on the statistical identification of dsge models abstract dynamic stochastic general equilibrium dsge models are now considered attractive by the profession not only from the theoretical perspective but also from an empirical standpoint as a consequence of this development methods for diagnosing the fit of these models are being proposed and implemented in this article we illustrate how the concept of statistical identification that was introduced and used by spanos spanos aris 1990 the simultaneous equations model revisited statistical adequacy and identification journal of econometrics 44 87 105 to criticize traditional evaluation methods of cowles commission models could be relevant for dsge models we conclude that the recently proposed model evaluation method based on the dsge var might not satisfy the condition for statistical identification however our application also shows that the adoption of a favar as a statistically identified benchmark leaves unaltered the support of the data for the dsge model and that a dsge favar can be an optimal forecasting model

majorization as a tool for optimizing a class of matrix functions the problem of minimizing a general matrix trace function possibly subject to certain constraints is approached by means of majorizing this function by one having a simple quadratic shape and whose minimum is easily found it is shown that the parameter set that minimizes the majorizing function also decreases the matrix trace function which in turn provides a monotonically convergent algorithm for minimizing the matrix trace function iteratively three algorithms based on majorization for solving certain least squares problems are shown to be special cases in addition by means of several examples it is noted how algorithms may be provided for a wide class of statistical optimization tasks for which no satisfactory algorithms seem available

approximating a symmetric matrix we examine the least squares approximationc to a symmetric matrixb when all diagonal elements get weightw relative to all nondiagonal elements whenb has positivityp andc is constrained to be positive semi definite our main result states that whenw 1 2 then the rank ofc is never greater thanp and whenw 1 2 then the rank ofc is at leastp for the problem of approximating a givenn n matrix with a zero diagonal by a squared distance matrix it is shown that the sstress criterion leads to a similar weighted least squares solution withw n 2 4 the main result remains true other related problems and algorithmic consequences are briefly discussed

an alternating least squares method for the weighted approximation of a symmetric matrix bailey and gower examined the least squares approximationc to a symmetric matrixb when the squared discrepancies for diagonal elements receive specific nonunit weights they focussed on mathematical properties of the optimalc in constrained and unconstrained cases rather than on how to obtainc for any givenb in the present paper a computational solution is given for the case wherec is constrained to be positive semidefinite and of a fixed rankr or less the solution is based on weakly constrained linear regression analysis

weighted least squares fitting using ordinary least squares algorithms a general approach for fitting a model to a data matrix by weighted least squares wls is studied this approach consists of iteratively performing steps of existing algorithms for ordinary least squares ols fitting of the same model the approach is based on minimizing a function that majorizes the wls loss function the generality of the approach implies that for every model for which an ols fitting algorithm is available the present approach yields a wls fitting algorithm in the special case where the wls weight matrix is binary the approach reduces to missing data imputation

minimization of a class of matrix trace functions by means of refined majorization a procedure is described for minimizing a class of matrix trace functions the procedure is a refinement of an earlier procedure for minimizing the class of matrix trace functions using majorization it contains a recently proposed algorithm by koschat and swayne for weighted procrustes rotation as a special case a number of trial analyses demonstrate that the refined majorization procedure is more efficient than the earlier majorization based procedure

bilinear mixed effects models for dyadic data this article discusses the use of a symmetric multiplicative interaction effect to capture certain types of third order dependence patterns often present in social networks and other dyadic datasets such an effect along with standard linear fixed and random effects is incorporated into a generalized linear model and a markov chain monte carlo algorithm is provided for bayesian estimation and inference in an example analysis of international relations data accounting for such patterns improves model fit and predictive performance

lower bounds for the discrepancy of inversive congruential pseudorandom numbers with power of two modulus the inversive congruential method with modulus m 2w for the generation of uniform pseudorandom numbers has recently been introduced the discrepancy z l of a tuples of consecutive pseudorandom numbers generated by such a generator with maximal period length m 2 is the cru cial quantity for the analysis of the statistical independence properties of these pseudorandom numbers by means of the serial test it is proved that for a pos itive proportion of the inversive congruential generators with maximal period length the discrepancy ij2 is at least of the order of magnitude m xl2 for all k 2 this shows that the bound d 2 2 0 m 1 2 logw 2 established by the second author is essentially best possible

