Statistical Models for the Inference of Within-person Relations: A Random Intercept Cross-Lagged Panel Model and Its Interpretation

1 Statistical M odels for the I nference of Within- person Relations: A Random I ntercept Cross-L ag g e d Panel M ode l and Its Interpretation Sa tos h i U sa mi G raduat e School of Educat i on, The U ni ver si t y of T okyo • Th is is a tr a n s la tio n o f a n a rtic le p u b lish e d in th e Japan ese Journal of De vel opment al P s ychol ogy , V ol . 33 ( o r ig in a l a rtic le is a v a ila b le fro m h ttp s://w w w .js ta g e . js t. g o . jp /a rtic le /jjd p /3 3 /4 /3 3 _ 2 6 7 /_ p d f/ - c h a r/ja ). • T hi s a rt i cl e has been f a i t hf ul l y t r ans l a t ed f r om t he or i gi na l J apane se , but a ny di s cr epa nci e s i n t he t r a nsl at i on a r e t he sol e r es ponsi bi l i t y of t he aut hor s . • Wh e n c itin g th is a r tic le , p le a s e p ro v id e t he f ol l ow i ng bi bl i ogr aphi c i nf or ma t i on f or Th e J apanes e J ournal of De vel opm ent al P syc hol ogy : U sam i , S . ( 2022) . S ta tis tic a l m o d e ls f o r th e in f e re n c e o f w ith in - per son re l at i ons: A r andom i nt er cept cr oss - l agged pane l model and i t s i nt er pr et at i on (I n Ja panes e) . Th e Japanes e Jour nal of De vel opment al P syc hol ogy , 33 , 267 - 286. 2 St at ist ical M odels for t h e I n feren ce of Wit h in -person Relat ion s: A Ran dom I n t ercept Cross-Lagged P an el M odel an d It s In t erpret at ion T he cross - l ag g ed pane l model ( C L P M ) ha s be en wi del y used, par t i cul ar l y i n ps ychol og y, t o i nf er l ong i t udi nal r el at i ons am ong var i ables . At t he same t i me, cont r ol l i ng f or betw een - p er son het er og enei t y and capt uri ng wi t hi n - per son r el at i ons as p r o ce sses o f wit h i n - p erso n cha ng e are r eg arded as key com ponent s t o causal i nf er en ce bas ed on l ong i t udi nal dat a. S i nce H ama ke r, Kui per , and Gr as man ( 2015) cr i t i ci zed t he CLPM f o r its lim ita tio n s in in f e rrin g with in - per son r el at i ons, t he r andom i nt er cept c ro ss- l ag g ed p anel mod el (R I - C LP M), w h ic h in c o r p o r a te s s ta b le tr a it f act o r s r epr ese nt i ng st abl e i ndi vi dua l di f f er e nc es, ha s r api dl y spr e ad , e spe c i al l y i n psyc hol og y. A t th e s a m e tim e , a lth o u g h m a n y s ta tis tic a l m o d e ls a r e a v a ila b le f o r in f e r rin g with in - per son r el at i ons, t he di st i nct i ons am ong t hem ha ve not been cl ear l y del i neat ed, and d i sc ussion s o v e r th e in te rp r e ta tio n an d s e le c tio n o f s ta tistic a l m o d e ls re m a in a c tiv e . In t hi s paper , I posi t i on t he R I - C L P M as one us ef ul met hod f or i nf er r i ng wi t hi n - per son re la tio n s , e x p la in its p ra c tic a l issue s , and org an i ze it s mat hemati cal and concep t u al re la tio n s h ip s w ith o th e r s ta tistic a l m o d e ls , a s w e ll a s p o te n tia l p ro b le m s th a t m a y a ris e i n t hei r appl ic a tio n . I n p a rtic u la r , I p o in t o u t th a t a d is tin c tiv e f e a tu r e o f th e s ta b le tr a it f a c to rs in th e RI - CL P M , i n repres enti ng b et ween - per son het er og enei t y, i s t he a s su m p tio n th a t th e y a r e u n c orre la ted with w ith in - per s on var i abi l i t y, a nd t hat th is poi nt s er ves as an i mpor t ant l i nk t o t he ma t hemat i cal re l at i onshi p w i t h t he dynam i c panel mode l , a not her pr omi si ng al t er nat i ve. [K eyword s ] L o ng it udina l da t a , W it hin - pe r s o n re la t io n, C a u sal i n fer en ce, C ro s s - l agged p an el m od el s, S tr uc t ur a l e qua t io n mo de ling 3 I nt r o duc t io n Des cri p t ive r esearch an d CL P M T her e ar e i ncr eas i ngl y many l ongi t udi nal st udi es eac h year , wi t h mor e t han te n t housa nd papers now bei ng publ i shed a nnual l y wor l dwi de. M ost quant i t at i ve l ongi t udi nal s t udi es r epor t ed i n devel opment al psyc hol ogy and r el at ed f i el ds a re pos i t i oned as desc r i pt i ve or e xpl ora t ory re se ar ch ( Ha mak er , M ul der , & van I Jz endoor n , 2020) . D esc r i pt i ve s t ud i es a im to un ders t and act ual t raj ect ori e s ( pat t erns) of ch anges i n devel opmen t , gr owt h , et c., or i ndi vi d ual or gr ou p di f f er e nc es, and l at ent gr owt h mode l s , l at en t growt h mi x ed mod el s, and h i erar chical l i near model s ( al so cal l ed mi x ed ef fect s m o d e ls, m u ltile v e l m o d e l s, et c.) are of t en u sed as s t at i st ical mod el s. On t he ot her hand, expl anat or y re se a r ch ai ms t o i dent i fy t he ca use s t hat br i ng abo ut chan ge and t o es t i mat e th e ir e ffe c ts q u a n tita tiv e ly ; th a t is, it is re s e a rc h o r ie n te d to wa rd c a u s a l in fe r e n c e . A cor e t heme i n caus al i nfe r ence i s t he i ss ue of conf oundi ng, a pr obl em t hat i s par t i cul ar l y pre val ent i n sur veys a nd obse r vat i onal r es ear ch. W hi l e t her e i s ongoi ng debate over how t o def i ne con f ou nd er s ( V and er W ee l e & S hpi t se r , 2013 ) , he r e w e def i n e a confoun der as “a vari ab l e t h at , whe n i nfe r r i ng an ef f ect fr om an i ndependent ( ex pl an at o r y) vari ab l e X t o a depe nde nt (obj ec t i ve) var i a bl e Y , aff ect s Y an d al so af fect s X ( or i s cova r i ant and has covar i ance with X ).” T her e ar e t wo t ypes of conf ounder s: t hose whos e val ues c an cha nge acro ss tim e poi nt s (tim e - var yi ng) and t hose w hose val ues remai n the sam e (t i me - i nvar i ant ) . I n th e s ta tistic a l a n a ly s is o f lo n g itu d in a l s t udi es, ef f ect s expl ai ne d by pas t mea sur e ment dat a (l agged var i abl es ) and u n it ef fect s (o r i ndi vi dual effe c ts) as la te n t v a r ia b le s a r e in tro d u c e d in to th e s ta tis tic a l m o d e l. By cont r ol l i ng for conf ounder s a nd, mor e br oadl y , unobs er ved het er ogenei t y , s uch pr ocedur es a llo w cl oser exami n at i ons of causal rel at i on s bet ween v ar i abl es. A common t ype of expl anat ory r es ear ch i n devel opment al psyc hol ogy r es ear ch i s i nfer en ces of reci proca l r el at i o ns, w hi ch are re la tio n s b etwe e n two o r m o re tim e - v a ryi ng vari ables . F or exam pl e, w e mi ght hyp ot h es i ze t hat i mpr oved sl eep habi t s af fe ct ment al heal t h, or conver se l y t hat i mpr oved ment al heal t h af f ect s s l eep habi t s, or t hat bot h re la tio n s e x is t. Th e c r o s s - la g g e d p a n e l m o d e l ( CLPM) is a s ta tis tic a l m o d e l th a t h a s been wi del y us ed si nce t he 199 0s t o make such i nf er ences , par t i cul ar l y i n psychol ogy . T he C L P M i s posi t i oned as a r egr ess i on model th a t cont rol s fo r l agged vari ables (a c ro ss- l agged r egr ess i on model ) t o addr es s t he p o te n tia l i ss ue of conf oundi ng. O ne r eas on f or t he i ncr eas ed use of t he C L P M i s c r i t i ci sms (e .g. , R ogosa, 1980) of t he us e of cor re l at i ons ( cr oss - l agged cor r el at i ons; e.g. , us i ng th e s ymbol s de sc r i bed bel ow , co r r el at i on bet ween X 1 and Y 2 o r X 2 and Y 1 ) o r th eir d iffere n c es to in fe r r eci procal rel at i on s bet ween t wo vari ab l es t hat wer e measured separ at el y in t i me. In addi t i on, t he subs equ en t r api d dev el op men t of s t r uct ur al equ at i on mode l i ng (S E M ) and s o ftw a re f o r its a p p lic a tio n a ls o c o n tribu te d to th e w i des pr ea d use of th e CLP M. In d e ed , 4 a l i t er at ure r evi ew of 270 ar t i cl es publ i she d si nce 2009 i n i nt er nat i on al medi c al j ourn a l s , i ncl udi ng psyc h ol og y j ou r nal s, i ndi cat ed t hat mor e t han 90% of st at i st i cal model s us ed to in fe r r e c ip r o c a l re la tio n s u tiliz e d th e C L P M ( Us am i , T odo, & M ur ayam a, 2019). T hus, i t can be sa i d t hat t he C L P M has es t abl i she d a f i r ml y ent r enche d posi t i on i n expl anat or y re se ar ch i n psychol ogy . M eanw hi l e, coi nci di ng r ough l y wi t h i ncr eas ed us e of t he C L P M , t he l at ent grow t h mode l ( L GM ) or l at ent cur ve mode l (L CM ) ( M er edi t h & T i sa k, 1984, 1990) al so became po pular , and several st ati st i cal mod el s comb i n i ng t h e L CM and C L P M wer e pr opos ed i n t he 2000s . However , rel at i vel y few user s hav e been awar e of opt i ons beyond t he C L P M . I n addi t i on, t hese model s wer e not pr opose d wi t h t he i nt ent of d ir e c tly avoi di ng us e of t he C L P M . In feren ce o f wi t h i n - pers on rel at i on s a nd the RI- CLPM However , rel at i vely recent l y , t he CL P M came un der s ub st ant i al cri t i ci sm, mar k i n g an i mpor t ant t urni ng poi nt i n bot h met h od ol og i cal and appl i ed r ese ar ch. T he wor k t hat t r i gger ed t hi s s hi f t was Ham aker et al . ( 2015). T he main i ssue add r es sed i n t h at paper i s w ith in - per son r el at i ons 1) , wh ic h a r e th e s u b je c t o f th is sp e c ia l iss u e p a p e r . With in - per son r el at i ons a r e t hose f oun d i n t he pr oce s s of i nt r ape r s on al chan ge , s uch as “w hen a per son s l eeps l onger , t hat per son bec omes mor e me nt al l y heal t hy .” W i t hi n - per son r el at i ons ar e of t en cont ra st ed wi t h gr oup - l evel or bet ween - per s on re l at i ons, and t hes e ar e n ot the same. Indeed, th e y can be e xact opposi t es i n ext re me s i t uat i ons, suc h as hi gh - in te n s ity e x e r c is e c a u sing h e art a tta c k s in s o m e p e o p le (a w ith in - per s on r el at i on) , despi t e rout i ne ex er ci se l o w eri ng t h e r i sk of hear t at t acks i n mo st peo pl e ( a b et ween - per son r el at i on) ( C ur r an & B auer , 201 1). I n g e n e ra l, w ith in - per son r el at i ons ar e not nece ss ar i l y equi va l ent t o caus al i t y , be caus e t hey do not al wa ys a dequat el y cont rol for pos si bl e conf oun de r ef fe ct s, but wi t hi n - per s on re l at i ons ar e nonet hel es s c onsi der ed t he c o re of c aus al i nf er ence ba se d on l ongi t ud i nal dat a . Hamak er her s el f l ong ar gu ed for t h e i mpor t ance of i nfe r ence of wi t hi n - per son r el at i ons , as i n Ha make r ( 2012) and ot her wo r k s. T he main cri t i ci sm of Hamaker et al . ( 20 15) i s t hat b et ween - v a r ia b le re la tio n s th a t t he CL P M can i nfer are a mi xt u r e of bet ween - a n d w ith in - p er so n r el at i ons , and t hus do n o t p u r e ly re fle c t w ith in - per son re l at i ons. As o n e s ta tistic a l m o d e l f o r e s tim a tin g with i n - per son r el at i ons, t hey pr es ent t he r andom - in te rc e p t CLPM (RI - C LPM ), in wh ic h s ta b le t r ai t fact o r i s added to t h e C L P M as a random - in te r c e p t th a t re p r e s e n ts s ta b le in d iv id u a l di ff erences. T hat pape r was publ i she d i n a s peci al “l ongi t udi nal t opi cs ” i ss ue of P s ychol ogi cal M et hods , one of t he l eadi ng i nt er na t i ona l j our nal s i n psyc homet ri cs , and us e of t he R I - C LPM expande d expl osi vel y , pa r t i cul ar l y fro m E ur ope , w here Hamaker and aut hors wer e based. I n f act , at t he t i me of wr i t i ng t hi s paper ( Augus t 202 2), t hei r paper had be e n 5 ci t ed mor e t han 1,600 t i mes , i ndi cat i ng t he magni t ude o f i t s i mpact. W e can ex pect appl i cat i ons of t he R I - C L P M t o i ncr eas e eve n f ur t he r i n t he f ut ur e, re pl aci ng use of t he C LPM. T he re ar e onl y mi nor mat h emat ical dif fer ences bet ween th e CL PM a n d th e RI- C LPM, but bot h mod el s ar e empi r i cal l y kno wn to oft en pr oduce s ig n if ic a n tly d iffe r ent re s u lts in te r m s o f d ire c tio n ( X → Y , Y → X , o r both) , ef fect si ze, and si gn ( posi t i ve or negat i ve ) in in fe r rin g re c ip ro c a l re la tio n s. T his pract ical aspect als o p l ay s a r o l e i n t he cr i t i que by H ama ker et al . ( 2015) st rong l y i mpac t in g p s y ch olo gy res earch ers (es p eci al l y t hose i nves t i gat i ng devel opment and per sonal i t y) . S ome at t empts hav e been mad e t o com par e t he r esul t s of dat a ana l ysi s a nd se condar y anal ysi s us i ng th e RI- C L P M and ot her met hods ( e.g . , Usa m i, T o d o , et al . 2019; Or t h, C l ar k, D on ne l l an, & R obi ns , 202 1) . T hen, a s hor t t i me af t er i t s popul ar i zat i on i n E urope , appl i cat i ons of t he R I - C LPM, w hi ch i s i nt ended t o i nf er re ci pr ocal re l at i ons a s pr oces se s of w ith in - per son chan ge, gr adual l y i ncr eas ed i n psychol ogy and r el at ed r ese ar ch f i el ds i n Ja pan. However , as descr i b ed abo ve, t here ar e alr ead y many st at i st i cal mod el s t hat can descr i be r eci p r ocal rel at i on s ( as a process of with in - per s on chang e) t hat have been pr opos ed i n psychol ogi cal re se ar ch and r el at ed f i el ds , and t hese model s ar e u s u a lly es t i mat ed t hrough S E M t o eval uat e t hei r goodn ess of fi t . T her e ar e t hus var i ous wa ys t o speci fy and cont r ol uno bse r ve d con f ou nd er s and fur t her m or e un ob ser v ed het er og en ei t y . In a d d itio n , w h ile w ith in - p er son rel ati on s an d mat ter s rel at ed t o t heir inf erence have been par t i cul ar l y act i ve i n psyc ho l ogy i n r ecent year s, wh en we l oo k at resear ch f i el ds such as econo mi cs, i nfer enc es abo u t reci p r o ca l and ot h er b id ire c tio n a l re la tio n s a re n o t necessar il y int en ded , but t here ar e st ati st i cal mo del s t hat ar e closel y rel at ed , such as the dynam i c pane l dat a mod el ( DP M ) ( C hi gi r a, H ayak aw a & Y amam ot o , 201 1 ; W ool dri d g e , 201 1; Hs i ao, 2014) . C onvent i onal d iff e re n c e s in a p p lie d s ta tistic a l m o d e ls a n d e s tim a tio n m e th o d s a r e t hus o f t en ob s erv ed i n d i ff erent res ea rch area s (e. g ., Hamak er & M ut hén, 2020) . T hat bei ng t he cas e, i s appl i cat i on of t he R I - CLPM a lwa y s ju s tifie d ? I f s o, w hy? As a r el at ed ques t i on, f r om t he per spe ct i ve of causal inf erence, w hat i s t h e s i gni f i canc e of cont rol l i ng f or s ta b le tra it fa c to r s as st ab l e indi vi du al di ff eren ces? T h es e ques t i ons w er e not neces sar i l y suf fi ci ent l y addr es se d in Hamaker et al . ( 20 15). W hi l e t he aut hor hi mse l f ha s, t o some ext ent , in v e s tig a te d t he com par i son a nd sel ect i on of s t at i st i cal mod el s used to i n f er r eci procal rel at i on s (U sami , M urayama, & H amake r , 2019; Us ami , 20 21 ) , in p a r t due t o th e m a n y ty p es o f sta tis tic a l m o d e ls , m a ny m a tte rs re la te d to th e se p o in ts a re s till b e in g d e b a te d , a n d psyc ho met ri cs res ear ch er s do not al ways shar e a c ommon vi ew . I n fa c t, c ritic ism s o f th e RI- C L P M have r ece nt l y be e n r ai sed ( L üdt ke & R obi t zsc h, 202 2 ) , and a l ongsi de t he di ver si t y of as sum ed dat a gener a tin g pr oce ss es an d act ua l r es ear ch hyp ot he se s, t he debat e over met h od ol ogi es for in f e rrin g re c ip ro c a l re la tio n s a s with in - per son r el at i ons c an be c onsi der ed t o have 6 becom e mor e di ver se and mor e comp l ex . T o t he aut hor ’ s kno wl edg e, t her e has been no l i t er at ur e r evi ew of suc h t opi cs , i ncl udi ng an over vi ew of t he R I - CLPM. S tr uc tu r e o f th is p a p e r T he re mai nder of t hi s pa per i s s t r uct ur ed as f ol l ows . W e fi rs t descri be t h e model r epr ese nt at i on of t he C L P M and t he R I - C L P M and s ome a nal ysi s exa mpl e s . N ext , f rom t he s t andpoi nt of pos i t i oni ng t he R I - C LP M as an ef fect i ve method for est i mati ng re c ip ro c a l re la tio n s a s w ith in - per son r el at i ons, w e di scus s and e xpl ai n some p r act i cal t opi cs r el at ed t o appl yi ng t he R I - CLPM ( number s of r equi r ed t i me poi nt s, in d ex fo r vari ance o f s ta b le tra it fa c to r s , mode l ext en si ons, m eas ure ment er r