Palgrave Handbook of Econometrics: Applied Econometrics

966 Continuous-Time Stochastic Volatility Models

Table 19.1 The parameter estimates (annualized) and thet-

statistics (reported in parentheses) are based on maximizing the

log-likelihood (LL). The data cover the period January 2, 1990, to

December 31, 2007, a total of 4,535 observations.

Parameter SV1 SV2 SV3 SV4 SV5

k

6.780 5.809 7.999 0.759 3.810

(7.71) (7.09) (11.13) (5.49) (5.80)

θ

0.044 0.044 0.022 0.105 −3.368

(19.96) (11.52) (24.05) (−29.85) (−27.78)

σ

0.123 0.447 0.359 3.29 1.96

(93.71) (93.88) (51.34) (22.21) (94.46)

γ –––

1.157

(7.89) –

λ ––

31.498

(9.14) ––

1/η ––

0.017

(4.56) ––

LL 15,459 17,457 17,694 18,446 18,390

AIC −30,911 −34,908 −35,379 −36,884 −36,773

BIC −30,892 −34,889 −35,347 −36,850 −36,754

All parameters are significant at the 1% level. According to all model selection
criteria, the best fit is provided by SV4, followed by, in order, SV5, SV3, SV2 and
SV1. Consistent with previous results in the literature, the Gaussian SV1 model
provides a very poor approximation to the data. The square root specification of
SV2 improves the fit substantially and the addition of a jump component improves
performance further. In the SV3 model, the mean reversion parameter increases so
as to pull back the process to its long run mean after a jump event. However, the
increase in the speed of mean reversion shows that jumps do not have a persis-
tent effect on volatility. All the statistical criteria suggest that SV5 outperforms the
square root specifications. As also reported in Psychoyios, Dotsis and Markellos
(2007), this result should not come as a surprise, since SV5 is capable of generat-
ing a large increase in volatility at high levels, followed by rapid mean reversion.
The best model overall is the constant elasticity of variance specification SV4. The
estimate of the exponent in the diffusion,γˆ=1.16, suggests that, as volatility
increases, its own volatility increases at an even faster rate (see also Jones, 2003).
The model also appears capable of capturing strong heteroskedasticity in volatility
changes. The empirical results thus point to the conclusion that, at least under this
simplified set up, non-affine specifications outperform affine processes.

19.5 Conclusions

The list of methods that we have discussed in this chapter is by no means com-
plete. For example, we have omitted econometric approaches such as the SMM of