Measurement with Minimal Theory

Transcription

1 Federal Reserve Bank of Minneapolis Quarerly Review Vol.33, No. 1, July 2010, pp Moneary Adviser Research Deparmen Federal Reserve Bank of Minneapolis and Adjunc Professor of Economics Universiy of Minnesoa Absrac Applied macroeconomiss ineresed in idenifying he sources of business cycle flucuaions ypically have no more han 40 or 50 years of daa a a quarerly frequency. Wih sample sizes ha small, idenificaion may no be possible even wih correcly specified represenaions of he daa. In his aricle, I invesigae wheher small samples are indeed a problem for some commonly used saisical represenaions. I compare hree a vecor auoregressive moving average (VARMA), an unresriced sae space, and a resriced sae space ha are all consisen wih he same prooype business cycle model. The saisical represenaions ha I consider differ in he amoun of a priori heory ha is imposed, bu all are correcly specified. I find ha he idenifying assumpions of VARMAs and unresriced sae space represenaions are oo minimal: he range of esimaes for saisics of ineres for business cycle researchers is so large as o be uninformaive. The views expressed herein are hose of he auhor and no necessarily hose of he Federal Reserve Bank of Minneapolis or he Federal Reserve Sysem.

2 FEDERAL RESERVE BANK OF MINNEAPOLIS QR * Moneary Adviser Research Deparmen Federal Reserve Bank of Minneapolis and Adjunc Professor of Economics Universiy of Minnesoa Applied macroeconomiss ineresed in idenifying he more han 40 or 50 years of daa a a quarerly frequency. represenaion of he daa. In his aricle, I invesigae wheher small samples are indeed a problem for some commonly used saisical represenaions applied o he same prooype business cycle model. The business cycle model is a prooype in he sense ha many models, wih various fricions and shocks, are observaionally equivalen o i. The saisical represenaions ha I consider differ in he amoun of heoreical deail ha is imposed a would correcly idenify he sources of business cycles and he conribuions of differen shocks o economic - All are consisen wih he same prooype business cycle model, bu he VARMA imposes few resricions based on he underlying economic environmen, and he resriced sae space imposes many. In paricular, he VARMA represenaion is a sysem of equaions in reduced form, whereas he resriced sae space repre- rade-offs ha economic agens face in he heory. and unresriced sae space represenaions are oo minimal o uncover saisics of ineres for business cycle research wih sample sizes used in pracice. I demonsrae his by simulaing 1,000 daa ses of lengh 200 quarers using he prooype business cycle model. For each daa se and each of he hree saisical represenaions of he daa, I apply he mehod of maximum likelihood o esimae parameers for ha represenaion and hen consruc saisics of ineres o business cycle analyss. The saisics include impulse responses, vari- daa. For he VARMA and unresriced sae space rep- biased and have large sandard errors. The errors are so large as o be uninformaive. Since he resriced sae space represenaion relies maximum likelihood parameers are economically in- *I hank Elmar Merens, Ed Presco, and Warren Weber for heir commens. Codes o replicae he resuls of his aricle are available a my web- 2