or as sum pt i ons, and i mpr oper sol ut i ons ). T hen, ba sed on r ece nt di sc uss i ons , i ncl udi ng t he aut hor ’ s ow n s t udi es (U sam i , M ur ayam a et al ., 2019; U sam i , 2021, 2023 ) , we summar i ze probl ems th a t c a n a ris e w h e n in fe r rin g with in - per son r el at i ons, al ong wi t h an over vi ew of ot her s ta tistic a l m o d e ls . W e nex t dis cuss t h e DP M as ano t h er maj or al t ernat ive t o t h e R I - C L P M , t oget her wi t h cons i der at i on s r egar di ng mode l se l ect i on . I n par t i cul ar , w e poi nt o u t th a t a d is tin c tiv e f e a tu re o f th e RI - C L P M , i n t erms of r epresent i n g bet ween - per son h e te r o g e n e ity , is its a s s u m p tio n th a t th e tr a it fa c to r is u n c o rre la te d w ith th e w ith in - per son r el at i ons, and t hat t hi s as sum pt i on al so c ons t i t ut es a n i mpor t ant poi nt of cont act l i nki ng t he R I - CLP M to th e DPM. F in a lly , w e su m m a riz e th is paper and di sc uss pr os pect s fo r fu tu re re se a rch . C L PM a nd R I - C L PM r e pr e s e nt a ti o ns a nd an al ysi s ex am p l es R epr es en t at i on of the C LPM I n t he f ol l owi ng, w e des cr i be ea ch mod el w ith a s s u m in g ty p ic a l s itu a tio n s w h e r e th e CLP M o r RI - CL P M ar e us ed. O ur m odel not at i on f ol l ows Us ami , M ur ayam a, et al . ( 2019) . S uppose t hat t wo c ont i nuous va ri abl es X and Y ar e s i mul t aneous l y and l ongi t udi nal l y meas ure d se ver al t i mes , and t hat we a r e i nt er es t ed i n i nf er r i ng a r eci procal rel at i on bet ween t hose vari ab l es. A s des cr i bed bel ow , t he number of t i me poi nt s T i s t ypi cal l y ar ound 2 t o 6, and onl y r ar el y 10 or m or e 2) . F or meas u r ement s   and   a t d is c re te tim e p o in t  (1     ) fo r i ndi vi dual  ( 1     ) , th e CLPM is express ed as   =   +     (  ) +     (   ) +    ,   =   +     (  ) +     (  ) +   , (1 ) fo r   2 . Her e,   and   are cr oss - l ag ged regr essi on i n t ercept ter ms, an d   and   are aut oregres si ve co ef fi ci ents t hat ex pr ess t he d egree t o whi ch t h e var i ables 7 (o rd e red a s   ,   ) i n t he pr ese nt (  ) can be e xpl ai ned f r om the sam e l ag ged v ar i abl es (o rd e red a s   (  ) ,   (  ) ) , whi ch are meas u r emen t s fr o m t h e past (   1 ). A ut or egr ess i ve coe ff i ci en t s i n t hi s equat i on ar e s ai d t o be of t he f i rs t order becau se w e ar e cons i der i ng re gr es si on t o t he l agged va r i abl e of t he pr evi ous t i me poi nt . S i mi l ar l y ,   and   a re first - ord e r c ro ss - l agg ed coeff i ci ents t hat repr esent t he d egree t o whi ch t he l agg ed vari ab l e (or d er ed as   (  ) ,   (  ) ) can e xpl ai n anot he r var i ab l e in th e pres ent (o rd e red a s   ,   ). Cro ss - l ag ged coeff i ci ents are cor e paramet ers for e s tim a tin g r e c ip ro c a l re la tio n s : t he l ar ge r t hei r abs ol ut e val ue, t he s t ronge r t he r el at i on bet w een t he cor r es pondi ng var i abl es. B y cont r ol l i ng for t he aut or egr ess i ve t er m, w hi ch i s pas t i nfor mat i on, t he C L P M a llo w s fo r in te rp re ta tio n s t hat go beyon d si mpl e cr oss - l agg ed corr el at i on s measured at d if fe re n t tim e p o in ts . F o r examp l e,   woul d be a quant i t y re pr es ent in g the i n cr ement of t he cur re nt   ex pect ed when a p a s t  measurement   (  ) is c o n tro lle d ( i.e . , as sum i ng a hypo t h et i cal p op ulat i o n who se v al u e s ar e exp ect ed t o be t he same) and t he p a s t  measuremen t   (  ) i ncr eas e s by one uni t (   (  )    (  ) + 1 ).    and   ar e res i du al ter ms of t h e cr oss - l agged r egr ess i o n, whi ch ar e us ua l l y as sum ed t o f ol l ow a b iv a ria te n o r m a l d is trib u tio n (w ith a zero mean vector ) , a n d th e tim e - var yi ng re si dual covari an ce i s est im at ed . The measur ement s (    ,   ) a t th e firs t tim e p o in t (  = 1 ) a re t r eat ed as exogenous var i abl es t hat do not r es ul t f rom t he ot her var i abl es , and t hei r mean and ( co ) vari an ce ar e assumed and est i mat ed. I n E q. ( 1), t he subs cr i pt  re fle c ts th e a ss u m p tio n o f tim e - var yi ng quant i t i es f or i nt ercept s , aut ore gr e ss i ve coef f i ci e nt s, cr os s - l agg ed co ef fi ci ent s, and o t her paramet ers . Th is re f le c ts th e fa c t th a t in ty p ic a l lo n g itu d in a l s tu d ie s a p p ly in g th e C LP M, it is n o t uncom mon f or i nt er val s bet wee n mea s ure m en t t i me poi nt s t o be on t he scal e of mont h s or year s , whi ch may l ead t o qual i t at i ve di ff er e nc es i n t he dynam i cs of change bet wee n measur ements , an d f urt h er t hat t h ose i n t erval s ar e no t al ways equ al l y spac ed . O f co u rs e , as sum i ng whet her tim e - va r yi n g o r tim e - in v a ria nt par amet ers w i l l depend on t he s t udy and be det er mi ned by t he ana l yst acco rdi n g t o fact o r s s u c h as t he nat ur e of t he vari ab l es bei ng ex ami n ed , t he mea sur eme nt per i od, t he t ype of r es ear ch hypot hesi s ( suc h a s w h e th e r th e m a in in te re s t is to e x a m in e te m p o ra l s ta b ility in dyna mi cs of c hanges ) , and th e e s tim a tio n - r el at ed re aso n s ( su ch a s a desi r e t o r ed uce t he n um b e r o f fr ee paramet ers to m a k e th e e s tim a tio n m o re s ta b le ) . Wh ile th e in f o rm a tio n c r ite rio n and ot her gener al - pur pos e met hods can b e a p p lie d to m o d e l se le c tio n , s in c e th e CLPM is ( as sum i ng a m u ltiv a ria te n o rm a l d is trib u tio n ) us ual l y es t i mat ed t hrough S E M , var i ous mode l fi t i ndices can als o b e r efer red t o (s ee, e.g . , T oyo t a ( 199 8) and K l i ne ( 2016 ) f or t he bas i cs) . F urt her mor e, i t i s hi ghl y wor t hwhi l e t o examine t he mag ni t udes o f each el emen t of t h e re s id u a l c o r re la tio n m a trix ( Kl i ne, 2016) . A s an a l t er nat i ve t o E q. ( 1) , t he C L P M can be e xpr es sed us i ng t he devi at i on fr om 8 th e gr oup mean at each t i me po i n t , as fo llo w s ( e.g., Hamaker et al ., 2015 ) :   = µ  +    ,   = µ  +   ,  (2  )    =     (  )  +     (  )  +     ,    =     (   )  +     (   )  +   ,  (2  ) w h er e µ  and µ  are t h e gr oup means at ti me  , a nd    and    ar e t he devi at i ons of i ndi vi dual  f rom t hose m eans . T he mean of t he devi at i on i s 0. In Eq . (2 b ), aut oregres si v e t erms , cross - l ag ged ter ms, and res i du a l s ar e specif ied f or t he deviat i on s , a s in Eq . (1) ; ho wever , ref l ect i ng t h e f act t hat t he d evi ati on s have a mean o f zer o , t he in te r c e p t te rm α i ncl uded i n E q . ( 1) i s not i ncl uded. I n ot her wor ds, t he di ff er ence bet w een E qs. (1) and ( 2) can be descri bed as t h e di f fer ence betw een ex pres si n g t he mean st ruct u r e of each vari able by th e in te r c e p t te rm α or t he gr oup mean µ, where µ can be expr ess ed as a f unct i on i ncl udi ng  (U s ami , 2021) . B y cont r as t , a ut ore gr es si ve coeff ici ents , cr oss - l ag ged co ef f i ci ents , and r esi du al (co)var i ance are t h e s ame, r egar dl ess of s uch di ff er ences i n expr es s i on. F i gure 1a s hows a pat h di agr am f or t he C L P M when us i ng t he r epr es ent at i on base d on Eq . (2 ). W hi l e t he mathemat i cal d i ff erences bet ween t hese repr esent at i on s a r e triv ia l, a s di sc uss ed bel ow th e y ar e u s e fu l in d is tin g u is h in g b e tw e e n e x is tin g s ta tistic a l m o d e ls in a concep t u al , mathemat i cal sense ( e.g., Us ami , M urayama et al ., 201 9). Re p re s e nta tio n o f the RI - CLPM T he cross - l agg ed coeff i ci ent γ in th e C LPM w a s a q u a n tity re f le c tin g re c ip r o c a l r el at i ons bet wee n var i abl es af t er cont r ol l i ng f or t he aut or egr ess i ve t er m ( e.g. , f or   , as sum i ng a hypot h et i cal pop ul at i o n wi t h sa me   (  ) v al u es) . T h at being t he case, can we consi d er t hese cross - lagg ed coeff i ci ent s as quanti t i es r ef l ect i ng r eci procal rel at i on s a s with in - per son r el at i ons? I t can be sa i d t hat t he i ncl usi on of aut or egr ess i ve t er ms al l ows t he model t o r epr ese nt re l at i ons t hat go beyond t he gr oup - le v e l re la tio n s ; how ever , t hes e s houl d be under st ood mer el y as cor re l at i ons w i t hi n a s ubpopul at i on def i ned by havi ng t he s ame val ue at t he pr evi ous t i me poi nt , and i n t hat sens e t hey cannot be r egar ded as quan t i t i es r epr ese nt i ng wi t hi n - per s on r el at i ons. I n ot he r w ords , i n mos t cas es , i t i s unl i kel y t hat unobse r ved het er ogenei t y i s a dequat el y cont rol l ed. H amake r et al . ( 2015) cr i t i ci zed t he C L P M as an i nappr opri at e met h od for i nf er ri ng w ith in - per son r el at i ons, ar gui ng t hat al t hough i t i s ne ces sar y t o cont rol for s ta b le tra it f act ors re pr es ent i ng st abl e i ndi vi dual di ff er ences i n orde r t o dra w i nfe r ence s a bout w ith in - per son r el at i ons, t he C L P M does not t ake s uch c omponent s i nt o acc ount , and t hus pr opos ed th e RI - C LPM a s a n a lte r n a tiv e s ta tis tic a l m o d e l. In th e RI - CL P M , t h e meas u r ement s   ,   a r e firs t s p lit in a m a n n e r sim ila r to Eq . 9 (2 a ), a s   = µ  +   +    ,   = µ  +   +    . (3  ) A s in th e C LPM , µ  and µ  are t h e gr oup means at ti me  . Also,   and   a re s ta b le t r ai t fact ors r ep r esent in g s t abl e i ndi vi dual di ff er ences for i ndi vi dual  , a nd t hei r ( co ) vari ances are as sumed an d e s tim a te d . Th is fa c t o r covari an ce r efl ect s a bet ween - per son r el at i on th a t is sta b le o v e r tim e . Assu me tha t    and    ar e devi at i on s f or a n i ndi vi d ual  an d are uncorr el at ed wi t h th e s ta b le tra it f a c to rs . Th e gr oup mean s of t he s ta b le tr a it fa c to r and t he devi at i on ar e bot h 0 , an d the expect ed s cores f o r   and   (c ondi t i oned on t he s ta b le tr a it fa c to r ) become µ  +   and µ  +   . B y cont r ol l i ng for t he s ta b le tra it f a c to r a s a tim e - i nvar i ant st abl e i ndi vi dua l di f f er enc e,    and    in th e RI - C L P M can be i nt er pre t ed not as devi at i ons f rom t he gr oup mea n lik e i n t he C L P M , but as de vi at i ons f rom th e expected scor es f or ea ch i ndi vi dual , or a s q u a n titie s re p re s e n tin g w ith in - per son v a r ia b ility th a t a re u n c o r re la te d with s ta b le in d iv id u a l d if fe r e n c e s . Th e la tte r in te r p re ta tio n im p lie s th a t Eq . ( 3a) or t hogonal l y decom pose s t he var i ance of t he observed measurement s i n t o bet ween - per son var i ance, r epr ese nt ed by t he s ta b le tra it fa c to r , a n d w ith in - per son va r i a nc e, r epr e sen t ed by w i t hi n - p er s on va ria b ility .    and    can t hen be repr esent ed as    =     (  )  +     (  )  +     ,    =     (   )  +     (   )  +    . ( 3  ) Th is is fo r m a lly s im ila r to Eq . (2 b ), b u t u n lik e th e CLPM it re p re s e n ts a r e g re s s io n m o d e l fo r with in - p e rs o n v a ria bility . T h er ef o r e, for examp l e, t he cross - la g ge d c oef fic ie nt   can be i nt erpr et ed as t he ex pec t ed i ncrea se i n t h e curr ent    associ at ed wi t h a one - u n it in c re a s e in th a t in d iv id u a l’ s   (   )  , c ondi t i on al on t he i ndi vi d ua l havi ng a pr evi ous w ith in - per son de vi at i on   (  )  and a gi ven sta b le tra it fa c to r s c o r e   . F i gur e 1b s hows a pat h di agr am f or t he R I - C LPM. S ince t he o nl y d i ff erence bet ween t he RI - CLPM a n d th e CLPM is th e RI- CL PM ’ s i ncl usi on of t he s ta b le tr a it f act or , i f t he (co)var i ance o f t he s ta b le tr a it fa c to r is 0 (i. e . , if   =   = 0 ), t hey ar e m a th e m a tic a lly e q u iv a le n t. I n th e R I - CLPM, a s in th e CLPM, th e w ith in - per son v a r ia b ility a t th e fir st tim e p o in t (    ,    ) i s t re at ed as an exogenous var i abl e, and t he mean and ( co)vari ance are ass umed and e s tim a t ed. Not e t hat as i n t h e case of th e C L P M , t he ch oi ce o f whet her t o ass ume tim e - var yi ng o r tim e - in v a r ia n t q u a n titie s fo r paramet ers such as gr oup mean s , aut oregres si v e coeff i ci en t s, and c ro ss - l agged coef f i ci ent s ( and whe t he r t o ass ume equi va le nc e onl y at cer t ai n tim e p o in ts ) is a m a tte r 10 f or anal yst s to deci de fro m t he nat ur e of t he var i abl e s be i ng ex ami n ed , t h e measu reme n t per i od, a nd t he t ype of r es ear ch hyp ot h es i s. W hen model sel ect i on i s req u ired , one can u se pr ocedur e s su c h a s m o del f it in d ic e s , in f o rm a tion c rite r i a , an d re s id u a l c o rr e la tio n s. C ompar i son of e s tim a tio n r e s ul ts a nd a na ly sis e xampl es If th e d a ta gener at i ng proc es s c or r e s po nd s t o th e RI - C L P M and th e s ta b le tr a it fa c tor has nonzer o vari an ce , t hen usi ng t he C L P M t o anal yze t he dat a mi ght pr oduce d r a m a tic a lly d iffe re n t re su lts fo r th e c ro ss - l agged c oef fi ci ent s and ot her in f e re n tia l re s u lt s f rom wha t woul d r es ul t f r om us e of th e RI - CL PM. In part icul ar , t h e l arger t he ( co ) vari ance o f t he s ta b le tr a it fa c to rs , th e m o r e lik e ly it is th a t th e s ta tistic a l s i gni f i canc e, magni t ude, a nd si gn of t he cr oss - la g g e d c o e ff ic ie n t e s tim a te s will d iffe r acr oss model s, and t he R I - C LP M w ill b e ju d g e d to b e a b e tte r f it. T h er e are m any r epor t s of e m p iric a l e s tim a tio n re s u lts , in c lu d in g H a m a k e r e t a l. ( 2 0 1 5 ) , U s a m i, M ura yama e t al . ( 2019), Us ami , T odo e t al . ( 2019) , and O rt h et al . ( 2021) . Mo r e speci fi call y , t ho se st ud i es sho wed t hat autor egress i ve co ef fi ci ent s est i mat ed b y t he RI - C L P M t end t o be s mal l er and t hat s t andar d er ror s f or aut or egr ess i ve coe ff i ci ent s and c ro ss- l ag ged coeff i ci ents t end t o be l ar ger , r efl ect i ng t h e f act t hat t he RI - C LPM c o n sid e rs w ith in - per son re la tio n s a f t er cont rol l i ng f or sta b le tr a it fa c to r s (Muld e r & H amake r , 2020) . G ener al l y spe aki ng, di f f er ent est i mat i on r es ul t s c an obvi ousl y be obt ai ned by appl yi ng m a th e m a tic a lly d iffe r e n t s ta tis tic a l m o d e ls . Ho wev er , s ta b le tra it f a c to rs th a t r efl ect st ab l e indi vi d ual d i ff erences ( e.g.,   ) co nt ri b ut e t o each el emen t of t h e vari ance – co var i a nce st ruc t ure of t he c orr es po nd i ng var i abl e (  ) by an equa l ma gni t ude ( speci fi cal l y , t h e vari ance  (   ) of t he s ta b le tra it f a c to r) re fle c tin g th e la rg e im p a c t o n th e e s tim a tio n re s u lts o f th e ful l set of paramet ers . A s pr epar at i on f or t he di scus si on of model sel ect i on pre se nt ed l at er , we pr ovi de a bri ef empi ri cal exampl e aimed at co mpari ng es t i mat i on r esul t s . T abl e 1 s hows t he r es ul t s of appl yi ng t he C L P M , t he R I - C LP M, a n d o th e r sta tis tic a l m o d e ls d e s c rib e d bel ow to l ongi t udi nal dat a (  = 6,  = 4 , 671 ) col l ect ed fro m th e Min n e s o ta A d ole s c e nt C omm uni t y C ohort ( M AC C ) on a dol esc ent s ’ pe r cei v ed de gr ee of e xposur e t o sm oki ng t hr ough movi es (  ) and ac t ual smoki ng i nt ensi t y (  ). W e us ed ma xi mum l i kel i hood e s tim a tio n bas ed on S E M ( ML - S E M ) wi t h t he “l avaan” packa ge i n R ( R oss eel , 2012) fo r th e p ar amet er est i mat i on. T he ana l ys i s c ode ca n be obt ai ned f rom t he aut hor ’ s w e b s ite (h ttp ://u s a m i - l ab.c om/ Us ami _2022_j sdp_code .docx) . N o te th a t Us a m i, M ura yama e t al . ( 2019) a l s o compa re d t he r esul t s of var i ous s t at i s tic a l m o de ls u s in g th e same dat a, b ut due t o t hei r r el at i on t o t he f ol l