tests for two outliers in a linear model the detection of two outliers in a linear model is considered the case of two outliers in the same direction is studied in detail the first bonferroni inequality is used to obtain nominal upper percentage points for a random sample from a normal distribution the proposed statistic reduces to the well known murphy s test statistic similarly for two outliers on either direction our statistic reduces to the studentized range statistic an approximate relationship between the critical points of murphy s test and the studentized range statistic is investigated

generalized blocks for symmetric groups we study via character theoretic methods an l analogue of the modular representation theory of the symmetric group for an arbitrary integer l 2 we find that many of the invariants of the usual block theory ie when l is prime generalize in a natural fashion to this new context

explicit expressions of the nth gradient of 1 r explicit expressions of the nth gradient tensor and partial derivatives of the function 1 r are given in the absolute and in the local bases associated respectively with the cartesian and the spherical coordinates of space they may be used in various multipolar developments of r 1 interaction energies

simulation based finite and large sample tests in multivariate regressions abstract in the context of multivariate linear regression mlr models it is well known that commonly employed asymptotic test criteria are seriously biased towards overrejection in this paper we propose a general method for constructing exact tests of possibly nonlinear hypotheses on the coefficients of mlr systems for the case of uniform linear hypotheses we present exact distributional invariance results concerning several standard test criteria these include wilks likelihood ratio lr criterion as well as trace and maximum root criteria the normality assumption is not necessary for most of the results to hold implications for inference are two fold first invariance to nuisance parameters entails that the technique of monte carlo tests can be applied on all these statistics to obtain exact tests of uniform linear hypotheses second the invariance property of the latter statistic is exploited to derive general nuisance parameter free bounds on the distribution of the lr statistic for arbitrary hypotheses even though it may be difficult to compute these bounds analytically they can easily be simulated hence yielding exact bounds monte carlo tests illustrative simulation experiments show that the bounds perform well our findings illustrate the value of the bounds as a tool to be used in conjunction with more traditional simulation based test methods e g the parametric bootstrap which may be applied when the bounds are not conclusive

monte carlo results on several new and existing tests for the error component model abstract the breusch pagan test is a popular test for testing zero variances in a random error component model realizing the one sided nature of this problem honda 1985 derived the one sided lm test for the one way error component model and suggested a handy one sided candidate for the two way error component model in this paper we derive a new one sided test for testing the two way error component model based upon the results of gourieroux holly and monfort 1982 next we derive a lm test for testing zero individual time specific effects assuming that the other time period individual effects are present this is in contrast to the usual lm test which assumes that under the null hypothesis both effects are absent exact power comparisons whenever possible as well as monte carlo experiments are employed to compare the performance of these tests with the corresponding likelihood ratio and anova f tests

exact tests for contemporaneous correlation of disturbances in seemingly unrelated regressions this paper proposes finite sample procedures for testing the sure specification in multi equation regression models i e whether the disturbances in different equations are contemporaneously uncorrelated or not we apply the technique of monte carlo mc tests see dwass and barnard respectively ann math statist 28 1957 181 j r statist soc ser b 25 1963 294 to obtain exact tests based on standard lr and lm zero correlation tests we also suggest a mc quasi lr qlr test based on feasible generalized least squares fgls we show that the latter statistics are pivotal under the null which provides the justification for applying mc tests furthermore we extend the exact independence test proposed by harvey and phillips bull econ res 34 2 1982 79 to the multi equation framework specifically we introduce several induced tests based on a set of simultaneous harvey phillips type tests and suggest a simulation based solution to the associated combination problem the properties of the proposed tests are studied in a monte carlo experiment which shows that standard asymptotic tests exhibit important size distortions while mc tests achieve complete size control and display good power moreover mc qlr tests performed best in terms of power a result of interest from the point of view of simulation based tests the power of the mc induced tests improves appreciably in comparison to standard bonferroni tests and in certain cases outperform the likelihood based mc tests the tests are applied to data used by fischer j monetary econ 32 485 1993 to analyze the macroeconomic determinants of growth

