Clayton Copula Solution

Running head: CLAYTON COPULA SOLUTION 1
Clayton Copula Solution
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Institution

CLAYTON COPULA SOLUTION 2
Clayton Copula Solution
Question 1
The Clayton Copula
C ( u , v ) = ( u−θ + v−θ−1 )
−1
θ for θ>0
1) The density function is derived as (Oakes, 2005);
c ( u , v ) = ∂2 C ( u , v )
∂ u ∂ v
But, ∂2 C ( u , v )
∂u ∂ v = ∂
∂ v ( ∂ C ( u , v )
∂u )
Now, ∂C ( u , v )
∂ u = ∂
∂u {( u−θ+ v−θ −1 )
−1
θ }
Let y = ( u−θ+ v−θ −1 )
−1
θ = t
−1
θ ..................................... (i)
Also, let t = u−θ + v−θ−1 ........................................... (ii)
Using Product rule
∂ y
∂u = ∂ y
∂t . ∂ t
∂u
From equation (ii), ∂ t
∂u =−θu−θ−1
And from (i), ∂ y
∂ t =−1
θ t
−1
θ −1
=−1
θ ( u−θ+ v−θ −1 )
−1
θ −1
Then,
∂ y
∂u =−1
θ ( u−θ + v−θ−1 )
−1
θ −1
(−θu−θ−1 )
∂ y
∂u =u−θ−1 ( u−θ +v−θ−1 )
−1
θ −1
= ∂C ( u , v )
∂ u
The second derivative with respect to v
∂
∂ v (u−θ−1 ( u−θ +v−θ−1 )
−1
θ −1
)