CENTRO DE ESTUDIOS MONETARIOS Y FINANCIEROS T J KEHOE MACROECONOMICS I WINTER PROBLEM SET #6 This quesion requires you o apply he Hodrick-Presco filer o he ime series for macroeconomic variables for he counry ha you sudied in problem ses #3 and #4 a) Firs consider a se of daa he includes he componens of real GDP ha is, consumpion, invesmen, governmen expendiures, expors, and impors and of labor inpus ha is, hours (if hese daa are available) and employmen Consruc a able of business cycle facs like ha on pp 9 33 of Cooley and Presco Do his boh for he counry ha you are sudying and for he Unied Saes If quarerly daa are available for he counry ha you are sudying, use hem and use quarerly daa for he Unied Saes Oherwise use annual daa Take he naural logarihm of all variables (If he rade balance is one of your variables, calculae he rade balance as a fracion of GDP You canno ake he logarihm of a negaive number!) If you have quarerly daa, use he smoohing parameer 6 If you only have annual daa, use he smoohing parameer Calculae he sandard deviaions and auocorrelaions of he variables and he correlaions of he real GDP wih he leads and lags of he oher variables If you have quarerly daa, follow Cooley and Presco in calculaing correlaions wih 5 leads and lags, as well as he conemporary correlaion If you only have annual daa, calculae correlaions wih 3 leads and lags, as well as he conemporary correlaion Discuss your resuls, sressing he comparison of he business cycle facs for he counry ha you are sudying wih hose for he Unied Saes Also compare he filer for he counry ha you are sudying wih ha for he Unied Saes Noe: If you have quarerly daa, make sure ha hey are seasonally adjused If hey are no seasonally adjused, and you canno find seasonally adjused daa, hen you have hree choices: ( Use annual daa () Use he raw quarerly daa In his case, he fourh leads and lags will display jumps, so only compue 3 leads and lags (3) Seasonally adjus he daa yourself Be explici abou wha you are doing b) Now focus on a subse of variables for which you will compare he imes series daa wih daa generaed by your model in quesion 3: real GDP, real invesmen (invesmen in curren prices deflaed by he GDP deflaor as in quesion on problem se #3), consumpion (GDP in curren prices minus invesmen in curren prices deflaed by he GDP deflaor), capial, and labor inpu (hour worked or, if hese daa are no available, employmen) Consruc a series for he variable ha corresponds o Also consruc a series for oal facor produciviy Consruc a able for he counry ha you are sudying and one for he Unied Saes Discuss your resuls, sressing he comparison of he business cycle facs for he counry ha you are sudying wih hose for he Unied Saes
Consider an economy like ha in quesion on problem se #4 in which he equilibrium allocaion is he soluion o he opimal growh problem max E log C ( )log( ) Nh L s ( ) z C K K e ( ) AK L C, K K K N N Here z is a random variable ha akes on wo values z, z, and whose evoluion is governed by he saionary, firs order Markov chain wih ransiion marix Assume, for he sake of specificiy, ha, a, z z a) Define an Arrow-Debreu equilibrium for his economy b) Define a sequenial markes equilibrium for his economy Redefine variables C and persons N N ( ) N K by dividing by he number of effecive working age Divide L by Consider he social planner s problem N : c C / N ( C / N) k K / N ( K / N) L / N max E log c ( )log( ) h z s c k ( ) k e Ak c, k, h k K / N,
