A Service of Leibniz-Informationszentrum econstor Wirtschaft Leibniz Information Centre Make Your Publications Visible. zbw for Economics Schneeweiss, Hans Working Paper Abraham Wald Discussion Paper, No. 439 Provided in Cooperation with: Collaborative Research Center (SFB) 386: Statistical Analysis of discrete structures - Applications in Biometrics and Econometrics, University of Munich (LMU) Suggested Citation: Schneeweiss, Hans (2005) : Abraham Wald, Discussion Paper, No. 439, Ludwig-Maximilians-Universität München, Sonderforschungsbereich 386 - Statistische Analyse diskreter Strukturen, München, http://nbn-resolving.de/urn:nbn:de:bvb:19-epub-1808-3 This Version is available at: http://hdl.handle.net/10419/31151 Standard-Nutzungsbedingungen: Terms of use: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Documents in EconStor may be saved and copied for your Zwecken und zum Privatgebrauch gespeichert und kopiert werden. personal and scholarly purposes. 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Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, If the documents have been made available under an Open gelten abweichend von diesen Nutzungsbedingungen die in der dort Content Licence (especially Creative Commons Licences), you genannten Lizenz gewährten Nutzungsrechte. may exercise further usage rights as specified in the indicated licence. www.econstor.eu Abraham Wald Hans Schneeweiss Department of Statistics, University of Munich Akademiestr. 1, 80799 MÄunchen, Germany Abstract This paper grew out of a lecture presented at the 54th Session of the International Statistical Institute in Berlin, August 13 - 20, 2003, Schneeweiss (2003). It intends not only to outline the eventful life of Abraham Wald (1902 - 1950) in Austria and in the United States but also to present his extensive scienti¯c work. In particular, the two main subjects, where he earned most of his fame, are outline: Statis- tical Decision Theory and Sequential Analysis. In addition, emphasis is laid on his contributions to Econometrics and related ¯elds. Abraham Wald is best known, indeed he is famous, for having founded Statistical Decision Theory and also for having developed the theory of se- quential sampling. But he also contributed to many other ¯elds of Statistics often giving decisive impulses or even originating new directions of research. In Statistics proper one might mention: asymptotic maximum likelihood theory, nonparametric statistics, tolerance intervals, optimal experimental designs, discriminance analysis, statistical quality control, random walks, the problem of incidental parameters, linear models with errors in the variables, and many more. Econometricians know him for his work on seasonal adjustment, on index number theory, on the identi¯cation problem of econometric models, on the problem of estimating such models, and on the famous Wald test as an alternative to the likelihood ratio test. But he also made major contributions to mathematical economic theory 1 and to game theory. Interesting enough, he started his academic career as a pure mathematician working in the ¯eld of geometry. This enormous width of interest is certainly due to an eventful life, a life typical for many emigrants from the German sphere of influence in the late thirties of the 20th century. Abraham Wald was born on October 31, 1902 as a citizen of the Austrian - Hungarian Empire in Klausenburg in a German speaking area (SiebenbÄurgen) then belonging to Hungary. After World War I this region fell to Romania, the city changed its name to Cluj, and Wald became a Romanian citizen. Born into an orthodox Jewish family, it was contrary to his convictions to attend a public school, where classes were given on Saturdays. He therefore was taught privately, but nevertheless got a degree from a secondary school, which enabled him in 1927 to take up studies at the University of Vienna. He studied Mathematics, but he did not attend many courses, actually only three. This was possible under the system of complete academic free- dom, which then prevailed in German and Austrian universities. Instead he tried his hand to solve open mathematical problems. In the course of these studies he came into contact with Karl Menger and sat in his lectures on geometry and later participated in his Mathematical Colloquium. This was the beginning of a very productive period in Wald's life, where he published a number of important papers on geometry and topology mostly in "Ergebnisse eines Mathematischen Kolloquiums" (Results of a Mathematical Colloquium). I cannot go into any details. Su±ces it to mention just a few of his discoveries. He contributed to Hilbert's "Grundlagen der Geometrie" (Foundations of Geometry); he was able to axiomatically characterize a concept of be- tweenness in metric spaces; he extended Steinitz's theorem on the sums of a series of vectors