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Estimation and forecasting of long memory stochastic volatility models

  • Omar Abbara ORCID logo EMAIL logo and Mauricio Zevallos
Published/Copyright: March 25, 2022
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Studies in Nonlinear Dynamics & Econometrics
From the journal Studies in Nonlinear Dynamics & Econometrics Volume 27 Issue 1

Abstract

Stochastic Volatility (SV) models are an alternative to GARCH models for estimating volatility and several empirical studies have indicated that volatility exhibits long-memory behavior. The main objective of this work is to propose a new method to estimate a univariate long-memory stochastic volatility (LMSV) model. For this purpose we formulate the LMSV model in a state-space representation with non-Gaussian perturbations in the observation equation, and the estimation of parameters is performed by maximizing the likelihood written in terms derived from a Kalman filter algorithm. We also present a procedure to calculate volatility and Value-at-Risks forecasts. The proposal is evaluated by means of Monte Carlo experiments and applied to real-life time series, where an illustration of market risk calculation is presented.

Keywords: mixtures; non-Gaussian errors; value-at-risk
JEL Classification: C22; C53; G15
Corresponding author: Omar Abbara, Canvas Capital, Sao Paulo, Brazil, E-mail:

Funding source: Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP) 10.13039/501100001807

Award Identifier / Grant number: 2018/04654-9

Funding source: Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) 10.13039/501100003593

Funding source: Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) 10.13039/501100002322

Acknowledgment

We thank a referee for valuable suggestions that led to an improvement in the paper. We also thank the support of the Centre for Applied Research on Econometrics, Finance and Statistics (CAREFS).

  1. Author contribution: All the authors have accepted responsibility for the entire content of this submitted manuscript and approved submission.

  2. Research funding: The first author acknowledges the financial support of Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq, DOI 10.13039/501100003593) and Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES, DOI 10.13039/501100002322) PhD scholarship and the second author acknowledges financial support of Fundação de Amparo á Pesquisa do Estado de São Paulo (FAPESP, DOI 10.13039/501100001807) grant 2018/04654-9.

  3. Conflict of interest statement: The authors declare no conflicts of interest regarding this article.

Appendix

In this Appendix we obtain the Kalman filter algorithm for the estimation procedure discussed in Sections 2.2 and 2.3. The demonstrations are based on Shumway and Stoffer (2006, Section 6.2).

Obtaining the Kalman filter algorithm for the AR representation

We start by calculating X t|t−1, which is the predicted value of X t given the information until t − 1, as follows:

(49) X t | t − 1 = E ( X t | y 1 : t − 1 ) = E ( Φ X t − 1 + H ω t | y 1 : t − 1 ) = Φ E ( X t − 1 | y 1 : t − 1 ) = Φ X t − 1 | t − 1

Now, X t|t is given by:

(50) X t | t = E ( X t | y 1 : t ) = E E ( X t | y 1 : t , I j t ) | y 1 : t = ∑ j = 1 m E ( X t | y 1 : t , I j t ) π j t ,

π jt = P(I jt = 1|y 1:t ). To obtain the analytic expression of (50), it is necessary to calculate E(X t |y 1:t , I jt ). For each possible value of j, η t = V jt and the innovation ϵ jt is equal to:

ϵ j t = y t − E ( y t | y 1 : t − 1 , I j t = 1 ) = y t − E [ Θ X t | y 1 : t − 1 , I j t = 1 ] − E ( V j t | y 1 : t − 1 ) = y t − E ( V j t ) − Θ X t | t − 1 = y t − μ j − Θ X t | t − 1 .

where μ 1 = 0.

