Variance estimation in the change analysis of a linear regression model

We study a “direct test” of Chu and White (1992) proposed for detecting changes in the trend of a linear regression model. The power of this test strongly depends on a suitable estimation of the variance of the error variables involved. We discuss various types of variance estimators and derive their asymptotic properties under the null-hypothesis of “no change” as well as under the alternative of “a change in linear trend”. A small simulation study illustrates the estimators' finite sample behaviour.     

Verbesserung des nicht-randomisierten exakten Testes von R.A. Fisher

Summary The so-called Exact Test of R. A. Fisher for comparing two probabilitiesp ₁ andp ₂ in a Fourfold-Table with small cell frequencies is known as a UMPU-Test. But in practice the test is used in a nonrandomized, often tabulated version. Given a certain level of significanceα it is shown: the critical region of this nonrandomized test, referred to as “Fisher 1”, can be enlarged considerably. For instance for all sample-size-sums up to 20 andα=0.01 the total number of points in the critical regions of “Fisher 1” is 552 whereas the analogous number of the new version “Fisher 2” is 788. The size of tables for “Fisher 2” can be reduced considerably because the main parts of the critical regions can be described by the aid of some Chi-square-test versions. In particular Yates’ continuity-correction turns out to be always conservative in the above mentioned region relative to “Fisher 2” whereas this is not strictly true relative to “Fisher 1”.      

Einige Bemerkungen zur statistischen Messung der Konzentration

Zusammenfassung Nach einer kritischen Analyse der Begriffe „absolute“ und „relative“ Konzentration werden einige Kriterien gegeben, die Ma?zahlen der absoluten oder der relativen Konzentration zu erfüllen haben. Hierauf wird untersucht, inwieweit die gebr?uchlichsten Ma?zahlen der Konzentration (Gini, Herfindahl, Münzner) diese Kriterien erfüllen. Dann wird gezeigt, da? sich eine vonAdam angegebene, sehr allgemeine Klasse von Ma?zahlen als Ma?zahlen der absoluten Konzentration qualifizieren; einige wichtige Spezialf?lle werden genauer untersucht. Im vorletzten Teil der Arbeit werden Ma?zahlen für die Ver?nderung der Konzentration diskutiert und schlie?lich an einem praktischen Beispiel erprobt.

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Summary After analysing the concepts of “absolute concentration” versus “relative concentration”, some criteria for coefficients of concentration are given to qualify them either as measures for absolute or for relative concentration. On the basis of these criteria, some well-known coefficients (Gini, Herfindahl, Münzner) are examined. Furthermore, it is shown that the measures of “predictivity” as given byAdam can be regarded as coefficients of absolute concentration; some important special cases (one of which the Herfindahl index) are investigated. Finally, measures for the change in concentration are developed and applied to a practical example.

     

Some models and measures for evaluating performances with DEA: past accomplishments and future prospects

This paper covers some of the past accomplishments of DEA (Data Envelopment Analysis) and some of its future prospects. It starts with the “engineering-science” definitions of efficiency and uses the duality theory of linear programming to show how, in DEA, they can be related to the Pareto–Koopmans definitions used in “welfare economics” as well as in the economic theory of production. Some of the models that have now been developed for implementing these concepts are then described and properties of these models and the associated measures of efficiency are examined for weaknesses and strengths along with measures of distance that may be used to determine their optimal values. Relations between the models are also demonstrated en route to delineating paths for future developments. These include extensions to different objectives such as “satisfactory” versus “full” (or “strong”) efficiency. They also include extensions from “efficiency” to “effectiveness” evaluations of performances as well as extensions to evaluate social-economic performances of countries and other entities where “inputs” and “outputs” give way to other categories in which increases and decreases are located in the numerator or denominator of the ratio (=engineering-science) definition of efficiency in a manner analogous to the way output (in the numerator) and input (in the denominator) are usually positioned in the fractional programming form of DEA. Beginnings in each of these extensions are noted and the role of applications in bringing further possibilities to the fore is highlighted.

On the inequality Σp i f(p i )≥ p i f(q i )

Summary We shall deal with the inequality (2) and we prove among others the following results. Every solution of the inequality (2) is monotone increasing. Every solution of the inequality (2) is differentiable ifn≥3. A functionf satisfies the inequality (3) if and only if its saltus part and absolutely continuous part satisfy also the inequality (3). We give the general solution of (3) in the field of saltus function. Finally by the help of [Muszély] we give the general solution of (3). This results are generalisations of one ofJ. Aczél andJ. Pfanzagl.

