Bootstrapping the Hausman Test in Panel Data Models

In the article, it is shown that in panel data models the Hausman test (HT) statistic can be considerably refined using the bootstrap technique. Edgeworth expansion shows that the coverage of the bootstrapped HT is second-order correct.

The asymptotic versus the bootstrapped HT are compared also by Monte Carlo simulations. At the null hypothesis and a nominal size of 0.05, the bootstrapped HT reduces the coverage error of the asymptotic HT by 10–40% of nominal size; for nominal sizes less than or equal to 0.025, the coverage error reduction is between 30% and 80% of nominal size. For the nonnull alternatives, the power of the asymptotic HT fictitiously increases by over 70% of the correct power for nominal sizes less than or equal to 0.025; the bootstrapped HT reduces overrejection to less than one fourth of its value. The advantages of the bootstrapped HT increase with the number of explanatory variables.

Heteroscedasticity or serial correlation in the idiosyncratic part of the error does not hamper advantages of the bootstrapped version of HT, if a heteroscedasticity robust version of the HT and the wild bootstrap are used. But, the power penalty is not negligible if a heteroscedasticity robust approach is used in the homoscedastic panel data model.     

A bayesian analysis of trend determination in economic time series

In this paper we provide a comprehensive Bayesian posterior analysis of trend determination in general autoregressive models. Multiple lag autoregressive models with fitted drifts and time trends as well as models that allow for certain types of structural change in the deterministic components are considered. We utilize a modified information matrix-based prior that accommodates stochastic nonstationarity, takes into account the interactions between long-run and short-run dynamics and controls the degree of stochastic nonstationarity permitted. We derive analytic posterior densities for all of the trend determining parameters via the Laplace approximation to multivariate integrals. We also address the sampling properties of our posteriors under alternative data generating processes by simulation methods. We apply our Bayesian techniques to the Nelson-Plosser macroeconomic data and various stock price and dividend data. Contrary to DeJong and Whiteman (1989a,b,c), we do not find that the data overwhelmingly favor the existence of deterministic trends over stochastic trends. In addition, we find evidence supporting Perron's (1989) view that some of the Nelson and Plosser data are best construed as trend stationary with a change in the trend function occurring at 1929.     

Dynamic Graphics for Exploring Spatial Data with Application to Locating Global and Local Anomalies

We explore the application of dynamic graphics to the exploratory analysis of spatial data. We introduce a number of new tools and illustrate their use with prototype software, developed at Trinity College, Dublin. These tools are used to examine local variability—anomalies—through plots of the data that display its marginal and multivariate distributions, through interactive smoothers, and through plots motivated by the spatial auto-covariance ideas implicit in the variogram. We regard these as alternative and linked views of the data. We conclude that the most important single view of the data is the Map View: All other views must be cross-referred to this, and the software must encourage this. The view can be enriched by overlaying on other pertinent spatial information. We draw attention to the possibilities of one-many linking, and to the use of line-objects to link pairs of data points. We draw attention to the parallels with work on Geographical Information Systems.     

A Recursive Formulation of the Old Problem of Obtaining Moments from Cumulants and Vice Versa

It seems difficult to find a formula in the literature that relates moments to cumulants (and vice versa) and is useful in computational work rather than in an algebraic approach. Hence I present four very simple recursive formulas that translate moments to cumulants and vice versa in the univariate and multivariate situations.     

How non-normality affects the quadratic discriminant function

The quadratic discriminant function is commonly used for the two group classification problem when the covariance matrices in the two populations are substantially unequal. This procedure is optimal when both populations are multivariate normal with known means and covariance matrices. This study examined the robustness of the QDF to non-normality. Sampling experiments were conducted to estimate expected actual error rates for the QDF when sampling from a variety of non-normal distributions. Results indicated that the QDF was robust to non-normality except when the distributions were highly skewed, in which case relatively large deviations from optimal were observed. In all cases studied the average probabilities of misclassification were relatively stable while the individual population error rates exhibited considerable variability.     

On a paper by kageyama and mohan

Kageyama Mohan (1984) have presented three methods of constructing new incomplete block designs from balanced incomplete block designs, They raise questions about the designs which come from each of their methods, These questions are answered, Another series of group divisible designs is derived as a special case of their second method.     