Thus, considering that cov(ϵ jt , y s ) = 0 for s < t, we have:

cov ( X t , ϵ j t | y 1 : t − 1 ) = cov ( X t , y t − μ j − Θ X t | t − 1 | y 1 : t − 1 ) = cov ( X t − X t | t − 1 , Θ X t − Θ X t | t − 1 + V j t | y 1 : t − 1 ) = cov ( X t − X t | t − 1 , Θ ( X t − X t | t − 1 ) | y 1 : t − 1 ) = Var ( X t − X t | t − 1 | y 1 : t − 1 ) Θ ′ = P t | t − 1 Θ ′

where P t|t−1 = Var(X t − X t|t−1|y 1:t−1). Thus, the joint distribution of X t and ϵ jt conditional on y 1:t−1 is given by:

(51) X t ϵ j t | y 1 : t − 1 ∼ N X t | t − 1 0 , P t | t − 1 P t | t − 1 Θ ′ Θ P t | t − 1 Σ j t ,

where Σ j t = Var ( ϵ j t | y 1 : t − 1 ) = Θ P t | t − 1 Θ ′ + σ j 2 . Using Property (B.9) of Shumway and Stoffer (2006, Appendix B), we have:

(52) X t | t = X t | t − 1 + k j t ϵ j t ,

(53) k j t = P t | t − 1 Θ ′ Σ j t = P t | t − 1 Θ ′ Θ P t | t − 1 Θ ′ + σ j 2 .

Combining the results of Eq. (52) in Eq. (50) we have:

X t | t = ∑ j = 1 m E ( X t | y 1 : t , I j t = 1 ) π j t = ∑ j = 1 m ( X t | t − 1 + k j t ϵ j t ) π j t , X t | t = X t | t − 1 + ∑ j = 1 m π j t k j t ϵ j t .

Now we obtain the expressions for matrices P t|t−1 and P t|t . So:

(54) P t | t − 1 = E ( X t − X t | t − 1 ) ( X t − X t | t − 1 ) ′ | y 1 : t − 1 = E ( Φ X t − 1 + H ω t − Φ X t − 1 | t − 1 ) ( Φ X t − 1 + H ω t − Φ X t − 1 | t − 1 ) ′ | y 1 : t − 1 = E Φ ( X t − 1 − X t − 1 | t − 1 ) ( X t − 1 − X t − 1 | t − 1 ) ′ Φ ′ | y 1 : t − 1 + E H H ′ ω t 2 | y 1 : t − 1 = Φ P t − 1 | t − 1 Φ ′ + σ ω 2 H H ′ .

Here, we obtain an expression for P t|t . Similar to the calculation of X t|t we have:

P t | t = E ( X t − X t | t ) ( X t − X t | t ) ′ | y 1 : t = E E ( X t − X t | t ) ( X t − X t | t ) ′ | y 1 : t , I j t | y 1 : t , = ∑ j = 1 m E ( X t − X t | t ) ( X t − X t | t ) ′ | y 1 : t , I j t π j t , = ∑ j = 1 m Var ( X t − X t | t | y 1 : t , I j t = 1 ) π j t .

For each j we can obtain the expression of Var(X t − X t|t |y 1:t , I jt ) from the multivariate distribution of X t and ϵ jt conditional on y 1:t−1. The multivariate distributions are given in (51). Using Property (B.10) of Shumway and Stoffer (2006), for each j we have:

(55) Var ( X t − X t | t | y 1 : t , I j , t = 1 ) = I K − k j t Θ P t | t − 1 .

Then, P t|t is equal to:

(56) P t | t = ∑ j = 1 m I k − k j t Θ P t | t − 1 π j t .

Calculating filtered and predicted values of Ψ t of the Chan and Petris representation

The calculation of Ψ t|t−1 and Ψ t|t are similar to the calculation of X t|t−1, as follows:

(57) Ψ t | t − 1 = E ( Ψ t | z 1 : t − 1 ) = E ( F Ψ t − 1 + u t | z 1 : t − 1 ) = F E ( Ψ t − 1 | z 1 : t − 1 ) + δ t ,

where δ t = E(u t |z 1:t−1). To calculate δ t , from (34) we have:

E ( u t | z 1 : t − 1 ) = − E ( η t − 1 | z 1 : t − 1 ) , E ( ω t | z 1 : t − 1 ) , 0 , … , 0 ′

Now, since ω t and z s are independent for z < t, then E(ω t |z 1:t−1) = 0. Additionally, from (30) η t = z t − φΨ t so E(η t |z 1:t ) = E(z t − φΨ t |z 1:t ) = z t − φΨ t|t . Therefore,

δ t = − z t − 1 + φ Ψ t − 1 | t − 1 0 … 0 ′ ,

and from (57) Ψ t|t−1 can be calculated by:

(58) Ψ t | t − 1 = F Ψ t − 1 | t − 1 + δ t .

