shape estimation from shading defocus and correspondence using light field angular coherence light field cameras are quickly becoming commodity items with consumer and industrial applications they capture many nearby views simultaneously using a single image with a micro lens array thereby providing a wealth of cues for depth recovery defocus correspondence and shading in particular apart from conventional image shading one can refocus images after acquisition and shift one s viewpoint within the sub apertures of the main lens effectively obtaining multiple views we present a principled algorithm for dense depth estimation that combines defocus and correspondence metrics we then extend our analysis to the additional cue of shading using it to refine fine details in the shape by exploiting an all in focus image in which pixels are expected to exhibit angular coherence we define an optimization framework that integrates photo consistency depth consistency and shading consistency we show that combining all three sources of information defocus correspondence and shading outperforms state of the art light field depth estimation algorithms in multiple scenarios

the degrees of freedom of partial least squares regression the derivation of statistical properties for partial least squares regression can be a challenging task the reason is that the construction of latent components from the predictor variables also depends on the response variable while this typically leads to good performance and interpretable models in practice it makes the statistical analysis more involved in this work we study the intrinsic complexity of partial least squares regression our contribution is an unbiased estimate of its degrees of freedom it is defined as the trace of the first derivative of the fitted values seen as a function of the response we establish two equivalent representations that rely on the close connection of partial least squares to matrix decompositions and krylov subspace techniques we show that the degrees of freedom depend on the collinearity of the predictor variables the lower the collinearity the higher the complexity in particular they are typically higher than the naive approach that defines the degree

the generalized dynamic factor model consistency and rates abstract a factor model generalizing those proposed by geweke in d j aigner and a s goldberger latent variables in socio economic models north holland amsterdam 1977 sargent and sims new methods in business research federal reserve bank of minneapolis minneapolis 1977 engle and watson j amer statist assoc 76 1981 774 and stock and watson j business econom statist 20 2002 147 has been introduced in forni et al rev econ statist 80 2000 540 where consistent as the number n of series and the number t of observations both tend to infinity along appropriate paths n t n estimation methods for the common component are proposed rates of convergence associated with these methods are obtained here as functions of the paths n t n along which n and t go to infinity these results show that under suitable assumptions consistency requires t n to be at least of the same order as n whereas an optimal rate of n is reached for t n of the order of n2 if convergence to the space of common components is considered consistency holds irrespective of the path t n thus can be arbitrarily slow the optimal rate is still n but only requires t n to be of the order of n

statistical analysis of a multiply twisted helix abstract a multiply twisted helix in which a one dimensional array of components is twisted producing a helix which if twisted itself produces a doubly twisted helix and so on in which there are couplings between adjacent rounds of helices was investigated using a cellular automaton and the spectral statistics of a quantum particle a useful distance between the components of the structure is measured using a cellular automaton the dynamics of which is simulated for a number of initial conditions in order to determine the degree of connectivity between the components the area of the sphere with radius r on the basis of the above distance is shown to be proportional to r2 when r is small the anderson transition was investigated based on the spectral statistics of a quantum particle in a multiply twisted helix with on site random potentials as the strength of the on site random potentials increases the anderson transition occurs both results support our conclusion that the number of dimensions is always three for every multiply twisted helix if the couplings between adjacent rounds are strong enough

anderson transition driven by running fractal dimensions in a fractal shaped structure abstract anderson transition in a fractal shaped structure is driven by running fractal dimension from two to three we analyzed the spectral statistics of a quantum particle confined in a fractal shaped structure generated using the dielectric breakdown model in three dimensions when the fractal dimension is almost three the spectral statistics of low energy states are similar to those of a gaussian orthogonal ensemble as the fractal dimension decreases to two the spectral statistics become poissonian this indicates the onset of quantum localization in the fractal shaped structure so the anderson transition takes place as the fractal dimension changes the effects of introducing random potentials are also analyzed when a random magnetic field affects quantum states extended over our fractal shaped structure the spectral statistics are similar to those of a gaussian unitary ensemble on the other hand a random on site potential encourages quantum localization in our fractal shaped structure

