Econometric Methods for Discretely-Sampled Continuous-Time Models

Princeton University, Princeton NJ

Investigators

Abstract

Interest rates have traditionally been modeled in the economics literature as following continuous-time Markov processes, and more specifically, diffusions. By contrast, recent term structure models often imply non-Markovian continuous-time dynamics. Can discretely sampled interest rate data help decide which continuous-time models are sensible? Within the Markovian world, diffusion processes are characterized by the continuity of their sample paths. It is immediately, obvious that this condition cannot be verified from the observed sample path. By nature, even if the sample path were continuous, the discretely sampled interest rate data will appear as a sequence of discrete change. This grant continues work begun under NSF award 970305 on this fundamental problem in financial economics. This project develops new likelihood-based estimation methods for discretely-sampled continuous-time models and extends our understanding of the properties of estimators in four related situations: 1. Since many realistic models in economics involve multiple state variables, the first part of the project develops a closed-form sequence of likelihood functions applicable to arbitrary multivariate diffusion models. 2. These functions are used to infer consistent dynamic models from market data. 3. Allowing now for jumps, it is shown that the lower the frequency of observation, the more difficult it is to disentangle from discrete data the respective effects of the jump and volatility components. The project makes this intuition rigorous by deriving Fisher's information matrix from an explicit expansion of the likelihood answering questions such as: How fast does the precision of the jump estimates decrease when the frequency of observation decreases? What is the influence on the identifiability of jumps of the relative magnitudes of the (continuous) volatility and (discontinuous) jump parts? 4. New issues arise when the data are not only discretely but also possibly randomly spaced in time. The project derives the properties of estimators based on maximum-likelihood with either full or partial information, the generalized method of moments, and discrete sampling schemes such as the Euler approximation. Studying the effect of the sampling randomness, the asymptotic distribution of the various estimators will be decomposed in terms that are due to the discreteness vs. terms due to the randomness of the sampling. This makes it possible to compare the relative costs of ignoring either the discreteness or the randomness of the sampling scheme producing the data. When the estimators are asymptotically biased, their biases will also be analyzed. The project integrates research and education by creating datasets and developing publicly available computer code (both made available through the web, as has been the case for past projects) for each of the main research endeavors funded by this proposal. The results will be disseminated broadly through presentations at seminars, conferences and professional association meetings.