Evaluating Co-movements with the Total Value of Import and Export with Exchange Rate in Yunnan Province - Innovative Management in Information and Production - page 41

3.2 The Time-Varying Copula for Dependence Parameters

We commonly use Pearson's correlation coefficient ρ t to describe the dependence

structure in the dynamic Gaussian copula and dynamic Student- t copula. On the
other hand, we often use the Kendall's

τ t on the Gumbel and Clayton copulas. We
assumed that the dependence parameters rely on past dependence and historical
information ( m t − 1 − 0 . 5 )( w t − 1 − 0 . 5 ) . The dependence process of the Gaussian and

Student- t are, therefore,

ρ t = Λ ( α c + β c ρ t − 1 + γ c ( m t − 1 − 0 . 5 )( w t − 1 − 0 . 5 )) . (11)

The dependence process of the Gumbel is

τ t = Λ ( α c + β c τ t − 1 + γ c ( m t − 1 − 0 . 5 )( w t − 1 − 0 . 5 )) . (12)

In the conditional dependence, we assumed that ρ t and τ t determined from its past

level, ρ ( t − 1 ) and τ ( t − 1 ) . The parameter of β c captures the persistent effect and γ c

can captures historical information. , denotes the logistic

transformation, which is used to ensure the dependence parameters fall within the
interval

Λ = − ln [( 1 − x t ) / ( 1 + x t )]

.
While we change the historical information

( − 1 , 1 )

( m 0 , t − 1 − 0 . 5 )( w e , t − 1 − 0 . 5 ) to

1
10

10
i

, captures the variability of the dependence.
We proposed time-varying dependence processes for Clayton copula as

| m t − 1 − w t − 1 |

∑

= 1

10
i = 1 | w t − 1 − m t − 1 |

1
10

τ t = Π α c + β c τ t − 1 + γ c (13)

e − x ) − 1 denotes the logistic transformation.

Π =( 1 +

3.3 The Estimation and Calibration of the Copula

IFM method is used to estimate copula-based GARCH mode and get the parameters.
The method function is

T
t = 1 ln f it ( k i , t , θ it )

ˆ

θ it = arg max (14)

T
t = 1 ln C it ( F it ( k 1 , t ) ,..., F nt ( k n , t ) , θ ct ,

ˆ ˆ

θ ct = arg max θ it ) . (15)