law of large numbers for the sir model with random vertex weights on erdős renyi graph abstract in this paper we are concerned with the sir model with random vertex weights on erd s renyi graph g n p the erd s renyi graph g n p is generated from the complete graph c n with n vertices through independently deleting each edge with probability 1 p we assign i i d copies of a positive r v on each vertex as the vertex weights for the sir model each vertex is in one of the three states susceptible infective and removed an infective vertex infects a given susceptible neighbor at rate proportional to the production of the weights of these two vertices an infective vertex becomes removed at a constant rate a removed vertex will never be infected again we assume that at t 0 there is no removed vertex and the number of infective vertices follows a bernoulli distribution b n our main result is a law of large numbers of the model we give two deterministic functions h s t h v t for t 0 and show that for any t 0 h s t is the limit proportion of susceptible vertices and h v t is the limit of the mean capability of an infective vertex to infect a given susceptible neighbor at moment t as n grows to infinity

inference in linear time series models with some unit roots this paper considers estimation and hypothesis testing in linear time series models when some or all of the variables have unit roots our motivating example is a vector autoregression with some unit roots in the companion matrix which might include polynomials in time as regressors in the general formulation the variable might be integrated or cointegrated of arbitrary orders and might have drifts as well we show that parameters that can be written as coefficients on mean zero nonintegrated regressors have jointly normal asymptotic distributions converging at the rate t 2 in general the other coefficients including the coefficients on polynomials in time will have nonnormal asymptotic distributions the results provide a formal characterization of which t or f tests such as granger causality tests will be asymptotically valid and which will have nonstandard limiting distributions

predictive ability with cointegrated variables in this paper we outline conditions under which the diebold and mariano dm j bus econom statist 13 1995 253 test for predictive ability can be extended to the case of two forecasting models each of which may include cointegrating relations when allowing for parameter estimation error we show that in the cases where either the loss function is quadratic or the length of the prediction period p grows at a slower rate than the length of the regression period r the standard dm test can be used on the other hand in the case of a generic loss function if p r as t 0 then the asymptotic normality result of west econometrica 64 1996 1067 no longer holds we also extend the data snooping technique of white econometrica 68 2000 1097 for comparing the predictive ability of multiple forecasting models to the case of cointegrated variables in a series of monte carlo experiments we examine the impact of both short run and cointegrating vector parameter estimation error on dm data snooping and related tests our results suggest that size is reasonable for r and p greater than 50 and power improves with p as expected furthermore the additional cost in terms of size distortion due to the estimation of the cointegrating relations is not substantial we illustrate the use of the tests in a nonnested cointegration framework by forming prediction models for industrial production which include two interest rate variables prices and either m1 m2 or m3

the survival of one dimensional contact processes in random environments consider the inhomogeneous contact process on z1 with recovery rate 8 k at site k and infection rates a k and p k at site k due to the presence of infected neighbors at k 1 and k 1 respectively a special case of the main result in this paper is the following suppose that the environment is chosen in such a way that the 8 k s a k s and p k s are all mutually independent with the 8 k s having a common distribution and the a k s and p k s having a common distribution then the process survives if

on the cut polytope the cut polytopep c g of a graphg v e is the convex hull of the incidence vectors of all edge sets of cuts ofg we show some classes of facet defining inequalities ofp c g we describe three methods with which new facet defining inequalities ofp c g can be constructed from known ones in particular we show that inequalities associated with chordless cycles define facets of this polytope moreover for these inequalities a polynomial algorithm to solve the separation problem is presented we characterize the facet defining inequalities ofp c g ifg is not contractible tok 5 we give a simple characterization of adjacency inp c g and prove that for complete graphs this polytope has diameter one and thatp c g has the hirsch property a relationship betweenp c g and the convex hull of incidence vectors of balancing edge sets of a signed graph is studied

consistency properties of least squares estimates of autoregressive parameters in arma models a unified treatment of the consistency properties of the ordinary least squares estimates in an autoregressive fitting of time series from nonstationary or stationary autoregressive moving average models is given for a given model the orders of autoregressions which produce consistent estimates are obtained and the limiting values hence the biases of the estimates of other autoregressions are investigated

the complete convergence theorem holds for contact processes on open clusters of ℤd ℤ we study contact processes on open clusters of half space our result shows that the complete convergence theorem holds

contact processes with random recovery rates and edge weights on complete graphs in this paper we are concerned with the contact process with random recovery rates and edge weights on the complete graph with n vertices we show that the model has a critical value which is inversely proportional to the product of the mean of the edge weight and the mean of the inverse of the recovery rate in the subcritical case the process dies out before an amount of time with order o log n with high probability as n rightarrow infty in the supercritical case the process survives for an amount of time with order exp o n with high probability as n rightarrow infty