owi ng di scus si on, w e focus ed onl y on cer t ai n model s , such as th e C L P M and th e RI- C LPM. Her e we pres ent t he resul t s of e s tim a te d au t o r egress i ve co ef f i ci ents , cr o s s - l ag ged co ef fi ci ent s, and res i du al ( co ) vari ance u nder tim e - var yi ng o r tim e - i nvar i ant as sum pt i ons. However , si nce 11 i mpr oper sol ut i ons oc c ur r ed und er t he t i me - v a r yi n g condi t i ons of t he R I - CL PM, in th is condi t i on onl y t h e r esi du al (co)var i ances ar e t reat ed as t i me - i nvar i ant for t he sa ke of comparat ive conv en i en c e. We e x p la in how t o deal wi t h i mpr oper sol ut i ons be l ow . S ee C hoi , F or st er , E r i ckson, L azovi ch, and S out hwe l l ( 2012) and Us ami , M ur ayam a et al . ( 2019) f or f ur t her det ai l s r el at ed t o t he dat a. Th e e s tim a tio n re s u lts in d ic a te th a t, re g a rd le ss o f w h e th e r tim e - var yi ng or t i me - i nvar i ant as sum pt i ons a r e i mpos ed, t he es t i mat e s , s ta tis tic a l sig n ific a n c e , a n d sig n s o f t he cross - l agg ed co e f fi ci en t s di ff er bet ween t he CL P M an d t he RI - C LPM . Es tim a te s o f aut oregres si v e coeff i ci en t s wer e smal l er in th e R I - C L P M , and s t andar d er r or s f or t he est i mat es of t hese coeff i cient s wer e l arger in th e RI - CL PM . I n te rm s o f m o d e l fit, th e RI- CLPM s h o w e d b e tte r f it . Th e re s u lts o b ta in e d fro m a p p ly in g o th e r sta tis tic a l m o d e ls w ill b e d is c u s s e d la te r . So me p ract i cal i ssu es con cern i n g t h e a p p l i ca t ion of t h e RI - CLPM The n umb e r o f tim e p o ints re q uire d B ot h t he C L P M and t he R I - CLP M are usua l l y e s tim a te d w ith in a n SEM fra m ew o rk , me ani ng t he obj ect i ve f unct i on i s s et and t he sol ut i on i s obt ai ned ba se d on maki ng t he mean st ruct ure and v ar i an ce – co va ri anc e s t ruc t ure of t he m odel cl ose t o t he sampl e mean (v e cto r) a n d th e sampl e v ar ian ce – co vari a nce matr ix of t he o bser v ed d at a. B ol l en ( 19 89) and T oyoda ( 1998) e x plain t he fun dam ent al s of S E M opt i mi zat i on, a nd T oyoda ( 1992, 2012) pr ovi de s d e ta ils a b o u t e s tim a tio n iss u e s . A s a neces sar y cond i t i on for mod el i dent i f i ca t i on, i n ot her wor ds, t o obt ai n a uni que s o lu tio n , th e CLPM re q u ire s lo n g itu d in a l d a ta w ith a t le a s t  = 2 . B y cont ra st , t he R I - C LPM assu me s a s ta b le tra it f a c to r and e s tim a te s i t s var i ance an d co vari ance, res ult i ng i n t hree more par amet er s. Re f le c tin g th is, th e RI - C L P M re qui r es l ongi t udi nal dat a with a t le a s t  = 3 . T hese di ff er en ces ari se r egard les s of whet h er o ne assumes ti me - var yi ng o r tim e - in v a r ia n t au t o r egress i ve and c ro ss- l ag ged coeff i ci ents . A ccor di ng t o Us ami , T odo, e t al . ( 2019) , w ho r evi ew e d 270 s t udi es i n i nt er nat i onal j our nal s i n medi ci ne, ps ychol ogy , and r el at ed f i el ds t hat re por t ed i nfe r en tia l re s u lt s about r eci proc al r el at i ons, 106 st udi es ( 39% ) us ed l ongi t udi nal dat a w i t h  = 2 , 89 ( 33% ) used  = 3 , 36 ( 13% ) us ed  = 4 , a nd 16 ( 6% ) us ed  = 5 . Onl y 24 (9% ) us ed lo n g itu d in a l d a ta w ith   6 . In H amake r et al . ( 2015), whi ch conduct ed a s i mi l ar r evi ew of ar t i cl es publ i she d i n psychol ogy i n 2012, i t was re por t ed t hat near l y hal f ( 45% ) of t he 1 15 st udi es a ppl yi ng t he C L P M use d l ongi t udi nal dat a wi t h  = 2 . Th is is cons i st ent wi t h t he r es ul t s of Us ami , T odo e t al . ( 2019) . A s not ed above , i n re spons e t o t he cr i t i ci sm by Ha make r et al . ( 2015), t her e has been gr owi ng moment um t owar d compa r at i ve exam i nat i ons of anal yt i c r es ul t s and secondary an a l yses t hat t ake the R I - C L P M i nt o consi der at i on; howe ver , th e R I - C LPM 12 cannot be id e n tifie d f ro m l ongi t udi nal dat a w ith  = 2 . T heref ore, t o mak e i n f er en ce a b o u t w ith in - per son r el at i ons bas ed on t he R I - CLPM p o s s ib le , it is firs t n e c e s s a ry to adopt a r ese ar ch des i gn t hat pre sup po se s t he col l ec t i on of l ong i t udi nal dat a w i t h   3 . M ore over , ot her t hi ngs be i ng equa l , i ncr eas i n g t he num be r of tim e p o in ts i s exp ect ed to yi el d mo r e st ab l e est i mat es of t h e model par ame t ers , esp eci al l y t he (co)var i ances of t he s ta b le tra it f a c to r s . As a general mat t er , ho wever , t h e magnit ud es o f autor eg res si v e and c ro ss- l agged c oe f f i ci en t s w i l l depend on t he l engt h s of i nt er val s bet wee n meas ure ment poi nt s (l ag; se e, e.g. , D orm ann & G r i ff i n, 2015) , s o t he l ag and num ber of t i me poi nt s s houl d be det er mi ned w h ile consi der in g act u al r esear ch h ypotheses and measur ement per i od s. I nd ex f or var i an ce of s ta b le tr a it f a c to r s A s de scr i bed above , i n E q. ( 3a) t he s ta b le tra it f a c to r and devi at i on ( wi t hi n - per son vari abil i t y) ar e no t co r r el at ed, so t h e var i ance of obs er vat i on at each t i me po i n t (e.g., vari ance  (   ) fo r v a riab le Y ) can be or t hogonal l y decom pose d as t he sum of t he vari ance o f t he s ta b le tr a it fa c to r ( betw een - pe rs on v ar ian ce:  (   ) ) an d t he v ar i ance o f th e w ith in - per son va r i abi l i t y ( with in - per son vari an ce:  (    ) ). W hen i nfe r r i ng wi t hi n - per son r el at i ons, t he magni t ude 3) , s ta tis tic a l sig n if ic a n c e , s ig n s , and s t a n d a rd error s (c onf i d enc e i nt er va l s ) fo r e stim a te d c r o ss - l agg ed co ef f i c ient s, as w el l as coeff ici ent s of det er mi nat i on in l agged r egr es si on s , a re o f p rim ary in tere st . Ho wev er , t he pr opor t i on of t he v ar i ance o f t he s ta b le tr a it fa c to r to th e vari ance o f obs er vat i on a t tim e t :    =  (   )  (   ) =  (   )  (   ) +  (    ) ( 4) can al so be a u sef u l in d ex . T he l arger t he v al u e of    , t h e great er t h e pr opor t i on o f bet ween - per son v ar i ance at each t i me po i n t , and w h e n th e tr a je c to rie s o f m u ltip le i ndi vi dual s ar e cons i de r ed , t he ext ent t o whi ch t hose t ra j ect or i e s ar e s epar a t ed f r om one anot her become s i ncr eas i ngl y pronounc ed. B y co nt ras t , t h e smal l er t he v al u e of    , t he l ar ger t he pr oport i on of w ith in - per son vari ance, so t he ex t en t t o whi ch eac h i ndi vi dual t r aj ect or y come s c l oser t oget her and over l aps becom es mor e pr onounced. T he magni t ude of    i s consi der e d t o depe nd on char act er i st i cs suc h as t he nat ur e of t he var i abl es bei ng ex ami n ed , t h e measur ement per i od , and even t h e meas urement m e th o d (e . g . , re lia b ility o f th e m e a s u reme nt). Al t hough t her e i s, of cour se, var i at i on i n degr ee, i n many ca se s i t i s na t ura l t o as sum e t hat , i n act ual l ongi t udi nal dat a, s ome i ndi vi dual s cons i st en t l y show r el at i vel y hi gh obser v at i ons , whe r eas ot her s consi st ent l y 13 s how r el at i vel y l ow val ues . T hi s s ugges t s t hat t he va ri an ce of t he s ta b le tr a it fa c to r as a s t abl e i ndi vi dual di ff er ence i s nonze r o. I ndeed, st udi es appl yi ng t he R I - CLPM o f te n sh ow v ar i ance e s tim a te s of a non - n e glig ib le m a gn itu de, a s illu s tra te d in T a b le 1 , a n d in such cases t he est i mat ed cross - l ag ge d coe ff i ci en t s and r el at ed resul ts are m o r e l i kel y to di f f er subs t ant i al l y bet we en t he C L P M and t he R I - CL PM . U si ng l ongi t udi nal dat a (  = 3 ; ages 10, 12 , a nd 14 ) on de p re ss i v e sym p t oms ( S hor t M ood and F eel i ngs Q ues t i onnai r e scor es ) , s l eep dur at i ons, bedt i mes , a nd body mas s in d ex (BMI) me a su re d i n t he T okyo T een C ohor t S t udy conduc t ed t o i nvest i gat e adol es cent ment al and phys i cal devel opment ( fo r th e pr ot ocol pap er , see Ando et al ., 2019) , U sam i ( 2023 ) e s tim a te d pr opor t i ons of var i an ce of s ta b le tr a it fa c to r s to th e vari ance o f each v ar i ab l e at t he fi rs t ti me po i n t (    ;  = 1 and age 10) . A s a re s u lt, th e est i mat ed v al u es obt ai ned were, i n o r der ,    = 0 . 245 , 0 . 545 , 0 . 482 , 0 . 748 . In ot her wo rd s, compa r ed wi t h t he ot her var i abl e s, dep r es si on exhi bi t ed re l at i vel y l ar ge wi t hi n - per son v a r ia b ility (w ith in - pers on vari ance), w hereas s l eep d urat i on and bed t ime as life s ty le - re l at ed var i abl es , and pa r t i cul ar l y B M I as an i ndi cat or of body compos i t i on, appea r ed t o be mor e s t rongl y i nf l uenced by s t abl e i ndi vi dual di f f er ences , w i t h    t endi ng t o be r el at i vel y l ar ge. S ome ext en si on s t o the RI- CLPM W hen t he ef f ect s of var i abl es ar e t hought t o per si st over a l onger per i od, one m ay cons i der spe ci f yi ng not onl y fi rs t - or der but al so s ec o nd - or hi gher - or der aut or egr ess i ve and cr oss - lagged ter ms (e.g., eff ect s of   (   )  o n    and    ). I f s uch ef fect s are i n deed pr es ent , i ncor por at i ng t hem m ay al so he l p addr es s c onf oundi ng, f or exam pl e by al l o wi ng t h e causal eff ect s of fi rs t - order l agg ed vari ab l es t o b e es t i mat ed mo r e accurat el y , and may i mp r ov e model fi t . O ne pr ocedur e to r e p re s e n t a m o d el th a t in c lu d e s a tim e - var yi ng conf ounder (   ) is t o as sum e l i near r el at i ons am ong t he var i abl es and t o ext end E qs . ( 3a) and ( 3b) i n a s t r ai ght for wa r d manne r t o t he t hr ee - var i abl e cas e i nvol vi ng  ,  , and  . W hen t he vari able  i s i nt r oduced as a medi ator i n such a model , th e medi at i on ef f ect can be e s tim a te d i f var i ous m odel as sumpt i ons hol d (e .g. , as sumpt i on of l i near i t y among var i abl es and appr o pri at e cont r ol of t he conf ounder s ) , f or exam pl e, t hrough t he se r i es            (ba se d on t he wi t hi n - per son r el at i ons eva l uat ed by cont rol l i ng f or t he s ta b le tra it f a c to r of each va ri abl e, r at her t han t he gr oup - le v e l r e la tio n th a t is o fte n exami ned i n pract i ce). A s ye t anot her ext ensi on of t he R I - C LPM, Mu ld e r and H ama ker (2020) i l l ust r at e and expl ai n model r ep r esent at i o ns f or ( a) t he case wher e t h er e ar e t i me - in v a ria n t observed vari ab les , ( b) t h e case of ext en si o n to a mul t i - group mode l , and ( c) t he cas e of d e a lin g w ith m u ltip le in d ic a to r s ( f or exam pl e, whe n a cons t r uct such a s de pr es si on i s me asur ed l ongi t udi nal l y usi ng ps ychol ogi cal sca l e one may specif y a measur ement 14 mode l f or eac h i ndi vi dual i t em [ in d ic a to r] r at her t han r el yi ng on a su m s c o re). A s su mpt i on of c orr el at i on bet w een s ta b le tr a it f a c to r s a nd within - p e r so n va r ia b ility Th e R I - C L P M ass umes t hat t here i s no co r rel at i on bet we en the sta b le tra it f a c to r as a st able bet ween - per son di ff er en ce and w i t hi n - p e rs o n v a r ia b ility . Th is is b e c a u s e th e r e i s no r eas on t o ass ume t hat wi t hi n - pe r s on var i abi l i t y , whi ch i s pos i t i oned as a t empor al devi at i on fr om ea ch i ndi vi d ual ’ s expec t ed sc ore ( wh ic h is a func t i on of t he s ta b le tra it f act or ), is c o rr e la te d w ith th e sta b le tra it f a c to r ( Usami , M urayama et al ., 2019 ) . Never t heles s, as F ig u re . 1c s how s, i t i s pr ocedur al l y poss i bl e t o as sum e a c or re la tio n , a n d sp e c ific a lly fo r d a ta with  = 4 o r mo re , suc h mod el can b e i d ent if ied regar d l ess o f w h e th e r tim e - in v a r ia n t coef fi ci ents and r esi d ual (co)var i ances are ass umed . U nder t hi s s et t i ng, how ever , th e e x te nt to wh ic h th e sta b le tra it fa c to r c o n trib u te s to t he vari an ce of t he obser vat i o ns v a r ie s a c r o s s tim e p o ints . C onse qu ent l y , t he conce pt ual me ani ng of t he s ta b le tr ait fact or as repr esent i ng st able i nd i vidual dif fer ences al so chang es . As a rel at ed matt er , t he fact that the var i ance o f t he obser vat i on is n o t or t hogonal l y par t i t i oned i nt o t he var i ance of t he s ta b le tr a it fa c to r an d t h e vari ance o f th e w ith in - per son va r i abi l i t y make s i t l es s m eani ngful t o r epor t t he i ndex    def i ned above. T he eme r gence of suc h di f f i cul t i es a l so s ugges t s t hat , i n l i ght of t he model ’ s o r ig in a l in te n t, a s p e c ific a tio n th a t a s su m e s a c o rr e la tio n b e tw e e n th e s ta b le tra it fa c to r a n d with in - per son v a r ia b ility ca n no l onger , s t ri ct l y spe aki ng, be r egar ded as an R I - C LPM. E ven s o, s uch a s peci f i cat i on const i t ut es an i mpor t ant poi nt of cont act t hat es t abl i she s a mat hemat i cal con nec t i on wi t h t he dynam i c pane l model ( DP M ; F i gur e 1d) di sc uss ed l at er (A nder se n, 202 2 ). A ssu mp t ion s o f m e asurement e rror s an d i mpr oper sol u t i on s A mong t he st at i s t i cal model s pr oposed pr i or t o Ham aker et al . ( 2015), one t hat i s mat h emat ical l y v er y cl o s e t o the R I - C LP M is th e s ta b le tra it, a u to re g re s s iv e tra it, a n d s t at e mode l (S T A R T S ; Ke nny & Z aut r a, 1995, 2001), whi ch wa s pr opose d i n per sonal i t y r e sear ch 4) . I n t he S T A R T S model , t he obse r vat i ons (   ,   ) ar e par t i t i oned i nt o t rue sc or es (    ,   ) a n d m ea su re m e n t erro rs (    ,   ) t hat ar e assumed to be un cor rel at ed wi t h t hose t rue scor es as   =    +    ,   =   +   . (5  ) B ecause t he measurem ent err ors hav e mea n zero , t he mean o f t he obser vat i ons is e q u a l t o t hat of t he t rue sc or es . The measur ement err o r s a t e a c h tim e p o in t are us ual l y as sum ed t o f ol l ow a bi var i at e nor mal di st ri but i on, and t he meas u r ement erro r ( co ) vari ance s are es t i mat e d. H o w e v e r , d u e to id e n tific a tio n c o n s tra in ts , tim e - in v a ria n t 15 measur ement er ror (co) vari an ces ar e u s u al l y assumed. T hen, a s i n t he R I - CLPM , w e d iv id e th e s e tru e sco res a t e a c h tim e p o in t in to th e gr oup mea n, sta b le tra it fa c to r , a nd devi at i on ( wi t hi n - per son va r i abi l i t y) , and s peci fy a r egr ess i on equat i on f or t he devi at i on as    = µ  +   +     ,   = µ  +   +    , ( 5  )     =     (  )  +     (  )  +     ,    =     (  )  +     (  )  +    . (5  ) A l t hough t her e a re d iffere n c es a c co r di n g t o whe t her t he equa t i ons ar e f or obs er ved or t r ue val ues , t he s p e c ific a tio n s for E qs. ( 5b ) and ( 5c) ar e basi call y the same as t ho s e fo r th e RI - C LPM ( E qs. ( 3a) and ( 3b) ). N o te th a t th e ST A R T S model oft en i mpose s no nl i ne ar cons t ra i nt s on t he paramet ers s o t hat , f or exam pl e, t he var i ance of each c omponent of t he s ta b le tra it f a c to r , th e w ith in - per son va r i abi l i t y ( cor re spondi ng t o t he “aut ore gr es si ve tra it ” in th e S TA R T S mode l ) , a nd t he m eas ur eme nt er r or ( si mi l ar l y cor r es pondi ng t o t he “s t at e”) sat i sf y s ta tio n a r ity 5) in th e s e n s e th a t th e y a re tim e - in v a ria n t ( Don n el l an , Kenny , T rzes ni ewski , L ucas , & C onger , 2012 ). Su c h c o n s train ts ar e no t usua l l y i mpose d i n t he C L P M or R I - C LPM. I n ( developmenta l) p s y ch olo gy resear ch , res ea rche rs fr eq uentl y att empt to measur e l at en t co nst ruct s t h at cann ot be d i rect l y obser ved ; h owever , becau se meas urement re lia b ility is o f te n im p e rf e c t, p rim a rily d u e to th e m e a s u re m e n t m e th o d s th e m s e lv e