a generalization of partial autocorrelations useful in identifying arma models statistics are proposed to help identify the orders of autoregressive moving average models these are functions of the sample autocorrelations and include partial autocorrelations as special cases the large sample variances of the statistics are derived and small sample properties are explored by simulation their interpretation is illustrated using wolfer s annual sunspot numbers and wind velocity data cleveland 1972

forecasting using principal components from a large number of predictors this article considers forecasting a single time series when there are many predictors n and time series observations t when the data follow an approximate factor model the predictors can be summarized by a small number of indexes which we estimate using principal components feasible forecasts are shown to be asymptotically efficient in the sense that the difference between the feasible forecasts and the infeasible forecasts constructed using the actual values of the factors converges in probability to 0 as both n and t grow large the estimated factors are shown to be consistent even in the presence of time variation in the factor model

contrast enhancement based on intrinsic image decomposition in this paper we propose to introduce intrinsic image decomposition priors into decomposition models for contrast enhancement since image decomposition is a highly ill posed problem we introduce constraints on both reflectance and illumination layers to yield a highly reliable solution we regularize the reflectance layer to be piecewise constant by introducing a weighted ell 1 norm constraint on neighboring pixels according to the color similarity so that the decomposed reflectance would not be affected much by the illumination information the illumination layer is regularized by a piecewise smoothness constraint the proposed model is effectively solved by the split bregman algorithm then by adjusting the illumination layer we obtain the enhancement result to avoid potential color artifacts introduced by illumination adjusting and reduce computing complexity the proposed decomposition model is performed on the value channel in hsv space experiment results demonstrate that the proposed method performs well for a wide variety of images and achieves better or comparable subjective and objective quality compared with the state of the art methods

semiclassical level spacings when regular and chaotic orbits coexist the authors calculate semiclassical limiting level spacing distributions p s for systems whose classical energy surface is divided into a number of separate region in which motion is regular or chaotic in the calculation it is assumed that the spectrum is the superposition of statistically independent sequences of levels from each of the classical phase space regions sequences from regular regions having poisson distributions and those from irregular regions having wigner distributions the formulae for p s depend on the sum of the liouville measures of all the classical regular regions and on the separate liouville measures of the significant chaotic regions

recovering intrinsic images from a single image interpreting real world images requires the ability distinguish the different characteristics of the scene that lead to its final appearance two of the most important of these characteristics are the shading and reflectance of each point in the scene we present an algorithm that uses multiple cues to recover shading and reflectance intrinsic images from a single image using both color information and a classifier trained to recognize gray scale patterns given the lighting direction each image derivative is classified as being caused by shading or a change in the surface s reflectance the classifiers gather local evidence about the surface s form and color which is then propagated using the generalized belief propagation algorithm the propagation step disambiguates areas of the image where the correct classification is not clear from local evidence we use real world images to demonstrate results and show how each component of the system affects the results

testing for autoregressive against moving average errors in the linear regression model abstract this paper presents a new approach to hypotheses testing problems which are non nested in the classical sense and which concern the covariance matrix of the disturbance vector of the linear regression model in particular the application of the approach to testing for ar 1 disturbances against ma 1 disturbances is explored in some detail practical difficulties are discussed and selected upper bounds for the test s five percent significance points are tabulated the small sample power of four versions of the new test are compared empirically and a clear conclusion is made in regard to the best overall test

critical helix of hierarchical structures abstract electronic structures in a hierarchical helix with inter round couplings i e a multiply twisted helix are investigated using the spectral statistics of a quantum particle the hierarchical structure is constructed using a procedure in which a helical chain of components is twisted producing a doubly twisted helix which if itself is twisted produces a triply twisted helix and so on we compare three types of inter round couplings which enable electrons to return to their original positions when the strength of inter round couplings is sufficiently smaller than that along a basal helix the type of inter round couplings can be distinguished by the behavior of spectral statistics the multiply twisted helix of inter round couplings through a single site is unique in the sense that its electronic structure is robust against any change in the basal helix so is called a critical helix

