An Exercise in GMM Esimaion: The Lucas Model Paolo Pasquariello* Sern School of Business New York Universiy March, 2 2000 Absrac This paper applies he Ieraed GMM procedure of Hansen and Singleon (982) and Hamilon (994) o he esimaion of he Lucas Single Agen General Equilibrium Model, under differen specificaions for he asses ses, he insrumenal variables and he represenaive invesor s preferences. The empirical resuls, over a sample of 39 years of observaions, from 959 o 998, seem o confirm he weak evidence for he Lucas Model already emphasized in several papers in he financial lieraure. Esimaes of he relaive risk aversion coefficien appear o be generally low, usually no higher han 4. The ime discoun facor bea is generally higher han 0.99 bu lower han one. A se of seleced macro-economic facors seems o fare beer han lagged reurn variables in explaining he cross-secional variabiliy of asse reurns and he iner-emporal consumpion profile of he represenaive American invesor. The use of a CARA uiliy funcion leads o esimaes of he sign of gamma ha confirm he empirical evidence of wealh elasiciy of demand for risky asses being lower han one. * Ph.D. candidae a he Sern School of Business. Please address commens o he auhor a he Leonard N. Sern School of Business, New York Universiy, Kaufman Managemen Educaion Cener, Suie 9-90, 44 Wes 4h Sree, New York, NY 002-26, or hrough email: ppasquar@sern.nyu.edu.
. Inroducion The relaionship beween asse prices and consumpion and invesmen decisions has long been invesigaed in boh he economic and financial lieraure, as an essenial feaure of any asse-pricing model. In he basic CAPM seing, asse prices are deermined by he porfolio selecion process of agens who are assumed o consume all heir wealh afer jus one period. This simplificaion ignores he complexiy of iner-emporal consumpion decisions and he ineracion beween consumpion and porfolio choices. Lucas (978) was he firs o provide a complee heoreical examinaion of he sochasic behavior of he equilibrium asse prices resuling from a pure exchange economy in which a sochasic echnological process describes he availabiliy of a single perishable good and idenical agens allocae among hemselves a se of coningen claims wrien on he oucome of he random producion process. Lucas shows ha he equilibrium of his sylized economy implies ha a sochasic facor M exiss such ha is expeced produc wih any asse real reurn is equal o one, i.e. ha: j [ M ( R, )] = E β [ -a ] + j + i + j Because equaion [-a] aggregaes all invesors ino a single represenaive agen, who derives his uiliy from he aggregae consumpion of he underlying economy, Lucas shows ha M is simply he ineremporal marginal rae of subsiuion beween curren and fuure consumpion for ha represenaive agen. Hence, [-a] represens he Euler Equaion, resuling from he iner-emporal uiliy-maximizaion problem faced by he represenaive agen, for he idenificaion of his opimal consumpion and porfolio choices. In shor, equaion [-a] relaes real and financial economy, i.e. asse reurns and consumpion. The specificaion of a paricular uiliy funcion in [-a] allows he economerician o esimae boh he risk aversion and he elasiciy of iner-emporal consumpion of he represenaive agen, i.e. his propension o adjus he consumpion profile in response o emporary or permanen shocks o his invesmen opporuniies. In paricular, when U(C ) is assumed o be a ime-separable power uiliy funcion, of he form: γ C U ( C ) =, [ -b ] γ hen a single parameer, γ, governs boh he agen s risk and consumpion pah preferences. In his case, [] becomes:
