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1 Sochasic discoun facor for Mexico and Chile... Esocásica Sochasic discoun facor for Mexico and Chile, a coninuous updaing esimaion approach Humbero Valencia Herrera Fecha de recepción: 2 de diciembre de 2013 Fecha de acepación: 22 de abril de 2014 * Insiuo Tecnológico de Esudios Superiores de Monerrey, Campus Ciudad de México. Deparameno de Conabilidad y Finanzas humbero.valencia@iesm.mx Volumen 4, número 2, julio - diciembre, 2014, pp
2 Esocásica Facor de descueno esocásico para México y Chile, un enfoque de esimación coninuamene acualizada. RESUMEN Se propone uilizar el esimador calculado por el méodo de momen- mercados accionarios de México y Chile en el período , que bas economías el facor de descueno esocásico muesra años en los - dos eapas y los del ieraivo, y se muesra la superioridad del esimador coninuamene acualizado sobre esas dos écnicas de esimación an Palabras clave ABSTRACT This paper proposes he use of an esimaor calculaed using he generalized mehod of momens coninuously updaed o characerize a linear sochasic discoun facor for a given economy. The esimaor is applied o he Mexican and Chilean sock markes for , his period includes economies, ook values of less han one and presened high marke volailiy values during several years. A comparison wih he resuls from he wo sages generalized mehods of momens and he ieraive one is also discussed, showing he superioriy of he coninuous updaing esimaor over hese wo frequenly used esimaion echniques. Key word of Momens 104 Volumen 4, número 2, julio - diciembre, 2014
3 Sochasic discoun facor for Mexico and Chile... Esocásica Inroducion coninuously updaed esimaor o idenify he linear sochasic discoun , which includes he period of he inernaional economic credi e al applicaions, i has been used, for example, o measure he performance of e al Volumen 4, número 2, julio - diciembre, 2014, pp
4 Esocásica 1. An analysis of equilibrium condiions value of a discouned ime-separable uiliy is maximized, Max j E[ U(C +j)] j=0 (1) where he subjeive discoun facor measures he personal ime preference, 0 1, C j is he invesor s consumpion in period j, and UC ( j) is he period uiliy of consumpion a j W a I W (( R R ) w R )( W C ) (2) 1 i, f, i, f, i1 where w i, is he proporion invesed in risky asse i of he oal wealh in period, R i, is he reurn of risky asse i in period and R f, is he reurn of he risk free asse in period i a ime ime 1 for Ri, 1, which is a subse of he available informaion a,, U( C ) E ( R U( C ) ) (3) i, ( ) (4) E Ri, 1m1 where he sochasic discoun facor m 1 is equal o he sochasic ineremporal rae of subsiuion U( C 1)/ U ( C) 106 Volumen 4, número 2, julio - diciembre, 2014
5 Sochasic discoun facor for Mexico and Chile... Esocásica n risky asses in he economy are he vecor R, and E Rm, (5) where R is ha Cov( R, m ) ER ( ) R, (6) f Em ( 1 ) f where he reurn on one period riskless bond is R 1/ E( m 1 ) and f R 1 For example, in he case of power uiliy, UC ( ) ( C 1)/(1 ), where he elasiciy of iner-emporal subsiuion,, is he reciprocal of he relaive because wih a fuure one, whereas 1 E( m R ) exp[ E( log( m r ) ), Var log m r Var( log( m 1r1) )] (7) 2 Volumen 4, número 2, julio - diciembre, 2014, pp
6 Esocásica 1 E( log( r 1) ) E( log( m 1) ) Var( log( m 1r1) ). (8) 2 f f m f ER ( ) R R ( ER ( ) ER ( )) (9) i i i where is a benchmark s risk premium, in equilibrium, he marke reurn m m has he form a br, R and R as f m m f 1 EmR ( ) EmR ( ) Em ( ) R (10) i 1.1 Esimaion of Euler Equaion of Consumpion facor m mus saisfy, condiioned o previous informaion 1 is ha he expeced produc of any reurn R equal o one, EmR ( ) 1 (11) 1 EmR ( ) 1. 1 (12) reurn on he asses as well as relaive expeced consumpion sream which 108 Volumen 4, número 2, julio - diciembre, 2014
7 Sochasic discoun facor for Mexico and Chile... Esocásica c E R E 1 Rm c (13) preference parameers of he Euler equaion, he consan relaive risk and discoun facor, he GMM echnique is. E R c ERm c (14) E h( x, ) 0, (15) o where x and o is a p 1 c 1 Ehx, o E R 1 E 1 Rm 10. (16) c g ( ) [ ( ; ; )] 0 E f x z o g Volumen 4, número 2, julio - diciembre, 2014, pp
8 Esocásica T 1 g f x, z,. (17) T T 1 gt ( ) a 0 of T o minimize he funcion J, T J ( ) g ( ) W g ( ) (18) T T T T where WT W T can be T 1 W ( f( x ; z ; ) f( x ; z ; )) (19) T T 1 W T is chosen so ha gt disribuion of he observed variables, unlike he maximum likelihood W T a consisen esimaor of o WT Ir r choice of JT ( ) of T WT new values of W, T T 110 Volumen 4, número 2, julio - diciembre, 2014