tests of stationarity against a change in persistence this paper considers testing against a change in the order of integration of a time series either from i 0 to i 1 or from i 1 to i 0 at some known or unknown point in the sample the null hypothesis is that the series is stochastically stationary around a deterministic trend function for the case of a known change point the locally best invariant lbi tests against the above changes in the order of integration are derived under the assumption of gaussianity when the change point is not known we construct our tests taking functions of the lbi statistics over all possible break dates sub sample implementations of existing stationarity tests are also considered we demonstrate by a series of simulation experiments that for a given direction of change the lbi based approach can deliver considerably more powerful tests than both the sub sample stationarity tests and the ratio based tests of kim et al j econom 109 2002 389 and busetti and taylor tests of stationarity against a change in persistence university of birmingham department of economics discussion paper 01 13 2001 moreover the power losses from an unknown breakpoint do not appear to be large we also find that standard stationarity tests have good power against both changes from i 0 to i 1 and vice versa while the ratio based tests are consistent only against a known direction of change a further test constructed in terms of the lbi based statistics for the two possible directions of change is shown to perform generally better than the standard stationarity tests when the direction of change under the alternative is not known finally we apply the tests discussed in the paper to the us inflation rate and find evidence for a change in persistence from i 1 to i 0 behaviour although significantly the timing of this change varies according to whether or not a simultaneous change in the level of the series is allowed

on the comprehensive method of testing non nested regression models abstract the paper derives the specific form of the exponentially combined likelihood function of two competing multivariate non linear regression models and shows that the application of the comprehensive approach to testing non nested regression models will in general be indeterminate it establishes that in the univariate case there exists a large number of tests of non nested regression models which are consistent in addition to having the same asymptotic distribution under the null hypothesis the paper then derives a set of conditions under which all these consistent tests are asymptotically equivalent not only under the null hypothesis but also under local alternatives as an application of this latter result the paper establishes the asymptotic equivalence of the tests recently proposed by davidson and mackinnon and fisher and mcaleer under local alternatives and shows that within the class of tests considered in the paper these proposed tests possess maximum local power the latter test has this property only when the number of explanatory variables of the true model is not more than that of the false model

optimal rank based procedures for time series analysis testing an arma model against other arma models the problem of testing a given arma model in which the density of the generating white noise is unspecified against other arma models is considered a distribution free asymptotically most powerful test based on a generalized linear serial rank statistic is provided against contiguous arma alternatives with specified coefficients in the case when the arma model in the alternative has unspecified coefficients the asymptotic sufficiency in the sense of le cam of a finite dimensional vector of rank statistics is established this asymptotic sufficiency is used to derive an asymptotically maximin most powerful test based on a generalized quadratic serial rank statistic the asymptotically maximin optimal test statistic can be interpreted as a rank based weighted version of the classical box pierce portmanteau statistic to which it reduces in some particular problems asymptotically and under gaussian assumptions

seasonal integration and cointegration this paper develops tests for roots in linear time series which have a modulus of one but which correspond to seasonal frequencies critical values for the tests are generated by monte carlo methods or are shown to be available from dickey fuller or dickey hasza fuller critical values representations for multivariate processes with combinations of seasonal and zero frequency unit roots are developed leading to a variety of autoregressive and error correction representations the techniques are used to examine cointegration at different frequencies between consumption and income in the u k

the statistical foundations of a class of parametric tests for heteroscedasticity abstract in this paper we examine the statistical foundations of the five tests for heteroscedasticity considered by griffiths and surekha 1986 we identify the precise specification of the alternative hypothesis as an important determinant of the power of the tests

marginal likelihood score based tests of regression disturbances in the presence of nuisance parameters abstract this paper is concerned with tests of the covariance matrix of the disturbances in the linear regression model that involves nuisance parameters which cannot be eliminated by the usual invariance arguments score based tests namely lagrange multiplier lm and locally most mean powerful lmmp tests are derived from the marginal likelihood applications considered include i testing for random regression coefficients ii testing for second order autoregressive ar 2 disturbances and iii testing for arma 1 1 disturbances each in the presence of ar 1 disturbances an empirical size and power comparison shows that the new tests typically have more accurate asymptotic critical values and slightly more power than their respective conventional counterparts

