Inverse Test Confidence Intervals for Turning points: A Demonstration with Higher Order Polynomials. J. N. Lye and J. G. Hirschberg.

Transcription

1 Deparmen of Economcs Workng Paper Seres Inverse Tes Confdence Inervals for Turnng pons: A Demonsraon wh Hgher Order Polynomals J. N. Lye and J. G. Hrschberg Sep 0 Research Paper Number 60 ISSN: ISBN: Deparmen of Economcs The Unversy of Melbourne Parkvlle VIC 300

2 Inverse Tes Confdence Inervals for Turnng pons: A Demonsraon wh Hgher Order Polynomals. Absrac: J. N. Lye and J. G. Hrschberg Sepember 0 In hs paper we demonsrae he consrucon of nverse es confdence nervals for he urnng pons n esmaed nonlnear relaonshps by he use of he margnal or frs dervave funcon. Frs, we oulne he nverse es confdence nerval approach. Then we examne he relaonshp beween he radonal confdence nervals based on he Wald es for he urnng-pons for a cubc, a quarc and fraconal polynomals esmaed va regresson analyss and he nverse es nervals. We show ha he confdence nerval plos of he margnal funcon can be used o esmae confdence nervals for he urnng pons ha are equvalen o he nverse es. We also provde a mehod for he nerpreaon of he confdence nervals for he second dervave funcon o draw nferences for he characerscs of he urnng-pon. Ths mehod s appled o he examnaon of he urnng pons found when esmang a quarc and a fraconal polynomal from daa used for he esmaon of an Envronmenal Kuznes Curve. The Saa do fles used o generae hese examples are lsed n he appendx along wh he daa. Key words: Inverse Tes Confdence Inervals, Lkelhood Profle, Quarc, Fraconal Polynomals, Saa, Margnal Effec Funcon, Feller Mehod. We would lke o hank he anonymous referee for a number of commens ha proved very helpful n he preparaon of hs verson of he paper. We also wsh o hank parcpans a he Economerc Socey Ausralasan Meeng held n July of 0. Ths research s suppored by research funds suppled by he Faculy of Busness and Economcs, Unversy of Melbourne. The responsbly for any errors s enrely ours. Deparmen of Economcs, Unversy of Melbourne, Melbourne, Vc 300, Ausrala.

3 . Inroducon Regresson models n many applcaons employ a funconal specfcaon whch allows for a nonlnear relaonshp beween a regressor and he dependen varable. The mos common of hese s he quadrac specfcaon wh a wde applcaon of more complex hgher order polynomals and sem-paramerc mehods where par of he specfcaon s lnear and par s allowed o be flexble funcons of x. Typcally hese models are of he form: M y ( x ) z 0 j j j In hs specfcaon we assume ha he nonlnear specfcaon s n he varable x whle he oher regressors (z) are assumed o have a lnear nfluence on he dependen varable. 3 An mporan characersc of hese models s ha hey possess he propery of a non-consan margnal effec of he x varable over he values of x. In hs paper we demonsrae ha he confdence nervals of he margnal effec funcon wh respec o x can be used for nferences concernng he urnng-pons (somemes referred o as he saonary pons). We also show ha hese nervals are equvalen o a class of prevously defned nferenal mehods. In addon we demonsrae how he margnal of he margnal effecs funcon, or more drecly he second order funcon can be used for nferences n hgh order models. In hs dscusson we follow he mplcaons of he consderaon of he dervave funcons mpled by hese nonlnear specfcaons. 3 In hs dscusson we assume ha here s only one varable o be enered n a nonlnear form. In mos cases he exenson o more han one can be accommodaed n a smlar manner o he echnques we dscuss here.

4 Margnal effecs or dervave funcons of nonlnear regresson specfcaons have a sgnfcan hsory n he nerpreaon of regresson resuls. The nerpreaon of he margnal effecs s cenral o many economerc applcaons and s exensvely covered n exs such as Wooldrdge (00, chaper ). Recenly, he sascs package Saa has nroduced a seres of pos-esmaon procedures o compue margnal effecs and funcons based on he parameers esmaed from a wde varey of models. The orgnal verson of he currenly avalable margns roune was a user wren roune as descrbed n Barus (005). An early exposon of he margnal effecs for polynomal regressons can be found n Wllams (959, chaper 6), however hs reamen has been rarely referenced and s only recenly ha he esmaon of hgh order polynomals has become wdespread. In he case of he quadrac specfcaon he locaon of he maxmum or mnmum s defned as -/ mes he rao of he parameer on he lnear erm dvded by he parameer on he squared erm. Thus o draw nferences concernng he locaon of he urnng-pon requres he consrucon of confdence nervals for he rao of regresson parameers. However, Dufour (997) n hs survey of Impossbly Theorems n Economercs observes ha he drec applcaon of he Wald es o he rao of parameers devaes arbrarly from any approxmang dsrbuon because s a funcon of he daa o whch s appled. He hen proposes ha a vald nerval can be found usng he Feller (944) approach as he nverson of he approprae lkelhood rao es and demonsraes ha he nerval s defned as he roos o a quadrac expresson. In defnng he general form of Feller s mehod for he consrucon of confdence nervals for raos of lnear funcons of regresson parameers Rao (973) and Zerbe (978) oban equvalen expressons (equaons 4b..6 and.5-.7 respecvely). Applcaons of he Feller 3

5 echnque for he nferences for varyng ncome elasces from Engle curve esmaon can be found n Hrschberg e al (008). The nverson of a wo sded es for a lnear combnaon of he regresson parameers s no only lmed o he consderaon of raos of parameer values. In he examnaon of he exsence of urnng-pons n nonlnear funcons esmaed by polynomal regressons, Wllams (chaper 6, 959) and Hellbronner (979) demonsraed ha he confdence nerval of he locaon of he urnng-pons for a cubc regresson could be consruced by nverng he es for he locaon of he value of he regressor where he frs dervave funcon was equal o zero. Because he frs dervave funcon of he cubc s a quadrac ha mples wo urnng-pons, he nverson es used o consruc her confdence nervals, nvolves he soluon of he four roos for a quarc o defne he nervals of neres. 4 In he parcular applcaon consdered by Hellbronner, he lmed hs neres o only he nerval ha lay n he range of he x varable. Wllams also employs he second dervave o locae he pon of maxmum rae of change and alhough he alludes o he applcaon of nverse ess, he does no dscuss he deals of s mplemenaon. In hs paper we demonsrae he uses for he nverson of he es for a sngle lnear combnaon of parameers o draw nferences for a se of relaed nonlnear specfcaons. Vandaele (98) has shown ha hese nverse -ess are equvalen o he profle lkelhood ess for hese cases snce hese ess are equvalen o he correspondng lkelhood rao ess. Frs we wll oulne he concep behnd he nverson es. Then we dscuss he applcaon of he Feller/lkelhood profle es ha nvolves he nverson of he - es for he locaon of he urnng-pons (n hs case an exremum) pon defned by 4 Wllams refers o he nverse es confdence nervals as fducal lms. 4

