A generalized nonlinear model for long memory conditional heteroscedasticity

Grublytė, Ieva; Škarnulis, Andrius

doi:10.1080/02331888.2016.1259813

Cited by 2 publications

(15 citation statements)

References 21 publications

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“…For the GQARCH model in , similar results were established by Doukhan et al () and Grublytė and Škarnulis (). Namely, assume that parameters γ , ω , a , b j and j ⩾1 in satisfy

b_{j} \sim c j^{d - 1} (\exists 0 < d < 1 / 2, c > 0),

γ ∈[0,1), a ≠ 0 and

6 B_{2} + 4 | μ_{3} | {falsefalse}_{\sum}^{j = 1} | b_{j} |^{3} + μ_{4} {falsefalse}_{\sum}^{j = 1} b_{j}^{4} < {(1 - γ)}^{2},

where

μ_{p} : = normalE ζ_{0}^{p}, p = 1, 2, \dots, B_{2} : = \sum_{j = 1}^{\infty} b_{j}^{2} .

Then (Grublytė and Škarnulis, , Theorems 2.5 and 3.1) there exists a stationary solution of with

normalE r_{t}^{4} < \infty

such that

cov (r_{0}^{2}, r_{t}^{2}) \sim κ_{1}^{2} t^{2 d - 1}, t \to \infty

and

n

…”

Section: Introductionsupporting

confidence: 79%

“…Proposition (Grublytė and Škarnulis ()) Let γ ∈[0,1) and { ζ t } be an i.i.d. sequence with zero mean, unit variance and finite moment

μ_{p} : = normalE ζ_{0}^{p} < \infty

, where p ⩾2 is an even integer.…”

Section: Stationary Solutionmentioning

confidence: 99%

“…Remark Sufficient conditions for the existence of stationary solution of with finite moment

normalE | r_{t} |^{p} < \infty

and arbitrary p > 0 are obtained by Grublytė and Škarnulis, (, Theorem 2.4). They extend the corresponding result of Doukhan et al, (, Theorem 1) from γ = 0 to γ > 0.…”

Section: Stationary Solutionmentioning

confidence: 99%

“…Recently, Doukhan et al () and Grublytė and Škarnulis () discussed a class of quadratic ARCH models of the form

r_{t} = ζ_{t} σ_{t}, σ_{t}^{2} = ω^{2} + 1.19em (a + {falsefalse}_{\sum}^{j = 1} b_{j} r_{t - j} {1.19em)}^{2} + γ σ_{t - 1}^{2},

where

{ζ_{t}, t \in double-struckZ}

is a standardized i.i.d. sequence,

normalE ζ_{t} = 0, normalE ζ_{t}^{2} = 1

, and γ ∈[0,1), ω , a , b j , j ⩾1 are real parameters satisfying certain conditions; see Proposition .…”

Section: Introductionmentioning

confidence: 99%

“…sequence,

normalE ζ_{t} = 0, normalE ζ_{t}^{2} = 1

, and γ ∈[0,1), ω , a , b j , j ⩾1 are real parameters satisfying certain conditions; see Proposition . In Grublytė and Škarnulis (), was called the generalized quadratic autoregressive conditional heteroskedasticity (GQARCH) model. By iterating the second equation in , the squared volatility in can be written as a quadratic form

σ_{t}^{2} = \sum_{ℓ = 0}^{\infty} γ^{ℓ} ({) ω^{2} + (() a + \sum_{j = 1}^{\infty} b_{j} r_{t - ℓ - j} {())}^{2} (})

in lagged variables r t − 1 , r t − 2 ,…, and hence, it represents a particular case of quadratic ARCH model by Sentana () with

p = \infty

.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

QMLE for Quadratic ARCH Model with Long Memory

Grublytė

Сургайлис

Škarnulis

2016

Journal Time Series Analysis

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We discuss parametric quasi‐maximum likelihood estimation for quadratic autoregressive conditional heteroskedasticity (ARCH) process with long memory introduced in Doukhan emphet al. (2016) and Grublytė and Škarnulis (2016) with conditional variance involving the square of inhomogeneous linear combination of observable sequence with square summable weights. The aforementioned model extends the quadratic ARCH model of Sentana () and the linear ARCH model of Robinson () to the case of strictly positive conditional variance. We prove consistency and asymptotic normality of the corresponding quasi‐maximum likelihood estimates, including the estimate of long memory parameter 0 < d < 1/2. A simulation study of empirical mean‐squared error is included.

