copula regression

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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