exact tests and confidence sets in linear regressions with autocorrelated errors this article proposes a general method to build exact tests and confidence sets in linear regressions with first order autoregressive gaussian disturbances because of a nuisance parameter problem we argue that generalized bounds tests and conservative confidence sets provide natural inference procedures in such a context given an exact confidence set for the autocorrelation coefficient we describe how to obtain a similar simultaneous confidence set for the autocorrelation coefficient and any subvector of regression coefficient conservative confidence sets for the regression coefficients are then deduced by a projection method for any hypothesis that specifies jointly the value of the autocorrelation coefficient and any set of linear restrictions on the regression coefficients we get exact similar tests for tesing linear hypotheses about the regression coefficients only we suggest bounds type procedures exact confidence sets for the autocorrelation coefficient are built by inverting autocorrelation tests the method is illustrated with two examples copyright 1990 by the econometric society

finite sample properties and asymptotic efficiency of monte carlo tests on donne une approche unifiee qui couvre le cas d une distribution nulle arbitraire afin d etudier les proprietes statistiques des tests de monte carlo

elementary divisors of the shapovalov form on the basic representation of kac moody algebras we calculate the invariant factors of the standard lattice vz inside the basic representation of untwisted kac moody algebras of ade type this in turn gives the invariant factors of cartan matrices for symmetric groups in prime characteristic as well as for the iwahori hecke algebra at roots of unity

median unbiased estimation of coefficient variance in a time varying parameter model abstract this article considers inference about the variance of coefficients in time varying parameter models with stationary regressors the gaussian maximum likelihood estimator mle has a large point mass at 0 we thus develop asymptotically median unbiased estimators and asymptotically valid confidence intervals by inverting quantile functions of regression based parameter stability test statistics computed under the constant parameter null these estimators have good asymptotic relative efficiencies for small to moderate amounts of parameter variability we apply these results to an unobserved components model of trend growth in postwar u s per capita gross domestic product the mle implies that there has been no change in the trend growth rate whereas the upper range of the median unbiased point estimates imply that the annual trend growth rate has fallen by 0 9 per annum since the 1950s

testing for block effects in regression models based on survey data abstract this article considers the problem of testing for intrablock or intracluster correlation in regression disturbances that may occur when cluster or two stage sampling data is used in regression analysis it points out that the one sided lagrange multiplier test is locally best invariant an empirical power comparison suggests that if the block structure is known this test should be used otherwise the durbin watson test provides a useful test especially in large samples

a further class of tests for heteroscedasticity abstract this paper considers the problem of testing for heteroscedasticity in the linear regression model when one is willing to postulate only the ranking of the disturbance variances under the alternative hypothesis a new class of tests is proposed and methods of finding the tests critical values are discussed an empirical power comparison with other tests that have found favor in the literature demonstrates the potential power advantage of these new tests and results in particular versions being recommended for practical use

distributional solutions of the hypergeometric differential equation abstract we present the distributional solutions to the hypergeometric differential equation these solutions are obtained in the form of infinite series of the dirac delta functions and its derivatives we employ these solutions to observe their interesting features furthermore the form of these solutions is the same as the ones required for the weight distributions for a certain class of orthogonal polynomials

nonlinear hypotheses inequality restrictions and non nested hypotheses exact simultaneous tests in linear regressions in the context of the classical linear model the problem of comparing two arbitrary hypotheses on the regression coefficients is considered problems involving nonlinear hypotheses inequality restrictions or non nested hypotheses are included as special cases exact bounds on the null distribution of likelihood ratio statistics are derived the bounds are based on the central fisher distribution and are very easy to use in an important special case a bounds test similar to the durbin watson test is proposed multiple testing problems are also studied the bounds obtained for a single pair of hypotheses are shown to enjoy a simultaneity property that allows one to combine any number of tests this result extends to nonlinear hypotheses a well known result given by scheffe for linear hypotheses a method of building bounds induced tests is also suggested