E γ j C+ β ( + Ri, j+ ) = [ 2 ] C In equaion [2], he elasiciy of iner-emporal consumpion is he reciprocal of risk aversion. Hence, if he esimaes of γ, he Relaive Risk Aversion (RRA) coefficien, are small, he corresponding ineremporal consumpion elasiciy esimaes are large, suggesing ha very risk averse individuals are unwilling o adjus he seleced consumpion profile in response o shocks o he invesmen echnology. Equaion [2] represens an equilibrium, insofar as i equaes he expeced marginal rae of subsiuion beween curren and fuure consumpion wih he expeced rae of ransformaion beween curren and fuure wealh. If ha was no he case, he represenaive agen could sill improve his welfare by reallocaing resources from presen o fuure consumpion and vice-versa. Equaion [2] represens he building block for he so-called Consumpion-CAPM (C-CAPM): asse j s expeced excess reurn resuls o be inversely correlaed o he covariance beween he asse s reurn and he sochasic discoun facor M. In oher erms, an asse whose covariance wih M is small ends o have low reurns when M is high. In urn, M is high when he marginal uiliy of fuure consumpion is high. Bu M is high when fuure consumpion is low, as U > 0 and U < 0. Hence, his asse is considered more risky by he represenaive invesor, as i ends o generae low payoffs in he fuure, ha is exacly when he agen desires hem he mos. Thus, he will require a higher risk premium o hold i. We can hen say ha a es of equaion [2] is also a es for he validiy of he C-CAPM. 2. Esimaion The esimaion of he dynamic raional expecaion model described above presens wo difficul challenges for he economerician. Firs, he ques for efficien esimaion of he unknown coefficiens of equaion [2], β and γ, hrough Maximum Likelihood esimaion imposes srong assumpions on he naure of preferences and he sochasic behavior of he underlying echnology. Hence, any es of he validiy of he specified model is also a join es of he resricions required o esimae i. Second, he specificaion of he variables o be used in he esimaion is no rivial. Alhough reurns on common socks and bonds are usually accuraely measured and easily available, repored consumpion daa, as esimaes of he relevan consumpion flows in he model, are ypically subjec o measuremen problems, as emphasized by Breeden e al. (989). In he nex wo secions, we describe how hese obsacles have been handled in his research. I.e. for perishable goods, hence Non-Durable Goods and/or Services. 2
3. The GMM Approach Hansen and Singleon (HS, 982) sugges an economeric sraegy ha permis o idenify and esimae he parameers of he represenaive agen s dynamic objecive funcion, as well as o es he over-idenified resricions imposed by he heoreical model, while a he same ime avoiding any disribuional assumpion regarding he sochasic echnology and he resuling equilibrium. The main idea of HS is ha economic models based on agens expecaions abou he fuure, hence by definiion no observable, can sill be esimaed if hose agens form heir expecaions raionally, i.e. if he error hey make in forecasing is uncorrelaed wih he informaion available o hem a he ime of he forecas. As long as i is possible o observe a subse of he informaion agens acually use, hen his Raional-Expecaion hypohesis suggess a se of orhogonaliy condiions ha we can adoperae in a GMM seing o esimae he unknown parameers of he model. In he case of Lucas model, he se of orhogonaliy condiions resuls from he Firs Order Condiions of he represenaive agen s iner-emporal uiliy maximizaion problem in an uncerain environmen. The Law of Ieraed Expecaions allows us o express he se of condiional momens condiions of [2] in uncondiional erms: E f h [ f ( φ, C, C+, Z )] = 0 ( φ, C, C, Z ) = h( φ, C, C ) + ( ) j C+ φ, C, C = β ( + R ) + C γ + Z i, + j [ 3 ] I is clear from [3] ha he assumed economic model generaes a family of orhogonaliy condiions h( ). The vecor h( ) is hen used o consruc a specific crierion funcion o minimize wih respec o β and γ. The resuling esimaors, as showed in Hansen (982), are consisen and asympoically normal. I is relevan, in he procedure devised by HS, ha he crierion funcion J T is designed in such a way as o allow he economerician o obain an opimal se of esimaes for he parameers of ineres. Hence, J T is he mean squared weighed disance of he se of orhogonaliy condiions from heir heoreical value of zero. Consequenly, he weighs in J T are seleced opimally so o obain a vecor of esimaes wih he smalles asympoic variance among all possible alernaive weighing-schemes. This opimal weighing marix is proven o be he inverse of he esimaed covariance marix for he orhogonaliy condiions 2, S(T). 