This PDF is a selection from an out-of-print volume from the National Bureau of Economic Research
Volume Title: Evaluation of Econometric Models
Volume Author/Editor: Jan Kmenta and James B. Ramsey, eds.
Volume Publisher: Academic Press
Volume ISBN: 978-0-12-416550-2
Volume URL: http://www.nber.org/books/kmen80-1
Publication Date: 1980
Chapter Title: Model Construction and Evaluation When Theoretical Knowledge Is Scarce
Chapter Author: Herman Wold
Chapter URL: http://www.nber.org/chapters/c11693
Chapter pages in book: (p. 47 - 74) Model Construction and Evaluation EVALUATION OF ECONOMETRIC MODELS When Theoretical Knowledge Is Scarce
Theory and Application of PartialHERMAN Least WOLDSquares
DEPARTMENT OF STATISTICS UNIVERSITY OF UPPSALA UPPSALA, SWEDEN INTRODUCTION thequirementsis use not of applicable much of of the the investigatorin contemporary all circumstances, are farsetDuring greaterof rigorousmainly the than past becausetools are decade available. of the many For econometricians have come to statistical analysis knowledge re- realize that example, relationshipvariablesa typicalof the relevantregression in thecan regression, be distributions problemidentified its will withfunctional up require to the estimableUnfortunately, theoretical form, parameterwhether concept, there or values.notare and few knowledge topics in economics where these assump- the researcher to know the relevant the observed variablesouraFirst,tions likely knowledge arein list manytenable, is of even causal topics of atless theleast variables incomplete. statistical economicsin full. for There Asome distributional related our are theories twoempirical areas are of difficultymerely knowledge is that specified set of effect variables. Second, properties of the relevant prescriptions of deficiency. usually the variables of direct interest cannot be observed and one must All rights of reproduction in any formCopyright reserved. © 1980 by Academic Press. Inc. ISBN 0-12-416550-8 rely on 47 48indicator (marker) variables which are assumed to have some degree of asso- HERMAN WOLD ciation with the variables of theoretical interest. Under such circumstances to proceed along conventional lines with numerous heroic assumptions is inferentially hazardous. Another less ambitious but less knowledge sensitive approach is needed. Two items are required: a less knowledge intensive formulation of theory and a robust statistical procedure for drawing infer- ences when one is ignorant about the relevant statistical distributions. The former requirement can be met by the development of what has come to be mentknown is met in bythe the sociological use of partial literature least squares. as "path models."1 The latter require-
PATH MODELS: THEIR USE IN THE MODELING ConsiderOF RUDIMENTARY the inferential THEORIES problem facedPath by Adelman models and& Morriss their use (1967, are best1973), explained in terms of an example. who wished to learn something about the "causes of economic growth" in noncommunist developing countries. Seventy-four countries provided ob- servations on aspects of economic growth together with data on 41 indicators of economic, social, and political factors such as rate of growth of real per capita income, abundance of natural resources, extent of social mobility, and alldegree 74 countries of competitiveness into three of phases political Theof economicparties; initial analysissee growth: Table of 1. theselow, sporadicdata used or principal components to classify gationtheoreticalabortive, of the and causes insight high; of beyondsee differential Fig. a la. list Adelman growth of potentially rates and wouldMorriss useful be feltindicators supported that their ofby investi- growthlittle and of the causes of growth differentials. Further, no detailed assumptions designedabout statistical a path distributionsmodel to explain could economic beIn made. a later growth investigation in terms based of economic, on the same data, Adelman et al. (1975a) social, and political conditions. Whatever conceptualtheoretical insight there is in the model is illustrated diagrammatically by its arrow scheme; see politicalFig. lb. conditions The elementary and economic notions growth are these: rates. Social Economic conditions levels also affect affect both bothconditions political themselves conditions clearlyand economic affect growth growth rates. rates. Apart Finally, from the these political general statements about the four1 categories Blalockof the (1964, modeleconomic 1971). levels, social MODEL CONSTRUCTION BY PARTIAL LEAST SQUARES 49 causalpredictiveforcatorstionand the politicalof for whatfour these blocksobservableconditions, categories. purposes of the variableseconomic modelThe of the39 are observables canmodel,growth listed be used one ratesin Tablethat aswould suitableare 1. selectedlike With markersto respect obtain as or tosome there is the pivotal ques- indicators indi- the initial,onmeasure growth albeit of therates. tentative, relative The developmenteffect answers of, say,to these economicof path questions. models levels and politicalenables conditions,one to provide THE FOUR BLOCKS OF INDICATOR VARIABLES IN THE ADELMAN MODEL TABLE 1 (E) Indicators of economic Levels21. Level of industryinfrastructure Block No. 1 (P) Indicators of political conditions21 Degree of administrativecentralization ofefficiency political Block No. 3 6543 StructurePerDualismLevel capita of agriculture of GNP trade 543 PredominantDegreeDemocraticpower of freedom tradition basis of the presspolitical 7 Abundance of natural resources 876 partyExtentFactionalization system of politicalleadership of stability parties commitment Block No. 2 1110 9 PoliticalStrength strengthof labor movementof the militarytraditional elite Block No. 4 (S) Indicators of social conditions 321 SizeModernization of middletraditional class of outlookagricultural sector (G) Indicators of economic growth321 InvestmentImprovementRate of change rate in in Iaxation per capita system GNP 7654 DegreeExtent of of literacysocial masssocial communicationmobility tension 654 Improvementindustrialization in financialdegreeagriculture of institutions 1110 98 UrbanizationCrudeEthnicNational fertilityhomogeneity integration rate 87 Improvement in infrastructureresourcesRate of improvement of human soft model reported here.1312 "Adelman et al. (1975a). Two ofSocialAgricultural the 41organization indicators organization in the data bank were dropped in the 74 cases (countries) (a) variablesindicator Social Social Block, S Economicvariablesindicator Economic Block, E Economictries levels (E), social indicatorsFig. 