to spaces of in¯nite dimension; and - according to Karl Menger (1952) his masterpiece - he gave a new, coordinate free, foundation 2 of di®erential manifolds using a novel concept of curvature. Of greater interest to statisticians might be his characterization of Lebesgue measure as a measure ¹ which assigns the value 1 to every unit cube. Abraham Wald might have become a great geometer had not fate in- tervened in his career. After having got his Ph. D. in 1931 he looked for a position at the University of Vienna, but due to the adverse political and eco- nomic situation of that time no such position was available for him. Through Karl Menger's intervention, Wald became a private lecturer in mathematics to Karl Schlesinger, a banker with great interest in the mathematical foun- dations of economics. It was here that Abraham Wald learned about the concept of a Walrasian equilibrium in a pure exchange economy and also in an economy with pro- duction facilities. The economy was described by a set of supply and demand functions for each commodity and for each trader, relating commodities sup- plied and demanded to a vector of prices for these commodities under the assumption of perfect competition. In equilibrium, demand and supply had to match, leading to a system of equations for the unknown quantities of the commodities traded and their prices . It turns out that the number of equations equals the number of unknowns. But this is certainly not enough to guarantee the existence of a solution. Wald was the ¯rst to give su±cient conditions for the existence of a unique solution with nonnegative prices. This early work (1936) in equilibrium theory was much later taken up by economists in the early ¯fties culminating in an extended theory of math- ematical economics. Nobel Laureate Gerard Debreu acknowledged Wald's work in this ¯eld in his Nobel Lecture 1983. Eventually Wald got a position as a consultant in the Austrian Institute for Business Cycle Research, which then was headed by Oskar Morgenstern. Morgenstern acquainted him with the problem of seasonal adjustment of time series thus starting o® Wald's ¯rst genuinely statistical contribution. When the Institute applied a then popular method of Person's to the 3 series of unemployment data, the result was plainly wrong. The allegedly adjusted series did not only not eliminate seasonal variations, it even turned them to an opposite seasonal movement. Wald was able to show that Per- son's method only worked correctly if the seasonal pattern was invariant over time. However, with a slowly changing seasonal pattern results such as those observed could easily turn up. Wald then designed a method that allowed for slow movements of the am- plitude in the seasonal component. Suppose a time series xij; i = 1; ¢ ¢ ¢ ; n; j = 1; ¢ ¢ ¢ ; 12, i denoting the year and j the month, is decomposable into a smooth, a seasonal, and a random component: xij = mij + sij + rij: Then for a constant seasonal pattern sij = sj,s ¹ = 0, but for a seasonal pattern with slowly changing amplitude sij = aijsj; where aij, the amplitude series, varies slowly around the value 1 and is almost constant over any period of twelve consecutive months. Wald's method for seasonal adjustment takes this particular seasonal model into account. After eliminating the smooth component by a 12-month ¤ moving average, the resulting series xij is averaged over the years i = 1; ¢ ¢ ¢ ; n ¤ for every month j yielding the meanx: ¹ j as an estimate ofa: ¹ jsj. Due to the slow variation of the amplitude series aij the averagesa: ¹ j are approximately all the same:a: ¹ j ¼ a¹, wherea ¹ is the overall average of the series aij. Without ¤ loss of generality we can takea ¹ = 1 and sox: ¹ j is an estimate of sj: ¤ x:¹ j ¼ sj: Finally, for any ¯xed i and j, the amplitude aij is estimated by a local least squares procedure over twelve months: Xj+5 ¤ ¤ 2 (xik ¡ aijx:¹ k) ! min; aij k=j¡6 4 which yields the estimate P x¤ x:¹ ¤ Pk ik k a^ij = ¤ 2 : k(¹x:k) The seasonal component then is approximately given by ¤ sij ¼ baijx:¹ j : Subtracting this from the original series xij results in a seasonally adjusted time series. In his book "Berechnung und Ausschaltung von Saisonschwankungen" (1936), Wald explains in depth every single step of this procedure and care- fully accounts for the various approximations that appear along the line of calculations. The presentation here is somewhat simpli¯ed. Since the time of Wald's book many other models of seasonal movement have been considered and corresponding adjustment methods have been de- veloped. It be¯ts to pay tribute to the genius loci by mentioning the Berlin procedure of seasonal adjustment. This method starts from a local harmonic decomposition of the seasonal component and assumes a local polynomial for the smooth component and is certainly quite di®erent from Wald's ap- proach.