on graded cartan invariants of symmetric groups and hecke algebras we consider graded cartan matrices of the symmetric groups and the iwahori hecke algebras of type a at roots of unity these matrices are mathbb z v v 1 valued and may also be interpreted as gram matrices of the shapovalov form on sums of weight spaces of a basic representation of an affine quantum group we present a conjecture predicting the invariant factors of these matrices and give evidence for the conjecture by proving its implications under a localization and certain specializations of the ring mathbb z v v 1 this proves and generalizes a conjecture of ando suzuki yamada on the invariants of these matrices over mathbb q v v 1 and also generalizes the first author s recent proof of the kulshammer olsson robinson conjecture over mathbb z

model selection test procedures in a single linear equation of a dynamic simultaneous system and their defects in small samples abstract a variety of asymptotically valid tests for orthogonality serial correlation predictive failure and of coefficient restrictions are presented and their rejection probabilities are assessed in linear structural models with lagged dependent and possibly jointly dependent variables by monte carlo methods for all test procedures the small sample distribution under the null usually deviates substantially from the asymptotic distribution this impedes their use in a reliable model selection strategy for econometric time series analysis despite the harassing dependence of type i errors on factors generally unknown to the practitioner inconsistencies originating from specification errors or from disregarded simultaneity may be detected by particular tests in particular situations from this study some clues emerge on how to interpret in significant values of the various test statistics

computing p values for the generalized durbin watson and other invariant test statistics abstract shively ansley and kohn 1990 give an o n algorithm for computing the p values of the durbin watson and other invariant test statistics in time series regression they do so by evaluating the characteristic function of a quadratic form in standard normal random variables and then numerically inverting it in this paper we obtain a new expression for the characteristic function which simplifies the handling of the independent regressors and so is easier to evaluate we also obtain general easily computable bounds on the integration and truncation errors which arise in the numerical inversion of the characteristic function empirical results are presented on the speed and accuracy of our algorithm

regularity conditions for cox s test of non nested hypotheses abstract in this article we provide for the first time general regularity conditions and a rigorous proof of the asymptotic normality of cox s statistic for testing separate families of hypotheses the cox test for choosing between competing linear regression models is discussed as an example

modified lagrange multiplier tests for problems with one sided alternatives abstract a modified lagrange multiplier test statistic is proposed which takes explicit account of the one sided nature of the alternative in problems where the null hypothesis specifies that the true value of the parameter vector lies on the boundary of the parameter space computation of this statistic requires only the constrained maximum likelihood estimator conditions for the consistency of tests based on this statistic are examined and it is shown that the distribution of the statistic is not affected of nuisance parameters are allowed to lie on the boundary of the parameter space

optimal tests when a nuisance parameter is present only under the alternative this paper derives asymptotically optimal tests for testing problems in which a nuisance parameter exists under the alternative hypothesis but not under the null for example the results apply to tests of structural change with unknown changepoint the testing problem considered is nonstandard and the classical asymptotic optimality results for the lagrange multiplier wald and likelihood ratio do not apply a weighted average power criterion is used here to generate optimal tests this criterion is similar to that used by a wald 1943 to obtain the classical asymptotic optimality properties of wald tests in regular testing problems copyright 1994 by the econometric society

testing for a slowly changing level with special reference to stochastic volatility abstract a test for the presence of a stationary first order autoregressive process embedded in white noise is constructed so as to be relatively powerful when the autoregressive parameter is close to one the test statistic is shown to have a cramer von mises distribution in large samples a comparison is made with some standard tests for serial correlation in the context of a stochastic volatility model and the proposed test is shown to have relatively high power for the parameter values typically found with daily financial time series it is recommended that tests on returns be carried out using absolute values

generalized function solutions of differential and functional differential equations recently there has been considerable interest in problems concerning the existence of solutions to differential and functional differential equations fde in various spaces of generalized functions many important areas in theoretical and mathematical physics theory of partial differential equations quantum electrodynamics operational calculus and functional analysis use the methods of the distributions theory yet for ordinary differential equations ode and especially fde research in this direction is insufficiently developed and remains restricted to isolated results for some secondorder equations or special higher order systems we note in particular papers l 6 which contain references to previous work it is weil known that normal linear homogeneous systems of ode with infinitely smooth coefficients have no generalized function solutions other than the classical solutions in contrast to this case for equations with singularities in the coefficients new solutions in generalized functions may appear and some classical solutions may disappear the number m is called the order of the distribution

