On Threshold ARCH Models with Gram-Charlier 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

is the delay parameter, r is the threshold value.

t 0 1 t-1 p t p t t-d

t 0 1 t 1 p t p t t d

2t

... ,

... ,

(0, )

y y y when y r

y y y when r

N

-ｙ

d

Threshold ARCH Model Li and Lam (1995) Threshold ARCH model

t 0 1 t-1 p t p t t-d

t 0 1 t 1 p t p t t d

2 2t 0 1 t 1 m t m

t

... ,

... ,

...

(0, )t

y y y when y r

y y y when r

h

N h

-ｙ

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

3 4( ) ( )(1 ( ) ( ))f x x s H x k H x

33

4 24

( ) ( 3 ) / 6,

( ) ( 6 3) / 24.

H x x x

H x x x

( )x is the standard normal density function, and

Properties of GC density

The mean and variance are

The skewness and excess the kurtosis are and .

We use GC(u,h,s,k) to denote a GC density with mean u, variance h, skewness s, and excess kurtosis k.

3 4

23 4

( )(1 ( ) ( )) 0

( )(1 ( ) ( )) 1

x s H x k H x xdx

x s H x k H x x dx

s k3

3 4

43 4

( )(1 ( ) ( ))

( )(1 ( ) ( )) 3

x s H x k H x x dx s

x s H x k H x x dx k

Why Gram-Charlier density?

It nests Gaussian density. It has explicit skewness and kurtosis.

Threshold ARCH Model with Gram Charlier Density

is the indicator of regimes.Skewness and excess kurtosis will vary with time. The

structure if different in different regimes.

( ) ( ) ( )0

1

( ) ( ) ( ) ( )

( ) ( ) 20

1

( ) ( )0 1 1

( ) ( )0 1 1

,

(0, , , )

pi i i i

t k t k tk

i i i it t

mi i

t k t kk

i it t

i it t

y y

GC h s k

h

s s

k k

i

Double Threshold ARCH model with GC density (DTARCHSK)

0 1 1 0 1 1... ( )( ... )t t p t p r t d t p t p tX X X I X X X

2 20 1 1

0 1

0 1

...

( ) ,

( ) .

t t m t m

t r t d

t r t d

h

s I X

k I X

(0, , , )t t t tGC h s k

( ) ( )r t d t dI X I X r is the indicator function

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

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.

Group two0 1

0 1

( ) ,

( ) .t r t d

t r t d

s I X

k I X

0 1 1 0 1 1

2 20 1 1

... ( )( ... )

...

t t p t p r t d t p t p t

t t m t m

X X X I X X X

h

Group one

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

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.

1

1

1

21

0.2

(0, ,0.4,1) 0,

(0, ,0.4,1) 0.

0.1 0.4

t t t

t t t

t t t

t t

y y

GC h if y

GC h if y

h

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:

The conditional likelihood function is:

0 0 1 1 0 1: 0, 0pH

2

4

1 1ln 1 ( )

2 2t t

t tt t

l h k Hh h

0 1( ) ,t r t ds I X

Lagrange Multiplier test The fish information matrix is

Score function is

2

'( )

lE

( ', ', ', ', ') '

0ˆ ˆ ˆ, , , 0, 0

lD

Because the conditional density is symmetric, we can show that (Engle’s (1982) theorem),

The information matrix is block diagonal.

Therefore, we can drop the second block of the matrix which is not related to the threshold structure in the test.

2

2

2

0( , ) ( , ) '

( , , , ) ( , , , ) '0

( , ) ( , ) '

lE

lE

lE

2 2{ / } 0, { / } 0E l E l 2 2{ / } 0, { / } 0E l E l

As the time series is stationary and ergodic, by the martingale central limit theorem, we can show

' 1( )r r r rT N C L C L

0 1 0 1

0 1 0 1

21/ 2 1

ˆ ˆˆ ˆ ˆ ˆ, , , 0, 0 , , , 0, 0

2 21 1

ˆ ˆˆ ˆ ˆ ˆ, , , 0, 0 , , , 0, 0

,'

,' '

r

r r

l lT n C n E

l lC n E L n E

Supreme Lagrange Multiplier Test

If r is given, define the Lagrange-Multiplier test statistic as

If r is unknown, we define the supreme Lagrange-Multiplier test statistic as

The distribution of S was proved to be related to an Ornstein-Uhlenbeck (O-U) process (Chan(1990)).

' ' 1 1

ˆsup{ ( ) }r r r r rr R

S T C L C L T

' ' 1 1( )r r r r rLM T C L C L T

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

21

21

1: 0.2 , 1, (0, ,0,3)

2 : 0.2 , 1, (0, ,0.2,3)

3 : 0.2 , 0.1 0.4 , (0, ,0,3)

4 : 0.2 , 0.1 0.4 , (0, ,0.2,3)

t t t t t

t t t t t

t t t t t t

t t t t t t

Model X h GC h

Model X h GC h

Model X h GC h

Model X h GC h

21

21

21 1

21 1

5 : 0.2 , 0.1 0.8 , (0, ,0,3)

6 : 0.2 , 0.1 0.8 , (0, ,0.3,3)

7 : 0.5 , 0.1 0.5 , (0, ,0,3)

8 : 0.5 , 0.1 0.5 , (0, ,0.

t t t t t t

t t t t t t

t t t t t t t

t t t t t t t

Model X h GC h

Model X h GC h

Model X X h GC h

Model X X h GC h

3,3)

Experimental results10% 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.