s , it is n o t uncom mon f or t he r es ul t i ng dat a t o cont ai n meas ure ment er ror . T hi s w oul d i nt r oduce bi as i n t he par amet er es t i mat es o f th e RI - C L P M , whi ch does not di r ect l y accoun t for meas urement er ror ( e sp e c ia lly a t th e in itia l tim e poi nt ). A l t hough t he t wo m odel s di ff er i n t hei r c oncept ual i nt er pr et at i ons , i t i s s omew hat i r oni c t hat , w her eas t he S T A R T S model wa s pr opose d bef ore t he R I - C L P M and pr ovi des a mor e gener al f or mul at i on, i t i s t he R I - C L P M t hat has r api dl y gai ned popul ar i t y i n re cent years . H o w e v e r , it is e m p ir ic a lly k n o w n th a t in c lu d i n g m e as u r emen t e rro r as in th e ST A R TS model com es at t he cos t of a t endenc y t o produc e i mpr oper s ol ut i ons , s uch as n egat i ve est i mat es for t he v ar i ances of t h e sta b le trait fac to rs o r fo r t he res i du al an d err o r vari an ce s, as wel l as t he fai l ure of t he v ar i ance – cov ari an ce mat r i x of t he s ta b le tr a it fa c to rs to b e p o s itiv e d e fin ite ( e.g., Hamaker et al ., 201 5; Us ami , M ura yama e t al ., 2019; Us ami , T odo et al ., 2019; Or t h et al ., 2021) . T o addr es s t hi s i ss ue, se ver al appr oac he s have been sugg es t ed , i ncl ud i ng i ncr eas i n g t he num ber of tim e poi nt s and usi ng mul t i pl e i ndi cat or s ( C ol e, M ar t i n, & S t ei ger , 2005; L uhmann, S chi mm ack, & E i d, 201 1), as w el l as conduct i ng B ayes i an es t i mat i on wi t h pr i or d is trib u tio n s sp e c ifi ed f or t he par am et er val u es ( L üdt ke , R obi t zsc h, & W agner , 2018) . I t 16 i s al so em pi ri cal l y known t hat i mpr oper sol ut i ons may a ris e in th e RI - C LPM , a s in th e anal ysi s exampl e above; howeve r , t hey occur mor e f re quent l y i n t he S T A R T S model ( e. g., Us ami , T odo et al ., 2019) . I n addi t i on, f r om t he s t andpoi nt of model i dent i f i cat i on, t he S T AR T S model i s s ubj ect t o s omew hat st ronge r cons t ra i nt s t han t he R I - CL P M . B ecau s e t he S T AR T S model i s a st at i st i cal mod el wi t h a lar g er nu mber of paramet ers and a mor e co mpl ex s t r uct ur e, l ongi t u d in a l d a ta with a t le a s t  = 4 waves ar e requi red f or i d ent i fi cat i on. T o ef fect i vely avoi d i mpr op er s ol u t i ons a n d to o bt ai n mo r e st abl e param et er es t i mat e s, dat a w i t h ar ound  = 10 wa ves may be r equ i re d ( Kenny & Z aut ra , 2001) . A s not ed above, how ever , st udi es t hat act ual l y col l ect l ongi t udi nal dat a on s uch a s cal e t o exami ne r eci procal rel at i on s ar e l i mi t ed ( Us ami , T odo et al ., 2019) . O th e r s ta tis tic a l m o d e ls a nd the ir r e la tio n to the R I - CLPM Ma n y o th e r s ta tis tic a l m o d e ls, in c lu d in g th e S TA RT S model , are avai labl e to exami n e r e c ip ro c a l re la tio n s a t th e w ith in - per son l evel , and s ome ha ve bee n pr opose d af t er Ha make r et al . ( 2015). W hi l e t hes e mode l s w er e pr opose d i n psychol ogy and re la te d fie ld s a n d e s tim a te d u s u a lly t hr ough S E M fra me wo rk , th e r e a re s till o th e r opt i ons if w e lo o k beyond psyc hol ogy t o ot her fi el ds such as econ omi cs . Th is fa c t means t h at ther e are var i ous proc edur e s t o model an d to cont ro l fo r unobse rve d conf ounder s , as wel l as unobser ved het er ogenei t y . Wh ile th e RI - C LPM s e e m s to p r o vi de mor e reas on abl e mod el repr esent at i on s to in f e r w ith in - per son r el at i ons , at l east when compared w ith th e CLPM, is a p p lic a tio n o f th e R I - CLPM a lwa y s ju s tifie d , d e sp ite th e a v a ila b ility o f so many ot her s ta tis tic a l mode l s? I f so, why? As a r el at ed i ss ue, w hat i s t he si gni f i cance of cont rol l i ng f or s ta b le tra it f a c to r s as s t able i nd i v i d ual di ff erences i n t erms of causal inf erence? The se issues do not appear t o have bee n di scus sed i n suf fi ci ent det ai l i n Ham aker et al . ( 2015) . I n my ow n wor k ( Us ami , M ur ayam a et al ., 2019; Us ami , 2021) , I pr es ent ed a uni f i ed f ra mew ork f or compa r i ng seve r al st at i st i cal model s and or gani zi ng t hei r conce pt ual and ma t hemat i cal re l at i onshi ps, and on t hat basi s exa mi ned di f f er ences i n th e in te rp r e ta tio n of cr oss - l agged c oef f i ci ent s, posi t i oni ng t he R I - C L P M as one use f ul appr oach f or i nfe r r i ng wi t hi n - p e rs o n re la tio ns. How ever , due i n par t t o t he many t ypes o f av ai labl e sta tis tic a l m o d e ls a n d th e c o m p le x m a th e m a tic a l re la tio n s a m o n g th e m , di sc uss i on re mai ns ongoi ng r egar di ng model se l ect i on and es t i mat i on met hodol ogy especi all y f rom t h e pers pecti ve o f causal inf erence. Th e issue o f in ferrin g re cip ro c al re la tio n s a s w ith in - per son r el at i ons i s t hus bec omi n g i ncr ea si ng l y com pl ex and di ver s e. E ven whe n t he di scus si on i s l i mi t ed t o compa r i sons bet we en t he C L P M and t he R I - C L P M , t here ar e cu r rent ly contr ast i n g view s: some argue f or t he us ef ul nes s of t he C LPM (th a t in c lu d es second - or hi gher - orde r l ags ) ( L üdt ke & R obi t zs ch, 202 2), 17 wher eas ot hers, l i ke Hamaker et al. (2015 ) , have o nce again pr ovi ded a c ritic a l r eap pr ai sal o f t he CL P M (L ucas, 20 2 3 ). A not her i mpor t ant f act or t hat compl i cat es t he di scus si on i s t he di f f i cul t y of det er mi ni ng whe t her co mmo n fa ct or s i ncl ude d in s ta tistic a l m o d e ls , su c h a s s ta b le tra it f act or s , ade quat el y cont ro l f or unobs er ved conf ounder s (a nd fur t her mor e, unobs er ved het erogeneit y), despi te t he act u al data ge ner at i ng pr oces s a nd t he t rue model bei ng gener al l y unknown t o t he r es ear cher . Th is m a k e s it d iff ic u lt to d ra w a cl ear - c u t concl usi on re gar di ng t he be st s e le c tio n of mode l f or i nfe r r i ng wi t hi n - per son r el at i ons and how cr oss - l agged coe f f i ci ent s shoul d be i nt er pr et ed. I n t hi s sec t i on, f r om t he s t andpoi nt t hat r egar ds t he R I - C L P M as one use f ul appr oach f or i nfe r r i ng wi t hi n - per son r el at i ons, and dr awi ng on re cent di sc uss i ons i ncl udi ng t he aut hor ’ s ow n wor k (U sam i , M ur ayam a et al ., 2019 ; Us ami , 202 1, 202 3 ), I d is c u s s th re e is s u e s : (1 ) a n o v e rv ie w o f s e v e r a l sta tis tic a l m o d e ls with in th e SEM f r ame wor k t hat have bee n prop ose d i n psyc h ol o gy and r el at ed fi el ds ( i .e. , t he L CM - SR , L C S , and G CL M ), t oget her wi t h t he i nfe r ent i al probl ems t hat may ar i se w hen us i ng t he m; (2) an over vi ew of t he dynami c panel model ( DP M ) , us ed pr i mar i l y i n e c o n o m ic s , a n d its m a th e m a tic a l r e la tio nsh ip to th e RI - C L P M ; and ( 3) t he r ol e of s ta b le tra it f a c to rs in th e RI - C LPM f r om t he vi ew of causal inf er en ce, as wel l as t he i ssue of m o d e l se le c tio n . I n p a r tic u la r , I p o in t o u t th a t a d is tin c tiv e fe a tu re o f th e s ta b le tr a it fa c to r in th e RI - C L P M , i n ter ms of r epres en t i ng bet ween - p e r so n h e te rog e neity, is th a t it i s a ss umed t o be uncor r el at ed w i t h wi t hi n - per son v a r ia b ility , a n d th a t th is fe a tu r e a ls o cons t i t ut es an i mpor t ant poi nt of cont act l i nki ng t he R I - C LP M to th e DPM. Ove rvi ew of exi st i n g st ati st i cal m od el s an d r el ate d i ss u es LCM - SR B esi d es t he s ta b le tr a it fa c to r s, th e RI- CL P M and th e ST A R T S model share t h e f eat ure t hat t hey do not empha si ze t he mode l i ng of t ra j ect or y of change s for gr oup or i ndi vi dual s. I ndeed, t he gr oup mean i s express ed as µ  , µ  , and the mean st ruct u r e a lw a y s p e rf e c tly fits to t he dat a ( unl es s w e i mpose as sum pt i ons s uch as equal i t y cons t ra i nt s bet wee n t i me poi nt s) . E xpect ed sc or es for i ndi vi dual s gi ven th e s ta b le tr a it f act or s ar e ex pres sed as µ  +   , µ  +   . T he t r aj ect ori es of eac h i ndi vi dual and gr oup ar e m u tu a lly p ar al l el , and t hese mo del s do not i nt end t o d escr ibe tra je c to rie s and th e ir i ndi vi dual di f f er ence s i n a s t ruc t ur ed m anner usi ng som e com mon ( e.g. , gr owt h) fa cto rs . I n ot her wor ds, t he di ffe ren c e b et ween the se t wo m odel s i s t her ef or e charact eri zed n ot by t h ei r mean st ruct u r es, but by t h ei r covari ance st ruct u r es . By c o n tr a st, in d e s c rip tiv e re s e a r c h u s in g lo n g itu din a l d a ta , th e p r im a ry in te re s t lie s i n under st andi ng t he ave r age t ra j ect or y of change and i ndi vi dual di f f er ences i n tra je c to ry. F rom t hi s pe r spe ct i ve, t he l at ent grow t h model ( L C M ) i s a met hod t hat 18 e m e rg e d w ith in p s y c h o m e tric s , w ith fa c to r a n a ly s is a s o n e o f its in te lle c tu a l lin e a g e s . M ore over , propos al s for i nt egr a t i ve st at i st i cal mod el s be gan t o appea r mai nl y f rom t he 2000s onwar d. T hes e mod el s, l i ke t he L CM , us e com mon f act o rs (r ef er r ed t o as grow t h f act ors ) t o pr ovi de a pa r si moni ous r epr es ent at i on of t he aver age t r aj ect ory a nd in d iv i du al d i ff er en ces , whi l e at t he same t ime, l i ke the C L P M , al lowi ng i n f er ences about t he r eci p r ocal rel at i on s amo ng var i ab l es t hr ou gh aut ore g re ss i ve and cr oss - l agged te r m s. Th e LCM w ith s tru c tu r e d re s id u a ls ( LC M - SR ) ( C urr an, Howar d, Bai nter , L ane, & M cGi nl ey , 2013 ; C hi & R ei nse l , 1989) i s one s uc h st at i st i cal model , and wa s pr opose d as a me t hod for in fe rring w ith in - pe rs on r el at i ons . Th e LC M - S R has a r ep r esent at i on t h a t separ at es t h e part descr i b i ng t h e t raj ector y of c hange f r om t he r egr es si on equat i on i ncl udi ng t he aut or egr ess i ve and c r oss - l agged t erms. S p eci fi call y , in a n LCM - SR ass uming a t raj ect ory of l i near chan ge, obs er vat i ons   and   are express ed l i k e th e LC M as   =   + (   1 )   +    ,   =   + (   1 )   +    , ( 6  ) wher e   and   are i n t ercept s o f tr a je c to r y for i ndi vi dual i or , more speci fi cal l y , i nt ercept fact o r s r e fle c tin g th e m a g nitu d e o f th e tru e v a lu e a t th e f irs t tim e p o in t (  = 1 ).   and   are sl op e f ac t ors re pr es ent i ng t he ma gni t ude of t he s l ope of i ndi vi dual i ’s tra je c to ry. U n lik e th e s ta b le tr a it fa c to r , t hese t wo f act o r s ar e e stim a te d wi t hout f i xi ng th e ir mean s to 0 . W e al so assume and est i mat e t he cov ar i ance o f t hese fact ors . A posi t i ve co vari ance i n di cat es t hat t he lar g er t he t rue val ue at th e fir st tim e p o in t ( th e in te r c e p t) , t he lar g er t he sub s eq uen t chan g e s (t he sl ope) . I n t he cont ext of t he L CM , t he i nt ercept an d sl o pe fact ors are s ome t i mes col l ec t i vely r efer red t o as t h e grow t h fac to rs.    and    ar e t he r esi dual t er ms ( or , c or r espo nd i ng t o t he pr ev i ous expr ess i on s , th e devi at i ons) , w hi ch can a l s o be des cr i bed as t he quant i t y obt ai ned by det re ndi ng t he obs er vat i ons , t hat i s, by re movi ng t hei r l i near l y expr es se d t re nd compone nt . A s in th e (RI-)CLP M , at   2 , th e re s id u a l te rm s a re fu rthe r express ed us i ng r egr ess i on equat i on s w i t h l agged var i abl e s , su ch as    =     (  )  +     (  )  +     ,    =     (   )  +     (   )  +    , ( 6  ) and    ar e t reat ed as exo geno us var i ab l es, t h ei r ( co)var i ances are es t i mat ed, and t h ey ar e f ur t her as sum ed t o be uncor r el at ed wi t h t he gr owt h fa ct or s . Th is re f le c ts th e 19 in te n tio n o f th e LCM - SR, sim ila r to th e RI - CLPM , to c a p tu re w ith in - per son r el at i o n s b y us i ng r es i dual t er ms t hat ar e s epar at ed f r om and a ss umed t o be uncor r el at ed w i t h t he tra je c to rie s . N o te th a t if w e a ss u m e tim e - var yi ng aut ore gr es si ve coef f i ci ent s, c r oss - l agged coeff ici ents , and resi du al (co) vari an ce s on t he r i ght si de of t he r egr ess i on e q u a tio n , lo n g itu din a l d a ta w ith a t le a s t  = 4 will b e re q u ir e d fo r m o d e l id e n tific a tio n . LCM - SR m a y se e m s u ita b le fo r in fe r rin g with in - pe rs o n r e la tio ns. H o w e v e r , w ith in - per son r el at i ons e xpr e ss ed as E q. ( 6b) a r e r el at i ons of t he r es i dual s (    and    ) “aft er” t he i ndi vi dual ’ s tr a je c to r y of chan ge i s c ont ro l l ed by t he gr o wt h f act or s. In p a rtic u la r , if one cont r ol s f or t he sl ope f act or  , whi ch i s of t en r egar ded as re f l ect i ng wi t hi n - per son change over t i me and i ndi vi dual di f f er ences t her ei n — bot h of w hi ch ar e l i kel y t o be ma j or compone nt s fo r th e reci p r ocal rel at i on s i n many cases — th e w ith in - per son re la tio n s o f s u b s ta n tiv e in te re s t a re lik e ly to b e d isto rte d . Th is in d ic a te s o ver - a d justm e nt know n i n causal i nfer ence lite ra tu r e . Th is m e a n s th a t w h ile th e i nt ent i s t o cont rol and det r end a qua nt i t y (t he sl ope f act or s co re S ) t hat i s cons i der e d unn ec ess ar y for i nf er ri ng w ith in - p er son rel at i on s, we ar e al so un j u st i f i ably exclu di n g qu an ti t i es t hat ar e n ecess ar y fo r th e in fe ren c e ( component s r ef l ect i ng wi t hi n - per son cha nge and th e ir i ndi vi dual di ff erences) , whi ch Us am i, M u r ayam a et al . ( 20 19) de scr i be as “ t hrow i ng t he baby out w ith th e b a th w a te r ” . E ven i f sepa r at i ng t he s l ope f act or f r om E q. ( 6b) c an be j ust i fi ed, a p o te n tia l lim ita tio n in a p p ly in g th e LCM - SR is th a t th e sh a p e o f th e tra je c to ry ( e . g . , lin e a r) m u s t be speci fi ed co r rect l y . I n ot her wor ds, i f m o del mis speci fi cat i on occurs and t h e act u al mean st ruct ure i s not accurat el y repres ent ed , t hereby p r odu ci n g bias ed est i mat es, and s u c h b ia s w ill b e tra n s m itte d to th e b ia s in c ro ss - l ag ged co ef fi ci ent s t h at li e at t he co r e of i nfe r ence s r egar di ng wi t hi n - per son r el at i ons . F or t hese r eas ons, i t i s t he aut hor ’ s pos i t i on t hat , w hen t he goal i s t o i nfe r wi t hi n - per son re la tio n s , th e R I - C L P M shoul d gener al l y be r egar ded as havi ng an advant age over t he L C M - SR . As not ed above, howe ve r , t he act ual dat a gen erat i n g proc es s a nd t he t r ue model ar e unknow n . If th e (   1 )  t erm does no t ref lect t he main com ponent s o f t he reci procal rel at i on s o f i nt eres t , bu t rat h er ser v es as a t i me - var yi ng conf ounder s u m m a riz in g var i ou s uno bse rve d conf o un di n g f act o r s, t hen t he use of th e LCM - SR may be j ust i f i ed ( Us ami , M ura yama et al . 