joint procrustes analysis for simultaneous nonsingular transformation of component score and loading matrices in component analysis solutions post multiplying a component score matrix by a nonsingular matrix can be compensated by applying its inverse to the corresponding loading matrix to eliminate this indeterminacy on nonsingular transformation we propose joint procrustes analysis jpa in which component score and loading matrices are simultaneously transformed so that the former matrix matches a target score matrix and the latter matches a target loading matrix the loss function of jpa is a function of the nonsingular transformation matrix and its inverse and is difficult to minimize straightforwardly to deal with this difficulty we reparameterize those matrices by their singular value decomposition which reduces the minimization to alternately solving quartic equations and performing the existing multivariate procedures this algorithm is assessed in a simulation study we further extend jpa for cases where targets are linear functions of unknown parameters we also discuss how the application of jpa can be extended in different fields

on the non existence of optimal solutions and the occurrence of degeneracy in the candecomp parafac model the candecomp parafac cp model decomposes a three way array into a prespecified number of r factors and a residual array by minimizing the sum of squares of the latter it is well known that an optimal solution for cp need not exist we show that if an optimal cp solution does not exist then any sequence of cp factors monotonically decreasing the cp criterion value to its infimum will exhibit the features of a so called degeneracy that is the parameter matrices become nearly rank deficient and the euclidean norm of some factors tends to infinity we also show that the cp criterion function does attain its infimum if one of the parameter matrices is constrained to be column wise orthonormal

fitting longitudinal reduced rank regression models by alternating least squares an alternating least squares method for iteratively fitting the longitudinal reduced rank regression model is proposed the method uses ordinary least squares and majorization substeps to estimate the unknown parameters in the system and measurement equations of the model in an example with cross sectional data it is shown how the results conform closely to results from eigenanalysis optimal scaling of nominal and ordinal variables is added in a third substep and illustrated with two examples involving cross sectional and longitudinal data

the serial test for congruential pseudorandom numbers generated by inversions two types of congruential pseudorandom number generators based on inversions were introduced recently we analyze the statistical independence properties of these pseudorandom numbers by means of the serial test the results show that these pseudorandom numbers perform satisfactorily under the serial test the methods of proof rely heavily on bounds for character sums such as the weil stepanov bound for character sums over finite fields

explicit inversive congruential pseudorandom numbers with power of two modulus an explicit version of the inversive congruential method with power of two modulus for generating uniform pseudorandom numbers is introduced statistical independence properties of the generated sequences are studied by means of the serial test the method of proof relies on a detailed analysis of certain exponential sums

maximization of sums of quotients of quadratic forms and some generalizations monotonically convergent algorithms are described for maximizing six constrained functions of vectors x or matrices x with columns x 1 x r these functions are h 1 x sigma k x a k x x c k x 1 h 1 x sigma k tr x a k x x c k x 1 h over tilde 1 x sigma k sigma l x l a k x l x l c k x l 1 with x constrained to be columnwise orthonormal h 2 x sigma k x a k x 2 x c k x 1 subject to x x 1 h 2 x sigma k tr x a k x x a k x x c k x 1 subject to x x i and h over tilde 2 x sigma k sigma l x l a k x l 2 x l c k x l 1 subject to x x i in these functions the matrices c k are assumed to be positive definite the matrices a k can be arbitrary square matrices the general formulation of the functions and the algorithms allows for application of the algorithms in various problems that arise in multivariate analysis several applications of the general algorithms are given specifically algorithms are given for reciprocal principal components analysis binormamin rotation generalized discriminant analysis variants of generalized principal components analysis simple structure rotation for one of the latter variants and set component analysis for most of these methods the algorithms appear to be new for the others the existing algorithms turn out to be special cases of the newly derived general algorithms

some clarifications of the tuckals2 algorithm applied to the idioscal problem kroonenberg and de leeuw have suggested fitting the idioscal model by the tuckals2 algorithm for three way components analysis in theory this is problematic because tuckals2 produces two possibly different coordinate matrices that are useless for idioscal unless they are equal kroonenberg has claimed that when idioscal is fitted by tuckals2 the resulting coordinate matrices will be identical in the present paper this claim is proven valid when the data matrices are semidefinite however counterexamples for indefinite matrices are also constructed by examining the global minimum in the case where the data matrices have the same eigenvectors similar counterexamples have been considered by ten berge and kiers in the related context of candecomp parafac to fit the indscal model

