Reasonable Finite Sample Properties Research Articles

Statistical inference in discretely observed fractional Ornstein–Uhlenbeck processes

In this paper, we develop a two-stage method for estimating all the unknown parameters in the fractional Ornstein–Uhlenbeck model from discrete observations. The estimation procedure is built upon the marriage of the power variation and the minimum contrast estimation. In the first stage, the Hurst coefficient is estimated by the increment Bernoulli statistic and the volatility parameter is estimated by power variations. The asymptotic theory of the proposed estimators in the diffusion term are established under an in-fill asymptotic scheme. In the second stage, two drift parameters are estimated based on the minimum contrast estimation. Their asymptotic theory are analyzed via use of a double asymptotic scheme. The results from Monte Carlo studies illustrate that the proposed estimators have reasonable finite sample properties. An empirical illustration based on realized volatility indicates that the realized volatility is rough.

Local SIML estimation of some Brownian and jump functionals under market micro-structure noise

This paper is a contribution to a special issue on Data Science: Present and Future, because the main topic has been and will be in an active area of contemporary data science. High-frequency financial data are commonly available by now. To estimate Brownian and jump functionals from high-frequency financial data under market micro-structure noise, we introduce a new local estimation method of the integrated volatility and higher order variation of Ito’s semi-martingale processes. Although extending the realized volatility (RV) estimation to the general diffusion-jump processes without micro-market noise is straightforward, estimating Brownian and jump functionals in the presence of micro-market noise may not be easy. In this study, we develop the local SIML (LSIML) method, which is an extension of the separating information maximum likelihood (SIML) method proposed by Kunitomo et al. (Separating information maximum likelihood method for high-frequency financial data, 2018) and Kunitomo and Kurisu (Jpn J Stat Data Sci (JJSD) 4(1):601–641, 2021). The new LSIML method is simple, and the LSIML estimator has some desirable asymptotic properties and reasonable finite sample properties.

Is the assumption of constant factor loadings too strong in practice?

The assumption of constant loadings when estimating factor models is implicit in all empirical applications used in macroeconomics. We test this assumption explicitly in a general smooth transition loadings setting using a well-studied macroeconomic dataset on the U.S. economy. Our proposed testing approach has reasonable finite sample properties relative to alternatives that allow for an abrupt change in the factor loadings. In an empirical application, we find evidence in support of non-constancy in factor loadings and show that it is mainly concentrated in certain sections of the economy (such as financial variables). Overall, our findings provide additional support toward developing factor models, which explicitly account for non-constant factor loadings.

Testing for jumps and jump intensity path dependence

In this paper, we develop a “jump test” for the null hypothesis that the probability of a jump is zero, building on earlier work by Aït-Sahalia (2002). The test is based on realized third moments, and uses observations over an increasing time span. The test offers an alternative to standard finite time span tests, and is designed to detect jumps in the data generating process rather than detecting realized jumps over a fixed time span. More specifically, we make two contributions. First, we introduce our largely model free jump test for the null hypothesis of zero jump intensity. Second, under the maintained assumption of strictly positive jump intensity, we introduce two “self-excitement” tests for the null of constant jump intensity against the alternative of path dependent intensity. These tests have power against autocorrelation in the jump component, and are direct tests for Hawkes diffusions (see, e.g. Aït-Sahalia et al. (2015)). The limiting distributions of the proposed statistics are analyzed via use of a double asymptotic scheme, wherein the time span goes to infinity and the discrete interval approaches zero; and the distributions of the tests are normal and half normal. The results from a Monte Carlo study indicate that the tests have reasonable finite sample properties. An empirical illustration based on the analysis of 11 stock price series indicates the prevalence of jumps and self-excitation.

The SIML Estimation of Integrated Covariance and Hedging Coefficient Under Round-off Errors, Micro-market Price Adjustments and Random Sampling

For estimating the integrated volatility and covariance by using high frequency data, Kunitomo and Sato (Math Comput Simul 81:1272–1289, 2011; N Am J Econ Finance 26:289–309, 2013) have proposed the separating information maximum likelihood (SIML) method when there are micro-market noises. The SIML estimator has reasonable finite sample properties and asymptotic properties when the sample size is large when the hidden efficient price process follows a Brownian semi-martingale. We shall show that the SIML estimation is useful for estimating the integrated covariance and hedging coefficient when we have round-off errors, micro-market price adjustments and noises, and when the high-frequency data are randomly sampled. The SIML estimation is consistent, asymptotically normal in the stable convergence sense under a set of reasonable assumptions and it has reasonable finite sample properties with these effects.

