A Study in Analysis of Stationary Time Series. by Herman Wold
Review by: J. N.
Journal of the Royal Statistical Society, Vol. 102, No. 2 (1939), pp. 295-298
Published by: Wiley for the Royal Statistical Society
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1939]
295
REVIEWS OF STATISTICAL AND ECONOMIC BOOKS
CONTENTS
PAGE
1.- Wold (Herman). Analysis
of StationaryTime Series ... 295
2.-Fisher (R. A.). Statistical
Research
for
Methods
... 298
Workers(7th ed.) ...
and Yates (F.).
3.Statistical Tables for Biological, Agriculturaland Medical
... 298
...
...
Research
Deprez(F.). Tables forCal-
f culatingbyMachineLog-
arithmsto 13 Places of
Decimals.
Cohn (B.). Tables of Addition and SubtractionLog... 299
...
arithms ...
5.-Terry (G. S.). Duodecimal
...
... 299
Arithmetic ...
6.-Ferenczi (I.). The Synthetic
Optimumof Population ... 300
7.-Edge (P. Granville). Medical and SanitaryReportsfrom
BritishColonies,Protectorates
and DependenciesfortheYear
... 301
...
...
1936 ...
4.
PAGE
8.-Crum (W. L.), Patton(A. C.).
and Tebbutt(A. R.). Introductionto EconomicStatistics 302
9.-Oxford Economic Papers,
... 30t
...
No. 1, 1938 ...
10.-Hicks (J. R.).
...
Capital
Value and
... 307
...
11.-Einarsen (Johan). Re-investment Cycles and their
Manifestation in the Norwegian ShippingIndustry ... 312
12.-Plummer (Alfred). International Combinesin Modern
... 313
...
...
Industry
13.-Hanson (S. C.). Argentine
Meat and the BritishMarket 315
The
14.-Lawley (F. E.).
Growth of Collective Econ... 316
...
...
omy ...
15.-Roll (Erich). A Historyof
... 317
Economic Thought ...
16.-Other New Publications...
318
1.-A Studyin Analysis ofStationaryTime Series. By Herman
Wold. Uppsala: Almqvist und Wiksell, 1938. 9" x 6". viii +
214 pp. Price kr. 6 (bound kr. 8.)
In a lecture I gave two years ago on the differentmethods of
time series analysis, I divided all the methods applied into two
categories, which I called " empirical" and " aprioristic," and
remarkedthat although the empirical approach is much the more
common and, at present,the more useful, the futurebelonged to
the aprioristicmethod.* If we did not apply it at present,it was
because we knew too littleabout the necessarymathematicaltoolsnamely, the branch of the theoryof probabilitydealing with what
might be called " random curves " or " random processes." That
theory,originatedby the Russian mathematicians,Markoff,Khintchine, and Kolmogoroff,was then in an early stage and was to be
found only in a rather abstract form in articles in mathematical
journals.
This book by Herman Wold fillsthe need fora publicationgiving
* J. Neyman: Lectures and Conferenceson Mathematical Statistics,
Washington,D.C., 1937.
VOL. CII.
PART II.
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M
[Part II,
Reviews of Statistical and Economic Books
296
the theoryin a formadapted to practical applications and it may be
warmly recommendedto economists and statisticians engaged in
the analysis of time series. As stated in the preface,Mr. Wold is
a student of H. Cramer,and derived inspirationfromthe writings
of G. U. Yule and A. Khintchine,as well as fromthe lecturesof his
immediate teacher. The combination of the names of Yule and
Khintchine may seem surprising,forMr. Yule would not claim to
be a pure mathematician, and Mr. Khintchine would never be
recognized as a statistician. Yet it appears that theywerestudying
essentiallythe same things.