6 he esmaed quadrac regresson. We hen nroduce he cubc and demonsrae ha nferences concernng he locaon of urnng-pons can be found va he nverson ess. We hen exend he cubc o consder he quarc specfcaon ha has recenly been found n more applcaons n economcs. Ths dscusson s followed by he consderaon of an example of he use of a quarc. The las secon nroduces a more general specfcaon referred o as a fraconal polynomal. Ths specfcaon dffers from a radonal polynomal n ha all he hgher order dervaves are no zero by assumpon. We hen demonsrae how hs mehod can be appled o he same daa examned usng he quarc. The Saa rounes for hese analyses are gven n he appendces along wh a lsng of he daa used n he applcaons.. The nverse es confdence nerval The nverse es confdence nerval s found by frs defnng a relaonshp for a crcal value of a es sasc for a varable of neres and hen solvng for he values of he varable ha defne he lms of he nerval. For example, a es of a lnear combnaon of he mean vecor of normally dsrbued random k dmensonal vecor ~ N c μ,ω defned as R kμ r we can propose he followng - k k kk sded hypohess es: H : R 0 k H : R k μ r μ r And he usual confdence nerval for he scalar r s gven by: CI Rc z RΩR () Where z s he crcal sandard normal value for a (-α) confdence nerval. Under H 0 we can consruc he es sasc defned by: T Rc r RΩR () 5

7 To esablsh f he H 0 can be rejeced or no we need o deermne f T z. Alernavely, one could nver he problem by deermnng he crcal values for r referred o as r, where he null hypohess would be rejeced a he level of sgnfcance as defned by sasc ( T ) s gven by: Rc r z z. Thus he mpled value of he es T (3) RΩR The nverson nvolves he soluon o a quadrac n r defned as: Rc r z RΩR 0 (4) The wo soluons for r defne he nerval. 5 ˆ U L r, r rz RΩR, where rˆrc (5) In hs case he soluon s rval. To paraphrase Molère's Bourgeos Genlhomme who dscovers ha he s already speakng prose, we can now observe ha we have been usng nverse confdence nervals all along. However, n a number of cases he soluon s no equvalen o he radonal Wald es approach where we add and subrac a mulple of he sandard error of he parameer esmae o he esmae. Consder a relaed case where he mulplcave marx R s a funcon of an unknown parameer and r s gven. In hs case we wsh o es he value(s) of for whch R( ) μ r usng he wo-sded es defned by: H 0 : R( ) μ r H : R( ) μ r One way o proceed s o use esmaes of μdefned by c and o solve R( ˆ ) c r for esmae(s) of referred o as ˆ when s possble o solve 5 We wll follow he convenon where we use he crcumflex (^) o denoe he esmaed locaon and he lda (~) o denoe he esmaed nerval lms. 6

8 R (,) r c. We could hen consruc a Wald es wh an esmae of he varance of ˆ he ˆ derved from he applcaon of he dela mehod o fnd a lnear approxmaon for R (,) r c. 6 In hs case he nverse es provdes an alernave soluon. Agan specfy he es sasc under he null (noe ha we expec up o wce he number of values for he value of o defne he nerval(s) for all he soluons for R (,) r c ). T z R( ) cr R( ) ΩR( ) (6) Thus we can wre hs expresson as a quadrac n R( ) gven by: R( ) ccz Ω R( ) rcr ( ) r 0, (7) whch can be solved for he confdence nerval(s). Snce hs s a quadrac form here are now poenally wo soluons for every ˆ. Noe ha for any value of, here s no guaranee ha hey wll be real valued. In hs paper we nvesgae a parcular se of cases where he null hypohess of neres can be defned as R( ) c 0 and where he resrcon vecor can be wren as a polynomal funcon n and c s he vecor of esmaed regresson parameers. Typcally we have he lnear funcon vecor defned as: k R( ) (8) j R0 R R Rj Rk Thus n he cases deal wh here here are k poenal confdence bounds defned by he roos o he polynomal defned by he soluon o: z R( ) cc Ω R( ) 0 (9) 6 Noe ha he analyc soluon may no be avalable as we fnd n our dscusson below. As an alernave o he dela mehod one could use a smulaon or emprcal boosrap consruced by draws from an esmae of N μ,ω. 7

9 The graphcal equvalen ha we demonsrae s found from he plo of he confdence nerval defned by: CI( f ) R( ) cz R( ) ΩR ( ) (0) From whch we can locae he values of where he upper and lower bounds cross he zero reference lne o defne he The Quadrac Specfcaon The quadrac funconal form s wdely used n regresson o denfy a U or nvered U-shape relaonshp beween a regressor and he dependen varable. Many auhors clam o have found such a relaonshp by focussng of he sgnfcance and sgns of he esmaed coeffcens correspondng o he quadrac (Clark e al. 996) and possbly also examnng wheher he poson of he exremum falls whn he range of he daa (see Rchmond and Kaufman 006). Dufour s (997) admonon nowhsandng, a number of auhors connue o use he dela mehod o calculae a sandard error for he exremum and hen consruc symmerc confdence nervals (e.g. Kearsley and Rddell 00). 8 Alhough here has been a consderable number of Mone Carlo sudes ha record he smlary of he confdence nervals obaned usng he Feller and he dela has been shown ha he coverage of he dela confdence nerval s relaed o he characerscs of he parcular sample used n esmaon (Hrschberg and Lye 00a). More recenly, auhors have used confdence nervals for he exremum based on he Feller mehod as well as he Sasabuch 7 The case where hey do no cross he zero reference lne wll ndcae he suaon where no real roo for he equvalen polynomal defned by (9). 8 We refer o he dela as he Wald es confdence nerval derved by combnaon of he dela approxmaon for he varance of he rao wh he defnon of he confdence nerval by addng and subracng a mulple of he esmaed sandard error. 8

