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# Nonparametric Transition-Based Tests for Jump Diffusions

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*in*Journal of the American Statistical Association 104(487):1102-1116 · July 2005

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DOI: 10.2139/ssrn.955820 · Source: RePEc

Cite this publicationAbstract

We develop a specification test for discretely-sampled jump-diffusions, based on a comparison of a nonparametric estimate of the transition density or distribution function to their corresponding parametric counterparts. As a special case, our method applies to pure diffusions. We propose three different discrepancy measures between the null and alternative transition density and distribution functions. We establish the asymptotic null distributions of proposed test statistics and compute their power functions. The finite sample properties are investigated via simulation studies and are compared with those of alternative tests.

- ... This observation lead to the development of tests based on transition densities since these fully characterize di¤usion models. Our transition-based tests are most related to the ones developed in Sahalia et al (2009) and Li and Tkacz (2006) where fully nonparametric and parametric estimators of the transition density are compared. In a similar spirit, Hong and Li (2004) propose a test where transformed versions of the transition densities are compared, while Chen, Gao and Tang (2009) employ empirical likelihood techniques. ...... In contrast, we are able to test the speci…cation of each of the two functions characterizing the model separately. Our local power analysis complements the one carried out in Sahalia et al (2009). They specify alternatives in terms of the transition densities and …nd that transition-based tests have the ability to detect local deviations form the null at a better rate than CvM type tests. ...... In particular, we show that they are not able to detect local alternatives at a higher rate compared to CvM type tests. These seemingly contradictory results are due to the fact that Sahalia et al (2009) specify their alternatives in terms of the transition density while we focus on deviations in terms of underlying drift and di¤usion functions. Since, as already noted above, the transition density involves integration over the drift and di¤usion function, local features in these get smoothed out in the transition density and therefore not easily detected. ...Article
- Mar 2010
- J ECONOMETRICS

We propose novel misspecification tests of semiparametric and fully parametric univariate diffusion models based on the estimators developed in Kristensen (Journal of Econometrics, 2010). We first demonstrate that given a preliminary estimator of either the drift or the diffusion term in a diffusion model, nonparametric kernel estimators of the remaining term can be obtained. We then propose misspecification tests of semparametric and fully parametric diffusion models that compare estimators of the transition density under the relevant null and alternative. The asymptotic distribution of the estimators and tests under the null are derived, and the power properties are analyzed by considering contiguous alternatives. Test directly comparing the drift and diffusion estimators under the relevant null and alternative are also analyzed. Markov Bootstrap versions of the test statistics are proposed to improve on the finite-sample approximations. The finite sample properties of the estimators are examined in a simulation study. - ... Numerous misspeci…cation tests of di¤usion models given low-frequency data have been proposed in the literature. However, most of these test a fully parametric speci…cation against a nonparametric Markov alternative (see e.g., Aït-Sahalia, 1996b; Aït-Sahalia et al, 2009; Corradi and Swanson, 2005; Hong and Li, 2004). Thus, they simultaneously test both the speci…cation of the drift and di¤usion term, and, as a consequence, these tests are not informative about the possible cause of a given rejection and do not give much guidance in the search for a correct speci…cation . ...... There are however a number of di¢ culties involved using the KL-distance as pointed out in Robinson (1991) so we'll here focus on the two other distances. Tests of fully parametric speci…cation testing of (jump-)di¤usion models using the modi…ed KL and the L 2 -distance have been proposed in Sahalia et al. (2009) where it is shown that T PC = d P (^ p np ; ^ p fp ) and T 2 = d 2 (^ p np ; ^ p fp ), where ^ p fp is the estimated transition density of the fully parametric model, follows a standard normal distribution when suitably normalized. The proofs of these results proceed in two steps: (i) Show T PC is asymptotically equivalent to T 0 PC = d PC (^ p np ; p) where p is the true transition density and (ii) Derive the asymptotic distribution of T 0 PC ; the same strategy is employed for T 2 . ...... for k = 1; 2, where expressions of PC , PC , 2 and 2 can be found in Sahalia et al. (2009). Next, we wish to test a fully parametric speci…cation, ...Article
- Sep 2009

Two classes of semiparametric diffusion models are considered, where either the drift or the diffusion term is parameterized, while the other term is left unspecified. We propose a pseudo-maximum likelihood estimator (PMLE) of the parametric component that maximizes the likelihood with a preliminary estimator of the unspecified term plugged in. It is demonstrated how models and estimators can be used in a two-step specification testing strategy of semiparametric and fully parametric models, and shown that approximate/simulated versions of the PMLE inherit the properties of the actual but infeasible estimator. A simulation study investigates the finite sample performance of the PMLE. - ... It is well known that the transition density of most continuous time models has no closed form. As a result, some techniques to approximate the transition density is required in the transition based tests(see Hong and Li (2005), Ait-Sahalia, Fan and Peng (2008)), for example, the simulation methods of Pedersen(1995) and Brandt and Santa-Clara(2002), the Hermite expansion approach of Ait-Sahalia (2002b), or for a¢ ne di¤usions, the closed-form approximation of Du¢ e, Pedersen, and Singleton(2003) and the empirical characteristic function approach of Singleton (2001) and Jiang and Knight(2002). Although the asymptotic distribution of some tests(like Hong and Li(2005)) is not a¤ected by the estimation uncertainty, the use of the transition density may not be computationally convenient and may a¤ect the …nite-sample performance of the test. ...... 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". ...... It is well known that the transition density of most continuous time models has no closed form. As a result, some techniques to approximate the transition density is required in the transition based tests(see Hong and Li (2005), Ait-Sahalia, Fan and Peng (2008)), for example, the simulation methods of Pedersen(1995(2005)) is not a¤ected by the estimation uncertainty, the use of the transition density may not be computationally convenient and may a¤ect the …nite-sample performance of the test. In contrast, the in…nitesimal operator always has an explicit closed-form expression which can be identi…ed by the drift and di¤usion terms. ...Article
- Jun 2011
- J ECONOMETRICS

I develop an omnibus specification test for diffusion models based on the infinitesimal operator. The infinitesimal operator based identification of the diffusion process is equivalent to a "martingale hypothesis" for the processes obtained by a transformation of the original diffusion model. My test procedure is then constructed by checking the "martingale hypothesis" via a multivariate generalized spectral derivative based approach that delivers a N(0,1) asymptotical null distribution for the test statistic. The infinitesimal operator of the diffusion process is a closed-form function of drift and diffusion terms. Consequently, my test procedure covers both univariate and multivariate diffusion models in a unified framework and is particularly convenient for the multivariate case. Moreover, different transformed martingale processes contain separate information about the drift and diffusion specifications. This motivates me to propose a separate inferential test procedure to explore the sources of rejection when a parametric form is rejected. Simulation studies show that the proposed tests have reasonable size and excellent power performance. An empirical application of my test procedure using Eurodollar interest rates finds that most popular short-rate models are rejected and the drift misspecification plays an important role in such rejections. - ... Aït-Sahalia (1996) introduced a density based test by comparing the parametric and nonparametric density estimates for model (1); see also Hong and Li (2005), Bosq (1998, and Gao and King (2004) for density based approaches. For Markov models, Sahalia et al. (2009) proposed specification tests based on transition densities. Sahalia et al. (2010) also used transition densities to test the Markov hypothesis. ...... where N t is a Poisson process with intensity λ(X t− ), and J t ~ N(0, η 2 ) is the independent jump size. As in Sahalia et al. (2009), consider the specification λ(x) = λ, σ(x) = ξ, such that where σ = 0.013 as in (42) and τ ∈ [0, 1], with τ = 0 being the null model (42). For all cases, we use b n = 0.0005 to limit computations. ...Article
- Jun 2011
- J ECONOMETRICS

We address the nonparametric model validation problem for hidden Markov models with partially observable variables and hidden states. We achieve this goal by constructing a nonparametric simultaneous confidence envelope for transition density function of the observable variables and checking whether the parametric density estimate is contained within such an envelope. Our specification test procedure is motivated by a functional connection between the transition density of the observable variables and the Markov transition kernel of the hidden states. Our approach is applicable for continuous time diffusion models, stochastic volatility models, nonlinear time series models, and models with market microstructure noise. - ... Also, inference strategies relying on maximum-likelihood or Bayesian methods require the transition density of the process. Specification testing procedures for stochastic processes also make use of the transition densities (see, e.g., [1] [3] [7] [8] [18] and [24]). All these models, estimation methods and tests assume that the process is Markovian. ...... It is straightforward to verify that T 14 = O p (nh 4 1 ) = o(1/h 1 ), T 15 = o p (1/ √ h 1 h 2 ). Using the same argument as for (B.2) in [3], we obtain T 12 = o p ( ...We propose several statistics to test the Markov hypothesis for $\beta$-mixing stationary processes sampled at discrete time intervals. Our tests are based on the Chapman--Kolmogorov equation. We establish the asymptotic null distributions of the proposed test statistics, showing that Wilks's phenomenon holds. We compute the power of the test and provide simulations to investigate the finite sample performance of the test statistics when the null model is a diffusion process, with alternatives consisting of models with a stochastic mean reversion level, stochastic volatility and jumps. Comment: Published in at http://dx.doi.org/10.1214/09-AOS763 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org)
- ... There have been other tests for univariate diffusion models in the recent literature. Ait-Sahalia, Fan and Peng (2005) propose some tests by comparing the model-implied transition density and distribution function with their nonparametric counterparts. Chen and Gao (2005) also propose a transition densitybased test using a nonparametric empirical likelihood approach. ...Article
- Nov 2005

