ESTIMATION OF DYNAMIC LINEAR MODELS IN SHORT PANELS WITH ORDINAL OBSERVATION

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1 ESTIMATION OF DYNAMIC LINEAR MODELS IN SHORT PANELS WITH ORDINAL OBSERVATION Sehen Pdne THE INSTITUTE FOR FISCAL STUDIES DEPARTMENT OF ECONOMICS, UCL cemma workng aer CWP5/5

2 Esmaon of dnamc lnear models n shor anels wh ordnal obseraon Sehen Pdne Cenre for Mcrodaa Mehods and Pracce Inse for Fscal Sdes and Inse for Socal and Economc Research Uners of Essex Seember 4, resed Ma 5 Absrac: We deelo a smlaed ML mehod for shor-anel esmaon of one or more dnamc lnear eqaons, where he deenden arables are onl arall obsered hrogh ordnal scales. We arge ha hs laen aoregresson (LAR) model s ofen more arorae han he sal sae-deendence (SD) rob model for adnal and neral arables. We roose a score es for asssng n he reamen of nal condons and a new smlaon aroach o calclae he reqred aral derae marces. An llsrae alcaon o a model of hoseholds erceons of her fnancal well-beng demonsraes he seror f of he LAR model. Kewords: Dnamc anel daa models, ordnal arables, smlaed maxmm lkelhood, GHK smlaor, BHPS. JEL classfcaons: C3, C5, C33, C35, D84 Address for corresondence: See Pdne, ISER, Uners of Essex, Wenhoe Park, Colcheser CO4 3SQ. E-mal: sdne@essex.ac.k. I am graefl o Sarah Brown and Karl Talor for allowng me o se her BHPS daase and o hem and o Manel Arellano, Olma Boer, Sehen Jenkns, Che Ncole and arcans a he ISER JESS semnar and CIDE conference, Vence 5, for helfl commens. Ths work was sored b he Economc and Socal research Concl hrogh rojec Ferl and oer n deelong conres (award no. RES346) and he UK Longdnal Sdes Cenre.

3 Inrodcon In dscree daa modellng here s an moran dsncon o be made beween nheren and obseraonal dscreeness. Inheren dscreeness refers o a case where he arables of neres are narall dscree. For examle, an nddal s eher emloed or no emloed; she has a ners degree or no; she s marred or no. Obseraonal dscreeness arses when he arables of neres are narall connos, b he sre nsrmen sed o obsere hem moses dscreeness a a re-secfed ordnal scale of allowable resonses. Ths ales o a wde range of adnal qesons, whch ask resondens o record her erceons or belefs on a Lker scale. Examles of economerc analss of adnal arables hae rolferaed n recen ears, wh he deelomen of he economc lerare on haness and sasfacon (see Van Praag and Ferrer--Carbonell, 4, for a recen sre). There has been moran work on economerc mehodolog n hs area, arclarl he choce beween random and fxed effecs modellng n anels (Ferrer-Carbonell and Frjers, 4), b here has so far been lle dscsson of he dnamcs of erceons or of he mos arorae e of dnamc model o se. Obseraonal dscreeness does no onl arse wh adnal daa. I ma also occr n sre qesons abo more objece enes lke ncome, when resondens are reqred o lace hemseles whn one of a nmber of gen ncome ranges. The dscreeness here s an essenall arfcal conseqence of qesonnare desgn. For examle, bsness sres ofen ask abo execaons of fre sales or nesmen nenons. The resonden s execed ale for sales or nesmen condonal on hs nformaon se s a connos arable b he sre qesons call ask for a resonse graded as, down or no change. Mos of he economerc lerare dealng wh dscree models for longdnal daa assmes nheren dscreeness. The oneerng work of Heckman (978, 98a,b) cenred on bnar resonse models of he form: α ( β' x > ) where (.) s he ndcaor fncon, x s a ecor of srcl exogenos coaraes, s an nobsered nddal effec ncorrelaed wh x and ε s a random resdal ncorrelaed across nddals and me. We refer o () as he sae deendence (SD) model. I was deeloed rmarl for alcaons n labor economcs, where ε () - -

4 dscreeness s nheren n he roblem and where as ocomes of reresen sae deendence. In hese alcaons, he laen arable s essenall an arfcal consrc and here s no reason wh shold aear n he model (). Howeer, ades, execaons and ncomes are no nherenl dscree and he se of models lke (), alhogh common n he aled lerare, s qesonable. If he dscree nare of s onl an arfcal consrc mosed b he qesonnare desgner, hen behaor cenres on he connos arable, raher han he obsered ndcaor. In hese cases, raher han -, shold carr he dnamc feedback f he dnamc eqaon s o be a descron of behaor. The aer has for man objeces. Frsl, (aboe and n secon ) we make he case for sng dnamcs n, raher han -, n alcaons where he dscreeness s obseraonal raher han nheren; and hen exlore he nerreaon of he model and s dnamc mlcaons. The second objece, whch s he sbjec of secons 3-4, s o consder denfcaon and roose a raccal mehod of esmaon. The hrd am s o se a rocedre for dealng wh he nal condons roblem, sng new secfcaon ess whch are roosed n secon 5. Forhl, we roose a new smlaon mehod of esmang he cross-derae marces reqred for hese ess; hs s descrbed n he aendx. Secon 6 of he aer resens an llsrae alcaon o a anel daa model of nddals fnancal execaons and secon 7 concldes. The model. The sascal srcre We work wh a behaoral model secfed n erms of he naral connos arables as follows: α β' x ε We refer o hs as he Laen Aoregresson (LAR) model. The ecor x s assmed srcl exogenos and nddals are samled ndeendenl from he nderlng olaon. We make he sandard assmon of Gassan random effecs so ha he nobserables and ε sasf he followng assmons: () - -

