02 Lent Lecture 2 - TSFE8p Ach34

8/12/2019 02 Lent Lecture 2 - TSFE8p Ach34

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Further Time Series

8. Correlations and CopulasTime Series

Andrew Harvey

February 2013

Andrew Harvey () Further Time Series 8. Correlations and Copulas Fe bru ary 2 013 1 / 4 8

Joint distributions, Dependence and CopulasBivariate distributions

Joint density function is f(y1 , ..., yN).Restrict to bivariate and continuous

F(y1 , y2) =Pr(Y1 y1, Y2 y2) =Z y1

Z y2

f(x1 , x2)dx1dx2

Joint density is

f(y1 , y2) =2F(y1, y2)

y1y2.

Marginal, conditional conditional, f(y1 j y2)

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Bivariate normal with zero means is

f(y1 , y2) = 1

212p12 exp 1

2(12

) y21

21

22y1y2

12

+y22

22

where is the correlation coecient.Replacey1 byy1 and similarly for y2 to get general bivariate normal.The marginal ofy1 is N(1 ,

21)

The conditional, f(y1 j y2), is normal with

=1+ (1/2)y2 , 2 =21(1

2)


33

22

11

0 00.00-1

xy

-1

-2 -2

-3 -30.05

0.15

z 0.10



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Asset allocation

Understanding and measuring the relationship between movements indierent assets plays a key role in designing a portfolio. Markovitz -

portfolio that gives minimum variance for a given expected return. Themultivariate normal distribution is not usually suitable for this task for tworeasons: asset returns are not normally distributed and their comovementsare not adequately captured by correlation coecients. More specically,fat tails occur in marginals. Thus higher probability of larger movementsthan with a normal. Similarly the probability of two markets bothexhibiting a relatively high movement (in same direction) may be higherthan with a bivariate normal. Thus there may be a high probability thatboth markets experience large falls at the same time. This has importantimplications for an asset allocation strategy.


Measures of association

In a bivariate normal, correlation coecient is . Sample correlation is

r= t(y1t y1)(y2t y2)

pt(y1t y1)2 t(y2t y2)2

Correlation coecient based on ranks r1t, r2t, t=1, , , .T, is Spearmansroh, denotedrS.Because of properties of ranks it can be shown to simplifyto

rS =1 6

Tt=1(r1t r2t)

T(T2 1)

Linear correlation - doesnt capture nonlinear. Figure shows an exactcomonotonic relationship. Calculating rfrom a set of points on this line

will not give unity. Plot of ranks,r1t, r2t, t=1, , , .T,will yield a 45 degreestraight line and a correlation of one.



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-2.0 -1.5 -1.0 -0.5 0.5 1.0 1.5 2.0

-2

-1

1

2

3

4

5

x

y

Figure: A comonotonic relationship


A pair of random variables are said to be concordant if large (small)values of one tend to be associated with large (small) values of the other.More precisely, let the pair (y1i, y2i) and(y1j, y2j) denote two sets ofobservations. Then (y1i, y2i) and(y1j, y2j) areconcordant ify1i < y1j and

y2i < y2j, or ify1i > y1j andy2i > y2j. Otherwise they are discordant.Note that concordance is also dened by (y1i y1j)( y2i y2j) > 0.The strength of the relationship between two variables can be measured byKendalls Tau. All pairs of observations - of which there areT!/((T 2)!2!) =T(T 1)/2 are compared and Kendalls Tau iscomputed as the number of concordant pairs minus the number ofdiscordant pairs, divided by the total.

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ctau= 2T(T 1)

T

i=1

T

j>i

sgn(y1i y1j)sgn(y2i y2j)

Thus if the ordering ofy1i andy1j is the same as that ofy2i andy2j, one

enters the summation, rather than minus one. Like the correlationmeasures,r andrS, it lies in the range [1, 1], and like rS it depends onlyon ranks.Blomqvists beta is calculated by subtracting one from twice theproportion of observations which, when the medians are subtracted, havethe same sign. In range [1, 1].The correlation coecient r is sensitive to outliers. Apparent in gure

which shows a scatter plot for GM and IBM. The correlation is r=0.377,while Kendalls tau is 0.216. However, in rst 500 - which includes theoutliers from the crash of 1987 - r= .74 and Kendalls tau is .37.


-25 .0 -22 .5 -20 .0 -17 .5 -15 .0 -12 .5 -10 .0 -7.5 -5.0 -2.5 0 .0 2 .5 5 .0 7 .5 10.0 12.5

-20

-15

-10

-5

0

5

10 GM IBM

Figure: Scatter plot for GM and IBM

Andrew Harvey () Further Time Series 8. Correlations and Copulas Fe bru ary 20 13 1 0 / 4 8


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The probability that an observation from rst series is less than

1quantile,(1), at the same time as the corresponding observationfrom the second series is below the 2quantile,(2), is

Pr(Y1 (1), Y2 (2)) =F((1), (2))

Such probabilities are given by the copula.


