DOUBLY STOCHASTIC MODELS WITH THRESHOLD GARCH INNOVATIONS

INNOVATIONS

MUHAMMAD SHERAZ

Communicated by the former editorial board

Recently, there has been a growing interest in the methods addressing volatility in computational finance and econometrics. Peiris et al. [8] have introduced doubly stochastic volatility models with GARCH innovations. Random coeffi- cient autoregressive sequences are special case of doubly stochastic time series.

In this paper, we consider some doubly stochastic stationary time series with GARCH and Threshold GARCH errors. Some general properties of process, like variance and kurtosis are derived.

AMS 2010 Subject Classification: 60Gxx, 60C10, 62M10, 91B84.

Key words:GARCH processes, TGARCH−(1,1), doubly stochastic time series, RCA(1), RCA-MA(1), Sign-RCA-MA(1).

1. INTRODUCTION

Recently, there has been a growing development in the use of nonlinear volatility models in financial literature. Black-Scholes option pricing model has some contradictory assumptions, such as constant volatility and normally dis- tributed log returns. In many financial time series, particular empirical studies reveal some facts such as stock returns and foreign exchange rates, exhibit leptokurtosis and stochastic volatility. Any distribution can be characterized by a number of features such as mean, variance, skewness and kurtosis. The measure of kurtosis is considered as whether the data are peaked or flat relative to the Normal distribution.

The concept of stochastic volatility for financial time series, the autore- gressive conditionally heteroskedastic (ARCH) model was first studied by Engle [4] and its generalization, the GARCH model, by Bollerslev [2]. The GARCH model assume symmetric effects on volatility that is, good and bad news have the same effect on volatility, which is shortfall of these models. The Thresh- old GARCH (TGARCH) model was introduced by Zakoian [12] and various other nonlinear GARCH extensions have been proposed to capture asymmetric effects [3, 5]. Random coefficient autoregressive (RCA) time series were intro- duced by Nicholls and Quinn [7] and some of their properties were studied by

MATH. REPORTS16(66),3(2014), 421–430

Appadooet al. [1]. In this paper, we derive the kurtosis for various classes of doubly stochastic models with GARCH and TGARCH innovations. In Section 2, some random coefficient autoregressive time series are discussed. In Section 3, some doubly stochastic models with GARCH and TGARCH innovations are introduced and we derive the formula for kurtosis in terms of model parameters.

2.RANDOM COEFFICIENT AUTOREGRESSIVE TIME SERIES

Random coefficient autoregressive sequences represent a special case of doubly stochastic time series. Some random coefficient autoregressive time series are given as follows:

(2.1) y_t= (φ+b_t)yt−1+ε_t , t∈Z (2.2) y_t= (φ+b_t)yt−1+ε_t+θεt−1, t∈Z (2.3) yt= (φ+bt+ Φst−1)yt−1+εt+θεt−1, t∈Z

From the equations (2.1)–(2.3) we have Random coefficient autoregres- sive (RCA-1), Random coefficient autoregressive-moving average (RCA-MA- 1), Sign-RCA-MA -1, respectively. The mean, variance and kurtosis for the above processes have been discussed in [8]. The two necessary and sufficient conditions for second order stationarity are given by:

1. ^b_ε^t

t

∼

0 0

,

σ_b² 0 0 σ_ε²

; 2. φ²+σ_b²<1.

We write forst as:

s_t=







+1 if yt>0 0 ify_t= 0

−1 ifyt<0

where {b_t} and {ε_t} are errors in the model respectively, φ,Φ and s_t are pa- rameters also E(s²_t) = 1. In order to calculate the kurtosis, E(s⁴_t) = 1 [11]. A process of the following form was considered in [10]:

(2.4) b_t+1 =ab_t+ (1 +b_t)υ_t+1.

Recently, Peiris et al. [8] have studied a doubly stochastic model of the following form:

(2.5) y_t= (φ+b_t)yt−1+ε_t

(2.6) b_t+1=ab_t+ (1 +b_t)υ_t+1

Let y_t be a stationary doubly stochastic time series satisfying (2.5) and (2.6). The moments for the model have been recently calculated in [8], whereεt

is an identically distributed independent sequence of variables with zero mean and varianceσ_ε²_t.

1. ^υ_ε^t

t

N

∼

0 0

,

σ_υ² 0 0 σ²_ε

2. 1−a²−σ_υ² <1

3. 1−a²−2σ²_υ−φ²+φ²a²+φ²σ_υ² <1

In 2.5 and 2.6 the observed random variable is modeled in two steps.

Firstly, the distribution of the observed outcome is represented in a standard way and in the second step,bt is treated as being itself a random variable.

