edgeworth correction by bootstrap in autoregressions we prove that the distribution of least squares estimates in autoregressions can be bootstrapped with accuracy o n 1 2 a s thereby improving the normal approximation error of o n

testing for the constancy of parameters over time abstract tests are proposed for detecting possible changes in parameters when the observations are obtained sequentially in time while deriving the tests the alternative one has in mind specifies the parameter process as a martingale the distribution theory of these tests relies on the large sample results that is only the limiting null distributions are known except in very special cases the main tool in establishing these limiting distributions is weak convergence of stochastic processes suppose that we have vector valued observations x 1 x n obtained sequentially in time or ordered in some other linear fashion their joint distribution is described by determining the initial distribution for x 1 and the conditional distribution for each x k given the past up to x k 1 suppose further that these distributions depend on a p dimensional parameter vector at least locally i e in a short time period this may be more or less legitimate in the long run however the possibility of some ch

the jackknife and the bootstrap for general stationary observations we extend the jackknife and the bootstrap method of estimating standard errors to the case where the observations form a general stationary sequence we do not attempt a reduction to i i d values the jackknife calculates the sample variance of replicates of the statistic obtained by omitting each block of l consecutive data once in the case of the arithmetic mean this is shown to be equivalent to a weighted covariance estimate of the spectral density of the observations at zero under appropriate conditions consistency is obtained if l l n rightarrow infty and l n n rightarrow 0 general statistics are approximated by an arithmetic mean in regular cases this approximation determines the asymptotic behavior bootstrap replicates are constructed by selecting blocks of length l randomly with replacement among the blocks of observations the procedures are illustrated by using the sunspot numbers and some simulated data

the use of subseries values for estimating the variance of a general statistic from a stationary sequence let z i infty i infty be a strictly stationary infty mixing sequence without specifying the dependence model giving rise to z i and without specifying the marginal distribution of z i we address the question of variance estimation for a general statistic t n t n z 1 z n for estimating var t n from just the available data z 1 z n we propose computing subseries values t m z i 1 z i 2 z i m 0 leq i i m leq n these subseries values are used as replicates to model the sampling variability of t n in particular we use adjacent nonoverlapping subseries of length m m n with m rightarrow infty and m n n rigtharrow 0 our variance estimator is just the usual sample variance computed amongst these subseries values after appropriate standardization this estimator is shown to be consistent under mild integrability conditions we present optimal i e minimum m s e choices of m n for the special case where t n overset z n and z i is a normal ar 1 sequence a simulation study is conducted showing that those same choices of m n are effective when t n is a robust estimator of location and z i is subject to contamination

on the correlation distribution for a niho decimation let p be a prime n 2m and d 3p m 2 with m geq 2 and gcd d p n 1 1 in this paper the correlation distribution between a p ary m sequence of period p n 1 and its d decimation sequence is investigated in a unified approach some results for the binary case are extended to the general case it is shown that the problem of determining the correlation distribution for d can be reduced to that of solving two combinatorial problems related to the unit circle of the finite field mathbb f p n for an arbitrary odd prime p it seems difficult to solve these two problems however for p 3 by studying the weight distribution of the ternary zetterberg code and counting the numbers of solutions of some equations over mathbb f 3 n the two problems are solved and thus the corresponding correlation distribution for d is completely determined it is noteworthy that this is the first time that the correlation distribution for a non binary niho decimation has been determined since 1976

a trend resistant test for structural change based on ols residuals abstract we discuss the null distribution and local power of an ols based version of the cusum test and show that this contrary to other tests for structural change like e g the ploberger kontrus kramer fluctuation test can also be applied to trending data

testing for co integration in vector autoregressions with non stationary volatility abstract many key macroeconomic and financial variables are characterized by permanent changes in unconditional volatility in this paper we analyse vector autoregressions with non stationary unconditional volatility of a very general form which includes single and multiple volatility breaks as special cases we show that the conventional rank statistics computed as in johansen 1988 johansen 1991 are potentially unreliable in particular their large sample distributions depend on the integrated covariation of the underlying multivariate volatility process which impacts on both the size and power of the associated co integration tests as we demonstrate numerically a solution to the identified inference problem is provided by considering wild bootstrap based implementations of the rank tests these do not require the practitioner to specify a parametric model for volatility or to assume that the pattern of volatility is common to or independent across the vector of series under analysis the bootstrap is shown to perform very well in practice

testing superexogeneity and invariance in regression models abstract this paper introduces tests of superexogeneity and invariance under the null hypothesis the conditional model exhibits parameter constancy while under the alternative shifts in the process of the independent variables induces shifts in the conditional model the test is sensitive to particular types of parameter nonconstancy especially with changing variances and covariances we relate the test to rational expectations models and the lucas critique an empirical example of money demand finds prices and interest rates superexogenous in a conditional model but when the inflation specification changes superexogeneity fails although standard specification tests do not