simple structure in component analysis techniques for mixtures of qualitative and quantitative variables several methods have been developed for the analysis of a mixture of qualitative and quantitative variables and one called pcamix includes ordinary principal component analysis pca and multiple correspondence analysis mca as special cases the present paper proposes several techniques for simple structure rotation of a pcamix solution based on the rotation of component scores and indicates how these can be viewed as generalizations of the simple structure methods for pca in addition a recently developed technique for the analysis of mixtures of qualitative and quantitative variables called indomix is shown to construct component scores without rotational freedom maximizing the quartimax criterion over all possible sets of component scores a numerical example is used to illustrate the implication that when used for qualitative variables indomix provides axes that discriminate between the observation units better than do those generated from mca

gensvm a generalized multiclass support vector machine traditional extensions of the binary support vector machine svm to multiclass problems are either heuristics or require solving a large dual optimization problem here a generalized multiclass svm is proposed called gensvm in this method classification boundaries for a k class problem are constructed in a k 1 dimensional space using a simplex encoding additionally several different weightings of the misclassification errors are incorporated in the loss function such that it generalizes three existing multiclass svms through a single optimization problem an iterative majorization algorithm is derived that solves the optimization problem without the need of a dual formulation this algorithm has the advantage that it can use warm starts during cross validation and during a grid search which significantly speeds up the training phase rigorous numerical experiments compare linear gensvm with seven existing multiclass svms on both small and large data sets these comparisons show that the proposed method is competitive with existing methods in both predictive accuracy and training time and that it significantly outperforms several existing methods on these criteria

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

statistical independence of a new class of inversive congruential pseudorandom numbers linear congruential pseudorandom numbers show several undesirable regularities which can render them useless for certain stochastic simulations this was the motiviation for important recent developments in nonlinear congruential methods for generating uniform pseudorandom numbers it is particularly promising to achieve nonlinearity by employing the operation of multiplicative inversion with respect to a prime modulus in the present paper a new class of such inversive congruential generators is introduced and analyzed it is shown that they have excellent statistical independence properties and model true random numbers very closely the methods of proof rely heavily on weil stepanov bounds for rational exponential sums 39 refs

a bayesian hierarchical factorization model for vector fields factorization based techniques explain arrays of observations using a relatively small number of factors and provide an essential arsenal for multi dimensional data analysis most factorization models are however developed on general arrays of scalar values for a class of practical data arising from observing spatial signals including images it is desirable for a model to consider general observations e g handling a vector field and non exchangeable factors e g handling spatial connections between the columns and the rows of the data in this paper a probabilistic model for factorization is proposed we adopt bayesian hierarchical modeling and treat the factors as latent random variables a markov structure is imposed on the distribution of factors to account for the spatial connections the model is designed to represent vector arrays sampled from fields of continuous domains therefore a tailored observation model is developed to represent the link between the factor product and the data the proposed technique has been shown effective in analyzing optical flow fields computed on both synthetic images and real life videoclips

tensor decompositions and applications this survey provides an overview of higher order tensor decompositions their applications and available software a tensor is a multidimensional or n way array decompositions of higher order tensors i e n way arrays with n geq 3 have applications in psycho metrics chemometrics signal processing numerical linear algebra computer vision numerical analysis data mining neuroscience graph analysis and elsewhere two particular tensor decompositions can be considered to be higher order extensions of the matrix singular value decomposition candecomp parafac cp decomposes a tensor as a sum of rank one tensors and the tucker decomposition is a higher order form of principal component analysis there are many other tensor decompositions including indscal parafac2 candelinc dedicom and paratuck2 as well as nonnegative variants of all of the above the n way toolbox tensor toolbox and multilinear engine are examples of software packages for working with tensors