10 (980) nersecon-unon es (see Schnabel, C. and J. Wagner 008; Huang e al 0). 9 The quadrac specfcaon s ypcally wren as: M y x x 0 jzj () j Agan we assume ha here s a sngle regressor x enered as a quadrac funcon, M lnearly relaed regressors denoed by z j, and N observaons. By seng he margnal effec or he frs dervave of E yxz, wh respec o x, yxz x x, equal E, o zero, we can derve he value of x ha defnes he exremum of he relaonshp beween y and x gven values for he oher regressors, as. s eher he maxmum (when 0) or he mnmum (when 0 ). These condons are esablshed va he sgns of he second order dervave of he funcon where x. In hs case E y, z x, hus he sgn of wll esablsh he naure of he exremum. 0 ˆ The usual esmae of he exremum s gven by ˆ, where ˆ and ˆ are he OLS esmaes of and n () respecvely. The pon defned by ˆ also defnes he value of x where he margnal effec of x on y changes sgn. 3. The Dela mehod Appled o he Quadrac The dela mehod for drawng nferences abou a nonlnear funcon such as, s o esmae s varance by use of a frs order Taylor seres expanson o lnearze ˆ 9 Lnd and Mehlum (00) observe ha he Sasabuch (980) es a he level of sgnfcance s equvalen o deermnng f he confdence nerval for he exremum s whn he range of he x varable. 0 Ths also jusfes he use of he es of he null hypohess ha 0 o esablsh wheher an nflecon pon s presen or no and o jusfy he consderaon of hs funconal form. 9

11 he non-lnear funcon. In hs case he esmaed varance of ˆ s gven by (see, for example, Rao 973, pp ): - ¼ - ˆ ˆ ˆ ˆ ˆˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ 4 ˆ ˆ 4ˆ ˆ () where ˆ s he esmaed varance of ˆ and ˆ j s he esmaed covarance beween ˆ and ˆ j. A 00( )% confdence nerval for s defned by: ˆ ˆ ˆ ˆ ˆ ˆ ˆ U L (3) dela, ˆ 4 4ˆ where s he value from he dsrbuon wh an level of sgnfcance and % N k degrees of freedom. Ths s commonly referred o as a dela mehod confdence nerval and can readly be compued usng pos-regresson sofware rounes avalable n mos sascs packages. 3. The Inverson mehod Appled o he Quadrac An alernave approach o consruc a confdence nerval for s found by nverng he -es assocaed wh he null hypohess: H : 0 0 H: 0 A 00( )% confdence nerval for s obaned by solvng T ˆ ˆ 4 4 (4) We wll subsue for he F wh one degree of freedom n he numeraor. 0

12 The soluon s found by solvng 0 where a 4ˆ ˆ, a bc b ˆˆ 4 ˆ and c ˆ ˆ ˆ parameer esmaes as: ˆ ˆ and ˆ ˆ, where we defne he -raos for he ˆ. Then, he roos can be defned as: ˆ ˆ ˆ ˆˆ ˆˆ ˆ ˆ ˆ ˆ ˆ, (5) U L Feller If ˆ, hen(5) has wo real roos and a fne confdence nerval s consruced. Ths condon corresponds o beng able o rejec H 0: 0when s he level of sgnfcance (Buonaccors 979). In addon o he fne nerval case, he resulng confdence nerval may be he complemen of a fne nerval when (b 4ac > 0, a < 0) or of he whole real lne when b 4ac < 0, a < 0. These condons are dscussed n Scheffé (970) and Zerbe e al (98). Ths approach resuls n a confdence nerval ha s equvalen o he Feller mehod (Feller 944) whch provdes a general procedure for consrucng confdence lms for he rao of wo parameers, esmaed by normally dsrbued random varables. Graphcally, (as shown n Hrschberg and Lye 00b) he 00( )% confdence nerval for followng he nverson mehod, s obaned by plong an esmae of he frs dervave as a funcon of x ˆ ˆ x wh s correspondng 00()% confdence nerval defned as: E yx, z CI ˆ ˆ ˆ ˆ ˆ x x + 4 x + 4x (6) An esmae of he exremum value, x ˆ, s found by solvng he equaon ˆ ˆ ˆ 0. Smlarly, he bounds ha defne a 00( )% confdence nerval on are found by solvng for x n he relaonshp:

13 ˆ ˆ x ˆ ˆ ˆ x x (7) whch s equvalen o solvng he roos of he equaon: ˆ ˆ x ˆ ˆ ˆ x x (8) By rearrangng he erms n (8), hs can be wren as he quadrac equaon, ax + bx + c = 0, where a, b and c are he erms defned above. 4. The Cubc Specfcaon Asde from he quadrac he nex mos commonly employed polynomal specfcaon s he cubc. Many applcaons ha use he cubc are accompaned by a plo of he esmaed funcon along wh descrpve commens on s shape (e.g. Baglan e al. 008). In a lmed number of cases, he defnon of he urnng-pons mpled by he roos of he frs dervave funcon and he deermnaon of he sgns of he second order dervave are used o deermne he naure of he urnng-pons (Hellbronner 979, Naghshpour 005; Valano and Held 99). In some cases Wald ess are employed o fnd sandard errors for he urnng-pons as n French and Zarkn (995). Alernave echnques for he consrucon of confdence nervals have been appled n he applcaon of he cubc; Plassmann and Khanna (007) use a Bayesan approach for he esablshmen of urnng-pons. The cubc polynomal specfcaon s defned as: M 3 y x x 3x 0 jzj (9) j In hs case he frs dervave funcon s gven by a quadrac equaon defned as: E y/ x, z 3 3 x x x (0) The soluon of hs equaon when se o zero s gven by he soluon o a quadrac. Thus here are wo urnng-pons from he wo roos (gven 3 0) defned as:

14 () 3 3, where 3 3,, and The second dervave of he cubc evaluaed a he h roo of he quadrac n () defned by s gven as y x 6 3. Boh roos of () are real f 0. If hs s he case he second dervave es ndcaes ha s a relave (or local) mnmum f 6 0 and s a relave maxmum when The Dela mehod Appled o he Cubc 3 The approxmae varance of ˆ based on he dela mehod s V( ˆ ˆ ˆ ˆ ˆ ˆ ) d dd dd3 3 d dd3 3 d3 3 () d d d where 3 3, evaluaed a he esmaes, hus : d ˆ d ˆ ˆ 3ˆ 3, and 3 d 3 6ˆ ˆ 3 ˆ ˆ 3 ˆ ˆ ˆ for =,. A 00( )% dela mehod based confdence nerval for s gven by:, ˆ V ( ). U L dela A number of programs n compuer packages can be used o calculae he dela varance esmaes such as nlcom n Saa. 4. The Inverson Mehod Appled o he Cubc A confdence nerval for can be consruced by nverng he -es assocaed wh he null hypohess H 0 : 33 0 H: A 00( )% confdence nerval for s obaned by solvng for he values of a whch he followng expresson holds. 3

