a numerical ode solver that preserves the fixed points and their stability in this paper we provide a one step predictor corrector method for numerically solving first order differential initial value problems with two fixed points the method preserves the stability behaviour of the fixed points which results in an efficient integrator for this kind of problem some numerical examples are provided to show the good performance of the method

higher order properties of gmm and generalized empirical likelihood estimators in an effort to improve the small sample properties of generalized method of moments gmm estimators a number of alternative estimators have been suggested these include empirical likelihood el continuous updating and exponential tilting estimators we show that these estimators share a common structure being members of a class of generalized empirical likelihood gel estimators we use this structure to compare their higher order asymptotic properties we find that gel has no asymptotic bias due to correlation of the moment functions with their jacobian eliminating an important source of bias for gmm in models with endogeneity we also find that el has no asymptotic bias from estimating the optimal weight matrix eliminating a further important source of bias for gmm in panel data models we give bias corrected gmm and gel estimators we also show that bias corrected el inherits the higher order property of maximum likelihood that it is higher order asymptotically efficient relative to the other bias corrected estimators

diagnostic tests for homoskedasticity in spatial cross sectional or panel models abstract we propose an adjusted quasi score aqs method for constructing tests for homoskedasticity in spatial econometric models we first obtain an aqs function by adjusting the score type function from the given model to achieve unbiasedness and then develop an outer product of martingale difference opmd estimate of its variance in standard problems where a genuine quasi score vector is available the aqs opmd method leads to finite sample improved tests over the usual methods more importantly in non standard problems where a genuine quasi score is not available and the usual methods fail the proposed aqs opmd method provides feasible solutions the aqs tests are formally derived and asymptotic properties examined for three representative models spatial cross sectional static and dynamic panel models monte carlo results show that the proposed aqs tests have good finite sample properties

central limit theorems for quadratic forms with time domain conditions we establish the central limit theorem for quadratic forms t s 1 n b t s p m n x t x s of the bivariate appell polynomials p m n x t x s under time domain conditions these conditions relate the weights b t and the covariances of the sequences p m n x t x s and x t the time domain approach together with the spectral domain approach developed earlier yields a general set of conditions for central limit theorems

cross section regression with common shocks this paper considers regression models for cross section data that exhibit cross section dependence due to common shocks such as macroeconomic shocks the paper analyzes the properties of least squares ls estimators in this context the results of the paper allow for any form of cross section dependence and heterogeneity across population units the probability limits of the ls estimators are determined and necessary and sufficient conditions are given for consistency the asymptotic distributions of the estimators are found to be mixed normal after recentering and scaling the t wald and f statistics are found to have asymptotic standard normal x 2 and scaled x 2 distributions respectively under the null hypothesis when the conditions required for consistency of the parameter under test hold however the absolute values of t wald and f statistics are found to diverge to infinity under the null hypothesis when these conditions fail confidence intervals exhibit similarly dichotomous behavior hence common shocks are found to be innocuous in some circumstances but quite problematic in others models with factor structures for errors and regressors are considered using the general results conditions are determined under which consistency of the ls estimators holds and fails in models with factor structures the results are extended to cover heterogeneous and functional factor structures in which common factors have different impacts on different population units

a boundary value technique for singular perturbation problems abstract a new way to solve singular perturbation problems is introduced it is designed for the practicing engineer or applied mathematician who needs a practical tool for these problems easy to use modest problem preparation and ready computer implementation as with other methods the original problem is partitioned into inner and outer solution differential equation systems however asymptotic solution techniques are not employed the method is distinguished by the following facts the inner solution problem is solved as a two point boundary problem where the terminal boundary conditions are supplied by the solution of the outer solution problem in turn the outer solution differential equations are solved as an initial value system where the initial conditions are supplied by the inner solution at its terminal point the method is iterative on the terminal point of the inner solution problem three numerical examples are included

instrumental variable estimation of an error components model on propose des estimateurs efficaces en considerant des hypotheses alternatives sur les sources d endogeneite et les proprietes de variance covariance des perturbations

initial conditions and moment restrictions in dynamic panel data models estimation of the dynamic error components model is considered using two alternative linear estimators that are designed to improve the properties of the standard first differenced gmm estimator both estimators require restrictions on the initial conditions process asymptotic efficiency comparisons and monte carlo simulations for the simple ar 1 model demonstrate the dramatic improvement in performance of the proposed estimators compared to the usual first differenced gmm estimator and compared to non linear gmm the importance of these results is illustrated in an application to the estimation of a labour demand model using company panel data

dynamic spatial panel models networks common shocks and sequential exogeneity this paper considers a class of generalized methods of moments gmm estimators for general dynamic panel models allowing for weakly exogenous covariates and cross sectional dependence due to spatial lags unspecified common shocks and time varying interactive effects we significantly expand the scope of the existing literature by allowing for endogenous time varying spatial weight matrices without imposing explicit structural assumptions on how the weights are formed an important area of application is in social interaction and network models where our specification can accommodate data dependent network formation we consider an exemplary social interaction model and show how identification of the interaction parameters is achieved through a combination of linear and quadratic moment conditions for the general setup we develop an orthogonal forward differencing transformation to aid in the estimation of factor components while maintaining orthogonality of moment conditions this is an important ingredient to a tractable asymptotic distribution of our estimators in general the asymptotic distribution of our estimators is found to be mixed normal due to random norming however the asymptotic distribution of our test statistics is still chi square