5 (, ε ) X (3) ε (4) ε ε s for eer s (5) ~ ε σ N, σ ε (6) where X (x,..., x T ). We onl obsere accordng o a gradng scale and hs: r ff [ Γr, Γr ), r... R (7) where Γ - and Γ R. Noe ha he hresholds Γ r wll be obserable n he case of neral censorng (sch as earnngs models for groed daa) or secfed as nknown arameers n he ordered rob case (sch as Lker resonses). In he laer case, he model s normalsed b omng he nerce from x and seng ar(ε ), whch s eqalen o ddng, α s no affeced b hs normalsaon.. Inerreaon of arameers, β, and ε hrogh b σ ε n (). Noe ha There are wo cases o consder. In models where he dscreeness arses hrogh neral censorng of a dnamc regresson (sch as an earnngs model aled o groed earnngs daa), he gradng hresholds Γ r are obsered and hs he scale of s deermned b he model. Conseqenl, he coeffcens β hae he sal regresson nerreaon as he nsananeos resonse E(, x, ) / x and he long-rn resonse s gen b E ( x, ) / x β /( α) as sal, where x s he long-rn sac ale of x for nddal. Boh of hese resonses are ndeenden of he ales aken b x and. The case of nobsered gradng hresholds s less smle. Here we are dealng wh arables lke he execed nflaon rae or he srengh of a sbjece resonse sch as job sasfacon or haness. In all sch cases he scale of s nobsered and we esmae β/σ ε raher han β. Conseqenl, he esmaed coeffcens are nerreable as E( [ / σ ε ], x, )/ x. In man alcaons ( haness, for examle) hs roblem s more fndamenal han a lack of denfcaon ndced b Exceons o hs general neglec of models nolng laen dnamcs are aers b Arellano e. al. (997) and Boer and Arellano (997). Howeer, he conex and models consdered n hose sdes s qe dfferen from he case consdered here, as s he aroach o esmaon

6 merfec obseraon: here s a lack of naral ns for haness or l whch renders he scale of β nherenl ambgos. In some cases, where here are naral ns of measremen for, we can fx σ ε a a reasonable hohecal ale. For examle, for an analss of sre resonses o a qeson abo execed nflaon we mgh reasonabl se σ ε a (sa) half a ercenage on o allow a rogh b drec nerreaon of he model n erms of he naral ns. Noe ha α s denfable ndeendenl of σ ε. As a conseqence, we can esmae nambgosl he seed of adjsmen. For examle, followng a shock, he rooron of dseqlbrm whch s elmnaed whn s erods s -α s and hs s naffeced b normalsaon..3 Dnamcs The SD and LAR rocesses () and () ml dfferen aerns of dnamc behaor. Consder he followng arfcal examle: SD model:.8 x ε (8) LAR model: x ε (9) where x.5, ε ~ N(,) and ( > ). The arameers of he LAR rocess (9) hae been chosen o rerodce exacl hree roeres of he SD rocess (8): () Pr( ).877; () Pr( x)/ x.46; () Pr( - ).7. Wh he LAR arameers chosen n hs wa, he dsrbons of rn lenghs n saes and are dencal for he wo rocesses. Howeer, he relaonsh beween sccesse rn lenghs s no. Ths s refleced n he aocorrelaon fncons (Fgre ). As we wold exec, he LAR model has mch hgher aocorrelaons han he SD model for. For he obsered, he ACF decas faser for he SD han he LAR rocess, dese he fac ha he hae he same s-order aocorrelaon b consrcon. Ths, an LAR model wll dsla greaer erssence han an obseraonall smlar SD model, n hs qe sble sense. Condons () and () are mosed analcall o deermne β and β for gen α; Mone Carlo smlaon was hen sed o fnd he ale of α o eqalse he s order aocorrelaons

7 SD model: ACF() SD model: ACF() LAR model: ACF() LAR model: ACF() LAG ORDER Fgre ACFs for he SD and LAR models The wo models also dffer n erms of he mled dnamc mller effecs of x on. To llsrae hs, consder agan he bnar case and focs on wo moran feares: he mac on Pr( -, X, ) of swchng he condonng een from - o - ; and he mac of he hsor of {x } on he robabl of a ose resonse, who condonng on -. For he former, he SD model s relael smle: Pr(, X, ) Pr(, X, ) Φ ( α β' x ) Φ( β' x ) where Φ(.) s he cdf of he N(,) dsrbon. For he LAR model, we hae nsead: () Pr(, X, ) Pr(, X, ) Pr( Pr(, X, ) X, ) Pr( X, ) Pr( Pr(, X, ) X, ) Pr(, Pr( X, ) Pr( X, ) X, ) Pr( [ Pr( X, )] X, ) () Assme he rocess () s sable and long-esablshed. Then: s s s α β ' x s α ε s () α s - 5 -