2019) . Mo re o v e r , it is c le a r th a t th e w ith in - per son r el at i on s r ep r esent ed b y t he RI - CL P M i n t his case cann ot be in te rp r e te d t o ref l ect caus a l r el at i on s, b ecause s uch t ime - va r yi ng conf ounder s ar e not cont r ol l ed f or . H o w e v e r , it al so seems rar e that such a s i mp lif ie d fu n c tio n c a n d e s c rib e a c tu a l tim e - var yi ng conf ounder s. LCS O ne mot i vat i on f or t he pr oposa l of t he lat ent chan ge score ( LC S ) mode l , o r th e l at en t d i ff erence scor e mod el , by M cAr dl e and H ama gami ( 2001) was t o pr ovi de , lik e 20 th e LC M - SR , an i nt egr at ed ext ens i on of th e LC M a n d th e C L PM. U n lik e th e LC M - SR , t hi s model wa s not ori gi nal l y propos ed wi t h t he expl i ci t obj ect i ve of i nfe r r i ng wi t hi n - per son re la tio n s , b u t it in c lu d es aut or egr ess i ve and c r oss - l agg ed t er ms and i s us ed t o exami ne reci p r ocal r elat i ons betw een v ar i ables . I n t he or i gi nal LC S r epres entat i on , as t he mo del ’ s name indi cat es, t he amou nt s o f change ( of t rue val ues ) bet we en t i me poi nt s (    =       (  ) ) is e x p lic itly i nt r oduced a nd r eg r es si o n equ at i on i s s peci fi ed f or t hi s c hange . Ho wev er , a s i n Usami, Hayes , and M cA r dl e ( 2015, 2016) and Us ami , M ura yama et al . ( 2019), a ma t hema t i cal l y equi val ent r epr ese nt at i on can be gi ven w i t hout usi ng t hi s cha nge , so f r om t he per spe ct i ve of compa r i son w i t h ot her st at i st i cal model s, t he f ol l owi ng expl anat i on i s bas ed on expr ess i on s t hat do not e x p lic itly us e am ount s of c hange. As wi th th e S TA RT S mode l , L CS decom pos es t he obse r va t i on in to a tru e s c o re f an d measur ement er ror ε, as   =    +    ,   =   +   . (7  ) B ecause of th e id e n tif ic a tio n r eason , tim e - i nv ar i ant er ror (co) v ar i an ces ar e u sual ly assu me d. U n lik e th e s ta tistic a l m o d e ls d e s c rib e d above , a r egr es si on equat i on t hat i ncl udes aut ore gr e ss i ve and cr oss - la g ge d te r m s is s e t d ir e c tly o n th e tr u e sc o re s in s te a d of devi at i ons as    =   +     (  ) +     (  ) +    ,   =   +     (  ) +     (  ) +   . (7  ) Al so , a common f act or A i s i ncl uded i n t hi s e quat i on. LCS u s u a lly a s su m e s tim e - i nvari an t au t o r egres si ve coeff i ci ents and cr oss - l agg ed coeff i ci ent s, due t o s cal e i nvari an ce wi t h respect t o the measur e men t s descr i b ed below , bu t her e we ass ume tim e - var yi ng coeff i ci en t s f o r t he sake of mode l compa ri son . F urt her mor e, bec aus e i mpr oper s ol ut i ons occur re l at i vel y f r equent l y i n t he L C S , s t rong a ss umpt i ons a r e s omet i mes i mpos ed t o avoi d t hem, such a s f i xi ng t he r esi dual (c o) var i ance s t o zer o (i .e. ,    =   = 0 ). Wh e n tim e - var yi ng aut or egr ess i ve coe f f i ci ent s, c r oss - lagged co ef fi ci ent s, and resi du al (co)var i an ces a re a s s u m e d , lo n gitu d in a l d a ta w ith a t le a s t  = 4 w aves ar e r equi r ed f or m odel i dent i f i cat i on. T he comm on f act ors   ,   are r and o m i n t erc ept s in t he r egr es si on equat i on s and ar e r el at ed t o t he amount of c hange be t wee n t i me poi nt s f or i ndi vi dual i (e.g., Usami et al ., 2015) . Al though b ot h are r efer red t o as “i nter cep t s” ,   and   ar e not equi val ent t o t he gr owt h f act ors (   and   ) in th e LCM - S R, because t h ey are i ncl ud ed i n the equat i ons t oget her wi t h t he aut ore gr es si ve and cr oss - l agg ed t erms. B y cont ra st , t he 21 means of   and   ar e n ot f i xed t o zer o; r ather , t h ei r means and (co) vari an ces ar e est i mat ed, as i s t he case for growt h f actor s. I n ad di ti on , t he tr u e sco r es at t h e in itia l tim e poi nt (    and    ) are t reat ed as exo geno us var i ab l es, and t heir mean s and ( co ) vari ances are es t i mat ed. M o r eov er , corr elat i on s (covari ances) bet ween thes e t rue s cor es and t he com mon f act or s   and   ar e al so assumed and est i mat ed. T h i s as sum pt i on of cor re l at i on st ands i n cont ra st t o t he as sum pt i on i n t he R I - C L P M and ST AR TS th a t th e s ta b le tr a it fa c to rs a n d d e v ia tio n s (w ith in - pe rs on var i abi l i t y), or i n t he LCM - S R t hat t he gr owt h f act ors and devi at i ons (w ith in - per son va r i abi l i t y) , ar e unco r rel at ed. S o how does t he com mon f act or A d i ffe r fro m the s ta b le tra it f a c to r I ? T hi s pap er pr i mar i l y foc use s on t he R I - CL P M , but t he o nl y mat hemati cal dif fer ence b et ween th e RI- C LPM and th e ST A R T S model i s t he as sumpt i on of mea su re me nt er r or s , s o h er e we w ill c o m p a r e th e L C S and th e S TA R T S model , bot h of whi ch i nc l ude meas u r ement e rro rs. Anot h er feat ure shared by t hese mod el s i s t hat, asi d e f rom measur ement err o r s , onl y a sin g le com mon f ac t or (  o r  ) i s i ncl uded i n t he mode l . Un lik e th e s ta b le tra it f act or I (a n d lik e w is e , th e gr owt h fa ct or in th e LC M - S R ) , t he com mon f act or A is i ncl uded i n t he r egr es si on equat i on al ong wi t h t he aut ore gr es si ve and cr oss - l agged te rm s . T hi s di ff erence con ce r n s h o w each co mmo n f act or contr i b ut es t o t h e t rue scor es (o r obser vat i ons ) . I n t he S T A R T S model , as show n i n E q . (5 b ) , th e s ta b le tra it f a c to r contr ibut es onl y as a d i rect eff ect on t he tr u e s co r e at each ti me po i n t , for examp l e i n t h e fo rm       . M or eover , t he magni t ude of t hi s ef f ect i s gi ven by   o r   and i s cons t ant acr oss t i me poi nt s. T he gr owt h f act ors i n t he L C M - SR a ls o shar e t h is prope r t y ( th is i s t h e r eason why t he s ame sym bo l ,  , i s us ed f or bot h i nt er cept fa ct or s a nd st abl e tra it f a c to rs ). B y cont r as t , w i t h r egar d t o t he comm on f act o r  i n t he L C S , cons i der , f or e xampl e, i t s cont ri but i on f rom   to    . Giv e n th e re c u rs iv e re la tio n in Eq . (7 b ) , it is c le a r th a t t here exi st s not onl y a dir ect ef fect such as       , bu t als o an i n di rect ef fect such as           t hr ough t he aut or egr es si ve pat h. F ur t her mor e, unl i ke t he st abl e t ra i t fa cto r  ,   can al so aff ect a di ff erent v ar i abl e (  ) t hr ough t he cr oss - l agged pat h, as i n          . M ore general ly , t h i s means t h at the ef fect of t h e common fact o r   on   pr opagat es t o t he t r ue s cor e at t he next t i me poi nt ,   (  ) , t hrough t he aut or egr ess i ve pat h , a nd t hen f ur t her t o   (  ) . I n t hi s w ay , t he cont r i but i on of   to the tru e s co re (o r obse r vat i on ) a t a gi ven t i me poi nt acc umul at es over subs equent t i me poi nt s . As a r es ul t , unl i ke a s t abl e t ra i t f act or , t he magni t ude of t he cont ri but i on of   di f fe r s a cr oss t i me poi nt s. Us ami , M ur ayam a et al . ( 2019) r ef er re d t o a comm on f act or w i t h t hi s pr oper t y as a n accum ul at i ng f act or , i n orde r t o di st i ngui sh i t fr om a st able t rai t fact or . R efl ect i ng thi s dif fer ence, t he S T AR T S mo del an d t he LC S hav e 22 di ff erent cov ar i ance st ruct u r es and t h er efor e yiel d d i ff erent est imat es of t he cross - l agged coe ff i ci en t s. M or eov er , comp ar i so n of t he pat h di agr ams f or t he dyna mi c panel mode l ( DP M ; F i gur e 1d) , whi ch, l i ke t he L C S , i ncl udes an ac cum ul at i ng f act or , a nd t he RI- C L P M (F i gu r e 1 b) makes t he ch ar act eri st i cs of t he accu mul at i ng fact o r descri bed abov e vi sual l y clear er . T hi s di ff erence betw een accu mul ati ng f act ors and st able t rai t fact ors i mpli es mat h emat ical and concep t ual di ff erences i n b et ween - per son het er ogenei t y bei ng cont r ol l ed f or , a nd E q . ( 7b) t her ef ore cannot be r egar ded as expr ess i ng wi t hi n - per son r el at i ons equi val ent t o t hose r epr ese nt ed i n t he R I - CLPM o r th e ST A R TS m o d e l. B eca use t he magni t ud e of t he cont r i bu t i on va r i es ove r t i me, wha t i s c ont rol l ed f or by a n accumulat i ng f act or can n o t be cons i der e d as a s ta b le tra it. A lth o u g h suc h tim e - var yi ng pr oper t y in th e m selve s is f le x ib le a n d ap p e a lin g , it s e e m s d iff ic u lt to j udge whether t he e ffec ts o f (time - i nvar i ant ) conf ound er s i n act ual l ongi t udi nal dat a can be adequa t el y r epr ese nt ed t hrough a r el at i vel y si mpl i fi ed f unct i on such a s t hat i n E q . (7 b ). Mo re im p o rta n tly, u n le s s tim e - i nva r i ant con st r ai nt s ar e i mpos ed on t he par ame t er s, t he L C S does not sat i sf y sca l e i nvar i ance . I n ot her wor ds, i f t he obse r ved var i abl es ar e l i near l y t r ans f or med, not onl y t he mea ns of t he acc umul at i ng fa ct or s but al s o t he es t i mat es of t h e aut o r egress i ve an d cross - lagged co ef fi ci ent s, as w el l as t he mo del fi t , w i l l change . T hi s i mpl i es t hat t he aut ore gr es si ve and c r oss - lagged co ef fi ci ent s i n t h e L C S are not s eparat ed f rom t he tr aject ory of chang e ( i .e., t he mean st ruct u r e). By c o n tra s t, if tim e - in v a ria n t paramet ers are ass umed i n o r der t o ensure s cal e i nv ar i an ce, t he mod el may become u n real i st ic, esp ecia ll y when examini n g wi thi n - pe rs on re l at i ons over a l ong per i od of t i me, becaus e qual i t at i ve di f f er en ce s m ay ar i se i n t he dynami cs of chang e. Sta tistic a l m o d e ls s u c h a s th e LCM - S R and L C S , w hi ch ar e i nt ended t o enabl e t he s i mul t aneous model i ng of t ra j ect or i es of change ( i .e ., mea n st ruc t ur es ) and t he e s tim a tio n o f re c ip ro c a l re la tio n s a s with in - per son r el at i ons, may a ppear r eas onabl e i n th a t th e y seem t o achi eve t wo ai ms at o nce. However , becau s e t hese are not , i n f act , mut ual l y i ndepend en t r es ear ch hyp ot he se s ( Us am i , M ur ayam a et al ., 2019), at t empt i ng to in v e s tig a te t hem s i mul t aneous l y may i nst ead hi nder t he in v e s tig a tio n o f each hypot hes i s, wi t h t he r es ul t t hat one who c has es t wo har es may c at ch nei t her . B y i nt r oduci ng gr owt h f act ors (  ) or accumu lat i ng fact ors (  ) in to th e m o d e l in st ead of t he gr oup mean  , whi ch s er ves onl y t o capt ure t he mea n st ruc t ure , t her e i s a r i sk t hat t he w ith in - per son r el at i ons t hat one t r ul y wi she s t o capt ur e t hr ough t he cr os s - l agged coef f i ci ent s w i l l be di st or t ed. T her ef ore , w hen one i s i nt er est ed i n bot h model i ng tra je c to rie s o f change and i nf er r i ng wi t hi n - per son r el at i ons, i t woul d be pr ef er abl e t o exami ne them s e par at el y , f or exampl e by usi ng t he L C M and t he R I - CLPM. A not her model pr opo sed wi t h t he i nt ent i on of pr ov i di ng an i nt egr at i ve ext ensi on of th e LC M a n d th e CLPM is th e a u to r e g re s s iv e la te n t tr a je c to r y m o d e l (A L T ; C u rr a n & 23 B ol l en, 2001) . I gnori ng meas ure ment er r or s , t he AL T cor re sponds t o an L C S model w i t h an addi t i onal acc umul at i ng fa ct or (  ) th a t h a s tim e - var yi ng wei ght s ( f act or l oadi ngs) ( Us ami , M ura yama et al ., 2019) . F or t he sa me r eas ons di sc uss ed above , i t i s h ig h ly lik e ly th a t, in th e A L T a s we ll, th e c ro ss - l ag ged co ef fi ci ents do not ref lect t he w ith in - per son r el at i ons t hat one act ual l y wi she s t o capt ur e , a nd t hey t her ef o r e ca nn ot be r eg ar ded as repr esen ti ng th e w ith in - pe rs o n re la tio ns e q u ivale nt to th a t o f th e RI - C LP M. G CLM T he gen er al cross - l agged pane l model ( GC L M ) w as pr opose d i n t he f i el d of or gani zat i onal r es ear ch by Z yphur , Al l i son et al . ( 2020) a nd Z yphur , V oel kl e et al . ( 2020) wi t h t he ai m of devel opi ng a me t hod t hat wou l d come cl ose r t o caus al i nf er ence t hr ough a gen e ra liz a tio n o f th e CLPM. A m o n g th e s ta tis tic a l m o d e ls d is c u s se d th u s fa r , i t i s t he onl y one pr oposed a f t er Ha make r et al . ( 2015) a nd Us ami , M ura yama et al . ( 2019) , a nd al t hough i t has not been appl i ed as ext ens i vel y as t he R I - C LPM, a cons i der abl e numbe r of empi ri cal appl i cat i ons have al r eady bee n r epor t ed. Z yphur , A lliso n e t a l. (2 0 2 0 ) p o s itio n th e G CLM a s a sta tis tic a l m o d e l th a t c a n re p re s e n t th e dynam i cs of change mor e f l exi bl y by i ncl udi ng “s t abl e t r ai t fa ct or s” 6) as un i t ef fect s and m ovi ng - average ( M A) t erms . With o u t lo s s o f g e n e r a lity , w e e x p la in th e G CLM a ss u m in g fir st - orde r l ag. F or   3 , th is m o d e l can b e r epres en t ed as 7)   =   +     +     (   ) +     (  ) +     (  ) +     (   ) +    ,   =   +     +     (   ) +     (   ) +     (  ) +     (  ) +   , (8 ) wher e   and   are i n t ercept t er ms r epresent ing t he eff ect at t i me po i n t  ( occasi on ef fect ) ,   an d   are t h e aut oregr essi ve (AR ) coeff i ci en t s, and   an d   are t h e c ro ss- l agged (CL) coeff i ci en ts .   a n d   ar e com mon f ac t or s r ef er r ed t o by Z yphur , A l l i son e t al . ( 2020) a nd Z yphur , V oel kl e et al . ( 2020) as uni t ef f ect s or “st abl e t ra i t f act ors,” and t hei r f actor means are f i xed at zer o . T he ini ti al obse r vat i ons (   and   ) ar e t reat ed as exo geno u s vari ab l es, and thei r means and ( co ) vari ances are est i mat ed. In addi t i on, a l t hough not st at ed expl i ci t l y by Z yphur , Al l i son e t al . ( 2020) and Z yphur , V o e lk le e t a l. ( 2 0 2 0 ), it is a s su m e d — s im ila rly to th e s ta b le tr a it fa c to r s in th e RI - C LPM — t hat t he com mon f act or s   a nd   ar e u n c o rr e la te d w ith th e in itia l obs er vat i ons   a n d   ( howe ver , be c au se of t he nat ur e of  as an accumu l at i ng fact o r , di scussed below , they are cor rel at ed wi t h the obs er vat i ons f rom t he se cond t i me poi nt onw ar d, a nd t hi s a ss ump t i on i t s el f can al so be r el axed) .   and   are wei gh t s ( fact or l oadi ngs) . Al t hough Z yphur , Al l i son et al . ( 2020) and Z yphur , V oel kl e et al . ( 2020) al l ow t hes e w ei ght s t o be f i xed at 1, t hey r ecom mend assu ming tim e - var yi ng w e ig h ts . 