models of intermediate spectral statistics based on numerical results it is conjectured that the spectral statistics of certain pseudointegrable billiards have a special form similar to that of the anderson model at the transition point a simple theoretical model where such statistics can be obtained analytically is briefly discussed a few models with similar behavior are considered in particular we analytically found the eigenvalue statistics of a poisson distributed matrix perturbed by a rank one matrix which is a good model for spectral statistics of a singular billiard

testing for autocorrelation in the presence of lagged dependent variables a specification error approach abstract hausman s specification error testing procedure is used to develop serial correlation tests in lagged dependent variable models properties of the tests are discussed and comparisons are made with existing tests a reparameterization of the model using the error structure under the alternative hypothesis results in an augmented regression equation standard tests of exclusionary restrictions on coefficients of the augmented regressors can be regarded as serial correlation tests monte carlo experiments are used to examine small sample properties several of the newtests are shown to perform well in comparison with existing tests

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

ols or gls in the presence of specification error an expected loss approach omitted variables in regression analysis can lead to the erroneous conclusion that autocorrelation or heteroscedasticity is present the common response is to use the suggested gls procedure even if it is suspected that the error is a non zero disturbance mean the question addressed here is whether one is better off with the gls or with the ols estimator when the omitted portion of the regression cannot be incorporated into the regression using a loss function this paper relates the seriousness of ols and gls loss to identifiable parameters with consistent estimators of these parameters the researcher can choose between ols and gls

a closed form solution to retinex with nonlocal texture constraints we propose a method for intrinsic image decomposition based on retinex theory and texture analysis while most previous methods approach this problem by analyzing local gradient properties our technique additionally identifies distant pixels with the same reflectance through texture analysis and uses these nonlocal reflectance constraints to significantly reduce ambiguity in decomposition we formulate the decomposition problem as the minimization of a quadratic function which incorporates both the retinex constraint and our nonlocal texture constraint this optimization can be solved in closed form with the standard conjugate gradient algorithm extensive experimentation with comparisons to previous techniques validate our method in terms of both decomposition accuracy and runtime efficiency

estimation of a linear regression model with stationary arma p q errors abstract it is well known that consistent estimation of a linear regression model with a stationary gaussian arma process in the errors can be carried out by maximum likelihood or alternatively by two stage procedures involving estimation of the nuisance parameters followed by feasible generalized least squares for the model parameters we show that the estimators coincide up to o p t 3 2 and derive the variance to o t 2 which up to terms of this order is the same for both estimators considering the form of the error covariance matrix for an arma p q process allows us to examine a computationally convenient algorithm for estimation of the parameters of the regression model finally we provide a monte carlo comparison of the small sample properties of ols and two versions of the proposed estimator

entropy minimization for shadow removal recently a method for removing shadows from colour images was developed finlayson et al in ieee trans pattern anal mach intell 28 59 68 2006 that relies upon finding a special direction in a 2d chromaticity feature space this invariant direction is that for which particular colour features when projected into 1d produce a greyscale image which is approximately invariant to intensity and colour of scene illumination thus shadows which are in essence a particular type of lighting are greatly attenuated the main approach to finding this special angle is a camera calibration a colour target is imaged under many different lights and the direction that best makes colour patch images equal across illuminants is the invariant direction here we take a different approach in this work instead of a camera calibration we aim at finding the invariant direction from evidence in the colour image itself specifically we recognize that producing a 1d projection in the correct invariant direction will result in a 1d distribution of pixel values that have smaller entropy than projecting in the wrong direction the reason is that the correct projection results in a probability distribution spike for pixels all the same except differing by the lighting that produced their observed rgb values and therefore lying along a line with orientation equal to the invariant direction hence we seek that projection which produces a type of intrinsic independent of lighting reflectance information only image by minimizing entropy and from there go on to remove shadows as previously to be able to develop an effective description of the entropy minimization task we go over to the quadratic entropy rather than shannon s definition replacing the observed pixels with a kernel density probability distribution the quadratic entropy can be written as a very simple formulation and can be evaluated using the efficient fast gauss transform the entropy written in this embodiment has the advantage that it is more insensitive to quantization than is the usual definition the resulting algorithm is quite reliable and the shadow removal step produces good shadow free colour image results whenever strong shadow edges are present in the image in most cases studied entropy has a strong minimum for the invariant direction revealing a new property of image formation