Multivariate continuous-time models have been playing important roles in finance and economics. We develop an omnibus specification test for multivariate continuous-time models using the conditional characteristic function, which often has a convenient closed form or can be accurately approximated for many multivariate continuous-time models in finance and economics. Unlike the existing methods in the literature, the proposed omnibus test fully exploits the information in the joint conditional distribution of underlying economic processes and hence is expected to have good power in a multivariate context. A class of easy-to-interpret diagnostic procedures is supplemented to gauge possible sources of model misspecification. Simulation studies show that the tests provide reliable inference for sample sizes often encountered in finance and economics. The omnibus test has all-round power against various model misspecifications and the diagnostic tests can reveal useful information about the nature and type of model misspecification. In an application, we find that there is room for further improving upon a popular class of multivariate affine term structure models for monthly U.S. Treasury bond yields. It is documented that while there exists little dynamic structure in conditional means, there exist neglected dynamic structures in conditional variances and conditional correlations of yields with different maturities. - ... For example, tests characterized by comparing model implied transition densities with their nonparametric estimated (e.g. using kernels) counterparts (see e.g. Aït-Sahalia (1996,2002), Aït-Sahalia, Fan and Peng (2009)); and tests involving the examination of generalized cross spectra (see e.g. Hong and Li (2005) and Chen 1 and Hong (2008)). ...Article
- May 2011
- J Empir Finance

We review and construct consistent in-sample specification and out-of-sample model selection tests on conditional distributions and predictive densities associated with continuous multifactor (possibly with jumps) and (non)linear discrete models of the short term interest rate. The results of our empirical analysis are used to carry out a "horse-race" comparing discrete and continuous models across multiple sample periods, forecast horizons, and evaluation intervals. Our evaluation involves comparing models during two distinct historical periods, as well as across our entire weekly sample of Eurodollar deposit rates from 1982 to 2008. Interestingly, when our entire sample of data is used to estimate competing models, the "best" performer in terms of distributional "fit" as well as predictive density accuracy, both in-sample and out-of-sample, is the three factor Chen (Chen, 1996) model examined by Andersen, Benzoni and Lund (2004). Just as interestingly, a logistic type discrete smooth transition autoregression (STAR) model is preferred to the "best" continuous model (i.e. the one factor Cox, Ingersoll, and Ross (CIR: 1985) model) when comparing predictive accuracy for the "Stable 1990s" period that we examine. Moreover, an analogous result holds for the "Post 1990s" period that we examine, where the STAR model is preferred to a two factor stochastic mean model. Thus, when the STAR model is parameterized using only data corresponding to a particular sub-sample, it outperforms the "best" continuous alternative during that period. However, when models are estimated using the entire dataset, the continuous CHEN model is preferred, regardless of the variety of model specification (selection) test that is carried out. Given that it is very difficult to ascertain the particular future regime that will ensue when constructing ex ante predictions, thus, the CHEN model is our overall "winning" model, regardless of sample period. - ... As mentioned in the introduction, specification tests for the conditional distribution of a diffusion, when its closed form is unknown, have also been recently suggested by Aït Sahalia et al. (2009) and by Bhardwaj et al. (2008). The former test is based on the integrated mean square error of the difference between a local polynomial estimator of the conditional CDF and an exact approximation, based on a Hermite expansion, of the parametric CDF under the null, constructed with historical data. ...Article
- Jan 2009
- J ECONOMETRICS

This paper develops tests for comparing the accuracy of predictive densities derived from (possibly misspecified) diffusion models. In particular, we first outline a simple simulation-based framework for constructing predictive densities for one-factor and stochastic volatility models. We then construct tests that are in the spirit of Diebold and Mariano (1995) and White (2000). In order to establish the asymptotic properties of our tests, we also develop a recursive variant of the nonparametric simulated maximum likelihood estimator of Fermanian and Salanié (2004). In an empirical illustration, the predictive densities from several models of the one-month federal funds rates are compared. - ... There have been a few studies on testing continuous-time diffusion models for the interest rate. Some develop formal tests to check the drift and diffusion specifications jointly by using the marginal density or transitional density (e.g., Aït-Sahalia (1996a), Hong andLi (2005), Gao and King (2004), Aït-Sahalia,Fan and Peng (2008)). These tests are not suitable to test the drift model because they cannot separate the sources of misspecification upon rejection. ...Article
- Oct 2009

Continuous-time models are important for investigating interest rate term structure and pricing fixed income derivatives. Economic theory often provides little guidance on the choice of the form of continuous-time models, and existing one-factor and multi-factor continuous-time interest rate models often assume a linear drift, among other things. Some studies, based smoothed nonparametric kernel estimation, suggest that the drift of the interest rate process is nonlinear, particularly at high interest rate levels. However, this has been doubted as an artifact of smoothed nonparametric estimation in comparison with highly persistent interest rate data. Whether the drift of the interest rate process is linear or nonlinear remains an unsolved issue in the literature. In this paper, we take a new approach to re-address this important issue by first considering a general continuous-time regression for the interest rate process and then testing it via a generalized spectral derivative approach of Hong and Lee (2005) which is tailored to the continuous-time setting. Our method avoids the undesirable features of smoothed nonparametric estimation for highly persistent financial time series data. Unlike the existing approaches to testing linearity in drift, we allow for stochastic volatility and jumps, which have been well documented for the interest rate process in the literature. An empirically realistic simulation study shows that the generalized spectral derivative provides reliable inference in finite samples for continuous-time models. Based on the widely used 7-day Eurodollar rates, we document strong evidence that the interest rate process has a nonlinear drift and such evidence is robust to the presence of level effect, stochastic volatility, jumps, and different methods of drift parameter estimation. We further document that such popular nonlinear drift models as Aït-Sahalia’s (1996a) nonlinear drift model and Ahn and Gao’s (1991) Inverse-Feller drift model can capture some nonlinear drift dynamics of the short rate, and Aït-Sahalia’s nonlinear model outperforms Ahn and Gao’s nonlinear drift model due to its flexibility to capture asymmetric mean-reverting feature. However, they are still firmly rejected, indicating room for further improving the modelling of the drift function of the interest rate. - ... 5.3 we revise some testing ideas in continuous time models. In this setting, Sahalia et al. (2009) provided some tests for the transition density of a jump-diffusion process (which also apply for pure diffusions), sampled in a discrete way. The authors noted that the direct estimation of local characteristics of the process with discrete data may lead to inconsistent estimates, but the discretization procedure cannot be avoided, and in this work and in subsequent papers (already mentioned in the review) the model specification is done using densities at the observed discrete frequency. ...First for all, we would like to thank the discussants for reading our paper and fortaking time to prepare such interesting and valuable contributions. The feeling, afterrevising the discussions and going back to the original version of the paper, is thatthere is still too much to say about Goodness-of-Fit (GoF) tests for regression modelsand the discussants have given a good proof of this. Although they qualify the reviewas
- ... The alternative hypothesis is that the parameters in the above diffusion process do not coincide with the true parameters. Instead of comparing parameters directly (see AitSahalia et al. (2006)), we compare the cumulative distribution function. The null and alternative hypotheses are: ...ArticleFull-text available
- May 2009

The purpose of this paper is to add to the empirical evidence on the efficacy of alternative simulation models of the short term interest rate. This is done by constructing consistent specification tests that allow us to carry out a "horse-race" comparing various one, two, and three factor models (possibly with jumps), across multiple historical sample periods. We begin by outlining a three factor version of the simulation based specification test of Bhardwaj, Corradi and Swanson (BCS: 2008), which is based on a comparison of simulated and true conditional distributions and confidence intervals. Our evaluation involves comparing six affine models of the short rate during four historical periods, referred to as: "Post Bretton-Woods"; "Pre-1990s"; "The Stable 1990s"; and "Post 1990s". Based on the examination of Eurodollar rate data, we find that the CIR model, which is often rejected in the literature, performs best among the candidate models in "The Stable 1990s", while there is little to choose between one and two factor models when considering the "Post 1990s" period. Examination of "Pre 1990s" data, on the other hand, suggests there is little to choose between 2 and 3 factor models, and the one factor CIR model performs poorly. Moreover, under the "Post Bretton-Woods" period, the "best" performer is the three factor CHEN (1996) model examined by Andersen, Benzoni and Lund (2004). We conclude that the choice of model for simulating the future distribution of short rates is highly sample dependent. - ... There is an extensive literature on specification testing but most existing works are concentrated on the case that data of interest are directly observable. Some representative works include pseudo-likelihood ratio test [Azzalini and Bowman (1993)], square distance between parametric and nonparametric estimate [Härdle and Mammen (1993)], residualsbased tests [ Fan and Li (1996); Hong and White (1995)], generalized likelihood ratio test [Fan et al. (2001)], and density based approaches [Aït-Sahalia (1996); Gao and King (2004); Hong and Li (2005); Aït Sahalia et al. (2009)]. In the above works, direct observations from the model of interest are available, a feature unfortunately not shared by (1). ...Article
- Sep 2014
- J MULTIVARIATE ANAL