24 Th e te r m s   a nd   ar e movi ng - av er age (M A) coeff ici ents repr esent i ng the ef fect s of res i du al s fr o m t he same vari able at t he p r evi o us t i me po i n t , whereas   and   are cr oss - l agged m ovi ng - av er age (C L M A) coeff i ci ents repr esent i ng the ef fect s of r es i dual s f rom a di f f er ent var i abl e a t t he pr evi ous t i me poi nt . M A t er ms have tra d itio n a lly b e e n w id e ly u se d , e s p e c ia lly in m o d e ls f o r tim e s e r ie s d a ta with a la rg e numbe r of t im e p oi nt s. I n t h e G CL M , eff ect s f rom t he same vari ab l e are r ep r esent ed by AR and M A ter ms, wher eas ef fect s f r o m a di ff erent v ar i ab l e are r ep r esent ed by C L an d CLMA te r m s. Wh e n tim e - var yi ng coef fi ci ent s an d t h e r esi d ual (co) vari an ces ar e assu me d , l ongi t ud in a l d a ta w ith a t le a s t  = 4 waves ar e requir ed f or model id e n tific a tio n . Z yphur , Al l i son et al . ( 2020) and Z yphu r , V oel kl e et al . ( 2020) i nt er pre t t he com mo n fa cto r  in th e s e n s e o f th e s ta b le tra it f a c to r (  ) in th e R I - C L P M , and e xpl ai n t hat t he mode l obt ai ned by r emo vi n g t he M A and C L M A t er ms f r om t he G C L M i s equi val ent t o th e R I - C LPM ( t h at assumes tim e - var yi ng wei ght s f or t he st abl e t ra i t f act or) . H owe ver , as i s cl ear fr om t he di sc uss i on t hus f ar i n t hi s pape r and f r om t he f orm of E q . (8 ), th e com mon f act or  i s i n f act an ac cumul at i ng f act or . I n t hi s sens e, t he GC L M does not pr ovi de a r epr es ent at i on of w i t hi n - per son r el at i ons e qui val ent t o t hat of t he R I - C LPM (Us am i , 2021) . M ore ove r , al t hough an acc umul at i ng fa ct or wi t h t i me - var yi ng wei ght s ma y provi de a mor e f l exi bl e r epr ese nt at i on f r om t he s t andpoi nt of m odel f i t , i t i s al so l i kel y t o over cont r ol , as i n t he L C M - S R , i nfor mat i on on wi t hi n - per son c hange a nd in d iv id u al d i ff erences t h er ein — ma j or c omponent s of re ci pr ocal r el at i ons — t her eby d is to r tin g th e with in - per son r el at i ons t hat one ac t ual l y wi shes t o exam i ne. I n addi t i on, U sam i (2021) poi nt ed out t hat t he r es i dual t er ms a ss oci at ed wi t h t he M A a n d C L M A te r m s ar e gener al l y vague i n t er ms of wha t t hey s ubst an t i vel y r epr ese n t , and t hat , be caus e t hey ar e vi r t ual l y con t r ol l ed f or m ul t i pl e t i mes ( e.g. , t hr ough bot h t he A R and M A t er ms , or t hrough bot h t he C L and C L M A t er ms) , t he me ani ng of t he cr oss - l agged coeff ici ent s i n t he G C L M become s m or e com pl i cat ed and ma y al so i nduce m u ltic o llin e a rity. O f c o u rs e , w h e th e r it is a p p r o p r ia te to in c lu d e MA a n d C LM A te rm s depends on th e tr u e d a ta - gener at i ng pr oces s, and t he compl exi t y of i nt er pre t at i on o f coeff ici ent s does not i n i t s el f i mpl y i nf er ent i al bi as or r educe d es t i mat i on acc ura cy; non et h el ess, t hese remai n p r act i cal probl ems t h at t h e G CL M may ent ai l . S u mmar y of t h i s s u bs ect i on an d a u n i f i ed f ram ew ork f or t h e m odel s Th e LC M - S R , L CS (or AL T ) , and G CL M propos ed i n psych ol o gy and r el at ed fi el ds al l i nt r oduce com mon f act or s i n orde r t o re pr es ent and cont r ol for l at ent conf ound er s and, mor e br oadl y , bet we en - pers on het erogenei t y . T hese mod el s are char act eri zed pr i mar i l y by t he t ype of common f act or t hey i ncl ude ( i .e. , a s t abl e t r ai t fa ct or or an accumulat i ng f act or) , t he p r esen ce o r abse nc e of a mean st ruct u re f or t h at fact o r , and th e pres ence o r absenc e of corr el at i o ns wi th wi t hi n - p e rs o n v a r ia b ility. A lth o u gh th e re is th e 25 fu n d a m e n ta l d if fic u lty th a t th e tru e d a ta - gener at i ng proc es s i s us ual l y unknown, t he pot ent i al pr obl ems in i nf er ence s a bout wi t hi n - per son r el at i ons di scus sed i n t hi s s ubsec t i on can be cl as si fi ed i nt o t hre e t ypes : probl ems of over adj ust ment ( L CM - SR , G CLM), p ro b le m s a ris in g fr o m a tte m p tin g to in fe r with in - per son r el at i ons a nd t r aj ect ori es of chang e ( mean st ruct u r e s) sim u lta n e o u s ly (LC M - S R, LCS ), a n d c o m p le x ity in th e in te rp re ta tio n o f c o e ff ic ie n ts (G CLM) . Th e LC M - SR, lik e th e RI - C L P M , en abl es i nfer ence regardi n g wi t h i n - per son r el at i ons t hr ough dev i at i ons t hat ar e as sum ed t o be unco r r el at ed wi t h t he gr owt h f act ors (  ,  ) ; howeve r , bec aus e i t ma y over adj ust for t he sl ope f act or  , w h ic h is us ual l y consi dered to re f le c t with in - per son cha nge and i ndi vi dual di ff er ences t her ei n — m a jo r com ponen t s of r eci proc a l re l at i ons — th e w ith in - pe r son r el at i ons one a ct ual l y wi she s t o exam i ne ar e l i kel y t o be di st or t ed, and t he cr oss - l agg ed coeff i ci ent s may t heref ore be bi ased. In t he LC S , t h e accumul at i ng f act or  i s i ncl uded i n t he r egr ess i on equat i ons t oget her wi t h t he aut or egr ess i v e and cr oss - l ag ge d t erms , and as a r esul t , un l i k e a s t able tra it f a c to r , it c o n trib u te s to th e tru e sc o re s (o r obser vat i ons ) in a tim e - var yi ng manner . A lth o u g h th is fe a tu r e is f le x ib le , it is d if fic u lt, f o r e x a m p le , to ju d g e w h e th e r th e e f fe c ts o f (tim e - i nvar i ant ) c onf ound er s i n act ual l ongi t udi nal dat a c an be ade quat el y r epr ese nt ed t hrough a si mpl i f i ed f unct i on such a s t hat spec i f i ed by t he model . I n addi t i on, be caus e t he L C S does not i ncl ude t he gr oup mean  , it fa ils to s a tis fy s c a le in v a ria n c e u n le s s tim e - i nvar i ant cons t ra i nt s a re i mpose d on par amet er s s uch as t he aut or egr ess i ve and c r oss - l agged c oef f i ci ent s. T he G C L M , w hi ch i ncl udes t he accumulat i ng f act or  , l i kew i se c ar ri es a r i s k of over adj ust i ng f or i nfor mat i on on w ith in - per son c hange a nd i ndi vi dual di f f er ences , a s in th e LCM - S R, because i t i nt r oduces an acc umul at i ng fa ct or wi t h t i me - var yi ng wei ght s ( f act or l oadi ngs) . M ore over , t he i ncl usi on of movi n g - av era ge ter ms who se sub st an t i ve mean ing i s ob s c u r e may comp l icat e t he i n t erpr et at i on o f t he cross - l ag ge d co ef fi ci ent s and may a l so i nduce m u ltic o llin e a rity. T hes e model s ar e al so cons i de r ed t o car ry a hi gher r i sk of i mpr oper sol ut i ons t han th e R I - C L P M ( e.g. , O rt h et al ., 2021) . Al t hough Or t h et al . ( 2021) di d not di r ect l y e x a m in e th e G CLM, th e c o m p le x ity o f its s tru c tu re s u g g e s ts th a t it m a y b e a t le a s t a s p ro ne t o i mpr oper sol ut i ons a s t he L CM - S R and L C S , i f not mor e s o. S i nce H ama ker et al . ( 2015) , m odel s ot her t han t he R I - CLPM — p a rtic u la rly th e G CLM — ha ve i ncr eas i ngl y bee n app l i ed. A s di s cuss ed abov e, ho wev er , t he me ani ng of th e w ith in - per son r el at i ons r epr es ent ed by t hese model s i s not t he s ame . M or eover , because t hese mo d e ls a ls o pr ovi de mat h emat i cal l y d i ff erent mean s t ruct u r es and covari an ce st ruct ure s, t h ey y i eld di ff erent est i mat es of t he cross - l agg ed coeff i ci en t s as w el l ( e. g., Us am i , M ur ayam a et al ., 2019; Or t h et al ., 2021; Us ami , 2021) . I n act ual appl i cat i ons, howe ver , i t se ems t hat i nsuf f i ci ent at t ent i on has of t en been pa i d t o t hese di f f er ence s a nd t o t he pot ent i al probl ems di scus s ed above. 26 B esi d es mat hemati cal di ff eren ce s such as t he type and numb er o f commo n fact ors, i ncl udi ng st abl e t ra i t f act or s a nd acc umul at i ng fa ct or s, di ff er ences i n t he not at i on and t er mi nol ogy adopt ed i n t he l i t er at ure i nt r oduci ng eac h model al s o made i t di ff i cul t t o di st i ngui s h cl ear l y among t he exi st i ng model s. Us ami , M urayama et al . ( 20 19) pr es ent ed a uni fi ed f r ame wor k f or expl ai ni ng t he ma t hema t i cal and conce pt ual di ff erences amon g v ar i o us st at i st ical mod e l s used to i n f er reci pro cal r elat i on s f rom t h r e e pers pecti ve s: ( i ) t he t ype of comm on fa ct or i n c lu d e d (i. e . , w h e th e r it is a n a c c u m u la tin g f act or) , (i i ) whet her t h er e i s an i nt eres t i n mo del i ng t raj ect o r i es of chan ge (mean s t r uct ur es ) , as i ndi cat ed by whet her t er ms suc h as t he gr oup mea n  o r in te r c e p t  ar e i ncl ud ed, and ( i ii ) w hether measur emen t err o rs a re as s umed. T hey f urt her show ed, t hr ough pat h di agr a ms and concept ual fi gure s, t hat each m odel can be posi t i oned as a s p e c ia l c a s e with in th is u n if ie d fr a m e wo rk — t hat i s , as a subm odel as sum i ng onl y cer t ai n co mmon fact o r s or p ar amet ers — an d a ls o illu str a te d th e mathemat i cal r el at i onshi ps a mong t he mode l s. M ore ov er , i f t he pre se nce of movi ng - av er age ter ms i s al so taken i nto accoun t , t he GC L M can l i kewi se be charact er ize d w ith in th e sa m e fra me wor k ( Us ami , 2021) . H owe ver , a l t hough t hose s t udi es r ef er r ed t o t he i ss ue i n connec t i on wi t h acc u mul a t i ng fa ct or s, t he dyn ami c pa nel mod el (D P M ), w hi ch has be e n us ed mai nl y i n economi cs, was not di re ct l y i ncl uded as a mode l for compa r i son. Ove rvi ew of t h e D P M an d i ts m at h em at i cal r el ati on to t h e R I - CLPM Over vi ew of t h e D PM A l t hough t he D P M ( C hi gi r a et al ., 201 1; W ool dri dge, 201 1; Hs i ao, 2014; Al l i son, Willia m s , & Mo r a l - B eni t o, 2017) i s t ypi cal l y use d t o anal y ze uni di re ct i onal r at her t han r eci proc al r el at i ons, f or pur pose s of compa r i son i t s f orm ul at i on f or t he t w o - v ar i abl e ca s e cons i der ed t hus f ar may be w ri t t en a s foll o ws ( e. g., Al l i son e t al ., 2017; Us ami , 202 3 ):   =   +   +     (   ) +     (   ) +    ,   =   +   +     (  ) +     (  ) +   . （ 9 ） Her e,   a nd   ar e i n t ercept ter ms repr esent i n g t he eff ect s of t i me po i nt  .   and   a r e ac cumul at i ng f act ors ( t houg h i n econ om i cs t hey ar e t ypi cal l y re f er r ed t o as uni t ef fect s or i nd i v i d ual ef fect s) , and t heir mean s ar e zero. F urt h er ,   a nd   ar e aut oregres si v e coeff i ci en t s,   and   are cr oss - l agg ed coeff i ci ents , and    a nd   are r esi d ual s. C o r r el at i on s ( co vari ances) are as sumed an d est imat ed b et ween t he in itia l obs er vat i ons (   and   ) , as wel l as bet ween those val ues and t he tw o accumulat i ng f act ors 8) . In a d d itio n , ju s t a s th e CLPM c a n b e r e w ritte n f ro m Eq . (1 ) to 27 Eq . (2 ), Eq . (9) can l i kewi se be r ef orm ul at ed usi ng t he gr oup mean  i nst ead of t he in te r c e p t  :   = µ  +    ,   = µ  +    , ( 10  )    =   +     (   )  +     (  )  +     ,    =   +     (   )  +     (   )  +    . ( 10  ) T he est imat es of t he au t oregres si v e coeff i ci en t s, cr oss - l agg ed coeff i cient s, and accumulat i ng f act ors, as wel l as t h e model fi t , ar e un af fect ed b y t his refo rm ul a tio n . F rom a compa r at i ve pe rs p ec t i ve, t he pat h di agr a m of t he D P M under t hi s f orm ul at i on i s s hown i n F i gur e 1d. I t shoul d be not ed, ho wev er , t hat t he t er m dynam i c pane l mode l en c omp ass es a wi d e var i et y of model s, i ncl udi ng t hose wi t h t re nd t er ms or m ov i ng - av e rag e te rm s . M o r e over , com par ed wi t h t he s t at i st i cal model s t hat have pr i ma ri l y been us ed i n psychol ogy and r el at ed f i el ds a nd di scus sed t hus f ar , dynam i c panel model s i n economi cs a r e of t en appl i ed i n set t i ngs w i t h a l ar ge number of t i me poi nt s ( e.g. , exce edi ng 100 or even 1,000) and r el at i vel y sho r t tim e in te rv a ls (la g s) , such as i n analyses of exchan ge rat es. I n addi t i on, econom i st s a r e r el at i vel y of t en i nt er es t ed i n for eca st i ng f ut ure val ues . F or t hes e and r el at ed r eas ons, i t i s com mon i n dynami c panel model i ng t o as sum e t i me - i nvari an t co ef fi ci ent s. T he same i s t rue fo r t he di sc uss i on in Ande r se n ( 202 2 ), refe rre d t o bel ow , who al so us es t he t er m D P M . I n t he pr ese nt paper , how ever , for pur pose s of mode l compa r i son, E qs . (9) and ( 10) a ss ume t i me - var yi ng coef f i ci ent s, whi l e al so t aki ng i nt o account t he t i me - i nvar i ant cas e. I t shoul d t her ef or e be not ed t hat , i n t hi s paper , t he t er m D P M i s us ed t o r ef er t o a l i mi t ed cl as s of model s t hat do not i ncl ude movi ng - av er age ter ms and t h at al l ow t i me - var yi ng coef f i ci ent s . I n addi t i on, i n economi cs, f or t he D PM (t hat assumes tim e - in v a ria n t c o e ff ic ie n ts ) , es t i mat i on of t en r el i es on m et hods ot her t han maxi mum l i kel i hood, s uch as t he gener al i zed me t ho d of mome nt s ( GM M ), beca us e t he f or m of t he es t i mat or depends on whet h er the un i t eff ect s are tr eat ed as r and om eff ects or f i xed eff ect s, as wel l as on t h e ass umpt i ons re gar di ng t he i ni t i al condi t i ons di scus sed be l ow ( Chi gi r a et al ., 201 1) . H oweve r , be caus e t he pr i mar y foc u s of t hi s s ect i on i s on com p ar i ng t he f or mul at i ons of d if fe re n t sta tistic a l m o d e ls , here I as sume t he u se of max i mum l i kel i hood base d on S E M ( ML - S E M ) , as wi t h t he pr ece di ng model s. F or t ypi cal model spe ci f i cat i ons and n o ta tio n s f o r dy nami c p an el mod el s, as w el l as det ail s of M L - SEM im p le m e n ta tio n a n d com par i sons o f e s tim a tio n re s u lts w ith G MM, s e e Mo r a l - Be ni t o ( 2013) and Al l i s on et al . ( 2017) . N ow , i gnori ng meas ure ment er r or s , th e D PM in Eq . ( 9) c or r esponds t o t he L CS when t he int ercept t er m  i s i ncl uded and t he mea n of t he acc umul at i ng f act or i s f i xed 28 t o zero. As a resul t, becau se t he mean st ruct ure can b e exp r esse d as a functi on of  , th e pr obl em of sca l e i nvar i ance obs er ved i n t he L C S w i t h t i me - var yi ng coef fi ci ent s i s r es ol ved. F r om anot her pe r spe ct i ve, t he DP M can a l so be under st ood as cor re spondi ng to th e GC LM w h e n th e tim e - var yi ng we i ght s a ss oci at ed w i t h t he acc umul at i ng fa ct or s ar e f i xed at 1 and t he M A and C L M A t er ms a r e s et t o 0. T her ef or e, a l t hough t hi s ul t i mat el y depends on t he t rue dat a - gen er a t i ng pr o ces s , t he r i sks of over adj ust ment and pr ob le m s o f in te rp re ta b ility a re lik e ly to b e s u b s ta n tia lly re d u c e d . Ma the m a tic a l r e la tio n with the RI- CLPM F or t hese r eas ons, t he D P M may be r egar ded as enabl i ng i nfe r ence r egar di ng w ith in - per son r el at i ons i n a mor e appr opr i at e way t han t he mode l s di sc uss ed i n t he pr evi ous s ubsec t i on ( L CM - S R , L C S , and G C L M ). T hi s t hen r ai ses t he ques t i on of w hi ch of t he t wo m odel s — t he DP M or t he R I - C L P M , whi ch has r api dl y gai ned popul ar i t y i n psychol ogy and r el at ed f i el ds — s houl d be cons i der ed pr ef er ab l e as a s ta tistic a l m o d e l. T o a d d r e ss th is q u e s tio n , it is fir st n e c e s sa r y to c la r ify th e m a th e matic a l r e la tio n sh ip bet ween t hese mod el s. At fi rs t glance, t h ey may app ear mat hemati cal l y un r el at ed, because t he RI - CLPM in Eq . ( 3) i s f or mul at ed i n t er ms of devi at i ons, w her ea s t he DP M in Eq . (9 ) is fo rmu la ted in term s o f obse r vat i ons . How ever , i t i s know n t hat , pr ovi ded t he aut or egr ess i ve and c r oss - lagged co ef fi ci ent s ar e t i me - i nvar i ant ( and t he abs ol ut e val ues of t h e ei g enval u e s of t he co ef fi ci e n t mat ri x are l ess t h an 1; e.g., Hamake r , 2005) , t he t wo mod el s yi el d mat hemati cal l y eq ui val en t repr esent at i on s and t he same coeff ici ents when cert ai n assump ti on s are added t o each mod el (Anders en, 2 02 2 ). O n e s uch as sum pt i on i s t hat , i n t he D P M , t o t ake i nt o account t hat t he pr oces s of cha nge has al r eady be en oper at i ng pri or t o  = 1 , a cons t ra i nt i s i mpos ed on t he w ei ght s ( fa ct or lo a d in g s ) a s s o c ia te d with th e a c c u m u la tin g fa c to r s a t th is tim e p o in t by a f unct i on of t h e par amet er s; under t hi s c onst ra i nt , t he D PM bec omes equi val ent t o t he R I - CLPM . Th is t ype o f speci fi cat i on i s somet imes ref err ed t o as t he co nst rai n ed app r oach ( e.g., J onger l i ng & Ha make r , 201 1). I t has al s o l ong been know n t hat , w hen t he coe f f i ci ent s a re tim e - i nvar i ant , i mposi ng anal ogous c onst ra i nt s on t he we i ght s i n t he AL T by t he sa me pr ocedur e r ender s i t equi val ent t o t he L C M - S R ( Ham aker , 2005; Us ami , M ura yama e t al ., 2019) . Th e o th e r a s s u m p tio n is th a t, if c o rr e la tio n s b e tw e e n th e s ta b le tr a it fa c to rs a n d th e in itia l obs er vat i ons are al l owed i n the R I - CLPM , i t becom es equi val ent t o t he D P M ( Ande r se n, 202 2 ). Th e e s tim a te s o f th e (tim e - in v a r ia n t) cro s s- l agg ed coeff i ci ent s al so become ident i cal . T his t yp e of speci fi cat i on i s somet i mes ref err ed to as t he pr edet er mi ned app roa ch . F ol l owi ng And er se n (202 2 ), t h e pres en t pap er ref ers t o t he R I - C L P M under t hi s sp e c ific a tio n as t he p r ed eter mined RI - C LPM , a n d its p a th d ia g r a m is s hown i n F i gur e 1c. As not ed i n t he pr evi ous se ct i o n, un der t hi s spe ci f i cat i on t he st abl e 29 t r ai t fact ors, li ke accumu l at i ng fact ors, b e com e co mmon fact ors t hat d o not exer t t i me - i nvari an t d i rect eff ect s, and t he v ar i ance of t h e obs er vat i ons ca n no l onger be or t hogonal l y decom pose d i nt o wi