the identification of arma models summary we present a new powerful method for determining the order of arma p q models having small sets of observations the procedure is based on an autoregressive order determination criterion and on linear estimation methods simulated data are used to demonstrate the capabilities of the approach for a survey nevertheless the reliability and adaptability of the methods remain at best tenuous in this paper we propose a simple but accurate procedure based on an autoregressive order determination criterion and linear estimation methods to identify the order of arma processes from a finite set of observations the paper is organized as follows in 2 we describe our approach in 3 we demonstrate the performance of our identification procedure based on simulations of several model structures with varying number of observations and 4 contains concluding remarks it is important that in reality the order of any process based on finite samples may never be correctly identified by any procedure all models are simplifications of reality their value and strength lie in their ability to explain the current situation and to predict the future our procedure therefore should be viewed as one which parsimoniously identifies the order of the arma model which reasonably depicts the underlying process governing the behaviour of the available data box jenkins 1976 ch 1 for brevity we assume that the reader is familiar with the characteristics and properties of arma models unless otherwise stated we use the notation and conventions of box

the three pass regression filter a new approach to forecasting using many predictors we forecast a single time series using many predictor variables with a new estimator called the three pass regression lter 3prf it is calculated in closed form and conveniently represented as a set of ordinary least squares regressions 3prf forecasts converge to the infeasible best forecast when both the time dimension and cross section dimension become large this requires only specifying the number of relevant factors driving the forecast target regardless of the total number of common including potentially irrelevant factors driving the cross section of predictors we derive inferential theory including limiting distributions for estimated relevant factors predictive coefcients and forecasts and provide consistent standard error estimators we explore two empirical applications forecasting macroeconomic aggregates with a large panel of economic indices and forecasting stock market aggregates with many individual assets price dividend ratios these combined with a range of monte carlo experiments demonstrate the 3prf s forecasting power

anderson transition in fractal based complexes abstract we investigated the anderson transition in fractal based complexes which were generated using the dielectric breakdown model when parameter in the model was changed from 1 to 2 at time 1 during the growth when 1 2 our nerve cell like complex can be considered a dendritic fractal grown on a somatic fractal on the other hand when 1 2 our nebula like complex cannot be divided into two regions the spectral statistics of a quantum particle in these fractal based complexes were analyzed and the result indicates the existence of an anderson transition an extended electron showing quantum chaos becomes localized when 1 decreases in a nerve cell like complex and when 1 increases in a nebula like complex it is shown that these two types of fractal based complexes can be distinguished by the type of anderson transition the features of which are characterized based on the berry robnik distribution

forecasting economic time series using targeted predictors abstract this paper studies two refinements to the method of factor forecasting first we consider the method of quadratic principal components that allows the link function between the predictors and the factors to be non linear second the factors used in the forecasting equation are estimated in a way to take into account that the goal is to forecast a specific series this is accomplished by applying the method of principal components to targeted predictors selected using hard and soft thresholding rules our three main findings can be summarized as follows first we find improvements at all forecast horizons over the current diffusion index forecasts by estimating the factors using fewer but informative predictors allowing for non linearity often leads to additional gains second forecasting the volatile one month ahead inflation warrants a high degree of targeting to screen out the noisy predictors a handful of variables notably relating to housing starts and interest rates are found to have systematic predictive power for inflation at all horizons third the targeted predictors selected by both soft and hard thresholding changes with the forecast horizon and the sample period holding the set of predictors fixed as is the current practice of factor forecasting is unnecessarily restrictive