Most existing works on specification testing assume that we have direct observations from the model of interest. We study specification testing for Markov models based on contaminated observations. The evolving model dynamics of the unobservable Markov chain is implicitly coded into the conditional distribution of the observed process. To test whether the underlying Markov chain follows a parametric model, we propose measuring the deviation between nonparametric and parametric estimates of conditional regression functions of the observed process. Specifically, we construct a nonparametric simultaneous confidence band for conditional regression functions and check whether the parametric estimate is contained within the band. - ... Under DGP A5, as shown in Sahalia et al. (2006), the transition density is, at the …rst order in ; a mixture of normal distributions: ...Article
- Aug 2010

We develop a nonparametric regression-based goodness-of-fit test for multifactor continuous-time Markov models using the conditional characteristic function, which often has a convenient closed form or can be approximated accurately for many popular continuous-time Markov models in economics and finance. An omnibus test fully utilizes the information in the joint conditional distribution of the underlying processes and hence has power against a vast class of continuous-time alternatives in the multifactor framework. A class of easy-to-interpret diagnostic procedures is also proposed to gauge possible sources of model misspecification. All the proposed test statistics have a convenient asymptotic N(0, 1) distribution under correct model specification, and all asymptotic results allow for some data-dependent bandwidth. Simulations show that in finite samples, our tests have reasonable size, thanks to the dimension reduction in nonparametric regression, and good power against a variety of alternatives, including misspecifications in the joint dynamics, but the dynamics of each individual component is correctly specified. This feature is not attainable by some existing tests. A parametric bootstrap improves the finite-sample performance of proposed tests but with a higher computational cost. - ... Li and Tkacz (2006) built a consistent bootstrap test for conditional density functions with time-series. Sahalia et al. (2009) developed a nonparametric specification test for the conditional density function of a Markovian process. Both the Li and Tkacz and Aït-Sahalia et al. tests can only be used to test the conditional density (distribution) function with compact support (Assumption A.4 in Li and Tkacz 2006, Condition 2 in Aït-Sahalia et al. 2009). ...Article
- Jan 2009
- ECONOMET REV

We propose a new test for a multivariate parametric conditional distribution of a vector of variables yt given a conditional vector xt. The proposed test is shown to have an asymptotic normal distribution under the null hypothesis, while being consistent for all fixed alternatives, and having non-trivial power against a sequence of local alternatives. Monte Carlo simulations show that our test has reasonable size and good power for both univariate and multivariate models, even for highly persistent dependent data with sample sizes often encountered in empirical finance. - ... It also requires being small to ensure the accuracy of the approximation. While specification testing on the diffusion process can be carried out by testing the transitional density function (Chen, Gao, and Tang, 2008;A¨ıtA¨ıt-Sahalia, Fan, and Peng, 2009), rejection via testing the transitional density specification may not provide information on which part of the process, the drift or the diffusion, is misspecified. In addition, correct parametric specification of the transition density function does not necessarily imply an explicit parametric form for each of the drift and diffusion functions. ...ArticleFull-text available
- Aug 2011

This paper proposes a nonparametric simultaneous test for parametric specification of the conditional mean and variance functions in a time series regression model. The test is based on an empirical likelihood (EL) statistic that measures the goodness of fit between the parametric estimates and the nonparametric kernel estimates of the mean and variance functions. A unique feature of the test is its ability to distribute natural weights automatically between the mean and the variance components of the goodness-of-fit measure. To reduce the dependence of the test on a single pair of smoothing bandwidths, we construct an adaptive test by maximizing a standardized version of the empirical likelihood test statistic over a set of smoothing bandwidths. The test procedure is based on a bootstrap calibration to the distribution of the empirical likelihood test statistic. We demonstrate that the empirical likelihood test is able to distinguish local alternatives that are different from the null hypothesis at an optimal rate. - Article
- Aug 2011
- ANN I STAT MATH

We derive nonparametric tests of symmetry using asymmetric kernels with either shrinking or fixed bandwidths. We show how to extend the approach to examine conditional symmetry by deriving conditions under which our tests are applicable to residuals from semiparametric models with a (sufficiently smooth) nonparametric link function. As a by-product, we prove the consistency of the asymmetric kernel estimator of the derivative of the density function. Simulations show that the asymptotic tests perform well even in very small samples, entailing better size and power properties than some of the existing symmetry tests. - Article
- Apr 2013
- OPER RES

Jump-diffusion processes are ubiquitous in finance and economics. They arise as models of security, energy and commodity prices, exchange and interest rates, and default timing. This paper develops a method for the exact simulation of a skeleton, a hitting time and other functionals of a one-dimensional jump-diffusion with state-dependent drift, volatility, jump intensity and jump size. The method requires the drift function to be C¹, the volatility function to be C², and the jump intensity function to be locally bounded. No further structure is imposed on these functions. The method leads to unbiased simulation estimators of security prices, transition densities, hitting probabilities, and other quantities. Numerical results illustrate its features. - Article
- Oct 2011
- J ECONOMETRICS

We develop an omnibus specification test for multivariate continuous-time models using the conditional characteristic function, which often has a convenient closed-form or can be accurately approximated for many multivariate continuous-time models in finance and economics. The proposed test fully exploits the information in the joint conditional distribution of underlying economic processes and hence is expected to have good power in a multivariate context. A class of easy-to-interpret diagnostic procedures is supplemented to gauge possible sources of model misspecification. Our tests are also applicable to discrete-time distribution models. Simulation studies show that the tests provide reliable inference in finite samples. - Article
- Apr 2011
- J Empir Finance

Recent studies of conditional factor models do not specify conditioning information but use data from small windows to estimate the time series of conditional alphas and betas. In this paper, we propose a nonparametric method using an optimal window to estimate time-varying coefficients. In addition, we offer two empirical tests of a conditional factor model. Using our new method, we examine the performance of the conditional CAPM and the conditional Fama-French three-factor model in explaining the return variations of portfolios sorted by size, book-to-market ratios, and past returns, for which recent literature has generated controversial results. We find that, although in general the conditional FF model outperforms the conditional CAPM, both models fail to explain well-known asset-pricing anomalies. Moreover, for both models, the failure is more pronounced for the equally-weighted portfolios than for the value-weighted ones. - Article
- Aug 2019
- OXFORD B ECON STAT

This paper proposes a simultaneous test for the specification of the conditional mean and conditional variance functions as well as the error distribution in nonlinear time series models. Constructed by comparing two density estimators for the response variable, the proposed test has a Gumbel‐limiting distribution under the null hypothesis and is consistent against a general class of alternative hypotheses. A parametric bootstrap procedure is proposed for practical implementation, and is shown to perform well in extensive simulations. The application to the continuous time diffusion model is illustrated via an analysis on the U.S. Federal fund rate data. - Chapter
- Feb 2009

In this chapter, we deal with nonparametric methods for discretely observed financial data. The main ideas of nonparametric kernel smoothing are explained in the rather simple situation of density estimation and regression. For financial data, a rather relevant topic is nonparametric estimation of a volatility function in a continuous-time model such as a homogeneous diffusion model. We review results on nonparametric estimation for discretely observed processes, sampled at high or at low frequency. We also discuss application of nonparametric methods to testing, especially model validation and goodness-of-fit testing. In risk measurement for financial time series, conditional quantiles play an important role and nonparametric methods have been successfully applied in this field too. At the end of the chapter we discuss Grenander’s sieve methods and other more recent advanced nonparametric approaches. - Article
- Apr 2009
- REV ECON STUD

This paper introduces a new class of parameter estimators for dynamic models, called simulated non-parametric estimators (SNEs). The SNE minimizes appropriate distances between non-parametric conditional (or joint) densities estimated from sample data and non-parametric conditional (or joint) densities estimated from data simulated out of the model of interest. Sample data and model-simulated data are smoothed with the same kernel, which considerably simplifies bandwidth selection for the purpose of implementing the estimator. Furthermore, the SNE displays the same asymptotic efficiency properties as the maximum-likelihood estimator as soon as the model is Markov in the observable variables. The methods introduced in this paper are fairly simple to implement, and possess finite sample properties that are well approximated by the asymptotic theory. We illustrate these features within typical estimation problems that arise in financial economics. - Article
- Jan 2009