1. (S), Data political matrix indicatorsand: Low arrow (L), (P), scheme Sporadic economic for (M) Adelman's growth and High rates model. (H) (G). degrees (a) Classification of economic of growth, 74 using 41 indicators (b) coun- influenceEconomicRef. Adelman the50 Economiclevels & Morriss and growth Social (1967; rates.conditions 1973) (b) influence Four-block the model Political designed conditions; by Adelman these etal.three (1 blocks Political Block, P Political HERMAN WOLD 975a) directlyprincipal observed) tool in econometrics variables were since initiated thePath 1930s. in modelssociology Path with models about manifest 1960.2 with latent (directlyConcep- observed) variables have been indicatorvariables Economic ----,. (in- a Block, Economic tually, the transition from manifest to latent variables in path modeling Growth G growth modelsstatisticalborrowed1973,ogy. Thewith 1978, inference.its transitionrationalelatent 1980), variables Statistical fromusing to latent the his have methodsclassical LISREL variables been developedformodels versions estimatingopened of of factorby a the K.newthe MLanalysisG. field Joreskog (Maximum ofin psychol-interest (1970, Like- in parameters of path variables lihood)1980b) approach, using PLS and (Partial by2 H. Cf. Least WoldDuncan Squares) (1974,(1966). 1975a,b,c,d,estimation. 1977a,b, 1978a,b, MODEL CONSTRUCTION BY PARTIAL LEAST SQUARES 51 THEviewmodels PLS of APPROACH withthe PLS latent procedure. TO variables. PATH MODELS The next WITHThe section presentLATENT begins paper with gives a brief exposition of the PLS approach to path a broad overall intendedVARIABLES: for AN multidisciplinary OVERVIEW OF ENDS and otherANDThe MEANS applicationsPLS approach where to path the problemsmodels with latent variables is primarily isticsexplored are involved:are complex (a) andcausalpredictive theoretical knowledge analysis, (b) is complexity of the prob- scarce. Three character- latentin lemseach variables: explored,of the four theoretical and phases (c) scarcity specification,of the of construction prior estimation,theoreticalIn this of section knowledge.testing, the andemphasis evaluation. is on how these characteristics a PLS path model with are reflected
arrowTheoretical scheme Definition and its of block the Model structure.The The design arrow of scheme a PLS path illustrates model basicwith latent variables is indicated by its eachvariablefeatures block of (LV).the the manifest Themodel. LVs, variablesThe which manifest theare structural assumedvariables tounits be ofindicators the model, of are grouped into blocks. In are inter- a latent tutesinformationindicatorsrelated the by theoretical innerconstitute in the relations. model. definition the Theblock formal of structure. theThe relationshipsmodel. specification The The arrows residualsbetween of indicate the thefor block LYsthe channels estimated structureand and the inner relations consti- their of arrowinner relations heads in andFig. relationships lb. A feature betweenof the model, LVs andwhich indicators is not explicitly illus- are marked by trated by the arrow scheme, is the specification of the causalpredictive ferencesof innerrelations.those relations,inLVs terms As that corollaries of they are indicators effectprovide of variables theand/or causalpredictive specification LVs in ofthe the inner of correspondinginferences the relations, block explanatory structureon the indicatorsand the namely, in- variables. The block structure, the inner relations, and the causalpredictive relations are called the structural relations of the model. tors.teristic EstimationThe weightsfeatures. of of theEach the Model indicatorsLV is estimated in each as Theaggregate estimation procedure for a PLS model with LVs has four charac- a weighted aggregate of its indica- are determined by the chooseweight relations among differentof the various modesA, blocks. B, The or Cformodel the builder design has of the the weight option to relations. The selection of estimation mode is guided by the subject matter of the model. The estimation proceeds in three stages. First, an iterative pro- cedure estimates the weights and the LVs. Second, the LYs estimated in the firsttoof stagenotethe model thatprovide in by the OLSregressors third (Ordinary stage for the estimating Least location Squares) theparameters other regressions. unknown are It is estimated. coefficientsimportant suchTesting as theconfidence Model intervals and goodnessWith respectof fit, which to the aretesting based and on evaluation of the model, classical methods distri- consistency.technicalscarcitybutionalSquares)-oriented ofproperties rather theoretical Under than regular ofbut real, knowledge.the distribution-free forconditionsobservables, ML aimsPLS ML formodelingmethods.are and optimalnot PLS available This insteadestimates accuracy parting becauseuses are butof LSco-consis- the ways PLS for of (Leastthe is tent, so that there is no substantial difference between the two set of esti- mates.ficancewith(1974) LVs. of cross-validation theAn model.overall The test procedure ofmethod. the modelTwo The uses istypesrobustness an provided adaptation of statistical byof thetheof the variouspredictivetests StoneGeisser have parameter been developed for PLS path models signi- 52 (1975b)).estimates is tested by random perturbation of the data (see Adelman et HERMAN WOLD al.
