On Threshold ARCH
Models with GramCharlier Density
Xuan Zhou and W. K. Li
Department of Statistics and Actuarial Science,HKU
Outline






Introduction to Threshold Model
Introduction to Gram Charlier Density
Threshold Model with Gram Charlier density
Estimation method
Testing method
Empirical Results
Motivation
 Some financial time series are asymmetric.

Investors are more nervous when the market is falling
than when it is rising. Negative shocks entering the
market lead to a larger return volatility than positive
shocks of a similar magnitude.

Many models have been proposed to investigate this
asymmetric feature, such as the ARCH-M model by
Engle (1987) and the EGARCH model by Nelson (1990).
 Innovation are believed to be non-Gaussian
Threshold AR Model
 Tong’s (1978) threshold autoregressive model
 y t   0   1 y t-1  ...   p y t  p   t , w hen y t-d  r

 y t   0  1 y t 1  ...   p y t  p   t , w hen ｙt - d  r
t

N (0,  )
2
d is the delay parameter, r is the threshold value.
Threshold ARCH Model

Li and Lam (1995) Threshold ARCH model
 y t   0   1 y t-1  ...   p y t  p   t , w hen y t-d  r

 y t   0  1 y t 1  ...   p y t  p   t , w hen ｙt - d  r
h t   0   1 t 1  ...   m  t  m
2
t
N (0, ht )
2
Conditional density
 It is generally accepted that the conditional
distribution of the asset return is not Gaussian
(Mills(1995)).
 The leptokurtosis has been found in most financial
time series.
Introduction to Gram-Charlier (GC)
density
 The Gram-Charlier density is
f ( x )   ( x )(1  s H 3 ( x )  k H 4 ( x ))
 ( x ) is the standard normal density function, and
H 3 ( x )  ( x  3 x ) / 6,
3
H 4 ( x )  ( x  6 x  3) / 24.
4
2
Properties of GC density
 The mean and variance are

  ( x )(1  s
H 3 ( x )  k H 4 ( x )) xdx  0



 ( x )(1  s H 3 ( x )  k H 4 ( x )) x dx  1
2
 The skewness and excess the kurtosis are s and k .


  ( x )(1  s
H 3 ( x )  k H 4 ( x )) x dx  s
3


  ( x )(1  s
H 3 ( x )  k H 4 ( x )) x dx  3  k
4
 We use GC(u,h,s,k) to denote a GC density with mean u,
variance h, skewness s, and excess kurtosis k.

Why Gram-Charlier density?
 It nests Gaussian density.
 It has explicit skewness and kurtosis.
Threshold ARCH Model with Gram
Charlier Density
p
yt   0 
i
  k yt  k   t ,
(i)
(i)
(i)
k 1
t
G C (0, ht , s
ht  
(i )
0
(i )
(i )
(i)
,k
(i)
)
m

  k  tk
(i )
2
k 1
s t   0   1 s t 1
(i )
(i)
k t   0   1 k t 1
i is the indicator of regimes.
Skewness and excess kurtosis will vary with time. The
structure if different in different regimes.
(i)
(i)
Double Threshold ARCH model with
GC density (DTARCHSK)
X t   0   1 X t 1  ...   p X t  p  I r ( X t  d )( 0  1 X t 1  ...   p X t  p )   t
t
G C (0, ht , s t , k t )
ht   0   1 t 1  ...   m  t  m
2
2
st   0  I r ( X t  d )  1 ,
k t   0  I r ( X t  d ) 1 .
I r ( X t  d )  I ( X t  d  r ) is the indicator function
Estimation
 Several problems in the estimation.
 The number of parameters is large.
 As pointed out by Bond (2001), MLE estimation is
quite sensitive to initial parameters, therefore it's
necessary to search over a wide set of initial
parameters before selecting the model with the
highest likelihood value.
Estimation method: ECM
 Step 1: For a given value of the skewness and kurtosis,
fit the model by MLE.
 Step 2: Conditional on the estimates, calculate the
residuals. Find the maximum likehood estimates of the
skewness and the kurtosis of the residuals.
 Step 3: Repeat until all the parameters converge.
X t   0   1 X t 1  ...   p X t  p  I r ( X t  d )( 0  1 X t  1  ...   p X t  p )   t
ht   0   1 t 1  ...   m  t  m
2
2
st   0  I r ( X t  d )  1 ,
k t   0  I r ( X t  d ) 1 .
Group one
Group two

The convergence is fast. Almost every simulation converges
in three iterations.

The first step of the ECM method is a quasi-maximum
likelihood estimation. It converges when the third and
fourth moments are assumed finite. Therefore, the
assumed value for the skewness and kurtosis would not
affect Step 1 much. The parameters for mean and variance
structure converge fast. All estimates converged within
three iterations.
Lagrange Multiplier Test

The threshold structure and GC density both can help
explain the asymmetric features, combination of them will
definitely enhance the model's ability to capture asymmetry.