The mathematical equivalent of the statistical conception of
the time series is the random process. This is definedas follows:
Imagine a finiteor infinitetimeintervaland denote by t any moment
in it. Consider next a random variable x(t) which, in a certain
sense, correspondsto the moment t. To each moment t there will
correspond some particular random variable x(t). The random
variables x(t') and x(t") correspondingto two, differentmoments
t' and t" need not be independent. We are thus led to consider
an infinitesystemof, in general,mutually dependentvariables and
theirjoint probabilitylaw. Such a systemof variables is just what
is called a random process. It may be continuous,if the variable
t is allowed to vary continuously,or discrete-if t may have only
a denumerable set of equidistant values. An actually observed
time seriesis consideredas a set of particularvalues of the variables
x(t).
Mr. Wold limits his considerationsto discrete and stationary
randomprocesses. Denote by F(t1,t2, .. . tn; Uj1, U2 ... Un) the
probability of the simultaneous fulfilmentof the n inequalities
x(t,)< ui fori -1, 2, - - -, n, as determinedby the joint probabilitylaw of the variables x(t) forminga discrete random process.
If this functionpossesses the propertythat F(t1 + k, t2 + k, . . ..
tn-+
k; ul,
U2,
. . . un)
_ :F(t1,t2, .
. ., tl;
u1, U2, . .
.,
U7O),
then
the random process is called stationary. We may explain this
verbally by saying that by a stationaryprocess the probabilityof
various situationsat some momentsdoes not depend on the position
of those particularmoments,but on theirmutual distances in time
and on what happened before. The processwhichis not stationary
is called evolutive.
The author gives the necessary definitionsand a sequence of
interestingtheorems concerningthe discrete stationary processes.
Some of these are taken fromcurrentliterature,but some are new.
Various remarkable types of the processes considered are distinguished. The simplest type is, of course, the purely random
process by which any variable x(t') is independent of any other
x(t") and all follow the same law of frequency. Next comes the
type described as the process of moving averages. If (t) is a
random variable belongingto a purelyrandom process, and if for
any t the variable ,(t) is definedby the relation
t(t)
cbom(t)+ bltv(t-1)
+ .(.+
bie(th- n),
thentheprocesscomposedof thevariablesi(ti)is calledtheprocess
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1939]
Reviews of Statistical and Economic Books
297
of moving averages. We recognize here a conception previously
discussed by Yule, and there are many more similarcontacts with
the work of Yule in the book.
Having discussed a number of types of random processes, the
author furnishesproofs of their various properties,and then gives
a theorem of considerable interest, concerningthe structure of
the most general discrete stationary process. This, in fact, can
always be presentedas a sum oftwo componentsthe nature ofwhich
is known. When dealing with stationary processes, the author
discusses what he calls stochasticaldifferenceequations, and proves
some of their fundamentalproperties. All this is done in Chapters
II and III. Chapter I is given to a review of some of the currerilt
methods of time series analysis. The fourthand last chapter will
be the most interestingto a non-mathematicalreader,forit contains
the applications of the theory to a number of actual time series.
The firstexample is the series of yearly wheat prices in Western
Europe, 1518-1869, compiled by Sir William Beveridge. Next
come two correlatedtime series of (i) the level of Lake Vaner and
(ii) the rainfall. Finally, there is a detailed discussion of a series
representingthe cost-of-livingindex in Sweden. Besides those
examples, the discussions of which amount almost to separate
studies, the author refersfrequentlyto various time series previously discussed by Mr. Yule, Sir G. Walker and others. The book
ends with two appendices on the Cramer-MisesCo2 test for goodness
of fit,and-a very interestingone-on the numericalsignificanceof
correlationcoefficients.
It seems probable that the publication of Mr. Wold's book,
which is, so far as I know, the firstof its kind, will considerably
influence the development of the time series analysis, but still
some of the most important problems involved remain unsolved
and, as the author says himself,at presentit is difficultto see how
to attack them. He describes them as sampling problems, but
really they are problems of testing hypothesesand of estimation.