the moments of a kloosterman sum and the weight distribution of a zetterberg type binary cyclic code in this correspondence we give the moments of a kloosterman sum over fq in terms of the frequencies of weights in the binary zetterberg code of length q 1 which are known by the work of schoof and van der vlugt the method is illustrated by giving explicit formulae for the moments up to the tenth moment as a corollary the weight distribution of a zetterberg type binary cyclic code is obtained

finite sample simulation based inference in var models with application to granger causality testing tests in vector autoregressive var models are typically based on large sample approximations involving the use of asymptotic distributions or bootstrap techniques after documenting that such methods can be very misleading even with fairly large samples we propose a general simulation based technique that allows one to control completely test levels in parametric var models in particular we show that maximized monte carlo tests dufour 2005 monte carlo tests with nuisance parameters a general approach to finite sample inference and nonstandard asymptotics in econometrics journal of econometrics forthcoming can provide provably exact tests for such models whether they are stationary or integrated applications to order selection and causality testing are considered as special cases the technique developed is applied to a var model of the u s economy

inference when a nuisance parameter is not identified under the null hypothesis many econometric testing problems involve nuisance parameters which are not identified under the null hypotheses this paper studies the asymptotic distribution theory for such tests the asymptotic distributions of standard test statistics are described as functionals of chi square processes in general the distributions depend upon a large number of unknown parameters we show that a transformation based upon a conditional probability measure yields an asymptotic distribution free of nuisance parameters and we show that this transformation can be easily approximated via simulation the theory is applied to threshold models with special attention given to the so called self exciting threshold autoregressive model monte carlo methods are used to assess the finite sample distributions the tests are applied to u s gnp growth rates and we find that potter s 1995 threshold effect in this series can be possibly explained by sampling variation

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

bootstrapping general empirical measures it is proved that the bootstrapped central limit theorem for empirical processes indexed by a class of functions mathscr f and based on a probability measure p holds a s if and only if mathscr f in mathrm clt p and int f 2 dp infty where f sup f in mathscr f f and it holds in probability if and only if mathscr f in mathrm clt p thus for a large class of statistics no local uniformity of the clt about p is needed for the bootstrap to work consistency of the bootstrap the bootstrapped law of large numbers is also characterized these results are proved under certain weak measurability assumptions on mathscr f

some new results on the cross correlation of m sequences the determination of the cross correlation between an m sequence and its decimated sequence has been a longstanding research problem considering a ternary m sequence of period 33r 1 we determine the cross correlation distribution for decimations d 3r 2 and d 32r 2 where gcd r 3 1 meanwhile for a binary m sequence of period 22lm 1 we make an initial investigation for the decimation d 22lm 1 2m 1 2s where l 2 is even and 0 s 2m 1 it is shown that the cross correlation takes at least four values furthermore we confirm the validity of two famous conjectures due to sarwate et al and helleseth in this case

niho type cross correlation functions via dickson polynomials and kloosterman sums suppose that n 2k is even we study the cross correlation function between two m sequences for niho type decimations d 2 sup k 1 s 1 we develop a new technique to study the value distribution of these cross correlation functions which makes use of dickson polynomials as a first application we derive here the distribution of the six valued cross correlation function for s 3 and odd k up to a term which depends on kloosterman sums in addition applying simpler methods we prove a theorem providing niho type decimations with four valued cross correlation functions and their distribution we conjecture that the latter result actually covers all such decimations

on the walsh transform of a class of functions from niho exponents in this paper a class of functions from niho exponents with four valued walsh transform is obtained for any prime by a uniform method and the distribution of the walsh transform values is also completely determined in particular this class of functions is proven to be bent for a special case although it is shown that the obtained bent functions are equivalent to the leander kholosha s class of bent functions a direct and much simpler proof for the bentness of this kind of niho functions is provided

invalidity of bootstrap for critical branching processes with immigration this aiticle considers a case of parametric bootstrap when the observations consist of generation sizes of the branching process with immigration together with the immigration component of each generation suppose we estimate the offspring mean m by the maximum likelihood estimator m l e it is then shown that the bootstrap version of the standardized m l e does not have the same limiting distribution as the standardized m l e under the assumption that m 1 critical case in other words the asymptoticvalidity does not hold for the parametric bootstrap in the critical case in fact given the sample the value of the conditional distribution function of the bootstrap version of standardized m l e defines a sequence of random variables whose limit in distribution is also shown to be a random variable when m 1 the approach used here is via a sequence of branching processes for which a general weak convergence in d 0 oo 1 result is established using operator semigroup convergence theorems