1. Model Formulation When KnowledgeTheoretical Is Scarce variablesmodelsnotationalof blocks with are details. LVsanddenoted Kwe In the ourshall by number general use the offormulas following Weindicators begin for notation. our thein thediscussionbasic jthJ denotesdesign of the formulation of soft block. The manifestmodels with some of PLSthe path number and the latent variables by Xfl. J = 1,... ,J; j=1,...,J. k = 1,... ,K (2)(1) MODELThe valuesCONSTRUCTION taken by BYthe PARTIAL manifest LEAST and latent SQUARES variables, which are also known 53 correspondingas "case values," estimates are denoted by theby appropriateA:s is customary, Roman we letters. denote The theoretical data bank entities by Greek lettersXJkfl,Jfl: and the = 1,... ,J, k = 1,... ,K, n = 1,... ,N. (3) catorsof Aclelman of economic, and Morrisssocial, and (1967, political 1973) conditions. has N = 74 Adelman's observations model, on given41 indi- byblocks, the arrow with scheme K1 = 7 shownindicators in Fig. of economic ib, groups levels, the observables E; K2 = 13 indicatorsinto J = 4 of K4social = 8 indicators conditions, of economic S; K3 = growth,11 indicators G; see ofTable political 1. conditions, P; and ablesinto1.1 THE ARROW blocks SCHEME of indicators, designatesThe arrow the scheme hypothetical groups latent the observablesthatvariables, is, the manifest vari- and shows the relationships between the LVs; these are assumed to form a causal chain system.3 The "outer" arrows between each LV and its in- dicators serve to illustrate the optional choice of estimation modes A, B, or C, which will be discussed in Section 2. Depending upon their position in the causal chain system, the blocks and their corresponding latent variables are either exogenous or endogenous. In Adelman's model the two blocks and LVs of economic levels and social conditions are exogenous, the two blocks and LVs of political conditions and economic growth rates are endogenous. assumed1.2. THE BLOCK to be linear: STRUCTURE In the basic design of PLS path models with LVs, the block structure is whereXjk isXjk the = LVjk + of7tjkj the + jth Ujk, block, J xJk 1,...is the,K, loading of the kth indicator k = 1,... ,K, (4) themodeling block structure.adheres to the exposition in Mosbaekin thejth & block, AsH. toWold the the (1970).general jts are rationale location of causalparameters chain systems and the and vs interdependentare residuals systems, in PLS In each relation in (4), the systematic part is assumed to be the condi- Thetionalhas "predictor" expectationa conditional assumption of expectation the indicator in (5) impliesof when zero thethatand latent eachis uncorrelated residualvariable in is eachgiventhatwith equation the latent is, E(xkJ) = IUjk + xJJ, j = 1,... ,J, k = 1,... ,K.4 (5) thevariable block occurring structure inin (4),that some equation. standardization Since both of xi,, scale and is c necessary in order are unknown in stantiveto avoid resultsambiguity. of the The statistical choice ofanalysis. standardization In PLS path does models not affect all LVsthe are sub- Itstandardized is a fundamental so as to principle have unit in variance: PLS path modeling that the information var() = 1, j= 1,.. .,J. (6) between the blocks and the ensuing causalpredictive inference is conveyed through the latent variables. Accordingly, it is assumed that the latent variableswhereas are, the in residuals general, ofintercorrelated, any block are say, assumed to be uncorrelated with the = i,j = 1,... ,J, (7) Inresiduals the basic of othermodel blocks design and it is with assumed, all latent furthermore, variables: that the residuals are ij, i,j=1,...,J, r(vh,vJk) = T(V,) = h==1,...,K, k=zl,...,K3. r(vh,J) = 0, (8) 54 mutuallyFormally, uncorrelated the block withinstructure blocks: as specified by Eqs. (4)(9) is the same for all r(vJh,vJk) = 0, i = 1,... ,J, h = 1,... ,K. HERMAN WOLD (9) PLS models. What differs between models is the specified number of blocks structure,and the number so also of are indicators the signs inof eachJust block.as the scales of the latent variables , and ic. Clearly, a change in the sign in the are ambiguous in the block jththe latent model. variable Thejk' model will change builder correspondingly defines thek manifest the signsvariables of all =Xjk 1,2,.. its loadings . K. The choice of sign is inherent in the subject matter of so that their . Thesignsand sign atare the inof same accordance time for consistent with the estimation nature of of the the parametersThe postulated "predictor" by OLS specification(S)latent regression, variable cf. provides H. Wold the basis for causalpredictive inferences from (4) then determines the signs of the loading iu in the block (1963). MODELstructure CONSTRUCTION in (4). Hence, BY when PARTIAL the loadingsLEAST SQUARES are estimated, their more or less 55 assumedhypotheses.11,k>complete to0,j be =agreement positively1, k = 1,2,. relatedwith .. , 7. the toIn the postulatedthe LVFor second for example, level signs of providesineconomic the first a blockpartial of test Adelman's model all block, the size of the traditional indicatorsdevelopment, are of agricultural1.3. THE INNER sector2,1 RELATIONS is negatively related to the<0,j LV = for 2, socialk = 1. conditions, endogenouswhich are linear variablessay, and form a Hcausal in numbertheThe chain LVs system. of innera PLS Using relations path subscripts model are are denoted j,, related for by a path of "inner" = + I3(i,)(i) + 8j, '* = J,. . . dH' relations, (10)the by whereand are specified by the conditional expectations (J) denotes the column vector of the E(J(J) = cx + '3(i)(i)' latent variables3* that appear as 3i' . . ,JH, shown in (10) (11) denoteandregressors those with latent nonzero variables coefficients with which in the ith equation as is the row vector of the corresponding coefficients. Also, let .. is directly connected, say, . , q, = (12) Adelman'swith tja model has two inner relations: j and a = 1,. . 