testing for stationarity with a break in this paper we investigate a test for the null hypothesis of trend stationarity with a structural change against a unit root we derive the limiting distribution of an lagrange multiplier lm test statistic and its characteristic function under a sequence of local alternatives the local limiting power of the lm test depends on the persistence of the stationary component of the process and the more persistent the process the less powerful is the test statistic we also propose a test statistic that does not depend on the fraction of the pre break points to the sample size under the null hypothesis which we call the ps test though it is convenient for the critical point not to depend on the break point the ps test is found to be less powerful than the lm test under the alternative close to the null hypothesis finite sample simulations show that when the break point is known the lm test tends to be oversized when the process is rather persistent while the size distortion of the ps test is not so pronounced on the other hand the empirical sizes of both tests are close to the nominal one when the break point is estimated by the least squares method though the power decreases compared with the known break point case

a distributional theory for asymptotic expansions the purpose of this chapter is to present the distributional theory of asymptotic expansions for functions of one variable this chapter and the next where the multidimensional expansions are studied are the central part of the book

the asymptotic expansion of certain multi dimensional generalized functions abstract we apply regularization of divergent integrals in the derivation of the asymptotic expansion of certain multi dimensional generalized functions we further present several illustrations to demonstrate that the asymptotic development of generalized functions provides a lucid formulation of many concepts in asymptotic analysis such as the expansion of oscillatory integrals and the expansion of certain series

testing for unit roots in seasonal time series abstract regression estimators of coefficients in seasonal autoregressive models are described the percentiles of the distributions for time series that have unit roots at the seasonal lag are computed by monte carlo integration for finite samples and by analytic techniques and monte carlo integration for the limit case the tabled distributions may be used to test the hypothesis that a time series has a seasonal unit root

small disturbance asymptotics and the durbin watson and related tests in the dynamic regression model abstract until recently it was thought inappropriate to apply the durbin watson dw test to a dynamic linear regression model because of the lack of appropriate critical values recently inder 1986 used a modified small disturbance distribution sdd to find approximate critical values this paper studies the exact sdd of statistics of the same general form as the dw statistic and suggests some changes to inder s result we show how to calculate true small disturbance critical values and bounds for these critical values that take into account the exogenous regressors our results give a justification for the use of the familiar tables of bounds when the dw test is applied to a dynamic regression model

generalized perfect isometries in some groups of lie rank one abstract in a recent paper b kulshammer j b olsson and g r robinson introduced notions of generalized blocks and generalized perfect isometries and studied them in the case of the symmetric group s n it is the purpose of this paper to investigate some families of groups of lie rank one exhibiting generalized perfect isometries where there are none in the sense given by m broue and thus proving a weaker analogue of one of broue s conjectures in these cases

selecting the best linear regression model a classical approach in this paper we apply the model selection approach based on likelihood ratio lr tests developed in vuong 1985 to the problem of choosing between two normal linear regression models which are not nested in each other first we compare our model selection procedure to other model selection criteria then we explicitly derive the procedure when the competing linear models are non nested and neither one is correctly specified some simplifications are seen to arise when both models are contained in a larger correctly specified linear regression model or when at least one competing linear model is correctly specified a comparison of our model selection tests and previous non nested hypothesis tests concludes the paper

optimal weighted average power similar tests for the covariance structure in the linear regression model this paper suggests a procedure for the construction of optimal weighted average power similar tests for the error covariance matrix of a gaussian linear regression model when the alternative model belongs to the exponential family the paper uses a saddlepoint approximation to construct simple test statistics for a large class of problems and overcomes the computational burden of evaluating the complicated integrals arising in the derivation of optimal weighted average power tests extensions to panel data models are considered applications are given to tests for error autocorrelation in the linear regression model and in a panel data framework