a cointegration approach to estimating preference parameters in this paper we estimate the long run intertemporal elasticity of substitution of non durable consumption which has often been estimated with the generalized methods of moments gmm the gmm estimator however is not consistent in the presence of liquidity constraints aggregation over heterogeneous consumers unknown preference shocks or a general form of time nonseparability we use engle and granger s cointegration methodology in order to develop an estimator which is consistent even in the presence of these factors we then form a formal test that compares the estimates obtained using cointegration techniques with those obtained using gmm

asymptotic normality when regressors have a unit root under fairly general conditions ordinary least squares and linear instrumental variables estimators are asymptotically normal when a regression equation has nonstationary right hand side variables standard formulas may be used to calculate a consistent estimate of the asymptotic variance covariance matrix of the estimated parameter vector even if the disturbances are conditionally heteroskedastic and autocorrelated so inference may proceed in the usual way the key requirements are that the nonstationary variables share a common unit root and that the unconditional mean of their first differences is nonzero copyright 1988 by the econometric society

formulation and estimation of dynamic models using panel data abstract this paper presents a statistical analysis of time series regression models for longitudinal data with and without lagged dependent variables under a variety of assumptions about the initial conditions of the processes being analyzed the analysis demonstrates how the asymptotic properties of estimators of longitudinal models are critically dependent on the manner in which samples become large by expanding the number of observations per person holding the number of people fixed or by expanding the number of persons holding the number of observations per person fixed the paper demonstrates which parameters can and cannot be identified from data produced by different sampling plans

generalized potlatch and smoothing processes we consider simple generalizations of the potlatch and smoothing processes which were introduced in 8 and studied in 5 these generalizations provide relatively simple examples of infinite interacting systems which exhibit phase transition the original potlatch and smoothing processes do not exhibit phase transition our results show that for the generalized processes phase transition does not usually occur in one or two dimensions but usually does occur in higher dimensions upper and lower bounds for the relevant critical values are obtained as one application of our results we obtain the limiting behavior of the critical values for the linear contact process in d dimensions as d thus answering a question we raised in 3 this is done by comparing the contact process with an appropriate generalized smoothing process

a canonical decomposition theory for metrics on a finite set abstract we consider specific additive decompositions d d 1 d n of metrics defined on a finite set x where a metric may give distance zero to pairs of distinct points the simplest building stones are the slit metrics associated to splits i e bipartitions of the given set x while an additive decomposition of a hamming metric into split metrics is in no way unique we achieve uniqueness by restricting ourselves to coherent decompositions that is decompositions d d 1 d n such that for every map f x r with f x f y d x y for all x y x there exist maps f 1 f n x r with f f 1 f n and f i x f i y d i x y for all i 1 n and all x y x these coherent decompositions are closely related to a geometric decomposition of the injective hull of the given metric a metric with a coherent decomposition into a weighted sum of split metrics will be called totally split decomposable tree metrics and more generally the sum of two tree metrics are particular instances of totally split decomposable metrics our main result confirms that every metric admits a coherent decomposition into a totally split decomposable metric and a split prime residue where all the split summands and hence the decomposition can be determined in polynomial time and that a family of splits can occur this way if and only if it does not induce on any four point subset all three splits with block size two

divide and conquer approximation algorithms via spreading metrics we present a novel divide and conquer paradigm for approximating np hard graph optimization problems the paradigm models graph optimization problems that satisfy two properties first a divide and conquer approach is applicable second a fractional spreading metric is computable in polynomial time the spreading metric assigns lengths to either edges or vertices of the input graph such that all subgraphs for which the optimization problem is nontrivial have large diameters in addition the spreading metric provides a lower bound t on the cost of solving the optimization problem we present a polynomial time approximation algorithm for problems modeled by our paradigm whose approximation factor is o min log t log log t log k log log k where k denotes the number of interesting vertices in the problem instance and is at most the number of vertices we present seven problems that can be formulated to fit the paradigm for all these problems our algorithm improves previous results the problems are 1 linear arrangement 2 embedding a graph in a d dimensional mesh 3 interval graph completion 4 minimizing storage time product 5 subset feedback sets in directed graphs and multicuts in circular networks 6 symmetric multicuts in directed networks 7 balanced partitions and p separators for small values of p in directed graphs

critical value for the contact process with random recovery rates and edge weights on regular tree abstract in this paper we are concerned with contact processes with random recovery rates and edge weights on rooted regular trees t n let and be two nonnegative random variables such that p m 1 for some m 0 for each vertex x on t n x is an independent copy of while for each edge e on t n e is an independent copy of an infected vertex x becomes healthy at rate x while an infected vertex y infects an healthy neighbor z at rate proportional to y z for this model we prove that the critical value under the annealed measure approximately equals n e e 1 1 as n grows to infinity furthermore we show that the critical value under the quenched measure equals that under the annealed measure when the cluster containing the root formed with edges with positive weights is infinite

