tests of equal forecast accuracy and encompassing for nested models we examine the asymptotic and finite sample properties of tests for equal forecast accuracy and encompassing applied to 1 step ahead forecasts from nested linear models we first derive the asymptotic distributions of two standard tests and one new test of encompassing and provide tables of asymptotically valid critical values monte carlo methods are then used to evaluate the size and power of tests of equal forecast accuracy and encompassing the simulations indicate that post sample tests can be reasonably well sized of the post sample tests considered the encompassing test proposed in this paper is the most powerful we conclude with an empirical application regarding the predictive content of unemployment for inflation

spectral theory of some non selfadjoint linear differential operators we give a characterization of the spectral properties of linear differential operators with constant coefficients acting on functions defined on a bounded interval and determined by general linea

a variance minimization problem for a markov decision process abstract in the steady state of a discrete time markov decision process we consider the problem to find an optimal randomized policy that minimizes the variance of the reward in a transition among the policies which give the mean not less than a specified value the problem is solved by introducing a parametric markov decision process with average cost criterion it is shown that there exists an optimal policy which is a mixture of at most two pure policies as an application the toymaker s problem is discussed

markov decision problems where means bound variances we identify a rich class of finite horizon markov decision problems mdps for which the variance of the optimal total reward can be bounded by a simple linear function of its expected value the class is characterized by three natural properties reward nonnegativity and boundedness existence of a do nothing action and optimal action monotonicity these properties are commonly present and typically easy to check implications of the class properties and of the variance bound are illustrated by examples of mdps from operations research operations management financial engineering and combinatorial optimization

mean variance tradeoffs in an undiscounted mdp the unichain case the problem analyzed here is the computation of pareto optima in the sense of high mean and low variance of the stationary distribution in the unichain undiscounted markov decision process mdp for short

robust out of sample inference this paper presents analytical empirical and simulation results concerning inference about the moments of nondifferentiable functions of out of sample forecasts and forecast errors special attention is given to the measurement of a model s predictive ability using the test of equal mean absolute error tests for equal mean absolute error and mean square error are used to evaluate predictions of excess returns to the s p 500 composite simulations indicate that appropriately constructed tests for equal mean absolute error can provide more accurately sized and more powerful tests than inappropriately constructed tests for equal mean absolute error and mean square error

the diffusion equation with nonlocal data abstract we study the diffusion or heat equation on a finite 1 dimensional spatial domain but we replace one of the boundary conditions with a nonlocal condition through which we specify a weighted average of the solution over the spatial interval we provide conditions on the regularity of both the data and weight for the problem to admit a unique solution and also provide a solution representation in terms of contour integrals the solution and well posedness results rely upon an extension of the fokas or unified transform method to initial nonlocal value problems for linear equations the necessary extensions are described in detail despite arising naturally from the fokas transform method the uniqueness argument appears to be novel even for initial boundary value problems

bootstrapping factor augmented regression models abstract this paper proposes and theoretically justifies bootstrap methods for regressions where some of the regressors are factors estimated from a large panel of data we derive our results under the assumption that t n c where 0 c n and t are the cross sectional and the time series dimensions respectively thus allowing for the possibility that the factor estimation error enters the limiting distribution of the ols estimator as an asymptotic bias term as was recently discussed by ludvigson and ng 2011 we consider general residual based bootstrap methods and provide a set of high level conditions on the bootstrap residuals and on the idiosyncratic errors such that the bootstrap distribution of a rotated ols estimator is consistent we subsequently verify these conditions for a simple wild bootstrap residual based procedure