t hi n - per son a nd bet we en - per son c omponent s. T her ef ore , a l t hough i nt r odu ci ng s uch cor r el at i o ns gi ve s r i se t o t he equi val en ce bet we en t he predet ermi ned R I - C L P M and t he D P M , t he pr edet er mi ned R I - C L P M s houl d no l onger be regar ded as t h e same ki n d of s t at i st ical mod el as t he RI - C LP M, b u t r a th e r a s one bas ed on a di f f er ent desi gn l ogi c. M ore over , whe n t i me - varyi n g coeff ici ents are ass umed , t h e equi va l enc e betw e en t he p r ed et erm ined RI - CL P M and t he DP M no l onger hol ds. I n t hat cas e, i n addi t i on t o t he R I - CLPM, th e re e x is t a t le a st tw o c a n d id a te s ta tistic a l m o d e ls (i .e., predet ermi n ed R I - C L P M and t he DP M ) t hat i nvol ve accumulat i ng ( o r, “ accumul at i ng - lik e ” ) fa c tors . T abl e 1 pre se nt s t he r es ul t s of appl yi ng t he pr edet er mi ned R I - C L P M and t he DP M to th e MACC d a ta exp l ained ear l i er . As i n t he pr evi ousl y re por t ed anal yses of t he CLP M a n d RI - CLPM, e s tim a tio n wa s c a rr ie d o u t u s in g ML - S EM. Th e ta b le s h o w s th a t, w h e n tim e - i n var i an t autor e g r es si ve co eff i ci en t s, cr o ss - l agg ed coeff i ci ents , and resi du al ( co ) vari ances are as sumed, t he predet ermi n ed R I - CLPM a n d th e D PM y ie ld id e n tic a l est i mat es of t he co ef f i ci en t s ( as wel l as t h ei r s t and ar d err ors) . By contr ast , t h i s r el at i onshi p does not hol d whe n t i me - va r yi ng coe ff i ci ent s ar e as sume d. I n addi t i on, re f le c tin g th e fa c t th a t th e s e tw o m o dels off er mor e fl exi b l e repr e sen t at i on s i n t h e sense th a t th e ir a c c u m u la tin g (o r a c c u m u la tio n - lik e ) fa c to r s c o n tr ib u te in a tim e - var yi ng ma nner , t hey show bet t er f i t t han t he R I - C L P M . M ore over , unde r t he t i me - var yi ng condi t i on, t he DP M exhi bi t s s omew ha t d if fe re n t e s tim a tio n r e su lt s f r om t hos e of t he R I - C L P M and t he pr edet er mi ned R I - CLPM, in th a t th e s ta tistic a l s ig n ific a n c e o f s o m e o f t he cross - l agg ed coeff i ci ent est imat es i s no t con si st en t across t h e model s. Recon si d eri n g t h e RI - CLPM : The r o le o f the s ta bl e t r a it f act or f r om t h e vi ew of c au sal i nfe re nc e a nd m o de l s el ecti o n I n l i ght of t he f or egoi ng di scus si on, I r econs i der wha t t he s t abl e t ra i t f act or cont r ol l ed f or i n t he R I - C LP M is , what it means fr o m t h e per spect i ve o f causal i nf er ence , a nd how mode l sel ect i on shoul d be conduc t ed . St a bl e t r a it f a c t o r , c o nfo unde r , a nd c e nt e r ing In th e RI - C L P M , t he s t abl e t r ai t fa ct or cont r i but es t o t he obs er ved var i abl es onl y as a ( tim e -i nvar i ant ) d ir e c t e ffe c t, w h ile it d o e s n o t c o n tr ib u te to th e d e viatio n s , th a t is, th e w ith in - per son va r i abi l i t y . I n t he t er mi nol og y of caus a l i nf er ence, t he obs er ved var i abl e is a c o llid e r th a t is th e jo in t c o n se q u e n c e o f b o th th e s ta b le tr a it fa c to r a n d th e w ith in - p e r so n v a r ia b ility, a n d th e p a th fro m th e s ta b le tra it fa c to r to w ith in - p e rs o n v a ria b ility is bl ocked by t he obs er ved var i abl e, w hi ch f unct i ons a s a col l i der . 30 F rom t hi s per spe ct i ve, U sa m i (202 3 ), w h ile e x p lic itly p re s e n tin g th e d a ta - ge ne r a t i ng p r o ce ss th a t ref lect s th e R I - C L P M ( F i gur e 1b) and t he i dent i f i cat i on condi t i ons f or caus al ef f ect s, ar gued t hat al t hough t he s t abl e t ra i t f act or c ont rol s f or unobs er ved bet ween - perso n h et erogeneit y , i t cannot be r egarded as a ( t i me - i nvar i ant ) conf ounder lik e an accumu l at i n g f act or 9) . Ho w e v e r , th e w ith in - per son r el at i ons r epr ese nt ed i n t he RI- C L P M are r el at i on s amon g dev i ati on s , whi ch are l at en t vari ab l es, and inf erence r egar di ng t hem mus t ul t i mat el y be bas ed on i nf or mat i on f r om t he obs er ved var i abl es. Th ere fo re , i gnor i ng and fa ilin g to a c c o u n t fo r th e s ta b le tra it f a c to r th a t lin k s th e obs er ved var i ab l es and t he wi t hi n - per son va r i abi l i t y i nf or mat i on, de spi t e i t s pr es ence , i nduces bi as i n t he es t i mat i on of t he cr oss - l agg ed coeff i ci ent s . Th is s u g g e sts th a t, if tim e - i nvar i ant conf ounder s a r e i n f act pr es ent i n t he dat a - gener at i ng proc es s, t he R I - CL P M canno t d i rect ly accommodate t hem, and t h ei r in f lu e n c e is o n ly p a rtia lly a b s o rb e d in to th e s ta b le tra it fa c to r s . Th e in c lu s i on of t he s ta b le tra it f a c to r s shoul d t her ef or e be under st ood not so m uch as cont rol l i ng for a conf ounder , but ra t her as cor r es pondi ng t o t he oper at i on of cent er i ng by t he ( t r ue) per son me an, as i s of t en di scus sed i n t he cont ext of hi er ar chi cal l i near model i ng (e .g. , Usam i , 2017; As par ouh ov & M ut hén, 20 18 ) . M ore ov er , c ent er i n g may be vi ewe d as t he key oper at i on t hat l i nks mode l s f or de vi at i ons s uch as t he R I - CL PM (o r LCM - S R) a n d mode l s f or obse r vat i ons such a s t he DP M ( or AL T ) . A l t hough W ang and M axw el l ( 2015) di scus sed t he dec o mpo si t i on of w i t hi n - perso n an d betw ee n - p er son r el at i ons a nd cent eri ng i n h i erar chical l i near model s ( t hat do not i ncl ude lagged v ar i ab l es ) , fu rt h e r in v e stig a t i on i s nee ded r egar di n g t he r el at i on bet we en t he i nt er pr et a t i on of t he comm on f act ors i n t he var i ous s t at i st i cal model s di sc uss e d i n t hi s pa per and w het her cent er i ng i s or i s not appl i ed t o each va r i abl e. Da ta - g e ne r a t ing pro c e s s a nd a s s umpt io ns a bo ut i nit ia l c o ndit io ns Usam i (20 2 3 ) a ls o e x p la in e d th a t th e tre a tm e n t o f th e v a r ia b le a t th e in itia l tim e poi nt (  = 1 ), r ef er r ed t o as t he i ni t i al condi t i on, di ff er s depe ndi ng on w het her t he com mon f act or a ss umed i n t he dat a - generat ing process i s a st able t rai t f act o r o r an acc umul at i ng fa ct or . I n sur vey and obs er vat i onal st udi es i n par t i cul ar , i t i s nat ura l t o ass ume that , i n many l ongi t udi nal dat as et s, t he act ual dat a - gener at i ng proc es s ha s al r eady be en oper at i ng pri or t o  = 1 . I n such a cas e, i f an accumul at i ng f act or i s ass umed i n t h e proces s, i ts i nfl uen c e has al read y acc umu l at ed i n the obser vat i ons a t th e in itia l tim e poi nt , and t her ef or e mode l i ng t hat t akes t hi s i nf l uence i nt o acc ount i s r equi r ed, as i n t he L C S , AL T , or DP M ( f or exam pl e, by us i ng t he pr edet er mi ned approach, whi ch assumes corr elat i ons betw een the i n i t i al obse r vat i ons and t he acc umul at i ng f act or; e. g ., Gi sc he , W es t , & V oel kl e, 2021) . B y cont ra st , i f onl y a st abl e t r ai t fact or i s assumed i n t h e process , i t cont ri bu t es only as a di rect eff ect on t he obs er vat i ons and does not exer t a cumul at i ve i nf l uence on wi t hi n - p e rs o n v a ria b ility a t 31 th e in itia l tim e p o in t. Th ere fo re , even i f t he pr oces s had a l re ady been ope r at i ng bef or e  = 1 , t here i s no need t o assume a corr el at i on bet ween the st abl e t rai t fact or and the d e v ia tio n ( o r with in - p e r so n v a ria b ility ) a t  = 1 (Usami, 20 2 3 ). In o th e r wo rd s , if it is pl aus i bl e t o ass ume t hat t he pr oces s does not cont ai n het er ogene i t y i n t he f or m of a conf ounder l i ke an a ccum ul at i ng f act or , t hen t he i n c lu sio n o f a sta b le tra it f a c to r u n c o rr e la te d w ith w ith in - per s on var i abi l i t y , t oge t her wi t h t he or t ho go nal dec o m posi t i on in to with in - p er son vari an ce and b et ween - person vari an ce as i n t he RI - C LPM, may be r egar ded as a val i d proc edur e . C r i ti que s o f the R I - C LPM a nd m o de l s e le c tio n L üdt ke and R obi t zs ch ( 202 2 ), t hrough s i mul at i on and anal yt i cal i nvest i gat i ons, ar gued t hat when unobs er ved t i me - i nvar i ant conf ounder s ar e pr ese nt , t he condi t i ons under w hi ch t he R I - CLPM c a n p ro p e rly d e a l w ith th e ir in flu e n c e a re q u ite lim ite d . A s d is c u s s e d a b o v e , th is re f le c ts th e fa c t th a t th e s ta b le tr a it fa c to r c o n tr o lle d fo r in th e R I - CLPM is n o t a c onf ounder and cont r i but es onl y i n t he f or m of a t i me - in v a ria n t d ir e c t ef fe ct . L üdt ke and R obi t zs ch ( 202 2 ) a ls o argued t he use f ul nes s of t he C L P M wi t h hi gher - or der l ags. How ev er , whet her t hi s i s appr opr i at e f r om t he per spec t i ve of caus al i nfer en ce agai n d ep en d s on the t rue data - g en erat in g proc ess, and as t he CL P M co nt ai ns nei t her comm on f act ors nor uni t ef f ect s, t he r ange o f ( uno bs er ved ) h e te ro g e n e ity th a t it c a n d ire c tly c a p tu r e is lik e ly to b e lim ite d . F urt her mor e, L üdt ke and R obi t zsc h (202 2 ) c r itic iz e d th e in fe r e n c e o f w ith in - per s on r el at i ons bas ed on var i a b ility a ro u n d th e w ith in - pe r son mean (expect ed val ue) , ar gui ng t hat suc h i nf er ence i s es t abl i shed onl y by i gnori ng t he pot ent i al l y i mpor t ant i nf l uences of f act ors expl ai ni ng bet we en - per son di ff er ences and t her ef or e depa r t s f rom t he in v e stig a t i on of t he or i gi nal caus al hypot hes i s. R el at edl y , O r t h et al . ( 2021) r ecom mende d t he use of t he C L P M for i nfe r r i ng bet wee n - per son r el at i ons a nd t he R I - CLP M fo r in f e rr in g w ith in - per son r el at i ons. As L üdt ke and R obi t zs ch ( 202 2 ) t hems el ves not ed, t hes e i ss ues c an ul t i ma t el y be un der s t ood as conc er ni n g di ff er e n c e s i n th e e s tim a n d , th a t is , th e ta rg e t o f e stim a tio n c o r re s p on d in g to th e sc ie n tific q u e s tio n o f in te r e st, wh ic h has re cent l y been e mphas i zed i n bi ost at i st i cs and epi demi ol ogy . As H amake r et al . ( 2015) ori gi nal l y poi nt ed out , t he r el at i on s exam i ned i n t he C L P M canno t u sual ly be cl ear l y char act eri zed as ei t her a pure w i t hin - pe r son r el at i on s o r a bet ween - p e rso n re l a tio n s . T her ef or e, w hen one i s i nt er est ed i n i nf er ri ng wi t hi n - per s on r el at i ons, i t i s gene r al l y pr ef er abl e t o i nt r oduce com mon f act or s ( or uni t ef fe ct s) under s ome as sumpt i ons , as i n t he R I - C LP M or DP M , so as t o separat e i nfer ence co ncerni n g w ith in - p er son rel at i ons f r om t hat conce r ni ng bet w een - per son r el at i ons. I n causal i nfer ence, i t i s general ly r equ i red t h at the speci fi ed st at i st ical mod el appropri atel y ref l ect s the as sumed d at a - gener at i ng pr oces s ( e. g., Gi sche et al ., 2021) . H oweve r , i n most cas es i t i s unknown w het he r t he t rue dat a - ge ner a t i ng pr o ces s cont a i ns 32 a s t abl e t ra i t f act or , a n acc umul at i ng f act o r , bot h, or nei t her , and t hr ough wha t pat hs or f unct i onal for ms t hese cont r i but e t o t he obse r vat i ons ( e. g., C ur r an & B auer , 201 1; A nder se n, 202 2 ; U sa m i, 2 0 2 3 ). M ore ove r , al t houg h an accum ul at i ng fa ct or i s f l exi bl e i n th a t it c a n re p re s e n t tim e - var yi ng cont r i but i ons t o obse r vat i ons th a t a r e n o t lim ite d to di rect eff ects , i t i s not neces sar i l y easy to j udge whe t her t he i nf l uence of ( t i me - i nvar i ant ) c onf ounder s i n act ual l ongi t udi nal dat a can be a ppr opr i at e ly re p res e nt ed t hr ough t he pat hs or func t i onal f or ms as sum ed, f or exam pl e, i n t he DP M . T ake n t oget her , w hen one i s i nt er es t ed i n i nf er ri ng wi t hi n - per son r el at i ons, one r eal i st ic approach may b e t o est i mat e a part icul ar s t at i st ical mod el — such as t h e R I - C L P M , t he D P M , or t he pr edet er mi ned R I - C LPM — bas ed on t he as sum ed dat a - gener at i ng proc e ss , w h ile a lso e s tim a tin g o th e r c a nd idate m o dels a s se n s itiv ity a na ly s es , as i n t he com par i son s hown i n T abl e 1, and exam i ni ng t he appr opri at enes s of model s e le c tio n a n d th e ro b u s tn e s s o f th e e s tim a tio n r e su lts with re fe r e n c e to fit in d ic e s , in f o rm a tio n c r ite ri a, and r esi dual cor r el at i ons ( e. g., Us am i , 202 3). Summa r y a nd f ut ur e d ir e c t io ns Summa r y o f t his pa pe r S even years h ave p as se d si n ce Hamaker et al . (2015 ) cr i t i ci zed the C L P M , and the RI- C L P M has r api dl y spr ead i n psyc ho l ogy and re l at ed f i el ds w i t h suc h mom en t um t ha t i t ma y r epl ace t he C L P M , l ong re gar ded as t he gol d s t andar d. Al t hough t hi s pape r has pres en t ed s ever al i s sue s s ur r ou nd i ng t he R I - C LP M , at leas t i n the compari son b et ween t he C L P M and R I - CLPM, th e CLPM — w hi ch does not i ncl ude a s t abl e t r ai t fa ct or r efl ect i ng uno bser v ed bet ween - pe rs on het er o ge nei t y — can no t be r egard ed as sui t ab l e f o r in f e rrin g w ith i n - per son r el at i ons, even i f hi gher - order l ags are as sumed. By cont ras t , al t hough t he R I - C L P M app ear s t o r epresent b et ween - per son het er ogenei t y i n a s omew hat l i mi t ed wa y , i t can neve rt hel ess be posi t i oned as one ef f ect i ve me t hod for in f e rrin g w ith in - per son re la tio n s . A t th e s a m e tim e , howeve r , man y ot her st at i st i cal mod el s are al so avai l ab l e. M ore over , a l t hough t he prec i se meani ng of t he wi t hi n - per son r el at i ons r epr es ent ed by each mod el an d t he est i mat es of t he cross - l agg ed coeff i ci ents usual l y d i ff er acr o ss models , t he mat hemati cal and concep t ual r elat i on s among t hem h ave no t n ecess ari l y been cl earl y organi zed , and i n actua l a p p lic a tio n s in s u f fic ie n t a tte n tio n se e m s to h a v e been pa i d t o t hese di ff er enc es . A l t hou gh t he di s cuss i on of st at i st i cal mode l s f or i nfe r r i n g r eci p r o c a l r e la tio n s a s with in - per son r el at i ons ha s bec ome i ncr eas i ngl y compl ex and di ver se , w i t h many cont ri bu t i ons cont i nu i ng t o appea r , i n t hi s pape r I have at t empt ed t o or gani ze t hes e i ss ues, dr aw i ng al so on my ow n wor k (U sam i , M ur ayam a et al ., 2019; U sam i , 2021, 202 3). B ecause t he actual d at a - gener at i ng pr oces s and t he t rue model ar e gene ra l l y 33 unknow n t o t he r es ear cher , i t i s di f f i cul t t o j udge w het her t he com mon f act ors or uni t e ff e c ts in c lu d e d in a s ta tis tic a l m o d e l a p p r o p r ia te ly c o n trol fo r la te n t c o n fo un d e rs ( m o r e broadl y , bet ween - pe rs on het er ogenei t y ) . Th is m a k e s it d if fic u lt to a rr iv e a t a d e f in itiv e concl usi on r egar d i ng t he bes t mode l choi ce for i nfe r r i ng wi t hi n - p ers o n rel ati on s and t he pr oper i nt er pr et at i on of t he cr oss - l agged coe ff i ci ent s. Agai nst t hi s ba ckgr ound, t hi s paper has a im e d to c la rify th e p o te n tia l p r o b le m s th a t m a y a r ise in in f e rrin g w ith in - p e r so n re la tio n s fr o m e x is tin g s ta tistic a l m o d e ls (LC M - SR, LCS , AL T , a n d GCLM ) f r om t he per spe ct i ves of over adj ust ment , t he pr obl em of si mul t aneous l y i nf er ri ng w ith in - p er son rel at i on s and t raj ect ori es of cha n g e (mean st ruct ure s) , an d t he co m p le x ity in in te rp re ta tio n o f co ef fi cient s . A st at i st i cal model t hat avoi ds t hes e pr obl e ms a nd i ncl udes a