Given a time series and a hypotheticalprobabilistical model, MO,
representedby a random process, shall we agree that the model
MO does representthe machinerywhich produced the time series,
or shall we reject the model Mo and look foranother? This is the
kind of question we must learn how to answer. The generalmethod
of attacking it seems to be that commonto all problemsof testing
hypotheses. If we expect that the original model may be wrong,
we must be clear about the ways in which it may be wrong. This
will involve a definitionnot of one single random process MO, but
of the whole family of them, that may be denoted by (M). If a
time series consists of n figures,then it can be representedby just
one point,E, in a space W ofn dimensionsand, whateverbe a region
w in that space, each of the models M will definea probabilitythat
the point E will fall withinw. The process of lookingfora criterion
appropriate to test the hypothesisthat Mo adequately represents
the actual machinerybehindthe time-seriesconsidered,is equivalent
to a search of a certainregionwo in the space W such that, (i) whenever Mo is true the point E is unlikelyto be found in wo and (ii)
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298
Reviews of Statistical and Economic Books
[Part II,
whenever Mo is not an adequate model of the phenomena considered,thenthe chance ofE beingfoundin wois as greatas possible.
We see that the questions to solve do involve the problems of
sampling. The amount of such problemsseems to be even greater
than is suggested by the author. But there is somethingbeyond
those problems, that of choosing a proper criterion. In view of
the author's manifestskill, we may confidentlyhope that some of
these problemswill be foundsolved in futureeditionsof Mr. Wold's
very interestingbook.
J. N.
for ResearchWorkers.By R. A. Fisher.
2.-StatisticalMethods
7th edition. Edinburgh: Oliver and Boyd, 1938. 8"'' X 5Y".
xv + 356 pp.
I5S.
In the seventheditionof this now classicalworkthe author
followshis previouspracticeof addingnew sectionson methods
that have recentlybeen evolved. In this editiona mostvaluable
of groupsby meansof
note has been added on the discrimination
multiplemeasurements.The statisticaltechniqueof the analysis
attention
receivedconsiderable
hasrecently
measurements
ofmultiple
froma numberof quarters,and the utilityand wideapplicationof
in makingpossiblethe exact treatmentof
functions
discriminant
insolubleproblemsare nowrecognized.
previously
has also been expanded
polynomials
The sectionon orthogonal
bywayoforthogonal
to theirtheory,
introduction
so as to givea fuller
a revisionwhichis made more
betweenobservations,
comparisons
are
coefficients
valuableby the factthat tablesof the polynomical
Tables.
nowavailablein Statistical
F. Y.
and Medical
3.-StatisticalTablesfor Biological,Agricultural
Research. By R. A. Fisherand F. Yates. London: Oliverand
Boyd,1938. 11-" x 8k". 90 pp. I2s. 6d.
tablesin thisbook,and it is onlynecessary
Thereare thirty-four
themto givean idea oftheirvalue. Thefirstsevenare
to enumerate
and associated
pointsofthenormaldistribution
tablesofsignificance
and
coefficient,
t, X2, z and the correlation
samplingdistributions,
Fisher'sStatisticalMethods.
willbe familiarto readersof Professor
tables,
2 x 2 contingency
Thencometablesfortestingsignificance
These are
and tables of profitsand angular transformations.
followedby tables of Latin squaresand associatedquantities; of
polynomials.The set
scoresforrankeddata; and of orthogonal
requiredin statistical
withsomeofthematerialcommonly
concludes
squares,reciprocals,sines,
naturallogarithms,
work,logarithms,
tangents,randomnumbers,and a veryusefultable of constants,
to theusualpeffunctory
weightsand measureswhichis farsuperior
conclude.
tablewithwhichvolumesofthiskindgenerally
can use a stockphrasewithoutexaggeration.
For oncea reviewer
to usersof the newermethodsin
The book will be indispensable
M. G. K.
statistics.
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