Finite Sample Correction Research Articles

Detecting misspecifications in autoregressive conditional duration models and non-negative time-series processes

We develop a general theory to test correct specification of multiplicative error models of non-negative time-series processes, which include the popular autoregressive conditional duration (ACD) models. Both linear and nonlinear conditional expectation models are covered, and standardized innovations can have time-varying conditional dispersion and higher-order conditional moments of unknown form. No specific estimation method is required, and the tests have a convenient null asymptotic N(0,1) distribution. To reduce the impact of parameter estimation uncertainty in finite samples, we adopt Wooldridge's (1990a) device to our context and justify its validity. Simulation studies show that in the context of testing ACD models, finite sample correction gives better sizes in finite samples and are robust to parameter estimation uncertainty. And, it is important to take into account time-varying conditional dispersion and higher-order conditional moments in standardized innovations; failure to do so can cause strong overrejection of a correctly specified ACD model. The proposed tests have reasonable power against a variety of popular linear and nonlinear ACD alternatives.

Finite Sample Correction Factors for Panel Cointegration Tests

Abstract: In this paper we present finite T mean and variance correction factors and corresponding response surface regressions for the panel cointegration tests presented in Pedroni (1999, 2004), Westerlund (2005), Larsson et al. (2001), and Breitung (2005). For the single equation tests we consider up to 12 regressors and for the system tests vector autoregression dimensions up to 12 variables. All commonly used specifications for the deterministic components are considered. The time dimension sample sizes are 10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 200, 500.;

Inference in the Presence of Stochastic and Deterministic Trends

The focus of this paper is inference about stochastic and deterministic trends when both types are present. We show that, contrary to asymptotic theory and the existing literature, the parameters of the deterministic components must be taken into account in finite samples. We analyze the ubiquitous Likelihood Ratio test for the rank of cointegration in vector processes. Here, we directly control the parameters of the data generating process so that a local-asymptotic framework accounts for the interactions between stochastic and deterministic trends. We show that the usual finite sample corrections are invalid as they take no account of the relative magnitudes of these two types of trends. In an empirical application to European GDP series, we show that using usual corrections leads to underestimating the number of stochastic trends. Additionally, block-local models provide embedding frameworks which provide a rationale for consistent estimation and testing of the whole set of parameters.

Detecting Misspecifications in Autoregressive Conditional Duration Models

We propose a new class of specification tests for Autoregressive Conditional Duration (ACD) models. Both linear and nonlinear ACD models are covered, and standardized innovations can have time-varying conditional dispersion and higher order conditional moments of unknown form. No specific estimation method is required, and the tests have a convenient null asymptotic N(0,1) distribution. To reduce the impact of parameter estimation uncertainty in finite samples, we adopt Wooldridge's (1990a) device to our context and justify its validity. Simulation studies show that the finite sample correction gives better sizes in finite samples and are robust to parameter estimation uncertainty. And, it is important to take into account time-varying conditional dispersion and higher order conditional moments in standardized innovations; failure to do so can cause strong overrejection of a correctly specified ACD model. The proposed tests have reasonable power against a variety of popular linear and nonlinear ACD alternatives.

Disability and Employment

Measurement error in health and disability status has been widely accepted as a central problem in social science research. Long-standing debates about the prevalence of disability, the role of health in labor market outcomes, and the influence of federal disability policy on declining employment rates have all emphasized issues regarding the reliability of self-reported disability. In addition to random error, inaccuracy in survey datasets may be produced by a host of economic, social, and psychological factors that can lead respondents to misreport work capacity. We develop a nonparametric foundation for assessing how assumptions on the reporting error process affect inferences on the employment gap between the disabled and nondisabled. Rather than imposing the strong assumptions required to obtain point identification, we derive sets of bounds that formalize the identifying power of primitive nonparametric assumptions that appear to share broad consensus in the literature. Within this framework, we introduce a finite-sample correction for the analog estimator of the monotone instrumental variable (MIV) bound. Our empirical results suggest that conclusions derived from conventional latent variable reporting error models may be driven largely by ad hoc distributional and functional form restrictions. We also find that under relatively weak nonparametric assumptions, nonworkers appear to systematically overreport disability.

A nonparametric test for similarity of marginals—With applications to the assessment of population bioequivalence

In this paper we suggest a completely nonparametric test for the assessment of similar marginals of a multivariate distribution function. This test is based on the asymptotic normality of Mallows distance between marginals. It is also shown that the n out of n bootstrap is weakly consistent, thus providing a theoretical justification to the work in Czado, C. and Munk, A. [2001. Bootstrap methods for the nonparametric assessment of population bioequivalence and similarity of distributions. J. Statist. Comput. Simulation 68, 243–280]. The test is extended to cross-over trials and is applied to the problem of population bioequivalence, where two formulations of a drug are shown to be similar up to a tolerable limit. This approach was investigated in small samples using bootstrap techniques in Czado, C., Munk, A. [2001. Bootstrap methods for the nonparametric assessment of population bioequivalence and similarity of distributions. J. Statist. Comput. Simulation 68, 243–280], showing that the bias corrected and accelerated bootstrap yields a very accurate and powerful finite sample correction. A data example is discussed.