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Zusammenfassung Wir betrachten die Ungleichung (2) und beweisen die folgenden Ergebnisse. Die allgemeine L?sung der Ungleichung (2) ist monoton wachsend. Die allgemeine L?sung der Ungleichung (2) ist differenzierbar im Falle wennn≥3 ist. Wir geben also die allgemeine L?sung der Ungleichung (3) mit Hilfe von [Muszély]. Diese Ergebnisse sind Verallgemeinerungen des Resultats vonJ. Aczél undJ. Pfanzagl.

     

A further note on the asymptotic efficiency of friedman’s x r 2 -test

Summary For independently distributed error components, the asymptotic relative efficiency (A.R.E.) ofFriedman’sx _r ² -tests with respect to the classical analysis of variance test has been studied byElteren andNoether andSen [1967]. The present note extends these results to the case of correlated errors arising in some random-effects or mixed-effects models. Work supported by the U.S. Army Research Office, Durham, Grant DA-ARO-D-31-124-G 746.     

Social networks and labour productivity in Europe: an empirical investigation

This paper uses firm-level data recorded in the Amadeus database to investigate the distribution of labour productivity in different European countries. We find that the upper tail of the empirical productivity distributions follows a decaying power-law, whose exponent α is obtained by a semi-parametric estimation technique recently developed by Clementi et al. [Physica A 370(1):49–53, 2006]. The emergence of “fat tails” in productivity distribution has already been detected in Di Matteo et al. [Eur Phys J B 47(3):459–466, 2005] and explained by means of a model of social network. Here we show that this model is tested on a broader sample of countries having different patterns of social network structure. These different social attitudes, measured using a social capital indicator, reflect in the power-law exponent estimates, verifying in this way the existence of linkages among firms’ productivity performance and social network.     

An outlier test for linear processes — II. Large contamination

In Flak/Schmid (1993) an outlier test for linear processes was introduced. The test statistic bases on a comparison of each observation with a one-step predictor. It was assumed that an upper bound for the total number of outlierss _n is known, wheren denotes the sample size. The asymptotic distribution of the test statistic was derived under the assumption thats _n/n → 0 ands _n → ∞ asn → ∞. This note deals with the asymptotic behaviour of this quantity, ifs _n/n →p ₀ ∈ (0, 1).     

About probability-like measures for entire theories

5. Summary It is discursively argued that a much closer rapport between the methodologies implicitly taught in Economic Theory and in Statistics is required and that the necessary interdisciplinary bridge can be provided, and clarification attained, through the study of logical measure functions of theKemeny-Carnap type. It is also argued that subjectivistic axiom systems of probability, while valuable in their own right and as bases for behavior theories, cannot by themselves render the study of logical measure functions superfluous. Elementary aspects of these functions are then explained, with somewhat more detailed references toKemeny’s measuresm ands and a proposed degree-of-theoretization measure, whose applicability to economics is negatively appraised. Finally,Carnap’s degree-of-confirmation functions are briefly dealt with, but throughout the paper the emphasis is on those properties of logical measure functions which donot primarily bear on statistical inference and estimation). “...I think, on the whole, one theory fits nearly everything. That is, if you admit one coincidence—and I think one coincidence is allowable. More than one, of course, is unlikely ...” Prof. Dr. E. M. Fels, 2126 CL, University of Pittsburgh, Pittsburgh 13, Penna., USA. The Murder at the Vicarage. New York: Dell, 1961, p. 197 (Dodd, Mead & Co., 1930).     

Competitive market equilibrium under asymmetric information

This paper studies the existence of a competitive market equilibrium under asymmetric information. There are two agents involved in the trading of the risky assets: an “informed” trader and an “ordinary” trader. The market is competitive and the ordinary agent can infer the insider information from the price dynamics of the risky assets. The insider information is considered to be the total supply of the risky assets. The definition of market equilibrium is based on the law of supply-demand as described by a rational expectations equilibrium of the Grossman and Stiglitz (Am Econ Rev 70:393–408, 1980) model. We show that equilibrium can be attained by linear dynamics of an admissible price process of the risky assets for a given linear supply dynamics.      

