On distribution-free goodness-of-fit testing of exponentiality

There is a need to test the hypothesis of exponentiality against a wide variety of alternative hypotheses, across many areas of economics and finance. Local or contiguous alternatives are the closest alternatives against which it is still possible to have some power. Hence goodness-of-fit tests should have some power against all, or a huge majority, of local alternatives. Such tests are often based on nonlinear statistics, with a complicated asymptotic null distribution. Thus a second desirable property of a goodness-of-fit test is that its statistic will be asymptotically distribution free. We suggest a whole class of goodness-of-fit tests with both of these properties, by constructing a new version of empirical process that weakly converges to a standard Brownian motion under the hypothesis of exponentiality. All statistics based on this process will asymptotically behave as statistics from a standard Brownian motion and so will be asymptotically distribution free. We show the form of transformation is especially simple in the case of exponentiality. Surprisingly there are only two asymptotically distribution free versions of empirical process for this problem, and only this one has a convenient limit distribution. Many tests of exponentiality have been suggested based on asymptotically linear functionals from the empirical process. We illustrate none of these can be used as goodness-of-fit tests, contrary to some previous recommendations. Of considerable interest is that a selection of well-known statistics all lead to the same test asymptotically, with negligible asymptotic power against a great majority of local alternatives. Finally, we present an extension of our approach that solves the problem of multiple testing, both for exponentiality and for other, more general hypotheses.     

HODGES-LEHMANN EFFICACIES FOR LIKELIHOOD RATIO TYPE TESTS IN CURVED BIVARIATE NORMAL FAMILIES

For hypothesis testing in curved bivariate normal families we compare various size a tests by means of their Hodges-Lehmann efficacies at fixed alternatives, in particular when these tests have equal optimal asymptotic power in the local Pitman sense. The locally most powerful tests and the likelihood ratio tests for the curve are both Pitman optimal, but the latter turn out to have higher Hodges-Lehmann efficacy. All the tests considered here, including the locally most powerful tests, are likelihood ratio tests against suitable (possibly enlarged) sets of alternatives, the curve itself being an important special case of such a subset. In passing we illustrate a general result in Brown (1971) concerning Hodges-Lehmann optimality obtained by enlarging the model.     

Breaking the curse of dimensionality in nonparametric testing

For tests based on nonparametric methods, power crucially depends on the dimension of the conditioning variables, and specifically decreases with this dimension. This is known as the “curse of dimensionality”. We propose a new general approach to nonparametric testing in high dimensional settings and we show how to implement it when testing for a parametric regression. The resulting test behaves against directional local alternatives almost as if the dimension of the regressors was one. It is also almost optimal against classes of one-dimensional alternatives for a suitable choice of the smoothing parameter. The test performs well in small samples compared to several other tests.     

Robust Non‐nested Testing for Ordinary Least Squares Regression when Some of the Regressors are Lagged Dependent Variables*

The problem of testing non‐nested regression models that include lagged values of the dependent variable as regressors is discussed. It is argued that it is essential to test for error autocorrelation if ordinary least squares and the associated J and F tests are to be used. A heteroskedasticity–robust joint test against a combination of the artificial alternatives used for autocorrelation and non‐nested hypothesis tests is proposed. Monte Carlo results indicate that implementing this joint test using a wild bootstrap method leads to a well‐behaved procedure and gives better control of finite sample significance levels than asymptotic critical values.     

Asymptotically distribution-free tests for the volatility function of a diffusion

This paper develops two tests for parametric volatility function of a diffusion model based on Khmaladze (1981)’s martingale transformation. The tests impose no restrictions on the functional form of the drift function and are shown to be asymptotically distribution-free. The tests are consistent against a large class of fixed alternatives and have nontrivial power against a class of root-n$n$ local alternatives. The paper also extends the tests of volatility to testing for joint specifications of drift and volatility. Monte Carlo simulations show that the tests perform well in finite samples. The proposed tests are then applied to testing models of short-term interest, using data of Treasury bill rate and Eurodollar deposit rate. The empirical results show that the commonly used CKLS volatility function of Chan et al. (1992) fits volatility function poorly and none of the parametric interest rate models considered in the paper fit data well.     

Granger causality in risk and detection of extreme risk spillover between financial markets

Controlling and monitoring extreme downside market risk are important for financial risk management and portfolio/investment diversification. In this paper, we introduce a new concept of Granger causality in risk and propose a class of kernel-based tests to detect extreme downside risk spillover between financial markets, where risk is measured by the left tail of the distribution or equivalently by the Value at Risk (VaR). The proposed tests have a convenient asymptotic standard normal distribution under the null hypothesis of no Granger causality in risk. They check a large number of lags and thus can detect risk spillover that occurs with a time lag or that has weak spillover at each lag but carries over a very long distributional lag. Usually, tests using a large number of lags may have low power against alternatives of practical importance, due to the loss of a large number of degrees of freedom. Such power loss is fortunately alleviated for our tests because our kernel approach naturally discounts higher order lags, which is consistent with the stylized fact that today’s financial markets are often more influenced by the recent events than the remote past events. A simulation study shows that the proposed tests have reasonable size and power against a variety of empirically plausible alternatives in finite samples, including the spillover from the dynamics in mean, variance, skewness and kurtosis respectively. In particular, nonuniform weighting delivers better power than uniform weighting and a Granger-type regression procedure. The proposed tests are useful in investigating large comovements between financial markets such as financial contagions. An application to the Eurodollar and Japanese Yen highlights the merits of our approach.     