The copula

A copula models the relationship between two variables independently oftheir marginal distributions. It therefore focuses on dependence orassociation.The copula is a joint distribution function of standard uniform randomvariables, that is

C(u1 , u2) =Pr(U1 u1 , U2 u2), 0 u1, u2 1

Since the PIT, F(Y), has a uniform distribution, the copula may becombined with the marginal distribution functions to give the full jointdistribution function.Specically, a copula computed at u1 =F1(y1), u2 =F2(y2) givesF(y1 , y2) since

C(F(y1), F(y2)) = Pr(U1 F1(y1), U2 F2(y2))= Pr(F11 (U1) y1 , F

12 (U2) y2)

= Pr(Y1 y1 , Y2 y2) =F(y1 , y2)

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Joint distribution

The Clayton copula is dened as

C(u1,

u2) = (u

1 + u

2 1)

1/,

2 [1,

),

6=0 (1)= u1u2 , =0

Suppose that the marginal distributions ofY1 andY2 , are bothexponential, that is thepdf is 1i exp(y/i), i=1, 2, and that they areconnected by a Clayton copula. The CDF for the marginals is

F(y) =1 exp(y/i), i=1, 2.

Hence the joint distribution function ofY1 andY2 is

F(y1 , y2) = ((1 exp(y1/1) + (1 exp(y2/2)

1)1/.


Sklars theorem

Sklars theorem states that ifF(y1 , y2) is a joint distribution function withcontinuous marginalsF1(y1) andF2(y2), then there exist a unique copula.Marginal distributions do not need to be of the same form, nor is thechoice of copula constrained by the choice of marginals.Hence, given the joint distribution function, the univariate marginals andthe dependence structure can be separated, with the dependence structurerepresented by the copula.Ify1 =(1) is the 1quantile and y2 =(2) is the 2quantile, thenwe can set u1 =1 andu2 =2 so enabling us to write

C(1 , 2) =F((1), (2)).

So the copula is the joint distribution with respect to the quantiles.



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Thecopula density is

c(u1 , u2) = 2

u1u2C(u1 , u2).

We may write

f(y1 , y2) =c(F(y1), F(y2)).f1(y1).f2(y2)

If the marginal densities are uniform, the joint density function is thecopula density. If not, its shape is stretched and contracted by the form ofthe probability density functions.Product copula

When the variables are independent

C(u1 , u2) =Pr(U1 u1). Pr(U2 u2) =u1u2 , 0 u1 , u2 1

andc(u1 , u2) is unity.


Gaussian copula

A Gaussian copula is constructed asCG(u1 , u2) =(1(u1),1(u2)), where (.) is the cdf of a standard normalvariate, and (.) is a bivariate normal distribution function in which thecorrelation is . Thus

CG(u1, u2) = 1

2p

12Z 1 (u2 )

Z 1 (u1 )

exp s2 22sr+ r22(12)

dsdrhttp://www.wired.com/print/techbiz/it/magazine/17-03/wp_quantClayton copula

Clayton copula was dened in1.Figures show scatter plots of 200 ranked observations generated fromClayton copulas with =1 and =5. The concentration of points in thelower left hand corner, particularly with =5 indicates tail dependence.

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0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0Clayton copul a (T = 200, = 1).

U2 U1

Figure: Scatter plot of 200 ranked observations from a Clayton copula with =1.


0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0Clayton copul a (T = 200, = 5).

U2 U1

Figure: Scatter plot of 200 ranked observations from a Clayton copula with =5.



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For =5, gure shows the proportions in the squares dened by thedeciles ( estimatingc(u1, u2)/100).

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

00.1

0.20.3

0.40.5

0.60.7

0.80.9

1

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

u1

2D h istogram of the data simulated from the Clay ton copula w ith=5

u2

em

piricaldistribution

Figure: Bivariate histogram of 200 observations simulated from a Clayton copulawith =5.


0.1 0.20.3 0.4 0.5

0.6 0.70.8 0.9

0.2

0.40.6

0.8

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

u1

Empirical C DF of the data simulated from the Clay ton copula w ith=5

u2

Fn

(u1,u2

)

Figure: Empirical copulaAndrew Harvey () Further Time Series 8. Correlations and Copulas Fe bru ary 20 13 2 2 / 4 8


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Survival copulas

Thesurvival function is dened by

C(u1 , u2) =Pr(U1 > u1 , U2 > u2).