monte carlo evidence on panel data regressions with ar 1 disturbances and an arbitrary variance on the initial observations abstract for the ar 1 model one implicitly assumes a specific variance on the initial observation or that the process started a long time ago to insure homoscedasticity this paper investigates anderson and hsiao s 1982 suggestion of an arbitrary variance on the initial observation of the autoregressive process this is done in a panel data framework regardless of when the ar 1 process started one can translate this starting date into an effective initial variance assumption this effective initial variance can be estimated with panel data and tested for departures from homoscedasticity the consequences of this departure on various estimators is studied via monte carlo experiments the following estimators are compared ols iterative cochrane orcutt ico and two maximum likelihood mle estimators one corresponding to the usual stationary assumption denoted by bm beach and mackinnon 1978 and another mle corresponding to the arbitrary variance assumption on the initial observations both mle s can be achieved as iterative gls the former using a prais winsten pw transformation and the latter using a generalized pw transformation gpw some of the major findings are the following bm performs poorly for large departures of the variance of the initial observations from the usual homoscedastic assumption whereas the mle corresponding to the gpw transformation is robust to various assumptions on the variance of the initial observations ico continues to perform poorly relative to estimators that use all time series observations even though the co estimator of performs well compared to that of gpw the t statistic which tests the true value of the parameter may be misleading for ols and bm depending upon the departure of the initial variance from the homoscedastic assumption however this t statistic performs well for the gpw estimator finally the likelihood ratio test which tests the usual homoscedastic assumption on the initial disturbances and a pre test estimator based on this test perform well in monte carlo experiments and are recommended

application of statistical mechanics to np complete problems in combinatorial optimisation recently developed techniques of the statistical mechanics of random systems are applied to the graph partitioning problem the averaged cost function is calculated and agrees well with numerical results the problem bears close resemblance to that of spin glasses the authors find a spin glass transition in the system and the low temperature phase space has an ultrametric structure this sheds light on the nature of hard computation problems

origins of the limited information maximum likelihood and two stage least squares estimators theil basmann and sargan are often credited with the development of the two stage least squares tsls estimator of the coefficients of one structural equation in a simultaneous equations model however anderson and rubin had earlier derived the asymptotic distribution of the limited information maximum likelihood liml estimator by finding the asymptotic distribution of what is essentially the tsls estimator

efficient estimation using panel data in an important recent paper hausman and taylor 1981 hereafter ht considered the instrumental variable estimation of a regression model using panel data when the individual effects may be correlated with a subset of the explanatory variables they provided a simple consistent estimator and an efficient estimator more recently amemiya and macurdy 1986 hereafter am have suggested an alternative estimator which is more efficient than the ht estimator under certain conditions and given stronger assumptions than ht made however the relationship between the ht and am papers is less clear than it might be in part because of notational differences between the two papers in this paper we clarify the relationship between the ht and am estimators and we show that the difference between these estimators lies in the treatment of the time varying explanatory variables which are uncorrelated with the effects ht use each such variable as two instruments means and deviations from means while am use such variables as t 1 instruments as deviations from means and also separately for each of the t available time periods this enables us to make clear the conditions under which the am estimator is more efficient than the ht estimator we also present each estimator in a form which allows it to be calculated using standard instrumental variables two stage least squares software following the am path one step further we then define a third bms estimator which under yet stronger assumptions is more efficient than the am estimator both ht and am use as instruments the deviations from means of the time varying variables which are correlated with the effects a more efficient estimator may be obtained by using separately the t 1 linearly independent values of these deviations from individual means consistency requires that these be legitimate instruments and whether this is so depends on why these time varying variables are correlated with the effects for example if such correlation arises solely because of a time invariant component which is removed in taking deviations from individual means these instruments are legitimate

two low accuracy methods for stiff systems two low accuracy explicit one step methods for stiff ordinary differential equations are extended directly to solve systems of equations some defects of the component form of these methods are avoided to perform these a new set of vector computations are introduced some numerical experiments are presented to show the superiority of the new methods

a technique for the numerical solution of initial value problems based on a class of birkhoff type interpolation method in this paper a class of birkhoff type interpolation problems on arbitrary nodal points is studied the explicit representation characterization the uniqueness and the error function are explicitly given furthermore we apply the obtained birkhoff type interpolation method to find i the numerical solution of high order initial value problems ivps and the corresponding error of this approximation ii the approximation of some special functions with their explicit error functions and iii new interpolatory type quadrature formulae of precision degree at least m n 1 and m kn 1 m n email protected n n k 2 numerical examples are included to demonstrate the validity and applicability of the technique proposed in this paper and a comparison is made with the existing results the results reveal that the new method is effective simple and accurate

some comments on preference order dynamic programming models abstract a simple deterministic dynamic programming model is used as a general framework for the analysis of stochastic versions of three classical optimization problems knapsack traveling salesperson and assembly line balancing problems it is shown that this model can provide an alternative to the preference order models proposed for these problems counterexample to the optimality of the preference order models are presented