Definition 2.1 ([12]).Let zt be an independently identically distributed (iid) sequence of random variables such thatE(z_t) = 0 andV ar(z_t) = 1 then εt is called a Threshold GARCH−(p, q) process if it satisfies:

σ_t = ω+

p

X

i=1

α_i,+ε⁺_t−i−αi,−ε⁻_t−i+

q

X

j=1

β_jσt−j

εt = σtzt

where ω,αi,+,αi,− and β are real numbers and through the coefficients αi,+, αi,− the current volatility depends on both the modulus and the sign of the past returns. The positive and negative components of ε_t are given by

ε⁺_t = max (εt,0), ε⁻_t = min (εt,0) andεt=ε⁺_t +ε⁻_t .

3. DOUBLY STOCHASTIC MODELS WITH THRESHOLD GARCH ERRORS

In this section, we present our results for doubly stochastic volatility models with Threshold GARCH errors. We consider equations (2.1) to (2.3) and provide the moments of doubly stochastic models with TGARCH−(1,1).

Theorem 3.1. Consider a doubly stochastic model of the following form with TGARCH−(1,1):

yt = (φ+bt)yt−1+εt

b_t+1 = ab_t+ (1 +b_t)υ_t+1

ε_t = σ_tz_t

σt = ω+α1,+ε⁺_t−1−α1,−ε⁻_t−1+β1σt−j

Then we have the following results:

1. E(y_t²) = ω²σ_z²(1 +a₁)(1−a²−σ_υ²)

(1−a₁)(1−a₂)(1−a²−2σ_υ²−φ²+φ²a²+φ²σ_υ²) 2. E(y_t⁴) = f1

f2

, where

f1 = 3σ_z⁴ω⁴(1 + 3a1+ 5a2+ 3a1a2+ 3a3+ 5a1a3+ 3a2a3+a1a2a₃₎ (1−a1)(1−a2)(1−a3)

+ 6ω⁴σ_z⁴(1 +a1)²[φ²(1−a²) +σ_υ²(1−φ²)]

(1−a1)²(1−a2)²(1−a²−2σ²_υ−φ²+φ²a²+φ²σ_υ²)

f2 = 1−φ⁴−3σ_υ⁴[ (−1−a²−5σ²_υ+a³+a⁵−16a³σ²_υ+ 3aσ_υ²−9aσ_υ⁴ (−1 +a³+ 3aσ_υ²)(−1 +a²+σ²_υ)(−1 + 6a²σ²_υ+a⁴+ 3σ_υ⁴)]

−6φ²[ σ²_υ

(1−a²−σ²_υ)]−4φ6a[ σ⁴_υ

(1−a³−3aσ_υ²)(1−a²−σ_υ²)] 3. The Kurtosis of the process follows the definition K^(y)= E(y⁴_t)

[E(y_t²)]². Proof. 1. First we calculate the value of E(y_t²)

E(y_t²) =E[(φ+bt)yt−1+εt]²=φ²E(y²_t−1) +E(b²_t)E(y_t−1² ) +E(ε²_t) E(y_t²)−φ²E(y²_t−1)−E(b²_t)E(y_t−1² ) =E(σ_t²)σ²_z

Using the second order stationarity condition and by putting values of E(σ²_t) andE(b²_t) we get:

E(y²_t) = E(σ²_t)σ_z² (1−φ²−E(b²_t))

E(y²_t) = ω²σ_z²(1 +a1)(1−a²−σ_υ²)

(1−a₁)(1−a₂)(1−a²−2σ_υ²−φ²+φ²a²+φ²σ²_υ) 2. Now we findE(y⁴_t)

E(y_t⁴) =E[(φ+b_t)yt−1+ε_t]⁴

E(y⁴_t) =φ⁴E(y_t−1⁴ )+E(b⁴_t)E(y_t−1⁴ )+E(ε⁴_t)+6φ²E(b²_t)E(y⁴_t−1)+4φE(b³_t)E(y_t−1⁴ ) + 6φ²E(ε²_t)E(y²_t−1) + 6E(ε²_t)E(b²_t)E(y_t−1² ) + 12φE(ε_t)E(b_t)E(y²_t−1)

E(y⁴_t) = 3σ⁴_zE(σ_t⁴) + 6σ²_zE(b²_t)E(σ²_t)E(y_t−1² ) + 6σ²_zφ²E(σ_t²)E(y_t−1² ) 1−φ⁴−E(b⁴_t)−6φ²E(b²_t)−4φE(b³_t)