The calculation of Ψ t|t is similar to that of X t|t . Defining ϵ ̃ j t = z t − μ j − φ Ψ t | t − 1 and P ̃ t | t − 1 = Var ( Ψ t − Ψ t | t − 1 | z 1 : t − 1 ) , the joint distribution of Ψ t and ϵ ̃ j t conditional on z 1:t−1 is given by:

(59) Ψ t ϵ ̃ j t | z 1 : t − 1 ∼ N Ψ t | t − 1 0 , P ̃ t | t − 1 P ̃ t | t − 1 φ ′ φ P ̃ t | t − 1 Σ ̃ j t ,

And, finally, we obtain:

Ψ t | t = ∑ j = 1 m E ( Ψ t | z 1 : t , I j t = 1 ) π ̃ j t = ∑ j = 1 m ( Ψ t | t − 1 + k ̃ j t ϵ ̃ j t ) π ̃ j t , Ψ t | t = Ψ t | t − 1 + ∑ j = 1 m π ̃ j t k ̃ j t ϵ ̃ j t .

Calculating the variance matrices P ̃ t | t − 1 and P ̃ t | t

Now, we calculate the matrices P ̃ t | t − 1 and P ̃ t | t . We start with P ̃ t | t − 1 .

P ̃ t | t − 1 = E ( Ψ t − Ψ t | t − 1 ) ( Ψ t − Ψ t | t − 1 ) ′ | z 1 : t − 1 , = E ( F Ψ t − 1 + u t − F Ψ t − 1 | t − 1 − δ t ) ( F Ψ t − 1 + u t − F Ψ t − 1 | t − 1 − δ t ) ′ | z 1 : t − 1 , = E F ( Ψ t − 1 − Ψ t − 1 | t − 1 ) + ( u t − δ t ) F ( Ψ t − 1 − Ψ t − 1 | t − 1 ) + ( u t − δ t ) ′ | z 1 : t − 1 = E F ( Ψ t − Ψ t | t − 1 ) ( Ψ t − Ψ t | t − 1 ) ′ F ′ | z 1 : t − 1 + E F ( Ψ t − Ψ t | t − 1 ) ( u t − δ t ) ′ | z 1 : t − 1 + E ( u t − δ t ) ( Ψ t − Ψ t | t − 1 ) ′ F ′ | z 1 : t − 1 + E ( u t − δ t ) ( u t − δ t ) ′ | z 1 : t − 1 P ̃ t | t − 1 = I + II + III + IV .

Expression I is equal to P ̃ t | t − 1 = F P ̃ t − 1 | t − 1 F and it can be obtained similarly to the P t|t−1 in the AR representation.

Now we calculate expression IV. First, the expression u t − δ t is equal to:

u t − δ t = − η t − 1 ω t ⋮ 0 − − z t − 1 + φ Ψ t − 1 | t − 1 0 ⋮ 0 = − η t − 1 + z t − 1 − φ Ψ t − 1 | t − 1 ω t ⋮ 0 .

Since z t = φΨ t + η t , then:

− η t − 1 + z t − 1 − φ Ψ t − 1 | t − 1 = − η t − 1 + φ Ψ t − 1 + η t − 1 − φ Ψ t − 1 | t − 1 = φ ( Ψ t − 1 − Ψ t − 1 | t − 1 ) .

Therefore:

(60) u t − δ t = φ ( Ψ t − 1 − Ψ t − 1 | t − 1 ) ω t … 0 ′ ,

and considering that ω t and Ψ t are independent, IV is equal to:

(61) I V = E ( u t − δ t ) ( u t − δ t ) ′ | z 1 : t = Ω t − 1 | t − 1 = φ P ̃ t − 1 | t − 1 φ ′ 0 … 0 0 σ ω 2 … 0 ⋮ ⋮ ⋱ ⋮ 0 0 … 0 .