arbitrage factor structure and mean variance analysis on large asset markets we examine the implications of arbitrage in a market with many assets the absence of arbitrage opportunities implies that the linear functionals that give the mean and cost of a portfolio are continuous hence there exist unique portfolios that represent these functionals these portfolios span the mean variance efficient set we resolve the question of when a market with many assets permits so much diversification that risk free investment opportunities are available ross 12 14 showed that if there is a factor structure then the mean returns are approximately linear functions of factor loadings we define an approximate factor structure and show that this weaker restriction is sufficient for ross result if the covariance matrix of the asset returns has only k unbounded eigenvalues then there is an approximate factor structure and it is unique the corresponding k eigenvectors converge and play the role of factor loadings hence only a principal component analysis is needed in empirical work

panel data models with multiple time varying individual effects abstract this paper considers a panel data model with time varying individual effects the data are assumed to contain a large number of cross sectional units repeatedly observed over a fixed number of time periods the model has a feature of the fixed effects model in that the effects are assumed to be correlated with the regressors the unobservable individual effects are assumed to have a factor structure for consistent estimation of the model it is important to estimate the true number of individual effects we propose a generalized methods of moments procedure by which both the number of individual effects and the regression coefficients can be consistently estimated some important identification issues are also discussed our simulation results indicate that the proposed methods produce reliable estimates

numerical treatment of singular initial value problems abstract an efficient extrapolation scheme whose basic integrator is the inverse euler scheme cf fatunla 1982 is proposed for nonlinear singular initial value problems y x y y 0 y 0 the automatic polynomial rational extrapolation code difexi deufhard 1983 1985 is modified to accomodate the basic integrator the new algorithm was implemented in variable step variable order mode and compares favourably with the earlier works

nonlinear multistep methods for initial value problems abstract some k step kth order explicit nonlinear multistep methods nmm are proposed for both stiff and singular initial value problems the algorithms are based on a local representation of the solution by the reciprocal of a polynomial for a k step scheme there is need to solve a linear system of equations for the k coefficients of the inverse polynomial which are apparently global for linear differential equations the resultant integration schemes are stable and convergent and compare favorably with the methods of lambert and shaw 2 luke fair and wimp 7 and lambert 1

generalized method of moments with many weak moment conditions using many moment conditions can improve efficiency but makes the usual generalized method of moments gmm inferences inaccurate two step gmm is biased generalized empirical likelihood gel has smaller bias but the usual standard errors are too small in instrumental variable settings in this paper we give a new variance estimator for gel that addresses this problem it is consistent under the usual asymptotics and under many weak moment asymptotics is larger than usual and is consistent we also show that the kleibergen 2005 lagrange multiplier and conditional likelihood ratio statistics are valid under many weak moments in addition we introduce a jackknife gmm estimator but find that gel is asymptotically more efficient under many weak moments in monte carlo examples we find that t statistics based on the new variance estimator have nearly correct size in a wide range of cases

linearization techniques for singularly perturbed initial value problems of ordinary differential equations linearization methods for singularly perturbed initial value problems in ordinary differential equations based on taylor series expansions and piecewise analytical solutions are presented if only the linear terms are retained it is shown that exponentially fitted techniques are obtained whereas if quadratic terms are kept then riccati s equations result both linear and quadratic linearization methods are shown to be nonuniformly convergent in the small perturbation parameter and the former are more accurate than the latter in equally spaced grids linearization methods are also used with three different adaptive mesh refinement methods that account for the magnitude of the jacobian the variation of the solution and the resolution of the initial layer and it is shown that mesh refinement methods based on the variation of the solution can provide very accurate solutions although they may yield large errors at the outer edge of the initial layer if the grid spacing increases rapidly it is also shown that for values of the perturbation parameter on the order of unity quadratic linearization methods are more accurate than linearization methods and can account for the presence of internal layers it is also shown that linearization methods on grids whose spacing obeys a geometric progression are not robust and may suffer from a loss in the order of convergence compared with adaptive methods that resolve the boundary layer or adapt the grid according to the numerical solution

contributions to the development of differential systems exactly solved by multistep finite difference schemes abstract the motivation underlying this contribution has been to complete some of the topics concerning exact schemes for numerically solving ordinary differential equations a procedure for obtaining differential systems exactly solved by a given finite difference method is described examples illustrating the application of the procedure for obtaining first order second order and systems of differential equations exactly solved by different numerical methods are given among the numerical methods considered there are the trapezoidal rule the two step adams bashforth method and the numerov method some numerical examples are presented to provide evidence that the procedure works properly

determining the number of factors in approximate factor models in this paper we develop some econometric theory for factor models of large dimensions the focus is the determination of the number of factors r which is an unresolved issue in the rapidly growing literature on multifactor models we first establish the convergence rate for the factor estimates that will allow for consistent estimation of r we then propose some panel criteria and show that the number of factors can be consistently estimated using the criteria the theory is developed under the framework of large cross sections n and large time dimensions t no restriction is imposed on the relation between n and t simulations show that the proposed criteria have good finite sample properties in many configurations of the panel data encountered in practice

another look at the instrumental variable estimation of error components models this article develops a framework for efficient iv estimators of random effects models with information in levels which can accommodate predetermined variables our formulation clarifies the relationship between the existing estimators and the role of transformations in panel data models we characterize the valid transformations for relevant models and show that optimal estimators are invariant to the transformation used to remove individual effects we present an alternative transformation for models with predetermined instruments which preserves the orthogonality among the errors finally we consider models with predetermined variables that have constant correlation with the effects and illustrate their importance with simulations

