This paper develops methodology for nonparametric estimation of a measure of the overlap of two distributions based on kernel estimation techniques. This quantity has been proposed as a measure of economic polarization between two groups, Anderson (2004) and Anderson, Ge, and Leo (2006). In ecology it has been used to measure the overlap of species. We give the asymptotic distribution theory of our estimator, which in some cases of practical relevance is nonstandard due to a boundary value problem. We also propose a method for conducting inference based on estimation of unknown quantities in the limiting distribution and show that our method yields consistent inference in all cases we consider. We investigate the finite sample properties of our methods by simulation methods. We give an application to the study of polarization within China in recent years using household survey data from two provinces taken in 1987 and 2001. We find a big increase in polarization between 1987 and 2001 according to monetary outcomes but less change in terms of living space.
This paper develops tests for inequality constraints of nonparametric regression functions. The test statistics involve a one-sided version of L_{p}-type functionals of kernel estimators. Drawing on the approach of Poissonization, this paper establishes that the tests are asymptotically distribution free, admitting asymptotic normal approximation. Furthermore, the tests have nontrivial local power against a certain class of local alternatives converging to the null at the rate of n^{-1/2}. Some results from Monte Carlo simulations are presented.
We develop a general class of nonparametric tests for treatment effects conditional on covariates. We consider a wide spectrum of null and alternative hypotheses regarding conditional treatment effects, including (i) the null hypothesis of the conditional stochastic dominance between treatment and control groups; (ii) the null hypothesis that the conditional average treatment effect is positive for each value of covariates; and (iii) the null hypothesis of no distributional (or average) treatment effect conditional on covariates against a one-sided (or two-sided) alternative hypothesis. The test statistics are based on L₁-type functionals of uniformly consistent nonparametric kernel estimators of conditional expectations that characterize the null hypotheses. Using the Poissionization technique of Gine et. al. (2003), we show that suitably studentized versions of our test statistics are asymptotically standard normal under the null hypotheses and also show that the proposed nonparametric tests are consistent against general fixed alternatives. Furthermore, it turns out that our tests have non-negligible powers against some local alternatives that are n^{-1/2} different from the null hypotheses, where n is the sample size. We provide a more powerful test for the case when the null hypothesis may be binding only on a strict subset of the support and also consider an extension to testing for quantile treatment effects. We illustrate the usefulness of our tests by applying them to data from a randomized, job training program LaLonde (1986) and by carrying out Monte Carlo experiments based on this dataset.
We propose a new test of the stochastic dominance efficiency of a given portfolio over a class of portfolios. We establish its null and alternative asymptotic properties, and define a method for consistently estimating critical values. We present some numerical evidence that our tests work well in moderate sized samples.
A formal test on the Lyapunov exponent is developed to distinguish a random walk model from a chaotic system. The test is based on the Nadaraya-Watson kernel estimate of the Lyapunov exponent. We show that the estimator is consistent: The estimated Lyapunov exponent converges to zero under the random walk hypothesis, while it converges to a positive constant for the chaotic system. The test is thus expected to have discriminatory powers. We derive the asymptotic distribution of the estimator, and make it possible to formally test for the null hypothesis of random walk against chaos. The proposed test statistic is a simple normalization of the estimated Lyapunov exponent. It is shown that the null distribution of the test statistic is given by the range of standard Brownian motion on the unit interval. We confirm through simulation that our test performs reasonably well in finite samples. We also apply our test to some of the standard macro and financial time series. For most of the series we considered, however, we find no significant empirical evidence of chaos. We also discuss some of the limitations of our empirical findings.
We propose non-nested hypotheses tests for conditional moment restriction models based on the method of generalized empirical likelihood (GEL). By utilizing the implied GEL probabilities from a sequence of unconditional moment restrictions that contains equivalent information of the conditional moment restrictions, we construct Kolmogorov-Smirnov and Cramér-von Mises type moment encompassing tests. Advantages of our tests over Otsu and Whang's (2007) tests are: (i) they are free from smoothing parameters, (ii) they can be applied to weakly dependent data, and (iii) they allow non-smooth moment functions. We derive the null distributions, validity of a bootstrap procedure, and local and global power properties of our tests. The simulation results show that our tests have reasonable size and power performance in finite samples.
