[PPT] - Vine Copulas as a Way to Describe and Main Idea: Using . . . PowerPoint Presentation

SLIDE 1

Need for Studying . . . Statistical Character . . . Copulas Case of Three of More . . . Main Idea: Using . . . D-Vine Copulas: Idea C-Vine Copulas: Idea Vine Copulas vs. Other . . . How Vine Copulas Are . . . Home Page Title Page ◭◭ ◮◮ ◭ ◮ Page 1 of 12 Go Back Full Screen Close Quit

Vine Copulas as a Way to Describe and Analyze Multi-Variate Dependence in Econometrics: Computational Motivation and Comparison with Bayesian Networks and Fuzzy Approaches

Songsak Sriboonchitta1, Jainxi Liu1, Vladik Kreinovich2, and Hung T. Nguyen1,3

1Department of Economics, Chiang Mai University

Chiang Mai, Thailand, songsak@econ.chiangmai.ac.th

2Department of Computer Science, University of Texas at El Paso

500 W. University, El Paso, TX 79968, USA, vladik@utep.edu

3Department of Mathematical Sciences, New Mexico State University

Las Cruces, New Mexico 88003, USA, hunguyen@nmsu.edu

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Need for Studying . . . Statistical Character . . . Copulas Case of Three of More . . . Main Idea: Using . . . D-Vine Copulas: Idea C-Vine Copulas: Idea Vine Copulas vs. Other . . . How Vine Copulas Are . . . Home Page Title Page ◭◭ ◮◮ ◭ ◮ Page 2 of 12 Go Back Full Screen Close Quit

1. Need for Studying Dependence in Economics

In physics, many parameters, many phenomena are in-

dependent.

So, we can observe (and thoroughly study) simple sys-

tems by a small number of parameters.

Based on these simple systems, we determine the laws
f mechanics, electrodynamics, thermodynamics, etc.
We then combine these laws to describe more complex

phenomena.

In contrast, in economics, most phenomena are inter-

related.

So, in econometrics, studying dependence is of utmost

importance.

SLIDE 3

Need for Studying . . . Statistical Character . . . Copulas Case of Three of More . . . Main Idea: Using . . . D-Vine Copulas: Idea C-Vine Copulas: Idea Vine Copulas vs. Other . . . How Vine Copulas Are . . . Home Page Title Page ◭◭ ◮◮ ◭ ◮ Page 3 of 12 Go Back Full Screen Close Quit

2. Statistical Character of Economic Phenomena

Most physical processes are deterministic.
If we repeatedly drop the same object from the Leaning

Tower of Pisa, we observe the same behavior.

In contrast, if several very similar restaurants open in

the same area, some of them will survive and some not.

It is practically impossible to predict which will sur-

vive.

At best, we can predict the probability of survival.
Thus, in economics, we need to study dependence be-

tween random variables.

SLIDE 4

Need for Studying . . . Statistical Character . . . Copulas Case of Three of More . . . Main Idea: Using . . . D-Vine Copulas: Idea C-Vine Copulas: Idea Vine Copulas vs. Other . . . How Vine Copulas Are . . . Home Page Title Page ◭◭ ◮◮ ◭ ◮ Page 4 of 12 Go Back Full Screen Close Quit

3. Copulas

The joint distribution can be described by cdf and

marginals: F(x1, x2)

def

= Prob(X1 ≤ x1 & X2 ≤ x2); Fi(xi)

def

= Prob(Xi ≤ xi).

Independence means that F(x1, x2) = F1(x1) · F2(x2).
A natural way to describe dependence is to describe a

function C(a, b) such that F(x1, x2) = C(F1(x1), F2(x2)).

Such functions C(a, b) are called copulas.
The pdf f(x1, x2) can also be described in terms of

copulas: f(x1, x2) = c(F1(x1), F2(x2))·f1(x1)·f2(x2); c(a, b)

def

= ∂2C(a, b) ∂a ∂b .