3 erpreable and can be consrained o lie in economically plausible ranges. In pracice, business cycle researchers may pu furher consrains on he ranges of hese parameers using independen micro or macroevidence. I also do his and compare resuls across experimens, varying consrains on he possible ranges of he maxi- he main resuls which are he saisics of ineres for business cycle analyss are no sensiive o varying plausible range. pare he small sample performances of he VARMA and unresriced sae space represenaions wih ha Srucural VARs wih Long-Run Resricions, for space represenaions, which impose much more heory. SVARs, and he sae space represenaion performs slighly beer han he SVARs. However, none of he represenaions hey consider yield precise esimaes for he saisics ha hese auhors highligh. In he following secion, I lay ou he prooype business cycle model. Then I summarize he hree saisical represenaions. The mehod of maximum likelihood used o esimae parameers of he hree represenaions is described, and I repor on he business cycle secion concludes. The Prooype Business Cycle Model I use a prooype growh model as he daa-generaing process for his sudy. The model is a prooype in he sense ha a large class of models, including hose wih various ypes of fricions and various sources of shocks, are equivalen o a growh model wih imevarying wedges ha disor he equilibrium decisions of agens operaing in oherwise compeiive markes. wedges are modeled like ime-varying produciviy, labor income axes, and invesmen axes. Since many idenifying one paricular configuraion does no any one deailed economy. Households in he economy maximize expeced uiliy over per capia consumpion c and labor l, E c l 1 1 N 1 subjec o he budge consrain and he capial accumulaion law, c 1 x 1 wl rk T x l 1 g k 1 k x n 1 where k denoes he per capia capial sock, x per capia invesmen, w he wage rae, r he renal rae on capial, he discoun facor, he depreciaion rae of capial, N he populaion wih growh rae equal o 1+ g n, and T he per capia lump-sum ransfers. The series l and x are sochasic and sand in for imevarying disorions ha affec he households inraemporal and ineremporal decisions. Chari, Kehoe, and l as he labor wedge and x as he invesmen wedge. FKZLwhere K and L are aggregae capial and labor inpus and Z is a labor-augmening echnology parameer which is assumed o be sochasic. Chari, Kehoe, and McGraan Z he and demonsrae an equivalence beween he prooype model wih ime- mies wih underlying fricions ha cause facor inpus o log Z is a uni-roo wih innovaion log z. The process for he exogenous sae vecor s [log z,, ] is 1 s P Ps Q 0 1 gz 1 l 1 x l x l 0 s l 0 x 0 0 x z l 0, 0 0 x 1 The assumpion ha he shocks are orhogonal is unrealisic for many acual approaches. 1 3

4 FEDERAL RESERVE BANK OF MINNEAPOLIS QR Srucural VARs wih Long-Run Resricions I have no included a commonly used saisical represenaion sudy. Alhough SVARs are widely used by business cycle analyss, considerable debae has been generaed recenly abou heir usefulness. One criique leveled by Chari, Kehoe, and be addressed by using a VARMA represenaion, as is done in he aricle. However, given he wide use of SVARs, reviewing he subsance of he criique may be helpful. The Procedure. I will focus aenion on he SVAR procedure wih long-run resricions, which is a simple ime series echnique ha uses minimal economic heory o idenify he paern of responses of economic aggregaes o possible shocks in he economy. Following his procedure, he analys esimaes a represenaion, and imposes cerain srucural assumpions To be more precise, le Y be an N-dimensional vecor conaining observaions a ime period Y on p lags, Y A0 AY 1 1 A2Y2 ApYp v. The second sep is o inver his VAR o ge he corresponding moving average, Y v Bv B v , where v Ev v. Mechanically, i is easy o recursively compue he Bimaes of he A is easily consruced from he VAR residuals. One more sep is needed o derive he srucural MA one ha has inerpreable shocks. The srucural MA is given by Y C e C e C e , where Eee, e C0 1 v, and Cj = BjC0 for j 1. The elemens of v are simple residuals in a VAR, bu he elemens of e are he shocks of ineres. To provide hese shocks wih an economic inerpreaion, we need o impose srucural resricions on he elemens of C 0 and. An Applicaion. To beer undersand hese resricions, i been a he cener of he recen debae concerned wih he usefulness of SVARs. In his case, Y is a wo-dimensional vecor conaining he change in he log of labor produciviy and he log of hours. The lag lengh p pracice, analyss choose small values for p because hey have sample sizes of roughly 200 quarers. For he srucural shocks, e is a echnology shock and he second elemen a demand shock. Nex, we need resricions o idenify elemens of C 0 and. Seven resricions ypically used are as follows. Three come from equaing variance-covariance marices C0C0 Three come from assuming ha he shocks are orhogonal I The las comes from he assumpion ha demand shocks have no long-run effec on labor produciviy jc j12 0 SVAR users call his las resricion a long-run resricion. I assumes ha echnology shocks have a permanen effec on he level of labor produciviy, whereas demand shocks do no. The SVAR Claims. lieraure is ha a posiive echnology shock leads o a fall in researchers o conclude ha a cerain class of business cycle models, referred o as or RBC models, is no promising for he sudy of business cycles, since mos RBC models predic a posiive response in hours. They furher claim ha models are more promising because hese models can produce he fall in hours afer a echnology shock. The CKM Criique. Users of he SVAR procedure claim ha beween promising and unpromising classes of models wih minimal assumpions abou he economic environmen. CKM evaluaed his claim wih a very simple es. They generaed daa from an RBC model, drawing a large number of samples wih Two problems are associaed wih he SVAR procedure ha lead o biased and uninformaive resuls. The source of runcaion bias. Theoreical business cycle VAR. In oher words, p p second problem is small sample bias. Mos saisical proce- Avoiding Truncaion Bias. One easy adapaion of he SVAR procedure is o use a VARMA represenaion ha allows for moving average erms. This is wha is done in his sudy. Using VARMAs akes care of he problem of runcaion bias. Then, I can deermine he exen of he problem of small sample sizes. 4