E(y⁴_t) = 3σ⁴_zE(σ_t⁴) + 6σ²_zE(σ_t²)[E(b²_t) +φ²]E(y_t−1² ) 1−φ⁴−E(b⁴_t)−6φ²E(b²_t)−4φE(b³_t)

Using the second order stationarity condition and by putting values of E(σ²_t), E(σ²_t) and E(b²_t), E(b³_t) and E(b⁴_t), we get the required value of E(y_t⁴) as given in the statement of the theorem and

E(σ²_t) = ω²(1 +a1) (1−a₁)(1−a₂)

E(σ⁴_t) = ω⁴(1 + 3a1+ 5a2+ 3a1a2+ 3a3+ 5a1a3+ 3a2a3+a1a2a3) (1−a₁)(1−a₂)(1−a₃)

a1= 1

√

2π(α1,++α1,−)+β1, a2 = 1

2(α²_1,++α²_1,−)+ 2

√

2πβ1(α1,++α1,−) +β₁² a₃=

r2

π(α³_1,++α³_1,−) +3

2β₁(α²_1,++α²_1,−) + 3

√2πβ₁²(α_1,++α1,−) +β³₁ 3. We can getK^(y)= E(y_t⁴)

[E(y_t²)]². The Kurtosis of the process follows from its definition ofK^(y).

Theorem 3.2. Consider the following doubly stochastic time series with given conditions:

y_t = (φ+b_t)yt−1+ε_t+θεt−1

bt+1 = abt+ (1 +bt)υt+1

a) ^υ_ε^t

t

∼N

0 0

,

σ²_υ 0 0 σ_ε²

b) 1−a²−σ_υ² <1

c) 1−a²−2σ_υ²−φ²+φ²a²+φ²σ_υ² <1

where ε_t is an identically distributed independent sequence of variables with mean zero and varianceσ_ε². Then we have results forE(y_t²), E(y_t⁴)and Kurtosis as follows:

1. E(y_t²) = ω²σ²_ε(1 +θ²)(1−a²−σ_υ²) (1−a²−2σ_υ²−φ²+φ²a²+φ²σ²_υ)

2. E(y_t⁴) = 3σ⁴_ε(1 + 6θ²+θ⁴) + 6σ_ε²(1 +θ²)[(φ²+E(b²_t)]E(y²_t−1) 1−φ⁴−E(b⁴_t)−6φ²E(b²_t)−4φE(b³_t)

3. The Kurtosis of the process follows the definition K^(y)= E(y⁴_t) [E(y_t²)]². Proof. 1. First we calculate the value of E(y_t²):

E(y_t²) =E[(φ+b_t)yt−1+ε_t+θεt−1]²

E(y_t²) = φ²E(y²_t−1) +E(b²_t)E(y_t−1² ) +E(ε²_t) +θ²E(ε²_t−1) +2E(φ+b_t)E(yt−1)E(ε_t) + 2θE(ε_t)E(εt−1) +2θE(φ+b_t)E(yt−1)E(εt−1)

E(y_t²)−φ²E(y²_t−1)−E(b²_t)E(y²_t−1) =E(ε²_t) +θ²E(ε²_t−1)

Using the second order stationarity condition and by replacing the values of E(σ²_t) and E(b²_t) we get:

E(y_t²) = σ_ε²(1 +θ²)

(1−φ²−E(b²_t)) = σ_ε²(1 +θ²) (1−φ²−_(1−a^σ2^2υ−σ_υ²)) E(y_t²) = σ_ε²(1 +θ²)(1−a²−σ_υ²)

(1−a²−2σ²_υ−φ²+φ²a²+φ²σ_υ²) 2. Now we calculate E(y_t⁴)

E(y_t⁴) =E[(φ+bt)yt−1+εt+θεt−1]⁴

Because the odd moments ofytare zero, we can write the above expression by expanding the above equation and applying E we get:

E(y⁴_t) =Eh

(φ+b_t)⁴y_t−1⁴ + 6(φ+b_t)²y²_t−1(ε_t+θεt−1)²+ (ε_t+θεt−1)⁴i Using the stationarity condition and E(εt) =E(εt−1) = 0

E(y_t⁴) = E(φ+bt)⁴E(y_t−1⁴ ) +E(ε⁴_t) +θ⁴E(ε⁴_t) + 6θ²E(ε⁴_t)

+6E(φ+b_t)²E(ε²_t)E(y_t−1² ) + 6θ²E(φ+b_t)²E(ε²_t)E(y²_t−1) E(y_t⁴) = 3σ_ε⁴(1 + 6θ²+θ⁴) + 6σ²_ε(1 +θ²)[(φ²+E(b²_t)]E(y_t−1² )

1−φ⁴−E(b⁴_t)−6φ²E(b²_t)−4φE(b³_t) 3. The Kurtosis of the process follows from its definition

K^(y) = E(y_t⁴) [E(y²_t)]²,

K^(y)=

3σ⁴_ε(1 + 6θ²+θ⁴) + 6σ_ε²(1 +θ²)[(φ²+E(b²_t)]E(y²_t−1) 1−φ⁴−E(b⁴_t)−6φ²E(b²_t)−4φE(b³_t)

σ_ε²(1 +θ²)(1−a²−σ²_υ) (1−a²−2σ_υ²−φ²+φ²a²+φ²σ_υ²)

2 .

