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BY
RAHUL A. PARSADRAKE UNIVERSITY
&STUART A. KLUGMAN
SOCIETY OF ACTUARIES
Copula Regression
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Outline of Talk
OLS Regression
Generalized Linear Models (GLM)
Copula Regression
Continuous case Discrete Case
Exaples
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!otation
!otation"
# $ Dependent %aria&le
'ssuption
# is related to s in soe functional for
VariablestIndependen,, 21 kXXX
),,(]|E[ 2111 nnn XXXfxXxXY ===
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OLS Regression
ikikiii XXXY +++++= 22110
Y is linearly related to Xs
OLS Model
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OLS Regression
2)(minii
YY
( )kikii
XXY
YXXXY
''
110
1
++=
=
Estimated Model
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OLSMulti*ariate !oral Distri&ution
'ssue
+ointl, follo- a ulti*ariate noral distri&ution
T.en t.e conditional distri&ution of # / X follo-s
noral distri&ution -it. ean and *ariance gi*en &,
kXXXY
,,,21
)()|( 1 xXXYXy xxXYE +==
YXXXYXYYVariance = 1
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OLS 0 M%!
#1.at 2 Estiated Conditional ean
3t is t.e MLE
Estiated Conditional %ariance is t.e error *ariance
OLS and MLE result in sae *alues
Closed for solution exists
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GLM
# &elongs to an exponential fail, of distri&utions
g is called t.e link functionx4s are not rando
#/x &elongs to t.e exponential fail,
Conditional *ariance is no longer constant5araeters are estiated &, MLE using nuerical
et.ods
)()|( 1101
kkxxgxXYE +++==
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GLM
Generalization of GLM" # can &e an, distri&ution(SeeLoss Models)
Coputing predicted *alues is difficult
!o con*enient expression conditional *ariance
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Copula Regression
# can .a*e an, distri&ution
Eac. i can .a*e an, distri&ution
T.e 6oint distri&ution is descri&ed &, a Copula
Estiate # &, E(#/X=x) $ conditional ean
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Copula
3deal Copulas -ill .a*e t.e follo-ing properties"
ease of siulation
closed for for conditional densit,
different degrees of association a*aila&le fordifferent pairs of *aria&les7
Good Candidates are"Gaussian ! MVN C"u#a
t1Copula
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M%! Copula
CD8 for M%! is Copula is
9.ere Gis t.e ulti*ariate noral cdf -it. zeroean: unit *ariance: and correlation atrixR7
Densit, of M%! Copula is
9.ere vis a *ector -it. it. eleent
)])([)],([(),,,( 111
21 nn xFxFGxxxF =
5.01
2121 *2
)(
exp)()()(),,,(
= R
vIRv
xfxfxfxxxf
T
nn
)]([1 ii xFv =
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Conditional Distri&ution inM%! Copula
T.e conditional distri&ution of xn gi*en x;
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Copula RegressionContinuous Case
5araeters are estiated &, MLE7
3f are continuous *aria&les: t.en-e use pre*ious e=uation to find t.e conditionalean7
one1diensional nuerical integration is needed tocopute t.e ean7
kXXY
,, 1
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Copula RegressionDiscrete Case
9.en one of t.e co*ariates is discrete
P!$#%"
deterining discrete pro&a&ilities fro t.e Gaussian
copula re=uires coputing an, ulti*ariatenoral distri&ution function *alues and t.uscoputing t.e likeli.ood function is difficult
S#u'in"
Replace discrete distri&ution &, a continuousdistri&ution using a unifor kernel7
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Copula Regression $ Standard Errors
>o- to copute standard errors of t.e estiates?
's n 1@ A: MLE : con*erges to a noraldistri&ution -it. ean and *ariance 3()1;: -.ere
3()$ 3nforation Matrix7
= )),(ln(*)(
2
2
XfEnI
n
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>o- to copute Standard Errors
Loss Models" BTo o&tain inforation atrix: it isnecessar, to take &ot. deri*ati*es and expected
*alues: -.ic. is not al-a,s eas,7 ' -a, to a*oid t.ispro&le is to sipl, not take t.e expected *alue7
3t is called BO&ser*ed 3nforation7
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Exaples
'll exaples .a*e t.ree *aria&les
R Matrix "
Error easured &,
'lso copared to OLS
( ).* ).*
).* ( ).*
).* ).* (
2)( ii YY
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E+a"#% (
Dependent $ 1 Gaa
T.oug. is siulated fro 5areto: paraeterestiates do not con*erge: gaa odel fit
Error"
Va!ia$#%s X(,Pa!%' X-,Pa!%' X,Gaa
5araeters : ;FF : FF : ;FF
MLE 7: ;H;7;; ;7F: ;;7FF 7II: JK7
Copula KFFF7K
OLS HI;I7J
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E+ ( , S'an/a!/ E!!!s
Diagonal ters are standard de*iations and off1diagonal ters are correlations
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Exaple ; 1 Cont
Maxiu likeli.ood Estiate of Correlation Matrix
( ).*(( ).011
F7I;; ; F7I;
F7H F7I; ;
R-hat =
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Exaple
Dependent $ 1 Gaa
; 0 estiated Epiricall,
Error"
Va!ia$#%s X(,Pa!%' X-,Pa!%' X,Gaa
5araeters : ;FF : FF : ;FFMLE 8(x) 2 xn $ ;n
f(x) 2 ;n8(x) 2 xn $ ;n
f(x) 2 ;n7F: J;7F
Copula KK:I7K
OLS HI:;I7J
GLM J;:H7IK
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Exaple
Dependent $ 1 Gaa
5areto for estiated &, Exponential
Error"
Va!ia$#%s X(,Pissn X-,Pa!%' X,Gaa5araeters K : FF : ;FF
MLE K7HK ;;7 7HI: JJ7J
Copula KI:HJ
OLS KJ:K7K
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Exaple
Dependent $ 1 Gaa
; 0 estiated Epiricall,
C 2 N of o&s x and a 2 (N of o&s 2 x)
Error"
Va!ia$#%s X(,Pissn X-,Pa!%' X,Gaa
5araeters K : FF : ;FF
MLE 8(x) 2 cn P anf(x) 2 an
8(x) 2 xn $ ;nf(x) 2 ;n
7H: J7J
Copula OLS GLM
KK:JJJ7J KJ:K7K HK:IFJ7J
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Exaple K
Dependent $ ; 1 5oisson
: estiated &, Exponential
Error"
Va!ia$#%s X(,Pissn X-,Pa!%' X,Gaa
5araeters K : FF : ;FF
MLE K7HK ;;7 7HH: JJ7J
Copula ;FJ7I
OLS ;;7HH
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Exaple H
Dependent $ ; 1 5oisson
0 estiated &, Epiricall,
Error"
Va!ia$#%s X(,Pissn X-,Pa!%' X,Gaa
5araeters K : FF : ;FF
MLE K7HI 8(x) 2 xn $ ;nf(x) 2 ;n
8(x) 2 xn $ ;nf(x) 2 ;n
Copula ;;F7F
OLS ;;7HH