asymptotics for out of sample tests of granger causality abstract this paper presents analytical monte carlo and empirical evidence concerning out of sample tests of granger causality the environment is one in which the relative predictive ability of two nested parametric regression models is of interest results are provided for three statistics a regression based statistic suggested by granger and newbold 1977 forecasting economic time series academic press inc london a t type statistic comparable to those suggested by diebold and mariano 1995 comparing predictive accuracy journal of business and economic statistics 13 253 263 and west 1996 asymptotic inference about predictive ability econometrica 64 1067 1084 and an f type statistic akin to theil s u since the asymptotic distributions under the null are nonstandard tables of asymptotically valid critical values are provided monte carlo evidence supports the theoretical results an empirical example evaluates the predictive content of the chicago fed national activity index for growth in industrial production and core pce based inflation

constrained undiscounted stochastic dynamic programming in this paper we investigate the computation of optimal policies in constrained discrete stochastic dynamic programming with the average reward as utility function the state space and action sets are assumed to be finite constraints which are linear functions of the state action frequencies are allowed in the general multichain case an optimal policy will be a randomized nonstationary policy an algorithm to compute such an optimal policy is presented furthermore sufficient conditions for optimal policies to be stationary are derived there are many applications for constrained undiscounted stochastic dynamic programming e g in multiple objective markovian decision models

optimal policies for controlled markov chains with a constraint abstract the time average reward for a discrete time controlled markov process subject to a time average cost constraint is maximized over the class of al causal policies each epoch a reward depending on the state and action is earned and a similarly constituted cost is assessed the time average of the former is maximized subject to a hard limit on the time average of the latter it is assumed that the state space is finite and the action space compact metric an accessibility hypothesis makes it possible to utilize a lagrange multiplier formulation involving the dynamic programming equation thus reducing the optimization problem to an unconstrained optimization parametrized by the multiplier the parametrized dynamic programming equation possesses compactness and convergence properties that lead to the following if the constraint can be satisfied by any causal policy the supremum over time average rewards respective to all causal policies is attained by either a simple or a mixed policy the latter is equivalent to choosing independently at each epoch between two specified simple policies by the throw of a biased coin

on finding optimal policies for markov decision chains a unifying framework for mean variance tradeoffs this paper proves constructively the existence of optimal policies for maximum one period mean to standard deviation ratio negative variance with bounded mean and mean penalized by variance markov decision chains by reducing them to a related mathematical program this program entails maximizing xb d xb c xb over x in a polytope and with given bounds on xb where c and d are convex and either d is constant or d is positive and nondecreasing c is nondecreasing and xb is nonpositive this program is in turn reduced to maximizing x b b over x in the polytope parametrically in along the way under the nonnegative initial distribution assumption we generalize the rule of constructing a stationary maximum average reward policy from an extreme optimal solution of the associated linear program the paper unifies and extends formulations and existence results for problems discussed by white 1987 filar and lee 1985 sobel 1985 kawai 1987 and filar kallenberg and lee 1989 and gives an ef

comparing k populations with linear rank statistics abstract there are many nonparametric tests available for the analysis of k independent samples for example the kruskal wallis test wilcoxon rank sum test when k 2 is widely used for detecting location differences and mood or ansari bradley or sigel tukey rank statistics are often suggested for detecting scale differences if more general alternatives are of interest then one can use omnibus tests such as the generalized kolmogorov smirnov and cramer von mises statistics see conover 1980 secs 6 3 and 6 4 or kiefer 1959 unfortunately significant results from these omnibus tests are hard to interpret instead i would like to propose some new tests based on a 4 k table of linear rank statistics that are simple to interpret because the four rows of the table are aimed at specific alternatives location scale skewness and kurtosis row summary statistics consist of weighted sums of squares of the individual entries the first two of these are the kruskal wallis statistic and mood s k sa

mean variance tradeoffs in an undiscounted mdp a stationary policy and an initial state in an mdp markov decision process induce a stationary probability distribution of the reward the problem analyzed here is generating the pareto optima in the sense of high mean and low variance of the stationary distribution in the unichain case pareto optima can be computed either with policy improvement or with a linear program having the same number of variables and one more constraint than the formulation for gain rate optimization the same linear program suffices in the multichain case if the ergodic class is an element of choice