Stock Return Predictability: Is it There?

We ask whether stock returns in France, Germany, Japan, the UK and the US are predictable by three instruments: the dividend yield, the earnings yield and the short rate. The predictability regression is suggested by a present value model with earnings growth, payout ratios and the short rate as state variables. We use this model imposing a constant risk premium to examine the finite sample evidence on predictability. Not only do we find the short rate to be a relevant state variable theoretically, it is also the only robust short-run predictor of equity returns. The evidence in Lamont (1998) on earnings and dividend yield predictability is not robust to our increased sample period, does not survive finite sample corrections and does not extend to other countries. We find no evidence of long-horizon predictability once we account for finite sample influence. Finally, cross-country predictability appears stronger than predictability using local instruments.

Prediction intervals for regression models

This paper presents simple large sample prediction intervals for a future response Y f given a vector x f of predictors when the regression model has the form Y i = m ( x i ) + e i where m is a function of x i and the errors e i are iid. Intervals with correct asymptotic coverage and shortest asymptotic length can be made by applying the shorth estimator to the residuals. Since residuals underestimate the errors, finite sample correction factors are needed. As an application, three prediction intervals are given for the least squares multiple linear regression model. The asymptotic coverage and length of these intervals and the classical estimator are derived. The new intervals are useful since the distribution of the errors does not need to be known, and simulations suggest that the large sample theory often provides good approximations for moderate sample sizes.

Some Second-Order Asymptotics for Extreme Value Linear Regression Models

ABSTRACT In this article we derive finite-sample corrections in matrix notation for likelihood ratio and score statistics in extreme-value linear regression models. We consider three corrected score tests that perform better than the usual score test. We also derive general formulae for second-order biases of maximum likelihood estimates of the linear parameters. Some simulations are performed to compare the likelihood ratio and score statistics with their modified versions and to illustrate the bias correction.

Size properties of cointegration tests in misspecified systems

The small sample size properties of three frequently used cointegration tests when a system has been misspecified are investigated. Specifically, the misspecification consists of one relevant variable being omitted from a system with one cointegrating vector. A Monte Carlo study shows that the Johansen (1991) trace test, adjusted by a simple finite sample correction, has the most robust behaviour when lag length in the test equations is chosen according to traditional information criteria.

Bias analysis in entropy estimation

We consider the problem of finite sample corrections for entropy estimation. New estimates of the Shannon entropy are proposed and their systematic error (the bias) is computed analytically. We find that our results cover correction formulas of current entropy estimates recently discussed in literature. The trade-off between bias reduction and the increase of the corresponding statistical error is analyzed.

A finite sample correction for the variance of linear efficient two-step GMM estimators

Monte Carlo studies have shown that estimated asymptotic standard errors of the efficient two-step generalized method of moments (GMM) estimator can be severely downward biased in small samples. The weight matrix used in the calculation of the efficient two-step GMM estimator is based on initial consistent parameter estimates. In this paper it is shown that the extra variation due to the presence of these estimated parameters in the weight matrix accounts for much of the difference between the finite sample and the usual asymptotic variance of the two-step GMM estimator, when the moment conditions used are linear in the parameters. This difference can be estimated, resulting in a finite sample corrected estimate of the variance. In a Monte Carlo study of a panel data model it is shown that the corrected variance estimate approximates the finite sample variance well, leading to more accurate inference.

Nearly Unbiased Maximum Likelihood Estimation for the Beta Distribution

We analyze the finite-sample behavior of three second-order bias-corrected alternatives to the maximum likelihood estimator of the parameters that index the beta distribution. The three finite-sample corrections we consider are the conventional second-order bias corrected estimator (Cordeiro et al ., 1997), the alternative approach introduced by Firth (1993) and the bootstrap bias correction. We present numerical results comparing the performance of these estimators for thirty-six different values of the parameter vector. Our results reveal that analytical bias corrections considerably outperform numerical bias corrections obtained from bootstrapping schemes.

Further Evidence on the Great Crash, the Oil-Price Shock, and the Unit-Root Hypothesis

Recently, Perron has carried out tests of the unit-root hypothesis against the alternative hypothesis of trend stationarity with a break in the trend occurring at the Great Crash of 1929 or at the 1973 oil-price shock. His analysis covers the Nelson–Plosser macroeconomic data series as well as a postwar quarterly real gross national product (GNP) series. His tests reject the unit-root null hypothesis for most of the series. This article takes issue with the assumption used by Perron that the Great Crash and the oil-price shock can be treated as exogenous events. A variation of Perron's test is considered in which the breakpoint is estimated rather than fixed. We argue that this test is more appropriate than Perron's because it circumvents the problem of data-mining. The asymptotic distribution of the estimated breakpoint test statistic is determined. The data series considered by Perron are reanalyzed using the test static. The empirical results make use of the asymptotics developed for the test statistic as well as extensive finite-sample corrections obtained by simulation. The effect on the empirical results of fat-tailed and temporally dependant innovations is investigated. In brief, by treating the breakpoint as endogenous, we find that there is less evidence against the unit-root hypothesis than Perron finds for many of the data series but stronger evidence against if for several of the series, including the Nelson–Plosser industrial-production, nominal-GNP, and real-GNP series.