fitting the off diagonal dedicom model in the least squares sense by a generalization of the harman and jones minres procedure of factor analysis harshman s dedicom model providesa framework for analyzing square but asymmetric materices of directional relationships amongn objects or persons in terms of a small number of components one version of dedicom ignores the diagonal entries of the matrices a straight forward computational solution for this model is offered in the present paper the solution can be interpreted as a generalized minres procedure suitable for handing asymmetric matrices

macropca an all in one pca method allowing for missing values as well as cellwise and rowwise outliers multivariate data are typically represented by a rectangular matrix table in which the rows are the objects cases and the columns are the variables measurements when there are many variables

on a nonlinear congruential pseudorandom number generator a nonlinear congruential pseudorandom number generator with modulus m 2 w is proposed which may be viewed to comprise both linear as well as inversive congruential generators the condition for it to generate sequences of maximal period length is obtained it is akin to the inversive one and bears a remarkable resemblance to the latter

on generalized inversive congruential pseudorandom numbers the inversive congruential method with prime modulus for generating uniform pseudorandom numbers has several very promising properties very recently a generalization for composite moduli has been introduced in the present paper it is shown that the generated sequences have very attractive statistical independence properties

a generalization of takane s algorithm for dedicom an algorithm is described for fitting the dedicom model for the analysis of asymmetric data matrices this algorithm generalizes an algorithm suggested by takane in that it uses a damping parameter in the iterative process takane s algorithm does not always converge monotonically based on the generalized algorithm a modification of takane s algorithm is suggested such that this modified algorithm converges monotonically it is suggested to choose as starting configurations for the algorithm those configurations that yield closed form solutions in some special cases finally a sufficient condition is described for monotonic convergence of takane s original algorithm

pseudorandom vector generation by the multiple recursive matrix method pseudorandom vectors are of importance for parallelized simulation methods in this paper we carry out an in depth analysis of the multiple recursive matrix method for the generation of uniform pseudorandom vectors which was introduced in an earlier paper of the author we study in particular the periodicity properties the lattice structure and the behavior under the serial test for sequences of pseudorandom vectors generated by this method

tests of randomness by the gambler s ruin algorithm abstract in the gambler s ruin algorithm on the cyclic group z n 0 1 n 1 we consider arrival time at 0 starting from a fixed point x 0 and use several versions of arrival time algorithm to test pseudorandom number generators this kind of test based on the exact probability density for a random walk on a finite group is done for the first time the test results show hidden defects in some generators such as combined multiple recursive generators and mersenne twister generators

a monotonically convergent algorithm for orthogonal congruence rotation brokken has proposed a method for orthogonal rotation of one matrix such that its columns have a maximal sum of congruences with the columns of a target matrix this method employs an algorithm for which convergence from every starting point is not guaranteed in the present paper an iterative majorization algorithm is proposed which is guaranteed to converge from every starting point specifically it is proven that the function value converges monotonically and that the difference between subsequent iterates converges to zero in addition to the better convergence properties another advantage of the present algorithm over brokken s one is that it is easier to program the algorithms are compared on 80 simulated data sets and it turned out that the new algorithm performed well in all cases whereas brokken s algorithm failed in almost half the cases the derivation of the algorithm is given in full detail because it involves a series of inequalities that can be of use to derive similar algorithms in different contexts

a multivariate reduced rank growth curve model with unbalanced data a multivariate reduced rank growth curve model is proposed that extends the univariate reducedrank growth curve model to the multivariate case in which several response variables are measured over multiple time points the proposed model allows us to investigate the relationships among a number of response variables in a more parsimonious way than the traditional growth curve model in addition the method is more flexible than the traditional growth curve model for example response variables do not have to be measured at the same time points nor the same number of time points it is also possible to apply various kinds of basis function matrices with different ranks across response variables it is not necessary to specify an entire set of basis functions in advance examples are given for illustration

on the lattice structure of a nonlinear generator with modulus 2 a abstract nonlinear congruential pseudorandom number generators based on inversions have been introduced and analysed recently these generators do not show the simple lattice structure of the widely used linear congruential generators which are too regular for certain simulation purposes in the present paper a nonlinear congruential generator based on inversions with respect to a power of two modulus is considered it is shown that the set of points formed by consecutive pseudorandom numbers has a more complicated lattice structure it forms a superposition of shifted lattices the corresponding lattice bases are explicitly determined and analysed