statistical mechanics of an np complete problem subset sum we study the statistical properties of an np complete problem the subset sum using the methods and concepts of statistical mechanics the problem is a generalization of the number partitioning problem which is also an npcomplete problem and has been studied in the physics literature the asymptotic expressions for the number of solutions are obtained these results applied to the number partitioning problem as a special case are compared with those which were previously obtained by a different method we discuss the limit of applicability of the techniques of statistical mechanics to the present problem

nearly efficient estimation of time series models with predetermined but not exogenous instruments particularly under the assumption of rational expectations a model may have serially correlated errors and those errors may be uncorrelated with contemporaneous and lagged values of a predetermined instrument yet the instruments may not be strictly exogenous this paper proposes a method for transforming such a model to one without serial correlation while keeping the instrument predetermined standard theory of instrumental variables estimation then applies furthermore it turns out that for transformations of the class proposed asymptotic distribution theory is the same whether the serial correlation properties of the errors are known a priori or estimated as the number of lagged values of the predetermined variables used as instruments increases the asymptotic variance of the standard instrumental variables estimator applied to the transformed model approaches that of the optimal estimator proposed by hansen and sargent 8 in a number of recently developed macroeconomic models behavioral equations arise in which error terms can be asserted on the basis of economic arguments to be uncorrelated with some set of instrumental variables at a certain set of dates but not to be uncorrelated with the instruments at all dates examples of such

efficient gmm estimation of spatial dynamic panel data models with fixed effects abstract in this paper we derive the asymptotic properties of gmm estimators for the spatial dynamic panel data model with fixed effects when n is large and t can be large but small relative to n the gmm estimation methods are designed with the fixed individual and time effects eliminated from the model and are computationally tractable even under circumstances where the ml approach would be either infeasible or computationally complicated the ml approach would be infeasible if the spatial weights matrix is not row normalized while the time effects are eliminated and would be computationally intractable if there are multiple spatial weights matrices in the model also consistency of the mle would require t to be large and not small relative to n if the fixed effects are jointly estimated with other parameters of interest the gmm approach can overcome all these difficulties we use exogenous and predetermined variables as instruments for linear moments along with several levels of their neighboring variables and additional quadratic moments we stack up the data and construct the best linear and quadratic moment conditions an alternative approach is to use separate moment conditions for each period which gives rise to many moments estimation we show that these gmm estimators are n t consistent asymptotically normal and can be relatively efficient we compare these approaches on their finite sample performance by monte carlo

an error corrected euler method for solving stiff problems based on chebyshev collocation in this paper we present error corrected euler methods for solving stiff initial value problems which not only avoid unnecessary iteration process that may be required in most implicit methods but also have such a good stability as all implicit methods possess the proposed methods use a chebyshev collocation technique as well as an asymptotical linear ordinary differential equation of first order derived from the difference between the exact solution and the euler s polygon these methods with or without the jacobian are analyzed in terms of convergence and stability in particular it is proved that the proposed methods have a convergence order up to 4 regardless of the usage of the jacobian numerical tests are given to support the theoretical analysis as evidences

consistent moment selection procedures for generalized method of moments estimation this paper considers a generalized method of moments gmm estimation problem in which one has a vector of moment conditions some of which are correct and some incorrect the paper introduces several procedures for consistently selecting the correct moment conditions the procedures also can consistently determine whether there is a sufficient number of correct moment conditions to identify the unknown parameters of interest the paper specifies moment selection criteria that are gmm analogues of the widely used bic and aic model selection criteria the latter is not consistent the paper also considers downward and upward testing procedures all of the moment selection procedures discussed in this paper are based on the minimized values of the gmm criterion function for different vectors of moment conditions the procedures are applicable in time series and cross sectional contexts application of the results of the paper to instrumental variables estimation problems yields consistent procedures for selecting instrumental variables

rational one step methods for initial value problems abstract first second and third order explicit nonlinear one step methods are proposed for singular and stiff initial value problems ivp s the algorithms are based on the representation of the solution by a finite continued fraction numerical examples are provided to illustrate the algorithms

convergence on error correction methods for solving initial value problems higher order semi explicit one step error correction methods ecm for solving initial value problems are developed ecm provides the excellent convergence o h 2 p 2 one wants to get without any iteration processes required by most implicit type methods this is possible if one constructs a local approximation having a residual error o h p on each time step as a practical example we construct a local quadratic approximation further it is shown that special choices of parameters for the local quadratic polynomial lead to the known explicit second order methods which can be improved into a semi explicit type ecm of the order of accuracy 6 the stability function is also derived and numerical evidences are presented to support theoretical results with several stiff and non stiff problems it should be remarked that the ecm approach developed here does not yield explicit methods but semi implicit methods of the rosenbrock type both ecm and rosenbrock s methods require to solve a few linear systems at each integration step but the ecm approach involves 2p 2 evaluations of the jacobian matrix per integration step whereas the rosenbrock method demands one evaluation only however it is much easier to get high order methods by using the ecm approach