We propose non-nested tests for competing conditional moment restriction models using a method of conditional empirical likelihood, recently suggested by Kitamura, Tripathi and Ahn (2004) and Zhang and Gijbels (2003). We use the implied conditional probabilities to define our test statistics, which take into account the full implications of conditional moment restrictions. We develop three types of non-nested 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 power proerties that are very distinct from some of the existing tests based on finite-dimensional unconditional moment restrictions and are consistent against alternatives that cannot be detected by the latter type tests. In particular, if the support of the moment function is bounded, our Cox-type test is consistent against all departures from the null hypothesis toward the non-nested alternative hypothesis under very mild conditions. On the other hand, the moment encompassing and efficient score encompassing tests require some additional assumptions for consistency which guarantee the non-centrality parameters to be non-zero. Simulation experiments show that our tests have reasonable finite sample properties.
This paper considers a semiparametric cointegrating regression model. As usual, the long-run economic relationship is modeled as a parametric cointegrating regression. The remaining error term is, however, further explained nonparametrically by a functional explanatory variable. The nonparametric component of our model is indeed specified as a function of the distribution, rather than the level, of other nonstationary covariate whose distribution changes over time. In the paper, we develop the statistical theories of this model. In particular, an efficient econometric estimator is proposed and its asymptotic distribution is obtained. A specification test for the model is also investigated. The model and methodology are applied to analyze how an aging population influences the consumption level and the savings rate in the U.S. We find that the impact of age distribution on the consumption level and the savings rate is consistent with the life-cycle hypothesis. This is in contrast with the previous studies based on parametric approaches, which repeatedly produced implausible impact curves for the savings rate.
We propose a new method of testing stochastic dominance that improves on existing tests based on the standard bootstrap or subsampling. The method admits prospects involving infinite as well as finite dimensional unknown parameters, so that the variables are allowed to be residuals from nonparametric and semiparametric models. The proposed bootstrap tests have asymptotic sizes that are less than or equal to the nominal level uniformly over probabilities in the null hypothesis under regularity conditions. This paper also characterizes the set of probabilities that the asymptotic size is exactly equal to the nominal level uniformly. As our simulation results show, these characteristics of our tests lead to an improved power property in general. The improvement stems from the design of the bootstrap test whose limiting behavior mimics the discontinuity of the original test's limiting distribution.
We propose a test of the hypothesis of stochastic monotonicity. This hypothesis is of interest in many applications in economics. Our test is based on the supremum of a rescaled U-statistic. We show that its asymptotic distribution is Gumbel. The proof is difficult because the approximating Gaussian stochastic process contains both a stationary and a nonstationary part and so we have to extend existing results that only apply to either one or the other case. We also propose a refinement to the asymptotic approximation that we show works much better in finite samples. We apply our test to the study of intergenerational income mobility.
We provide a test of the Monday effect in daily stock index returns based on the stochastic dominance criterion. We apply our test to a number of stock indexes including large caps and small caps as well as UK and Japanese indexes. We find strong evidence of Monday effect in some cases under this stronger criterion. However, we also confirm previous studies that the effect is concentrated in the second half of the month and on days when the previous Friday return was negative. The effect is also reversed or weakened in the big US indices post 1987. Overall the evidence in support of a single Monday effect is weak.
We propose a simple method of measuring directional predictability and testing for the hypothesis that a given time series has no directional predictability. The test is based on the correlogram of quantile hits. We provide the distribution theory needed to conduct inference, propose some model free upper bound critical values, and apply our methods to stock index return data. The empirical results suggests some directional predictability in returns especially in mid range quantiles like 5%-10%
This paper considers an empirical likelihood method to estimate the parameters of the quantile regression (QR) models and to construct confidence regions that are accurate in finite samples. To achieve the higher-order refinements, we smooth the estimating equations for the empirical likelihood. We show that the smoothed empirical likelihood (SEL) estimator is first-order asymptotically equivalent to the standard QR estimator and establish that confidence regions based on the smoothed empirical likelihood ratio have coverage errors of order n^(-1) and may be Bartlett-corrected to produce regions with errors of order n^(-2), where n denotes the sample size. Our result is an extension of the previous result of Chen and Hall (1993) to the regression context. Monte Carlo experiments suggest that the smoothed empirical likelihood confidence regions may be more accurate in small samples than the confidence regions that can be constructed from the smoothed bootstrap method recently suggested by Horowitz (1998).
