Advances in Risk Management

316

Table 16.6

Continued

Panel C: post-September 11 period

S&P500 conditional variance equationh11,

=t

1,82

×

10

−^6

+

0,9110

h11,

t−

+ 1

0,0569

h12,

t−

+ 1

0,0002

h22,

t−

+ 1

0,0188

(^2) ε1,
t−
1
−
0,0494
ε1,t
−^1
ε2,t
−^1

• 0,0193
  (^2) ε2,
  t−


• 1
  0,0875
  (^2) η1,
  t−


• 1
  0,0172
  η1,
  t−
  η 1
  2,t
  +− 1
  0,0008
  (^2) η2,
  t−
  1
  6,05
  ×
  10
  −^7
  0,0018
  0,0043
  0,00003
  0,0036
  0,0049
  0,0055
  0,0042
  0,0544
  0,0013
  (3,0171)
  (517,13)
  (13,0777)
  (0,6622)
  (5,2752)
  (−
  10,069)
  (3,4925)
  (20,966)
  (0,3165)
  (0,6577)
  IBEX35 conditional variance equationh22,
  =t
  1,24
  ×
  10
  −^6


• 
  

  0,0009
  h11,t
  −^1

  
  


• 
  

  0,0571
  h12,
  t−

  
  


• 1
  0,9186
  h22,
  t−


• 1
  0,0325
  (^2) ε1,
  t−
  1
  −
  0,01596
  ε1,t
  −^1
  ε2,t
  −^1


• 
  

  0,0019
  (^2) ε2,
  t−

  
  


• 1
  0,0003
  (^2) η1,
  t−
  − 1
  0,0066
  η1,
  t−
  η 1
  2,t
  −^1


• 
  

  0,0319
  (^2) η2,
  t−
  1
  4,82
  ×
  10
  −^7
  0,0001
  0,0043
  0,0013
  0,0037
  0,0099
  0,0025
  0,0005
  0,0049
  0,0032
  (2,5799)
  (6,6162)
  (13,1568)
  (709,68)
  (8,6504)
  (1,6033)
  (0,7843)
  (0,6382)
  (−
  1,3466)
  (9,8872)
  Notes
  :h
  11
  and
  h^22
  denote the conditional variance for the S&P500
  and IBEX35 return series, respectively. Below
  the estimated coefficients are the standard errors,
  with the corresponding
  t-values given in
  parentheses.
  The expected value is obtained taking expectations
  to the non-linear functions, therefore involving
  the estimated variance-covariance matrix
  of the parameters. In order to calculate the standard
  errors,
  the function must be linearized using first-order
  Taylor series expansion. This is sometimes
  called the “delta method”. When a variable
  Y
  is a function of a variable
  X, i.e.,
  Y=
  F(
  X), the delta method allows
  us to obtain approximate formulation of the
  variance of
  Y
  if: (1)
  Y
  is differentiable with respect to
  X
  and (2) the variance of
  X
  is known. Therefore:
  V(Y)
  ≈
  (
  Y)
  2 ≈
  (
  ∂Y ∂X
  )^2
  (
  X)
  2 ≈
  (
  ∂Y ∂X
  )^2
  V(
  X)
  When a variable
  Y
  is a function of variables
  X
  and
  Z
  in the form of
  Y=
  F(
  X,
  Z), we can obtain approximate formulation of the
  variance of
  Y
  if: (1)
  Y
  is differentiable with respect to
  X
  and
  Z
  and (2) the
  variance of
  X
  and
  Z
  and the covariance between
  X
  and
  Z
  are known. This is:
  V(
  Y)
  ≈
  (∂
  )Y ∂X
  2 V
  (X
  )+
  (
  ∂Y ∂Z
  )^2
  V(
  Z)

  
  


• 
  

  ( 2
  ∂Y ∂X
  )(
  ∂Y ∂Z
  )
  Cov
  (X
  ,Z
  )
  Once the variances are calculated it is straightfor
  ward to calculate the standard errors.