On a lemma associated with Box,Jenkins and Granger

The lemma that the aggregate of independent moving average processes is also moving average is discussed, and a simple proof presented. This proof depends on the fact that, for any order q, the vectors of MA autocorrelations (?₁…?_q) occupy a convex space.     

On the structure of moving average processes

This paper uses Hilbert space methods to develop a rigorous proof that the sum of two uncorrelated moving average processes of order q₁ and q₂ is an MA process of order q ≦ max (q₁, q₂). The methods establish the existence of suitable random shocks for the summed process, they illuminate relationships between the coefficients of such processes and their random shocks, and they provide means for proving that the random shocks of the summed processes are normal when the shocks of the underlying processes are normal. The role of the Wold decomposition is examined in terms of multiple representations of an MA process.     

Existence of equilibria with infinitely many commodities: Banach lattices reconsidered

The purpose of this paper is twofold. The first aim is to present an extension of the results on the existence of Walrasian equilibrium to the infinite dimensional setting. The result depends on two crucial assumptions. These are the compactness of the collection of feasible allocations and the non-emptiness of the interior of the production set. The proof is a direct generalization of Bewley's (1972) proof for the L_∞ case. The second purpose of this paper is to show that the recent result of Mas-Colell (1986) on the existence of equilibrium for exchange economies on Banach lattices can be obtained through an argument based on the result outlined above. That is, exchange economies on Banach lattices with ‘uniformly proper’ preferences behave as though they were production economies in which the production sets have non-empty interior.     

Note on the existence and modulus of continuity of the {\textit{SLE}}_8 curve

We review one method for estimating the modulus of continuity of a Schramm–Loewner evolution (SLE) curve in terms of the inverse Loewner map. Then we prove estimates about the distribution of the inverse Loewner map, which underpin the difficulty in bounding the modulus of continuity of SLE for $\kappa =8$ . The main idea in the proof of these estimates is applying the Girsanov theorem to reduce the problem to estimates about one-dimensional Brownian motion.     

Blocked-Error-R 2: A Conceptually Improved Definition of the Proportion of Explained Variance in Models Containing Loops or Correlated Residuals

Bentler and Raykov (2000, Journal of Applied Psychology 85: 125–131), and Jöreskog (1999a, http://www.ssicentral.com/lisrel/column3.htm, 1999b http://www.ssicentral. com/lisrel/column5.htm) proposed procedures for calculating R ² for dependent variables involved in loops or possessing correlated errors. This article demonstrates that Bentler and Raykov’s procedure can not be routinely interpreted as a “proportion” of explained variance, while Jöreskog’s reduced-form calculation is unnecessarily restrictive. The new blocked-error-R ² (beR ²) uses a minimal hypothetical causal intervention to resolve the variance-partitioning ambiguities created by loops and correlated errors. Hayduk (1996) discussed how stabilising feedback models – models capable of counteracting external perturbations – can result in an acceptable error variance which exceeds the variance of the dependent variable to which that error is attached. For variables included within loops, whether stabilising or not, beR ² provides the same value as Hayduk’s (1996) loop-adjusted-R ². For variables not involved in loops and not displaying correlated residuals, beR ² reports the same value as the traditional regression R ². Thus, beR ² provides a conceptualisation of the proportion of explained variance that spans both recursive and nonrecursive structural equation models. A procedure for calculating beR ² in any SEM program is provided.     

An exact non-cooperative support for the sequential Raiffa solution

This article provides an exact non-cooperative foundation of the sequential Raiffa solution for two-person bargaining games. Based on an approximate foundation due to Myerson (1991) for any two-person bargaining game (S, d) an extensive form game G^S,d is defined that has an infinity of weakly subgame perfect equilibria whose payoff vectors coincide with that of the sequential Raiffa solution of (S, d). Moreover all those equilibria share the same equilibrium path consisting of proposing the Raiffa solution and accepting it in the first stage of the game.By a modification of G^S,d the analogous result is provided for subgame perfect equilibria. These results immediately extend to implementation of a sequential Raiffa (solution based) social choice rule in subgame perfect equilibrium.     