8 and herefore Pr(, - X, ) Φ (µ, µ - ; α) and Pr( X, ) Φ(µ ), where Φ (.,.;α) s he barae sandard normal cdf wh correlaon α and µ s he scaled condonal mean (-α ) / [ s α s β x -s /(-α)]. Ths: Pr(, X, ) Pr( Φ( µ, µ ; α) Φ( µ ) Φ( µ, X, ) Φ( µ ) [ Φ( µ )] ) (3) The moran dfference beween () and (3) s ha he former deends onl on he crren ecor x, whereas he laer deends on he enre hsor of x. Consder now he alernae smmar measre, Pr( X, ). The LAR rocess ges a relael smle form: Pr( X, ) Φ( µ ) (4) mlng ha he lagged margnal resonse decas geomercall: X Pr(, ) s φ( µ ) α α β x s (5) where φ(.) s he sandard normal df. For he sae-deendence model, we can wre: Pr( X, ) Pr( Pr( Rearrange and wre hs as a recrson: P P where: P Pr( X, ) δ Φ(αβ x ) - Φ(β x ) ρ Φ(β x ). Solng back o an arbrar erod : P, X, X, ) Pr(, ) Pr( X X, ), ) (6) δ ρ (7) P j j j j j s s s δ ρ δ (8) where we se he conenon δ. On reasonable assmons abo he x- rocess, solng back ndefnel leads o he followng reresenaon: j j P s s s ρ δ (9) j j - 6 -

9 Ths: Pr( X x s, ) ρ x s s j s δ j k s s j δ φ ρ δ j ( β' x ) k s s k j k s δ j ρ δ s k β k j δ j δ x [ φ( α β' x ) φ( β' x )]β () The rofle of Pr( X, )/ x -s s hs consderabl more comlcaed han he geomerc deca mled b he SD model (). s s s s 3 Esmaon 3. Inal condons In he SD model, here are wo alernae aroaches for dealng wh he random effecs. Heckman (98b) secfes an aroxmaon o he dsrbon of X,, and hen deres he dsrbon of... T, X, sng seqenal condonng. The random effecs are hen negraed o b nmercal qadrare. The alernae aroach, sed b Wooldrdge () s o secf nsead he dsrbon of, X. A sem-aramerc aran de o Arellano and Carrasco (3) noles he seqence of condonal means E(...,x... x ) λ, whch are esmaed as nsance arameers. The laer aroach has man adanages n models lke () b s roblemac n LAR models, where he lagged deenden arable s no obserable and canno be condoned on. Condonng on s obserable conerar comlcaes maers enormosl. For hs reason, we se he Heckman reamen of nal condons, ogeher wh an exlc hohess esng rocedre o conrol he bas ndced b aroxmaon error n he assmed dsrbon of X,. Assme ha we obsere and x oer a erod T. The LAR rocess () mles he followng dsrbed lag reresenaon: 3 3 In he case where he Γ r are no obserable, we mose he normalsaon σ ε and henceforh and σ are re-nerreed accordngl., β - 7 -

10 α s s α s α β ' x α ε () s α s s Ths s a sefl bass for esmaon f eher s sffcenl large and α decas sffcenl radl wh or f we can fnd a good emrcal aroxmaon for Wre hs aroxmaon o X, as:. δ' w γ η () r ff [ Γr, Γr ), r... R (3) where w s a ecor consrced from X ; δ and γ are arameers and, n he ordered rob case, Γ r ma dffer from Γ r. The random erm η sasfes he followng assmons: η X (4) η ε X for eer > (5) X N(, σ η ) (6) Noe ha, een n he ordered rob case, η s no normalsed o hae n arance. In rncle, he ecor w ma conan all dsnc elemens of {x, X }. Howeer, n racce ma be fond ha w x s adeqae, or ha lmed T smmares, sch as w {, x T x }, work well. Ths s essenall an emrcal sse. Wh aroxmaon ()-(3), eqaon () becomes: s s δ ' w α β' xs c α εs α η (7) s s α where c ( - α )/ ( - α) α γ. The model now consss of eqaon () and a se of eqaons (7) for an collecon of erods >. In racce, he nal condons model () s onl an aroxmaon and s a oenal sorce of secfcaon error. Howeer, f α < so ha α as, hen he nflence of he nal condons declnes as we consder laer erods. There s, herefore, a case for leang a ga (of S erods) beween he nal erod and he sbseqen erods sed o esmae he LAR model. Conseqenl, we work wh a ssem of (T-S) eqaons conssng of () and (7) for S T. Daa on { S } are no sed. The choce of S noles a rade-off beween ossble mssecfcaon bas and effcenc, snce ncreasng S - 8 -