As an extension of the article by NÃºÃ±ez, De la Cruz and Ortega (2007), different parametric models with jumps are tested with the methodology developed by Ait-Sahalia and Peng (2006), based on the transition function. Data analyzed are the peso-dollar exchange rate. The idea is to implement continuous-time parametric models for the peso-dollar exchange rate. The results confirm that the proposed continuous time models are not good enough to explain the behavior that describes the peso-dollar exchange rate. However, considering some continuous time models with Poisson jumps is possible to describe such behavior. - Article
- Mar 2010
- J ECONOMETRICS

This article studies density and parameter estimation problems for nonlinear parametric models with conditional heteroscedasticity. We propose a simple density estimate that is particularly useful for studying the stationary density of nonlinear time series models. Under a general dependence structure, we establish the root n consistency of the proposed density estimate. For parameter estimation, a Bahadur type representation is obtained for the conditional maximum likelihood estimate. The parameter estimate is shown to be asymptotically efficient in the sense that its limiting variance attains the Cramér-Rao lower bound. The performance of our density estimate is studied by simulations. - ArticleFull-text available
- Jan 2010

Testing for structural breaks and identifying their location is essential for econometric modeling. In this paper, a Hidden Markov Model (HMM) approach is used in order to perform these tasks. Breaks are defined as the data points where the underlying Markov Chain switches from one state to another. The estimation of the HMM is conducted using a variant of the Iterative Conditional Expectation-Generalized Mixture (ICE-GEMI) algorithm proposed by Delignon et al. (1997), that permits analysis of the conditional distributions of economic data and allows for different functional forms across regimes. The locations of the breaks are subsequently obtained by assigning states to data points according to the Maximum Posterior Mode (MPM) algorithm. The Integrated Classification Likelihood-Bayesian Information Criterion (ICL-BIC) allows for the determination of the number of regimes by taking into account the classification of the data points to their corresponding regimes. The performance of the overall procedure, denoted IMI by the initials of the component algorithms, is validated by two sets of simulations; one in which only the parameters are permitted to differ across regimes, and one that also permits differences in the functional forms. The IMI method performs well in both sets. Moreover, when it is compared to the Bai and Perron (1998) method its performance is superior in the assessing the number of breaks and their respective locations. Finally, the methodology is applied for the detection of breaks in the monetary policy of United States, the di erent functional form being variants of the Taylor (1993) rule. - Article
- Dec 2011
- COMPUT STAT DATA AN

A new test for the goodness of fit of parametric forms of the drift and volatility functions of interest rate models is proposed. The test is based on a marked empirical process of the residuals. More specifically, a marked empirical process is constructed using estimators of the integrated regression function and the integrated conditional variance function for the drift function and the volatility function, respectively. Distributions of these processes are approximated using bootstrap techniques. This test is then applied to simulated classical financial models and is illustrated in an empirical application to a EURIBOR data set. - Article
- Feb 2013
- BERNOULLI

Markov processes are used in a wide range of disciplines including finance. The transitional densities of these processes are often unknown. However, the conditional characteristic functions are more likely to be available especially for Lévy driven processes. We propose an empirical likelihood approach for estimation and model specification test based on the conditional characteristic function for processes whose sample paths can be either continuous or discontinuous with jumps. An empirical likelihood estimator for the parameter of a parametric process, and a smoothed empirical likelihood ratio test for the parametric specification of the process are proposed, which are shown to have good theoretical properties and empirical performance. Simulations and empirical case study are carried out to confirm the effectiveness of the estimator and the test. - Article
- Jan 2010

In this study, two GMM type estimators of drift parameters are proposed for both univariate and multivariate semi-parametric diffusion models with unrestricted volatility. The conditional moment restriction, through which the estimators are constructed, follows from a characterization of diffusion processes based on the infinitesimal operator, which is equivalent to transition density in terms of identifying the complete dynamics. The infinitesimal operator of the diffusion process enjoys the nice property of being a closed-form expression of drift and diffusion terms in an essentially separate manner, which makes the proposed estimators robust to the mispecification of the diffusion function. The first estimator is obtained by integrating out the diffusion function via the quadratic variation (covariation), which is estimated by the realized volatility(covariance) in a first step using high frequency data. The second estimator is constructed based on the separate identification condition and is actually applicable for a general instantaneous conditional mean model in continuous time, which covers the stochastic volatility and jump diffusion models as special cases. Our estimators for both univariate and multivariate models are unified in the same framework and particularly easy-to-implement. Simulation studies show that they possess fairly good finite sample performances. - Article
- Mar 2013
- J ECONOMETRICS

We develop a nonparametric test to check whether a process can be represented by a stochastic differential equation driven only by a Brownian motion. Our testing procedure utilizes the infinitesimal operator-based martingale characterization combined with a generalized spectral approach. Such a testing procedure is feasible and convenient because the infinitesimal operator of the diffusion process has a closed-form expression. The proposed test is applicable to both univariate and multivariate processes and has an N(0,1)N(0,1) limit distribution under the diffusion hypothesis. Simulation and empirical studies show that the proposed test has reasonable performance in small samples. - ArticleFull-text available
- Feb 2018

The hallmark of bipolar disorder is a clinical course of recurrent manic and depressive symptoms of varying severity and duration. Mathematical modeling of bipolar disorder holds the promise of an ability to personalize diagnoses, to predict future mood episodes, to directly compare diverse datasets, and to link basic mechanisms to behavioral data. Several modeling frameworks have been proposed for bipolar disorder, which represent competing hypothesis about the basic framework of the disorder. Here, we test these hypotheses with self-report assessments of mania and depression symptoms from 178 bipolar patients followed prospectively for 4 or more years. Statistical analysis of the data did not support the hypotheses that mood arises from a rhythmic process or multiple stable states (e.g., mania or depression) or that manic and depressive symptoms are highly anti-correlated. Alternatively, it is shown that bipolar disorder could arise from an inability for mood to quickly return to normal when perturbed. This latter concept is embodied by an affective instability model that can be personalized to the clinical course of any individual with chronic disorders that have an affective component. - Article
- Jan 2017
- J Empir Finance

This paper develops nonparametric specification tests for stochastic volatility models by comparing the nonparametically estimated return density and distribution functions with their parametric counterparts. Asymptotic null distributions of the tests are derived and the tests are shown to be consistent. Extensive Monte Carlo experiments are performed to study the finite sample properties of the tests. The proposed tests are applied in a number of empirical examples. - Article
- Oct 2015
- J BUS ECON STAT

This article proposes a simulation-based density estimation technique for time series that exploits information found in covariate data. The method can be paired with a large range of parametric models used in time series estimation. We derive asymptotic properties of the estimator and illustrate attractive finite sample properties for a range of well-known econometric and financial applications. - ArticleMuch of economic theory, especially macro-economics and the study of commodity, financial, and other markets, relies on the use of non-linear structural models to study medium-term and long-run dynamic behaviour of an economy. Continuous-time econometrics is based on the argument that as economic systems are largely continuous they can be better represented and estimated by differential equation rather than difference equation systems. This paper reviews the development of full-information Gaussian estimators of non-linear systems which may then be extended to the estimation of models of intertemporally optimizing agents and other boundary point models, and models where the parameters of the stochastic innovation process enter the deterministic part of the model or vice versa. The long-properties of these models may be studied by calculating the Lyapunov exponents which give information on the form of the attractor the model, the dynamic stability of the model for given parameter values and whether it is structurally stable. The critical dependence of some attractors, and particularly strange attractors, on parameter values emphasizes the need for consistent, efficient estimation. A structural approach provides a rigorous alternative to using single time series to determine whether economic systems exhibit aperiodic or chaotic dynamical behavior.
- Article
- Mar 2015
- J ECONOMETRICS

This paper develops a specification test for stochastic volatility models by comparing the nonparametric kernel deconvolution density estimator of an integrated volatility density with its parametric counterpart. distance is used to measure the discrepancy. The asymptotic null distributions of the test statistics are established and the asymptotic power functions are computed. Through Monte Carlo simulations, the size and power properties of the test statistics are studied. The tests are applied to an empirical example. - Article
- Jul 2014

In this paper, a generalized residual goodness of fit test is proposed as an improvement on an existing test method for one-dimension diffusion models. By comparative analysis, we find that the generalized residual goodness of fit test is an effective test method. When diffusion models are used to describe stock prices, interest rates and exchange rates, the parameters of their drift function and diffusion function will change over time rather than be constants due to changeful economic environment. Based on this idea, we propose a diffusion model with segmented time-varying parameters. The test proposed above is used to examine the segmented time-varying characteristic of the model parameters. - Article
- Dec 2007
- TEST