The survival function gives the probability that both observations lie abovetheir pre-assigned quantiles, that is

C(1 , 2) =Pr(y1t> 1(1), y2t > 2(2)) =1 1 2+ C(1, 2);(2)

This depends on C(1 , 2) as

C(1 , 2) =1 1 2+ C(1, 2)

Note that C(0.5, 0.5) =C(0.5, 0.5).The probabilities of being in the other two quadrants are 1 C(1 , 2)and2 C(1 , 2). Similar relationships hold for the correspondingsample proportions.


Proof.

From Cherubini et al (2004, p75) or McNeil et al (2005, p196), C(1, 2)is

Pr (y1t> 1(1) andy2t > 2(2))

= Pr (y1t> 1(1)) +Pr(y2t> 2(2)) (Pr (y1t> 1(1) ory2t> 2(2))

= 1 1+1 2 (1 C(1 , 2)) =1 1 2+ C(1, 2)

Similarly,

Pr (y1t > 1(1) andy2t 2(2))

= 2 1+Pr (y1t 1(1) andy2t > 2(2))

Since the four probabilities must sum to one, we nd

Pr (y1t 1(1) andy2t > 2(2))=1 C(1, 2)

Pr (y1t> 1(1) andy2t 2(2))=2 C(1, 2)



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Dependence

The copula provides a exible way of capturing dependence.The quadrant association,

C(1

, 2

) + C(1

, 2

), 0 1

, 2 1

gives a measure of dependence in the range [0, 1].It can be seen from (2) that quadrant association just depends onC(1 , 2) and is equal to

1 1 2+2C(1 , 2)

There ispositive quadrant dependency ifC(1 , 2) 12.Blomqvists beta is the quadrant association at 1 =2 =0.5,standardized so as to lie in the range [1, 1] and to be zero when theseries are independent. It is given by2(C(0.5, 0.5) + C(0.5, 0.5)) 1= 4C(0.5, 0.5) 1.


Dependence

Conditional probabilities for measuring dependence depend on the copula.The probability that an observation from rst series is less than a givenquantile,(1), given that the corresponding observation from the secondseries is below a given quantile, (2), is

F((1), (2))/F((2)) =C(1 , 2)/2 .

For the Clayton copula with 1 =2 =

C(, )/=

2 1/

(3)

Figure plotsC(, )/for three values of. When =1 the tail

dependence for = .10 is 0.526, but if=5 it goes up to 0.870; compareearlier gures.

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0.0 0.1 0.2 0.3 0.4 0.5

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Tau

C/Tau

Figure: Lower tail dependence, C(, )/, for Clayton copula for =1 (solidline), =5 (upper) and =0.5 (dashed) and independent (lower).


Dependence

The coecients of tail dependence are measures of pairwise dependencethat depend on the copula; see McNeil et al (2005, p208). The coecientof lower (left) tail dependence, or lower tail index, is

L = lim!0 C(, )/,

while the coecient of upper (right) tail dependence is

U = lim!1

C(, )/(1 ).

If two variables have a bivariate normal distribution, they areasymptotically independent in the tails as the coecients of tail

dependence are both zero unless =1 in which case they are both one.On the other hand, a t-copula does exhibit tail dependence.



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Dependence

For > 0, the Clayton copula exhibits lower tail dependence, withL =21/, as is easily seen from (3).For =1, C(, )/' 1/2 for small andL =0.5. For =5,

L =0.

870,

the same as was calculated for =0.

1.

As !,

C(, )/! 1.The practical implications are that with a small , such as 0.05 or 0.01,C(, ) may be close to and the probability of one variable being belowits quantile given that the other is below its quantile is close tounity.Using LH

bopitals rule shows that the upper tail dependence for the

Clayton copula is zero. An example of positive upper tail dependence is

provided by the bivariate Gumbel family of copulas

C(u1 , u2) =exp([( ln u1) + ( ln u2)

]1/, 1

ThenU =2 21/ andU > 0 for > 1.


It is possible to derive the relationship between Kendalls Tau and thetheoretical copula

au=4E[C(U1 , U2)] 1

Corresponding relationships may be derived for other measures such asSpearmans roh.For copulas based on a single parameter, there is usually a relationshipbetween this single parameter and measures of dependence. eg for theClayton copula, Kendalls Tau is /(+2); see Embrechts et al (2003,p35).