SLIDE 5

Need for Studying . . . Statistical Character . . . Copulas Case of Three of More . . . Main Idea: Using . . . D-Vine Copulas: Idea C-Vine Copulas: Idea Vine Copulas vs. Other . . . How Vine Copulas Are . . . Home Page Title Page ◭◭ ◮◮ ◭ ◮ Page 5 of 12 Go Back Full Screen Close Quit

4. Case of Three of More Variables

F(x1, . . . , xn) = Prob(X1 ≤ x1 & . . . & Xn ≤ xn) can

also be described as F(x1, . . . , xn) = C(F1(x1), . . . , Fn(xn)).

The copula C(a, . . . , b) has to be determined from the

data.

To describe a function C(a, . . . , b) of n variables with

accuracy h > 0, we need h−n values C(i1 · h, . . . , in · h).

For n ≥ 3, we usually do not have that much data.
So, we need to describe the general dependence in

terms of functions of one and two variables.

SLIDE 6

Need for Studying . . . Statistical Character . . . Copulas Case of Three of More . . . Main Idea: Using . . . D-Vine Copulas: Idea C-Vine Copulas: Idea Vine Copulas vs. Other . . . How Vine Copulas Are . . . Home Page Title Page ◭◭ ◮◮ ◭ ◮ Page 6 of 12 Go Back Full Screen Close Quit

5. Main Idea: Using Conditional Probabilities

We started with a formal definition of independence

F(x1, x2) = F1(x1) · F2(x2).

A more intuitive definition is F1|2(x1 | x2) = F1(x1),

where F1|2(x1 | x2)

def

= Prob(X1 ≤ x1 | X2 = x2).

In general, F1|2(x1 | x2) = C1|2(F1(x1), F2(x2)), where

C1|2(a, b)

def

= ∂C12(a, b) ∂b .

For densities, we have

f1|2(x1 | x2) = c12(F1(x1), F2(x2))·f1(x1); c12(a, b) = ∂2C12(a, b) ∂a ∂b .

SLIDE 7

Need for Studying . . . Statistical Character . . . Copulas Case of Three of More . . . Main Idea: Using . . . D-Vine Copulas: Idea C-Vine Copulas: Idea Vine Copulas vs. Other . . . How Vine Copulas Are . . . Home Page Title Page ◭◭ ◮◮ ◭ ◮ Page 7 of 12 Go Back Full Screen Close Quit

6. D-Vine Copulas: Idea

For two variables, we have F(x1, x2) = C12(F1(x1), F2(x2)).
For three variables, we similarly have

F12|3(x2, x2 | x3) = C12|3(F1(x1 | x3), F2(x2 | x3), x3).

In general, for different values x3, we can have different

copulas C(a, b) = C12|3(a, b, x3).

It often makes sense to assume that the dependence

between X1 and X2 does not depend on X3: F12|3(x1, x2 | x3) = C12|3(F1|3(x1 | x3), F2|3(x2 | x3)).

We already know how to describe F1|3(x1 | x3) and

F2|3(x2 | x3) in terms of bivariate copulas and marginals.

Thus, we can describe F12|3(x1, x2 | x3) (and so,

F123(x1, x2, x3)) in terms of bivariate copulas and marginals.

SLIDE 8

Need for Studying . . . Statistical Character . . . Copulas Case of Three of More . . . Main Idea: Using . . . D-Vine Copulas: Idea C-Vine Copulas: Idea Vine Copulas vs. Other . . . How Vine Copulas Are . . . Home Page Title Page ◭◭ ◮◮ ◭ ◮ Page 8 of 12 Go Back Full Screen Close Quit

7. C-Vine Copulas: Idea

Main idea: we use probability densities instead of prob-
abilities. In general:

f(x1, x2, x3) = f1|23(x1 | x2, x3) · f2|3(x2 | x3) · f3(x3).

We know that f2|3(x2 | x3) = c23(F2(x2), F3(x3))·f2(x2).
Similarly,

f1|23(x1 | x2, x3) = c12|3(F1|3(x1 | x3)), F2|3(x2 | x3), x3)·f1|3(x1 | x3).