5 where [ z, l, x] is he vecor of shocks hiing he economy a dae. Approximae equilibrium decision funcions can be - has he form k ˆ log ˆ + 1 = k + log z + k z l l + + x x 0 k log k ˆ s s 0, where k ˆ k / Z1 is derended capial. From he saic for oupu, invesmen, and labor which I use laer, namely, log ˆ log ˆ log xˆ log k ˆ s xk xs log = log kˆ s, l lk ls where ˆ = /, xˆ = x / Z, and he coefficien vecors, xs, and ls ha muliply s in equaions - of preferences and echnology ha appear in he original - Observables In all represenaions laer, I assume ha he economic modeler has daa on per capia oupu, labor, and invesmen. Because oupu and invesmen grow over ime, he vecor of observables is aken o be The elemens of Y are he growh rae of log labor produciviy, he log of he labor inpu, and he log of he invesmen share. 2 All elemens of Y are saionary. For he prooype model, hese observables can be wrien as funcions of S = [log k ˆ, s, s1, 1 ]. To see his, noe ha he change in log produciviy is a funcion of he sae oday (log k ˆ, s, 1 ) and he sae yeserday (logk ˆ, s, 1 ). The capial sock a he beginning of 1 1 he las period log ˆk 1 can be wrien in erms of log ˆk and s 1 only on oday s sae (logk ˆ, s, 1 ). Thus, all of he observables can be wrien as a funcion of S, which is an vecor. Three Saisical Represenaions I use he form of decision funcions for he prooype model o moivae hree differen bu relaed saisical represenaions of he economic ime series. A Resriced Sae Space Represenaion The sae space represenaion for he prooype model has he form S AS B E I 1 1 Y CS where he parameer vecor is ig,, g,,,,,,,,,,. n z l x l x l x Here, i is he ineres rae and is used o se he discoun facor = gz 1 + i I use o compue an equilibrium and hen consruc k s P 0 P Q A 0 B 0 I C 1, where 1 is a 31 zeros oherwise, and 0 is a 3 1vecor of zeros. Recall ha P and Q are 3 3 marices. Thus, A is an marix, B an, and C a. Elemens of 2 I have chosen variables ha business cycle researchers ypically do, bu oher variaions ha I ried such as using oupu growh raher han labor produciviy growh did no affec my resuls. 5

6 FEDERAL RESERVE BANK OF MINNEAPOLIS QR Esimaes ˆ are found by applying he mehod of maximum likelihood. The exac likelihood funcion For he resriced sae space represenaion, I consider hree ses of resricions on he parameer space. In wha I refer o as he loose consrains case, I assume ha he parameers in can ake on any value as long as an equilibrium can be compued. In wha I refer o as he modes consrains case, I assume ha he parameers in are consrained o be economically plausible. Finally, I consider a igh consrains conroversial for business cycle heoriss. They are he ineres rae i, he growh raes g n and g z, he depreciaion rae, he capial share, and he mean ax raes l and x. In he igh consrains case, I only esimae he parameers affecing key elasiciies, namely, and, and parameers affecing he sochasic processes for he shocks. There is no consensus on he values for hese parameers. An Unresriced Sae Space Represenaion In he resriced sae space represenaion, all crossequaions resricions are imposed. This necessiaes making many assumpions abou he economic environmen. Suppose insead ha I assume only ha he sae preferences and echnologies. I do, however, need o impose some minimal resricions ha imply he S = [log k, s, s1], where log k = log ˆ log ˆ / k k) z z z z z z l l l l x x x x and s log z, l, x. Then he unresriced sae space represenaion can be wrien as = [ ] S A S B, E I wih 1 u u 1 Y C S A u u k 1 l x l x 0 0 0, B u and C u on s 1 of A u is l = ll zz x = x x z z The vecor o be esimaed,, is herefore given by,,,,,, vecc k l x l x u where he C u includes only he elemens ha are no a priori se o 0. As in he case of he resriced sae space represenaion, esimaes are found by applying he mehod of maximum likelihood. From his, I ge ˆ. PROPOSITION 1. The sae space represenaion, equaion. Proof. 3 if ( A 1, B 1, C 1 ) and ( 2, 2 A B, 2 C ) are observaionally u u u u u u 3