Central limit theorem for linear eigenvalue statistics of the Wigner and the sample covariance random matrices

We consider n × n real symmetric random matrices n ^?1/2 W with independent (modulo symmetry condition) entries and the (null) sample covariance matrices n ^?1 A ^T A with independent entries of m × n matrix A. Assuming first that the 4th cumulant (excess) κ ₄ of entries of W and A is zero and that their 4th moments satisfy a Lindeberg type condition, we prove that linear statistics of eigenvalues of the above matrices satisfy the central limit theorem (CLT) as n → ∞, m → ∞, ${m/n\rightarrow c\in[0,\infty)}$ with the same variance as for Gaussian matrices if the test functions of statistics are smooth enough (essentially of the class ${\mathbb{C}^5}$ ). This is done by using a simple “interpolation trick”. Then, by using a more elaborated techniques, we prove the CLT in the case of non-zero excess of entries for essentially ${\mathbb{C}^4}$ test function. Here the variance contains additional term proportional to κ ₄. The proofs of all limit theorems follow essentially the same scheme.     

Nonlinear generalization of the Frobenius theorem: A symmetric approach

A further generalization of the Samuelson-Solow's results for the Frobenius theorem is presented. We ask the existence of a semi-positive vector x with a non-negative scalar λ for an equation F(x;λ)=0, where F is a vector function from Rⁿ into itself with the parameter value λ. The nature of our extension as well as economic interpretation of our assumptions is also discussed in the paper.     

Third-order optimum properties of estimator-sequences

SequencesT ⁽ⁿ⁾ ,n∈N, are considered, whereT ⁽ⁿ⁾ estimates a vector parameter ?∈R ^p from an i.i.d. sample of sizen, and such sequences are compared on the basis of their risks ∫L(n ^1/2(T ⁿ (x)?θ))P _θ ⁿ (dx) relative to loss functionsL:R ^p →R. A characterization is given for sequencesT ^*(n) ,n∈N, which generate an essentially complete class in the following sense: For any sequenceT ⁽ⁿ⁾ ,n∈N, there exist functions Φ _n ,n∈N, such that forn→∞ we have $$\begin{gathered} \smallint L (n^{1/2} (T^{*(n)} + n^{ - 1} \Phi _n (T^{*(n)} ) - \theta )) dP_\theta ^n \leqslant \hfill \\ \leqslant \smallint L (n^{1/2} (T^{(n)} - \theta )) dP_\theta ^n + o (n^{ - 1} ), \hfill \\ \end{gathered} $$ for every ? and everyL satisfying certain conditions. If the estimator-sequences are compared by their risks ∫W(T ⁽ⁿ⁾ d P _θ ⁿ ,θ) with respect to loss functionsW:R ^p ×Θ→R then a similar result on asymptotically complete classes is valid. The results are obtained under the assumption thatT ^*(n) ,n∈N, andT ⁽ⁿ⁾ ,n∈N, admit stochastic expansions which are sufficiently regular, that the loss functionsL andW are sufficiently smooth and bounded by polynomials, and that the estimator-sequences have asymptotically bounded moments; the latter condition is not needed for bounded functionsL andW.     

Tests of significance for the mean of a finite lot

Let ξ₁, ξ₂, ξ₃, ... be independent identically distributed random variables each with normal distribution with mean μ and variance σ². Tests for the process mean μ are well-known elements of statistical analysis: the Gauß test under known process variance σ², Student’st-test under unknown process variance σ². Let the process be partitioned in lots (ξ₁, ..., ξ _N ), (ξ _N+1, ..., ξ_2N ), ... of sizeN. Consider (ξ₁, ..., ξ _N ) as a stochastic representative of this lot sequence and let the lot be characterized by the lot mean $\frac{1}{N}\sum\limits_{i = 1}^N {\xi _i } $ . The lot mean can be considered as a parameter of the joint conditional distribution function of the lot variables under $\frac{1}{N}\sum\limits_{i = 1}^N {\xi _i } = z$ . The present paper investigates the analogies of the Gauß test and Student’st-test for the lot situation, i.e. tests of significance for the lot meanz under known and unknown process variance σ². This approach is of special interest for the statistical control of product quality in situations where the quality of a lot of items 1, 2, ...,N with quality characteristics ξ₁, ξ₂, ..., ξ _N is identified with the lot average $\frac{1}{N}\sum\limits_{i = 1}^N {\xi _i } = z$ .     

The effects of autocorrelation among errors on the consistency property of OLS variance estimator

In a linear regression model the ordinary least squares (OLS) variance estimator (S²) converges in probability to E(S²) even when the errors are autocorrelated. Of interest, however, is the rate of convergence. In this paper we shed some light on this question for the case of a linear trend model. In particular the relation between the rate of convergence and the correlation property of the errors is explored. It is shown that the retardation of the rate of convergence is not appreciable if the correlation is moderate, but it can be severe for extreme correlations.     