2 The circulariy inheren o his choice (he weighing marix needed o esimae β and γ depends on he esimaed values of β and γ hemselves) is avoided wih he esimaion sraegy described below, from Hamilon (994). 3
Then, given a seleced se of insrumenal variables and he vecor of momen condiions, he esimaion sraegy adoped in his paper is he following: = An iniial weighing marix W(0) = S(0) = I is seleced; = An iniial se of esimaes for β and γ, β(0) and γ(0), is obained applying he GMM procedure o J T wih W(0); = β(0) and γ(0) are hen used o esimae he covariance marix of he momen condiions and he resuling new weighing marix W(). We use a Newey-Wes esimaor wih 5 lags: S () i = R( 0) + [ R() l + R() l '] T () = h(,, γ ) h( j, β, γ )' R l W i T = + l () = S() i 5 l= β [ 4 ] = A new se of esimaes for esimaes for β and γ, β() and γ(), is obained applying he GMM procedure o J T wih W(); = The procedure coninues unil convergence 3. 4. Daa Selecion As suggesed in he inroducion, several sudies have invesigaed he empirical validiy of he C-CAPM and, consequenly, faced he difficuly of selecing a reasonable se of asse reurns, properly measured consumpion of perishable asses and adequae insrumenal variables for he esimaion of β and γ. In 989, Breeden, Gibbons and Lizenberger idenified he basic disincion beween he appropriae heoreical definiion of aggregae consumpion per capia and he consumpion series repored by he Deparmen of Commerce. While he C-CAPM prices asses wih respec o changes in aggregae consumpion beween wo poins in ime, he available daa on aggregae consumpion provide oal expendiures on goods and services over a period of ime. This concepual difference beween flow and sock measures suggess wo problems. Firs, goods and services need no o be consumed in he same period when hey are purchased, unless hey are really perishable. Second, measured aggregae consumpion is closer o an inegral of consumpion over a period of ime han o spo consumpion a a specific poin in ime. Moreover, while reurns on socks and mos financial asses are available on an inraday basis, monhly consumpion daa reporing, he mos frequency available a he momen, begins 3 Hamilon (994) suggess his ieraive procedure o guaranee ha he final esimaes for β and γ do no depend on he iniial choices for he GMM algorihms. As we will see laer, an addiional sep is needed o selec he proper se of iniial values. 4
only in January of 959. Finally, pure sampling error, alhough random and hopefully uncorrelaed wih economic variables, may affec he repored consumpion daa. How imporan are hen he resuling biases in he esimaion of he parameers of he Lucas model? As suggesed by Breeden e al., sampling errors in repored consumpion do no appear o affec esimaes and significance ess. The mos serious shorcomings arise from he fac ha, while changes in he spo consumpion are assumed o be uncorrelaed, changes in repored inerval consumpion have posiive auo-correlaion. Real monhly per-capia consumpion raio firs order auo-correlaion for a sample from January 959 o December 998 is close o 0.22, and persisen up o he enh lag. Breeden e al. compue a heoreical quarer-o-quarer firs-order auo-correlaion of 0.25. Real quarerly per-capia consumpion raio auo-correlaion in our sample is abou 0.2786, i.e. insignificanly differen from he heoreical value. This suggess ha he use of quarerly consumpion daa, sampled from monhly daa, should provide more accurae esimaes of he parameers of ineres, if we ignore he