Theorem 3.3. Consider a doubly stochastic model of the following form with TGARCH−(1,1)

yt = (φ+bt)yt−1+εt+θεt−1

b_t+1 = ab_t+ (1 +b_t)υ_t+1 ε_t = σ_tz_t

σ_t = ω+α_1,+ε⁺_t−1−α1,−ε⁻_t−1+β₁σt−j

Then we have the following results:

1. E(y_t²) = ω²σ_z²(1 +θ²)(1 +a1)(1−a²−σ_υ²)

(1−a1)(1−a2)(1−a²−2σ_υ²−φ²+φ²a²+φ²σ_υ²) 2. E(y_t⁴) = f1

f2

,where

f₁=σ_z⁴ω⁴(1+ 3a1+ 5a2+ 3a1a2+ 3a3+ 5a1a3+ 3a2a3+a1a2a₃₎

(1−a₁)(1−a₂)(1−a₃) (1+6θ²+θ⁴) + 6ω⁴σ_z⁴(1 +θ²)²(1 +a₁)²[φ²(1−a²) +σ_υ²(1−φ²)]

(1−a₁)²(1−a₂)²(1−a²−2σ²_υ−φ²+φ²a²+φ²σ_υ²) f₂= 1−φ⁴−3σ⁴_υ[ (−1−a²−5σ²_υ+a³+a⁵−16a³σ²_υ+ 3aσ_υ²−9aσ_υ⁴

(−1 +a³+ 3aσ_υ²)(−1 +a²+σ_υ²)(−1 + 6a²σ_υ²+a⁴+ 3σ_υ⁴)]

−6φ²[ σ²_υ

(1−a²−σ²_υ)]− −4φ6a

σ_υ⁴

(1−a³−3aσ²_υ)(1−a²−σ²_υ)

3. The Kurtosis of the process follows from its definition K^(y)= E(y_t⁴) [E(y²_t)]² Proof. 1. First we calculate the value of E(y_t²)

E(y_t²) =E[(φ+bt)yt−1+εt+θεt−1]²

E(y²_t)=φ²E(y_t−1² )+E(b²_t)E(y²_t−1)+E(ε²_t)+θ²E(ε²_t−1)+2E(φ+bt)E(yt−1)E(εt) +2θE(εt)E(εt−1) + 2θE(φ+bt)E(yt−1)E(εt−1)

Using the second order stationarity condition and by replacing the values of E(σ²_t) and E(b²_t) we get:

E(y²_t) = σ_z²(1 +θ²)E(σ²_t) (1−φ²−E(b²_t))

E(y²_t) = ω²σ_z²(1 +θ²)(1 +a₁)(1−a²−σ_υ²)

(1−a1)(1−a2)(1−a²−2σ_υ²−φ²+φ²a²+φ²σ²_υ)

2. Now we calculate E(y_t⁴)

E(y_t⁴) =E[(φ+b_t)yt−1+ε_t+θεt−1]⁴ Taking expectation E we get:

E(y_t⁴) = E(φ+b_t)⁴E(y_t−1⁴ ) +E(ε⁴_t) +θ⁴E(ε⁴_t) + 6θ²E(ε⁴_t)

+6E(φ+b_t)²E(ε²_t)E(y_t−1² ) + 6θ²E(φ+b_t)²E(ε²_t)E(y²_t−1) Using the second order stationarity condition and by replacing the values of E(ε⁴_t) and E(ε²_t)

E(y_t⁴) = 3σ_z⁴E(σ⁴_t)(1 + 6θ²+θ⁴) + 6σ²_zE(σ_t²)(1 +θ²)[(φ²+E(b²_t)]E(y²_t) 1−φ⁴−E(b⁴_t)−6φ²E(b²_t)−4φE(b³_t)

Now by replacing the values of E(σ_t⁴), E(σ_t²) andE(b²_t), E(b³_t), E(b⁴_t) we obtain the required value ofE(y⁴_t) as:

E(y⁴_t) = f₁

f₂, wheref₁ and f₂are given in the statement of the theorem.