It often makes sense that assume that the correspond-

ing copula does not depend on x3; then: f1|23(x1 | x2, x3) = c12|3(F1|3(x1 | x3)), F2|3(x2 | x3))·f1|3(x1 | x3).

Hence, f(x1, x2, x3) = c12|3(F1|3(x1 | x3)), F2|3(x2 | x3))·

f1|3(x1 | x3) · c23(F2(x2), F3(x3)) · f2(x2) · f3(x3).

SLIDE 9

Need for Studying . . . Statistical Character . . . Copulas Case of Three of More . . . Main Idea: Using . . . D-Vine Copulas: Idea C-Vine Copulas: Idea Vine Copulas vs. Other . . . How Vine Copulas Are . . . Home Page Title Page ◭◭ ◮◮ ◭ ◮ Page 9 of 12 Go Back Full Screen Close Quit

8. Vine Copulas vs. Other Techniques

Bayesian networks assume that some conditional dis-

tributions are independent.

Thus, the Bayesian network approach can be viewed

as a particular case of the vine copula approach.

In fuzzy logic, we estimate P(A & B) as f&(P(A), P(B))

for an appropriate t-norm f&(a, b).

In particular, for X1 ≤ x1 and X2 ≤ x2, we get

F12(x1, x2) = f&(F1(x1), F2(x2)); F123(x1, x2, x3) = f&(F12(x1, x2), F3(x3)).

Here, we use the same operation to combine probabil-

ities corresponding to different variables.

In contrast, in vine copulas, we can use different cop-

ulas for different pairs of variables.

SLIDE 10

Need for Studying . . . Statistical Character . . . Copulas Case of Three of More . . . Main Idea: Using . . . D-Vine Copulas: Idea C-Vine Copulas: Idea Vine Copulas vs. Other . . . How Vine Copulas Are . . . Home Page Title Page ◭◭ ◮◮ ◭ ◮ Page 10 of 12 Go Back Full Screen Close Quit

9. Vine Copulas vs. Other Techniques (cont-d) General copulas ↓ Vine copulas ւ ց Bayesian Fuzzy networks techniques

Both Bayesian networks and fuzzy techniques have nu-

merous successful applications.

The more general vine copula eliminates disadvantages
f both approaches:

– in contrast to Bayesian techniques, vine copula can handle dependence between variables; – in contrast to fuzzy, vine copulas use different “and”-

perations (copulas) to combine different variables.

SLIDE 11

Need for Studying . . . Statistical Character . . . Copulas Case of Three of More . . . Main Idea: Using . . . D-Vine Copulas: Idea C-Vine Copulas: Idea Vine Copulas vs. Other . . . How Vine Copulas Are . . . Home Page Title Page ◭◭ ◮◮ ◭ ◮ Page 11 of 12 Go Back Full Screen Close Quit

10. How Vine Copulas Are Used in Econometrics

Economic processes are highly dynamic.
One of the most adequate models for the dynamics of

each variable r is the ARMA(p, q)-GJR(k, ℓ) model rt = c +

p

i=1

ϕi · rt−i + εi

q

j=1

ψj · εt−j, εt = ht · ηt, h2

t = ω + k

i=1

αi · ε2

t−i +

i: εt−i<0

γi · ε2

t−i + ℓ

j=1

βj · h2

t−j.

Here, residuals ηt corresponding to different moments

t are independent.

Copulas are used to describe the joint distribution of

the residuals ηt corresponding to different variables r.

SLIDE 12

Need for Studying . . . Statistical Character . . . Copulas Case of Three of More . . . Main Idea: Using . . . D-Vine Copulas: Idea C-Vine Copulas: Idea Vine Copulas vs. Other . . . How Vine Copulas Are . . . Home Page Title Page ◭◭ ◮◮ ◭ ◮ Page 12 of 12 Go Back Full Screen Close Quit

11. Acknowledgments

We are greatly thankful to the Faculty of Economics
f Chiang Mai University for the financial support.
This work was also supported in part:
by the National Science Foundation grants:
HRD-0734825 and HRD-1242122 (Cyber-ShARE