On robust properties of the SIML estimation of volatility under micro-market noise and random sampling

For estimating the integrated volatility and covariance by using high frequency data, Kunitomo and Sato (2008, 2011) have proposed the Separating Information Maximum Likelihood (SIML) method when there are micro-market noises. The SIML estimator has reasonable finite sample properties and asymptotic properties when the sample size is large under general conditions with non-Gaussian processes or volatility models. We shall show that the SIML estimator has the asymptotic robustness property in the sense that it is consistent and has the stable convergence (i.e. the asymptotic normality in the deterministic case) as well as reasonable finite sample properties when there are micro-market noises and the observed high-frequency data are sampled randomly with the underlying (continuous time) stochastic process. We also discuss some implications of our results on public policy and risk managements in financial markets.

Separating Information Maximum Likelihood estimation of the integrated volatility and covariance with micro-market noise

For estimating the integrated volatility and covariance by using high frequency financial data, we propose the Separating Information Maximum Likelihood (SIML) method when there are possibly micro-market noises. The resulting estimator, which is represented as a specific quadratic form of returns, is simple and their properties have been investigated by Kunitomo and Sato (2008a, 2008b, 2010, 2011). We show that the SIML estimator has reasonable asymptotic properties; it is consistent and it has the asymptotic normality when the sample size is large and the integrated volatility is deterministic under general conditions including some non-Gaussian and volatility models. Based on simulations, we find that the SIML estimator has reasonable finite sample properties and it would be useful for practice. The SIML estimator has the asymptotic robustness properties in the sense it is consistent when the noise terms are weakly dependent and they are endogenously correlated with the efficient market price process. We illustrate the use of SIML by analyzing Nikkei-225 futures, which are the derivatives of the major stock index in Japan.

TESTING FOR NONNESTED CONDITIONAL MOMENT RESTRICTIONS VIA CONDITIONAL EMPIRICAL LIKELIHOOD

We propose nonnested tests for competing conditional moment restriction models using the method of conditional empirical likelihood, recently developed by Kitamura, Tripathi, and Ahn (2004) and Zhang and Gijbels (2003). To define the test statistics, we use the implied conditional probabilities from conditional empirical likelihood, which take into account the full implications of conditional moment restrictions. We propose three types of nonnested tests: the moment-encompassing, Cox-type, and efficient score-encompassing tests. We derive the asymptotic null distributions and investigate their power properties against a sequence of local alternatives and a fixed global alternative. Our tests have distinct global power properties from some of the existing tests based on finite-dimensional unconditional moment restrictions. Simulation experiments show that our tests have reasonable finite sample properties and dominate some of the existing nonnested tests in terms of size-corrected powers.

Penalized empirical likelihood estimation of semiparametric models

We propose an empirical likelihood-based estimation method for conditional estimating equations containing unknown functions, which can be applied for various semiparametric models. The proposed method is based on the methods of conditional empirical likelihood and penalization. Thus, our estimator is called the penalized empirical likelihood (PEL) estimator. For the whole parameter including infinite-dimensional unknown functions, we derive the consistency and a convergence rate of the PEL estimator. Furthermore, for the finite-dimensional parametric component, we show the asymptotic normality and efficiency of the PEL estimator. We illustrate the theory by three examples. Simulation results show reasonable finite sample properties of our estimator.

Testing linearity in cointegrating smooth transition regressions

Summary This paper develops statistical tests that can be used to test linearity in cointegrating smooth transition regression models. These tests extend previous similar tests by considering I(1) regressors instead of stationary or mixing regressors and they also allow for more general transition mechanisms than in previous studies. As is typical in cointegrating regressions, the regressors and errors of the model can be serially and contemporaneously correlated. In order to allow for this feature, an endogeneity correction based on a leads-and-lags approach is employed. The proposed tests are very simple to use because ordinary least squares techniques and standard chi-square limiting distributions apply. Simulation experiments indicate that the tests have reasonable finite sample properties. Empirical applications to a U.K. money demand function illustrate the practical usefulness of the tests.