show abstract

“…For the GQARCH model in , similar results were established by Doukhan et al () and Grublytė and Škarnulis (). Namely, assume that parameters γ , ω , a , b j and j ⩾1 in satisfy

b_{j} \sim c j^{d - 1} (\exists 0 < d < 1 / 2, c > 0),

γ ∈[0,1), a ≠ 0 and

6 B_{2} + 4 | μ_{3} | {falsefalse}_{\sum}^{j = 1} | b_{j} |^{3} + μ_{4} {falsefalse}_{\sum}^{j = 1} b_{j}^{4} < {(1 - γ)}^{2},

where

μ_{p} : = normalE ζ_{0}^{p}, p = 1, 2, \dots, B_{2} : = \sum_{j = 1}^{\infty} b_{j}^{2} .

Then (Grublytė and Škarnulis, , Theorems 2.5 and 3.1) there exists a stationary solution of with

normalE r_{t}^{4} < \infty

such that

cov (r_{0}^{2}, r_{t}^{2}) \sim κ_{1}^{2} t^{2 d - 1}, t \to \infty

and

n

…”

Section: Introductionsupporting

confidence: 79%

“…Proposition (Grublytė and Škarnulis ()) Let γ ∈[0,1) and { ζ t } be an i.i.d. sequence with zero mean, unit variance and finite moment

μ_{p} : = normalE ζ_{0}^{p} < \infty

, where p ⩾2 is an even integer.…”

Section: Stationary Solutionmentioning

confidence: 99%

“…Remark Sufficient conditions for the existence of stationary solution of with finite moment

normalE | r_{t} |^{p} < \infty

and arbitrary p > 0 are obtained by Grublytė and Škarnulis, (, Theorem 2.4). They extend the corresponding result of Doukhan et al, (, Theorem 1) from γ = 0 to γ > 0.…”

Section: Stationary Solutionmentioning

confidence: 99%

“…Recently, Doukhan et al () and Grublytė and Škarnulis () discussed a class of quadratic ARCH models of the form

r_{t} = ζ_{t} σ_{t}, σ_{t}^{2} = ω^{2} + 1.19em (a + {falsefalse}_{\sum}^{j = 1} b_{j} r_{t - j} {1.19em)}^{2} + γ σ_{t - 1}^{2},

where

{ζ_{t}, t \in double-struckZ}

is a standardized i.i.d. sequence,

normalE ζ_{t} = 0, normalE ζ_{t}^{2} = 1

, and γ ∈[0,1), ω , a , b j , j ⩾1 are real parameters satisfying certain conditions; see Proposition .…”

Section: Introductionmentioning

confidence: 99%

“…sequence,

normalE ζ_{t} = 0, normalE ζ_{t}^{2} = 1

σ_{t}^{2} = \sum_{ℓ = 0}^{\infty} γ^{ℓ} ({) ω^{2} + (() a + \sum_{j = 1}^{\infty} b_{j} r_{t - ℓ - j} {())}^{2} (})

in lagged variables r t − 1 , r t − 2 ,…, and hence, it represents a particular case of quadratic ARCH model by Sentana () with

p = \infty

.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

QMLE for Quadratic ARCH Model with Long Memory

Grublytė

Сургайлис

Škarnulis

2016

Journal Time Series Analysis

Self Cite

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show abstract

A nonlinear model for long-memory conditional heteroscedasticity*

2016

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Abstract. We discuss a class of conditionally heteroscedastic time series models satisfying the equation r t = ζ t σ t , where ζ t are standardized i.i.d. r.v.'s and the conditional standard deviation σ t is a nonlinear function Q of inhomogeneous linear combination of past values r s , s < t with coefficients b j . The existence of stationary solution r t with finite pth moment, 0 < p < ∞ is obtained under some conditions on Q, b j and the pth moment of ζ 0 . Weak dependence properties of r t are studied, including the invariance principle for partial sums of Lipschitz functions of r t . In the case when Q is the square root of a quadratic polynomial, we prove that r t can exhibit a leverage effect and long memory, in the sense that the squared process r 2 t has long memory autocorrelation and its normalized partial sums process converges to a fractional Brownian motion.

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A generalized nonlinear model for long memory conditional heteroscedasticity

Cited by 2 publications

References 21 publications

QMLE for Quadratic ARCH Model with Long Memory

QMLE for Quadratic ARCH Model with Long Memory

A nonlinear model for long-memory conditional heteroscedasticity*

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