Indirect Inference for Lévy‐driven continuous‐time GARCH models

Abstract

We advocate the use of an Indirect Inference method to estimate the parameter of a COGARCH(1,1) process for equally spaced observations. This requires that the true model can be simulated and a reasonable estimation method for an approximate auxiliary model. We follow previous approaches and use linear projections leading to an auxiliary autoregressive model for the squared COGARCH returns. The asymptotic theory of the Indirect Inference estimator relies on a uniform strong law of large numbers and asymptotic normality of the parameter estimates of the auxiliary model, which require continuity and differentiability of the COGARCH process with respect to its parameter and which we prove via Kolmogorov's continuity criterion. This leads to consistent and asymptotically normal Indirect Inference estimates under moment conditions on the driving Lévy process. A simulation study shows that the method yields a substantial finite sample bias reduction compared with previous estimators. DO RÊGO SOUSA ET AL. with parameter (to be specified in Section 2), L is a Lévy process with Lévy measure L ≢ 0 and having càdlàg sample paths. The volatility process ( s ( )) s ≥ 0 is predictable and its stochasticity depends only on L. The COGARCH process satisfies many stylized features of financial time series and is suited for modeling high-frequency data (see In many practical problems, one observes the log-price process (P iΔ ( 0 )) n i=1 on a fixed grid of size Δ > 0 and the question of interest is how to estimate the true parameter 0 . The data used for estimation are returnsSeveral methods have been proposed to estimate the parameter of a COGARCH process. A method of moments was proposed in Haug et al. (2007), Bibbona and Negri (2015) used prediction based estimation as developed in Sørensen (2000), and Maller et al. (2008) proposed a pseudo maximum likelihood (PML) method, which also works for nonequally spaced observations. Both momentand prediction-based estimators are consistent and asymptotically normal under certain regularity conditions. The asymptotic properties of the PML estimator were studied in Iannace (2014) and in Kim and Lee (2013), which require that Δ ↓ 0 as n → ∞. For the COGARCH process, Bayracı and Ünal (2014) used Indirect Inference with an auxiliary discrete-time GARCH model with Gaussian residuals. No theoretical results were proved, but their simulation study suggests that Indirect Inference estimators (IIEs) achieve a similar performance as the PML estimator of Maller et al. (2008) for fixed Δ > 0. Furthermore, Müller (2010) proposed a Markov chain Monte Carlo method, when L is a compound Poisson process.In this paper, we advocate an Indirect Inference method, different to the one suggested in Bayracı and Ünal (2014), to estimate the COGARCH parameter and derive the asymptotic properties of the estimator. Such methods were introduced in Smith (1993) and generalized in Gouriéroux, Monfort, and Renault (1993), and they offer a way to overcome many estimation problems by a clever simulation method. In sho...

Fractionally Integrated COGARCH Processes*

“…More sophisticated estimation procedures are envisaged like the estimation of d in a preliminary step by an appropriate estimator. Then in a second step the FICOGARCH parameters can be estimated by modified COGARCH estimators as suggested in Bibbona and Negri (2015), Haug et al (2007), Maller et al (2008), or Do Rego Sousa et al (2017. This is a version of the semiparametric approach suggested in Robinson (1994).…”

Moment‐based estimation for the multivariate COGARCH(1,1) process

“…Therefore, an important question is how to estimate the true parameter based on observations . In the univariate case, several methods have been proposed to estimate the parameters of the COGARCH process (Bayracı & Ünal, 2014; Bibbona & Negri, 2015; do Rêgo Sousa et al, 2019; Haug et al, 2007; Maller et al, 2008). All these methods rely on the fact that the COGARCH process is, under certain regularity conditions, ergodic and strongly mixing.…”

Robust estimation of stationary continuous‐time arma models via indirect inference

“…The authors applied the indirect inference method to macroeconomics, microeconomics, finance, and auction models; see as well Monfort (1996) and Phillips and Yu (2009) for applications to continuous-time models, Gouriéroux et al (2000) and Kyriacou et al (2017) for applications to time series models, and Monfardini (1998) for applications to stochastic volatility models. In addition indirect inference is used for bias reduction in finite samples as, for example, in Gouriéroux et al (2000), Gouriéroux et al (2010), Yu (2011), Kyriacou et al (2017), and do Rêgo Sousa et al (2019). An alternative approach for bias correction is given in Wang et al (2011) for univariate and multivariate diffusion models.…”