120oW). It represents the warm phase of the ENSO cycle, and is sometimes referred to as a Pacific warm episode. El Niño originally referred to an annual warming of sea-surface temperatures along the west coast of tropical South America

of the ENSO cycle, and is sometimes referred to as a Pacific cold episode.

The results of La Niña are usually the opposite of those of El Niño; for example,

- El Niño would cause a dry period in the Midwestern U.S., while La Niña

ld i ll i d i hwould typically cause a wet period in that area.

- La Niña often causes drought conditions in the western Pacific; flooding in

northern South America; mild wet summers in northern North America, and

drought in the southeastern United States

- El Niño would cause warmer and drier winters in Canada

- In Africa, El Niño would cause a wetter –than-normal conditions in during

spring in East Africa and drier-than-normal during winter in South- central

- In Asia, El Niño would cause a drier conditions in parts of south east and

wetter conditions in parts of south west (Iran)

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ProblemProblem definitiondefinition and Objectivesand ObjectivesEnso is a climate trigger of the global climate.

The changing climate is becoming more variable or more extremes?The changing climate is becoming more variable or more extremes?How about the volatility of ENSO?

ObjectiveObjectiveLooking at the variability of atmospheric-oceanic index, ENSO

How the volatility of ENSO is changing ? Is it becoming more volatile?

Enso is going to be more variable or more extreme?

Data andData and MethodologyMethodologyData and Data and MethodologyMethodology

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DATADATA

Southern Oscillation Index 1940 2011

The Southern Oscillation is an oscillation in surface air pressure between the tropical eastern and the western Pacific Ocean waters. The strength of the Southern Oscillation is measured by the Southern Oscillation Index (SOI). 

Southern Oscillation Index 1940-2011

2

-1

0

1

2

3

4

SO

I

La Nina

-5

-4

-3

-2

Time

El Niño http://www.cpc.ncep.noaa.gov/data/indices/soihttp://www.srh.noaa.gov/jetstream/tropics/enso.htm

MéthodologieMéthodologie

ARMA-GARCH error modeling GARCH modeling

SOI change point detection

ARMA GARCH error modeling GARCH modeling

1 – ARMA modeling for conditional variance- Model identification- Model estimation- Testing model adequacy

-GARCH modeling for conditional variance- Model identification- Model estimation

2- Testing the residuals for inconstant variance( Heteroscedasticity, conditional variance or volatility)

Testing Procedures

Comparison before and afterchange point

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ARMA (p,q) model

tqtp BSOIB )()(

q

jjtj

p

iitit SOIcSOI

11

1- Identification of p and q

2- parameter estimation with maximum likelihood method

Ljung-Box test for the residuals

L

kkrkNNNQ

1

21 )()()2(

3- Testing adequacy of an ARMA model

2 parameter estimation with maximum likelihood method

Test for the ARCH effect (inconstant variance)

Test the ACF of squared residuals

Lagrange multiplier test of Engle (1982)

GARCH model

In hydrology, for example for streamflow or precipitation, GARCH model isfitted to the residuals of a linear model such as ARMA

ttt e MV

222 ttt e

)1,0( ~ Normalet

j

jtji

itit1

2

1

22

This model is called ARMA-GARCH error model where the conditional meanis modeled by an ARMA model while the conditional variance of the residualsare modeled by a GARCH model.

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Testing procedure for change detection in SOI

Nonparametric tests

Test for stationaryAugmented Dickey Fuller (ADF)

- Test for change in the men (Wilcoxon rank sum based)

- Test for change in the variance (levene)- Test for change in the distribution function (Kolmogorov-Smirnov)

- Augmented Dickey Fuller (ADF)

Test for nonlinearity- Brock-Dechert-Scheinkman (BDS)