11 redces boh he nflence of nal condons and he amon of daa sed for esmaon. Increasng S also redces he scale of he comaonal roblem. Ths ssem s nonlnear n s arameers θ {α, β, δ, γ, σ, σ ε, σ η }, where σ ε n he case of obserable neral bondares. 3. Idenfcaon Consder he model wh nobsered gradng hresholds. Paron he coaraes no a common se of me-naran arables ζ and a seqence of mearng coaraes ξ, so ha x (ζ, ξ ). Assme a fll secfcaon of he nal condon (9), so ha w (ζ, ξ... ξ T ). Make he frher assmon ha he marx lm(n - w w ) s ose defne. An ordered rob model for on w wll conssenl esmae he normed coeffcen ecor δ/, where σ η γ σ. Consder eqaon (7), for an erod, >. Rewre n sandardsed form: α ( α ) β ω ( α) ζ ' ζ β ξ ' ξ αβ c ξ ' ξ s α ε α β... s s ξ α η / ' ξ (8) where β (β ζ, β ξ ), c σ (-α )/(-α ) α σ η and ω s he arable δ w / whch can be consrced from he coeffcens of he nal condons model (). Rewre (8) n smlfed noaon as: / ω b ξ υ (9) a ' ζ d ' ξ d ' ξ... d, ' Noe ha he coaraes (ω, ζ, ξ... ξ ) are (asmocall) non-collnear. Ths, ordered rob esmaon of (9) wll generae conssen esmaes of he scaled coeffcens (a, b, d,..., d -, ). Idenfcaon hen roceeds as follows. Frs, he ale of α can be consrced as an elemen of an of he ecors of raos d s /d s-,. If α s zero, he model becomes a sac random effecs ordered rob, so here s no new denfcaon sse; we consder he case α henceforh. Wh α known, β can be nferred o scale as g / g where g [b (-α)/(-α ), d ]. Ths, he ke behaoral arameers α and he drecon of he ecor β are essenall denfable from onl wo waes of he anel. The rao, R, of a o α ges he ale /, hs: - 9 -

12 - - R (3) The correlaon beween he random errors n eqaons () and (7), whch can be esmaed conssenl b jon esmaon or from he generalsed resdals, s ρ sasfng he followng: η σ α σ γ ρ c (3) Eqaons (3) and (3) are clearl nsffcen o deermne he hree remanng nknowns, γ, σ and σ η, so fll denfcaon reqres a leas wo waes of daa, n addon o wae. Consder he 3-wae case, where we hae daa for,,. Calclae each of he raos ( / ) as α /a. Usng he defnon (3), afer some manlaon he qan γ σ / can be exressed as: a a ρ ρ α σ γ (3) Noe ha a for α, so (3) s well-defned. Now exress as (A α γ) σ B α σ η, where A (-α )/(-α) and B (-α )/(-α ). Ths: B γ A A η σ σ α γσ α σ (33) We know he ale of γ σ / from (3) and we know a ror ha (γ σ σ η )/ s eqal o. Ths ges he followng ar of eqaons wh known rgh-hand sdes: A B A α γσ α σ,, (34) Noe ha he marx ) ( ) ( α α B A B A s non-snglar for all α, so here s a nqe solon for (σ / ) and (/ ). From hese, σ and are deermned. The ale of γ s hen gen b (3) and σ η b - γ σ, so all arameers are denfed. 3.3 SML esmaon Ths denfcaon argmen does no lead o an effcen esmaor, snce does no mose all he resrcons on he coeffcens (a, b, d,..., d -, ) n (9), nor does exlo he relaonsh beween he resdal correlaon ρ and he model

13 arameers. Insead we se a smlaed ML rocedre. Le he obsered ocome for be r, mlng [ Γ, Γ ). The lkelhood for hs se of eens s: r r ( r, r,..., r ) Pr( υ A, υ A,..., A ) Pr S S T T X S S υt T (35) where s υ c α ε s α η, s µ α δ' w α β' x s and A s he neral Γ µ, Γ ). The resdal ecor υ (υ, υ S υ T ) has a [ r r µ coarance marx wh elemens: ω γ σ σ (36) ω η γ cσ σ ηα, S < T (37) ω s mn(s,) s s cscσ α σ ε α σ η, S < (s, ) T (38) The robabl (35) s a (T-S)-dmensonal recangle robabl. Under normal, robables of hs knd can be calclaed sng he GHK smlaor (Hajasslo and Rd, 994), wh anhec acceleraon sed o mroe smlaon recson. We consrc he followng smlaed log-lkelhood fncon: n ln Lˆ( θ ) ln ˆ ( θ) (39) where P ˆ ( θ ) s he redced robabl (35) for nddal, esmaed sng he GHK algorhm. The smlaed lkelhood s maxmsed nmercall wh resec o θ. P 4 The exenson o hgher-order and ml-eqaon models Mos alcaons of he mehod roosed here wll be o sngle-eqaon models. Howeer, here s no dffcl n he generalsaon o a general J-dmensonal ssem of he redced-form eqaons 4 : J j α jk k β j ' x j ε j, j... J (4) k One moran wa n whch ml-eqaon ssems ma arse s hrogh hgherorder lags. Consder he model: 4 Noe ha he case of a srcral form wh conemoraneos feedback can be n he redced form (38) n he sal wa and hen esmaed sbjec o he nonlnear srcral resrcons on he redced form coeffcens α jk and β j. - -