The advance of technology facilitates the collection of statistical data. Flexible and refined statistical models are widely sought in a large array of statistical problems. The question arises frequently whether or not a family of parametric or nonparametric models fit adequately the given data. We give a selective overview on nonparametric inferences using generalized likelihood ratio (GLR) statistics. We introduce generalized likelihood ratio statistics to test various null hypotheses against nonparametric alternatives. The trade-off between the flexibility of alternative models and the power of the statistical tests is emphasized. Well-established Wilks’ phenomena are discussed for a variety of semi- and nonparametric models, which sheds light on other research using GLR tests. A number of open topics worthy of further study are given in a discussion section. - Article
- Jul 2014
- BERNOULLI

We study the problem of the efficient estimation of the jumps for stochastic processes. We assume that the stochastic jump process $(X_t)_{t\in[0,1]}$ is observed discretely, with a sampling step of size $1/n$. In the spirit of Hajek's convolution theorem, we show some lower bounds for the estimation error of the sequence of the jumps $(\Delta X_{T_k})_k$. As an intermediate result, we prove a LAMN property, with rate $\sqrt{n}$, when the marks of the underlying jump component are deterministic. We deduce then a convolution theorem, with an explicit asymptotic minimal variance, in the case where the marks of the jump component are random. To prove that this lower bound is optimal, we show that a threshold estimator of the sequence of jumps $(\Delta X_{T_k})_k$ based on the discrete observations, reaches the minimal variance of the previous convolution theorem. - Article
- Dec 2012
- COMPUT STAT DATA AN

We develop a hypothesis testing approach to checking model misspecification on parametric structures in continuous-time stochastic diffusion models. The key idea behind the development of our test statistic is rooted in a ratio of two types of information matrices, the negative sensitivity matrix and the variability matrix, in the context of martingale estimating equations. We propose a bootstrap resampling method to implement numerically the proposed diagnostic procedure. Through intensive simulation studies, we compare the proposed approach with several currently popular methods and show that our approach is advantageous in the aspects of type I error control, power improvement as well as computational efficiency. Two real-world data examples are included to illustrate the practical use of our proposed testing procedure. - Article
- May 2019
- J AM STAT ASSOC

This article develops statistical tools for testing conditional independence among the jump components of the daily quadratic variation, which we estimate using intraday data. To avoid sequential bias distortion, we do not pretest for the presence of jumps. If the null is true, our test statistic based on daily integrated jumps weakly converges to a Gaussian random variable if both assets have jumps. If instead at least one asset has no jumps, then the statistic approaches zero in probability. We show how to compute asymptotically valid bootstrap-based critical values that result in a consistent test with asymptotic size equal to or smaller than the nominal size. Empirically, we study jump linkages between US futures and equity index markets. We find not only strong evidence of jump cross-excitation between the SPDR exchange-traded fund and E-mini futures on the S&P 500 index, but also that integrated jumps in the E-mini futures during the overnight period carry relevant information. Supplementary materials for this article are available as an online supplement. - ArticleFull-text availableThis paper makes three contributions. First, we develop a simple simulation based framework for construct- ing predictive densities for one-factor and stochastic volatility diusion processes (at arbitrary prediction horizons), that is suitable for the case in which the functional form of the conditional distribution is not known. Second, we outline a simulation and bootstrap based methodology that enables one to construct tests for pairwise and multiple comparison of possibly misspecied diusion processes, in terms of their out sample predictive ability. The tests involve comparing the distributional mean square error (MSE) of alternative models, along the lines of Diebold-Mariano and White's Reality Check tests. Third, we establish consistency and asymptotic normality of Simulated Generalized Method of Moments and Nonparametric Simulated Quasi Maximum Likelihood estimators, and therst order validity of their bootstrap analogs, in a recursive setting. This enables one to construct asymptotically valid bootstrap based critical values, which properly take into account the location bias due to recursive estimation and the contribution of parameter estimation error. As an empirical illustration, several predictive density models for the one-month Eurodollar deposit rate are compared.
- ArticleFull-text available
- Jan 2005

This paper makes two contributions. First, we develop a simple simulation based framework for constructing predictive densities for one-factor and stochastic volatility diusion processes (at ar- bitrary prediction horizons), that is suitable for the case in which the functional form of the condi- tional density is not known. Second, we outline a simulation and bootstrap based methodology that yields tests for pairwise comparison as well as tests for multiple comparison of possibly misspecified diusion processes, in terms of their out sample predictive ability. - We discuss nonparametric tests for parametric specifications of regression quantiles. The test is based on the comparison of parametric and nonparametric fits of these quantiles. The nonparametric fit is a Nadaraya-Watson quantile smoothing estimator. An asymptotic treatment of the test statistic requires the development of new mathematical arguments. An approach that makes only use of plugging in a Bahadur expansion of the nonparametric estimator is not satisfactory. It requires too strong conditions on the dimension and the choice of the bandwidth. Our alternative mathematical approach requires the calculation of moments of Bahadur expansions of Nadaraya-Watson quantile regression estimators. This calculation is done by inverting the problem and application of higher order Edgeworth expansions. The moments allow estimation bounds for the accuracy of Bahadur expansions for integrals of kernel quantile estimators. Another application of our method gives asymptotic results for the estimation of weighted averages of regression quantiles.
- ArticleFull-text available
- Jan 2011

In applied density estimation problems, one often has data not only on the target variable, but also on a collection of covariates. In this paper, we study a density estimator that incorporates this additional information by combining parametric estimation and conditional Monte Carlo. We prove an approximate functional asymptotic normality result that illustrates convergence rates and the asymptotic variance of the estimator. Through simulation, we illustrate the strength of its finite sample properties in a number of standard econometric and financial applications. - Article
- Oct 2012
- ECONOMET J

We propose a nonparametric approach to the estimation and testing of structural change in time series regression models. Under the null of a given set of the coefficients being constant, we develop estimators of both the nonparametric and parametric components. Given the estimators under null and alternative, generalized F and Wald tests are developed. The asymptotic distributions of the estimators and test statistics are derived. A simulation study examines the fi?nite-sample performance of the estimators and tests. The techniques are employed in the analysis of structural change in US productivity and the Eurodollar term structure. - Article
- Sep 2012
- J ECONOMETRICS

This paper gauges volatility transmission between stock markets by testing conditional independence of their volatility measures. In particular, we check whether the conditional density of the volatility changes if we further condition on the volatility of another market. We employ nonparametric methods to estimate the conditional densities and model-free realized measures of volatility, allowing for both microstructure noise and jumps. We establish the asymptotic normality of the test statistic as well as the first-order validity of the bootstrap analog. Finally, we uncover significant volatility spillovers between the stock markets in China, Japan, UK and US. (c) 2012 Elsevier B.V. All rights reserved. - Article
- Jan 2014
- J ECONOMETRICS

Modeling conditional distributions in time series has attracted increasing attention in economics and finance. We develop a new class of generalized Cramer–von Mises (GCM) specification tests for time series conditional distribution models using a novel approach, which embeds the empirical distribution function in a spectral framework. Our tests check a large number of lags and are therefore expected to be powerful against neglected dynamics at higher order lags, which is particularly useful for non-Markovian processes. Despite using a large number of lags, our tests do not suffer much from loss of a large number of degrees of freedom, because our approach naturally downweights higher order lags, which is consistent with the stylized fact that economic or financial markets are more affected by recent past events than by remote past events. Unlike the existing methods in the literature, the proposed GCM tests cover both univariate and multivariate conditional distribution models in a unified framework. They exploit the information in the joint conditional distribution of underlying economic processes. Moreover, a class of easy-to-interpret diagnostic procedures are supplemented to gauge possible sources of model misspecifications. Distinct from conventional CM and Kolmogorov–Smirnov (KS) tests, which are also based on the empirical distribution function, our GCM test statistics follow a convenient asymptotic N(0,1)N(0,1) distribution and enjoy the appealing “nuisance parameter free” property that parameter estimation uncertainty has no impact on the asymptotic distribution of the test statistics. Simulation studies show that the tests provide reliable inference for sample sizes often encountered in economics and finance. - ArticleFull-text available
- Apr 2007

This paper makes two contributions. First, we develop a simple simulation based framework for constructing predictive densities for one-factor and stochastic volatility diffusion processes (at arbitrary prediction hori-zons), that is suitable for the case in which the functional form of the conditional distribution is not known. Second, we outline a simulation and bootstrap based methodology that enables one to construct tests for pairwise and multiple comparison of possibly misspecified diffusion processes, in terms of their out sample predictive ability. The tests involve comparing the distributional mean square error (MSE) of alternative models; and are based in principle on the point model selection methodologies discussed in Diebold and Mariano (1995), White (2000), and Corradi and Swanson (2007a). Critical values for the tests are based on block bootstrap statistics that are adjusted for a location bias arising due to recursive estimation; and that are constructed using versions of recursive simulated generalized method of moments and nonparamet-ric simulated quasi maximum likelihood parameter estimators that are recentered in order to ensure that paremeter estimation error is properly captured. JEL classification: C22, C51. Keywords: block bootstrap, diffusion processes, nonparametric simulated quasi maximum likelihood, param-eter estimation error, recursive estimation, simulated generalized method of moments, stochastic volatility. - Article
- Feb 2012