ResultsResults

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Change point detection

120

1976

40

60

80

100

CU

M_s

td_S

OI

1940-1976 1977-2011

-20

0

20

Jan-

40

Jan-

43

Jan-

46

Jan-

49

Jan-

52

Jan-

55

Jan-

58

Jan-

61

Jan-

64

Jan-

67

Jan-

70

Jan-

73

Jan-

76

Jan-

79

Jan-

82

Jan-

85

Jan-

88

Jan-

91

Jan-

94

Jan-

97

Jan-

00

Jan-

03

Jan-

06

Jan-

09

C

Time

15

Preliminary analysis

Series mean Standarddeviation

Skewness Kurtosis maximum minimum

1940-2011 -0.11 1.09 -0.19 3.54 2.9 -4.6

1940-1976 0 0009 1 01 0 076 2 82 2 9 -2 7

Descriptive statistics

1940 1976 0.0009 1.01 0.076 2.82 2.9 2.7

1977-2011 -0.23 1.16 -0.31 3.75 2.9 -4.6

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20

30

40

50

60

1940-1976

KS test: p=0.65JB test: p= 0.50

-4 -3 -2 -1 0 1 2 3 40

10

20

50

60

70

1977-2011

Normality is accepted

-5 -4 -3 -2 -1 0 1 2 3 40

10

20

30

40

KS test: p=0.003JB test : p=0.0028

Normality is rejected

17

-1

0

1

2

3

4

ntil

es

of

Inp

ut

Sa

mp

le

Quantile-Quantile plot

1940-1976

-4 -3 -2 -1 0 1 2 3 4-4

-3

-2

Standard Normal Quantiles

Qu

an

1

2

3

4

ut

Sam

ple

-4 -3 -2 -1 0 1 2 3 4-5

-4

-3

-2

-1

0

Standard Normal Quantiles

Qu

an

tile

s of

Inp

u

1977-2011

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1

2

3

-3

-2

-1

0

SOI

-4

1 21977-20111940-1976

1940-1976

1977-2011

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0.4

0.6

0.8

e A

uto

corr

ela

tion

Sample Autocorrelation Function ACF

ocor

rela

tion

1940-1976

0 10 20 30 40 50-0.2

0

0.2

Lag

Sa

mpl

0 6

0.8

ela

tion

Sample Autocorrelation Function

on

Au

to

Lag

0 10 20 30 40 50-0.2

0

0.2

0.4

0.6

Lag

Sa

mp

le A

uto

corr

eA

uto

corr

elat

io

Lag

1977-2011

21

ACF for squared SOI

0.4

0.6

0.8

Sam

ple

Aut

ocor

rela

tion

ocor

rela

tion

1940-1976

0 2 4 6 8 10 12 14 16 18 20-0.2

0

0.2

Lag

S

0.6

0.8

atio

nti

on

Au

to

Lag

0 2 4 6 8 10 12 14 16 18 20-0.2

0

0.2

0.4

Lag

Sa

mp

le A

uto

corr

ela

Au

toco

rrel

at

Lag

1977-2011

22

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Nonparametric test

Wilcoxon rank sum based nonparametric test for equality in mean

Equality rejectedp= 0.0053

Nonparametric Levence test for variance equality

Equality rejectedp= 0.0043

Equality rejected

Nonparametric K-S test for equality of distribution

q y jp= 0.0357

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ARMA(p,q) model selection for SOI, 1940-1976