gmm estimation of social interaction models with centrality this paper considers the specification and estimation of social interaction models with network structures and the presence of endogenous contextual correlated and group fixed effects when the network structure in a group is captured by a graph in which the degrees of nodes are not all equal the different positions of group members as measured by the bonacich 1987 centrality provide additional information for identification and estimation in this case the bonacich centrality measure for each group can be used as an instrument for the endogenous social effect but the number of such instruments grows with the number of groups we consider the 2sls and gmm estimation for the model the proposed estimators are asymptotically efficient respectively within the class of iv estimators and the class of gmm estimators based on linear and quadratic moments when the sample size grows fast enough relative to the number of instruments

consistent model and moment selection procedures for gmm estimation with application to dynamic panel data models this paper develops consistent model and moment selection criteria for gmm estimation the criteria select the correct model specification and all correct moment conditions asymptotically the selection criteria resemble the widely used likelihood based selection criteria bic hqic and aic the latter is not consistent the gmm selection criteria are based on the j statistic for testing over identifying restrictions bonus terms reward the use of fewer parameters for a given number of moment conditions and the use of more moment conditions for a given number of parameters the paper also considers a consistent downward testing procedure the paper applies the model and moment selection criteria to dynamic panel data models with unobserved individual effects the paper shows how to apply the selection criteria to select the lag length for lagged dependent variables to detect the number and locations of structural breaks to determine the exogeneity of regressors and or to determine the existence of correlation between some regressors and the individual effect to illustrate the finite sample performance of the selection criteria and the testing procedures and their impact on parameter estimation the paper reports the results of a monte carlo experiment on a dynamic panel data model

an efficient variable step size rational falkner type method for solving the special second order ivp in this paper firstly a rational one parameter family of falkner type explicit methods is presented for directly solving numerically special second order initial value problems in ordinary differential equations the proposed family of methods has second algebraic order of convergence imposing that the principal term of the local truncation error of the proposed family vanishes we get an expression for the free parameter at the grid point xn yn by substituting this value of the free parameter in the family a new rational third order method is obtained further by combining the third order method with any member of the second order family their variable step size formulation as an embedded pair is considered some numerical experiments are given to illustrate the performance and efficiency of the proposed methods

limit theory for panel data models with cross sectional dependence and sequential exogeneity abstract the paper derives a general central limit theorem clt and asymptotic distributions for sample moments related to panel data models with large n the results allow for the data to be cross sectionally dependent while at the same time allowing the regressors to be only sequentially rather than strictly exogenous the setup is sufficiently general to accommodate situations where cross sectional dependence stems from spatial interactions and or from the presence of common factors the latter leads to the need for random norming the limit theorem for sample moments is derived by showing that the moment conditions can be recast such that a martingale difference array central limit theorem can be applied we prove such a central limit theorem by first extending results for stable convergence in hall and heyde 1980 to non nested martingale arrays relevant for our applications we illustrate our result by establishing a generalized estimation theory for gmm estimators of a fixed effect panel model without imposing i i d or strict exogeneity conditions we also discuss a class of maximum likelihood ml estimators that can be analyzed using our clt

confidence intervals for diffusion index forecasts and inference for factor augmented regressions we consider the situation when there is a large number of series n e ach witht observations and each series has some predictive ability for some variable of interest a methodology of growing interest is first to estimate common factors from the panel of data by the method of principal components and then to augment an otherwise standard regression with the estimated factors in this paper we show that the least squares estimates obtained from these factor augmented regressions are t consistent and asymptotically normal if t n 0 the conditional mean predicted by the estimated factors is min t n consistent and asymptotically normal except when t n goes to zero inference should take into account the effect of estimated regressors on the estimated conditional mean we present analytical formulas for prediction intervals that are valid regardless of the magnitude of n t and that can also be used when the factors are nonstationary

non linear one step methods for initial value problems abstract a one step first order non linear method is proposed for initial value problems the algorithms are based on a representation of the solution by the reciprocal of a polynomial

a numerical study of multistep methods based on continued fractions abstract we investigate rational multistep methods of both osculatory and adams type for solving singular initial value problems formulas that permit variable steplength are constructed to avoid possible difficulties arising from the rational interpolation in two test examples it is found that explicit formulas of the adams type perform generally better than those based on the osculatory method the most accurate solutions are obtained from predictor corrector formulas based on the osculatory method

given a one step numerical scheme on which ordinary differential equations is it exact a necessary condition for a non autonomous ordinary differential equation to be exactly solved by a one step finite difference method is that the principal term of its local truncation error be null a procedure to determine some ordinary differential equations exactly solved by a given numerical scheme is developed examples of differential equations exactly solved by the explicit euler implicit euler trapezoidal rule second order taylor third order taylor van niekerk s second order rational and van niekerk s third order rational methods are presented

estimating systems of equations with different instruments for different equations abstract for applications where different instruments are required for different equations this paper offers a relatively primitive set of assumptions under which the nonlinear three stage least squares estimator is the asymptotically efficient generalized method of moments estimator these conditions can be verified for a general class of panel data models under reasonable assumptions about the dynamics in the model on the other hand the 3sls estimator is necessarily inefficient for models such as hedonic price systems and for certain simultaneous equations models with measurement error or omitted variables in some equations