14 α β' x ε α (4) Ths s eqalen o he -eqaon ssem: α α β' Ssem (4) s a secal case of (4), wh one nonsochasc eqaon and herefore a x ε snglar error coarance marx. We rern o hs examle below. where In marx noaon, he general ssem (4) becomes: A Bx ε (4) (43) ( J ). The coeffcen marces are A {α jk } and B (β β J ). The aroxmaon o he nal ales dsrbon s generalsed o: Dw G η (44) where D and G are coeffcen marces. The corresondng gradng hresholds are Γ and Γ, j... J, r... R j. jr jr The ndeendence assmons (3), (4), (4) and (5) are exended o he ecor case and we assme: X N(, Σ ) (45) ε X N(, Σ ε ) for eer (46) η X N(, Σ η ) (47) The jh dagonal elemen of Σ ε s normalsed o n f j has nobserable hresholds. The analoge of (7) s: s A Dw A Bxs C s s A A η s ε s (48) where C (I - A) - (I - A ) A G. Le he obsered ocome for j be r j, mlng j Γ, Γ ). The lkelhood for hs se of eens s: [ j, rj j, r j Pr ( r for j... J, Τ w, x,..., x ) j j Pr T ( [ Γ µ, Γ µ ) for j... J Τ) j j, rj j j, rj j, where Τ s he ndex se {, S T}, j and µ j are he jh elemens of he ecors s s C A ε s A η and A Dw A Bx s (49) µ resecel. The - -

15 coarance marx of he resdal ecor (, S T ) has a block srcre, where blocks (, ), (, ) and (s, ) are resecel: Ω GΣ G' (5) Σ η Ω GΣ C ' Σ ( A )' (5) η Ω s mn( s, ) s s C Σ C ' A Σ ( A )' A Σ ( A )', S < (s, ) T (5) s ε η The robabl (49) s a J(T-S)-dmensonal recangle robabl, ha can agan be aroxmaed b he GHK smlaor n moderael-szed ssems. In he secal case where mlle eqaons hae arsen from an orgnal model wh dnamcs of order hgher han here are redndances among he se of neqales defnng he robabl (49). For examle, n he model (4), he een r mles he een r - r wh robabl one. The T-S redndances of hs knd hales he dmensonal of he robabl (49). 5 Secfcaon ess How do we choose he nmber of anel waes, S, o sk? Consderaons of esmaon effcenc sgges a small ale for S, whle worres abo mssecfcaon bas nrodced b he nal condon aroxmaon sggess a large ale. To resole hs sse, I sgges se of a es whch examnes he conssenc of esmaes based on he waes S... T wh he obsered ocomes n wae S. If no sgnfcan conflc s fond for wae S, we hen redce he sk rae from S o S- waes and reesmae o mroe effcenc. Ths can be done seqenall nl a sasfacor on on he bas-effcenc radeoff s reached. We consder an aroach based on he score ecor for wae S-. 5 Ths aroach allows wae-secfc arameers sch as me dmmes. Wre he log-lkelhood fncon based on -daa for waes, S... T as L(ψ, τ ), where ψ s he sbecor of θ whch s common o all waes and τ s he ecor of an frher arameers denfable from he esmaon samle (call me dmmes for erods S... T-). Le L(ψ, τ ) be he log-lkelhood for an esmaon samle 5 Anoher ossbl s a Hasman arameer conras es, comarng he arameer esmaes reslng from skng S and S- waes. In racce, hs ofen enconers roblems arsng from non-osedefneness of he esmaed arance marx of he conras ecor. The Hasman es also reqres wo major esmaon ses

16 coerng onl waes and S. The ecor τ wll sall conan onl he coeffcen of a dmm for erod S, whch s essenall an nerce erm. Now maxmse L ( ψˆ, τ ) wh resec o he nknown wae-s arameer o ge ~ τ. Ths s a lowdmensonal (sall scalar) omsaon and relael eas o erform. Exandng he frs-order condon for ~ τ abo (ψ, τ ) ges: ~ l L ψˆ ψ L ( ~ τ lτ τψ ( ) ττ τ τ ) O () (53) Dfferenang L ( ψˆ, ~ τ ) wh resec o ψˆ ges: ~ l ( ˆ ) ( ~ ψ lψ n Lψψ ψ ψ n Lψτ τ τ ) O () (54) ~ ~ where: l τ and l ψ are he aral deraes of L ( ψˆ, ~ τ ) wh resec o ψˆ and ~ τ ; l τ and l ψ are he derae ecors of L(ψ, τ ); and L are he ψψ, Lψτ Lτψ ' and Lττ second-derae marces of n - L ealaed a he re arameer ales. Sandard lkelhood resls ml: -/ n (ˆ θ θ) n H( θ ) l ( θ) o () (55) where H(θ) s he Hessan marx of he mean log-lkelhood L/n; and l(θ) l (θ) s he score ecor. Usng (53)-(55), he normed second-sage score ecor for ψˆ s: ( ) ( ) [ ] / / / n l L L n l L L L L H( θ) ( n l( θ) ) () ~ / l ψ ψ ψτ ττ τ ψψ ψτ ττ τψ n o (56) Under H, he ecors -/ l τ n, -/ l ψ n and n -/ l(θ) conerge n dsrbon o a ~ lmng zero-mean normal dsrbon. Conseqenl, n / l ψ conerges o a normal dsrbon wh zero mean ecor and coarance marx esmaed conssenl b: where: n ~ V ξ ξ ' (57) n ~ ~ ~ ~ [ L L L L ] H(ˆ) θ l (ˆ θ) ~ ~ ~ ~ ξ l L L l (58) ψ ψτ ττ τ ψψ ψτ ττ τψ Here he sbscr denoes he score conrbon of he h obseraon and he lde denoes deraes ealaed a he on ( θ ˆ, ~ τ ). The score es sasc s hen: ~ ~ ~ LM n l ' V l (59) ψ whch has a χ dsrbon nder H wh degrees of freedom eqal o he dmenson of ψ. Ths es can be ewed eher as a secfc es for he resence of bas ndced ψ - 4 -