generalised cartan invariants of symmetric groups kulshammer olsson and robinson developed an analogue of modular representation theory of symmetric groups where is not necessarily a prime they gave a conjectural combinatorial description for invariant factors of the cartan matrix in this context we confirm their conjecture by proving a more precise blockwise version of the conjecture due to bessenrodt and hill

simulation based finite sample tests for heteroskedasticity and arch effects tests for heteroskedasticity in linear regressions are typically based on asymptotic approximations we show that the size of such tests can be perfectly controlledin nite samples through monte carlo test techniques with both gaussian and non gaussian heavy tailed disturbance distributions the procedures studied include standard heteroskedasticity tests e g glejser bartlett cochran hartley breusch pagan godfrey white szroeter as well as tests for arch type heterosked asticity sup type andcombinedtests are also proposedto allow for unknown breakpoints in the variance the fact that the proposedproced ures achieve size control andhave goodpower is demonstrated in a monte carlo simulation c 2003 elsevier b v all rights reserved

fast evaluation of the distribution of the durbin watson and other invariant test statistics in time series regression abstract a method is given for evaluating p values in o n operations for a general class of invariant test statistics that can be expressed as the ratio of quadratic forms in time series regression residuals the best known of these is the durbin watson statistic although several others have been discussed in the literature the method is numerically exact in the sense that the user specifies the error tolerance at the outset as with existing exact methods the problem is reexpressed in terms of the distribution function of a single quadratic form in independent normals which is evaluated by numerically inverting its characteristic function existing methods however calculate the characteristic function by reducing the matrix defining the quadratic form to either eigenvalue or tridiagonal form each of which requires o n 3 operations for sample size n whereas the new method uses a modification of the kalman filter to do it in o n operations moreover the new method has minimal storage requiremen

monte carlo tests with nuisance parameters a general approach to finite sample inference and nonstandard asymptotics abstract the technique of monte carlo mc tests dwass 1957 annals of mathematical statistics 28 181 187 barnard 1963 journal of the royal statistical society series b 25 294 provides a simple method for building exact tests from statistics whose finite sample distribution is intractable but can be simulated when no nuisance parameter is involved we extend this method in two ways first by allowing for mc tests based on exchangeable possibly discrete test statistics second by generalizing it to statistics whose null distribution involves nuisance parameters maximized mc mmc tests simplified asymptotically justified versions of the mmc method are also proposed these provide a simple way of improving standard asymptotics and dealing with nonstandard asymptotics

the alternative durbin watson test an assessment of durbin and watson s choice of test statistic abstract this paper examines durbin and watson s 1950 choice of test statistic for their test of first 0order autoregressive regression disturbances attention is focused on an alternative statistic d theoretical and empirical power properties of the d test are compared with those of the durbin watson test the former is found to be locally best invariant while the latter is approximately locally best invariant the d test is also found to be more powerful than its counterpart against negative autocorrelation and for small values of the autocorrelation coefficient against positive autocorrelation selected bounds for significance points of d are tabulated

markovian processes two sided autoregressions and finite sample inference for stationary and nonstationary autoregressive processes in this paper we develop finite sample inference procedures for stationary and non stationary autoregressive ar models the method is based on special properties of markov processes and a split sample technique the results on markovian processes intercalary independence and truncation only require the existence of conditional densities they are proved for possibly nonstationary and or non gaussian multivariate markov processes in the context of a linear regression model with ar 1 errors we show how these results can be used to simplify the distributional properties of the model by conditioning a subset of the data on the remaining observations this transformation leads to a new model which has the form of a two sided autoregression to which standard classical linear regression inference techniques can be applied we show how to derive tests and confidence sets for the mean and or autoregressive parameters of the model we also develop a test on the order of an autoregression we show that a combination of subsample based inferences can improve the performance of the procedure an application to u s domestic investment data illustrates the method

exact tests in single equation autoregressive distributed lag models abstract for hypotheses on the coefficient values of the lagged dependent variables in the arx class of dynamic regression models test procedures are developed which yield exact inference for given up to an unknown scale factor distribution of the innovation errors they include exact tests on the maximum lag length for structural change and on the presence of seasonal or multiple unit roots i e they cover situations where usually asymptotic and non exact t f aoc adf or hegy tests are employed the various procedures are demonstrated and compared in illustrative empirical models and the approach is critically discussed