Abstract The Markov property is a fundamental,property in time series analysis and is often assumed in economic,and …nancial modelling.,We develop,a test for the Markov,property,using the conditional characteristic function embedded in a frequency domain approach, which checks the implication of the Markov property,in every conditional moment,(if exist) and over many,lags. The proposed,test is applicable to both univariate and multivariate time series with discrete or continuous distributions. Simulation studies show that with the use of a smoothed,nonparametric transition density-based bootstrap procedure, the proposed test has reasonable sizes and all- around,power against non-Markov alternatives in …nite samples. We apply the test to a number of high-frequency …nancial time series and …nd strong evidence against the Markov property. Key words: Markov property, Conditional characteristic function, Generalized cross-spectrum, - Article
- Jan 2008

Financial econometrics has become an increasingly popular research field. In this paper we review a few parametric and nonparametric models and methods used in this area. After introducing several widely used continuous-time and discrete-time models, we study in detail dependence structures of discrete samples, including Markovian property, hidden Markovian structure, contaminated observations, and random samples. We then discuss several popular parametric and nonparametric estimation methods. To avoid model mis-specification, model validation plays a key role in financial modeling. We discuss several model validation techniques, including pseudo-likelihood ratio test, nonparametric curve regression based test, residuals based test, generalized likelihood ratio test, simultaneous confidence band construction, and density based test. Finally, we briefly touch on tools for studying large sample properties.

- Article
- Jun 1996
- J AM STAT ASSOC

Traditional nonparametric tests, such as the Kolmogorov—Smirnov test and the Cramér—Von Mises test, are based on the empirical distribution functions. Although these tests possess root-n consistency, they effectively use only information contained in the low frequencies. This leads to low power in detecting fine features such as sharp and short aberrants as well as global features such as high-frequency alternations. The drawback can be repaired via smoothing-based test statistics. In this article we propose two such kind of test statistics based on the wavelet thresholding and the Neyman truncation. We provide extensive evidence to demonstrate that the proposed tests have higher power in detecting sharp peaks and high frequency alternations, while maintaining the same capability in detecting smooth alternative densities as the traditional tests. Similar conclusions can be made for two-sample nonparametric tests of distribution functions. In that case, the traditional linear rank tests such as the Wilcoxon test and the Fisher—Yates test have low power in detecting two nearby densities where one has local features or contains high-frequency components, because these procedures are essentially testing the uniform distribution based on the sample mean of rank statistics. In contrast, the proposed tests use more fully the sampling information and have better ability in detecting subtle features. - Article
- Jun 1996
- J AM STAT ASSOC

Traditional nonparametric tests, such as the Kolmogorov-Smirnov test and the Cramér-Von Mises test, are based on the empirical distribution functions. Although these tests possess root-n consistency, they effectively use only information contained in the low frequencies. This leads to low power in detecting fine features such as sharp and short aberrants as well as global features such a high-frequency alternations. The drawback can be repaired via smoothing-based test statistics. We propose two such kinds of test statistics based on the wavelet thresholding and the Neyman truncation. We provide extensive evidence to demonstrate that the proposed tests have higher power in detecting sharp peaks and high frequency alternations, while maintaining the same capability in detecting smooth alternative densities as the traditional tests. Similar conclusions can be made for two-sample nonparametric tests of distribution functions. In that case, the traditional linear rank tests such as the Wilcoxon test and the Fisher-Yates test have low power in detecting two nearby densities where one has local features or contains high-frequency components, because these procedures are essentially testing the uniform distribution based on the sample mean of rank statistics. In contrast, the proposed tests use more fully the sampling information and have better ability in detecting subtle features. - ArticleFull-text availableWe extend the adaptive and rate-optimal test of Horowitz and Spokoiny (2001) for specication of parametric regression models to weakly dependent time series regression models with an empir- ical likelihood formulation of our test statistic. It is found that the proposed adaptive empirical likelihood test preserves the rate-optimal property of the test of Horowitz and Spokoiny (2001).
- Article
- Mar 2000
- J PORTFOLIO MANAGE

The Chicago Board Options Exchange's Market Volatility Index (VIX) is called the "investor fear gauge." To understand why, it is necessary to understand the index's construction. To understand how VIX performs its role, it is necessary to examine its history and its relation to stock market returns. In this article, the author describes the construction of the volatility index and examines its movements over the past fourteen years. - Article
- Jan 1993
- Math Meth Stat

This expository paper treats asymptotically minimax problems of testing a simple hypothesis against nonparametric alternatives obtained by removing some neighborhoods of the hypothesis from a given infinite-dimensional set of probability measures. We consider the problems of detecting a signal in a Gaussian white noise and of testing for uniformity of a probability density. In Part I asymptotically exact conditions of distinguishability are stated for alternatives described by function sets of certain degree of smoothness with a removed L p -ball. For alternatives described by ellipsoids in l p with a removed l p -ball, the exact asymptotics of the minimax risk and of the minimax error probabilities along with the form of asymptotically minimax tests are presented. In Part II the methods of obtaining lower bounds for the minimax risk, and in Part III the methods of construction of asymptotically minimax tests are described. The major part of results is due to the author. For the results stated in Part I, in Part II and Part III the proofs or detailed outlines of the proofs are given. - Article
- Jan 1995

When estimating a mean regression function and its derivatives, locally weighted least squares regression has proven to be a very attractive technique. The present paper focuses on the important issue of how to select the smoothing parameter or bandwidth. In the case of estimating curves with a complicated structure, a variable bandwidth is desirable. Furthermore, the bandwidth should be indicated by the data themselves. Recent developments in nonparametric smoothing techniques inspired us to propose such a data-driven bandwidth selection procedure, which can be used to select both constant and variable bandwidths. The idea is based on a residual squares criterion along with a good approximation of the bias and variance of the estimator. The procedure can be applied to select bandwidths not only for estimating the regression curve but also for estimating its derivatives. The resulting estimation procedure has the necessary flexibility for capturing complicated shapes of curves. This is illustrated via a large variety of testing examples, including examples with a large spatial variability. The results are also compared with wavelet thresholding techniques, and it seems that our results are at least comparable, i.e. local polynomial regression using our data-driven variable bandwidth has spatial adaptation properties that are similar to wavelets. - ArticleFull-text available
- Jan 2003

Summary The conditional density function is very useful for forecasting and statistical inferences, particularly in financial economics. Yet, its bandwidth selection has not yet been sys- tematically studied. In this article, we extend the idea of cross-validation (CV) for choosing the smoothing parameter of the "double-kernel" local linear regression for es- timating a conditional density. Our selection rule optimizes the estimated conditional density function by minimizing the integrated square error (ISE). We also discuss three other bandwidth selection rules. The first is an ad-hoc method used by Fan, Yao and Tong (FYT, 1996). The second rule, as suggested by Hall, Wol and Yao (HWY, 1999), employs the idea of bootstrap for the bandwidth selection in the estimation of condi- tional distribution functions. We modify the HWY approach to suit the bandwidth selection for the conditional density function. The last is the rule of thumb approach proposed by Hyndman and Yao (2002). The performance of the newly proposed CV approach is compared with these three methods by simulation studies, and our method performs outstandingly. The method is also illustrated by application to two sets of time series. - Article
- Dec 1995

Local least squares kernel regression provides an appealing solution to the nonparametric regression, or "scatterplot smoothing," problem, as demonstrated by Fan, for example. The practical implementation of any scatterplot smoother is greatly enhanced by the availability of a reliable rule for automatic selection of the smoothing parameter. In this article we apply the ideas of plug-in bandwidth selection to develop strategies for choosing the smoothing parameter of local linear squares kernel estimators. Our results are applicable to odd-degree local polynomial fits and can be extended to other settings, such as derivative estimation and multiple nonparametric regression. An implementation in the important case of local linear fits with univariate predictors is shown to perform well in practice. A by-product of our work is the development of a class of nonparametric variance estimators, based on local least squares ideas, and plug-in rules for their implementation. - ArticleFull-text availableThis paper evaluates the use of the nonparametric kernel method for testing specification of diffusion models as originally considered in At-Sahalia (1996). A serious doubt on the ability of the kernel method for diffusion model testing has been cast in Pritsker (1998), who observes severe size distortion of the test proposed by At-Sahalia and finds that 2755 years of data are required in order for the kernel density estimator to attain a level of accuracy achieved with 22 years of independent data. We introduce in this paper a set of additional measures to the kernel method and show that the severe size distortion observed by Pritsker (1998) can be overcome by implementing these measures. Our simulation for both the Vasicek and Cox-Ingersoll-Ross diffusion models indicates that the proposed test has reasonable size and power under various degrees of data persistence for as little as 10 years of data. We apply the proposed test to a monthly Federal Fund rate data and find there are empirical supports for some of the one-factor diffusion models proposed in the literature.
- Article
- Sep 1982
- PROBAB THEORY REL