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

asymptotic distribution of auc nris and idi based on theory of u statistics the change in area under the curve auc the integrated discrimination improvement idi and net reclassification index nri are commonly used measures of risk prediction model performance some authors have reported good validity of associated methods of estimating their standard errors se and construction of confidence intervals whereas others have questioned their performance to address these issues we unite the auc idi and three versions of the nri under the umbrella of the u statistics family we rigorously show that the asymptotic behavior of auc nris and idi fits the asymptotic distribution theory developed for u statistics we prove that the auc nris and idi are asymptotically normal unless they compare nested models under the null hypothesis in the latter case asymptotic normality and existing se estimates cannot be applied to auc nris or idi in the former case se formulas proposed in the literature are equivalent to se formulas obtained from u statistics theory if we ignore adjustment for estimated parameters we use sukhatme randles dewet condition to determine when adjustment for estimated parameters is necessary we show that adjustment is not necessary for ses of the auc and two versions of the nri when added predictor variables are significant and normally distributed the ses of the idi and three category nri should always be adjusted for estimated parameters these results allow us to define when existing formulas for se estimates can be used and when resampling methods such as the bootstrap should be used instead when comparing nested models we also use the u statistic theory to develop a new se estimate of auc copyright 2017 john wiley sons ltd

mean variance criteria in an undiscounted markov decision process abstract in the steady state of an undiscounted markov decision process we consider the problem of finding an optimal stationary probability distribution that minimizes the variance of the reward in a transition among the stationary probability distributions which give a mean not less than a specified value the problem consists of a mathematical program with linear constraints and a non linear objective the solution technique replaces the non linear part of the objective with a constant inserts the constant as a constraint and then parametrically analyzes the resulting linear program three numerical examples are discussed

forecasting with factor augmented regression a frequentist model averaging approach abstract this paper considers forecast combination with factor augmented regression in this framework a large number of forecasting models are available varying by the choice of factors and the number of lags we investigate forecast combination across models using weights that minimize the mallows and the leave h out cross validation criteria the unobserved factor regressors are estimated by principle components of a large panel with n predictors over t periods with these generated regressors we show that the mallows and leave h out cross validation criteria are asymptotically unbiased estimators of the one step ahead and multi step ahead mean squared forecast errors respectively provided that n t however the paper does not establish any optimality properties for the methods in contrast to well known results in the literature this result suggests that the generated regressor issue can be ignored for forecast combination without restrictions on the relation between n and t simulations show that the mallows model averaging and leave h out cross validation averaging methods yield lower mean squared forecast errors than alternative model selection and averaging methods such as aic bic cross validation and bayesian model averaging we apply the proposed methods to the us macroeconomic data set in stock and watson 2012 and find that they compare favorably to many popular shrinkage type forecasting methods

indirect methods of imputation of missing data based on available units one of the most difficult problems confronting investigators who analyze data from surveys is how to treat missing data many statistical procedures cannot be used immediately if any values are missing imputation of missing data before starting statistical analysis is then necessary this paper proposes imputation methods of the mean based on indirect estimators of available cases a complete simulation study was performed to test the proposed techniques

tests of equal accuracy for nested models with estimated factors abstract in this paper we develop asymptotics for tests of equal predictive ability between nested models when factor augmented regressions are used to forecast we provide conditions under which the estimation of the factors does not affect the asymptotic distributions developed in clark and mccracken 2001 and mccracken 2007 this enables researchers to use the existing tabulated critical values when conducting inference despite the presence of estimated predictors as an intermediate result we derive the asymptotic properties of the principal components estimator over recursive windows we provide simulation evidence on the finite sample effects of factor estimation and apply the tests to the case of forecasting excess returns to the s p 500 composite index