inference for unit roots in dynamic panels where the time dimension is fixed this paper derives similar asymptotic unit root tests for first order autoregressive panel data models assuming that the time dimension of the panel is fixed it is shown that the limiting distributions of the test statistics are normal the assumption that the time dimension is fixed allows us to derive analytical expressions for the moments of the distributions similarity with respect to the initial conditions of the data generating process is achieved by including fixed effect dummy variables in the regression model while similarity with respect to fixed effects in the data generating process is achieved by including a linear deterministic trend for each individual unit of the panel when fixed effects or individual trends are included as regressors the least squares estimator of the autoregressive parameter is inconsistent and thus the test statistics must be appropriately adjusted monte carlo evidence suggests that the proposed tests have empirical size that is very close to the nominal five percent level and substantially more power than the corresponding unit root tests for the single time series case

critical value for contact processes on clusters of oriented bond percolation in this paper we are concerned with contact processes on open clusters of oriented bond percolation in zd where the disease spreads along the direction of open edges we show that the two critical values in the quenched and annealed cases are equal with probability one and are asymptotically equal to dp 1 as the dimension d grows to infinity where p is the probability that the edge is open

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

mean estimation bias in least squares estimation of autoregressive processes abstract estimation of the parameters of an autoregressive process with a mean that is a function of time is considered approximate expressions for the bias of the least squares estimator of the autoregressive parameters that is due to estimating the unknown mean function are derived for the case of a mean function that is a polynomial in time a reparameterization that isolates the bias is given using the approximate expressions a method of modifying the least squares estimator is proposed a monte carlo study of the second order autoregressive process is presented the monte carlo results agree well with the approximate theory and generally speaking the modified least squares estimators performed better than the least squares estimator for the second order process we also considered the empirical properties of the estimated generalized least squares estimator of the mean function and the error made in predicting the process one two and three periods in the future

on a class of metrics related to graph layout problems we examine the metrics that arise when a finite set of points is embedded in the real line in such a way that the distance between each pair of points is at least 1 these metrics are closely related to some other known metrics in the literature and also to a class of combinatorial optimization problems known as graph layout problems we prove several results about the structure of these metrics in particular it is shown that their convex hull is not closed in general we then show that certain linear inequalities define facets of the closure of the convex hull finally we characterize the unbounded edges of the convex hull and of its closure

mirror image distributions and the dickey fuller regression with a maintained trend abstract a puzzle of many empirical studies where dickey fuller regressions with a maintained trend are run is that the t test of the ar root and the trend t statistic frequently appear to be of equal magnitude but with opposite sign in this paper i demonstrate that when the true process contains a drift then the asymptotic distributions of the two statistics will be the mirror image of each other and even in finite samples this property will be predominant as long as the standardized drift is sufficiently strong

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

phase transition for the large dimensional contact process with random recovery rates on open clusters in this paper we are concerned with the contact process with random recovery rates on open clusters of bond percolation on mathbb z d let xi be a random variable such that p xi ge 1 1 which ensures mathrm e frac 1 xi infty then we assign i i d copies of xi on the vertices as the random recovery rates assuming that each edge is open with probability p and the infection can only spread through the open edges then we obtain that begin aligned limsup d rightarrow infty lambda d le lambda c frac 1 p mathrm e frac 1 xi end aligned where lambda d is the critical value of the process on mathbb z d i e the maximum of the infection rates with which the infection dies out with probability one when only the origin is infected at t 0 to prove the above main result we show that the following phase transition occurs assuming that lceil log d rceil vertices are infected at t 0 where these vertices can be located anywhere then when the infection rate lambda lambda c the process survives with high probability as d rightarrow infty while when lambda lambda c the process dies out at time o log d with high probability

new approximation techniques for some linear ordering problems we describe logarithmic approximation algorithms for the np hard graph optimization problems of minimum linear arrangement minimum containing interval graph and minimum storage time product this improves upon the best previous approximation bounds of even naor rao and schieber j acm 47 2000 pp 585 616 for these problems by a factor of omega log log n we use the lower bound provided by the volume w of a spreading metric for each of the ordering problems above as defined by even et al in order to find a solution with cost at most a logarithmic factor times w for these problems we develop a divide and conquer strategy where the cost of a solution to a problem at a recursive level is c plus the cost of a solution to the subproblems at this level and where the spreading metric volume on the subproblems is less than the original volume by omega c log n ensuring that the resulting solution has cost o log n times the original spreading metric volume we note that this is an existentially tight bound on the relationship between the spreading metric volume and the true optimal values for these problems for planar graphs we combine a structural theorem of klein plotkin and rao proceedings of the 25th acm symposium on theory of computing 1993 pp 682 690 with our new recursion technique to show that the spreading metric cost volumes are within an o log log n factor of the cost of an optimal solution for the minimum linear arrangement and the minimum containing interval graph problems