A test of goodness-of-fit based on extreme spacings with some efficiency comparisons

Tests for the goodness-of-fit problem based on sample spacings, i.e., observed distances between successive order statistics, have been used in the literature. We propose a new test based on the number of “small” and “large” spacings. The asymptotic theory under close alternative sequences is also given thus enabling one to calculate the asymptotic relative efficiencies of such tests. A comparison of the new test and other spacings tests is given.     

Sampling studies on a test against trend in binomial data

Summary A series of 375 “Monte-Carlo” realizations were performed on a number of groupings of 100 samples each fromt distinct binomial populations (t=3, 4, 5, 7, 9) in order to obtain estimates of the power function of theV andX ² index of dispersion tests in detection of a “trend” in the binomial probabilities.     

A general class of estimators for estimating population mean using auxiliary information

Summary A general class of estimators for estimating the population mean of the character under study which make use of auxiliary information is proposed. Under simple random sampling without replacement (SRSWOR), the expressions of Bias and Mean Square Error (MSE), up to the first and the second degrees of approximation are derived. General conditions, up to the first order approximation, are also obtained under which any member of this class performs more efficiently than the mean per unit estimator, the ratio estimator and the product estimator. The class of estimators in its optimum case, under the first degree approximation, is discussed. It is shown that it is not possible to obtain optimum values of parameters “a”, “b” and “p”, that are independent of each other. However, the optimum relation among them is given by (b−a)p=ρ C _y/C _x. Under this condition, the expression of MSE of the class is that of the linear regression estimator.     

Testing whetherF is more IFR thanG

A distributionF is said to be “more IFR” than another distributionG ifG ⁻¹ F is convex. WhenF(0) =G(0) = 0, the problem of testingH ₀ :F(x) =G (θx) for someθ > 0 andx ⩾ 0, against the alternativeH _A:F is more IFR thanG, is considered in this paper. Both cases, whenG is completely specified (one-sample case) and when it is not specified but a random sample form it is available (two-sample case) are considered. The proposed tests are based onU-statistics. The asymptotic relative efficiency of the tests are compared with several other tests and the test statistics remain asymptotically normal under certain dependency assumptions. Research supported in part by a grant from the US Air Force Office of Scientific Research.     

Bestimmung kostenoptimaler Parameter für den kontinuierlichen Stichprobenplan vonDodge

Summary Dodge's sampling plan for a continuous production process is determined by two constantsi andk. Under some assumptions about the statistical properties of the production process the average loss due to the costs for inspection and for replacing defective units by good ones is described by a loss function. It is shown that there are always constantsi andk which minimize the loss function in a special sense. Tables of these optimum parameters are computed.     

On a noncentral chi-square distribution in the analysis of weapon systems effectiveness

Summary A natural conjugate prior distribution for the parameters involved in the noncentral chi-square leads to many known distributions. The applications of the distributions thus obtained are briefly pointed out in evaluating the ‘kill’ probability in the analysis of weapon systems effectiveness. The ‘kill’ probabilities or the expected coverage are obtained associated with a gamma prior distribution and compared with those obtained byMcnolty. This paper is read in a symposium on Mathematical Sciences held under the auspices of Delhi University, Delhi im January 1966.     

Tests for departure from normality in the case of linear stochastic processes

Summary In this paper the asymptotic distributions of the third and the fourth sampling moments of a discrete-parameter linear process are derived together with the asymptotic distributions of the sampling skewness and the excess of kurtosis. The knowledge of these distributions allows us, in the case of “large” samples, to test the departure from normality, a problem which can be regarded as important in various practical applications, but which cannot be treated with the aid of classical tests based on the assumption that the sample values are independent. Some numerical examples illustrate the applications of the proposed tests in practice.

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Zusammenfassung In diesem Beitrage sind die asymptotischen Stichprobenverteilungen der dritten und vierten Momente eines linearen stochastischen Prozesses und auch die asymptotischen Stichprobenverteilungen der Schiefe und des Exzesses der Kurtosis hergeleitet. Die Kenntnis dieser Verteilungen erlaubt für “gro?e” Stichproben die Abweichung der Proze?verteilung von der Normalverteilung zu prüfen, ein Problem, das in vielen F?llen der praktischen Anwendungen wichtig ist, aber mit den klassischen Testen, welche die Unabh?ngigkeit der Stichprobenwerte annehmen, nicht behandelt werden kann. Die praktische Anwendung der vorgelegten Methode ist mit einigen numerischen Beispielen illustriert.