MODEL p q AIC BIC

AR(1) 1 0 2.39 2.4

AR(2) 2 0 2.52 2.53

AR(3) 3 0 2.61 2.62

MA(1) 0 1 2 58 2 59

ARMA-GARCH error model

MA(1) 0 1 2.58 2.59

MA(2) 0 2 2.64 2.65

MA(3) 0 3 2.70 2.71

ARMA(1,1) 1 1 2.30 2.31

ARMA(2,1) 2 1 2.36 2.38

ARMA(2,2) 2 2 2.48 2.50

ARMA(3 1) 3 1 2 49 2 51ARMA(3,1) 3 1 2.49 2.51

ARMA(3,2) 3 2 2.56 2.58

ARMA(4,1) 4 1 2.48 2.50

ARMA(5,1) 5 1 2.51 2.53

ARMA(6,1) 6 1 2.52 2.54

ARMA(7,1) 7 1 2.57 2.5824

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0.4

0.6

0.8

Sam

ple

Aut

ocor

rela

tion

Sample Autocorrelation Function

ocorrelation

ACF for the residuals

0 5 10 15 20 25 30 35 40 45 50-0.2

0

0.2

Lag

S

0.6

0.8

on

Sample Autocorrelation FunctionLag

ion

Auto

0 5 10 15 20 25 30 35 40 45 50-0.2

0

0.2

0.4

Lag

Sam

ple

Aut

ocor

rela

ti

Lag

Autocorrelat

ACF for squared residuals

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Test de ARCH pour ARMA (1,1) des résidus

0.9

1

0 2

0.3

0.4

0.5

0.6

0.7

0.8

pval

ueP-v

alu

e

Critical p‐value (0.05)

0 2 4 6 8 10 12 14 16 18 200

0.1

0.2

LagLag

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Test for stationarity (ADF test) Statistic: -5.88P-value= 0.00001

Test for the residuals

Test for nonlinearity (BDS test)

dimension BDS statistic P-value

m=2 0.00008 0.76

m=3 0.00126 0.78

m=4 0.00048 0.92

m=5 -0.00082 0.88

m=6 -0.00284 0.59

ARMA (p, q) model selection for SOI, 1977-2011

MODEL p q AIC BIC

AR(1) 1 0 2.57 2.58

AR(2) 2 0 2.68 2.69

AR(3) 3 0 2.81 2.82

MA(1) 0 1 2.84 2.85

MA(2) 0 2 2 84 2 85MA(2) 0 2 2.84 2.85

MA(3) 0 3 2.90 2.91

ARMA(1,1) 1 1 2.47 2.49

ARMA(2,1) 2 1 2.53 2.55

ARMA(2,2) 2 2 2.68 2.70

ARMA(3,1) 3 1 2.68 2.69

ARMA(3,2) 3 2 2.68 2.70( , )

ARMA(4,1) 4 1 2.75 2.77

ARMA(5,1) 5 1 2.77 2.79

ARMA(6,1) 6 1 2.80 2.82

ARMA(7,1) 7 1 2.81 2.83

ARMA(8,1) 8 1 2.82 2.84

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0.2

0.4

0.6

0.8

Sam

ple

Au

toco

rrel

atio

n

Sample Autocorrelation Function

ocor

rela

tion

ACF for the residuals

0 5 10 15 20 25 30 35 40 45 50-0.2

0

0.2

Lag

S

0.6

0.8

latio

n

Sample Autocorrelation Function

Au

t

Lag

atio

n

0 5 10 15 20 25 30 35 40 45 50-0.2

0

0.2

0.4

Lag

Sam

ple

Aut

ocor

rel

Lag

Au

toco

rrel

aACF for the squared

residuals

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0.9

1

ARCH test for residuals of ARMA (1,1)

0.3

0.4

0.5

0.6

0.7

0.8

pval

ueP

-val

ue

Critical p‐value (0 05)

0 2 4 6 8 10 12 14 16 18 200

0.1

0.2

LagLag

Critical p value (0.05)

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MODEL V M AIC BIC

GARCH(1,0) 1 0 2.49 2.53

GARCH(2,0) 2 0 2.51 2.56

ARCH(3,0) 3 0 2.50 2.56

GARCH(0,1) 0 1 2.53 2.57

Selecting GARCH(v,m) modelFor SOI, 1977-2011

GARCH(0,2) 0 2 2.49 2.54

GARCH(0,3) 0 3 2.50 2.56

GARCH(1,1) 1 1 2.47 2.52

GARCH(2,1) 2 1 2.51 2.57

GARCH(2,2) 2 2 2.51 2.58

GARCH(3,1) 3 1 2.49 2.56

( )GARCH(3,2) 3 2 2.52 2.60

GARCH(4,1) 4 1 2.53 2.61

Test of stationarity (ADF test)Statistic: -5.11P-value= 0.00001

Test for the residuals

Test for nonlinearity (BDS)

dimension BDS statistic P-value

m=2 0.0088 0.0109

m=3 0.0192 0.0005

m=4 0.0260 0.0001

m=5 0.0272 0.0001

m=6 0.0262 0.0001

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MODEL V M AIC BIC

GARCH(1,0) 1 0 2.49 2.53

GARCH(2,0) 2 0 2.51 2.56

GARCH (v, m) model selection for SOI, 1977-2011

( , )