fac ha his implies a smaller amoun of daa for he GMM esimaion iself. We focus on a sample inerval of approximaely fory years, from January 959 o December 998 and, o approximae he spo consumpion, use annual aggregae consumpion daa for Nominal (Seasonally Adjused) Non-Durable Goods and Services as repored monhly by he Federal Reserve Bank of S. Louis on is research web-sie (F.R.E.D.) and compue from hem he corresponding monhly per-capia nominal amouns. This allows us o aenuae he sock-flow bias described above. As suggesed earlier in his paragraph, quarerly observaions are sampled from monhly daa. All quaniies in he Lucas model are assumed o be real, as he agens are supposed o inerac in a pure exchange economy, wih no money or wealh-preserving ools for he perishable goods. Moreover, alhough he producion echnology is assumed o be subjec o sochasic shocks, he naure of hese goods remains unchanged over ime, hardly a reasonable assumpion for he evoluion of he American economy in he pas half cenury or so. This means ha he represenaive agen may change his consumpion paern overime no simply because of permanen shocks o his income or wealh, bu also because he se of goods available o him for consumpion changes. We can be sure ha he represenaive agen would have bough a compuer for his personal use, if compuers were he commodiy goods ha hey represen in oday s economy. This is unforunaely no simply a philosophical issue: real consumpion daa are usually obained by deflaing he nominal series wih a CPI indicaor P ha is buil on bundles of consumpion ha are no necessarily consisen wih he se of goods and/or services included in he C series. As a reference poin, we use he CPI Index in January 959 as a proxy for he nominal price of he bundle of goods ha we assume being included in he consumpion series. Then, we deflae each of he nominal per-capia consumpion series in our daabase according o he following formula: 5
~ C 0 C = P 0 0 ~ ~ C = C * ( + g ) ( + π ) C g = Ln C P π = Ln P [ 5 ] In a similar fashion, nominal reurns for each of he asses used in our analysis are ransformed ino real reurn using he following algorihm: + ri Ri = + π, [ 6 ] Pi + Di ri = P i where R i is he real reurn of asse i a ime. In his paper we use differen ses of asses for he esimaion of he Lucas Model. We sar wih a se comprising Value-Weighed monhly reurns compued from all NYSE-AMEX-NASDAQ socks and including all disribuions, an Equally-Weighed reurn series from he same index and a Treasury-Bill reurn series, from January 959 o December 998. The T-Bill rae is calculaed from he average Sree Convenion quoed yield repored in he Fama-Bliss Daabase for -monh bills, according o he following formula: B R R D = 00 R 360 Annual = 00 365 = B D 2 Annual ( + R ) D = duraion, in days Sree avg, [ 7 ] We hen exend he available reurn se by including four Cap-Based porfolios made of NYSE-AMEX- NASDAQ socks ordered in erms of marke capializaion. Hence he firs porfolio includes he op 25% socks in he hree indices for marke cap, and he fourh he boom quarer. We also consider he effec of adding o he original asses se six Cross-Secional bond porfolios. These porfolios, also available in he CRSP Daabase, are buil by including all bonds wihin a cerain mauriy from he compuaion dae in seven caegories: 0-2 Monhs, 2-24 Monhs, 24-36 Monhs, 36-48 Monhs, 48-60 Monhs, 60-20 Monhs and > 20 Monhs. 6