15 T ˆ ˆ (3) By rearrangemen we defne a quarc equaon whose roos are he lms of he mpled confdence nervals for he urnng-pons of he orgnal cubc funcon. Ths quarc expresson ha s equvalen o (3) s gven below. ˆ 3 ˆ ˆ ˆ ˆ ˆ ˆ 3 3 ˆ 3 3 ˆ ˆ 3 +4 ˆˆ ˆ ˆ ˆ (4) The correspondng four roos o hs quarc defne he confdence nervals for he urnng pons and can be found numercally usng wdely avalable sofware such as he polyroos funcon n Saa and oher wdely used sofware ha mplemen he algorhm proposed by Jenkns and Traub (970) for he numercal dervaon of he roos for hgh order polynomals. The roos can hen be assocaed wh he correspondng confdence bounds for These can also be obaned graphcally by plong an esmae of he frs dervave as a funcon of x ˆ ˆ ˆ x 3 3x and a 00( ) confdence nerval. ˆ ˆ ˆ x x x E y/ x, z ˆ 4 x ˆ ˆ 4x CI x ˆ ˆ ˆ 3 x 3 9x 3 (5) The esmaed values of correspond o he values a whch ˆ ˆ x ˆ x and he correspondng bounds are gven by he values where he confdence nerval nersecs he zero reference lne. Noe ha he CI s defned for all possble values of x however f he nerval never cus he zero reference lne we can conclude ha for hs value of ha he nerval s nfne. The possbly of an nfne bound s one of he characerscs of he nverson es and corresponds o he case where we oban magnary roos for (4). 4

16 In order o clearly denfy he naure of he roo as eher a maxmum or a mnmum we nver he es ha he second dervave s equal o zero as we have done for he confdence nervals of he urnng-pons. The funcon of he parameers defned by he second dervave funcon s gven as y 6 x 3 hus we can wre he es as: ˆ 6ˆ 3 T (6) 36ˆ 4ˆ 4ˆ 3 3 We can solve for he roos of he quadrac defned below n (7) o esablsh he nerval over whch he urnng-pons can be denfed as a maxmum or a mnmum ˆ 3 ˆ ˆ ˆ ˆ ˆ ˆ In analogous fashon o he nervals found for he frs dervave funcon, we also fnd ha hs nerval s equvalen o plong he second dervave funcon wh he correspondng 00() confdence nerval evaluaed a dfferen values of x s gven by: y x z x E /, (7) CI ˆ ˆ x x ˆ ˆ x ˆ (8) Thus, he plo of he confdence nerval for he second dervave funcon can be used o defne he characerscs of he roos of he frs dervave funcon. 5. The Quarc Specfcaon Followng he rend n he appled economcs leraure o esmae hgher order polynomals a number of sudes repor he esmaon of a quarc or a fourhorder polynomal (Coae and Fry 0; Kumar and Vswanahan 007; Wong 0). As n he applcaon of lower order polynomals, he mos common approach o descrbng he esmaed funcon s o plo he esmaed funconal form. In hs secon we demonsrae ha he nverson es confdence nerval for he urnng- Here we use o denoe confdence bounds for he nd dervave funcon. 5

17 pons can be found from he confdence nerval of he frs dervave funcon. We follow he approach oulned by Wllams (Secon 6.4, 959). For he quarc or fourh order polynomal he regresson s specfed as: M 3 4 y x x 3x 4x 0 jzj (9) j where he frs dervave funcon s gven by a cubc equaon defned as: y 3 x x 33x 4 4x (30) The urnng-pons can be found from he soluon of hs equaon when se o zero: (3) 3 a a a3 0 Where a 3 3, a 4 4 4, and a3 and defne he nermedae erms: 4 4 a Q= 3a 9 and R Defnng arccos 3 Q If R a 9aa 7a R= If Q R 0hen here are 3 real roos. 3 hen he roos are gven by he equaons: a Qcos 3 3 (3) a Qcos 3 3 (33) 4 a 3 Qcos. 3 3 (34) Q 0hen here s only real roo gven by he equaon: a sgn(r) R Q R Q R Q R. (35) 3 For he quarc he second order dervave gven he value of he roos as defned by he quadrac equaon: y x (36) where represens he roo beng examned as esmaed by ˆ based on he esmaes of he parameers of he model. 6

18 To deermne wheher a parcular roo s a relave maxmum or mnmum pon we would es f he value s < 0, or > 0 respecvely. Noe ha alhough he sgn of he second dervave funcon s suffcen o deermne f a urnng-pon s a relave maxmum or mnmum, he second dervave beng equal o zero s only a necessary condon for he locaon of an nflecon pon. 5. The Dela mehod Appled o he Quarc The expressons defned by (3) (35) for he roos of he quarc hough hey are complex can be used o derve dela sandard errors when a program s avalable o oban he approxmae sandard errors for non-lnear funcons of regresson parameers. An example s shown n he Saa roune provded n Appendx A. for he applcaon n Secon 5.3. Noe ha due o he complexy of hese expressons may be necessary o derve he esmaed varance covarance marx for he nermedae expresson such as he a defned above. The dela confdence nerval of he second dervave funcon can also be deermned by esmang he varance of he roos o (36) evaluaed a he urnngpons defned by ˆ derved n (3)-(35) above. An example of hs s descrbed n Secon The Inverson Mehod Appled o he Quarc Alernavely we can consruc he nervals from he nverson of he es usng he ess for he frs dervaves and he second dervaves. Agan he frs dervave funcon for he quarc s defned as: dy 3 x 3 3x + 4 4x (37) dx 7