Bootstrap Procedures for Recursive Estimation Schemes with Applications to Forecast Model Selection

In recent years it has become apparent that many of the classical testing procedures used to select amongst alternative economic theories and economic models are not realistic. In particular, researchers have become more aware of the fact that parameter estimation error and data dependence play a crucial role in test statistic limiting distributions, a role which had hitherto been ignored to a large extent. Given the fact that one of the primary ways for comparing different models and theories is via use of predictive accuracy tests, it is perhaps not surprising that a large literature on the topic has developed over the last 10 years, including, for example, important papers by Diebold and Mariano (1995), West (1996), and White (2000). In this literature, it is quite common to compare multiple models (which are possibly all misspecified - i.e. they are all approximations of some unknown true model) in terms of their out of sample predictive ability, for given loss function. Our objectives in this paper are twofold. First, we introduce block bootstrap techniques that are (first order) valid in recursive estimation frameworks. Thereafter, we present two applications where predictive accuracy tests are made operational using our new bootstrap procedures. One of the applications outlines a consistent test for out-of-sample nonlinear Granger causality, and the other outlines a test for selecting amongst multiple alternative forecasting models, all of which may be viewed as approximations of some unknown underlying model. More specifically, our examples extend the White (2000) reality check to the case of non vanishing parameter estimation error, and extend the integrated conditional moment (ICM) tests of Bierens (1982, 1990) and Bierens and Ploberger (1997) to the case of out-of-sample prediction. Of note is that in both of these examples, it is shown that appropriate re-centering of the bootstrap score is required in order to ensure that the tests are properly sized, and the need for such re-centering is shown to arise quite naturally when testing hypotheses of predictive accuracy. The results of a Monte Carlo investigation of the ICM test suggest that the bootstrap procedure proposed in this paper yield tests with reasonable finite sample properties for samples with as few as 300 observations.

A Test for Comparing Multiple Misspecified Conditional Distributions

This paper introduces a conditional Kolmogorov test, in the spirit of Andrews (1997), that allows for comparison of multiple misspecifed conditional distribution models, for the case of dependent observations. A conditional confidence interval version of the test is also discussed. Model accuracy is measured using a distributional analog of mean square error, in which the squared (approximation) error associated with a given model, say model i, is measured in terms of the average over U of E((Fi(u|Zt,0i†)-F0(u|Zt0o))squared), where U is a possibly unbounded set on the real line, Zt is the conditioning information set, Fi is the distribution function of a particular candidate model, and F0 is the true (unkown) distribution function. 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). Valid asymptotic critical values are obtained via a version of the block bootstrap which properly captures the effect of parameter estimation error. The results of a small Monte Carlo experiment indicate that the conditional confidence interval version of the test has reasonable finite sample properties even for samples with as few as 60 observations.

Lagrange Multiplier Tests for Normality Against Seminonparametric Alternatives

In this article, I derive the Lagrange multiplier test of the null hypothesis that a stationary random vector has a (possibly heteroscedastic) normal distribution against the alternative that the distribution is a member of the family with seminonparametric probability density functions considered by Gallant and Tauchen (1989). The test is shown to contain special cases of the moment tests proposed by Newey (1985) and Tauchen (1985). Evidence from a small simulation study is reported, showing that the test has reasonable finite-sample properties in moderately sized samples. The test is applied to the change of price in a treasury-bill data series analyzed by Tauchen and Pitts (1983) and Tauchen (1985).

Calibration as estimation

Calibration in macroeconomics involves choosing fre parameters by matching certain moments of simulted models with those of data. We formally examine this method by treating the process of calibration as an econometric estimator. A numerical version of the Mehra-Prescott (1985) economy is the setting for an evaluation of calibration estimators via Monte Carlo methods. While these estimators sometimes have reasonable finite-sample properties they are not robust to mistakes in setting non-free parameters. In contrast, generalized method-of-moments (GMM) estimators have satisfactory finite-sample characteristics, quick convergence, and informational requirements less stringent than those of calibration estimators. In dynamic equilibrium models in which GMM is infeasible we offer some suggestions for improving estimates based on calibration methodology.