As we focus in his analysis on he sample period 959-998 and no daa are available for he longes mauriy porfolio from 962 o 97 4, we aggregae he las wo porfolios ino a single equally-weighed porfolio of bonds wih mauriy higher han 60 monhs. Hence his porfolio is characerized by a longer duraion saring from 97. Finally, fundamenal for he GMM procedure described in secion 3 is he choice of Insrumenal Variables Z. We use wo differen ses of insrumens along his analysis. Firs, following he suggesions of Hansen and Singleon, we include in Z lagged reurns and he lagged consumpion raio index. Then, we insead selec a group of insrumens for which we idenify an economic raionale in explaining he variabiliy of he asse reurns we consider. The insrumens are Real GDP per-capia, Real Federal Expenses per-capia, he Personal Savings Rae and he Real M3 MoM Growh Rae. Permanen changes in he real income should affec he consumpion profile of he represenaive agen. The demand for resources from he Federal Governmen is supposed o exer a pressure on he domesic bond marke. Changes in he personal saving rae reflec more fundamenal changes in he iner-emporal consumpion paerns of he economy. Finally, moneary policy is widely considered effecive, in he shor erm, in condiioning he behavior of real variables. In he nex secion we repor and commen on he resuls of our analysis. 5. The Empirical Resuls Along he lines of Hansen and Singleon, we sar by esimaing he model of equaion [3] using he original se of asses and lagged asse reurns and consumpion raio as insrumens. Table repors he resuls of he GMM procedure 5 described in secion 3, for he case in which he Real Consumpion is measured wih Non-Durables and Services (NDS), and for he case in which Non-Durables (NDS) alone proxy for he perishable good described in Lucas economy (Case ). Table also shows he esimaes we obained when we run he GMM procedure wihou including any insrumenal variable. This esimaion is possible, for he model being over-idenified, i.e. for he number of momen condiions being higher han he numbers of unknown parameers o esimae. In order o obain an esimae of he parameers of ineres corresponding o a value of he crierion funcion J T as close as possible o is global minimum, we perform a Grid-Search on J T by repeiively changing he iniial values for β and γ, unil a global minimum appeared o have been achieved. However, we also repor he esimaion resuls obained by using he Hansen-Singleon esimaes as iniial values for our ieraed GMM procedure. 4 During which no long-mauriy noes were issued by he Treasury. 5 In he Appendix, we aach a sample code for he GMM Esimaion of he Lucas Model for some of he cases described in he ex. 7
Figure shows an example of his Grid-Search for he resuls repored in he firs row of able. Figure : Grid-Search Mehod for he GMM Ieraive procedure NDS wih 3 asses and 4 insrumens J T for differen iniial values of γ 0.088 0.086 0.084 0.082 0.08 0.078 0 5 0 5 20 Several ineresing facs are eviden from his firs se of resuls. Firs, esimaes of he discoun facor β appear o be reasonably lower han one and relaively precise, i.e. wih a low sandard error. Table : Esimaion of Lucas Model Lagged Variables as Insrumens 6 Saring Values Esimaion of he Lucas Model: GMM (Case : Lagged Variables as Insrumens) Saring Dae β γ C # # # β s.e. γ s.e. J T χ 2 DF Prob Ins Asses Lags Feb-59 0.993 6.00 NDS 4 3 0.9966 0.0025 3.588.4378 0.0785 37.5240 3 0.9997 Feb-59 0.993 0.9457 NDS 4 3 0.9975 0.0004 0.4462 0.520 0.0866 4.4037 3 0.9999 Feb-59 0.993 0.00 NDS 0 3 0.030 0.0733 3.3 53.253 0.0005 0.2389 0.3750 Feb-59 0.993 0.9457 NDS 0 3 0.030 0.0733 3.3 53.2507 0.0005 0.2389 0.3750 Feb-59 0.993 20.00 ND 4 3 0.9992 0.009 3.744.3477 0.0759 36.2588 3 0.9995 Feb-59 0.993 0.9457 ND 4 3 0.9973 0.0003 0.2936 0.0794 0.086 4.697 3 0.9999 Feb-59 0.993 0.00 ND 0 3 0 0.8534 0.2274 87.723 58.49 0.0038.8363 0.8246 Feb-59 0.993 0.9457 ND 0 3 0 0.8534 0.2274 87.724 58.434 0.0038.837 0.8247 6 The degrees of freedom for he Chi-Squared Tes of Over-idenificaion are calculaed in he fashion repored by Hansen and Singleon (982) and Hamilon (994), i.e. by including he consan erm in he se of insrumenal variables. If m is he number of momen condiions, q is he number of insrumens (excluding he consan erm), and L is he number of parameers o esimae, hen DF is given by m(q+) L. 8