Bubbles detection for inter-war European hyperinflation: A threshold cointegration approach

This paper undertakes an empirical investigation into the existence of inflationary bubbles during the inter-war European hyperinflation for Germany, Hungary, Poland and Russia. Our Monte Carlo simulations show that the residual-based threshold cointegration methodology of Caner and Hansen (2001) is better able to detect periodically collapsing bubbles. Moreover, this methodology possesses greater power against nonlinear stationary alternatives in a finite sample than several commonly used cointegration tests that do not allow for multiple regime shifts. The empirical results of the threshold cointegration tests provide evidence of stationary, regimeswitching processes in money demand dynamics, but suggest that there are no inflationary bubbles in any of the countries.     

A NONPARAMETRIC TEST AGAINST ORDERED ALTERNATIVES FOR DATA DEFINED BY INTERVALS

Abstract. The paper deals with a generalization of the Jonckheere test against ordered alternatives which allows for arbitrary censoring. It includes some previously published tests as special cases.     

Tests for changing mean with monotonic power

Several widely used tests for a changing mean exhibit nonmonotonic power in finite samples, due to “incorrect” estimation of nuisance parameters under the alternative. In this paper, we study the issue of nonmonotonic power in testing for changing mean. We investigate the asymptotic power properties of the tests, using a new framework where alternatives are characterized as having “large” changes. The asymptotic analysis provides a theoretical explanation to the power problem. Modified tests that have monotonic power against a wide range of alternatives of structural change are proposed. Instead of estimating the nuisance parameters based on ordinary least squares residuals, the proposed tests use modified estimators, based on nonparametric regression residuals. It is shown that tests based on the modified long-run variance estimator provide an improved rate of divergence of the tests under the alternative of a change in mean. Tests for structural breaks based on such an estimator are able to remain consistent, while still retaining the same asymptotic distribution under the null hypothesis of constant mean.     

Performance of conditional Wald tests in IV regression with weak instruments

We compare the powers of five tests of the coefficient on a single endogenous regressor in instrumental variables regression. Following Moreira [2003, A conditional likelihood ratio test for structural models. Econometrica 71, 1027–1048], all tests are implemented using critical values that depend on a statistic which is sufficient under the null hypothesis for the (unknown) concentration parameter, so these conditional tests are asymptotically valid under weak instrument asymptotics. Four of the tests are based on k-class Wald statistics (two-stage least squares, LIML, Fuller's [Some properties of a modification of the limited information estimator. Econometrica 45, 939–953], and bias-adjusted TSLS); the fifth is Moreira's (2003) conditional likelihood ratio (CLR) test. The heretofore unstudied conditional Wald (CW) tests are found to perform poorly, compared to the CLR test: in many cases, the CW tests have almost no power against a wide range of alternatives. Our analysis is facilitated by a new algorithm, presented here, for the computation of the asymptotic conditional p-value of the CLR test.     

Goodness-of-fit tests based on series estimators in nonparametric instrumental regression

This paper proposes several tests of restricted specification in nonparametric instrumental regression. Based on series estimators, test statistics are established that allow for tests of the general model against a parametric or nonparametric specification as well as a test of exogeneity of the vector of regressors. The tests’ asymptotic distributions under correct specification are derived and their consistency against any alternative model is shown. Under a sequence of local alternative hypotheses, the asymptotic distributions of the tests are derived. Moreover, uniform consistency is established over a class of alternatives whose distance to the null hypothesis shrinks appropriately as the sample size increases. A Monte Carlo study examines finite sample performance of the test statistics.     

Constructing smooth tests without estimating the eigenpairs of the limiting process

Based on the well known Karhunen–Loève expansion, it can be shown that many omnibus tests lack power against “high frequency” alternatives. The smooth tests of  Neyman (1937) may be employed to circumvent this power deficiency problem. Yet, such tests may be difficult to compute in many applications. In this paper, we propose a more operational approach to constructing smooth tests. This approach hinges on a Fourier representation of the postulated empirical process with known Fourier coefficients, and the proposed test is based on the normalized principal components associated with the covariance matrix of finitely many Fourier coefficients. The proposed test thus needs only standard principal component analysis that can be carried out using most econometric packages. We establish the asymptotic properties of the proposed test and consider two data-driven methods for determining the number of Fourier coefficients in the test statistic. Our simulations show that the proposed tests compare favorably with the conventional smooth tests in finite samples.     