a model of fractional cointegration and tests for cointegration using the bootstrap abstract the paper proposes a framework for modelling cointegration in fractionally integrated processes and considers methods for testing the existence of cointegrating relationships using the parametric bootstrap in these procedures arfima models are fitted to the data and the estimates used to simulate the null hypothesis of non cointegration in a vector autoregressive modelling framework the simulations are used to estimate p values for alternative regression based test statistics including the f goodness of fit statistic the durbin watson statistic and estimates of the residual d the bootstrap distributions are economical to compute being conditioned on the actual sample values of all but the dependent variable in the regression the procedures are easily adapted to test stronger null hypotheses such as statistical independence the tests are not in general asymptotically pivotal but implemented by the bootstrap are shown to be consistent against alternatives with both stationary and nonstationary cointegrating residuals as an example the tests are applied to the series for uk consumption and disposable income the power properties of the tests are studied by simulations of artificial cointegrating relationships based on the sample data the f test performs better in these experiments than the residual based tests although the durbin watson in turn dominates the test based on the residual d

how to implement the bootstrap in static or stable dynamic regression models test statistic versus confidence region approach by combining two alternative formulations of a test statistic with two alternative resampling schemes we obtain four different bootstrap tests in the context of static linear regression models two of these are shown to have serious size and power problems whereas the remaining two are adequate and in fact equivalent the equivalence between the two valid implementations is shown to break down in dynamic regression models then the procedure based on the test statistic approach performs best at least in the ar 1 model similar finite sample phenomena are illustrated in the arma 1 1 model through a small scale monte carlo study and an empirical example

sample splitting and threshold estimation threshold models have a wide variety of applications in economics direct applications include models of separating and multiple equilibria other applications include empirical sample splitting when the sample split is based on a continuously distributed variable such as firm size in addition threshold models may be used as a parsimonious strategy for nonparametric function estimation for example the threshold autoregressive model tar is popular in the nonlinear time series literature threshold models also emerge as special cases of more complex statistical frameworks such as mixture models switching models markov switching models and smooth transition threshold models it may be important to understand the statistical properties of threshold models as a preliminary step in the development of statistical tools to handle these more complicated structures despite the large number of potential applications the statistical theory of threshold estimation is undeveloped it is known that threshold estimates are super consistent but a distribution theory useful for testing and inference has yet to be provided this paper develops a statistical theory for threshold estimation in the regression context we allow for either cross section or time series observations least squares estimation of the regression parameters is considered an asymptotic distribution theory for the regression estimates the threshold and the regression slopes is developed it is found that the distribution of the threshold estimate is nonstandard a method to construct asymptotic confidence intervals is developed by inverting the likelihood ratio statistic it is shown that this yields asymptotically conservative confidence regions monte carlo simulations are presented to assess the accuracy of the asymptotic approximations the empirical relevance of the theory is illustrated through an application to the multiple equilibria growth model of durlauf and johnson 1995

bootstrapping cointegrating regressions abstract in this paper we consider bootstrapping cointegrating regressions it is shown that the method of bootstrap if properly implemented generally yields consistent estimators and test statistics for cointegrating regressions for the cointegrating regression models driven by general linear processes we employ the sieve bootstrap based on the approximated finite order vector autoregressions for the regression errors and the first differences of the regressors in particular we establish the bootstrap consistency for ols method the bootstrap method can thus be used to correct for the finite sample bias of the ols estimator and to approximate the asymptotic critical values of the ols based test statistics in general cointegrating regressions the bootstrap ols procedure however is not efficient for the efficient estimation and hypothesis testing we consider the procedure proposed by saikkonen 1991 asymptotically efficient estimation of cointegration regressions econometric theory 7 1 21 and stock and watson 1993 a simple estimator of cointegrating vectors in higher order integrating systems econometrica 61 783 820 relying on the regression augmented with the leads and lags of differenced regressors the bootstrap versions of their procedures are shown to be consistent and can be used to do asymptotically valid inferences a monte carlo study is conducted to investigate the finite sample performances of the proposed bootstrap methods

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

bootstrapping the hegy seasonal unit root tests this paper proposes bootstrap versions of the seasonal unit root tests of inter alia hylleberg engle granger and yoo j econometrics 44 1990 215 238 hegy we report a simulation study of the properties of both the conventional and bootstrapped seasonal unit root tests when applied to series having higher order serial correlation and or periodic heteroscedasticity both of which are known to severely distort the significance level of the conventional tests our results demonstrate that the bootstrap provides good approximations to the statistics null distributions moreover the bootstrap corrects the adverse effects of data dependent lag selection seen in the conventional augmented hegy tests the bootstrapped tests have comparable power to infeasible exactly significance level corrected lag augmented hegy tests and their use is recommended