n accum ul at i ng f act or capable of repr esent i n g t i me - va r yi ng cont ri but i ons t o obse r vat i ons th a t a r e n o t lim ite d t o dir ect ef fect s i s t h e D P M exp r essed i n E q s. ( 9) or ( 10) . B y cont ra st , t he st abl e t ra i t fa c to r in th e RI - CLPM co n trib u te s to th e obse r vat i ons o n ly a s a tim e - in v a ria n t d ir e c t ef fect repr esent i ng st abl e indi vidu al d i ff eren ce s. The RI - C L P M and t he DP M may bot h b e s tro n g c a n d id a te s ta tistic a l m o d e ls fo r in f e rr in g w ith in - p er so n r el at i on s, but it c a n n o t necessar il y b e st at ed c l earl y which of t h em mor e accurat ely capt ures t he actual d at a - gener at i ng pr oce ss and yi el ds l ess bi as ed i nf er enc e r egar di ng wi t hi n - per son r el at i ons. A l t hough t he ac cumul at i ng f act or i n t he D P M i s f l exi bl e, i t i s not neces sar i l y eas y t o j udge whe t her t he i nfl uenc e of ( t i me - i nva ri ant ) con founde r s i n act ual l ongi t udi nal dat a can be appr opri at el y r epr ese nt ed t hr oug h t he pat h s and fun ct i on al for ms as sum ed by t he mode l . I n addi t i on, t hi s pape r has des cr i bed anot her model s peci f i c a tio n , te rm e d th e predet ermi n ed RI - C LP M, in w h ic h c o rr e la tio n s a re a ss u m e d b e tw e e n th e in itia l obs er vat i ons a n d th e sta b le tra it f a c to rs , a lth o u g h u n d e r th is sp e c ific a tio n th e sta b le tr a it f act ors can no longer be i nt erpr et ed in t h e same w ay as t hose in th e R I - C L P M . W hen tim e - i nvari an t au t o r egres si ve coeff i ci ents , cr oss - l ag ged coeff i ci ents , and resi du al ( co ) vari ances are as sumed, t he p r edeter mi n ed R I - C L P M and t h e DP M are ma t hema t i cal l y equi val ent ( Ande r se n, 202 2 ) , but when s uch as sum pt i ons a r e not i mpos ed t h e y y ie ld d if fe re n t e s tim a tio n r e su lts (T a b le 1 ) . A lth o u g h th e a v a ila b le o p tio n s fo r s ta tistic a l m o d e ls a re th u s d iv e rs e , o n e r e a listic s tra te g y m a y b e to e s tim a te a p a r tic u la r s ta tis tic a l m o d e l — such as t he RI - C LP M, th e D P M , or even t he pr edet er mi ned R I - CLPM — base d on t he as sum ed dat a - gene r at i ng p r o ce ss, w hi l e al so e st i mat i ng ot her candi dat e mode l s as sens i t i vi t y anal yses and exam i ni ng t he appr o pr i at e nes s of model s el ect i on and t he r obust ness of t he e s tim a tio n r es ul t s by r ef er r i ng t o f i t i ndi ces , i nfor mat i on cr i t er i a, and r es i dual cor r el at i ons ( e.g. , Usam i , 202 3 ). Fut ure dire c t io ns RI- C L P M or D P M , or som e ot her st at i st i cal model such a s t he pr edet er mi ned R I - 34 C L P M ? Regar dles s of t h es e d i ff erences i n model ch oi ce, h owever , the need remai ns unchange d t o i dent i f y and col l ect conf ounder s i n advanc e — p a rtic u la rly tim e - var yi ng conf ounde r s — a n d to c o n trol fo r th e m a p p r o p r ia te ly w ith in th e s ta tistic a l m o d e l . M any of t he s t at i st i cal model s i nt roduc e d i n t hi s pa per , i ncl udi ng t he R I - C LP M, a re ty p ic a lly es t i mat ed t hrough S E M , and w hen t here ar e obse r ved conf ounder s t hey ar e of t en r epr ese nt ed und er t he as sum pt i on of l i near r el at i ons wi t h t he var i ab l es of i nt er es t (  ,  ) in th e in f e re n c e o f w ith in - per son r el at i on s. Al t hou gh such pr o ced ur e s a r e us ef u l t o som e ext ent, es peci al l y when co nt i nu ou s v ar i ab l es ar e an al y zed, t h e l inear i t y assumed i n th e s t andar d S E M and pat h anal ysi s ha s of t en bee n cr i t i ci zed fr om t he per spe ct i ve of caus al i nf er ence ( e.g. , Hong, 2015). At the sam e t i me, t he cau s al i n f er ence li t erat ure ha s propo sed some app r oaches that do n ot assume such l i neari t y , as w el l as r ob ust ap proaches t hat ar e les s af fect ed b y m o d e l m iss p e c if ic a tio n . Fro m th is p e r sp e c tiv e , U sa m i ( 2 02 3 ), und er th e d a ta - ge n er a ti n g pr oces s as sume d by t he R I - CLPM , p ro v id e d m a them atica l d e f in itio n s o f th e s ta b le tra it fa c to r a n d w ith in - per son var i abi l i t y and pr opo se d a met hod for i nf er ri ng w i t hi n - per son r el at i ons bas ed on st epwi se e st i mat i on t hat does not neces sar i ly a s su m e lin e a r ity w ith tim e - var yi ng conf ounder s. M ore spe ci fi cal l y , s cor es r epr es ent i ng wi t hi n - per son v a r ia b ility ( with in - person vari a bi l i t y scores ) ar e fi rs t predi cted f or each v ar i abl e t h r ough a measurement mod e l based on S E M (or fact or anal ysi s) , and these ar e t h en t reat ed as obs er ved var i abl es. M et hods known ma i nl y i n epi demi ol ogy as m ar gi nal st ruc t ura l mode l s ( M S M s) and s t ruc t ura l nest ed model s ( S NM s) (e .g. , R obi ns & He r nán, 2009; Her nán & Robins, 20 21 ) ar e t h en app l i ed to est i mat e t he eff ects . T his app r o ach can als o be ext ended t o i nfe r enc e und er t he as sum pt i on t hat an acc umul at i ng f act or such as t hat i ncl uded i n t he D P M i s pr ese nt . M et hodol ogi cal devel opment f or i nf er r i ng wi t hi n - per son r el at i ons, i n s p ir e d by appr oache s or i gi nat i ng i n caus al i nf er ence and not necessar il y conf i ne d t o psyc homet ri cs, i s l i kel y t o be an i mpor t ant di r ect i on f or f ut ure r esear ch. T he pr obl em of sepa r at i ng or di st i ngui s hi ng wi t hi n - group r e la tio n s fr om b et ween - gr oup re la tio n s , o r with in - per s on re l at i ons f r om bet we en - per s on r el at i ons, has l ong b e e n of i nt er est i n psyc hol ogy and r el at ed f i el ds. I n t hat sens e, di s cus si on sur r oundi ng re c ip ro c a l re la tio n s a s w ith in - per son r el at i ons — an ol d yet new i ss ue — ent ered a new phas e af t er Ha make r et al . ( 2015). At pre se nt , t hi s t opi c i s be i ng act i vel y di sc uss ed, es peci al l y i n connec t i on wi t h t he R I - CLPM , b u t th e d is c u s s io n is s till c e n te r e d la rg e ly a m ong psycho me t ri ci ans a n d cann ot yet be sa i d t o i nvo l ve, f or exampl e, r es ear cher s i n ot her rel at ed fi elds s uch as eco nom etr ics , epidem i ol o g y , or causal inf eren ce . I hope th a t t hi s paper wi l l hel p or gani ze t he var i ous i ss ues sur r oundi ng t he i nf er ence of wi t hi n - per son r el at i ons a nd st i mul at e new quest i ons, t her eby pr omot i ng heal t hi er di scus si on and m et hodol og i cal dev el o pme nt a m o ng no t onl y psy cho me t ri ci ans bu t al so s ubst an t i v e r es ear cher s i n psychol ogy and r el at ed f i el ds w ho make us e of t hes e me t hods. 35 Ref eren ces A lliso n , P . D. , Willia m s , R. , & Mo ra l - Be ni t o, E . ( 2017) . M axi mum l i kel i hood for cr oss - l agged pane l model s w i t h fi xed ef fe ct s. Soc i us , 3 , 1 – 17. A nder se n, H .K. ( 2022) . E qui val ent appr oaches t o deal i ng wi t h unobs er ved het erog en ei ty i n cr oss - l ag g ed panel model s: Inve st i g at i ng t he bene f i t s a nd dr aw backs of t he l at ent cur ve mode l wi t h st r uct ur ed r esi dual s and t he r andom i nt ercept cross - l ag g ed panel model . 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O rgani z at i onal R ese arc h Met hods, 23 , 688 – 716. 40 F oot n ot es 1 ) W ithin - per s on and betwe en - p er son r el at io ns ar e somet i mes r ef er r ed to as intra -indivi dua l and inte r - indi vidual rela tion s , r especti vel y . 2 ) Howe ver , intens ive longitudi nal da ta — cha r ac t erized not only by a la r ge number of indi vidua l s but also by a l ar ge number of time p oint s ( with t he number a nd ti ming of mea s ur eme nts often di f f er i ng acr oss ind iv i dual s) , as in cas es wher e au t om ated as sess m en ts f r om wear ab le d evi ces ar e ut i li zed — h ave al so in cr easin gl y been col l ected in r ecent year s. F o r such d ata, app r oach es bas ed on continuo us - time models ( V oelkle, Oud, Davidov , & S chmidt, 2012; Rya n, Kuiper , & Ha maker , 2018) , whic h tr ea t t ime in a c ontinuous fo r m , h ave r ecen tl y att r act ed gr owi ng attention, bec aus e they ca n addres s the problem that the magnitude of the ( cros s - lagge d) coe f f icients may vary depe nding on time i nt er v als ( lag s) bet ween time po int s . T he deve lopment and c ompari s on of continuous - tim e models c or res ponding to t he RI - C L PM a nd the other s tatistica l models disc us se d in thi s paper appea r to be import a nt t opics f or fut ure res earch. 3 ) Orth e t al. ( 20 24) atte mpte d to e s tablis h e mpirica l guide lines f or the ef fec t s ize s o f c ros s - lagge d coef f icients obtained whe n applying t he C L P M and RI - CL P M. 4 ) T he STA RT S m ode l or igi na ted as an exte nsion of the latent s tate - trait mode l, whic h w as des igned to decompos e a s ingl e longi tudinally meas ur ed va r iable into tr a it a nd sta te compone nts. Although th e S TA R TS model wa s initia lly for mula ted for a s ingle variab le, it wa s s ubs eque ntl y exte nded to ac commodate two or mor e va r iables , at which point cros s -lagged term s were incorpor ated into the model. 5 ) Us ami, Murayama et al. (2019) us e d the te r m “ s tatio na r ity” to r e fer to the s etting of time - i nvar i an t au to r egr essive co ef f i cien ts, cr o ss - l agg ed coef f i cien ts, an d r esid ual ( co) v ar i ance s . 6 ) Z yphur , Alli s on et al. ( 2020) r ef er r ed to s table tr a it f a cto rs in their mode l as “s tab le fac tor s ” . 7 ) Z yphur , Alli s on et al. ( 2020) and Z yphur , Voelkle e t al. ( 2020) a ss umed time - invariant aut or eg r ess i ve co ef f ici ent s   , cr o ss - lag ged coef f i ci ent s   , moving- aver age co ef f i ci ent   , and cr o ss - lag ged moving- aver ag e coef f i ci ent   , but an e xte ns ion to time - va r ying coe f f icie nts is als o poss i ble. 8 ) I f accum u lat in g f act o r s ar e v iew ed as r ando m i n ter cep ts, t hey may app ear sim il ar t o t he hierar c hical linear models commonly use d in psyc hological res ea r ch for longitudi na l data, name ly t hos e that include lagge d variables and a s s ume r andom e f f ec ts. How ever , h ier ar ch i cal line ar mode ls do not in clud e in terce pt terms r e pres entin g time - specif ic eff ects, n or d o t hey us ually allo w c oef ficie nts to vary a cros s time p oints . In add ition, time - varyi ng res idual var i ances ar e n ot ty pi call y assum ed, an d a m aj or di f f er en ce is that th e ran dom inte r c ept is as sume d to be uncorr ela ted with the explana tor y variables . T hat s aid, t he distinction betwe en r a ndom- ef f ect s and f i xed - eff ects mo del s is also r el at ed t o t he i ss ue o f cent er i ng t h e var i abl es. For a disc us s ion of thi s poi nt i n t he case of mo del s wit ho ut lag ged var i abl es, see Hamak er and M uthén (2020) . 9 ) I n Us ami, M ur aya ma et a l. ( 2019) , the s t a ble tr ait f a ctor wa s des cr ibed a s a confounder, but this point wa s corr ec ted in Us ami ( 2023). 41 Ta b le 1 : C ompar i son of r e s u lts a c ross s ta tis tic a l m odel s . CL P M RI - CL P M P r e de te r m ine d RI - CL PM DPM t ime - invar i a nt t ime - va rying t ime - invar i a nt t ime - va rying* t ime - invar i a nt t ime - va rying t ime - invar i a nt t ime - va rying Est. SE Est. SE Est. SE Est. SE Est. SE Est. SE Est. SE Est. SE                                                   . 82 . 01 . 91 . 03 . 70 . 01 . 22 . 43 . 53 . 02 . 59 . 05 . 53 . 02 . 45 . 04 . 04 . 01 . 05 . 03 - . 01 . 02 . 08 . 06 - . 01 . 02 - .04 . 05 - .01 . 02 - .02 . 03 . 51 . 01 . 52 . 01 . 17 . 01 . 29 . 14 . 17 . 01 . 24 . 02 . 17 . 01 . 20 . 02 . 01 . 00 . 05 . 02 - . 01 . 01 2. 4 1. 7 - . 01 . 01 - .03 . 02 - .01 . 01 . 06 . 03 . 82 . 01 . 85 . 02 . 70 . 01 . 63 . 03 . 53 . 02 . 31 . 07 . 53 . 02 . 47 . 03 . 04 . 01 . 09 . 03 - . 01 . 02 . 07 . 05 - . 01 . 02 . 01 . 05 - .01 . 02 . 03 . 03 . 51 . 01 . 47 . 01 . 17 . 01 . 12 . 02 . 17 . 01 . 14 . 02 . 17 . 01 . 15 . 02 . 01 . 00 . 05 . 01 - . 01 . 01 . 06 . 02 - . 01 . 01 . 01 . 03 - .01 . 01 . 04 . 02 . 82 . 01 . 85 . 02 . 70 . 01 . 70 . 03 . 53 . 02 . 11 . 11 . 53 . 02 . 50 . 03 . 04 . 01 . 04 . 03 - . 01 . 02 . 02 . 05 - . 01 . 02 . 14 . 07 - .01 . 02 . 04 . 04 . 51 . 01 . 53 . 01 . 17 . 01 . 11 . 02 . 17 . 01 . 11 . 02 . 17 . 01 . 17 . 02 . 01 . 00 - .01 . 01 - . 01 . 01 . 00 . 01 - . 01 . 01 . 00 . 04 - .01 . 01 . 00 . 01 . 82 . 01 . 82 . 01 . 70 . 01 . 76 . 02 . 53 . 02 . 58 . 04 . 53 . 02 . 54 . 02 . 04 . 01 - .03 . 03 - . 01 . 02 - .06 . 05 - . 01 . 02 - .02 . 06 - .01 . 02 - .02 . 04 . 51 . 01 . 49 . 01 . 17 . 01 . 11 . 02 . 17 . 01 . 08 . 03 . 17 . 01 . 11 . 02 . 01 . 00 . 01 . 01 - . 01 . 01 - .01 . 01 - . 01 . 01 . 01 . 02 - .01 . 01 . 01 . 01 . 82 . 01 . 79 . 01 . 70 . 01 . 66 . 02 . 53 . 02 . 51 . 03 . 53 . 02 . 53 . 02 . 04 . 01 . 02 . 03 - . 01 . 02 - .06 . 05 - . 01 . 02 - .01 . 05 - .01 . 02 . 03 . 04 . 51 . 01 . 55 . 02 . 17 . 01 . 20 . 03 . 17 . 01 . 18 . 03 . 17 . 01 . 17 . 02 . 01 . 00 . 00 . 01 - . 01 . 01 - .01 . 01 - . 01 . 01 . 01 . 01 - .01 . 01 . 01 . 01     ,    |     ,    . 05 . 01 . 04 . 01 . 05 . 01 . 04 . 01 . 03 . 01 . 01 . 01  (   ,   )   (   ,   ) |   (   ,   )   (   ,   ) |   (   ,   )     ,    |     ,      (   ,   ) |   (   ,   ) . 05 . 02 . 04 . 02 - . 02 . 01 - .01 . 01 - . 05 . 02 - .04 . 02 . 03 . 02 . 04 . 02 - . 02 . 01 - .01 . 02 . 04 . 01 . 01 . 01 . 01 . 01 - .01 . 01 . 13 . 01 . 12 . 01 - . 60 . 05 - .55 . 06 . 26 . 02 . 30 . 02 . 04 . 01 . 04 . 02 . 04 . 01 . 04 . 01  (   ) |  (   )     |     . 15 . 01 . 15 . 01 . 15 . 01 . 15 . 01 . 10 . 01 . 40 . 01 . 58 . 04 . 61 . 03 1. 1 . 04 1. 1 . 06 . 25 . 03 . 63 . 03  (   )  (   ) . 41 . 01 . 41 . 01 . 25 . 01 . 26 . 01 . 25 . 01 . 27 . 01 . 41 . 01 . 10 . 01 . 57 . 03 . 62 . 03 . 03 . 03 . 01 . 01 . 64 . 06 . 59 . 06 . 58 . 03 . 26 . 03 d f 68 40 65 49 61 33 61 33 CFI 0. 875 0. 895 0. 959 0. 966 0. 964 0. 991 0. 964 0. 988 AIC 73465. 64 73192. 02 72224. 17 72125. 29 72151. 05 71778. 16 72151. 05 71818. 56 BI C 73607. 51 73514. 47 72385. 39 72389. 69 72338. 07 72145. 75 72338. 07 72186. 15 RM SE A 0. 077 0. 091 0. 045 0. 047 0. 043 0. 030 0. 043 0. 034 SRM R 0. 113 0. 101 0. 066 0. 060 0. 066 0. 030 0. 066 0. 031 * Unde r the “ t i me - i nva r i a n t” c o ndi ti on, e q ua l it y c on st r a i nt s a r e im p ose d a c r oss ti m e p oi nt s o n t he a ut or e gr e ss iv e c oe f f ic ie n ts , c r os s - la gge d c o e f f i c ie nt s, a nd r e sidu a l ( c o )v ar i anc e s. Ho w e v er , un der t he ti m e - va r y ing c o n dit io n f o r t he R I - C LP M , e q ua li ty c ons tr a in ts a r e im po se d o n t he r e s idu a l ( c o) v a r ianc e s be c a use of im p r o per so lu ti o ns . T he e stim a te s of the r e sidua l ( c o) va r ia nc e s a nd the m e a ns/inte r c e pts a r e om itte d. B oldf a c e indic a te s sta tistic a l sig nif ic a nc e a t the two - ta ile d 5% le ve l, a nd ita lic s indic a te tha t the c or r e sponding value s a r e ide ntic a l a c r oss m ode ls. 42 Fi gu r e 1 : Pa th d ia gra m s o f th e s ta tistic a l m o d e ls . * Pat h s rep r es en ti n g r es idu al cov ar i an ces ar e om itt ed . Th e m ean o f each co m mo n facto r is f i x ed at zero , an d its (co ) var iance is es timated. In (c) , the co m m on f act o r (  ) in th e p red eterm i n ed R I - C L P M d iff ers in bo th con cep tual an d m ath e m atica l r o le f r o m th e s tab le tr ait f a cto r in th e R I - C LP M shown i n (b), w hi c h exe rt s ti me - in v ariant d irect ef fects on t he obser v atio n s; nor doe s th is co mm o n f actor or tho g o n ally d eco mp o s e the v ar i an ce o f th e ob s er v a tio n s in to b etw een - pe rson a nd wi t hin - per s on v ar i an ce s . Ho wev er, wh en tim e - in var ian t auto reg r es si v e coef ficients , cr o s s - l a gge d co efficien t s , and r es idu al (co )v arian ces ar e as s u med , th e co rres p o n d in g es timates in (c) the p r edeterm ined RI - CL PM an d ( d ) the DPM are id en ti cal. (E quat i on 2 ) (E quat i on 3 ) (E quat i on 10 )