cartan determinants and shapovalov forms we compute the determinant of the gram matrix of the shapovalov form on weight spaces of the basic representation of an affine kac moody algebra of ade type possibly twisted as a consequence we obtain explicit formulae for the determinants of the cartan matrices of p blocks of the symmetric group and its double cover and of the associated hecke algebras at roots of unity

on generalized blocks for alternating groups abstract in a recent paper kulshammer olsson robinson gave a d analogue for the nakayama conjecture for symmetric groups where d 2 is an arbitrary integer we prove that there is a natural d analogue of the nakayama conjecture for alternating groups whenever d is 2 or an arbitrary odd integer greater than 1 this generalizes an old result of kerber

testing for individual sphericity in heterogeneous panels this manuscript considers locally best invariant tests for sphericity in heterogeneous panels a new integral representation for the characteristic function of the test statistic under the null is presented along with an algorithm for inverting it to obtain the distribution function a saddlepoint approximation to the null distribution addresses the need to quickly compute approximate p values in empirical work the approximation shows substantial improvements over the normal approximation when the cross sectional dimension is small

exact confidence sets and goodness of fit methods for stable distributions abstract usual inference methods for stable distributions are typically based on limit distributions but asymptotic approximations can easily be unreliable in such cases for standard regularity conditions may not apply or may hold only weakly this paper proposes finite sample tests and confidence sets for tail thickness and asymmetry parameters and of stable distributions the confidence sets are built by inverting exact goodness of fit tests for hypotheses which assign specific values to these parameters we propose extensions of the kolmogorov smirnov shapiro wilk and filliben criteria as well as the quantile based statistics proposed by mcculloch 1986 in order to better capture tail behavior the suggested criteria compare empirical goodness of fit or quantile based measures with their hypothesized values since the distributions involved are quite complex and non standard the relevant hypothetical measures are approximated by simulation and p values are obtained using monte carlo mc test techniques the properties of the proposed procedures are investigated by simulation in contrast with conventional wisdom we find reliable results with sample sizes as small as 25 the proposed methodology is applied to daily electricity price data in the us over the period 2001 2006 the results show clearly that heavy kurtosis and asymmetry are prevalent in these series

some impossibility theorems in econometrics with applications to structural and dynamic models general characterizations of valid confidence sets and tests in problems which involve locally almost unidentified lau parameters are provided and applied to several econo metric models two types of inference problems are studied i inference about parame ters which are not identifiable on certain subsets of the parameter space and ii inference about parameter transformations with discontinuities when a lau parameter or parametric function has an unbounded range it is shown under general regularity conditions that any valid confidence set with level 1 a for this parameter must be unbounded with probability close to 1 a in the neighborhood of nonidentification subsets and will have a nonzero probability of being unbounded under any distribution compatible with the model no valid confidence set which is almost surely bounded does exist these properties hold even if identifying restrictions are imposed similar results also obtain for parameters with bounded ranges consequently a confidence set which does not satisfy this characterization has zero coverage probability level this will be the case in particular for wald type confidence intervals based on asymptotic standard errors furthermore wald type statistics for testing given values of a lau parameter cannot be pivotal functions i e they have distributions which depend on unknown nuisance param eters and even cannot be usefully bounded over the space of the nuisance parameters these results are applied to several econometric problems inference in simultaneous equations instrumental variables iv regressions linear regressions with autoregressive errors inference about long run multipliers and cointegrating vectors for example it is shown that standard asymptotically justified confidence intervals based on iv estimators such as two stage least squares and the associated standard errors have zero coverage probability and the corresponding t statistics have distributions which cannot be bounded by any finite set of distribution functions a result of interest for interpreting iv regressions with weak instruments furthermore expansion methods e g edgeworth expansions and bootstrap techniques cannot solve these difficulties finally in a number of cases where wald type methods are fundamentally flawed e g iv regressions with poor instruments it is observed that likelihood based methods e g likelihood ratio tests and confidence sets combined with projection techniques can easily yield valid tests and confidence sets