GARCH(3,0) 3 0 2.50 2.56

GARCH(0,1) 0 1 2.53 2.57

GARCH(0,2) 0 2 2.49 2.54

GARCH(0,3) 0 3 2.50 2.56

GARCH(1,1) 1 1 2.47 2.52

GARCH(2,1) 2 1 2.51 2.57

GARCH(2,2) 2 2 2.51 2.58

GARCH(3,1) 3 1 2.49 2.56

GARCH(3,2) 3 2 2.52 2.60

GARCH(4,1) 4 1 2.53 2.61

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AIC BIC

GARCH(1 1) 0 62 0 28 ( 0 0002) 0 02 ( 0 82) 2 47 2 52

GARCH parameters

GARCH(1,1) 0.62 0.28 (p=0.0002) 0.02  (p=0.82) 2.47 2.52

GARCH(1,0) 0.63 0.33 (p=0.0003) ‐‐‐‐‐‐‐‐‐‐‐‐‐ 2.49 2.53

GARCH(2,0) 0.66 0.06 (p=0.11)0.22 (p=0.0031)

‐‐‐‐‐‐‐‐‐‐‐‐‐ 2.51 2.56

GARCH(3,0) 0.44 0.25 (p=0.02)0.16 (p=0.06)0.17 (p=0.08)

‐‐‐‐‐‐‐‐‐‐‐‐‐‐ 2.50 2.56

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4

5

e

Feb. 05

1

2

3

Con

dit

ion

al V

aria

nc

0

Time

35

R² = 0.6351

3

3.5

4

e

0 5

1

1.5

2

2.5

Con

dit

ion

al V

aria

nce

0

0.5

‐4 ‐3 ‐2 ‐1 0 1 2 3

Lagged Innovations

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1

2

3

4

Con

diti

onal

Var

ianc

e

Feb. 74Jan. 51

1940-1976

0

C

Time

8

10

12

Var

ianc

eMar. 05

Mar. 83

1977-2011

GARCH(1,1)

0

2

4

6

Con

diti

onal

V

Time

1977 2011

GARCH(1,1)

37

1940-1976

2

3

4

ion

al v

aria

nce

1977-2011

JAN FEB MAR APR MAY JUN JUL AUG SEP PCT NOV DEC0

1

Month

Con

dit

i

8

10

12

ian

ce

JAN FEB MAR APR MAY JUN JUL AUG SEP OCT NOV DEC0

2

4

6

8

Month

Con

dit

ion

al v

ari

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Test of stationarity (ADF test)

1940-1976Statistic: -8.73P-value= 0 00001

Test for conditional variance

1977-2011Statistic: -7.77P-value= 0 00001

Test of nonlinearity (BDS)

Dimension1940-1976 1977-2011

BDS statistic P-value BDS statistic P-value

2 0 081 0 0001 0 101 0 0001

P-value= 0.00001 P-value 0.00001

m=2 0.081 0.0001 0.101 0.0001

m=3 0.135 0.0001 0.166 0.0001

m=4 0.163 0.0001 0.204 0.0001

m=5 0.172 0.0001 0.218 0.0001

m=6 0.170 0.0001 0.217 0.0001

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ConclusionsConclusionsSOI characteristics have been changed in recent years:

Di t ib ti i b i k d d l Distribution is becoming more skewed and non-normal

The frequency of La Nina phase has increased (more extreme)

SOI is becoming less stationary (more variability)

SOI is becoming more nonlinear (more uncertainty)

The volatility of SOI is becoming more significant

Short run persistence in the volatility is observed in recent decades (less remember)

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Thank You !

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