19 A confdence nerval for he nervals can be consruced by nverng he F-es assocaed wh he null hypohess ha he frs dervave s equal o zero versus he wo-sded alernave: H : H: A 00( )% confdence nerval for s obaned by solvng for he soluons o T 3 ˆ ˆ 3 ˆ 3 4 ˆ ˆ + 4 ˆ + ˆ 3ˆ 4 3ˆ ˆ 9ˆ 6ˆ 4ˆ 6ˆ (38) The soluon can be found by fndng he roos of he 6 h order polynomal defned by: ˆ 4 ˆ 4 ˆ 3 ˆ 4 ˆ34 ˆ 3 ˆ ˆ 4 ˆ3 ˆ 4 ˆˆ ˆˆ 3 4 ˆ ˆ 3 4 ˆˆ ˆ 3 ˆ ˆ 3 ˆˆ ˆ ˆ ˆ Noe ha alhough a general analyc soluon for hese roos s no avalable we can oban numercal soluons as dscussed above. An esmae of he frs dervave as a funcon of x can be ploed wh a confdence nerval as: (39) Y ˆ ˆ ˆ ˆ x x + 3 x ˆ ˆ x 3 4 ˆ ˆ 3 4 ˆ ˆ ˆ ˆ x x x x CI ˆ x 6ˆ x 34 4 (40) and he equvalen bounds for he frs dervave funcon can be locaed where he confdence nerval ncludes zero. In hs case he second order dervave funcon can also be used as a es of whch soluons are relave exremum pons. Thus we are neresed n he regon n whch he null hypohess can be rejeced. 8

20 H 0: H: Usng he es for he second order dervave funcon a he esmaed roo we fnd ha he soluon becomes he roos of a quarc. T ˆ 6ˆ ˆ ˆ 4ˆ 348ˆ ˆ ˆ ˆ (4) For he lms of he second dervave funcon where we are unable o rejec he null hypohess we can solve for he soluons defned by of he quarc defned n (4): 44 ˆ ˆ 44 ˆ ˆ ˆ 36ˆ ˆ 3 ˆ ˆ ˆ ² ˆ ˆ ˆ 4 ˆ ˆ The four roos of (4) defne he lms of he regons over whch we canno rejec he hypohess ha he sgn of he second dervave s neher posve nor negave. Agan he alernave o he soluon o hs polynomal s o nspec he plo of he (00-α)% confdence nerval of second dervave funcon defned by: (4) 4ˆ 4ˆ 3 Y ˆ ˆ ˆ ˆ ˆ 4 36 x 33 CI x x x 44ˆ x ˆ x x (43) By esablshng f he CI for he second dervave funcon a he locaon of esmaed roos of he frs dervave funcon (37) defned by he values of ˆ s greaer or less han zero we can nfer he sgn of he second order dervave a he urnng-pon. 9

21 5.3 Example of a Quarc 3 U e al. (00) esmae an Envronmenal Kuznes Curve (EKC) for he USA usng annual observaons on CO per capa emssons from 99 o 996. Usng a quarc polynomal hey esmae an EKC wh wo peaks a abou US$8,000 and US$8,000 and a rough a abou US$0,000 (as measured n consan 990 dollars). 4 In hs case he regresson s specfed wh only one ndependen varable whch s ncluded as a quarc of he form: y x x x x (44) Where y s he CO emssons per capa as repored n U e al (00) and x s he real GDP per capa (n consan US$000) from Wner e al. (008). Alhough hese daa are me seres n naure and hey exhb he characerscs of poenal un-roos we appeal o he analyss by Lu e al (009) n whch hey demonsrae ha he properes of he asympoc nferenal mehods work well when such a hgh order polynomal s esmaed n he levels. Consequenly, for hs example we wll apply OLS o hs sample and employ he asympoc sandard errors. Coeffcen Esmae Sd Error. -Sa ˆ ˆ ˆ ˆ ˆ Adjused R 0.86 N 68 Table. The parameer esmaes for he quarc funcon appled o he U e al daa seres as defned n Appendx B. Table presens he resulng parameer esmaes and her sandard errors. In Fgure we have ploed he scaer plo of he observaons along wh he predced 3 The Saa roune o generae he resuls n hs secon s gven n Appendx A.. Noe ha he fgures have been eded o add commens. 4 The daa used n hs analyss are lsed n Appendx B. 0

22 funcon. Noe ha we have ncluded addonal observaons wh x values from $30,000 o $35,000 ha are ousde he range of he observaons. 5 Usng he frs dervave approach we esmae hree urnng-pons. Two maxma a US$8076 and US$7,045 and a mnmum a $US8,476. Lsed n Table are he esmaed sandard errors for hese non-lnear funcons of he daa as esmaed by he use of he dela mehod appled o he non-lnear equaons defned n (3) o (34). Also lsed n Table are he Confdence nervals obaned usng boh he sandard errors esmaed va he dela mehod and usng he nverson es approach based on he soluon o he sxh order polynomal as defned n (39). The laer esmaes were obaned numercally hrough he applcaon of he Saa roune polyroos. CO per capa Real GDP per capa ($0,000) Fgure. The scaer plo of he observaons and he quarc funcon f by he regresson. 5 Here we explo a common feaure n many regresson sofware packages o predc ou-of-sample when he dependen varable s mssng bu he regressors are avalable.

23 95% Confdence Inerval. Turnng-pon Esmae Sd. Err. Dela Inverse es ˆ ˆ ˆ Table The esmaed urnng-pons and he sandard error esmaed va he applcaon of he dela mehod wh he confdence nervals based on he wo mehods. All values are n $0,000. From Table we noe ha he mos of he bounds are smlar excep for he upper bound on he las urnng-pon ha appears o be far ou of he sample range for he x varable. The maxmum value for real GDP per capa n he sample s $9,000 whch s well whn he nerval over whch we are unable o rejec he null hypohess ha he frs dervave funcon s zero hus mplyng a fla relaonshp up o $30,600 mpled by he dela CI or $34,00 found va he nverson es. Effecs on Lnear Predcon Real GDP per capa ($0,000) Fgure The frs dervave funcon wh a 95% confdence nerval and he confdence nervals found usng he nverse es.