a point optimal test for heteroscedastic disturbances abstract a new test for additive heteroscedasticity in the disturbances of the linear regression model is proposed power functions of various forms of the new test are compared empirically with those of currently favoured tests for a range of heteroscedastic models the results highlight the power advantage of a test which is mp1 at a central point in the alternative hypothesis parameter space the main conclusion is that the recommended version of the new test is generally more powerful than existing tests against medium and severe heteroscedasticity whereas the king and szroeter tests perform better against weak heteroscedasticity

isometries and extra special sylow groups of order p3 abstract a new type of isometry between the sets of irreducible characters of blocks is introduced and a conjecture is stated using the classifications of finite simple groups and saturated fusion systems over an extra special group p of order p 3 and exponent p we check that the conjecture is affirmatively answered for the principal blocks of finite groups with sylow p subgroup p

testing for linearity in a semiparametric regression model abstract we propose a new exact test for nonlinearity in a regression model with one possibly nonlinear component the test is based on the stochastic interpretation of spline smoothing given in wahba 1978 and is a point optimal invariant test as defined in king 1988 based on this stochastic model we show that the power of the exact test compares favourably with several other exact tests for nonlinearity proposed in the literature and that its level and power are robust to long and short tailed symmetric error distributions we also present an o n algorithm based on a state space approach to compute exact p values for the point optimal invariant test whereas previous implementations required o n3 operations an example is presented to illustrate the theory

exact inference methods for first order autoregressive distributed lag models methods are proposed to build exact tests and confidence sets in the linear first order autoregressive distributed lag model with i i d disturbances for general linear hypotheses on the regression coefficients inference procedures are obtained which have known level the tests proposed are either similar i e they have constant rejection probability for all data generating processes consistent with the null hypothesis or use bounds which are free of nuisance parameters correspondingly the confidence sets are either similar with known size i e they have constant coverage probability or conservative we also develop exact tests and confidence sets for various nonlinear transformations of model parameters such as long run multipliers and mean lags the practical usefulness of these exact methods which are also asymptotically valid under weak regularity conditions is illustrated by some power comparisons and with applications to a dynamic trend model of money velocity and a model of money demand

exact tests for structural change in first order dynamic models abstract several finite sample tests of parameter constancy against the presence of structural change are proposed for a linear regression model with one lagged dependent variable and independent normal disturbances the procedures derived include analysis of covariance cusum cusum of squares and predictive tests the approach used to obtain the tests involves the application of three techniques derivation of an exact confidence set for the autoregressive parameter based on using an appropriately extended regression a union intersection technique and when required randomization the tests proposed are illustrated with some artificial data and applied to a dynamic trend model of gross private domestic investment in the u s

an analysis of tests for regression coefficient stability abstract an exact small sample point optimal test is developed for testing the hypothesis that a regression coefficient is constant against the alternative that it is generated by a first order autoregressive process in addition it is shown how watson and engle s 1985 asymptotic test can be modified so that its small sample level can be computed and its small sample power properties investigated simulation results show that the point optimal test developed in this paperoutperforms the small sample watson engle test as well as other tests given in the literature a point optimal test is also constructed for a stochastic coefficient generated by an arima 1 1 0 process

perfect isometries and murnaghan nakayama rules acknowledgement the authors wish to thank the referee for a very careful and precise reading of several earlier versions of this manuscript they are grateful for the number quality and helpfulness of comments and suggestions received

on asymptotic expansions of twisted products the series development of the quantum mechanical twisted product is studied the series is shown to make sense as a moment asymptotic expansion of the integral formula for the twisted product either pointwise or in the distributional sense depending on the nature of the factors a condition is given that ensures convergence and is stronger than previously known results possible applications are examined

locally best invariant tests of the error covariance matrix of the linear regression model summary this paper considers a class of hypothesis testing problems concerning the covariance matrix of the disturbances in the classical linear regression model a test that is locally best invariant against one sided alternative hypotheses is constructed and shown to be identical to a one sided version of the lagrange multiplier test