alternative bootstrap procedures for testing cointegration in fractionally integrated processes abstract this paper considers alternative methods of testing cointegration in fractionally integrated processes using the bootstrap the investigation focuses on a choice of statistic b use of bias correction techniques and c designing the simulation of the null hypothesis three residual based tests are considered two of the null hypothesis of non cointegration the third of the null hypothesis that cointegration exists the tests are compared in monte carlo experiments to throw light on the relative roles of issues a c in test performance

weight formulas for ternary melas codes in this paper we derive a formula for the frequencies of the weights in ternary melas codes and we illustrate this formula by computing a table of examples

testing for structural change in conditional models in the past decade we have seen the development of a new set of tests for structural change of unknown timing in regression models most notably the supf statistic of andrews 1993 econometrica 61 825 856 the expf and avef statistics of andrewsploberger 1994 econometrica 62 1383 1414 and the statistic of nyblom 1989 journal of american statistical association 84 223 230 the distribution theory used for these tests is primarily asymptotic and has been derived under the maintained assumption that the regressors are stationary this excludes structural change in the marginal distribution of the regressors as a result these tests technically cannot discriminate between structural change in the conditional and marginal distributions this paper attempts to remedy this de ciency by deriving the large sample distributions of the test statistics allowing for structural change in the marginal distribution of the regressors we nd that the asymptotic distributions of the supf expf avef and statistics are not invariant to structural change in the regressors to solve the size problem we introduce a xed regressor bootstrap which achieves the rst order asymptotic distribution and appears to possess reasonable size properties in small samples our bootstrap theory allows for arbitrary structural change in the regressors including structural shifts polynomial trends and exogenous stochastic trends it allows for lagged dependent variables and heteroskedastic error processes 2000 elsevier science s a all rights reserved

an open problem on the distribution of a niho type cross correlation function in this paper let n 2k and d 3 cdot 2 k 2 with k geq 3 and gcd d 2 n 1 1 based on some analysis of certain equations over finite fields and the number of codewords with hamming weight five in zetterberg code the correlation distribution between a binary m sequence of period 2 n 1 and its d decimation sequence is completely determined this solves a ten year old open problem proposed by dobbertin et al

bootstrap unit root tests in panels with cross sectional dependency we apply bootstrap methodology to unit root tests for dependent panels with n cross sectional units and t time series observations more specifically we let each panel be driven by a general linear process which may be different across cross sectional units and approximate it by a finite order autoregressive integrated process of order increasing with t as we allow the dependency among the innovations generating the individual series we construct our unit root tests from the estimation of the system of the entire n cross sectional units the limit distributions of the tests are derived by passing t to infinity with n fixed we then apply bootstrap method to the approximated autoregressions to obtain critical values for the panel unit root tests and establish the asymptotic validity of such bootstrap panel unit root tests under general conditions the proposed bootstrap tests are indeed quite general covering a wide class of panel models they in particular allow for very general dynamic structures which may vary across individual units and more importantly for the presence of arbitrary cross sectional dependency the finite sample performance of the bootstrap tests is examined via simulations and compared to that of commonly used panel unit root tests we find that our bootstrap tests perform relatively well especially when n is small

on contracted semigroup rings sufficient conditions are given for a contracted semigroup ring in which the two sided ideals have a certain property to be a semiprime b semiprimitive c prime d primitive the results are applied to the contracted semigroup rings of inverse semigroups where they provide new proofs of theorems of domanov

a bootstrap theory for weakly integrated processes abstract this paper develops a bootstrap theory for models including autoregressive time series with roots approaching to unity as the sample size increases in particular we consider the processes with roots converging to unity with rates slower than n 1 we call such processes weakly integrated processes it is established that the bootstrap relying on the estimated autoregressive model is generally consistent for the weakly integrated processes both the sample and bootstrap statistics of the weakly integrated processes are shown to yield the same normal asymptotics moreover for the asymptotically pivotal statistics of the weakly integrated processes the bootstrap is expected to provide an asymptotic refinement and give better approximations for the finite sample distributions than the first order asymptotic theory for the weakly integrated processes the magnitudes of potential refinements by the bootstrap are shown to be proportional to the rate at which the root of the underlying process converges to unity the order of boostrap refinement can be as large as o n 1 2 for any 0 our theory helps to explain the actual improvements observed by many practitioners which are made by the use of the bootstrap in analyzing the models with roots close to unity

some results on cross correlation distribution between a p ary m sequence and its decimated sequences for an odd prime p and two positive integers m k such that m gcd k m 3 is odd let d be a positive integer satisfying d p k 1 2 mod p m 1 i n this paper the cross correlation between a p ary m sequence and its d decimated sequences is investigated and the cross correlation distribution is completely determined the relationship between the decimations d considered in this paper and some known ones is also studied this paper generalizes some previous results and also gives new decimations which lead to low cross correlation