Efficient estimation of general dynamic models with a continuum of moment conditions

Abstract

There are two difficulties with the implementation of the characteristic function-based estimators. First, the optimal instrument yielding the ML efficiency depends on the unknown probability density function. Second, the need to use a large set of moment conditions leads to the singularity of the covariance matrix. We resolve the two problems in the framework of GMM with a continuum of moment conditions. A new optimal instrument relies on the double indexing and, as a result, has a simple exponential form. The singularity problem is addressed via a penalization term. We introduce HAC-type estimators for non-Markov models. A simulated method of moments is proposed for nonanalytical cases. r

Realized Laplace transforms for estimation of jump diffusive volatility models

“…Given our choice of c, the lowest weight corresponds to the density of a normal distribution at 2 standard deviations and this is exactly the region where the normal density has curvature (and hence weighs differently the different points (u, v)). This is similar to the use of the Gaussian kernel in empirical characteristic function based estimation in Jiang and Knight (2002) and Carrasco et al (2007). From a practical point of view, using dirac deltas would probably not lead to much loss of information as the joint Laplace transform is typically rather smooth.…”

“…In the infeasible case when the variance process V t is directly observed (at integer times), one can match the empirical and model-implied joint Laplace transform at a given lag K. As shown in Feuerverger and Mureika (1977), see also Carrasco et al (2007), appropriate weighting of these moments can lead to asymptotic equivalence to the estimation equations…”

“…Within the regions we weight the distance L(u, v; k) − L(u, v; k|ρ) by the weight measure ω(du, dv), while we let the data determine (through the optimal weight matrix W ) the relative importance of each pair regionlag. This approach can be viewed as a feasible alternative to the use of a continuum of moment conditions based on L(u, v; k|ρ) (over u, v and k) as in the case when the stochastic process of interest (here V t ) is fully observable considered in Carrasco et al (2007). While for many models, e.g., the affine jump-diffusion class, the joint Laplace transform is known analytically, this is not the case generally for the integrated one, L(u, v; k|ρ), and hence the latter has to be evaluated by (one-dimensional) numerical integration which is quick and easy.…”

“…3 Thus some aspects of our estimation methodology are in common with the long tradition of estimation based on the characteristic function (Parzen, 1962) and conditional characteristic functions as in Feuerverger and Mureika (1977), Singleton (2001), Jiang and Knight (2002), Yu (2004), Bates (2006) and Carrasco et al (2007). For example, we face issues similar to those of the continuum of moment conditions and also the numerical difficulties associated with computing inverse integral transforms.…”

Parallelization strategies for continuum-generalized method of moments on the multi-thread systems

“…This, of course, takes a very long time, especially in the calculation which is processed sequentially. This increases the latency of calculation process, particularly impact on the arbitrage problem in asset pricing [4,5,6]. Meanwhile, all modern computers are now supported by hierarchical memory systems and hyperthreading technology, which allowing for parallel computing.…”