9 Sochasic discoun facor for Mexico and Chile... Esocásica if he insrumens are weak and has beer small sample properies han he wo-sep General Mehod of Momens and insrumenal variables esimaors merical opimizaion because he beas and he esimae of he variance-covariance marix,, which depends on he beas are calculaed simulaneously 1 Tg argmin J argmin ( ) [ S( )] g( ) cue 2. The Mexican and Chilean economies 2.1 The Mexican economy Volumen 4, número 2, julio - diciembre, 2014, pp
10 Esocásica 2.2 The Chilean economy a consequence of he inernaional economic crisis and he conra cyclical 3. Discussion and Analysis 112 Volumen 4, número 2, julio - diciembre, 2014
11 Sochasic discoun facor for Mexico and Chile... Esocásica If equaion (13) is esimaed for each reurn and he reurn for he risk free rae is subraced for each of he reurns, he following momen f e i i E m R R E mr, (20), 0 e where R i, is he excess reurn of asse i m can be wrien e as a br i, a E mr E 1bR R E R be R R 0 (21) e e e e e e i, m, i, i, i, m, can be differen from zero, in his case he momen condiion becomes 1 E mr a E br R a e e e i, 0 m, i, 0 e e e i i m E R, a0 be R, R, 0 (22) Volumen 4, número 2, julio - diciembre, 2014, pp
12 Esocásica For he Chilean economy, in he wo parameer model, he discoun facor measures in he Chilean economy, which decreased he sensiiviy of he 114 Volumen 4, número 2, julio - diciembre, 2014
13 Sochasic discoun facor for Mexico and Chile... Esocásica for Chile was small for hose years compared wih oher years, in which he Conclusions and recommendaions more reliable esimaes of he linear sochasic discoun facor han he wo Volumen 4, número 2, julio - diciembre, 2014, pp
14 Esocásica Bibliografía Economerica Economerica Journal of Business Economerics Journal Journal of Poliical Journal of Poliical Economy Economerica Polyechnic Insiue 116 Volumen 4, número 2, julio - diciembre, 2014
15 Sochasic discoun facor for Mexico and Chile... Esocásica Invesigaciones Económicas Revisa de Economía Aplicada Invesigaciones Económicas Economic Modeling Bell Journal of Economics Journal of Moneary Economics Volumen 4, número 2, julio - diciembre, 2014, pp
16 Esocásica Table 1. Mean and sandard deviaion of he daily marke index reurns in Mexico and Chile Counry Chile: IPSA México: IRT Index Year Mean Sd. Dev. Mean Sd. Dev Daily reurns. Source: Own elaboraion Table 2. Mean and sandard deviaion of he daily reurns in Mexico and Chile Mexico Chile Year Mean Sd. Dev. Mean Sd. Dev Based on asses wih a leas 60 quoes in he year. Source: Own elaboraion 118 Volumen 4, número 2, julio - diciembre, 2014
17 Sochasic discoun facor for Mexico and Chile... Esocásica Table 3. Cue esimaor, wo parameer model México Chile 2007 re_me *** *** _cons *** *** 2008 re_me *** *** _cons *** *** 2009 re_me ** *** _cons ** *** 2010 re_me *** *** _cons *** *** 2011 re_me ** *** _cons * *** 2012 re_me ** *** _cons *** *** ***, **, * saisically significan a he 99%, 95% and 90%. Source: Own elaboraion Table 4. Hansen overidenificaion es of all insrumens, CUE esimaor, wo parameer model Mexico Chile Hansen Chi-sq(1) Hansen J Chi-sq(1) J saisic P-val saisic P-val Source: Own elaboraion Volumen 4, número 2, julio - diciembre, 2014, pp
18 Esocásica Table 5. Cue esimaor, one parameer model Mexico Chile re_me Coef. Z Coef. z *** *** *** *** *** *** *** ** *** ** *** Source: Own elaboraion Figure 1: Sochasic discoun facor for Mexico and Chile, one parameer model Source: Own elaboraion 120 Volumen 4, número 2, julio - diciembre, 2014
19 Sochasic discoun facor for Mexico and Chile... Esocásica Table 6. Hansen overidenificaion es of all insrumens, CUE esimaor, one parameer model Mexico Chile Hansen Chi-sq(1) Hansen Chi-sq(1) J saisic P-va J saisic P-va Source: Own elaboraion Table 7. Two seps and IGMM esimaors for Mexico and Chile, one parameer model México Chile Two seps Coef. Z Coef. z *** *** ** *** ** ** ** Igmm Coef. Z Coef. z *** *** ** *** ** ** ** The firs hree lags of he excess marke reurn were used as insrumens. *,**, *** saisically significan a 90, 95 and 99 percen. Source: Own elaboraion Volumen 4, número 2, julio - diciembre, 2014, pp
20 Esocásica 122 Volumen 4, número 2, julio - diciembre, 2014
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