24 Alernavely, he nverse es confdence nerval from he CI of he frs dervave funcon defned by (40) can be found from Fgure. 6 We have ncluded he locaon of he roos obaned from he numerc soluon o demonsrae ha hey concde wh he values found by locang he pons where he 95% CI of he s dervave funcon cu he zero reference lne. In he case ha he polynomal defned by (39) does no have 6 real roos we would fnd ha he equvalen fgure o Fgure would have CIs ha would no cross he zero lne for hese magnary roos. Thus may be possble for he equvalen plo o dsplay open ended nervals for some roos n whch case he upper or lower lm may be or - respecvely. In order o deermne he naure of hese urnng-pons we now can nvesgae he nd dervave funcon of hese esmaed values. The second order dervave funcon for he quarc s gven by (36) and we can now evaluae hs funcon a he urnng-pons defned by he roos of he cubc mpled by he frs dervave funcon defned n (3). The dela mehod would nvolve he subsuon of he esmaes for he roos no he second dervave funcon o deermne he sgn. The Saa ˆ roune n he appendx has been se o evaluae he second dervaves a hese pons. The resulng nervals for each urnng-pon are gven n Table 3. From hs able we can nfer ha he frs urnng-pon has a negave second dervave ndcang a local maxmum, he second urnng-pon has a posve second dervave ndcang a local mnmum and he hrd has a posve second order dervave whch mples anoher local maxmum. In addon, we also fnd ha he esmaed sandard errors for he frs wo urnng-pons ndcae ha each of hese values of y x ˆ are sgnfcanly 6 The frs dervave plo can be provded by he margnplo roune n Saa as shown n he program provded n he appendx. Alernavely one can use he nlcom roune wh a lle more programmng hs mehod s also shown n he Saa roune n he appendx. 3

25 dfferen from zero. However for he hrd urnng-pon we are unable o rejec he null hypohess ha he second dervave s equal o zero for hs pon whch s necessary bu no suffcen o ndcae ha he las urnng-pon (locaed a $7,045) s no a relave exrema. Turnngpon y x ˆ Sandard error 95% Confdence Inerval ($0,000) Dela Inverse ˆ ˆ ˆ Table 3. The 95% CI for he second order dervave evaluaed a each of he esmaed urnng-pons wh he correspondng dela and Inverse confdence nervals. Alernaely, we can use he nverse es o compue he confdence nerval for he mpled urnng-pons by compung he CI for he second dervave funcon evaluaed a each urnng-pon. The plo of he 95% confdence nerval for he second dervave funcon can be used o oban equvalen nervals for Table 3. Fgure 3 plos he frs dervave funcon as he sold lne and he 95% CI for he second dervave funcon as defned n (36) as he doed lnes. Noe ha he urnng-pons have been locaed by he vercal lnes. The nervals for he second dervave as shown n Fgure 3, can be found by locang he values along he reference lnes defned a he urnng-pons of he s dervave funcon, ha are cu by he CI of he nd dervave funcons. These nervals are lsed n Table 3. Fgure 3 provdes an ndcaon of he regons for whch he sgn of he second dervave s posve, negave and ndeermnae. Thus one can locae he CI for he nd dervave a hose values of he regressor ha concde wh he roos of he quarc defned n (4). 4

26 nd Derv CI Fgure 3 s Dervave Funcon wh nd Dervave 95% CI. The CI for he nd dervave funcon can be read a each urnng-pon. In sum he locaon and classfcaon of he urnng-pons can be esablshed by use of he approprae plo of he frs and second dervaves of he esmaed funcon along wh he correspondng confdence nervals. Thus s possble o mplemen he nverson es by he use of he approprae plos of he s and nd dervaves and her assocaed CIs. 6. Fraconal Polynomals nd Derv CI nd Derv CI Real GDP per capa ($0,000) In order o demonsrae he general naure of he nverse es nervals we consder an exenson of sandard polynomals referred o as fraconal polynomals proposed by Royson and Alman (994). In he economcs leraure hese specfcaons are smlar o he applcaon of he mnflex Lauren expanson employed by Barne (983) o generalze he use of a second order Taylor seres for demand sysem esmaon. Fraconal polynomals nvolve he use of he posve and negave neger powers as well as he log, square-roo and nverse of he square roo of he varable of neres. A prmary applcaon of hs approach has been where regresson models canno be f usng he usual leas squares approach (e.g. lmed 5

27 dependen varables and oher maxmum lkelhood models). Due o he more complex naure of he esmaon needed for such mehods as smoohng splnes (as dscussed n Rupper e al 003) and oher rue non-paramerc echnques, fraconal polynomals have been used where lnear specfcaons can be accommodaed by exsng sofware such as n he applcaon of he Cox proporonal hazards model (see Krsansen e al 008). A ypcal specfcaon for a 3 rd order fraconal polynomal as denfed by he powers 3,,, ½,0,½,,,3 would be defned as: y x x x x ln x 3 / x x x x / (45) As n he cases for he regular polynomals we can defne a vecor of regressors ha are funcons of he covarae of neres (x). 7 y X where X x x x x ln( x ) x x x x (46) 3 / / 3 And 0 9. We can defne can be wren n erms of neger powers: w x so ha he vecor of regressors X w w w w ln( w) w w w w (47) To deermne he locaons of he exrema for hs funcon we can evaluae he frs order and second order dervaves usng he vecors for he frs and second dervaves defned as: w 0 6w 4w w w w w 4 w 6w (48) X Here we drop all oher covaraes o smplfy our expressons. They would no nfluence he remander of hs secon as long as hey were no neraced wh he x varable. 6

28 (49) X w 0 w 6w w w 0 w 30w w Thus he esmaed frs dervave funcon s defned as X w ˆ can be wren as a h order polynomal defned as: ˆ ˆ ˆ ˆ ˆ X 6 7 9w ˆ w w w 5w w w ˆ 5 ˆ 4 ˆ ˆ 4w 3w 4 w 6 (50) Thus, (50) can be used o solve for he locaons of he exrema n erms of he esmaed parameers ˆ as ˆ X when ˆ w 0 hus we have: 6ˆˆ 4ˆˆ ˆˆ ˆˆ ˆˆ ˆˆ ˆˆ 4ˆˆ 6ˆ 0 (5) However, he roos of a h order polynomal have no analyc soluon, hus he applcaon of he radonal dela mehod would requre he use of numerc dervaves for he approxmaon of he varance of he real valued roos. Or alernavely, one could derve confdence nervals for he roos va he use of a paramerc boosrap where one draws values of he coeffcens from a mulvarae normal pseudo-random number generaor defned by usng he esmaed parameers and covarance. 8 The es ha we nver o locae he confdence nervals for he exrema s he es ha he value of he frs dervave o be equal o zero s defned as: X ˆ w X cov( ˆ X w ) w (5) The value of he covarae (w) ha sasfes hs expresson can be found by he soluon of he polynomal n w defned by: X ˆˆ cov( ˆ X w ) 0 w (53) 8 Ths echnque may prove problemac when roos are only derved as real for a subse of smulaons. 7

