α = relative risk aversion

Transcription

1 BRAV CONSTANTNDES AND GECZY (BCG) Lucas model [ ] E R = Whils providin rea inuiion has enerally been rejeced by he empirical sudies. BCG do wo hins. ) Relax assumpion of complee consumpion insurance No loner able o use per capia consumpion as if here was a sandin-in consumer. base on household consumpion raher han per-capia assumpion. use a equally weihed averae of household marinal raes of consumpion Ci define β ( C ) = i i= where β = rae of ime preference = number of house holders α = relaive risk aversion C Per Capial = β ( C ) where C = C i ) Keep assumpion of complee consumpion insurance bu examine he quesion of Limied Paricipaion of households in capial markes. i.e. look a per capial consumpion bu only for here households ha paricipae in capial markes.

2 DATA SET Consumer Expendiure Survey by Bureau of Labour saisics. Quarerly daa form 980; Random sample of 5,000 households. Also has households holdins of oal asses and hence we can sraify accordin o capial marke paricipaion Three main quarerly Tranches- January, February, arch. Presen resuls for each ranches as well as overall. Resuls Use equiy risk premium (Rm - R f) boh EW & V and hih book/marke Vs. Low book/marke, RH -R L houh o be a value /rowh disincion. Use a number of measures of (eqn 4) is an equally weihed averae of household RS C ( C ) α i = β i (eqn 5) is a hird order Taylor series of eqn 4 around i = = and = Ci where ( C ) i T i i [ ( ) ( ) i α α = β + + ( ) 3 i ( )( ) αα α i + + Goin form eqn (4) o eqn (5). Ci ( C ) = β β i = i i= Take a Taylor series expansion around i = s Term nd Term 3rd Term 4h Term f ( θ ) = f ( θ ) + f '( θ )( θ θ ) + f "( θ )( θ θ ) + f '''( θ )( θ θ ) ! 0 0 3! 0 0

3 s Term ( ) α f θ0 = β = β = β nd Term ( α + ) ( θ ) θ θ = β ( ) f' 0 ( 0) i Since = i = 0 3rd Term ( ) ( ) ( ) α + = + ( ) f ''( θ )( θ θ ) β α α 0 0 4h Term ( ) * ( ) ( α + ) = β α α + ( ) i ( ) i = + βα( α ) i f ''( θ )( ) 0 θ θ0 3! 3 = βα α + + ( α + 3) 3 ( )( α ) ( i ) ( α ) = βα ( α + )( α + ) ( ) 3 ( + 3) i * 3 ( α + 3) 3 ( )( α ) ( ) 3 i = βα α+ +

4 ( ) 3 i βα( α )( α ) = + + Puin hese all oeher and noin ha β is common o all we e. i i ( ) ( ) ( ) ( ) ( + ) β α α α α α Which is eqn (5) (eqn ) is a nd order Taylor series of eqn 4 around i = = β + α α + ( ) ( ) i (eqn 7) akes he assumpion ha income shocks are i.i.d lo normal C = β e Ci i i i ( ) αα+ lo( ) lo( ) (eqn 8) assumes consumpion rowh rae households and hence uses = C i i C i i = is equal across all = β (eqn 9) Ci = β which is he normal per capial consumpion Ci eqn (8) & (9) are boh forms of marke compleeness.

5 arke Compleeness Resuls See Table, 3, 4 (Equiy Risk Premium) Table uses eqn 4 Table 3 uses eqn 5 Table 4 uses eqn eqn 9 Table 7 uses eqn (4), 5 & 9 (for Value/Growh socks) boom line is ha eqn 4 & 5 do a prey ood job of sein: [ ] E R = for Equiy Risk Premium E ( ( R R) } = 0 m f for Value/Growh socks E [ ( R R )] = 0 H L Resuls are reasonable in erms of sinificance and also relaively low values of Relaive Risk Aversion. Paricipaion in Equiy ks (eqn 9) Here we reain per capia consumpion bu only for hose ha paricipae in capial markes. For quarerly daa he model is rejeced. For mh daa as he marke paricipaion level is raised form zero o $k, $0k, $0k, $30k, $40k he model is NOT rejeced bu values of RRA 0 are implied (see Table 5).

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10 Exensions. Why no es he implicaions re equiy risk premium and value/rowh sock joinly. i.e. ( R R ) E ( R H R L m f = 0 The ess in he paper have assumed β =. Here β and α could be esimaed as we have equaions in unknowns.. Why no use condiionin info i.e. he model implies ( ) Rm Rf E = ( R H R L 0 We can use insrumens in he info se e.. pas, pas equiy premium and pas value/rowh Z =, Rm Rf, RH RL,. Now have *4 = 8 equaions in unknown ivin a ( ) Rm Rf ( RH RL ) ( Rm Rf )( Rm R f) ( RH RL )( Rm Rf ) E ( R m R f )( R H R L ) ( RH RL )( RH RL ) ( Rm Rf ) ( RH R ) L χ disribuion. 3. Why NOT combine marke incompleeness and limied capial marke paricipaion. i e use eqn 4 for RS ( ) and hen see how i does as marke paricipaion level is raised.