a nonlinear congruential pseudorandom number generator with power of two modulus a nonlinear congruential pseudorandom number generator is studied where the modulus is a power of two investigation of this generator was suggested by knuth 7 a simple necessary and sufficient condition is given for this generator to have the maximal period length

pseudorandom vector generation by the compound inversive method pseudorandom vectors are of importance for parallelized simulation methods in this paper a detailed analysis of the compound inversive method for the generation of k dimensional uniform pseudorandom vectors a vector analog of the compound inversive method for pseudorandom number generation is carried out in particular periodicity properties and statistical independence properties of the generated sequences are studied based on the discrete discrepancy of s tuples of successive terms in the sequence the results show that the generated sequences have attractive statistical independence properties for pseudorandom vectors of dimensions k 4

multivariate analysis with linearizable regressions we study the class of multivariate distributions in which all bivariate regressions can be linearized by separate transformation of each of the variables this class seems more realistic than the multivariate normal or the elliptical distributions and at the same time its study allows us to combine the results from multivariate analysis with optimal scaling and classical multivariate analysis in particular a two stage procedure which first scales the variables optimally and then fits a simultaneous equations model is studied in detail and is shown to have some desirable properties

estimating linear statistical relationships this paper on estimating linear statistical relationships includes three lectures on linear functional and structural relationships factor analysis and simultaneous equations models the emphasis is on relating the several models by a general approach and on the similarity of maximum likelihood estimators under normality in the different models in the first two lectures the observable vector is decomposed into a systematic part and a random error the systematic part satisfies the linear relationships estimators are derived for several cases and some of their properties given estimation of the coefficients of a single equation in a simultaneous equations model is shown to be equivalent to estimation of linear functional relationships

techniques for rotating two or more loading matrices to optimal agreement and simple structure a comparison and some technical details matrices of factor loadings are often rotated to simple structure when more than one loading matrix is available for the same variables the loading matrices can be compared after rotating them all separately to simple structure an alternative procedure is to rotate them to optimal agreement and then compare them in the present paper techniques are described that combine these two procedures specifically five techniques that combine the ideals of rotation to optimal agreement and rotation to simple structure are compared on the basis of contrived and empirical data for the contrived data it is assessed to what extent the rotations recover the underlying common structure for both the contrived and the empirical data it is studied to what extent the techniques give well matching rotated matrices to what extent these have a simple structure and to what extent the most prominent parts of the different loading matrices agree it was found that the simple procedure of combining a generalized procrustes analysis gpa with varimax on the mean of the matched loading matrices performs very well on all criteria and for most purposes offers an attractive compromise of rotation to agreement and simple structure in addition to this comparison some technical improvements are proposed for bloxom s rotation to simple structure and maximum similarity

a simple general procedure for orthogonal rotation a very general algorithm for orthogonal rotation is identified it is shown that when an algorithm parameter is sufficiently large the algorithm converges monotonically to a stationary point of the rotation criterion from any starting value because a sufficiently large is in general hard to find a modification that does not require it is introduced without this requirement the modified algorithm is not only very general but also very simple its implementation involves little more than computing the gradient of the rotation criterion while the modified algorithm converges monotonically from any starting value it is not guaranteed to converge to a stationary point it however does so in all of our examples while motivated by the rotation problem in factor analysis the algorithms discussed may be used to optimize almost any function of a not necessarily square column wise orthonormal matrix a number of these more general applications are considered empirical examples show that the modified algorithm can be reasonably fast but its purpose is to save an investigator s effort rather than that of his or her computer this makes it more appropriate as a research tool than as an algorithm for established methods

inversive congruential pseudorandom numbers avoid the planes nonlinear congruential pseudorandom number generators based on inversions have recently been introduced and analyzed these generators do not show the lattice structure of the widely used linear congruential method in the present paper it is proved that the points formed by d consecutive pseudorandom numbers of an inversive congruential generator with prime modulus possess an even stronger property any hyperplane in space contains at most d of these points that is to say the hyperplane spanned by d arbitrary points of an inversive congruential generator contains no further points this feature makes the inversive congruential method particularly attractive for simulation problems where linear structures within the generated points should be avoided

