A Martingale Approach for Testing Di¤usion Models Based on Innitesimal Operator

I develop an omnibus speci…cation test for di¤usion models based on the in…nitesimal operator instead of the already extensively used transition density. The in…nitesimal operatorbased identi…cation of the di¤usion process is equivalent to a "martingale hypothesis" for the new processes transformed from the original di¤usion process. The transformation is via the celebrated "martingale problems". My test procedure is to check the "martingale hypothesis" via a multivariate generalized spectral derivative based approach which enjoys many good properties. The in…nitesimal operator of the di¤usion process enjoys the nice property of being a closedform expression of drift and di¤usion terms. This makes my test procedure capable of checking both univariate and multivariate di¤usion models and particularly powerful and convenient for the multivariate case. In contrast checking the multivariate di¤usion models is very di¢ cult by transition density-based methods because transition density does not have a closed-form in general. Moreover, di¤erent transformed martingale processes contain di¤erent separate information about the drift and di¤usion terms and their interactions. This motivates us to suggest a separate inference-based test procedure to explore the sources when rejection of a parametric form happens. Finally, simulation studies are presented and possible future researches using the in…nitesimal operator-based martingale characterization are discussed. Keywords: Di¤usion; Jump Di¤usion; Markov; Martingale; Martingale problem; Semi-group; Drift; In…nitesimal operator; Transition density; Generalized spectrum. Correspondence: Zhaogang Song, Uris Hall 445, Department of Economics, Cornell University, Ithaca, NY 14850, USA. Email: [email protected]. 1 1 Introduction Di¤usion processes have proven to be mostly successful in …nance over the past three decades in modeling the dynamics of for instance interest rates, stock prices, exchange rates and option prices. On the one hand, the continuous ‡ow of information into …nancial markets makes it intuitive and necessary to use continuous-time models among which di¤usion models may be the most extensively used. On the other hand, the development of stochastic calculus o¤ers us elegant mathematical tools for solving many important problems in …nance when di¤usion models are used. However, while economic theories have implications about the relationship between economic variables, they usually do not suggest any concrete functional form for the processes; the choice of a model is somewhat arbitrary. As a result, a great number of parametric di¤usion models have been proposed in the literature, see for example Ait-Sahalia(1996a), Ahn and Gao(1999), Chan, Karolyi, Longsta¤ and Sanders(1992), Cox, Ingersoll and Ross(1985), and Vasicek(1977). Generally a parametric speci…cation of a di¤usion model will essentially specify the whole dynamics of the underlying process, for example, the transition density or the in…nitesimal operator of it as a Markov process. Therefore, model misspeci…cation may yield misleading conclusions about the dynamics of the process by rendering inconsistent parameter estimators and their variance-covariance matrix estimators. In practice, such a mis-speci…ed model may result in large errors in pricing, hedging and risk management. Therefore, the development of reliable speci…cation tests for di¤usion models is necessary to tackle such problems. However, although in the past decade or so substantial progress has been made in developing estimation methods1, both parametrically and non or semi-parametrically(see Ait-Sahalia(2002b), Jiang and Knight(1997), Kristensen (2008a,b), Stanton(1997). Bandi and Phillips(2003), for example), relatively little e¤ort has been devoted to speci…cation and evaluation of di¤usion models. In this study, we will develop an omnibus test for the speci…cation of di¤usion models based on the in…nitesimal operator which is an alternative characterization of the whole dynamics of the process to transition function or transition density used by Ait-Sahalia, Fan and Peng(2008), Chen and Hong(2008a), Corradi and Swanson (2005), and Hong and Li(2005). By the celebrated "martingale problems" developed by Strook and Varadhan(1969), the identi…cation of the di¤usion process is equivalent to a "martingale hypothesis" for the processes which come from the transformation of the original di¤usion process implied by the "martingale problems". I then check the "martingale hypothesis" for these processes by extending Hong’s(1999) generalized spectral approach to a multivariate generalized spectral derivative based test which is particularly powerful against alternatives with zero autocorrelation but a nonzero conditional mean and which has a convenient one-sided N(0; 1) limit distribution2. The in…nitesimal operator of the di¤usion process has the nice property of being a closed-form expression of drift and di¤usion terms. This makes my test procedure enjoy many good properties which I will discuss in the following. Since the speci…cation of a parametric di¤usion model usually refers to specifying the so-called drift and di¤usion terms, we can roughly summarize the existing researches on speci…cation test of parametric di¤usions3 Sundaresan (2001) points out that “perhaps the most signi…cant development in the continuous-time …eld during the last decade has been the innovations in econometric theory and in the estimation techniques for models in continuous time.”For other reviews of this literature, see (e.g.) Tauchen (1997). Hong’s(1999) test of martingale hypothesis is in spirit similar to Bierens’s(1982) and Bierens and Ploberger’s(1997) integrated conditional moment tests for model speci…cation. But the latter two has null limit distributions which are a sum of weighted chi-squared variables with weights depending on the unknown data generating process and cannot be tabulated. 3 In recent years, there have been some researches which check the generic properties of a continuous time process and which are naturally nonparmatric, including the tests of markov property(Ait-Sahalia, 1997; Chen and Hong 2008b), of jumps(Ait-Sahalia