7 equivalen sae space represenaions, hen hey are relaed by Au T AuT, Bu T 2 1 Bu, and Cu = CuT. T saisfying hese equaions is T = 1. I is simple algebra o show ha his is he case for he unresriced sae space rep- Q.E.D. I is useful o compare he marices for he resriced - model s deep srucural parameers and mus saisfy he cross-equaion resricions imposed by he heory. On he oher hand, he only srucure imposed on coef- resricions. I am no imposing anyhing more. A Vecor Auoregressive Moving Average Represenaion Saring from he sae space represenaion, equaion observables in Y is easily derived by recursive subsiuion. In paricular, i is given by 2 Y CB CAB1 CA B2. Assume ha CB is inverible and le e = CB. Then I Y e CABCB e CA BCB e e C e C e Assuming he moving average is inverible, Y can also Y BY 1 1B2Y2 e, where B C BC B C. j j 1 j1 j1 1 PROPOSITION 2. VAR in equaion M B j B j 1 1 and herefore can be represened as a vecor auoregressive moving average represenaion of order Y B M Y e Me Eee 1 1 1, wih CBBC. ProofQ.E.D. Le denoe he vecor of parameers o be esimaed for he VARMA via maximum likelihood, which are all of he elemens of marices B 1, M, and. If I allow hese parameers o ake on any values, i is possible ha he sysem would be nonsaionary or noninverible. I reparameerize he VARMA as described inveribiliy. I also need o check ha B 1 has nonzero elemens and ha [ B1 + M, M ] has full rank o ensure I now have hree saisical represenaions ha are space represenaion, which makes explici use of he deails of he underlying model and imposes hese in cross-equaion resricions; he unresriced sae space represenaion, which imposes zero resricions on he sae space bu no cross-equaion resricions; and he informaion abou he reduced form of he sysem. For all hree, applying he mehod of maximum likelihood is a sraighforward procedure. 4 Seing Up he Laboraory erae 1,000 samples of daa { Y } using he prooype business cycle model. Each sample is 200 quarers in lengh, which is ypical for acual applicaions. This is done by randomly drawing sequences for he shocks { varepsilon }. These shocks, along wih an iniial value of he sae s 0, imply a sequence of exogenous saes { s } ˆk 0 and he sequence { s }, for he business cycle model. For each sample, he rue given by.,.,.,.,.,.,.,.,.,.,.,,,. Using he resriced sae space represenaion, I apply he mehod of maximum likelihood o each of he 1,000 samples. This procedure yields 1,000 esimaes ˆ of 4 on esimaing dynamic linear economies.