alternative algorithms for the estimation of dynamic factor mimic and varying coefficient regression models this paper provides a general approach to the formulation and estimation of dynamic unobserved component models after introducing the general model two methods for estimating the unknown parameters are presented both are algorithms for maximizing the likelihood function the first is based on the method of scoring the second is the em algorithm a derivative free method each iteration of em requires a kalman filter and smoother followed by straightforward regression calculations the paper suggests using the em methods to quickly locate a neighborhood of the maximum scoring can then be used to pinpoint the maximum and calculate the information matrix

effects of variance function misspecification in analysis of longitudinal data the approach of generalized estimating equations gee is based on the framework of generalized linear models but allows for specification of a working matrix for modeling within subject correlations the variance is often assumed to be a known function of the mean this article investigates the impacts of misspecifying the variance function on estimators of the mean parameters for quantitative responses our numerical studies indicate that 1 correct specification of the variance function can improve the estimation efficiency even if the correlation structure is misspecified 2 misspecification of the variance function impacts much more on estimators for within cluster covariates than for cluster level covariates and 3 if the variance function is misspecified correct choice of the correlation structure may not necessarily improve estimation efficiency we illustrate impacts of different variance functions using a real data set from cow growth

testing for omitted variables and non linearity in regression models for longitudinal data when fitting regression models to investigate the relationship between an outcome variable and independent variables of primary interest there is often concern whether omitted variables or assuming a different functional relationship could have changed the conclusion or interpretation of the results in longitudinal studies of aging the concern with omitted variables is well known in the context of cohort and period effects which refer to unmeasured variables systematically related to the individual s year of birth and secular trends in outcome respectively we present and compare three approaches to detecting omitted confounders and non linearity in the random effects model for longitudinal data laird and ware 1982 with random slope and intercept across individuals the first approach compares simple unweighted within and between regression coefficients the second is the hausman specification test for regression models and the third approach involves testing directly the significance of functions of individual specific covariate means means i in the random effects regression model this last approach is motivated by the models that arise when cohort or period effects are ignored we compare the three approaches and illustrate their application

a variational framework for retinex retinex theory addresses the problem of separating the illumination from the reflectance in a given image and thereby compensating for non uniform lighting this is in general an ill posed problem in this paper we propose a variational model for the retinex problem that unifies previous methods similar to previous algorithms it assumes spatial smoothness of the illumination field in addition knowledge of the limited dynamic range of the reflectance is used as a constraint in the recovery process a penalty term is also included exploiting a priori knowledge of the nature of the reflectance image the proposed formulation adopts a bayesian view point of the estimation problem which leads to an algebraic regularization term that contributes to better conditioning of the reconstruction problem based on the proposed variational model we show that the illumination estimation problem can be formulated as a quadratic programming optimization problem an efficient multi resolution algorithm is proposed it exploits the spatial correlation in the reflectance and illumination images applications of the algorithm to various color images yield promising results

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

least squares regression when the independent variable follows an arima process abstract in the regression model yt xt ut where the disturbances ut may be autocorrelated and xt follows an arima p 1 q process it is shown that the asymptotic distributions of the ordinary and generalized least squares estimates of are identical

self organized network of fractal shaped components coupled through statistical interaction a dissipative dynamics is introduced to generate self organized networks of interacting objects which we call coupled fractal networks the growth model is constructed based on a growth hypothesis in which the growth rate of each object is a product of the probability of receiving source materials from faraway and the probability of receiving adhesives from other grown objects where each object grows to be a random fractal if isolated but connects with others if glued the network is governed by the statistical interaction between fractal shaped components which can only be identified in a statistical manner over ensembles this interaction is investigated using the degree of correlation between fractal shaped components enabling us to determine whether it is attractive or repulsive

generalized least squares with misspecified serial correlation structures summary the regression literature contains hundreds of studies on serially correlated disturbances most of these studies assume that the structure of the error covariance matrix q is known or can be estimated consistently from data surprisingly few studies investigate the properties of estimated generalized least squares gls procedures when the structure of q is incorrectly identified and the parameters are inefficiently estimated we compare the finite sample efficiencies of ordinary least squares ols gls and incorrect gls igls estimators we also prove new theorems establishing theoretical efficiency bounds for igls relative to gls and ols results from an exhaustive simulation study are used to evaluate the finite sample performance and to demonstrate the robustness of igls estimates vis a vis ols and gls estimates constructed for models with known and estimated but correctly identified q some of our conclusions for finite samples differ from established asymptotic results