3 = 3 + fl311where q =. q(i) and varies with i. For example, + fl322 + C3, (13) andThe similarlyconditional for mean Eq. of(14). Eq. In (13) terms is of the subscript notation in Eqs. (10) E(3I1,2)= 4 + /3411 = c + /422 + + fl311 + /3322 /3433 + C4. (15)(14) and (12), we have relatedwhere the to LVsformula 3 and for 4, 1, the political block and the economic growth block, 1,,, = 3,4, = 1,2, 2 = 1,2,3, indicates= 3,4 that LV 1(economic block) is directly 3, = 1,2,4, 4 = 1,2,3, (17)(16) respectively. See Fig. lb. 56 HERMAN WOLD Elimination1.4.models THE CAUSALPREDICTIVE with of the LVs Latent provides Variable RELATIONS causalpredictiveFor (SELV) the indicators from relationsthe correspondingof any by endogenous Substitutive block LV the PLS approach to path structure (4) and inner relation (10). Hence in the basic design, the causal predictive relations read Xjk = /1jk + + + VJk, (18) theand PLSthe prediction approach errors gives are causalpredictive givenIn byAdelman's relations model, thefor thirdthe indicators and fourth ofblocks are endogenous, and so = Djk + 7tJkJ, 3 = Ii,. IH k = 1,. . . , K. (19) these blocks, namely, X3k = /23k + lt3kcz3 + 3k(fl311 + fl322) + V3k, with prediction errors given by X4k = 4k + 4k4 + 4k(411 + 422 + fl433) + V, k=1,...,8, (21)(20) 2. Model Fitting and Parameter Estimation with PLS V3k = V3k + lt3kE3, V4k = V4k + 7t4k84. (22) ofgive PLSraw a formaldataestimation or expositioncross-product can use of the thedata. observed procedure TheProceeding estimates data and defined to then theare PLSnumerically discussin (1.3) estimation itseither rationale. the ofin same pathterms models with LVs, we shall first except for rounding-off errors. We shall set forth the case of raw data input. Postponing the estimation of the location parameters, we assume that all Untilobservables further are notice, standardized furthermore, to zero wemean, shall say, drop theXk,J prime = XJkn superscripts, - Xjk, I = 1,... ,J, k = 1,.. . ,K, n = 1,... ,N. (1) denoting the standardized observables as before by
Xjk,1 now with x = 0, j = 1, . . . ,J, k = 1,... ,K, n = 1,. . . ,N. (2) MODELfirst stage CONSTRUCTION of the estimation BY PARTIAL process LEASTEach estimates SQUARESLV is the to beLYs estimated and the requisiteas a weighted sets aggregate of its indicators; the 57 of weights for the indicators. If we denote the estimate of by X, with case values Xi,,, the aggregate formula,where fwhich is a scalar, is the Wiksame are for the all weights, LVs in a and PLS Xjkfl path are model, the case is values= J for the (wjxjfl), j = 1,. . . ,J, k = 1,. . . ,K, n = 1,.. . ,N, (3) bymanifest the requirement variables. Since that Xi,,c have unit variance over the N cases, i.e., is standardized to unit variance, f, is determined The weight relations serve to determine the weights Wik in the PLS = 1 1/2 , = 1,.. . , J. (4) estimationconnectedbetweenXjk. ofLV inEq. the (3) arrow for each scheme. LV HenceBy the the inner weight relations relations of arethe designedmodel, there to is an exchange of information of the jth block and those LVs as a weighted aggregate of its with which indicatorsis directly provide ancillary information on the weights WJk from these same s, using involvenectedmodesOLS regressionsa ofinsum weightthe of arrow those forrelations, scheme,theLVs purpose. with say a whichAsum Foror B.with eachthe We LV signblock, note of factors thein the advance block model (+1 is or thatbuilderdirectly 1) bothdenoted has con-modes the by option to choose between two weightwhere therelations subscripts are defined j as follows. and defined by = sgn were defined in Eq. (1.12). Modes A and B j 1,. . . ,j, , jq, for the (5)
2.1. MODE A. SIMPLE OLS REGRESSIONS WITHTHE INDICATORS AS REGRESSANDS + . + LVs,where Wik Xjk are theare weightsthe manifest to be estimated,variables, andX, are di,, theare predictedthe residuals. values Here for and the Xjk = WJk(SJ.!X.I + sj.iJ2Xi,2 Sj,jjqXjjq) + djk k=1,...,K, (6) in droppedEq. (7) the subscriptregressions n in are the over manifest n = 1,.variables, . the LVs, and the residuals. . , N; for simplicity, we have 2.2. MODE B. MULTIPLE OLS REGRESSION WITH arrows between the LVs and theirThe indicators arrow schemeTHE in theINDICATORS illustrates various ASblocks; the REGRESSORS weight see relations mode A or B by "outer"s1X1 + s, 2X12 + . . + Sj,ijqXijq = (wxj) + d. (7) Fig. lb. The simple OLS regressions of mode A are illustrated by as many arrows directed outwards from the LV to its indicators. The multiple OLS regression of mode B is illustrated by a bundle of arrows directed inwards from the indicators to the LV. In the arrow scheme of a PLS path model, the "outer" arrow heads illustrate the residuals of the corresponding weight relations in (6) or (7). We shall say that the estimation is mode C if the model builder makes leastminingAcombined andone B:block. the If use weighta Theblock of modesfollowing relations represents A and rulein anthe B,of Whenexogenous variouseachthumb theof is blocksthemodel offeredLV, two use ofbuilder modesfor theB, the otherwise,model, has choicebeing settled the usedbetween use iterative the forA. choice at of mode A or B for deter- procedure of estimation of LVs and indicator weights is performed by 58alternating between the weighted aggregates shown in Eqs. (3) and (4) and HERMAN WOLD scriptsandthe weight mark (') andthe relations proxy("), Eqs. shownestimates (3)(7) in giveEqs. obtained the(5)(7).If we values inwrite steps of the LVs and of the weights r = 1,2,. . . for the consecutive steps in the iterative procedure r and (r + 1) by the super- in step (r + 1) in terms of the values obtained in step r as follows: X,=f7>(wkxjkfl), = ± j= 1,...,J, n= 1,...,N 1/2, (9)(8) XJk = wJk(sj,j,.X11 + { + SJ,jjqXq) + dJkfl, [ (wx)]} (w(kxJkfl) + k = 1,. . . , K, (11)(10) Sfa = sgn T(XmnXqn) + + S;ijqXjq = r = 1 a = 1,. . . , q. (12) equal, say, To start the iterative procedure with W,k 1, j=1,... ,J, k= 1,... ,K. all weights are taken to be (13) MODELwith the CONSTRUCTION linear aggregates BY PARTIAL shown LEASTAfter in (8) SQUARESthe and first the step, standardizing we use the weights factors f' Wk obtained in the rth step, together 59 determined by (9), in order to obtain the case values X:/, for the LVs in step (r + 1). Then, according as the weight relations for the jth block are decided to be mode A or B, the weights w/,, in step (r + 1) are computed by the simple OLS regressions shown in (10) and (11), respectively. In either case we use the latent variables X obtained in (8) and the signs in (12) at the (r + 1)th step. weights and the LVs. Thus the estimatedThe passage weights to theand limit LVs asare r given by Wik = r-'x Wk, cc gives the PLS estimates of both the = r-'oD uses the LYs estimated in the firstWhen stage the to estimatefirst stage the of blockthe PLS structure algorithm (1.4) is completed, the second stage j = 1,... ,J, lim k = 1,..., K, n = 1,... ,N. lim X,,, (14) by simple OLS regressions and the inner relations (1.10) by simple or multiple OLS regressions, as the case may be. The causalpredictive relations (1.18) relations.ofare the then LYs, Inestimated the the PLS block by algorithm structure,substitution. the theThe estimation inner third relations, stage of of location andthe PLSthe parameterscausalpredictive algorithm estimatesis the location parameters immediatereadily transformedmatter, as always in order in LS to estimation.Equations be able to (2)(14) use data for input the PLS in the algorithm form of with raw data input (1) are thecross-product same except data for rounding-offinput. The ensuing errors. parameter Combined estimates use of the are parameternumerically estimates and the raw data then provides estimates of the case values of the LVs and of the residuals in the block structure, the inner relations, and the causalpredictive relationsestimates that are important for the inter- pretation,input as testing, well as andfor cross-productfurther evolutionThe data PLSof input. the algorithm model. Computer is easily programs implemented allowing on the computer for raw data optional use of raw data input or cross-product data input are available toin runseveral the basic versions.5 PLS algorithm. In the program of HuiThe (1978) computer the PLS program procedure packages is combined MIDAS and MINITAB need no further implementation with the Fix-Point algorithm to estimate soft models that have interdependencies in the path of inner relations. The program of Apel & Lohmöller (1980) combines PLS with StoneGeisser cross-validation for testing the predictive significance of soft modeling. 2.3.the THREE first principalSPECIAL componentCASES OF PLS is numerically MODELINGTwo PLS models equivalent, are well-known up to a scale from factor, multivariate analysis. To begin, to a one-block PLS model estimated using the mode A procedure.6 The classical model standardizes the loadings Thlk to unit square sum. The iterative procedure (8)(12) in this case shrinks to Eqs. (8)(10) with (10) givesreducing the tofirst canonical coefficientThe as two-block the estimated PLS correlation path model between estimated the using theXlk,, mode = W'l'kXYfl B procedure + dlkfl, k = 1,... ,K1, n = 1,... ,N. (15) two latent variables.6 The iterative procedure now shrinks to the use of
Eqs. (8), (9), and (11), with (11) given by = I' (wkx1k) + djkfl, k = 1,. . . ,K1, to be used in Adelman's model areAs mode a third C since case, theconsider model the contains Adelman bothX'k,, model. = The appropriate relations k (wkxzkfl) ± d2k,,, k= 1,... ,K2, n= 1,... ,N. (16) endogenous and exogenous variables. Consequently, the weight relations 60are formed by mode B for the two first blocks, these being exogenous, and HERMAN WOLD by mode A for the two last blocks, which are endogenous. The weight rela- tions for Adelman's model are s31X3 + s41X4 = >(wlkxlk,J) + d1, k (17) s32X3, + s42X4,, = X3kflx4Ifl == '3k(3l-lnW4k(S41XI6 + 32X2 + s42X2,, + s4X6) + s43X36) + d3k,1, + d4k, k (W2kX2k,,) + d26, (19)(18)(20) LVswhereK =obtained 7,n =13, 1,. 11, .in andthe first8, respectively. stage to estimateThe second the structural stage of relations.the PLS estimationFirst, the procedure makes use of the . ,74 throughout and k = 1,. . . , K in Eqs. (17)(20) with blockof the structure manifest as shownvariables in Eq.Xjk6 (1.4) on the is estimateestimatedH. Wold X by (1966).of use thejth of an LV. OLS The regression regression MODEL CONSTRUCTION BY PARTIAL LEAST SQUARES 61 equation without the location parameters is as follows: . . Thegiving sign chosen for the limiting factorIn Eq. f, will, (9), bythe Eq. sign (8), of determinethe standardization thePk sign factor j7 was left ambiguous.as the estimateXjkfl for = PJkXJfl + Ujk,,, itik in Eq. (1.4). I = 1,. . , J, k = 1,. . , K, (21) for the array of estimated case values X(n = 1,. . . , N) of the latent variable shouldpjk(kchoosing choose the that sign sign of, X3for andX3 whichof the agreesarrayand of withhence loadings most by of Eq. =the 1,. (21) presumed . the sign signs for the array of estimated loadings . ,K). Thus for the jth block, there are two possibilities for The model builder for the theoretical loadings. That is, he should choose that sign so as to give JkP jk) (22) two different ways: either, say, by modeHowever, A or mode suppose C. The rulethat inthe the investigator previous estimates the model in one of Mode C .766 .779 23 (sgn i 24 .699 13 > 0. .797r14 r34 .481 .918 12 model draw from ApelMode & H. BA Wold (1980). Here and in the text the numerical results on Adelman's .842.671 .889 .810 .776 .705 .878.784 .709.488 .934.912 paragraph should then be applied to determineH. Wold (1978b).the signs for the mode A estimate of Xi,,, say, X5(A). For mode C, the sign of f should thereafter be chosen so as to give positive correlation over n of X,(A) and X,1(C), sixrjj values = r(x, are x) shownfor the oncorrelations the first line between Inof Adelman'sTable the 2. estimated For model, comparison, LVs, the the mode we resulting also C procedure was chosen. Writing r[X,1(A), X111(C)] > 0. (23) give the corresponding mode A and mode B estimates. Since Adelman's arrow scheme has no arrow between the first two LVs, there is nothing in the SIMPLE CORRELATIONS BETWEEN LVs GIVEN ALTERNATIVE ESTIMATION PROCEDURES TABLE 2 PLS estimation procedure that makes for close agreement between the estimatesthe differences are larger. The innerThe relations correlations shown mode in Eq. C (1.10) and A are show estimated close agreement. Relative to mode B 1 2; hence this column has been placed by itself. thewherewithout inner b(J) relations locationis the OLS inparameters Adelman'sestimate forin