Scandinavian Journal of Statistics, Vol. 38: 288–310, 2011 doi: 10.1111/j.1467-9469.2010.00705.x © 2010 Board of the Foundation of the Scandinavian Journal of Statistics. Published by Blackwell Publishing Ltd. Goodness-of-Fit based on Downsampling with Applications to Linear Drift Diffusions JULIE L. FORMAN Department of Biostatistics, University of Copenhagen BO MARKUSSEN Department of Basic Sciences and Environment, University of Copenhagen HELLE SØRENSEN Department of Mathematical Sciences, University of Copenhagen ABSTRACT. A goodness-of-fit test for continuous-time models is developed that examines if the parameter estimates are consistent with another for different sampling frequencies. The test compares parameter estimates obtained from estimating functions for downsamples of the data. We prove asymptotic results for stationary and ergodic processes, and apply the downsampling test to linear drift diffusions. Simulations indicate that the test is quite powerful in detecting non- Markovian deviations from the linear drift diffusions. Key words: continuous-time model, estimating function, goodness of fit test, linear drift diffu- sion, stationary process 1. Introduction Diffusion-based models have a wide range of applications. In biology, chemistry and physics, the models are used to represent phenomena that evolve continuously and randomly in time. In finance, diffusions and stochastic volatility models are used to model various price processes. The analysis of these models, however, is complicated: the process is observed in discrete time only and the likelihood of the discrete-time observations is rarely explicitly known. Through the last few decades, the estimating problem has received much attention, see Sørensen (2004) for a review, and Bibby et al. (2010) on estimating functions, Genon- Catalot et al. (2003) on stochastic volatility models, Gallant & Tauchen (2010) on indirect inference, and Aït-Sahalia et al. (2010) on operator methods. On the other hand, the literature on goodness-of-fit testing is limited. Nevertheless, goodness-of-fit is a matter of importance; in case a model is misspecified the related estimators may be inconsistent and the conclusions of the statistical analysis may be invalid. Aït-Sahalia (1996b) was perhaps the first to consider goodness-of-fit for stationary diffusion models, comparing the estimated stationary density implied by the model to a kernel-density estimator. In particular, the test aims at the marginal distribution of the data rather than the dependence structure of the model. Fan & Zhang (2003) suggested to compare parametric and non-parametric estimates of the drift and diffusion coefficient derived from a discretization scheme, and hence seeked to detect model deviations of any kind. The goodness-of-fit test from Hong & Li (2005) is based on uniform residuals, that is, observations transformed with the conditional distribution function given the past observations. If the model is true then the residuals are independent and uniformly distributed, so the idea is to check if this is true. A related test aimed at multivariate data based on non-parametric estimation of the conditional