wih he associaed Bellman s equaion V( k, z) max log c( ) log( h ) EV( k', z') z s c k' ( ) k e Ak c, k', h k, z given c) Suppose ha you have solved his dynamic programming problem and have found he policy funcions k' g( k, z), c c( k, z), and ( kz, ) Explain how you can use hese policy funcions o calculae he Arrow-Debreu equilibrium Explain how you can use hese policy funcions o calculae he sequenial markes equilibrium 3 (Opional) This quesion requires you o use he parameers,,, A, and and, if you have sufficien daa, and ha you calibraed in quesion in problem se #4, he compuer program ha you developed in quesion 3 in problem se #4, and he H-P filered daa from quesion o build a DSGE (dynamic, sochasic general equilibrium) model of he economy ha you are sudying Noe: If you are using quarerly daa, you should modify he parameers,,, : /4 /4 /4 /4 ( ) a) Find he sample sandard deviaion and auocorrelaion for he H-P filered logarihm of he oal facor produciviy ha you calculaed in par b of quesion Use hese saisics o calibrae he parameers and for he Markov chain for he produciviy shock z [Hin: There are analyical expressions for he sandard deviaion and auocorrelaion of z in is invarian disribuion] b) Find he seady sae capial socks for wo deerminisic versions of your economy: ˆk when z is consan a z z and ˆk when z is consan a z z This requires you o solve z eak z ( ) eak z h e Ak k( ) k 3
for k and c) Rewrie he Bellman s equaion in quesion as he sysem of wo funcional equaions V( k, z ) max log c( )log( h ) ( ) V( k', z ) V( k', z ) z s c k' ( ) k e Ak c, k', h k given V( k, z ) max log c( )log( h ) V( k', z ) ( ) V( k', z ) z s c k' ( ) k e A k c, k', h k given Choose a maximal capial sock k such ha k k ˆ Choose a grid of capial socks in he inerval (, k ] Modify he compuer program ha you developed o solve quesion 3 in problem se #4 o solve for V( k, z and V( k, z ) Also calculae he policy funcions gkz (, and gkz (, ) Noe: I is efficien o calculae he reurn funcion vkk (, ', z) max log c( )log( h ) z s c k' ( ) k e A k c, k', h k, k ' given a he beginning of he program, before you do any value funcion ieraions I is also useful o calculae he opimizing values of ckk (, ', z ) and ( kk, ', z) Noice ha, when you have obained he policy funcion for he nex period capial sock k' g( k, z), he policy funcions for consumpion and labor supply are c c( k, g( k, z), z) and ( kgkz, (, ), z) d) Graph he wo policy funcions gkz (, and gkz (, ) in ( kk, ') space along wih he 45 line Explain how you could use hese wo funcions along wih a random number generaor ha generaes uniform random numbers on he inerval [,] o produce realizaions of an equilibrium pah of capial socks for his economy e) Wrie a compuer program o simulae he DSGE model ha you have buil Choose a value of k such ha k ˆ k ˆ k Se z z Simulae he economy for n 4
periods Here n is he number of periods for which you have daa in quesion Discard he firs periods,,,,9 of resuls (This is o eliminae any sensiiviy of your resuls o he choice of ( k, z ) ) For every period,,,,9 n, calculae he logarihms of real GDP, real invesmen, capial, labor inpu, and oal facor produciviy Apply he Hodrick-Presco filer o hese daa Calculae he same saisics for hese daa as you did in he able ha you consruced in par b of quesion Do his sor of simulaion 5 imes and ake averages of hese saisics Produce a able like ha in par b of quesion and compare he resuls from your model wih he daa for he counry ha you are sudying 4 (Opional) This quesion guides you in producing a more professional DSGE model han ha in quesion 3, bu i requires you o wrie a compuer program o apply he Hodrick-Presco filer and o seasonally adjus he daa if hey are no seasonally adjused o sar wih a) If you are working wih quarerly daa, make sure ha i is seasonally adjused Tha is, if he daa are no seasonally adjused, seasonally adjus hem [Hin: You can find programs for doing his in Malab on he inerne] b) Modify your program for solving he dynamic programming problem in par b of quesion 3 o solve for models wih a Markov chain wih 3 saes ( z, z, z3) (,, ) and ransiion marix c) Calibrae he parameers and for he Markov chain for he produciviy o mach he sample sandard deviaion and auocorrelaion for series for he H-P filered logarihm of he oal facor produciviy ha you calculaed in par b of quesion [Hin: I is here ha you need o include he program for he Hodrick-Presco filer You can find such programs in Malab on he inerne] d) Simulae he model imes, for n periods each simulaion Calculae he same saisics as in he ables in par c of quesion 3 and par b of quesion Compare hese ables and discuss 5