We study the estimation of a regression function by the kernel method. Under mild conditions on the window, the bandwidth and the underlying distribution of the bivariate observations {(X i , Y i)}, we obtain the weak and strong uniform convergence rates on a bounded interval. These results parallel those of Silverman (1978) on density estimation and extend those of Schuster and Yakowitz (1979) and Collomb (1979) on regression estimation. - A data-based local bandwidth selector is proposed for nonparametric regression by local fitting of polynomials. The estimator, called the empirical-bias bandwidth selector (EBBS), is lather simple and easily allows multivariate predictor variables and estimation of any order derivative of the regression function. EBBS minimizes an estimate of mean squared error consisting of a squared bias term plus a variance term. The variance term used is exact, not asymptotic, though it involves the conditional variance of the response given the predictors that must be estimated. The bias term is estimated empirically, not from an asymptotic expression. Thus EBBS is similar to the ''double smoothing'' approach of Hardle, Hall, and Marron and a local bandwidth selector of Schucany, but is developed here for a far wider class of estimation problems than what those authors considered. EBBS is tested on simulated data, and its performance seems quite satisfactory. Local polynomial smoothing of a histogram is a highly effective technique for density estimation, and several of the examples involve density estimation by EBBS applied,to binned data.
- Article
- Jun 2003

This paper discusses specification tests for diffusion processes. In the one-dimensional case, our proposed test is closest to the nonparametric test of Aı̈t-Sahalia (Rev. Financ. Stud. 9 (1996) 385). However, we compare CDFs instead of densities. In the multidimensional and/or multifactor case, our proposed test is based on comparison of the empirical CDF of actual data and the empirical CDF of simulated data. Asymptotically valid critical values are obtained using an empirical process version of the block bootstrap which accounts for parameter estimation error. An example based on a simple version of the Cox et al. (Econometrica 53 (1985) 385) model is outlined and related Monte Carlo experiments are carried out. - We suggest two improved methods for conditional density estimation. The first is based on locally fitting a log-linear model, and is in the spirit of recent work on locally parametric techniques in density estimation. The second method is a constrained local polynomial estimator. Both methods always produce non-negative estimators. We propose an algorithm suitable for selecting the two bandwidths for either estimator. We also develop a new bootstrap test for the symmetry of conditional density functions. The proposed methods are illustrated by both simulation and application to a real data set.
- Book
- Jan 2003

Introduction.- Stationary Time Series.- Smoothing in Time Series.- ARMA Modeling and Forecasting.- Parametric Nonlinear Time Series Models.- Nonparametric Models.- Hypothesis Testing.- Continuous Time Models in Finance.- Nonlinear Prediction. - Book
- Jan 1986

Incluye bibliografía e índice - Article
- Jun 1952
- Ann Math Stat

The statistical problem treated is that of testing the hypothesis that $n$ independent, identically distributed random variables have a specified continuous distribution function $F(x)$. If $F_n(x)$ is the empirical cumulative distribution function and $\psi(t)$ is some nonnegative weight function $(0 \leqq t \leqq 1)$, we consider $n^{\frac{1}{2}} \sup_{-\infty<x<\infty} \{| F(x) - F_n(x) | \psi^\frac{1}{2}\lbrack F(x) \rbrack\}$ and $n\int^\infty_{-\infty}\lbrack F(x) - F_n(x) \rbrack^2 \psi\lbrack F(x)\rbrack dF(x).$ A general method for calculating the limiting distributions of these criteria is developed by reducing them to corresponding problems in stochastic processes, which in turn lead to more or less classical eigenvalue and boundary value problems for special classes of differential equations. For certain weight functions including $\psi = 1$ and $\psi = 1/\lbrack t(1 - t) \rbrack$ we give explicit limiting distributions. A table of the asymptotic distribution of the von Mises $\omega^2$ criterion is given. - Article
- Oct 2004
- ANN STAT

We provide a general method to analyze the asymptotic properties of a variety of estimators of continuous time diffusion processes when the data are not only discretely sampled in time but the time separating successive observations may possibly be random. We introduce a new operator, the generalized infinitesimal generator, to obtain Taylor expansions of the asymptotic moments of the estimators. As a special case, our results apply to the situation where the data are discretely sampled at a fixed nonrandom time interval. We include as specific examples estimators based on maximum-likelihood and discrete approximations such as the Euler scheme. - Article
- Jun 2002
- ANN STAT

This paper investigates the statistical relationship of the GARCH model and its diffusion limit. Regarding the two types of models as two statistical experiments formed by discrete observations from the models, we study their asymptotic equivalence in terms of Le Cam's deficiency distance. To our surprise, we are able to show that the GARCH model and its diffusion limit are asymptotically equivalent only under deterministic volatility. With stochastic volatility, due to the difference between the structure with respect to noise propagation in their conditional variances, their likelihood processes asymptotically behave quite differently, and thus they are not asymptotically equivalent. This stochastic nonequivalence discredits a general belief that the two types of models are asymptotically equivalent in all respects and warns against the common financial practice that applies statistical inferences derived under the GARCH model to its diffusion limit. - Article
- Apr 1999
- BERNOULLI

We study the problem of testing a simple hypothesis for a nonparametric ''signal + white-noise'' model. It is assumed under the null hypothesis that the ''signal'' is completely specified, e.g., that no signal is present. This hypothesis is tested against a composite alternative of the following form: the underlying function (the signal) is separated away from the null in the L<sub>2</sub> norm and, in addition, it possesses some smoothness properties. We focus on the case of an inhomogeneous alternative when the smoothness properties of the signal are measured in an L<sub>p</sub> norm with p<2. We consider tests whose errors have probabilities which do not exceed prescribed values and we measure the quality of testing by the minimal distance between the null and the alternative set for which such testing is still possible. We evaluate the optimal rate of decay for this distance to zero as the noise level tends to zero. Then a rate-optimal test is proposed which essentially uses a pointwise-adaptive estimation procedure. - Article
- Jan 1997
- Rev Financ Stud

I study the finite sample distribution of one of Aït-Sahalia's (1996c) nonparametric tests of continuous-time models of the short-term riskless rate. The test rejects true models too often because interest rate data are highly persistent but the asymptotic distribution of the test (and of the kernel density estimator on which the test is based) treats the data as if it were independently and identically distributed. To attain the accuracy of the kernel density estimator implied by its asymptotic distribution with 22 years of data generated from the Vasicek model in fact requires 2755 years of data. - Using locally polynomial regression, we develop nonparametric estimators for the conditional density function and its square root, and their partial derivatives. Two measures of sensitivity to initial conditions in nonlinear stochastic dynamic systems are proposed, one of which relates Fisher information with initial-value sensitivity in dynamical systems. We propose estimators for these, and show asymptotic normality for one of them. We further propose a simple method for choosing the bandwidth. The methods are illustrated by simulation of two well-known models in dynamical systems.
- Motivated by the problem of setting prediction intervals in time series analysis, we suggest two new methods for conditional distribution estimation. The first method is based on locally fitting a logistic model and is in the spirit of recent work on locally parametric techniques in density estimation. It produces distribution estimators that may be of arbitrarily high order but nevertheless always lie between 0 and 1. The second method involves an adjusted form of the Nadaraya--Watson estimator. It preserves the bias and variance properties of a class of second-order estimators introduced by Yu and Jones but has the added advantage of always being a distribution itself. Our methods also have application outside the time series setting; for example, to quantile estimation for independent data. This problem motivated the work of Yu and Jones.
- Article
- May 1998
- J STAT PLAN INFER

The primary aim of the paper is to place current methodological discussions in macroeconometric modeling contrasting the ‘theory first’ versus the ‘data first’ perspectives in the context of a broader methodological framework with a view to constructively appraise them. In particular, the paper focuses on Colander’s argument in his paper “Economists, Incentives, Judgement, and the European CVAR Approach to Macroeconometrics” contrasting two different perspectives in Europe and the US that are currently dominating empirical macroeconometric modeling and delves deeper into their methodological/philosophical underpinnings. It is argued that the key to establishing a constructive dialogue between them is provided by a better understanding of the role of data in modern statistical inference, and how that relates to the centuries old issue of the realisticness of economic theories. - ArticleFull-text available
- Feb 2005