One-sided and multi-sided rank tests against trend in the case of concordant observers

Summary Optimal rank tests are derived for testing homogeneity ofk populations observed bym independent andconcordant observers against oriented respectively non-oriented contiguity alternatives with respect to a location parameter.By means of the theorems obtained appropriate rank tests are proposed for testing against a one-sided respectively a multi-sided trend.     

Convenient specification tests for logit and probit models

We propose several Lagrange multiplier tests of logit and probit models, which may be inexpen- sively computed by means of artificial linear regressions. These maybe used to test for various forms of model inadequacy, including the omission of specified variables and heteroskedasticity of known form. We perform a number of sampling experiments, in which we compare the small-sam- ple properties of these tests and of likelihood ratio tests. One of the LM tests turns out to have better small-sample properties than any of the others. We then investigate the power of the tests against local alternatives, and conduct a further series of sampling experiments to compare the power of various tests.     

Testing for the bivariate Poisson distribution

This paper studies goodness-of-fit tests for the bivariate Poisson distribution. Specifically, we propose and study several Cramér–von Mises type tests based on the empirical probability generating function. They are consistent against fixed alternatives for adequate choices of the weight function involved in their definition. They are also able to detect local alternatives converging to the null at a certain rate. The bootstrap can be used to consistently estimate the null distribution of the test statistics. A simulation study investigates the goodness of the bootstrap approximation and compares their powers for finite sample sizes. Extensions for testing goodness-of-fit for the multivariate Poisson distribution are also discussed.     

Inflation forecasts and core inflation measures: Where is the information on future inflation?

This paper uses the forecast from a random walk model of inflation as a benchmark to test and compare the forecast performance of several alternatives of future inflation, including the Greenbook forecast by the Fed staff, the Survey of Professional Forecasters median forecast, CPI inflation minus food and energy, CPI weighted median inflation, and CPI trimmed mean inflation. The Greenbook forecast was found in previous literature to be a better forecast than other private sector forecasts. Our results indicate that both the Greenbook and the Survey of Professional Forecasters median forecasts of inflation and core inflation measures may contain better information than forecasts from a random walk model. The Greenbook's superiority appears to have declined against other forecasts and core inflation measures.     

On Tests for a Normal Mean with Known Coefficient of Variation

Hinkley (1977) derived two tests for testing the mean of a normal distribution with known coefficient of variation (c.v.) for right alternatives. They are the locally most powerful (LMP) and the conditional tests based on the ancillary statistic for μ. In this paper, the likelihood ratio (LR) and Wald tests are derived for the one‐ and two‐sided alternatives, as well as the two‐sided version of the LMP test. The performances of these tests are compared with those of the classical t, sign and Wilcoxon signed rank tests. The latter three tests do not use the information on c.v. Normal approximation is used to approximate the null distribution of the test statistics except for the t test. Simulation results indicate that all the tests maintain the type‐I error rates, that is, the attained level is close to the nominal level of significance of the tests. The power functions of the tests are estimated through simulation. The power comparison indicates that for one‐sided alternatives the LMP test is the best test whereas for the two‐sided alternatives the LR or the Wald test is the best test. The t, sign and Wilcoxon signed rank tests have lower power than the LMP, LR and Wald tests at various alternative values of μ. The power difference is quite large in several simulation configurations. Further, it is observed that the t, sign and Wilcoxon signed rank tests have considerably lower power even for the alternatives which are far away from the null hypothesis when the c.v. is large. To study the sensitivity of the tests for the violation of the normality assumption, the type I error rates are estimated on the observations of lognormal, gamma and uniform distributions. The newly derived tests maintain the type I error rates for moderate values of c.v.     

Specification diagnostics based on Laguerre alternatives for econometric models of duration

Score tests of the null hypothesis of exponentially distributed durations (conditional on regressors) against alternatives in a family of approximations to arbitrary distributions for non-negative random variables are developed. The test statistics take a simple, easily calculated and interpreted form. The alternatives considered are expansions of various orders in Laguerre polynomials. An economic model generating exponential unemployment duration is presented. The statistics are applied in a look at unemployment durations in the Denver Income Maintenance Experiment data.     

Local Powers of Optimal One‐sample and Multi‐sample Tests for the Concentration of Fisher‐von Mises‐Langevin Distributions

One‐sample and multi‐sample tests on the concentration parameter of Fisher‐von Mises‐Langevin distributions on (hyper‐)spheres have been well studied in the literature. However, only little is known about their behaviour under local alternatives, which is due to complications inherent to the curved nature of the parameter space. The aim of the present paper therefore consists in filling that gap by having recourse to the Le Cam methodology, which has recently been adapted from the linear to the spherical setup. We obtain explicit expressions of the powers for the most efficient one‐ and multi‐sample tests. As a nice by‐product, we are also able to write down the powers (against local Fisher‐von Mises‐Langevin alternatives) of the celebrated Rayleigh test of uniformity. A Monte Carlo simulation study confirms our theoretical findings and shows the empirical powers of the above‐mentioned procedures.