We propose a procedure for estimating the critical values of the extended Kolmogorov-Smirnov tests of Stochastic Dominance of arbitrary order in the general K-prospect case. We allow for the observations to be serially dependent and, for the first time, we can accommodate general dependence amongst the prospects which are to be ranked. Also, the prospects may be the residuals from certain conditional models, opening the way for conditional ranking. We also propose a test of Prospect Stochastic Dominance. Our method is based on subsampling and we show that the resulting tests are consistent and powerful against some N^{-1/2} local alternatives. We also propose some heuristic methods for selecting subsample size and demonstrate in simulations that they perform reasonably. We describe an alternative method for obtaining critical values based on recentering the test statistic and using full sample bootstrap methods. We compare the two methods in theory and in practice.
This paper proposes a statistical test of the martingale hypothesis. It can be used to test whether a given time series is a martingale process against certain non-martingale alternatives. The class of alternative processes against which our test has power is very general and it encompasses many nonlinear non-martingale processes which may not be detected using traditional spectrum-based or variance-ratio tests. We look at the hypothesis of martingale, in contrast with other existing methods which test for the hypothesis of martingale difference. Two different types of test are considered: one is a generalized Kolmogorov-Smirnov test and the other is a Cramer-von Mises type test. For the processes that are first order Markovian in mean, in particular, our approach yields the test statistics that neither depend upon any smoothing parameter nor require any resampling procedure to simulate the null distributions. Their null limiting distributions are nicely characterized as functionals of a continuous stochastic process so that the critical values are easily tabulated. We prove consistency of our tests and further investigate their finite sample properties via simulation. Our tests are found to be rather powerful in moderate size samples against a wide variety of non-martingales including exponential autoregressive, threshold autoregressive, markov switching, chaotic, and some of nonstationary processes.
This paper introduces specification tests for linear quantile regression models. The tests are applicable to dependent as well as independent observations. The proposed tests are Kolmogorov-Smirnov and Cramer-von Mises type tests and they are consistent against all alternatives to the null hypothesis, powerful against 1/?N alternatives, are not dependent on any smoothing parameters, and simple to compute. A subsampling procedure is suggested to approximate the critical values and is justified asymptotically under very weak conditions on the data generating processes. A small scale Monte Carlo experiments show that our test has good finite sample performance compared to an existing test.
This paper considers a test of the random walk hypothesis by comparing jointly the variance ratios at multiple observation intervals with unity. We suggest a subsampling procedure to approximate the asymptotic null distribution. Simulation results provide some favorable finite sample performance.
We introduce a kernel-based estimator of the density function and regression function for data that have been grouped into family totals. We allow for a common intra-family component but require that observations from different families be independent. We establish consistency and asymptotic normality for our procedures. As usual, the rates of convergence can be very slow depending on the behaviour of the characteristic function at infinity. We investigate the practical performance of our method in a simple Monte Carlo experiment.
This paper introduces specification tests for conditional moment restrictions. The proposed tests are generalizations of the Kolmogorov-Smirnov and Cramer-von Mises tests and they are consistent against all alternatives to the null hypothesis, powerful against 1/?n local alternatives and not dependent on any smoothing parameter. A nonparametric bootstrap procedure based on recentered criterion function is suggested to obtain critical values for the tests and is justified asymptotically.
This paper introduces specification tests of parametric mean-regression models. The null hypothesis of interest is that the parametric regression function is correctly specified. The proposed tests are generalizations of the Kolmogorov-Smirnov and Cramer-von Mises tests to the regression framework. They are consistent against all alternatives to the null hypothesis, powerful against 1/?n local alternatives, not dependent on any smoothing parameters and simple to compute. A wild-bootstrap procedure is suggested to obtain critical values for the tests and is justified asymptotically. A small scale Monte Carlo experiment shows that our tests (especially Cramer-von Mises test) have outstanding small sample performance compared to some of the existing tests.
This paper derives the asymptotic distribution of a smoothing-based estimator of the Lyapunov exponent for a stochastic time series under two general scenarios. In the first case, we are able to establish root-T consistency and asymptotic normality, while in the second case, which is more relevant for chaotic processes, we are only able to establish asymptotic normality at a slower rate of convergence. We provide consistent confidence intervals for both cases. We apply our procedures to simulated data.