Next, we obtain expression II. We define matrix A as:

(62) A = F ( Ψ t − 1 − Ψ t − 1 | t − 1 ) ( u t − δ t ) ′ .

From (60) we have:

A = F ( Ψ t − 1 − Ψ t − 1 | t − 1 ) φ ( Ψ t − 1 − Ψ t − 1 | t − 1 ) ω t 0 K C × 1 … 0 K C × 1 = F ( Ψ t − 1 − Ψ t − 1 | t − 1 ) φ ( Ψ t − 1 − Ψ t − 1 | t − 1 ) F ( Ψ t − 1 − Ψ t − 1 | t − 1 ) ω t 0 K C × 1 … 0 K C × 1 .

Since the quantity φ(Ψ t−1 − Ψ t−1|t−1) is a scalar, it is equal to its transpose. Then:

A = F ( Ψ t − 1 − Ψ t − 1 | t − 1 ) ( Ψ t − 1 − Ψ t − 1 | t − 1 ) ′ φ ′ F ( Ψ t − 1 − Ψ t − 1 | t − 1 ) ω t … 0 K C × 1 .

In consequence:

(63) II = E ( A | z 1 : t − 1 ) = Q t − 1 | t − 1 = F P ̃ t − 1 | t − 1 φ ′ 0 K C × K C .

Since matrix III is the transpose of matrix II we have:

(64) P ̃ t | t − 1 = F P ̃ t − 1 | t − 1 F ′ + Q t − 1 | t − 1 + Q t − 1 | t − 1 ′ + Ω t − 1 | t − 1 ,

where:

Q t − 1 | t − 1 = F P ̃ t − 1 | t − 1 φ ′ 0 K C × 1 … 0 K C × 1 , Ω t − 1 | t − 1 = φ P ̃ t − 1 | t − 1 φ ′ 0 … 0 0 σ ω 2 … 0 ⋮ ⋮ ⋱ ⋮ 0 0 … 0 .

The expression for P ̃ t | t − 1 is obtained in a very similar way to that of P t|t . For each value of j we can obtain:

(65) Var ( Ψ t − Ψ t | t | z 1 : t , I j , t = 1 ) = I K C − k ̃ j t φ P ̃ t | t − 1 .

Thus, P ̃ t | t is equal to:

(66) P ̃ t | t = ∑ j = 1 m I K C − k ̃ j t φ P ̃ t | t − 1 π ̃ j t .

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Supplementary Material

The online version of this article offers supplementary material (https://doi.org/10.1515/snde-2020-0106).

Received: 2020-08-30
Accepted: 2022-03-04
Published Online: 2022-03-25

© 2022 Walter de Gruyter GmbH, Berlin/Boston

Articles in the same Issue

  1. Frontmatter
  2. Research Articles
  3. Estimation and forecasting of long memory stochastic volatility models
  4. Uncertainty and realized jumps in the pound-dollar exchange rate: evidence from over one century of data
  5. Bidirectional volatility transmission between stocks and bond in East Asia – The quantile estimates based on wavelets
  6. A threshold model for the spread
  7. A Gini estimator for regression with autocorrelated errors
  8. State price density estimation with an application to the recovery theorem
  9. Testing for random coefficient autoregressive and stochastic unit root models
https://doi.org/10.1515/snde-2020-0106
Keywords for this article
mixtures; non-Gaussian errors; value-at-risk

Articles in the same Issue

  1. Frontmatter
  2. Research Articles
  3. Estimation and forecasting of long memory stochastic volatility models
  4. Uncertainty and realized jumps in the pound-dollar exchange rate: evidence from over one century of data
  5. Bidirectional volatility transmission between stocks and bond in East Asia – The quantile estimates based on wavelets
  6. A threshold model for the spread
  7. A Gini estimator for regression with autocorrelated errors
  8. State price density estimation with an application to the recovery theorem
  9. Testing for random coefficient autoregressive and stochastic unit root models
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