This paper introduces a test for the comparison of multiple misspecified conditional interval models, for the case of dependent observations. Model accuracy is measured using a distributional analog of mean square error, in which the approximation error associated with a given model, say, model i, for a given interval, is measured by the expected squared difference between the conditional confidence interval under model i and the true one.When comparing more than two models, a benchmark model is specified, and the test is constructed along the lines of the reality check of White (2000, Econometrica 68, 1097 1126). Valid asymptotic critical values are obtained via a version of the block bootstrap that properly captures the effect of parameter estimation error. The results of a small Monte Carlo experiment indicate that the test does not have unreasonable finite sample properties, given small samples of 60 and 120 observations, although the results do suggest that larger samples should likely be used in empirical applications of the test.The authors express their gratitude to Don Andrews and an anonymous referee for providing numerous useful suggestions, all of which we feel have been instrumental in improving earlier drafts of this paper. The authors also thank Russell Davidson, Clive Granger, Lutz Kilian, Christelle Viaroux, and seminar participants at the 2002 UK Econometrics Group meeting in Bristol, the 2002 European Econometric Society meetings, the 2002 University of Pennsylvania NSF-NBER time series conference, the 2002 EC2 Conference in Bologna, Cornell University, the State University of New York at Stony Brook, and the University of California at Davis for many helpful comments and suggestions on previous versions of this paper. - ArticleFull-text available
- Jan 2004

We propose a simultaneous model specification procedure for the conditional mean and conditional variance in nonparametric and semiparametric time series econometric models. An adaptive and optimal model specification test procedure is then constructed and its asymptotic properties are investigated. The main results extend and generalize existing results for testing the mean of a fixed design nonparametric regression model to the testing of both the conditional mean and conditional variance nonparametric and semiparametric time series econometric models. In addition, we develop computer-intensive bootstrap simulation procedures for the selection of an interval of bandwidth parameters as well as the choice of asymptotic critical values. An example of implementation is given to show how to implement the proposed simultaneous model specification procedure in practice. Moreover, finite sample studies are presented to support the proposed test procedure - Article
- Feb 2002
- Econometrica

When a continuous-time diffusion is observed only at discrete dates, in most cases the transition distribution and hence the likelihood function of the observations is not explicitly computable. Using Hermite polynomials, I construct an explicit sequence of closed-form functions and show that it converges to the true (but unknown) likelihood function. I document that the approximation is very accurate and prove that maximizing the sequence results in an estimator that converges to the true maximum likelihood estimator and shares its asymptotic properties. Monte Carlo evidence reveals that this method outperforms other approximation schemes in situations relevant for financial models. Copyright The Econometric Society 2002. - Article
- Feb 1999
- J FINANC

This paper has been reprinted from J. Finance 54, 1361-1395 (1999). It applies to interest rate models the theoretical method developed by the author [“Maximum-likelihood estimation of discretely sampled diffusion: A closed-form approach”, Working Paper, Princeton Univ. (1998)] to generate accurate closed form approximations to the transition function of an arbitrary diffusion. While the main focus of this paper is on the maximum-likelihood estimation of interest rate models with otherwise unknown transition functions, applications to the valuation of derivative securities are also briefly discussed. - Article
- Feb 2004
- J Am Stat Assoc

Many practical problems, especially some connected with forecast- ing, require nonparametric estimation of conditional densities from mixed data. For example, given an explanatory data vector X for a prospective customer, with components that could include the customer’s salary, occupation, age, sex, marital status and address, a company might wish to estimate the density of the expendi- ture, Y , that could be made by that person, basing the inference on observations of (X,Y ) for previous clients. Choosing appropriate smoothing parameters for this problem can be tricky, not least because plug-in rules take a particularly complex form in the case of mixed data. An obvious diculty,is that there exists no general formula for the optimal smoothing parameters. More insidiously, and more seri- ously, it can be dicult to determine which components of X are relevant to the problem of conditional inference. For example, if the jth component of X is inde- pendent of Y then that component is irrelevant to estimating the density of Y given X, and ideally should be dropped before conducting inference. In this paper we show that cross-validation overcomes these diculties.,It automatically determines which components are relevant and which are not, through assigning large smooth- ing parameters to the latter and consequently shrinking them towards the uniform distribution on the respective marginals. This eectively,removes irrelevant com- ponents from contention, by suppressing their contribution to estimator variance; they already have very small bias, a consequence of their independence of Y . Cross- validation also gives us important information about which components are rele- vant: the relevant components are precisely those which cross-validation has chosen to smooth in a traditional way, by assigning them smoothing parameters of conven- tional size. Indeed, cross-validation produces asymptotically optimal smoothing for relevant components, while eliminating irrelevant components by oversmoothing. In the problem of nonparametric estimation of a conditional density, cross-validation comes into its own as a method,with no obvious peers. KEYWORDS. Bandwidth choice, binary data, categorical data, continuous data, - Time-homogeneous diffusion models have been widely used for describing the stochastic dynamics of the underlying economic variables. Recently, Stanton proposed drift and diffusion estimators based on a higher-order approximation scheme and kernel regression method. He claimed that " higher order approximations must outperform lower order approximations" and concluded nonlinearity in the instantaneous return function of short-term interest rates. To examine the impact of higher-order approximations, we develop general and explicit formulas for the asymptotic behavior of both drift and diffusion estimators. We show that these estimators will reduce the numerical approximation errors in asymptotic biases, but their asymptotic variances escalate nearly exponentially with the order of approximation. Simulation studies also coné rm our asymptotic results. This variance inè ation problem arises not only from nonparametric é tting, but also from parametric é tting. Stanton' s work also postulates the interesting question of whether the short-term rate drift is nonlinear. Based on empirical simulation studies, Chapman and Pearson suggested that the nonlinearity might be spurious, due partially to the boundary effect of kernel regression. This prompts us to use the local linear é t based on the é rst-order approximation, proposed by Fan and Yao, to ameliorate the boundary effect and to construct formal tests of parametric é nancial models against the nonparametric alternatives. Our simulation results show that the local linear method indeed outperforms the kernel approach. Furthermore, our nonparametric " generalized likelihood ratio tests" are indeed versatile and powerful in detecting nonparametric alternatives. Using this formal testing procedure, we show that the evidence against the linear drift of the short-term interest rates is weak, whereas evidence against a family of popular models for the volatility function is very strong. Application to Standard & Poor 500 data is also illustrated.
- Article
- May 2001
- J FINANC

We estimate and compare a variety of continuous-time models of the short-term riskless rate using the Generalized Method of Moments. We find that the most successful models in capturing the dynamics of the short-term interest rate are those that allow the volatility of interest rate changes to be highly sensitive to the level of the riskless rate. A number of well-known models perform poorly in the comparisons because of their implicit restrictions on term structure volatility. We show that these results have important implications for the use of different term structure models in valuing interest rate contingent claims and in hedging interest rate risk. - Likelihood ratio theory has had tremendous success in parametric inference, due to the fundamental theory of Wilks. Yet, there is no general applicable approach for nonparametric inferences based on function estimation. Maximum likelihood ratio test statistics in general may not exist in nonparametric function estimation setting. Even if they exist, they are hard to find and can not be optimal as shown in this paper. We introduce the generalized likelihood statistics to overcome the drawbacks of nonparametric maximum likelihood ratio statistics. A new Wilks phenomenon is unveiled. We demonstrate that a class of the generalized likelihood statistics based on some appropriate nonparametric estimators are asymptotically distribution free and follow x2-distributions under null hypotheses for a number of useful hypotheses and a variety of useful models including Gaussian white noise models, nonparametric regression models, varying coefficient models and generalized varying coefficient models. We further demonstrate that generalized likelihood ratio statistics are asymptotically optimal in the sense that they achieve optimal rates of convergence given by Ingster. They can even be adaptively optimal in the sense of Spokoiny by using a simple choice of adaptive smoothing parameter. Our work indicates that the generalized likelihood ratio statistics are indeed general and powerful for nonparametric testing problems based on function estimation.
- Article
- Feb 1970
- J FINANC

Stanton (1997) and At-Sahalia (1996) use nonparametric estimators applied to short term interest rate data to conclude that the drift function contains important nonlinearities. We study the #nite-sample properties of their estimators by applying them to simulated sample paths of a square-root diffusion. Although the drift function is linear, both estimators suggest nonlinearities of the type and magnitude reported in Stanton (1997) and At-Sahalia (1996). Combined with the results of a weighted least squares estimator, this evidence implies that nonlinearity of the short rate drift is not a robust stylized fact. ForthcomingintheJournal of Finance ------------------------------------------------------------------ # Chapman is from the University of Texas at Austin and Pearson is from the University of Illinois at Urbana-Champaign. We thank Yacine At-Sahalia, Jonathan Berk, Murray Carlson, John Cochrane, Andy Filardo, Eric Hughson, Narasimhan Jegadeesh, Chris Jones, George P...