panel models with interactive effects abstract the multiplication of individual specific effects and time specific effects provides a more general formulation than the traditionally used additive form to capture the unobserved heterogeneity in panel data modeling it is also a useful approach for dimension reduction for modeling cross section dependence however and are unobservable we explore the implications for econometric modeling under various formulations of the interactive effects models and suggest a quasi likelihood approach as a common framework to study issues of estimation and statistical inference when regressors are either strictly exogenous or predetermined and under different combinations of the data size of cross sectional dimension n and time series dimensions t we also suggest some computationally simpler estimation methods in light of the quasi likelihood approach monte carlo studies are conducted to highlight the issues involved

common correlated effects estimation of heterogeneous dynamic panel data models with weakly exogenous regressors abstract this paper extends the common correlated effects cce approach developed by pesaran 2006 to heterogeneous panel data models with lagged dependent variables and or weakly exogenous regressors we show that the cce mean group estimator continues to be valid but the following two conditions must be satisfied to deal with the dynamics a sufficient number of lags of cross section averages must be included in individual equations of the panel and the number of cross section averages must be at least as large as the number of unobserved common factors we establish consistency rates derive the asymptotic distribution suggest using covariates to deal with the effects of multiple unobserved common factors and consider jackknife and recursive de meaning bias correction procedures to mitigate the small sample time series bias theoretical findings are accompanied by extensive monte carlo experiments which show that the proposed estimators perform well so long as the time series dimension of the panel is sufficiently large

panel data and unobservable individual effects abstract an important purpose in pooling time series and cross section data is to control for individual specific unobservable effects which may be correlated with other explanatory variables e g latent ability in measuring returns to schooling in earnings equations or managerial ability in measuring returns to scale in firm cost functions using instrumental variables and the time invariant characteristics of the latent variable we derive 1 1 a test for the presence of this effect and for the over identifying restriction we use 2 2 necessary and sufficient conditions for identification of all the parameters in the model and 3 3 the asymptotically efficient instrumental variables estimator and conditions under which it differs from the within groups estimator we calculate efficient estimates of a wage equation from the michigan income dynamics data which indicate substantial differences from within groups and balestra nerlove estimates particularly a significantly higher estimate of the returns to schooling

estimation of dynamic models with error components abstract observations on n cross section units at t time points are used to estimate a simple statistical model involving an autoregressive process with an additive term specific to the unit different assumptions about the initial conditions are a initial state fixed b initial state random c the unobserved individual effect independent of the unobserved dynamic process with the initial value fixed and d the unobserved individual effect independent of the unobserved dynamic process with initial value random asymptotic properties of the maximum likelihood and covariance estimators are obtained when t and when n the relationship between the pseudo and conditional maximum likelihood estimators is clarified a simple consistent estimator that is independent of the initial conditions and the way in which t or n is also suggested

a class of inverse runge kutta schemes for the numerical integration of singular problems a class of methods for the numerical solution of initial value problems whose solution has singular point s shall be discussed they are a and l stable and were found to perform efficiently well when faced with singularity problem but perform poorly when applied to non singular problems

covariance matrix estimation and the power of the overidentifying restrictions test class of heteroscedasticity and autocorrelation consistent covariance hacc matrix estimators that are now routinely used by practitioners in the calculation of the two step gmm estimator in applications it is customary to assume the model is correctly specified during the estimation and only to assess the specification after the second step using a statistic such as the overidentifying restrictions test one important consequence of this methodology is that the hacc estimator is calculated under the assumption that the model is correctly specified in this paper we examine the implications of model misspecification for the hacc covariance matrix estimator and the overidentifying restrictions test it is shown that the hacc estimator is asymptotically equivalent to the sum of two matrices one of these matrices is nonsingular and o 1 the other is of rank one and ob where b is the tt bandwidth used in the hacc estimator it is shown that this structure implies the inverse of the hacc estimator converges to a singular matrix as b with t and this t limiting matrix has rank q y 1 where q is the dimension of the moment condition it is shown that this limiting behavior translates into an overidentifying restrictions test that is ot rb in contrast it is shown that the overidentifying restrictions test is consistent pt and ot if the covariance matrix estimator is consistent under both null and alterna p tive this can be achieved by constructing the hacc estimator using the sample moment in mean deviation form 3

statistical mechanics of the random k satisfiability model the random k satisfiability problem consisting in verifying the existence of an assignment of n boolean variables that satisfy a set of m ensuremath alpha n random logical clauses containing k variables each is studied using the replica symmetric framework of diluted disordered systems we present an exact iterative scheme for the replica symmetric functional order parameter together for the different cases of interest k 2 kg 3 and k ensuremath gg 1 the calculation of the number of solutions which allowed us phys rev lett 76 3881 1996 to predict a first order jump at the threshold where the boolean expressions become unsatisfiable with probability one is thoroughly displayed in the case k 2 the rigorously known critical value ensuremath alpha 1 of the number of clauses per boolean variable is recovered while for kg 3 we show that the system exhibits a replica symmetry breaking transition the annealed approximation is proven to be exact for large k

a numerical algorithm for singular perturbation problems exhibiting weak boundary layers abstract a class of singularly perturbed two point boundary value problems bvp for second order ordinary differential equations is considered here to avoid the numerical difficulties in the solution to these problems we divide the domain into two subdomains the first bvp is a layer domain problem and the second bvp is a regular domain problem error estimates are derived for the numerical solution numerical examples are provided in support of the proposed method