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Estimation

The log-likelihood function of the observations y1t, y2t, t=1, ..., T, is

ln L(; y1 , y2) =T

t=1

ln c(F(y1t), F(y2t)) +T

t=1

ln f1(y1t) +T

t=1

ln f2(y2t)

where includes the parameters of both copula and marginals.The calculations may be simplied by rst estimating the marginals andthen the copula. This is called the inference for the margins (IFM)method. According to CLV (2004), it entails very little loss in eciency.In canonical ML (CML), the copula parameters are estimated from theranks without specifying the marginal distributions. The criterion function

to be maximised with respect to the copula parameters, c,

is

ln L(c; r1, r2) =T

t=1

ln c(r1t/(T+1), r2t/(T+1); c)


Empirical copula

Theempirical copula is dened on a lattice, the domain of the empiricalcopula, in which each axis in the unit square is broken into Tequal spacesdelineated by the points 0, 1/(T+1), 2/(T+1), ..., 1. ie for each axis

the points are i1/(T+1) andi2/(T+1),with i1 , i2 =0, .., T. Given pairsof ranks(r1,t, r2,t), t=1, ..., T, the empirical copula is

bC(i1/(T+1), i2/(T+1)) = 1T

T

t=1

I(r1,t i1)I(r2,t i2)

where I(r1,t i1) is the indicator function. This estimator converges tothe underlying copula, just as the univariate empirical DF, obtained

directly from the order statistics, converges to the true DF.



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Empirical copula

Theempirical copula frequency,

bc(i1/(T+1), i2/(T+1)), is 1/(T+1)

if the the observations rankedi1 and i2 occur at the same time, ie (i1,i2) is

an element of the sample, and is zero otherwise. It is similar to the scatterdiagram.It is more useful to construct a grouped empirical copula - this is like abivariate histogram. For example the gure for the data generated fromthe Clayton copula showed the proportions of ranked observations in thesquares dened by the deciles.** A changing copula can be tracked by using a time series lter, such asan EWMA, to estimate the copula probabilities. The application describedin Harvey (2010) shows how the association between the Hong Kong(Hang Seng) and Korean stock market indices increased in the late 1990s.


Dynamic copulas

Time-varying copulas can be modeled using the conditional score to drivea dynamic equation for the shape parameter. Since the conditional scoretakes account of the specication of the copula, it would seem to be abetter way of proceeding than the essentially ad hocapproach of Patton(2006). Creal et al(2012) illustrate the viability and relevance of the DCS

approach in an application of dynamic Gaussian copulas to exchange ratedata.The conditional score for the Clayton copula is

ut = ln(1t2t) + (1+tjt1)1 +2 ln(

tjt11t +

tjt12t 1)

+

1+2tjt1

tjt1

!(

tjt11t ln 1t+

tjt12t ln 2t)

tjt11t +

tjt12t 1

,

whereit=F(yit), i=1, 2.The response to a pair of observations is not as readily interpretable as itis for the bivariate normal distribution.



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Dynamic copulas

However, the rst term involves the product 1t2t, and so is a little likethe productx1tx2t. In the Gaussian model the score modies the impact ofx1tx2tby taking account of how the product was formed and the currentparameter value. The same is true here.Figure shows the response of the score when 2 varies, but 1 is xed.Two points are worth noting.Firstly, as expected, the response is asymmetric in the sense that thebehaviour when 1 xed at 0.9 is not a mirror image of the behaviour for1 xed at 0.1.

Secondly, when 1 =0.1, the score is only positive for vaues of2 close to0.1, the eect being more pronounced when =5, as opposed to =1.


0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

-3

-2

-1

0

1

tau

u

Figure: Response of score for =1 when 2 varies, but 1=0.1 (thick line) or0.9 (thin). Dashed line shows response for =5 and 1=0.1.

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Dynamic copulas

A full maximum likelihood approach can, in principle, be used to jointlyestimate dynamic volatility and copula parameters.However, a two-step procedure may be more appealing in practice. If aunivariate Beta-t-EGARCH is tted to each series, the PITs can becomputed using a subroutine for a regularized incomplete beta function.


REFERENCES

Cherubini, U., Luciano, E. and W. Vecchiato (2004). Copula methods inFinance. John Wiley and Sons: Chichester.Embrechts P, Lindskog F, and A.J. McNeil (2003). Modelling dependencewith copulas and applications to risk management. In: Handbook ofHeavy Tailed Distributions in Finance, ed. S. Rachev, Elsevier, Chapter 8,pp. 329-384Harvey, A. C. (2010) Tracking a changing copula. Journal of EmpiricalFinance,17, 485-500.McNeil, A.J., Frey, R. and P. Embrechts (2005). Quantitative RiskManagement. Princeton Series in Finance.Nelsen (1999). An introduction to copulas. Springer-Verlag. NewYork.Patton, A. J. (2006). Modelling asymmetric exchange rate dependence.

International Economic Review, 47, 527-56.Rodriguez, J.C. (2007). Measuring nancial contagion: a copula approach.Journal of Empirical Finance, 14, 401-23.



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02 Lent Lecture 2 - TSFE8p Ach34