Second, esimaes of he Relaive Risk Aversion coefficien, alhough economically plausible, are significanly affeced by he inclusion or exclusion of insrumenal variables in he analysis. The exclusion of insrumenal variables generaes unexpeced esimaes for bea, much larger values for gamma, and generally much higher uncerainy around he parameers we obain hrough he ieraed GMM procedure. Third, when lagged variables are included as insrumens, he resuling esimaes for gamma are relaively small and close o he values obained by Hansen and Singleon. These resuls seem o sugges, along he lines of wha we observed in secion, ha agens, as characerized by low Risk Aversion, are willing o adjus heir iner-emporal consumpion paers in response o shocks o he producion echnology. We hen repea he same analysis of able using NDS, ND and he se of macro-economic variables described in he previous secion as insrumenal variables (Case 2). Resuls are repored in Table 2 below. Table 2: Esimaion of Lucas Model Macro-Economic Variables as Insrumens Saring Values Esimaion of he Lucas Model: GMM (Case 2: Macroeconomic Variables as Insrumens) Saring Dae β γ C # # # β s.e. γ s.e. J T χ 2 DF Prob Ins Asses Lags Feb-59 0.993 20.00 NDS 4 3 0 0.9986 0.008 3.2329.9836 0.0740 35.3837 3 0.9993 Feb-59 0.993 0.9457 NDS 4 3 0.0000 0.0004.0848 0.442 0.058 27.7548 3 0.9902 Feb-59 0.993 20.00 ND 4 3 0 0.9993 0.0022 3.5867.907 0.046 22.0333 3 0.9452 Feb-59 0.993 0.9457 ND 4 3 0 0.9990 0.0004 0.987 0.2523 0.0570 27.2345 3 0.9884 Again, esimaes for bea are reasonably lower han one, and gamma values are close o 3, i.e. relaively low. I is worh noing, a his poin in he analysis, ha he Chi-Squared es, applied o each of he models we esimae and repored in he las hree columns of Table and 2 7, rejec in all case he adoped over-idenifying resricions. No surprisingly, he rejecion is sronger when more insrumenal variables are included in he analysis, i.e. he number of over-idenifying resricions being esed increases 8. Moreover, he macro-economic insrumens appear o fare beer han he lagged reurns and 7 Prob is he probabiliy ha a χ 2 (DF) random variable is less han he compued value of he es saisic under he Null hypohesis ha he resricions imposed o he se of momen condiions are saisfied. 8 Similar resuls were obained by Hansen and Singleon as well, especially in Table I 9
consumpion raio, a leas in erms of how srong he over-idenifying resricions are rejeced. Hence, we decide o use hem as he se of insrumens 9. Nex, we increase he se of original momen condiions, by including in our analysis in one case four Cap- Based porfolios (Case 3) and in anoher six Cross-Secional Bond porfolios (Case 4). As in he las wo examples, we repor, in Table 3, he resuls of he GMM procedure when insrumens are included, bu also cross-secional esimaes of he parameers of ineres, i.e. when jus he original momen condiions implying from he Lucas model are used. Figure 2 below repors anoher example of he Grid-Search mehodology for he analysis in he hird row of able 3. Figure 2: Grid-Search Mehod for he GMM Ieraive procedure NDS wih 3 asses and 4 insrumens J T for differen iniial values of γ 0.37 0.36 0.35 0.34 0.33 0.32 0.3 0.3 0.29 0 5 0 5 20 The mos sriking resul of Table 3 is he difference in he uncerainy surrounding he esimaes for gamma and bea for he cases in which insrumens are included or excluded in he analysis. The assumpions underlying he GMM procedure devised by Hansen and Singleon leaves enough laiude for he researcher o choose he componens of he vecor Z. Thus, ignoring he informaion conen of he Chi-Squared es, hese findings sugges ha when he sample used by he economerician is large, as in our case (wih 478 observaions), as many insrumens as possible should be used, in order o reduce he sandard error for he esimaes. Then, he GMM procedure will simply aribue a low weigh o saisically meaningless insrumens. The bes insrumens appear o be he ones ha ex-ane seem o help explain he mos of he cross-secional variabiliy of reurns and he iner-emporal consumpion allocaion decision process. However, when he daa-sample is relaively small, his selecion process has o be conduced hrough a proper idenificaion of economically meaningful insrumens. 9 All of he analysis repored in his paper have been replicaed wih he original se of lagged variables as insrumens, and none of he qualiaive resuls we presen here were affeced by his choice. 0