IMPROVED SCORE TESTS FOR VON MISES REGRESSION MODELS

We present, in matrix notation, a finite-sample correction formula to improve score tests in von Mises regression models with concentration covariates. The formula only requires simple operations on matrices and can be used to obtain analytically closed-form corrections for score test statistics in a variety of special von Mises models. The paper also provides a numerical comparison of the size of two score test statistics with bootstrap-based critical values.

On improving the robustness and reliability of Rao's score test

The results of misspecification tests, based on Rao's score principle, are now routinely reported in applied econometric work. This paper draws together some important recent results which are designed to improve: (a) the robustness of standard score tests; and (b) the reliability of the asymptotic approximations used for inferential purposes. The discussion of robustness includes (i) parametric, (ii) distributional, and (iii) higher-order moment robustness. The issue of finite sample reliability focuses on controlling the size of the score test using (i) different variance estimators in conjunction with standard asymptotic theory, (ii) finite sample corrections obtainable from higher-order asymptotic analysis, and (iii) bootstrap procedures.

Sampling properties of the spectrum and coherency of sequences of action potentials.

The spectrum and coherency are useful quantities for characterizing the temporal correlations and functional relations within and between point processes. This article begins with a review of these quantities, their interpretation, and how they may be estimated. A discussion of how to assess the statistical significance of features in these measures is included. In addition, new work is presented that builds on the framework established in the review section. This work investigates how the estimates and their error bars are modified by finite sample sizes. Finite sample corrections are derived based on a doubly stochastic inhomogeneous Poisson process model in which the rate functions are drawn from a low-variance gaussian process. It is found that in contrast to continuous processes, the variance of the estimators cannot be reduced by smoothing beyond a scale set by the number of point events in the interval. Alternatively, the degrees of freedom of the estimators can be thought of as bounded from above by the expected number of point events in the interval. Further new work describing and illustrating a method for detecting the presence of a line in a point process spectrum is also presented, corresponding to the detection of a periodic modulation of the underlying rate. This work demonstrates that a known statistical test, applicable to continuous processes, applies with little modification to point process spectra and is of utility in studying a point process driven by a continuous stimulus. Although the material discussed is of general applicability to point processes, attention will be confined to sequences of neuronal action potentials (spike trains), the motivation for this work.

Stock Return Predictability: Is It There?

We ask whether stock returns in France, Germany, the UK and the US are predictable by three instruments: the dividend yield, the earnings yield and the short rate. The predictability regression is suggested by a present value model with earnings growth, payout ratios and the short rate as state variables. We use this model imposing a constant risk premium to examine the finite sample evidence on predictability. Not only do we find the short rate to be a relevant state variable theoretically, it is also the only robust short-run predictor of equity returns. The evidence in Lamont (1998) on earnings and dividend yield predictability is not robust to our increased sample period, does not survive finite sample corrections and does not extend to other countries. We find no evidence of long-horizon predictability once we account for finite sample influence. Finally, cross-country predictability appears stronger than predictability using local instruments.

Broadband Semiparametric Estimation of the Memory Parameter of a Long‐Memory Time Series Using Fractional Exponential Models

We consider a fractional exponential, or FEXP estimator of the memory parameter of a stationary Gaussian long‐memory time series. The estimator is constructed by fitting a FEXP model of slowly increasing dimension to the log periodogram at all Fourier frequencies by ordinary least squares, and retaining the corresponding estimated memory parameter. We do not assume that the data were necessarily generated by a FEXP model, or by any other finite‐parameter model. We do, however, impose a global differentiability assumption on the spectral density except at the origin. Because of this, and its use of all Fourier frequencies, we refer to the FEXP estimator as a broadband semiparametric estimator. We demonstrate the consistency of the FEXP estimator, and obtain expressions for its asymptotic bias and variance. If the true spectral density is sufficiently smooth, the FEXP estimator can strongly outperform existing semiparametric estimators, such as the Geweke–Porter‐Hudak (GPH) and Gaussian semiparametric estimators (GSE), attaining an asymptotic mean squared error proportional to (log n)/n, where n is the sample size. In a simulation study, we demonstrate the merits of using a finite‐sample correction to the asymptotic variance, and we also explore the possibility of automatically selecting the dimension of the exponential model using Mallows’CL criterion.

A Finite Sample Correction for the Variance of Linear Two-Step GMM Estimators

Monte Carlo studies have shown that estimated asymptotic standard errors of the efficient two-step generalised method of moments (GMM) estimator can be severely downward biased in small samples. The weight matrix used in the calculation of the efficient two-step GMM estimator is based on initial consistent parameter estimates. In this paper it is shown that the extra variation due to the presence of these estimated parameters in the weight matrix accounts for much of the difference between the finite sample and the asymptotic variance of the two-step GMM estimator that utilises moment conditions that are linear in the parameters. This difference can be estimated, resulting in a finite sample corrected estimate of the variance. In a Monte Carlo study of a panel data model it is shown that the corrected variance estimate approximates the finite sample variance well, leading to more accurate inference.