The women of Cairo: Equilibria in large anonymous games

In this article, I characterize Nash equilibria of large anonymous games by providing the following neccessary and sufficient condition for an equilibrium distribution: for no subset K of actions more players play actions in K than have a best response in K to the given distribution. While neccessity is trivial the proof for sufficiency relies on a theorem by [Math. Proc. Camb. Philos. Soc. 78 (1974) 323] which is an extension of Hall’s theorem or the marriage lemma well known from graph theory. The veiling problem for the women of Cairo serves as an illustrating heuristic example explaining the nature of the result.     

A general statistical model for characterizing the growth patterns and projecting the level of performance of family planning clinics

A statistical model for characterizing the growth patterns and projecting the level of performance has been developed for family planning clinics. The model developed is y = β (1 ? e^?αt where y is the number of continuing contraceptive patients at time t, and α and β are parameters to be estimated. Results of a typical application of the model are presented and discussed.     

Existence of a convex extension of a preference relation

In this paper we say that a preference (i.e. irreflexive) relationP isregular (or aweak order) if both it and its non-comparability relation are transitive; we also say that a preference relationP ^* is aconvex extension of another preference relationP ifP?P ^* holds andP ^* is regular and convex-valued. We prove that a convex extension ofP exists if and only if every non-empty and finite set of alternativesA is not included in the convex hull of ∪ _x∈A P(x).     

Semiparametric analysis of discrete response: Asymptotic properties of the maximum score estimator

This paper is concerned with the estimation of the model MED(y|x) = xβ from a random sample of observations on (sgn y, x). Manski (1975) introduced the maximum score estimator of the normalized parameter vector $\text{β}{}^1\text{= β/?β?}$. In the present paper, strong consistency is proved. It is also proved that the maximum score estimate lies outside any fixed neighborhood of β¹ with probability that goes to zero at exponential rate.     

A theory of coalition formation in committees

Let G = (N,W) be a strong weighted majority game and let A be a set of alternatives. Denote by L the set of linear orders on A. A social choice function F:L^N → A is a representation of G if the simple game G¹(F) associated with F equals G. A coalition S is determining in G if it satisfies the following condition. Let F be a representation of G and let R^N ? L^N. Then, if a simple majority of the members of S consider an alternative x to be their best choice, then S can ‘enforce’ x to be a Nash equilibrium payoff in the resulting non-cooperative voting game g(F,R^N). In this paper we generalize the above notion of a determining coalition to committees (i.e., proper and monotonic simple games), and give a complete characterization of the set of determining coalitions of a committee. Furthermore, we discuss our notion of a determining coalition in the light of some real-life data on formation of coalitions in town councils in Israel.     

Utility functions for Debreu's ‘excess demands’

Given an arbitrary function x: $\text{R}$^l→$\text{R}$^l satisfying Walras law and homogeneity, Debreu decomposed x into the sum of l ‘individually rational’ functions x(p)=Σ^l_k=1[uvbar|x]^k(p). Here we find explicit utility functions u^k, constructed on the basis of a simple geometric intuition, which give rise to Debreu's excess demands [uvbar|x]^k(p).     

Investigating the incremental validity of employee engagement in the prediction of employee effectiveness: A meta-analytic path analysis

The current study used meta-analytic estimates and path analysis to examine whether the construct of employee engagement (EE) shows incremental validity in the prediction of employee effectiveness (a broad measure of performance-related behaviors) over other job attitudes such as job satisfaction, job involvement, and organizational commitment. Meta-analytic estimates between EE and various employee effectiveness indicators were computed from 49 published correlations representing a total of 22,090 individuals. We combined these estimates with published meta-analytic estimates between employee effectiveness and job attitudes to produce a meta-matrix representing 1,161 unique correlations. Using this meta-matrix, a series of path model comparisons produced two results: (1) EE bears low to moderate incremental validity over individual job attitudes (R² change of 0.02 to 0.06), and (2) EE bears low incremental validity over a higher-order job attitude construct representing the combination of other job attitudes in the prediction of a higher-order employee effectiveness construct (R² change of 0.01). Given the brevity of popular EE measures, the results suggest EE is better conceptualized as a higher-order measure of job attitudes that is an effective and concise predictor of employee effectiveness.