A simplified proof of a theorem on the difference of the moore–penrose inverses of two positive semi–difinte matrices

Simplified proofs are given of a standard result that establishes positive semi–definiteness of the difference of the inverses of two non–singular matrices, and of the extension of this result by Milliken and Akdeniz (1977) to the difference of the Moore–Penrose inverse of two singular matrices.     

Kernel estimation of a distribution function

A distribution function is estimated by a kernel method with

a poinrwise mean squared error criterion at a point x. Relation- ships between the mean squared error, the point x, the sample size and the required kernel smoothing parazeter are investigated for several distributions treated by Azzaiini (1981). In particular it is noted that at a centre of symmetry or near a mode of the distribution the kernei method breaks down. Point- wise estimation of a distribution function is motivated as a more useful technique than a reference range for preliminary medical diagnosis.     

Refined Inference on Long Memory in Realized Volatility

There is an emerging consensus in empirical finance that realized volatility series typically display long range dependence with a memory parameter (d) around 0.4 (Andersen et al., 2001 Andersen , T. G. , Bollerslev , T. , Diebold , F. X. , Labys , P. ( 2001 ). The distribution of realized exchange rate volatility . Journal of the American Statistical Association 96 ( 453 ): 42 – 55 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]; Martens et al., 2004 Martnes , M. , Van Dijk , D. , De Pooter , M. ( 2004 ). Modeling and forecasting S&P 500 volatility: Long memory, structural breaks and nonlinearity. Tinbergen Institute Discussion Paper 2004-067/4 . [Google Scholar]). The present article provides some illustrative analysis of how long memory may arise from the accumulative process underlying realized volatility. The article also uses results in Lieberman and Phillips (2004 Lieberman , O. , Phillips , P. C. B. ( 2004 ). Expansions for the distribution of the maximum likelihood estimator of the fractional difference parameter . Econometric Theory 20 ( 3 ): 464 – 484 . [Google Scholar], 2005 Lieberman , O. , Phillips , P. C. B. ( 2005 ). Expansions for approximate maximum likelihood estimators of the fractional difference parameter . The Econometrics Journal 8 : 367 – 379 . [Google Scholar]) to refine statistical inference about d by higher order theory. Standard asymptotic theory has an O(n ^?1/2) error rate for error rejection probabilities, and the theory used here refines the approximation to an error rate of o(n ^?1/2). The new formula is independent of unknown parameters, is simple to calculate and user-friendly. The method is applied to test whether the reported long memory parameter estimates of Andersen et al. (2001 Andersen , T. G. , Bollerslev , T. , Diebold , F. X. , Labys , P. ( 2001 ). The distribution of realized exchange rate volatility . Journal of the American Statistical Association 96 ( 453 ): 42 – 55 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) and Martens et al. (2004 Martnes , M. , Van Dijk , D. , De Pooter , M. ( 2004 ). Modeling and forecasting S&P 500 volatility: Long memory, structural breaks and nonlinearity. Tinbergen Institute Discussion Paper 2004-067/4 . [Google Scholar]) differ significantly from the lower boundary (d = 0.5) of nonstationary long memory, and generally confirms earlier findings.     

Lag Length Selection for Unit Root Tests in the Presence of Nonstationary Volatility

A number of recent papers have focused on the problem of testing for a unit root in the case where the driving shocks may be unconditionally heteroskedastic. These papers have, however, taken the lag length in the unit root test regression to be a deterministic function of the sample size, rather than data-determined, the latter being standard empirical practice. We investigate the finite sample impact of unconditional heteroskedasticity on conventional data-dependent lag selection methods in augmented Dickey–Fuller type regressions and propose new lag selection criteria which allow for unconditional heteroskedasticity. Standard lag selection methods are shown to have a tendency to over-fit the lag order under heteroskedasticity, resulting in significant power losses in the (wild bootstrap implementation of the) augmented Dickey–Fuller tests under the alternative. The proposed new lag selection criteria are shown to avoid this problem yet deliver unit root tests with almost identical finite sample properties as the corresponding tests based on conventional lag selection when the shocks are homoskedastic.