modelterms ofare, the in regression this stage, equation: without location As before using mode C estimation, (24) parameters: var(e3) = .413, (25) A procedure, For comparison we give the inner relations when X3estimated = - .026X1 by the + mode .790X2 + e3, = .514X + .517X2 - .274X3 + e4, var(e4) = .3 19. (26) We see that for the inner coefficients b(J), the difference between modes A and K4K3 = .554X1.537X1 + .166X2.338X2 +- .151X3e3, + e4, var(e3) = .351..498, (28)(27) 62mationC is larger robustness than for is naturalthe simple since total the correlations bs are partial correlations and are The difference in esti- HERMAN WOLD nonlinear functions of the simple correlationsAlthough the residual variances are somewhat smaller for mode C, which estimatedtheopment,commonwe haveexplanation by has chosensense mode a coefficient maythat Afor are X1,thebe more theestimation the b31 difference plausible.LV that that isof measures insignificantinForAdelman's robustness example, the model, or levelinjust evenmode referred of negative. C it to.is against Another Part the inner relations economic devel- of part of the story is that for the group of 74 countries, the model structure is fornonlineargrowth;and the canthree with besee groups illustrated Fig.regard lb.of tocountriesUsing inthe terms level the (L, mode ofof This M, theeconomic H) C innerlast procedure,classified point relations growth. has with the been as estimatedregardestimated emphasized to economic innerseparately by rela- Adelman & Morriss (1967, 1973) tions for the 28 countries with low (L) economic growth are X4X3 = .093K1.390X1 + .665X2.950X2 +- .672X3e3, + e4, (30)(29) and for the 25 countries with high (H) economic growth, they read The coefficient b31 is positive for the intermediate (M) group of 21 countries X3X4 = .159X1.356X + + .596X2 1.158X2 + -e3, .601X3 + e4. (32)(31) withsample avalue size of is.151. only about one thirdWhen of the the total countries sample. are Hence grouped in Eqs.with (29) regard to economic growth, the and (30), the estimates of the inner coefficients b(J) are less robust than for the relations in Eqs. (25) and (26) for the entire group of 74 countries. Finally, of estimatedletthe us inner again relations.by examine mode A,For a areshift the countriesin estimation group mode (H), with the innerregard relations, to the robustness when X3 = .373X + .530X2 + e3, (33) ficientshand, band with to regardthe nonlinearity to the shift of in the estimationWe model are structurenow from in mode a onposition theC to other. A, to on compare It the appears one the robustness of the inner coef- X4 = .522X1 - .261X2 + .397X3 + e4. (34) that the robustness is more pronounced for the shift in estimation mode than for the shift from total sample to subgroup (H). This is so whether the comparison is based on estimation Mode C as applied to the total sample or the subgroup (H).
parameters2.4. THE CASUALPREDICTIVE and LVs as RELATIONSThe relations shown in Eq. (1.18) can be written in terms of the estimated with prediction error i =h. Xjkn = Pjkhi(j)X(j)n + Vik,,, 'fF1, k = 1,. . . ,K, n = 1,.. . (35) Alternatively, one can write Xjk,, = PJkb( i) Vjkn = Ujk,, + pJkeJk,,. U)n + Vjk,,, (37)(36)
MODEL CONSTRUCTION BY PARTIAL LEAST SQUARES 63
J, kwhere the simple the "reestimated OLS regression loadings" of Xjk,J Pjk on are the obtained composite by computingvariable bu)X(j)fl, for fixed 64 HERMAN WOLD the estimation of the causalpredictiveIt will relations. suffice for The our results purposes for tothe illustrate first only the first two versions of x33 = .287X11, + .480X21, + U33,,, j = = j = = 1 x33version as applied to countries (H) are for = .136X1, + .989X2, -.313X1 + .523X2,, + V33n, k (var 1)3k) 3, k =3 and .580 V4k) = .796. 4, k (42)(41) Thex41,, second version with re-estimated loadings is .514X3,, + v41,,, (var i3k) k = (var .576, (43) Notex41,, = that both the residual variances and the coefficients of the LYs decrease.112X11, + .813X2 - .422X3 + i41,,, - (var p4k) = .779. (44) residualsa regression Vjkfl, over n = 1,. . . , N. In virtue of the least squares property of the obtainedfrom Eqs. from (41)(42) Eqs. to(35) Eqs. or (43)(44).(37) by Causalpredictiveusing This (3)is perhaps to substitute surprising relations for the atin right-hand first terms sight of the manifest variablesvar(f5Jk) can be var(vJk), 3 ii. . . k = 1,... ,K. (38) XJk = Pjk i 2.6.under FLEXIBILITY analysis due to the flexibility Theof the basic arrow design scheme. of PLS When "soft the modeling"observ- is malleable to the problem composition,numberables are and grouped number, structure into and of blocks thesize inner of for the relations. indirect blocks areThemeasurement flexible flexibility as wellofis heightenedthe as LVs,are the the by the choice of estimation mode A, B, or C, wherein two LS extremum prin- A.LVsindicatorsciples The andfirst come is canonicalwhen estimatedinto play.the LVcorrelation by (the mode component) B.gives InMore the maximum passageprecisely,is given correlation fromand the is twofirst estimated toprincipal between three by orcomponent two modemore gives LS prediction of the blocks in the PLS model, the extremum principles of principal components theyand canonical are given correlations by Eq. (6) and carry provide over,When albeit ancillary the in weight a predictivequalified relations sense. relations of the for jth the block are estimated by mode A, observables in the block in terms of estimates of the LVs with which the jth ButLVcedures,the iswhen predictiondirectly the they weightconnected are errors given relations of in bythe the Eq. relations ofarrow (7)the and jthscheme. shown block provide Byinare (6) estimatestheestimated have least minimum squaresfor by the mode correlationsprinciple, variance. B pro- of the jth LV and the LVs with which the jth LV is directly connected in the arrow scheme. By virtue of the least squares principle, the sum of the absolute thesevalues last ofmentioned these intercorrelations LVs. will be maximum for given estimates of used2.7. "INSTANT in the estimation ESTIMATION" can be writtenThis down description directly from of PLS the estimationarrow scheme is suggested because the relations estimation;and because