gmm estimation with cross sectional dependence this paper presents a spatial model of dependence among agents using a metric of economic distance measurements of this economic distance provide cross sectional data with a structure similar to that provided by the time index in time series data generalized method of moments estimators using such dependent data are shown to be consistent and asymptotically normal this paper presents a class of non parametric positive semi definite covariance matrix estimators that allow for general forms of dependence characterized by economic distance these covariance matrix estimators are shown to remain consistent when economic distances are not precisely observed

phase transition in the number partitioning problem universit at magdeburg institut fu r theoretische physik universit atsplatz 2 d 39106 magdeburg germany february 1 2008 number partitioning is an np complete problem of com binatorial optimization a statistical mechanics analysis re veals the existence of a phase transition that separates theeasy from the hard to solve instances and that re ects thepseudo polynomiality of number partitioning the phase di agram and the value of the typical ground state energy arecalculated 64 60 cn 02 60 pn 02 70 lq 89 80 h

gmm estimation of linear panel data models with time varying individual effects abstract this paper considers models for panel data in which the individual effects vary over time the temporal pattern of variation is arbitrary but it is the same for all individuals the model thus allows one to control for time varying unobservables that are faced by all individuals e g macro economic events and to which individuals may respond differently a generalized within estimator is consistent under strong assumptions on the errors but it is dominated by a generalized method of moments estimator this is perhaps surprising because the generalized within estimator is the mle under normality the efficiency gains from imposing second moment error assumptions are evaluated they are substantial when the regressors and effects are weakly correlated

panel data models with interactive fixed effects this paper considers large n and large t panel data models with unobservable multiple interactive eects these models are useful for both micro and macro econometric modelings in earnings studies for example workers motivation persistence and diligence combined to influence the earnings in addition to the usual argument of innate ability in macroeconomics the interactive eects represent unobservable common shocks and their heterogeneous responses over cross sections since the interactive eects are allowed to be correlated with the regressors they are treated as fixed eects parameters to be estimated along with the common slope coecients the model is estimated by the least squares method which provides the interactive eect s counterpart of the within estimator we first consider model identification and then derive the rate of convergence and the limiting distribution of the interactive eect s estimator of the common slope coecients the estimator is shown to be p nt consistent this rate is valid even in the presence of correlations and heteroskedasticities in both dimensions a striking contrast with fixed t framework in which serial correlation and heteroskedasticity imply unidentification the asymptotic distribution is not necessarily centered at zero biased corrected estimators are derived we also derive the constrained estimator and its limiting distribution imposing additivity coupled with interactive eects the problem of testing additive versus interactive eects is also studied we also derive identification conditions for models with grand mean time invariant regressors and common regressors it is shown that there exists a set of necessary and sucient identification conditions for those models given identification the rate of convergence and limiting results continue to hold

quasi maximum likelihood estimators for spatial dynamic panel data with fixed effects when both n and t are large this paper investigates the asymptotic properties of quasi maximum likelihood estimators for spatial dynamic panel data with xed eects when both the number of individuals n and the number of time periods t are large we consider the case where t is asymptotically large relative to n the case where t is asymptotically proportional to n and the case where n is asymptotically large relative to t in the case where t is asymptotically large relative to n the estimators are p nt consistent and asymptotically normal with the limit distribution centered around 0 when n is asymptotically proportional to t the estimators are p nt consistent and asymptotically normal but the limit distribution is not centered around 0 and when n is large relative to t the estimators are consistent with rate t and have a degenerate limit distribution the estimators of the xed eects are p t consistent and asymptotically normal we also propose a bias correction for our estimators we show that when t grows faster than n 1 3 the correction will asymptotically eliminate the bias and yield a centered con dence interval

typical solution time for a vertex covering algorithm on finite connectivity random graphs we analytically describe the typical solution time needed by a backtracking algorithm to solve the vertex cover problem on finite connectivity random graphs we find two different transitions the first one is algorithm dependent and marks the dynamical transition from linear to exponential solution times the second one gives the maximum computational complexity and is found exactly at the threshold where the system undergoes an algorithm independent phase transition in its solvability analytical results are corroborated by numerical simulations

estimating long run relationships from dynamic heterogeneous panels in panel data four procedures are widely used pooling aggregating averaging group estimates and cross section regression in the static case if the coefficients differ randomly all four procedures give unbiased estimates of coefficient means in the dynamic case when the coefficients differ across groups pooling and aggregating give inconsistent and potentially highly misleading estimates of the coefficients though the cross section can provide consistent estimates of the long run effects the theoretical results on the properties of the four procedures are illustrated by uk labour demand functions for 38 industries over 30 years

a non standard explicit integration scheme for initial value problems in this paper we present the construction of a non standard explicit algorithm for initial value problems the method results to be of second order and a stable this new algorithm has been proven to be suitable for solving different kind of initial value problems specifically non singular problems singular problems stiff problems and singularly perturbed problems some numerical experiments are considered in order to check the behaviour of the method when applied to a variety of initial value problems