     

Conditional nonparametric frontier models for convex and nonconvex technologies: a unifying approach

The explanation of productivity differentials is very important to identify the economic conditions that create inefficiency and to improve managerial performance. In the literature two main approaches have been developed: one-stage approaches and two-stage approaches. Daraio and Simar (2005, J Prod Anal 24(1):93–121) propose a fully nonparametric methodology based on conditional FDH and conditional order-m frontiers without any convexity assumption on the technology. However, convexity has always been assumed in mainstream production theory and general equilibrium. In addition, in many empirical applications, the convexity assumption can be reasonable and sometimes natural. Lead by these considerations, in this paper we propose a unifying approach to introduce external-environmental variables in nonparametric frontier models for convex and nonconvex technologies. Extending earlier contributions by Daraio and Simar (2005, J Prod Anal 24(1):93–121) as well as Cazals et al. (2002, J Econometrics 106:1–25), we introduce a conditional DEA estimator, i.e., an estimator of production frontier of DEA type conditioned to some external-environmental variables which are neither inputs nor outputs under the control of the producer. A robust version of this conditional estimator is proposed too. These various measures of efficiency provide also indicators of convexity which we illustrate using simulated and real data. Cinzia Daraio received Research support from the Italian Ministry of Education Research on Innovation Systems Project (iRis) “The reorganization of the public system of research for the technological transfer: governance, tools and interventions” and from the Italian Ministry of Educational Research Project (MIUR 40% 2004) “System spillovers on the competitiveness of Italian economy: quantitative analysis for sectoral policies” which are acknowledged. Léopold Simar received Research support from the “Interuniversity Attraction Pole”, Phase V (No. P5/24) from the Belgian Government (Belgian Science Policy) is acknowledged.     

The queuing system GI/M/I with finite waiting space

Summary Time dependent solution of the queuing system characterized by a general independent input, exponential service time distribution and a finite waiting space, has been first investigated by using the “phase method”. On finding the waiting room full, the customers then arriving may be turned away or the first customer may wait outside and the input process may be stopped till the customer then being served, completes its service. Steady state solutions of both these problems have been obtained and the difference in the operational behaviour of the two systems has been pointed out. For a 2-Erlang arrival distribution, the queuing parameters have been evaluated for different values ofρ _r andN.     

Selezione degli investimenti e costo del denaro

Alcuni Autori ritengono che non abbia senso il costo del denaro, in uno schema lineare in cui si può ripartire il proprio denaro in tutte e sole le operazioni che determinano il tasso di sconto. La conclusione di tali Autori appare azzardata: infatti il loro risultato discende da una ipotesi estranea al modello lineare. In questo lavoro io interpreto lo stesso modello, mostro che in esso ha senso il costo del denaro e propongo delle impostazioni alternative.

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An interesting economic problem consist in the search of the optimal allocation of one’s monetary resources in different financial transactions. We often consider as “optimal” the allocation which makes the sum of the actual values of investiments the highest possible, within the limits of all feasible operations. These values are reckoned according to the discount rates established by the financial market. This type of abstraction can be acceptable when we can split our capital into a number of operations that are on the whole negligible, if compared with the movement of the market. In case of a considerable turn over that may therefore affect the cost of money, it follows that we can’t determine an appropriate discount rate before establishing the distribution policy; on the other hand we cannot determine the distribution policy before defining the discount rate. As a border-line case we have that of an “enclosed” economy, where funds can be rationed into all and only those operations which determine the cost of capital. That is the case of “pure capital rationing”. To solve the problem of reckoning the cost of capital under pure capital rationing, some Authors have formulated the two dual linear programs (1) and (2) and have deduced that f is the vector of market prices if and only if the optimal dual solution is equal to f. As that happens if and only if f=0, Burton and Damon [2] consider their “main result a rigorous proof that there does not exist a meaningful solution for the pure capital rationing problem” and conclude “that if there exist a solution to the problem it is not to be found by the traditional linear programming formulations”. On the contrary I demonstrate that f=0 is the only possible capital cost because of the hypotesis =f, which is not related to the linear pattern and is not acceptable from an economic-financial point of view. Then I demonstrate that the market prices are all and only those based on which the actual values of the operations considered are 0. Nor was it right to expect more sophisticated conclusions of such an elementary scheme. Finally I express an alternative linear formulation, where the dual optimal solutions are exactly the market prices.

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Lavoro effettuato nell’ambito del G.N.A.F.A. del C.N.R.