On the other hand, they will also interact with each other
and prevent us to distinguish them.
Example
 Consider the model


 t

 t


y t  0.2 y t 1   t
G C (0, ht , 0.4,1) if y t 1  0,
G C (0, ht , 0.4,1) if y t 1  0.
ht  0.1  0.4  t 1
2
 When the previous data is positive, the conditional
density is skewed to the positive side. When the
previous data is negative, the conditional density is still
skewed to the positive side. Therefore, the behavior of
the series is asymmetric even the mean structure is
symmetric.
 Wong and Li’s (1997) test does not take into account
the conditional density. Their nominal 5% test on the
model reject 8% of the experiments.
Lagrange Multiplier test
 The null hypothesis is:
H 0 :  0  1 
  p   1  0,  0   1  0
st   0  I r ( X t  d )  1 ,
 The conditional likelihood function is:
l 

2


t
1 t
 1
 ln 1  k t H 4 (
  ht 
2
2 ht
ht





) 
 

Lagrange Multiplier test
 The fish information matrix is
 l
2
  E(
 
 Score function is
D 
'
)
  ( ',  ',  ',  ',  ') '
l

ˆ ,ˆ , ˆ0 ,  0 ,   0
Because the conditional density is symmetric, we can show
that (Engle’s (1982) theorem),
E { l /     }  0, E { l /     }  0
2
2
E { l /     }  0, E { l /     }  0
The information matrix is block diagonal.
2
2
2


 l
0
E

2

   ( ,  )  ( ,  ) '
 l

E

2

 l
  ( ,  ,  ,  )  ( ,  ,  ,  ) ' 
0
E


 (  ,  )  ( ,  ) ' 

Therefore, we can drop the second block of the matrix which is
not related to the threshold structure in the test.
As the time series is stationary and ergodic, by the martingale
central limit theorem, we can show
Tr  N ( C r  L r C
'
Tr  n
1 / 2
l

ˆ ,ˆ , ˆ0 ,  0 ,  1  0
2


l
1
Cr  n E 
 
1
Lr )
2


l
1
,C  n E 
 


'  ˆ ,ˆ ,ˆ ,  0 ,  0
0
1
2



l
1
, Lr   n E 

'  ˆ ,ˆ ,ˆ ,  0 ,  0
 
0
1


'  ˆ ,ˆ ,ˆ ,  0 ,  0
0
1
Supreme Lagrange Multiplier Test
 If r is given, define the Lagrange-Multiplier test statistic
as
1
1
L M  Tr ( C r  L r C L r ) Tr
'
'
 If r is unknown, we define the supreme LagrangeMultiplier test statistic as
S  su p{T r ( C r  L r C
'
r  Rˆ
'
1
1
L r ) Tr }
 The distribution of S was proved to be related to an
Ornstein-Uhlenbeck (O-U) process (Chan(1990)).
Test Comparison
Effect of the skewness in the testing
 When the skewness is included in the GC density, the
information matrix is no longer a block diagonal
matrix. Therefore we can not just drop the second
block of the matrix. As a result, the form of the
Lagrange Multiplier test will be more complicated to
handle.
 However, the critical value of the test for the models
with skewed density are almost the same as the test
of the corresponding models with non-skewed density.
Simulation Models
M odel 1 : X t  0.2   t , ht  1,  t  G C (0, ht , 0, 3)
M odel 2 : X t  0.2   t , ht  1,  t  G C (0, ht , 0.2, 3)
M odel 3 : X t  0.2   t , ht  0.1  0.4  t 1 ,  t  G C (0, ht , 0, 3)
2
M odel 4 : X t  0.2   t , ht  0.1  0.4  t 1 ,  t  G C (0, ht , 0.2, 3)
2
M odel 5 : X t  0.2   t , ht  0.1  0.8  t 1 ,  t  G C (0, ht , 0, 3)
2
M odel 6 : X t  0.2   t , ht  0.1  0.8  t 1 ,  t  G C (0, ht , 0.3, 3)
2
M odel 7 : X t  0.5 X t 1   t , ht  0.1  0.5  t 1 ,  t  G C (0, ht , 0, 3)
2
M odel 8 : X t  0.5 X t 1   t , ht  0.1  0.5  t 1 ,  t  G C (0, ht , 0. 3, 3)
2
Experimental results
10%
5%
2.5%
1.0%
Model 1
3.14
4.42
5.62
7.77
Model 2
3.12
4.38
5.74
7.78
Model 3
3.25
4.51
5.73
7.83
Model 4
3.21
4.47
5.75
7.82
Model 5
3.38
4.55
5.76
7.85
Model 6
3.33
4.54
5.76
7.86
Model 7
3.40
4.46
5.70
7.90
Model 8
3.43
4.51
5.72
7.94
Sample size = 400, replication = 1000. Estimate of r is
obtained by searching between 10% and 90% quantiles of
the data.
Empirical results

We apply our model to several foreign exchange rates
series, including British Pound (GBP/USD), Japanese
Yen/USD (JPY/USD), German Mark (GEM/USD) from Jan 2,
1990 to Dec 29, 2000.

Fit the data with four models: ARCH, ARCHSK, DTARCH and
DTARCHSK model.
Test results

The test of DTARCHSK and the test of DTARCH generate
very different conclusions on the existence of threshold
structure. The test of DTARCH is more likely to reject the
null hypothesis while the proposed one prefers the null
hypothesis. This is because the ARCHSK model has
captured most of the asymmetric features and need not to
further assign threshold structure.
Q&A
Some Conditional density Model

Engle and Gonzalez- Rivera (1991) Semiparametric ARCH
models.

Harvey and Siddique (1999) Autoregressive conditional
skewness

Brooks et al.(2005) Autoregressive conditional kurtosis

The skewness and the kurtosis of Student’s t-density have
to be calculated from the distributional parameters. How
skewed and how leptokurtic the density is can not be
conveyed directly.