17 b he nal condons aroxmaon or more generall as a es of he secfed dnamc srcre relang sccesse waes. The man echncal dffcl wh he es s he comaon of he second ~ ~ derae marces L and. The las of hese s arclarl roblesome, owng ψτ L ψψ o s hgh-dmensonal. These marces are er comlcaed o calclae hrogh analcal formlae and nmercal aroxmaons o large Hessan marces end o ~ be er naccrae. In he mlemenaon descrbed below, we hae comed L ψψ ~ b means of a smlaon algorhm (descrbed n Aendx ) and L sng recrse cenral dfference aroxmaons. ψτ 6 An alcaon o nddal execaons daa The Brsh Hosehold Panel Sre (BHPS) s he rncal sorce of naonallreresenae hosehold- and nddal-leel anel daa n he UK. Ths alcaon s based on he frs waes, relang o he ears 99-. Each ear, BHPS arcans are asked a seres of qesons abo her ades. Here we analse resonses o he followng qeson, sng a samle of,77 male hosehold heads: How well wold o sa o orself are managng fnancall hese das? Resonses hae been recoded as: Fndng er dffcl ; Fndng qe dffcl ; 3 Js abo geng b, 4 Dong alrgh ; 5 Lng comforabl. Under he LAR model, he nddal s nderlng assessmen of hs fnancal oson a me s a narall connos arable,, whch we assme o be generaed accordng o he anel aoregresson (). The resonden s hen assmed o ranslae no a resonse o he caegorcal sre qeson accordng o he rle (7). The fnal arameer esmaes for hs LAR model are gen n Table. Comaon was done sng he GHK smlaor, sng sccesse asses, nall wh 5 relcaons (wh anhec arance redcon), rsng o 5 once he neghborhood of he omm was reached. Followng conergence, a sngle eraon was erformed wh relcaons as a check on conergence and he omsed lkelhood ale. Followng nal exermenaon wh alernae secfcaons, we sed a sbse of he x-arables from wae for he nal condons model. Esmaon was - 5 -

18 hen done seqenall, sarng wh S 8, so ha nall noled onl he - obseraons from waes, 9 and. The sk rae S was hen redced seqenall whle he score es remaned nsgnfcan. We enconered no rejecon a an sage, so or fnal secfcaon ses all aalable waes of daa. Conseqenl, he recangle robables noled n SML esmaon are -dmensonal. The analogos SD model s: where d ( m ) m 4 α md m β' x ε m (6). Esmaed arameers for he SD model are gen n Aendx Table A.. The were comed sng 48-on Gass-Herme qadrare. Dese he fac ha he SD model has 3 more arameers han he LAR model, he laer achees a sbsanall hgher log-lkelhood. In he model, economc crcmsances are reresened b he leel of hosehold ncome er caa, he rooron of hosehold ncome earned b he resonden hmself, and a dmm for owner-occaon, ogeher wh he esmaed ale of he eq n he hose. As execed, he leel of hosehold er caa ncome has a sgnfcan ose effec on erceons of fnancal well-beng. The magnde of he resonden s ersonal conrbon o hosehold fnances has a sgnfcan ose nflence on hs reored erceons. There s srong edence o sor he wdel-held ew ha homeowners erceons resond o rsng hose ales. Hman caal also aears o be an moran elemen n erceed fnancal well-beng, snce here s a srong ose nflence of edcaonal aanmen. Recen changes n crcmsances are reresened b he frs dfferences n er caa hosehold ncome, he resonden s own ncome and he esmaed hose ale. None of hese s sascall sgnfcan. Howeer, hese objece fnancal facors are no sffcen o exlan he deermnaon and eolon of erceed fnancal well-beng. Oher exlanaor arables are mosl are me-naran. The small nmber of me-arng coaraes are nclded n he form of crren leels and changes from he reos ear. Ehnc s reresened b dmmes for he Black and Asan gros and here s edence of a negae dfference, whch s sascall sgnfcan for he laer gro. The effec of maral sas s cared b dmmes for beng marred/cohabng, dorced/searaed or wdowed. A frher dmm denfes hose - 6 -

