We provide a comprehensive analysis of the out-of-sample performance of a wide variety of spot rate models in forecasting the probability density of future interest rates. While the most parsimonious models perform best in forecasting the conditional mean of many financial time series, we find that the spot rate models that incorporate conditional heteroskedasticity and excess kurtosis or heavy-tails have better density forecasts. GARCH significantly improves the modeling of the conditional variance and kurtosis, while regime switching and jumps improve the modeling of the marginal density of interest rates. Our analysis shows that the sophisticated spot rate models in the existing literature are important for applications involving density forecasts of interest rates.

We propose two nonparametric transition density-based speciÞcation tests for continuous-time models. Unlike the marginal density used in the literature, the transition density can capture the full dynamics of a continuous-time process. To address the concerns of the Þnite sample perfor- mance of nonparametric methods in the literature, we introduce an appropriate data transfor- mation and correct the boundary bias of kernel estimators. As a result, our tests are robust to persistent dependence in data and provide reliable inferences for sample sizes often encountered in empirical Þnance. Simulation studies show that even for data with highly persistent depen- dence, our tests have reasonable size and good power against a variety of alternatives in Þnite samples. Besides one-factor diffusion models, our tests can be applied to a broad class of dynamic models, including discrete-time dynamic models, time-inhomogeneous diffusion models, stochas- tic volatility models, jump-diffusion models, and multi-factor diffusion models. When applied to Eurodollar interest rates, our tests overwhelmingly reject a variety of popular one-factor diffusion models. We Þnd that introducing nonlinear drift does not signiÞcantly improve the goodness of Þt, and the main reason for the rejection of one-factor diffusion models is the violation of the Markov assumption. Some popular non-Markovian models with GARCH, regime switching and jumps perform signiÞcantly better than one-factor diffusion models, but they are still far from being adequate to fully capture the interest rate dynamics. Our study shows that, contrary to the general perception in the literature, nonparametric methods are a reliable and powerful tool for analyzing Þnancial data.

We propose a new diagnostic test for linear and nonlinear time series models,using a generalized spectral approach+ Under a wide class of time series models that includes autoregressive conditional heteroskedasticity (ARCH) and autoregressive conditional duration (ACD) models, the proposed test enjoys the appealing“nuisance-parameter-free” property in the sense that model parameter estimation uncertainty has no impact on the limit distribution of the test statistic+ It is consistent against any type of pairwise serial dependence in the model standardized residuals and allows the choice of a proper lag order via data-driven methods. Moreover, the new test is asymptotically more efficient than the correlation integral?based test of Brock, Hsieh, and LeBaron (1991, Nonlinear Dynamics, Chaos, and Instability: Statistical Theory and Economic Evidence) and Brock, Dechert, Scheinkman, and LeBaron (1996, Econometric Reviews 15, 197?235), the well-known BDS test, against a class of plausible local alternatives (not including ARCH). A simulation study compares the finite-sample performance of the proposed test and the tests of BDS, Box and Pierce (1970, Journal of the American Statistical Association 65, 1509?1527), Ljung and Box (1978, Biometrika 65, 297?303), McLeod and Li (1983, Journal of Time Series Analysis 4, 269?273), and Li and Mak (1994, Journal of Time Series Analysis 15, 627? 636). The new test has good power against a wide variety of stochastic and chaotic alternatives to the null models for conditional mean and conditional variance. It can play a valuable role in evaluating adequacy of linear and nonlinear time series models. An empirical application to the daily S&P 500 price index highlights the merits of our approach.