a general solution for a class of weakly constrained linear regression problems this paper contains a globally optimal solution for a class of functions composed of a linear regression function and a penalty function for the sum of squared regression weights global optimality is obtained from inequalities rather than from partial derivatives of a lagrangian function applications arise in multidimensional scaling of symmetric or rectangular matrices of squared distances in procrustes analysis and in ridge regression analysis the similarity of existing solutions for these applications is explained by considering them as special cases of the general class of functions addressed

the serial test for a nonlinear pseudorandom number generator let m 2 and g m 1 3 m 1 a sequence y n y n g m is obtained by the formula y n 1 ay n b cy n mod m the sequence x n x n y n m is a sequence of pseudorandom numbers of the maximal period length m 2 if and only if a c 1 mod 4 b 2 mod 4 in this note the uniformity is investigated by the 2 dimensional serial test for the sequence we follow closely the method of papers by eichenauer herrmann and niederreiter

the simultaneous equations model revisited statistical adequacy and identification abstract the paper proposes a reinterpretation of the simultaneous equations model in order to allow for the nature and structure of the observed data to play a role in the empirical analysis of structural models without extenuating the importance of the theory this suggests a revised view of the identification problem separating it into statistical and structural identification re lative to a given data the proposed re interpretation provides a framework wherein structural modeling and the data based specifications such as the var and the dynamic data based reduced forms are compliments and not substitutes

dynamic identification of dynamic stochastic general equilibrium models this paper studies dynamic identification of parameters of a dynamic stochastic general equilibrium model from the first and second moments of the data classical results for dynamic simultaneous equations do not apply because the state space solution of the model does not constitute a standard reduced form full rank of the jacobian matrix of derivatives of the solution parameters with respect to the parameters of interest is necessary but not sufficient for identification we use restrictions implied by observational equivalence to obtain two sets of rank and order conditions one for stochastically singular models and another for nonsingular models measurement errors mean long run and a priori restrictions can be accommodated an example is considered to illustrate the results

an alternating least squares algorithm for fitting the two and three way dedicom model and the idioscal model the dedicom model is a model for representing asymmetric relations among a set of objects by means of a set of coordinates for the objects on a limited number of dimensions the present paper offers an alternating least squares algorithm for fitting the dedicom model the model can be generalized to represent any number of sets of relations among the same set of objects an algorithm for fitting this three way dedicom model is provided as well based on the algorithm for the three way dedicom model an algorithm is developed for fitting the idioscal model in the least squares sense

dynamic structural systems under indirect observation identifiability and estimation aspects from a system theoretic perspective in this partly expository paper the parameter identifiability and estimation of a general dynamic structural model under indirect observation will be considered from a system theoretic perspective the general dynamic model covers dynamic factor analytic models dynamic mimic models and joreskog s linear structural model as special cases its reduced form is under a slightly different specification known in system theory and econometrics as the stochastic stationary version of the state space model by using concepts and methods from system theory such as the observability and controllability concept the steady state kalman filter and a general nonlinear ml estimation procedure known as prediction error estimation the general dynamic model will be identified it will be shown that joreskog s lisrel procedure is a special case of the prediction error estimation procedure

bayesian model choice asymptotics and exact calculations abstract model determination is a fundamental data analytic task here we consider the problem of choosing amongst a finite with loss of generality we assume two set of models after briefly reviewing classical and bayesian model choice strategies we present a general predictive density which includes all proposed bayesian approaches we are aware of using laplace approximations we can conveniently assess and compare asymptotic behavior of these approaches concern regarding the accuracy of these approximation for small to moderate sample sizes encourages the use of monte carlo techniques to carry out exact calculations a data set fit with nested non linear models enables comparison between proposals and between exact and asymptotic values

a monotonically convergent algorithm for factals takane young and de leeuw proposed a procedure called factals for the analysis of variables of mixed measurement levels numerical ordinal or nominal mooijaart pointed out that their algorithm does not necessarily converge and nevels proposed a new algorithm for the case of nominal variables in the present paper it is shown that nevels procedure is incorrect and a new procedure for handling nominal variables is proposed in addition a procedure for handling ordinal variables is proposed using these results a monotonically convergent algorithm is constructed for factals of any mixture of variables