Moreover, he esimaes for bea and gamma in Case 3 of Table 3, i.e. when Cap-Based porfolios are included in he analysis, seem o confirm he range of values we had previously idenified hrough Tables and 2. Bea is again higher han 0.99 bu lower han one. A low gamma, of around 3.5, is sill in he 95 % confidence inerval of he esimae obained when insrumens are added o he procedure. No so economically meaningful appear o be he resuls of Case 4, when Cross-Secional bond porfolios are added o increase o nine he number of unresriced momen condiions resuling from he Lucas model. Table 3: Esimaion of Lucas Model New ses of Unresriced Momen Condiions Saring Values Esimaion of he Lucas Model: GMM (Case 3: Adding Cap-Based Porfolios) Saring Dae β γ C # # # β s.e. γ s.e. J T χ 2 DF Prob Ins Asses Lags Feb-59 0.993 0.00 NDS 0 7 0 0.9994 0.073 0.6056 42.6694 0.0505 24.29 5 0.9998 Feb-59 0.993 0.9457 NDS 0 7 0 0.9994 0.072 0.6927 42.6460 0.0505 24.54 5 0.9998 Feb-59 0.993 8.75 NDS 4 7 0 0.9985 0.007 4.886 0.8737 0.30 54.0087 33 0.9880 Feb-59 0.993 0.9457 NDS 4 7 0 0.9970 0.0005 0.838 0.974 0.36 54.2989 33 0.9888 β γ Esimaion of he Lucas Model: GMM (Case 4: Adding Cross-Secion of Bond Reurns) Feb-59 0.993 20.00 NDS 0 9 0.0738 0.022 83.530 33.4860 0.0629 30.079 7 0.9999 Feb-59 0.993 0.9457 NDS 0 9 0.0758 0.007 78.437 33.2659 0.0634 30.2824 7 0.9999 Feb-59 0.993 20.00 NDS 4 9 0.0002 0.0033 4.32.840 0.237 59.45 43 0.9485 Feb-59 0.993 0.9457 NDS 4 9 0 0.9966 0.0005.0575 0.224 0.268 60.6079 43 0.9606 In ha case, esimaes for bea are higher han one, meaning ha he represenaive agen would agree o receive less han a uni of good omorrow for a uni of good available o him oday. Bea in fac corresponds o / ( + R d). Hence, a bea higher han one implies a negaive personal discoun rae, i.e. a negaive personal rae of ransformaion beween curren and fuure wealh. This means ha he represenaive agen would no aribue value o ime, a fac ha clearly does no appear o happen in he American sociey of he pas fify years.! We finally explore wo differen direcions for he esimaion of he Lucas model, in he aemp o improve he performance of he model in explaining he observed cross-variaion of reurns. The poweruiliy funcion adoped so far assumes ha Relaive Risk Aversion is consan over ime and over any levels of consumpion for he represenaive agen. If we relax his assumpion, by allowing RRA o change in response o changes in he amoun of consumpion available, as i is reasonable o imagine for
an individual invesor 0, he resuling model should fare beer in he empirical invesigaion. For his purpose, we assume ha he represenaive agen s uiliy is of he following CARA form: ( C ) γc U = e U ' ' RA = = γ U ' RRA = C RA = γc drra = γ dc Equaion [8] describes a negaive exponenial uiliy funcion. This choice implies ha, a leas heoreically, he represenaive agen s demand for risky asses is unaffeced by changes in iniial wealh, wih riskless borrowing and lending absorbing he observed change. However, and his is he main difference wih he model in equaion [-b], he proporion of wealh invesed in he risky asse is no invarian wih respec o changes in wealh. The sign of gamma measures he sign of he wealh elasiciy of demand for he risky asse. If gamma is higher han zero, hen he wealh elasiciy is negaive, i.e. he fracion of iniial wealh invesed in he risky asse would decrease, as he agen s iniial wealh increases. [ 8 ] Table 4: Esimaion of Lucas Model Cap-Based Porfolios, CARA Uiliy or Quarerly Daa Saring Values Esimaion of he Lucas Model: GMM (Case 5: CARA Uiliy Funcion) Saring Dae β γ C # # # β s.e. γ s.e. J T χ 2 DF Prob Ins Asse Lag Feb-59 0.993 0.00 NDS 0 7 0 0.9972 0.0706-0.20 6.253 0.0504 24.080 5 0.9998 Feb-59 0.993 0.9457 NDS 0 7 0 0.9966 0.0700-0.056 6.92 0.0508 24.2658 5 0.9998 Feb-59 0.993 20.00 NDS 4 7 0.0555 0.099 0.902.848 0.067 50.992 33 0.9764 Feb-59 0.993 0.9457 NDS 4 7 0 0.9996 0.004 0.6284 0.0958 0.9 53.4897 33 0.9865 β γ Esimaion of he Lucas Model: GMM (Case 6: CRRA and Quarerly Daa) Mar-59 0.993 0.00 NDS 0 7 0.086 0.0283 3.873 8.5533 0.0874 4.7665 5.0000 Mar-59 0.993 0.9457 NDS 0 7 0 0.9827 0.034-9.303 9.4354 0.657 79.2065 5.0000 Mar-59 0.993 0.00 NDS 4 7 0 0.9938 0.0003-0.60 0.665 0.5 72.283 33 0.9999 Mar-59 0.993 0.9457 NDS 4 7 0 0.9939 0.0003 0.6205 0.857 0.57 72.502 33 0.9999 0 As a maer of fac, i appears from mos of he available empirical lieraure ha he wealhier segmens of he populaion are he ones ha inves more heavily in socks, i.e. in riskier securiies, bu also ha he proporion of heir wealh invesed in risky asses decreases for increasing wealh, hus suggesing a marginally increasing Relaive Risk Aversion. The CARA model we describe below does capure his effec, as RRA is increasing in C. 2
We also follow Breeden e al. s suggesion ha he use of quarerly consumpion daa (i.e. j= 3), sampled from monhly daa, should provide more accurae esimaes of he parameers of ineres, if we ignore he fac ha his implies a smaller amoun of daa for he GMM esimaion iself and makes he selecion of he proper insrumens a more criical issue. Table 4 repors our resuls for gamma and bea for he CARA uiliy funcion (Case 5) and Quarerly daa esimaion for model [-b] (Case 6). As eviden from he able, he use of a negaive exponenial uiliy funcion does no improve he performance of he model. When no insrumens are included in he analysis, gamma is slighly negaive, bu saisically no significanly differen from zero, given he big sandard errors resuling from he esimaion procedure. However, when he macro-economic insrumens are added, gamma is again posiive, and RRA is increasing, again suggesing he ineresing conclusion ha he wealh elasiciy of demand for risky asses is less han one. The use of quarerly daa appears o improve he saisical performance of he model, as he sandard errors of he esimaed parameers are significanly reduced wih respec o any of he analysis we repored in his secion. The resuls of his case seem o confirm our iniial conclusion ha bea is higher han 0.99 bu lower han one, and ha gamma is small. 6. Conclusions This paper applies he Ieraed GMM procedure of Hansen and Singleon (982) and Hamilon (994) o he esimaion of he Lucas Single Agen General Equilibrium Model, under differen specificaions for he asses ses, he insrumenal variables and he represenaive invesor s preferences. The empirical resuls, over a sample of 39 years of observaions, from 959 o 998, seem o confirm he weak evidence for he Lucas Model already emphasized in several papers in he financial lieraure. Esimaes of he relaive risk aversion coefficien appear o be generally low, usually no higher han 4. The ime discoun facor bea is generally higher han 0.99 bu lower han one. A se of seleced macro-economic facors seems o fare beer han lagged reurn variables in explaining he cross-secional variabiliy of asse reurns and he iner-emporal consumpion profile of he represenaive American invesor. The use of a CARA uiliy funcion leads o esimaes of he sign of gamma ha confirm he empirical evidence of wealh elasiciy of demand for risky asses being lower han one. The β 3 s esimaed from he use of quarerly daa (for j = 3) are very similar o he ones obained for monhly daa in Tables and 2, bu he corresponding sandard error is now much smaller. 3
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