identification of nonzero elements in the polynomial matrices of mixed varma processes on presente des methodes pour identifier des elements non nuls dans des modeles varma 1 1 et on montre comment utiliser ces procedures pour construire des modeles d ordre superieur

anderson localization in a superstring model representation abstract anderson localization has been considered via the nonlinear model we consider the possibility that the anderson localization is formulated in a superstring model based upon the finding that the function of anderson localization coincides with the function of the space time supersymmetric nonlinear model we recover the recent four loop result by wegner by the equivalence of the function of anderson localization to the supersymmetric nonlinear model a possible conjecture about higher order terms is discussed the difference between the orthogonal case and the unitary case is investigated in terms of the supersymmetric nonlinear model the advantage of the formulation in terms of superstring model is emphasized for the study of anderson localization

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

filters random fields and maximum entropy frame towards a unified theory for texture modeling this article presents a statistical theory for texture modeling this theory combines filtering theory and markov random field modeling through the maximum entropy principle and interprets and clarifies many previous concepts and methods for texture analysis and synthesis from a unified point of view our theory characterizes the ensemble of images i with the same texture appearance by a probability distribution f i on a random field and the objective of texture modeling is to make inference about f i given a set of observed texture examples in our theory texture modeling consists of two steps 1 a set of filters is selected from a general filter bank to capture features of the texture these filters are applied to observed texture images and the histograms of the filtered images are extracted these histograms are estimates of the marginal distributions of f i this step is called feature extraction 2 the maximum entropy principle is employed to derive a distribution p i which is restricted to have the same marginal distributions as those in 1 this p i is considered as an estimate of f i this step is called feature fusion a stepwise algorithm is proposed to choose filters from a general filter bank the resulting model called frame filters random fields and maximum entropy is a markov random field mrf model but with a much enriched vocabulary and hence much stronger descriptive ability than the previous mrf models used for texture modeling gibbs sampler is adopted to synthesize texture images by drawing typical samples from p i thus the model is verified by seeing whether the synthesized texture images have similar visual appearances to the texture images being modeled experiments on a variety of 1d and 2d textures are described to illustrate our theory and to show the performance of our algorithms these experiments demonstrate that many textures which are previously considered as from different categories can be modeled and synthesized in a common framework

asymptotics of the principal components estimator of large factor models with weakly influential factors abstract this paper introduces a drifting parameter asymptotic framework to derive accurate approximations to the finite sample distribution of the principal components pc estimator in situations when the factors explanatory power does not strongly dominate the explanatory power of the cross sectionally and temporally correlated idiosyncratic terms under our asymptotics the pc estimator is inconsistent we find explicit formulae for the amount of the inconsistency and propose an estimator of the number of factors for which the pc estimator works reasonably well for the special case when the idiosyncratic terms are cross sectionally but not temporally correlated or vice versa we show that the coefficients in the ols regressions of the pc estimates of factors loadings on the true factors true loadings are asymptotically normal and find explicit formulae for the corresponding asymptotic covariance matrix we explain how to estimate the parameters of the derived asymptotic distributions our monte carlo analysis suggests that our asymptotic formulae and estimators work well even for relatively small n and t we apply our theoretical results to test a hypothesis about the factor content of the us stock return data

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

the relationship among the specification tests of hausman ramsey and chow abstract this article describes a relationship between the tests for specification error in regression models introduced by ramsey 1969 and hausman 1978 it also shows that the chow 1960 test for structural shift can be viewed as a special case of specification error tests this article shows the common structure of these tests the common basis is the comparison of alternative estimators of regression coefficients under the null hypothesis one estimator is efficient and consistent and the other is consistent the probability limits of the two estimators are different under the alternative hypothesis