namely, of the easy the explicit and speedy estimation computerTwo offeatures the work. LVs in bythe the PLS use approach of weighted are instrumental in making for instant aggregates of their indicators and the direct estimation of the weight relations by mode A or B. Thanks to the explicit estimation of the LVs, no identifica- tion problems arise in "soft" modeling. The weight relations avoid the non- linear side relations that would arise if the aggregate relations were treated as side conditions and had to be taken into account through Lagrange multi- pliers. However, the weight relations are not deduced from the PLS model as defined by the block structure and the inner relations but are ancillary informationadjacent LVs gathered in the arrowfrom thescheme. intercorrelations between each LV and its almost2.8. CONVERGENCE certainly.8 For PLS models Forwith one- three and or two-blockmore blocks, models convergence the PLS estimation procedure converges of the estimation procedure has never been a problem in applications to real-world models and data. For example, in the numerical estimation of Adelman's model, convergence of the iterative procedure was reached for 66modes A, B, and C in three, six, and four steps, respectively. HERMAN WOLD 2.9. PLS OPTIMIzATION In PLS estimation, each regression step of the procedure is an LS esti- asmation theora unifying more weight of oneother condition. relationsor unknowns; more unknowns that nofor total, the in Thevarious termsor overall, PLS of blocks procedureproxy extremum ofor finalthe minimizes conditionmodel estimates are in is givenforits imposed firstone by stage each residual variance in firstEqs.mation stage(6) or mode of(7). the This APLS or is algorithmBdone for forthat each isblock. nothing block InLyttkens, depending eachother step than Areskoug r onthe = the1,2,. &conventional H. choice WoId. (1975).of esti- LS . the iterative procedure,MODEL CONSTRUCTION the residual variance BY PARTIAL of a LEAST weight SQUARES relation being minimized with 67 regard to the free parameters which are the weights. Mode A uses simple OLS regressionsmode B uses to minimize multiple OLSwith regressionrespect to theto minimize parameters with one regard by one, to thewhereas param- eters jointly. The conformity with the conventional LS procedure carries is performedstageover toto theestimate bylimit minimizing asthe r parameters-* the residual of theThe variancesstructural second stage inrelations the of block the of PLS thestructure, model. algorithm theThis uses the LVs obtained in the first . inner relations, and the causal-predictive relations SELY with reestimation. Thus the PLS procedure is partial LS in the sense that each step of the pro- cedure minimizes a residual variance with respect to a subset of the free parameters,limit the PLS given estimation proxy or procedure final estimates is coherent for thein the other sense parameters. that all the Inresidual the variances are minimizedjointly. However, the PLS procedure remains partial in the sense that no total residual variance or other overall criterion is set up for optimization.9 as2.10. the CONSISTENCY sample size increasesAND CONSISTENCY (consistency)These AT LARGE limitand as properties the block of size PLS increases estimation as refer to the passage to the limit well (consistency at large). Both properties hold good for PLS estimation under mild supplementary conditions to the model structure defined in Eqs. (l.3)(1.11). However, consistency does not imply that the estimator is unbiased in the limit; a well-known example is the canonical correlation coefficient. While consistency is a simple corollary of predictor specification, see Footnote 4, consistency at large is a straightforward implication of the classical probability theorem known as the Law of Large Numbers.1° since3. Construction 1971, as documented and Evaluation in a seriesof PLSThe of Models PLSprogress approach reports, to and path it ismodels even more with LVs has developed gradually recently that its foundations have been consolidated enough to support the crucial transition from a pioneering device to a useful tool for applied work. Table 3 refers to some10 20 applications toH. real-world Wold (1978a,(1980b). problems 1980b). and data that science (1977) Apel environmental Economics, pollution activity, Economic activity State 332A 3 10 science political (1975a) al. et Adelman sociology, Economics, conditions social and Economic conditions Political 331C 3 9 1980a,b) (1977, Wold H. & Noonan (b) al. (1975) al. et Noonan (a) sociology Education, conditions school and Home achievements Pupils' 331C 3 8 psychology social conditions sanctions economic (1978b) Wold H. economics, science, Political economic and Political of Outcome 211A 2 7 (1978) Wold S. & Dunn chemistry Medicine, agent Chemical Carcinogenicity .,11b 2 6 (1975a) Wold H. Sociology status Social participation Social 221C 2 5 (1936) Hotelling Anthropology measures Brother's measures Head 221B 2 4 etc. temperatures, al. (1978) al. et Wold S. Technology oven material, Raw quality Steel 11OA 1 3 equation) (1978) Wold S. chemistry Organic (Hammett substances Reacting speed Reaction 11OA 1 2 (1904) Spearman Psychology intelligence General traits Mental 100A 1 1 Authors application of Discipline indicators Explanatory indicators Criterion code blocks no. PLS of No. Item VArunLEs LATENT wim MODELS PATH PLS OF APPLICATIONS 3 TABLE Michigan. of University the of Bookstein L. F. from communication personal 'A as (1975b). Wold H. cf. estimates; profile" "low to referred have I parameters ensuing The Using regressors. as components principal its with regression OLS by estimated is block causal the of LV the (1967), Harman by proposed approach the as no. item in as one just has block LV. an as counted not therefore is and indicator same this to reduces (2.3), by estimated LV, the 7, indicator, PLS code PLS the in figures "The a If C. or B, A, mode, estimation the indicates letter capital the relations; inner and LVs, blocks, of number the indicate (1980a) Wold H. & Noonan 8(b) No. of Extension 17 19 17 18 (1978) Hui 14 No. of Extension of labor of supply and opportunity work of 17 (1980) Knepel & Cremer