29 In hs case would be necessary o solve he 4 h order polynomal. We ls par of hs 4 order polynomal n (54) o demonsrae ha such applcaons can lead o hghly complex expressons ha would rarely be used for compuaon / / / 79 6/ / / 3 6/ / / Agan he roos of hs polynomal could be deermned by usng a numerc soluon as n Secon 4.4. Furhermore, may be dffcul o nerpre he mplcaons for magnary roos. The alernave graphc soluon can be found by defnng he confdence nervals from he plos of he esmaed s dervave funcon for any nonlnear specfcaon. We only requre he esmaon of a lnear combnaon of he esmaed (54) parameers o defne he frs dervave funcon as nerval s gven by: X w ˆ and he confdence ˆ Y X X CI cov( ˆ X x ) w w w (55) Ths nerval can be used o locae he confdence bounds for he urnng pons of he polynomal by locang he pons where he CI of he frs dervave cu he zero reference lne. 9 Ths expresson was obaned usng he MuMah compuer algebra roune. 8

30 The second order dervave funcon can also be ploed n order o esablsh he nervals for he classfcaon of he urnng-pons. In hs case we would use he confdence nervals for he second dervave model defned by: ˆ Y X X CI ˆ X cov( ) x w w w (56) In he same manner as we have dscussed n he case of he applcaon o he quarc descrbed n Secon An applcaon of he Fraconal Polynomal 0 Followng n analogous fashon o he applcaon n Secon 5.3 for he quarc we reanalyse he U e al. (00) daa usng a 3 rd order fraconal polynomal as defned n (45). The resulng esmaed funcon wh he daa for esmaon s ploed n Fgure 4. CO per capa Real GDP per capa ($0,000) Fgure 4 The scaer plo and fed funcon based on 3 rd order fraconal polynomal. By comparng Fgures and 4 one can noe he wo prmary dfferences n he esmaed funcons. Frs, he fraconal polynomal esmaes a local maxmum and a mnmum beween. and.4 ha was mssed by he quarc. Secondly, he fraconal 0 The Saa roune o generae he resuls n hs secon s gven n Appendx A.. 9

31 esmae does no exhb he pronounced fall n he exrapolaed values pas he maxmum value of he daa ha was presen n he esmaed quarc funcon. The esmaed parameers for he 3 rd order fraconal polynomal are lsed n Table 4. Noe ha no all are sgnfcanly dfferen from zero bu he f s beer han he quarc. Coeffcen Esmae Sd Error. -Sa ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ Adjused R 0.95 N 68 Table 4. Parameer esmaes for he 3 rd order fraconal polynomal regresson. Due o he complcaons n he esmaon of he urnng-pons and he confdence nervals based on he dela mehod and he nverson es, we wll only use plos of he frs and second dervaves and her correspondng confdence nervals o esablsh he naure of he urnng-pons n hs case. Fgure 5 s a plo of he frs dervave funcon wh he correspondng 95% confdence nerval. From hs plo we can locae he 5 pons where he frs dervave funcon s equal o zero whch correspond o he frs maxmum a $,00 hen a mnmum a approxmaely $,00 followed by a maxmum a $7,300, a mnmum a $6,700 and a maxmum a $8,000. From hs plo we noe ha he confdence nervals for he frs dervave ndcae ha once he level of GDP per capa exceeds $,00 he frs dervave s never sgnfcanly dfferen from zero, hus we can conclude ha for values of GDP per capa over,00 here s no furher The Saa program used o creae hese resuls s lsed n appendx A. 30

32 varaon n CO wh respec o GDP per capa. The second order funcon can be used o esablsh he naure of he urnng-pons denfed by he frs dervave. Fgure 5 The frs dervave funcon of he 3 rd order fraconal polynomal funcon wh he correspondng 95% CI nd Derv CI nd Derv CI Real GDP per capa ($0,000) Fgure 6 The frs dervave funcon and he second dervave confdence nerval for GDP per capa from 0 o $7,000. In order o more accuraely deermne he locaon of he urnng pons and he confdence nervals we have ploed he frs dervave funcon and he confdence nerval for he second dervave funcon n wo fgures defned for wo 3

33 dfferen ranges of he regressor. Fgure 6 shows he second dervave confdence nerval and he frs dervave funcon for values of he per capa ncome under $7,000. Fgure 7 has he same plo from $7,000 o $30,000. From Fgure 6 we fnd ha he second dervave es ndcaes ha he frs urnng-pon a $,00 s a relave maxmum, he second urnng-pon a $,00 s a relave mnmum. From Fgure 7 we noe ha he hrd urnng-pon a $7,000 s a maxmum and he fourh urnng-pon a $6,700 s a mnmum. However, we can also read from Fgure 7 ha he confdence nerval for he second dervave funcon evaluaed a he poson of he las urnng-pon a $8,000 ncludes zero hus we canno rejec he null hypohess ha he urnng-pon s neher a maxmum nor a mnmum nd Derv CI nd Derv CI nd Derv CI Real GDP per capa ($0,000) Fgure 7 The frs dervave funcon and he second dervave confdence nerval for GDP per capa from $7,000 o $30, Conclusons In hs paper we have presened a mehod for he deermnaon of he locaon and properes of urnng-pons esmaed for non lnear models specfed n regressons. By he consrucon of he nverse es we can locae he lms for 3

34 confdence nervals of hgh order polynomals as well as for fraconal polynomals as he roos o hgh order polynomals. We hen demonsrae ha he resuls for he frs dervave funcon can be exended o he second dervave funcon whch can provde ess for he deermnaon of he naure of he urnng-pons denfed by he frs dervave. Alernavely, we have shown ha graphc mehods can be used o provde equvalen resuls and ha hey can be much more easly employed especally for models ha may be specfed ha do no have an analycal soluon found from hgher order polynomals. Fuure research n hs area could be dreced o he consderaon of he more complex resuls obaned when mulple regressors are allowed o nerac n an exenson of he general response surface model. In addon o he fraconal polynomals would also be possble o use he mehods proposed here o evaluae he urnng-pons for oher polynomal relaed specfcaons such as he Fourer Flexble Form specfcaon as proposed by Gallan (98) and he class of Chebyshev polynomals (see Smyh 005 for an example). These specfcaons are smlar o he fraconal polynomal n ha hey also allow for he defnon of hgher order dervave funcons ha are lnear n he regresson parameers hus hey can be ploed n a smlar manner o he fraconal polynomal examned here. 33