efficient estimation of models for dynamic panel data abstract in this paper we consider a dynamic model for panel data we show that under standard assumptions there are more moment conditions than are currently exploited in the literature some of these are linear but others are quadratic so that nonlinear gmm is required we also show that exogenous regressors generate a larger number of relevant moment conditions in a dynamic model than they would in a static model

phase transitions of subset sum and shannon s limit in source coding abstract we consider statistical properties of an np complete problem the subset sum using the methods and concepts of statistical mechanics after introducing the statistical mechanical treatment of the problem the phase transition behaviors are discussed we also introduce a source coding scheme which is based on the idea of the subset sum it is shown that the phase transitions in the subset sum is related to shannon s limit for source coding

efficiency bounds for semiparametric regression efficiency bounds for conditional moment restrictions with a nonparametric component are derived there is a given function of the data a random sample from a distribution f and a parameter the restriction is that a conditional expectation of this function is zero at some point in the parameter space the parameter has two parts a finite dimensional component and a general function evaluated at a subset of the conditioning variables an example is a regression function that is additive in parametric and nonparametric component as arises in sample selection models copyright 1992 by the econometric society

solving first order initial value problems by using an explicit non standard a stable one step method in variable step size formulation this paper presents the construction of a new family of explicit schemes for the numerical solution of initial value problems of ordinary differential equations odes the one parameter family is constructed by considering a suitable rational approximation to the theoretical solution resulting a family with second order convergence and a stable imposing that the principal term in the local truncation error vanishes we obtain an expression for the parameter value in terms of the point xn yn on each step with this approach the resulting method has third order convergence maintaining the characteristic of a stability finally combining this last method with other of order two in order to get an estimation for the local truncation error an implementation in variable step size has been considered the proposed method can be used in a wide range of problems for solving numerically a scalar ode or a system of first order odes several numerical examples are given to illustrate the efficiency and performance of the proposed method in comparison with some existing methods in the literature

inferring the rank of a matrix abstract this paper considers methods of inference concerning the rank of matrix a based on an asymptotically normal estimate of and some identifiable specification for one such specification is 0 in which case one is interested in the rank of we first propose and examine the properties of a test of the hypothesis that the rank is of a given size against the alternative that the rank is larger we then look at the problem of estimating the rank of this matrix using model selection procedures and sequential hypothesis testing conditions for consistency of such procedures are obtained some economic applications are discussed and the various methods are compared in a monte carlo experiment

inferential theory for factor models of large dimensions this paper develops an inferential theory for factor models of large dimensions the principal components estimator is considered because it is easy to compute and is asymptotically equivalent to the maximum likelihood estimator if normality is assumed we derive the rate of convergence and the limiting distributions of the estimated factors factor loadings and common components the theory is developed within the framework of large cross sections n and a large time dimension t to which classical factor analysis does not apply we show that the estimated common components are asymptotically normal with a convergence rate equal to the minimum of the square roots of n and t the estimated factors and their loadings are generally normal although not always so the convergence rate of the estimated factors and factor loadings can be faster than that of the estimated common components these results are obtained under general conditions that allow for correlations and heteroskedasticities in both dimensions stronger results are obtained when the idiosyncratic errors are serially uncorrelated and homoskedastic a necessary and sufficient condition for consistency is derived for large n but fixed t

large panels with common factors and spatial correlation this paper considers the statistical analysis of large panel data sets where even after condi tioning on common observed eects the cross section units might remain dependently distrib uted this could arise when the cross section units are subject to unobserved common eects and or if there are spill over eects due to spatial or other forms of local dependencies the paper provides an overview of the literature on cross section dependence introduces the con cepts of time speci c weak and strong cross section dependence and shows that the commonly used spatial models are examples of weak cross section dependence it is then established that the common correlated eects cce estimator of panel data model with a multifactor error structure recently advanced by pesaran 2006 continues to provide consistent estimates of the slope coe cient even in the presence of spatial error processes small sample properties of the cce estimator under various patterns of cross section dependence including spatial forms are investigated by monte carlo experiments results show that the cce approach works well in the presence of weak and or strong cross sectionally correlated errors we also explore the role of certain characteristics of spatial processes in determining the performance of cce estimators such as the form and intensity of spatial dependence and the sparseness of the spatial weight

on the asymptotic distribution of the moran i test statistic with applications by far the most popular test for spatial correlation is the one based on moran s 1950 i test statistic despite this the available results in the literature concerning the large sample distribution of this statistic are limited and have been derived under assumptions that do not cover many applications of interest in this paper we 3rst give a general result concerning the large sample distribution of moran i type test statistics we then apply this result to derive the large sample distribution of the moran i test statistic for a variety of important models in order to establishth ese results we also give a new central limit theorem for linear quadratic forms 2001 elsevier science s a all rights reserved jel classi cation c12 c21

probabilistic analysis of the number partitioning problem given a sequence of n positive real numbers the number partitioning problem consists of partitioning them into two sets such that the absolute value of the difference of the sums of over the two sets is minimized in the case in which the s are statistically independent random variables uniformly distributed in the unit interval this np complete problem is equivalent to the problem of finding the ground state of an infinite range random antiferromagnetic ising model we employ the annealed approximation to derive analytical lower bounds to the average value of the difference for the best constrained and unconstrained partitions in the large n limit furthermore we calculate analytically the fraction of metastable states i e states that are stable against all single spin flips and found that it vanishes like