sociology Economics, demand, Productivity, quality and Quantity 553C 5 14 asinNo. Same 4 16 (1978) Hui model the in Feedbacks observables downstreams (1979) Wold H. & 15 Kowalski, Gerlach, Chemistry upstreams pollution, Water pollution, Water 441C 4 (1980) Wold H. & pupils' Hui, Boardman, (b) expectations peers' motivation Pupils' 442C 4 14 (1978) Hui (a) psychology Education, expectations, Parents' achievements, (1980) Wold H. & Apel (c) political (1977a) Wold H. (b) science political conditions political conditions growth, 442C 4 13 (1975a) al. et Adelman (a) sociology, Economics, and social levels, Economic Economic communication)' Son's status Son's 331B 3 12 (personal Bookstein sociology Education, status Parents' behavior (1978) Andreoli & Rossa, 11 Hopple, Wilkenfeld, economics science, Political factors external and Internal policy Foreign 331B 3 theofwere ofshowevolution. scope PLSavailable that modeland the The limitations whenPLS construction purpose approach this of chapterof the thisand is PLSgaining finaltowas discuss approach. sectiondrafted. momentum the is Theaccumulating to summarize datesbut is instill the experience inthe last an evolution earlycolumn about stage Tablevariables.predictive3.1. 3WHEN gives analysis ComplexTO,evidence How ofTO, ofproblems problems the AND wide WHY withapplicabilityAs TO stated highlow USE informationcomplexity, atPLS ofthe PLS outset modeling butoccur PLS low modelingin with information. most latent or is all primarily designed for causal sciences and the list includes examples ranging from the natural and medical formatto the social or the and data. political The available sciences. dataBeing may distribution-free, be time series or PLS cross estimation sections. imposes no restrictions on the The observations on the indicators may be quantitative measurements, ordinal ranks, or records of occurrence, nonoccurrence, or of low versus high levels of the indicator. The arrow scheme is the conceptualtheoretical blueprint for the model. onmeasurementThebetweenindicators the arrow joint schemethe basis are ofLVs. the toof indicates Speaking beLVs,his used rudimentary and whichand broadly, which their LVs theoretical"inner the grouping enter researcher relations" the knowledge,model,into designs the are which blocks assumed the his observables- arrow experience for to indirect schemeexist and intuition about the problems explored, and the data that are at his thedisposal.evolutionary data, and the process. model isThe improved empiricalThe by content arrowinteractions scheme of the through model is usually isthe extracted tentativeestimation from since the model construction is an procedureConsequently, between the the researcher model and should the data begin and the with reactions a generous of the number researcher. of ob- fortheservablesindicators rich PLS empirical estimation content inprocedure. the of various the model In theblocks. andinteraction isTo favorable use betweenmany to observables the the accuracy data and makes of the whichoriginal should model be omitted. it will become The researcher apparent in which planning indicators his empirical are relevant analysis and 70must choose between the various modes of estimation. To use estimation HERMAN WOLD mode A for the whole model gives minimum residual variances in the block structure; but to use mode B for the whole model leads to maximum inter- correlations between the LVs that are directly connected in the arrow scheme. The hybrid mode C strikes a balance in these respects. MODEL CONSTRUCTION BY PARTIAL LEAST SQUARES 71 betweenance of thean LVvarious and its LVs indicators hinges theupon modelIn their the gives causeeffect indicators. two measures andIn the predictive for the relev-performance of a soft model the relev- relationship whereasoranalogous regression.anceindividual of its toan weight the contributionindicator,Thus difference Wik for themeasures namely, of indicatorbetween XJk theitsto theXJk,loadingcontribution coefficientsrelevance its loadingand of itsof fromXJk weight.PJkthe toLVmeasuressimple the of joint the and the jthrelevance multiple separate The difference is block, of the indicators in the jth block and thereby to the relevance of the jth LV. Both measures are of importance; hence in applied work with soft modeling coefficient,both measures on theshould assumption be used that and the reported. otherThe causalpredictive variables remain accuracy constant, ofhinges a specific explanatory variable and its andrelationsbringupon complete. whetherdrastic are Forchangesalways the example, selection predictive,in the a coefficients.of dormant explanatory butAs predictive variableto the variables distribution relationsthat comes is sufficiently are between into not playnecessarily causal realistic may and predictive inferences, causal causal. The question whether an explanatory variable is not only predictive, but also causal, belongs to the subject matter of the model. and appreciation are due to Dr. HeinoMy Apelresearch for collaborationfor this paper onhas Adelman's been supported model by and the for Stiftung the Volkswagenwerk. My thanks ACKNOWLEDGMENTS numerical results reported in Section 2.3. I would like to express my deep gratitude to Professors Kmenta and Ramsey for inviting me to include in their book an exposition of soft modeling based on Cahier 1979:06, Department of Econometrics, University of Geneva, and to Professor Ramsey for editing an abridgement of Cahier 1979:06 for this purpose. Adelman, I., & Morriss, C. T. Society, politics, andapproach. economic Baltimore, development: Md.: Johns A quantitative Hopkins Press, 1967. Revised edition 1971. REFERENCES Adelman, I., &el Morriss,al. Applications C. T. Economic of classification growthAdelmanMorriss'Stanford, tionsand and socialwith pathCal.: soft equity modelsStanford inforuzation. data. in with developing Univ.In H.latent Toronto:Press,Wold variables countries. 1973.(Ed.), Third Group toworld report:congress Modeling of econometrics, in complex August situa- 21-26, Adelman, I., et al. Perturbation analysis of NIPALS4.(a)Group1975. parameter Researchreport: Modeling estimates. 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