Theorem 3.4. Consider a doubly stochastic model of the following form with TGARCH(1,1) errors as:

yt = (φ+bt+ Φst)yt−1+εt+θεt−1

b_t+1 = ab_t+ (1 +b_t)υ_t+1 ε_t = σ_tz_t

σ_t = ω+α_1,+ε⁺_t−1−α1,−ε⁻_t−1+β₁σt−j

Then we have the following results:

1. E(y_t²) = ω²σ_z²(1 +θ²)(1 +a1)(1−a²−σ_υ²)

(1−a₁)(1−a₂)(1−a²−2σ_υ²−Φ²+Φ²a²+Φ²σ²_υ−φ²+φ²a²+φ²σ²_υ) 2. E(y_t⁴) =f1

f2

,where

f1 = 3σ⁴_zω⁴(1 +θ⁴+ 6θ²)(1+3a₁+5a₂+3a₁a₂+3a₃+5a₁a₃+ 3a₂a₃+a₁a₂a₃₎ (1−a₁)(1−a₂)(1−a₃)

+ 6σ_z⁴ω⁴(1 +θ²)²(1 +a₁)²

(φ²+ Φ²)[(1−a²−σ_υ²) +σ_υ² (1−a₁)²(1−a₂)²(1−a²−2σ²_υ−Φ²+Φ²a²+Φ²σ_υ²−φ²+φ²a²+φ²σ²_υ) f₂ = 1−φ⁴−3σ⁴_υ[ (−1−a²−5σ_υ²+a³+a⁵−16a³σ_υ²+ 3aσ_υ²−9aσ⁴_υ

(−1 +a³+ 3aσ²_υ)(−1 +a²+σ_υ²)(−1 + 6a²σ_υ²+a⁴+ 3σ⁴_υ)]

−6(φ²+ Φ²)[ σ²_υ

(1−a²−2σ²_υ)]−Φ⁴−6φ²Φ²

3. The Kurtosis of the process follows from its definition K^(y)= E(y_t⁴) [E(y²_t)]² Proof. 1. First we calculate the value of E(y_t²)

E(y_t²) =E[(φ+bt+ Φst)yt−1+εt+θεt−1]²

E(y²_t) = φ²E(y²_t−1) + Φ²E(y_t−1² ) +E(b²_t)E(y_t−1² ) +E(ε²_t) +θ²E(ε²_t−1) +2E(φ+b_t+ Φs_t)E(yt−1)E(ε_t) + 2θE(ε_t)E(εt−1)

+ 2θE(φ+b_t+ Φs_t)E(yt−1)E(εt−1)

Using the second order stationarity condition and by replacing the values of E(σ²_t) and E(b²_t) we get:

E(y²_t) = σ_z²(1 +θ²)E(σ²_t) (1−φ²−Φ²−E(b²_t))

E(y²_t) = ω²σ_z²(1 +θ²)(1 +a₁)(1−a²−σ_υ²)

(1−a₁)(1−a₂)(1−a²−2σ²_υ−Φ²+Φ²a²+Φ²σ_υ²−φ²+φ²a²+φ²σ²_υ) 2. Now we calculate E(y_t⁴)

E(y_t⁴) =E[(φ+b_t+ Φs_t)yt−1+ε_t+θεt−1]⁴

By applying expectation E, the simplification of the above step is the same as the one in the previous theorem and we can write:

E(y⁴_t) = 3ε⁴_t(1 + 6θ²+θ⁴) + 6ε²_t(1 +θ²)[(φ²+ Φ²+E(b²_t)]E(y_t−1² ) 1−φ⁴−3E(b⁴_t)−6(φ²+ Φ²)E(b²_t)−Φ⁴−6φ²Φ²

By replacing the values of ε⁴_t, ε²_t, E(b²_t), E(b⁴_t) and then using the second order stationarity condition and the values ofE(σ⁴_t), E(σ²_t) andE(y²_t) we obtain the final expression of E(y_t⁴) in terms of f₁/f₂ ,where f₁ and f₂ are given in the statement of the theorem.

3. The Kurtosis of the process follows from its definitionK^(y)= E(y⁴_t)

[E(y_t²)]².

4.CONCLUSIONS

In this paper, Kurtosis of doubly stochastic models with TGARCH in- novations are derived. Statistical inferences to these doubly stochastic models with other asymmetric GARCH errors and state space modeling can be viewed as a special case for nonlinear time series.

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Received 5 July 2012 University of Bucharest,

Faculty of Mathematics and Computer Science, Str. Academiei 14,

0100144 Bucharest, Romania