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38 Appendx A. Saa program o generae he resuls n Secon 5.3 /* */ Read he Carbon daa from U eal and run he quarc regresson Save he esmaed parameers, he predced values and plo he funcon and he daa. Use he mono colour scheme for all graphs. se scheme smono use "C:\daa\old_v\cubc\carbon_sock\u.da", clear label varable co "CO per capa" label varable g "Real GDP per capa ($0,000)" regress co g c.g#c.g c.g#c.g#c.g c.g#c.g#c.g#c.g esmaes sore eq_ predc p_co label varable p_co "Predced CO" woway (scaer co g, msymbol(+)) (lne p_co g, sor lwdh()), ylabel(,nogrd) /// yle("co per capa") legend(off) name(funconfrac, replace) /* */ Defne he roos of he quarc and apply he dela mehod o esmae confdence nervals Usng nlcom o compue he non-lnear funcon of he parameers o fnd he roos of he quarc and he dela sandard errors wh he confdence nerval. By usng sequenal calls o nlcom wh he pos opon we can buld up he complex funcons for hese roos. quely nlcom (a: (3*_b[c.g#c.g#c.g])/(4*_b[c.g#c.g#c.g#c.g])) /// (a:(_b[c.g#c.g])/(*_b[c.g#c.g#c.g#c.g]) ) /// (a3:(_b[g])/(4*_b[c.g#c.g#c.g#c.g]) ) /// (b:_b[c.g#c.g] ) (b3:_b[c.g#c.g#c.g]) (b4:_b[c.g#c.g#c.g#c.g]), pos quely nlcom (a: (_b[a]) ) /// (q: ( (_b[a]^-3*_b[a])/9) ) /// (q: (*_b[a]^3-9*_b[a]*_b[a] + 7*_b[a3])/54 ) /// (b:_b[b] ) (b3:_b[b3]) (b4:_b[b4]), pos nlcom (x: -*sqr(_b[q])*cos( acos((_b[q])/sqr((_b[q]^3))) /3) - (_b[a]/3)) /// (x: -*sqr(_b[q])*cos((acos((_b[q])/sqr((_b[q])^3))+ *_p)/3) - (_b[a]/3)) /// (x3: -*sqr(_b[q])*cos((acos((_b[q])/sqr((_b[q])^3))+ 4*_p)/3) - (_b[a]/3)) /// (a: *_b[b] + 6*_b[b3]* *_b[b4]*.8076^ ) /// (b: *_b[b] + 6*_b[b3]* *_b[b4]*.7045^ ) /// (3c: *_b[b] + 6*_b[b3]* *_b[b4]*.8476^ ) /* */ /* */ Rereve he orgnal quarc regresson parameer and covarance esmaes esmaes resore eq_ Sore he parameers from he quarc as b-b4, and he elemens of he covarance marx as Vj scalar b=_b[g] scalar b=_b[c.g#c.g] scalar b3=_b[c.g#c.g#c.g] scalar b4=_b[c.g#c.g#c.g#c.g] /* */ marx V = ge(vce) scalar v34 = V[3,4] scalar v4 = V[,4] scalar v4 = V[,4] scalar v3 = V[,3] scalar v3 = V[,3] scalar v = V[,] scalar v = V[,] scalar v = V[,] scalar v33 = V[3,3] scalar v44 = V[4,4] Defne he 6h order polynomal wh roos ha are he equvalen 95% nverse CIs for each nflexon pon. Se _s o he square of he crcal value for wo aled -es. scalar _ =.96 scalar _s = _^ scalar c0 = b^ - _s*v scalar c = 4*b*b - 4*_s*v scalar c = 6*b*b3 + 4*b^ - 6*_s*v3-4*_s*v scalar c3 = *b*b3 + 8*b*b4 - *_s*v3-8*_s*v4 scalar c4 = 9*b3^ + 6*b*b4-9*_s*v33-6*_s*v4 37

39 scalar c5 = 4*b3*b4-4*_s*v34 scalar c6 = 6*b4^ - 6*_s*v44 /* */ /* */ Use he maa program polyroos o compue he roos of hs polynomal. s_numscalar nsers he scalar values o he polyroo program maa polyroos((s_numscalar(("c0")),s_numscalar(("c")),s_numscalar(("c")), /// s_numscalar(("c3")), s_numscalar(("c4")),s_numscalar(("c5")),s_numscalar(("c6")))) Use he margns program o esmae he frs dervave funcon and he esmaed 95%CI hen plo wh margnsplo along wh he ndcors for he soluon found from he polyroos command o show hey concde wh he Feller ype bounds. que margns, dydx(g) a(g=(0.5(0.05)3.5)) margnsplo, recas(lne) recasc(rlne) cops(color(*.7)) ylne(0, lsyle(foreground)) /// name(frsderv, replace) ploregon(fcolor(gs6)) ylabel(,nogrd) /// xlne( , lpaern(".") lwdh(. )) /* */ Plo ou he s and nd dervave funcons o fnd he regons for nflexon pons as opposed o maxma and mnma usng predcnl predcnl d = _b[c.g] + *_b[c.g#c.g] * g + 3 *_b[c.g#c.g#c.g] * g^ /// + 4 *_b[c.g#c.g#c.g#c.g] * g^3, se(d_se) generae up = d + _ * d_se generae lw = d - _ * d_se lne d up lw g, lwdh(.4.3.3) lpaern( "" "-#-" "-#-") sor /// xlne( , lpaern(".") lwdh(. ) ) /// name(d, replace)ylne(0, lsyle(foreground)) /// ylabel(,nogrd) yle("s Dervave Funcon") legend(off) predcnl d = *_b[c.g#c.g] + 6 *_b[c.g#c.g#c.g] * g + *_b[c.g#c.g#c.g#c.g] * g^, se(d_se) generae up = d + _ * d_se generae lw = d - _ * d_se lne d up lw g f lw > -5, lwdh(.4.3.3) lpaern( "" "-#-" "-#-") sor /// xlne( , lpaern(".") lwdh(. ) ) /// name(d_d, replace)ylne(0, lsyle(foreground)) /// ylabel(,nogrd) le("s Dervave Funcon wh nd Dervave 95% CI") legend(off) woway (scaer co g) (lne p_co g, sor), name(funcon, replace) /// xlne( , lpaern(".") lwdh(. )) /// ylabel(,nogrd) le("funcon wh urnng-pon nervals") legend(off) lne p_co d d up lw g, /// name(all, replace) lsyle(p4 p p p) ylne(0,lsyle(foreground)) /// xlne( , lpaern(".") lwdh(. )) /// ylabel(,nogrd) le("funcon wh urnng-pon nervals") legend(off) 38