19 who hae made a ranson no he dorced/searaed gro whn he las ear. Oher sas ransons were nsgnfcan or oo few n nmber o erm relable esmaon. There s a sgnfcan ose nflence of a maral or cohabaon relaonsh and of wdowhood. Dorce or searaon redces erceed well-beng, wh a frher emorar redcon n he ear of searaon. Labor marke sas s reresened b dmmes for emlomen and self-emlomen, wh sgnfcan ose effecs. The sas of nemlomen has a sgnfcan negae effec, wh a frher emorar effec n he ear of ranson no nemlomen. Dfferences n hosehold sze and srcre are moran. There s a sgnfcan ose coeffcen for he nmber of hosehold members and a nearl offseng negae coeffcen for he nmber of chldren n he hosehold. There s no deecable mac of a new brh on fnancal erceons. These effecs are n addon o he er caa eqalsaon sed for he hosehold ncome arables. The relaonsh beween erceed well-beng and age s nerse U-shaed wh a eak a he 5-65 age gro, reflecng a sandard lfe-ccle aern of asse accmlaon. Healh sas has no sgnfcan mac. The ear dmmes show a srongl rsng rend from wae (99) o wae 9 (), followed b an abr fall n. These ear effecs reflec qe closel he macroeconomc rend n aerage earnngs growh. Dnamc adjsmen s cared b he aoregresse coeffcen α, whch s ose and srongl sgnfcan. On hs edence, here s a sgnfcan degree of erssence n erceons. The comarson beween he LAR and SD esmaes shows ha he laer generaes oo lle erssence hrogh he nheren dnamcs and comensaes for hs mssecfcaon b oeresmang he arance of he nddal effec. In or alcaon, he nra-erson correlaon, σ /(σ ), s esmaed o be.73 for he LAR model, comared o.46 for he SD model. The lagged resonses of o x deca raher faser n he SD model. For examle, consder he robabl of a good or er good resonse ( > 3). Le δ(s) be he derae Pr( >3 X)/ β x -s ealaed a he on x x - x -... x and consder he scaled seqence δ (s) δ(s)/δ(). We fnd δ ().33 and.79 for he LAR and SD models resecel, decang o δ (). and.76 and δ (3).36 and.. These are sbsanal dfferences. For alcaons o daa dslang greaer erssence han s aaren here, he dfference beween SD- and LAR-esmaed dnamcs cold be er moran ndeed

20 Table Esmaes and ess (sandard errors n arenheses) Dnamc model Inal condons Coarae βˆ Sd. err. ˆδ /σˆ η Sd. err. α.33. Black Asan In relaonsh Dorced/searaed Wdowed Newl dorced/searaed/wdowed Emloed Self-emloed Unemloed Newl nemloed Degree or oher frher edcaon A-leel O-leel / GCSE / CSE / oher qalfcaon Hosehold sze Nmber of chldren n hosehold nmber of chldren Homeowner Annal hosehold ncome er head ( -3 ) hosehold ncome er head ( -3 ).5.6 own ncome ( -3 ) Own ncome as share of hosehold ncome Vale of hose ( -5 ) Prooronae change n ale of hose..5 Age Age Age Age Poor healh Newl-deeloed ll-healh.36.8 Wae dmm Wae dmm Wae 3 dmm Wae 4 dmm Wae 5 dmm.4.5 Wae 6 dmm.7.5 Wae 7 dmm.8.5 Wae 8 dmm Wae 9 dmm -..5 Γ Γ Γ Γ σ γ.5.8 σ η Score es for he S model χ (59) (P.368) Log-lkelhood -4,

21 7 Conclsons We hae consdered an alernae o he dscree sae deendence (SD) model for dnamc modellng of ordnal arables from anel daa. The alernae LAR model noles ordnal obseraon of a laen aoregresson, raher han lagged feedback of he reos erod s dscree ocome. I s arged ha hs secfcaon s more arorae for a range of alcaons nolng obseraonal, raher han nheren, dscreeness. Examles nclde neral regressons and models of execaons, and sasfacon. We hae deeloed a smlaed maxmm lkelhood esmaor and an assocaed es rocedre desgned o asss n handlng he nal condons roblem. As ar of hs rocedre, a noel smlaon algorhm has been mlemened for comng a reqred nmercal Hessan marx. The mehod has been aled o a smle model of nddal erceons of fnancal well-beng, aled o UK hosehold anel daa. The LAR model rodes a robs descron of he eolon of fnancal erceons oer me, wh a sgnfcan role for lagged adjsmen. The LAR model fs he daa consderabl beer han he conenonal SD model and has qe dfferen eqlbrm and dnamc roeres. In arclar, he SD model generall dslas less erssence han he LAR model, and when mssed o model hghl-erssen daa, he esmaed arance of he nddal effec s based wards o comensae. References Arellano, M. and Carrasco, R. (3). Bnar choce anel daa models wh redeermned arables, Jornal of Economercs 5, Arellano, M., Boer, O. and Labeaga, J. M. (997). Aoregresse models wh samle selec for anel daa, nblshed workng aer (h://rhfd.eresmas.ne/obmajl97.df) Boer, O. and Arellano, M. (997). Esmang dnamc lmed deenden arable models from anel daa, Inesgacones Economcas, Chow, G. S. (96). Tess for eqal beween ses of coeffcens n wo lnear regressons, Economerca 8, Ferrer--Carbonell, A. and Frjers, P. (4) How moran s mehodolog for he esmaes of he deermnans of haness? Economc Jornal 4,

