Abstract
This paper discusses the Monte Carlo maximum likelihood method of estimating stochastic volatility (SV) models. The basic SV model can be expressed as a linear state space model with log chi-square disturbances. The likelihood function can be approximated arbitrarily accurately by decomposing it into a Gaussian part, constructed by the Kalman filter, and a remainder function, whose expectation is evaluated by simulation. No modifications of this estimation procedure are required when the basic SV model is extended in a number of directions likely to arise in applied empirical research. This compares favorably with alternative approaches. The finite sample performance of the new estimator is shown to be comparable to the Monte Carlo Markov chain (MCMC) method.
| Original language | English |
|---|---|
| Pages (from-to) | 271-301 |
| Number of pages | 31 |
| Journal | Journal of Econometrics |
| Volume | 87 |
| Issue number | 2 |
| DOIs | |
| Publication status | Published - 8 Sept 1998 |
| Externally published | Yes |
Funding
We are indebted to Jim Durbin and Andrew Harvey as well as seminar participants of the ESEM 96 Conference in Istanbul for helpful comments and suggestions. The comments of the Associate Editor and an anonymous referee helped greatly improve the quality of the paper. The second author is grateful to the ESRC for financial support as part of the project `Interrelationships in Economic Time Series', grant No. R000235330. All remaining errors remain our responsibility. Appendix A 1. The general univariate state space model ( Harvey, 1989 ) is: η t=1,…, T y t =Z t α t +ε t , ε H t ∼ N (0, t ), α t =T t α t−1 +c t +η t , η Q t ∼ N (0, t ), E (ε t , t )=G t , where y t is a scalar observation, α t is the ( m ×1) state vector, the covariance matrices H t (1×l) and Q t ( m × m ) are nonsingular, while the measurement and transition equation disturbances may be contemporaneously correlated with an ( m ×1) nonzero covariance matrix G t . In case of the SV model, the elements (or functions of the elements) of the parameter vector ψ =( φ , σ η , ρ ) enter into the appropriate elements of Q t , T t , G t and c t . The long-run volatility level, σ ̄ (along with any explanatory variables) enters the state vector α t which reduces the dimensionality of the nonlinear optimization problem of maximizing the likelihood function with respect to the parameter vector; see below. For instance, an correlated SV model with k explanatory variables, z k t grouped into z z t =(z 1 t ,…, k t )′ is put into the state space form by defining the matrices α , T , Q , Z t = h t γ t t = φ e k ′ e k i k t = σ 2 η −A 2 e k ′ e k e k e k ′ t =[1 z t ′] H , G , c , A=0.7979ρσ t = π 2 2 t = Bs t e k t = As t e k η , B=1.1061ρσ η where γ t is ( k ×1), e k is a ( k ×1) vector of zeros, I k is the ( k × k ) identity matrix, and s t is the sign of the return at time t . The basic SV model is obtained as a special case by setting ρ =0 and k =1, z 1 t =1, ∀ t . The Kalman filter is given by ν t =y t −Z t a t , F t =Z t P t Z t ′+H t , K t =(T t+1 P t Z t ′+G t )F −1 t , L t =T t+1 −K t Z t , a t+1 =T t+1 a t +c t +K t ν t , P t+1 =T t+1 P t L t ′+Q t+1 −G t K t ′, for t =1,…, T . The recursions are initialized with α 0 =N( a 0 , P 0 ) where P 0 is the unconditional variance matrix of the state vector which may contain diffuse elements. The parameter estimates for ψ are obtained by numerically optimizing the Gaussian log likelihood function as given by L 2π+ ln G (ψ)=− 1 2 ∑ t=1 T ( ln ln |F t |+ν t ′F −1 t ν t ). The estimate of ln σ ̄ 2 is given by the relevant element of a t while the standard errors are obtained from the relevant diagonal elements of P t ( Harvey, 1989 , p. 367). 2. The Kalman smoother ( 1, with de Jong, 1988 ; Koopman, 1993 ) is used to construct: ε ̂ t = E (ε t |y)=H t e t +G t ′r t , C | y)+H t = Var (ε t t −H t D t H t −G t ′N t G t +J t +J t ′, where y is the vector of all observations, J t = H t K t ′ N t G t , and the remaining quantities are obtained from the backwards recursions: e t =F −1 t ν t −K t ′r t , D t =F −1 t +K t ′N t K t , r t−1 =Z t ′F −1 t ν t +L t ′r t , N t−1 =Z t ′F −1 t Z t +L t ′N t L t for t = T ,…, r T =0 and N T =0. The prediction errors, ν t , their variances, F t , and the Kalman gain matrix, K t are outputs of the Kalman filter. 3. A special version of de Jong and Shephard's (1995) simulation smoother is used to give draws of 1 with ε (i) from p | y, ψ) G (ε : ε u (i) t =H t ε ̃ t +G t ′ r ̃ t +u (i) t , (i) t ∼ N (0, C ̃ t ) where the quantities ε ̃ t and C ̃ t are obtained from ε ̃ t =F −1 t ν t −K t ′ r ̃ t , D ̃ t =F −1 t +K t ′ N ̃ t K t , J t =H t K t ′ N ̃ t G t , M t =H t ( D ̃ t Z t −K t ′ N ̃ t T t+1 )+G τ ′ N ̃ t L t , C ̃ t =H t −H t D ̃ t H t −G t ′ N ̃ t G t +J t +J t ′ and the backwards recursions: r ̃ t−1 =Z t ′F −1 t ν t −M t ′ C ̃ −1 t u (i) t +L t ′ r ̃ t , N ̃ t−1 =Z t ′F −1 t Z t +M t ′ C ̃ −1 t M t +L t ′ N ̃ t L t for t = T ,…, r ̃ t =0 and N ̃ T =0 . Note that when a set of samples is required, the Kalman filter and the recursions for M D ̃ t , t , C ̃ t , N ̃ t−1 need only be applied once since these quantities remain the same for each sample.
Keywords
- GARCH model
- Importance sampling
- Kalman filter smoother
- Monte Carlo simulation
- Quasi-maximum likelihood
- Stochastic Volatility
- Unobserved components