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

markov decision problems and state action frequencies consider a controlled markov chain with countable state and action spaces basic quantities that determine the values of average cost functionals are identified under some regularity conditions these turn out to be a collection of numbers one for each state action pair describing for each state the relative number of uses of each action these conditional frequencies which are defined pathwise are shown to determine the state action frequencies that in the finite case are known to determine the costs this is extended to the countable case allowing for unbounded costs the space of frequencies is shown to be compact and convex and the extreme points are identified with stationary deterministic policies conditions under which the search for optimality in several optimization problems may be restricted to stationary policies are given these problems include the standard markov decision process as well as constrained optimization both in terms of average cost functionals and variability sensitive optimization an application to a queueing problem is given where these results imply the existence and explicit computation of optimal policies in constrained optimization problems the pathwise definition of the conditional frequencies implies that their values can be controlled directly moreover they depend only on the limiting behavior of the control this has immediate application to adaptive control of markov chains including adaptive control under constraints

algorithmic aspects of mean variance optimization in markov decision processes we consider finite horizon markov decision processes under performance measures that involve both the mean and the variance of the cumulative reward we show that either randomized or history based policies can improve performance we prove that the complexity of computing a policy that maximizes the mean reward under a variance constraint is np hard for some cases and strongly np hard for others we finally offer pseudopolynomial exact and approximation algorithms

a galerkin procedure for the diffusion equation subject to the specification of mass continuous and discrete galerkin procedures are derived and analyzed for the numerical solution of the diffusion equation u 1 u xx 0 x 1 0 t leqq t u x 0 f x u x 1 t g t and the specification of the mass int 0 b u x t dx m t 0 b 1 some numerical experiments are also presented

principal components estimation and identification of static factors a b s t r a c t it is known that the principal component estimates of the factors and the loadings are rotations of the un derlying latent factors and loadings we study conditions under which the latent factors can be estimated asymptotically without rotation we derive the limiting distributions for the estimated factors and factor loadings when n and t are large and make precise how identification of the factors affects inference based on factor augmented regressions we also consider factor models with additive individual and time effects the asymptotic analysis can be modified to analyze identification schemes not considered in this analysis

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

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

equivalence between out of sample forecast comparisons and wald statistics we demonstrate the equivalence between a commonly used out of sample test of equal predictive accuracy and the dierence between two wald statistics this equivalence greatly simplies the computational burden of calculating recursive out of sample tests and evaluating their critical values next we show that the limit distribution which has previously been expressed through stochastic integrals has a simple representation in terms of 2 distributed random variables and we derive the density of the limit distribution we also generalize the limit theory to cover local alternatives and characterize the power properties of the test our results shed new light on the test and establish certain weaknesses associated with using out of sample forecast comparison tests to conduct inference about nested regression models

estimating distribution functions from survey data summary a simple method for estimating population distribution functions and associated quantiles from sample survey data is described and some asymptotic theory for it presented the method assumes a model based approach to survey estimation and allows auxiliary population information to be directly incorporated into the estimation process monte carlo results comparing the proposed method with conventional design based methods are given these suggest that the model based approach offers significant gains when the auxiliary population information is linearly related to the survey variables of interest

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 linear kdv equation with an interface the interface problem for the linear korteweg de vries kdv equation in one dimensional piecewise homogeneous domains is examined by constructing an explicit solution in each domain the location of the interface is known and a number of compatibility conditions at the boundary are imposed we provide an explicit characterization of sufficient interface conditions for the construction of a solution using fokas s unified transform method the problem and the method considered here extend that of earlier papers to problems with more than two spatial derivatives

a unified approach to linearization variance estimation from survey data after imputation for item nonresponse variance estimation after imputation is an important practical problem in survey sampling when deterministic imputation or stochastic imputation is used we show that the variance of the imputed estimator can be consistently estimated by a unifying linearize and reverse approach we provide some applications of the approach to regression imputation fractional categorical imputation multiple imputation and composite imputation results from a simulation study under a factorial structure for the sampling response and imputation mechanisms show that the proposed linearization variance estimator performs well in terms of relative bias assuming a missing at random response mechanism