22 Hajasslo, V. and Rd, P. (994). Classcal esmaon mehods for LDV models sng smlaon, n Engle, R. F. and McFadden, D. L. (eds.) Handbook of Economercs 4, Amserdam: Norh-Holland. Heckman, J. J. (978). Smle sascal models for dscree anel daa deeloed and aled o es he hohess of re sae deendence agans he hohess of sros sae deendence, Annales de l INSEE 3, Heckman, J. J. (98a). Sascal models for dscree anel daa, n Srcral Analss of Dscree Daa wh Economerc Alcaons, Mansk, C. F. and McFadden, D. (eds.), Cambrdge MA: MIT Press. Heckman, J. J. (98b). The ncdenal arameers roblem and he roblem of nal condons n esmang a dscree me-dscree daa sochasc rocess, n Srcral Analss of Dscree Daa wh Economerc Alcaons, Mansk, C. F. and McFadden, D. (eds.), Cambrdge MA: MIT Press. Van Praag, B. and Ferrer--Carbonell, A. (4). Haness Qanfed. A Sasfacon Calcls Aroach. Oxford: Oxford Uners Press. Wooldrdge, J.M. (), A framework for esmang dnamc, nobsered effecs anel daa models wh ossble feedback o fre exlanaor arables, Economcs Leers 68,

23 Aendx : A smlaon aroxmaon o L ~ ψψ Or am s o esmae he robabl lm of he aral Hessan n L ( ψˆ, ~ τ ) / ψˆ ψˆ '. Le V be an asmocall ald aroxmaon o he coarance marx of ψˆ : for examle, we mgh se V n H(ˆ) θ. Snce V reflecs he arabl of ψˆ and s O (n - ), rodes a good merc for he aroxmaon of deraes. Le K be he Cholesk facor sch ha V KK. Noe ha K O (n -/ ). Now generae a seqence of ndeenden sedo-random N(, I) ecors M and m m consrc z n ( L ( ψˆ K, ~ τ ) L ( ψˆ, ~ τ )) µ, where µ s a selengh n arameer. Noe ha z m, and an coarance (wh resec o he dsrbon of m ) of z m wh owers of m, are O (n -/ µ ). Exand z m n a Talor seres: z m µ m m m d D q q n q> n n µ µ D m m ( ) ζ where d and D q are elemens of d K' ( n L ( ψˆ, ~ τ ) / ψˆ ) D n K ( n L ( ψˆ, ~ τ ) / ψˆ ψˆ ') K ' are O (). We hen hae he followng resls: E E E (A) n and, and ζ s a remander erm. Noe ha d and D m m µ m m ( z ) d E( ζ ) (A) n m m m m m m ( z ) D E( ζ ), q q µ n q m m m m ( z ( ) ) 3D D E ( ) ζ ) µ qq n q q (A3) (A4) where he execaons are aken wh resec o m. If necessar, he rocess for generang he araes m shold be arorael m rncaed o ensre ha L ( ψˆ µ K, ~ n τ ) alwas exss and ha erms of he form 3 ( ( L ψ ψ ψ ) E exs for an fncon ψ () bonded b j / q r ψ ψ ( ) ( ψ ˆ, ψˆ µ K). In racce, hs reqres ensrng ha smlaon of m ˆ aods ψ µk regons where he Γ r are non-ordered. A sffcenl small ale for µ (for examle - -

24 .) wll generall achee hs. Under hese crcmsances, he remander erms n (A)-(A4) are o (). Ths we can esmae he cross-arals conssenl as follows: Dˆ q n µ M m m m M m z q, q (A5) To calclae he remanng second deraes, we sole he smlaed analoge of he ssem of eqaons (A4), o ge: Dˆ n M µ Now ransform D back o ψ-sace: M P m m z m P q m q (A6) Lˆ ψψ n K' D K (A7) One adanage of hs mehod s ha relcaons can be made seqenall and he rocedre soed when he qanes (A5) and (A6) aear o hae reached conergence. The smlaed analoge of exresson (A) can be comared wh a conenonal nmercal derae o check on he neglgbl assmon for he remander erms. Anhec arance redcon can be sed o mroe smlaon recson n (A5) and (A6). - -

25 Aendx : SD model esmaes Table A. Esmaes for he SD model (sandard errors n arenheses) Dnamc model Inal condons Coarae βˆ Sd. err. ˆδ /σˆ η Sd. err. α.86.8 α.63.7 α α Black Asan In relaonsh Dorced/searaed Wdowed Newl dorced/searaed/wdowed Emloed Self-emloed Unemloed Newl nemloed Degree or oher frher edcaon A-leel O-leel / GCSE / CSE / oher qalfcaon Hosehold sze Nmber of chldren n hosehold nmber of chldren Homeowner Annal hosehold ncome er head ( -3 ) hosehold ncome er head ( -3 ) -..6 own ncome ( -3 ) -..9 Own ncome as share of hosehold ncome Vale of hose ( -5 ) ale of hose ( -5 )..5 Age Age Age Age Poor healh Newl-deeloed ll-healh.9.8 Wae dmm Wae dmm Wae 3 dmm Wae 4 dmm Wae 5 dmm.3.5 Wae 6 dmm..5 Wae 7 dmm..5 Wae 8 dmm Wae 9 dmm Γ Γ Γ Γ σ.684. γ Log-lkelhood -4,