8 FEDERAL RESERVE BANK OF MINNEAPOLIS QR he parameer vecor. For each esimae, I can consruc he coefficiens of he model s equilibrium equaions (4) (8). Wih numerical values for hese coefficiens, I can hen consruc he saisics ha business cycle analyss care abou, which will be discussed laer. Similarly, I can apply he mehod of maximum likelihood in he case of he oher wo saisical represenaions. For he unresriced sae space, he procedure yields esimaes for ˆΓ and, in urn, for A u ( ˆΓ ) and C u ( ˆΓ ) of (11). For he VARMA(1,1), he procedure yields esimaes for ˆΛ and, in urn, B ˆ 1, M ˆ, and ˆ of equaion (15). As before, once I have numerical values for he coefficiens in hese equaions, I can consruc he saisics of ineres for business cycle analyss. For he resriced sae space represenaion, hree levels of consrains on he parameer vecor are invesigaed. Recall ha he only resricion in he loose consrains case is ha an equilibrium exiss. In he modes consrains case, I assume ha he parameer consrains are Θ< Θˆ < Θ, where (16) Θ=[. 0075, 0,. 0025, 0,. 25,. 01,. 01,. 15,. 1, 1, 1000,,, ] Θ=[. 0125,. 0075,. 0075,. 025,. 45, 10, 10,. 35,. 1, 1, 1, 10, 10, 10. This implies an annual rae of ineres beween 3 and 5 percen; an annual growh rae of populaion beween 0 and 3 percen; an annual growh rae of echnology beween 1 and 3 percen; an annual depreciaion rae beween 0 and 10 percen; a capial share beween 25 and 45 percen; and beween 0.01 and 10; he mean labor wedge beween 0.25 and 0.35; he mean invesmen wedge beween 0. 1and 0.1; serial correlaion coefficiens beween 1 and 1; and sandard deviaions of he shocks beween 0 and 10 percen. In he igh consrains case, I fix he ineres rae, he growh raes, he depreciaion rae, he capial share, and he means of he ax raes during esimaion and use bounds in equaion (16) for he oher parameers. Business Cycle Saisics Saisics of ineres for business cycle analyss include impulse response funcions, variance decomposiions, auocorrelaions, and cross-correlaions. In his secion, I use he hree represenaions (9), (11), and (15) o consruc hese saisics. ] The firs se of saisics are impulse responses of he hree observables growh in labor produciviy, he log of labor, and he log of he invesmen share o 1 percen shocks in each of he hree shocks in. Here, I repor only he responses in he period of impac of he shock. In he resriced sae space represenaion, he impac of he shock is summarized by he elemens of CB. Similarly, he impac responses are summarized by CB u u for he unresriced sae space represenaion. For he VARMA, one needs addiional informaion o idenify CB from he variance-covariance = ( CB)( CB ). A ypical assumpion made in he lieraure o idenify he responses o a echnology shock ( z ) is o assume ha echnology shocks have a long-run effec on labor produciviy, whereas demand shocks ( l, x ) do no. This assumpion allows me o infer he firs column of CB. (See Chari, Kehoe, and McGraan 2008.) However, i does no imply anyhing for he relaive impacs of l and x. Since hese are no idenifiable, hey are no repored. The impac coefficiens of he impulse responses are repored in Table 1. The firs row shows he rue value of each saisic. For example, in he model, produciviy rises by 0.58 percen in response o a 1 percen increase in z, labor rises by 0.27 percen, and he invesmen share rises by 0.88 percen. Responses o shocks in l are shown in he middle hree columns, and responses o shocks in x are shown in he las hree columns. In he nex hree rows, I repor saisics based on he resriced sae space represenaion wih varying degrees of ighness in he consrains imposed during maximum likelihood esimaion. The las wo rows are he resuls for he unresriced sae space represenaion and he VARMA(1,1) represenaion. In each case, he firs number displayed is he mean esimae of he saisic averaged over he 1,000 daa ses. The second number displayed below in parenheses is he roo mean squared error (RMSE), which is defined as N 1 2 RMSE = ˆ i. N i= 1 ( ) In his formula, ˆ i is he ih esimae of he saisic, i = 1,, N, and is he rue value. If here is no bias due o small samples, hen is equal o he mean of he esimaes ˆ i, i = 1,, N, and he RMSE is equal o he sandard deviaion. 8

9 Table 1 Impac Coefficiens of Impulse Responses (Means and Roo Mean Squared Errors) Wha Happens afer 1% z Shock? Wha Happens afer 1% l Shock? Wha Happens afer 1% x Shock? log y /l log l logx / y log y /l logl logx / y log y /l logl logx / y True _ 1.52 _ _ 1.06 _ 3.52 Resriced SS Tigh _ 1.50 _ _ 1.04 _ 3.47 consrains (.03) (.05) (.18) (.05) (.15) (.46) (.07) (.21) (.31) Modes _ consrains (.04) (.08) (.22) (.07) (.18) (.61) (.10) (.30) (.45) Loose consrains (.06) (.16) (.35) (.08) (.29) (.96) (.16) (.44) (.63) Unresriced SS (.36) (.68) (1.44) (.36) (.75) (1.66) (.31) (.76) (1.63) VARMA (.52) (.96) (1.88) Noes: For each model, parameers are esimaed by he mehod of maximum likelihood. This is done for 1,000 daa ses of lengh 200 periods. The esimaed parameers are used o compue he impac coefficiens repored in he able. The erm logy /l is he growh in labor produciviy, y is oupu, l is labor, and x is invesmen. SS indicaes sae space model, and VARMA indicaes vecor auoregressive moving average model of order (1,1). For he Tigh consrains case of he resriced sae space model, only,, and he sochasic processes of he exogenous shocks are esimaed. For he Modes consrains case, all parameers are esimaed, bu he parameers are consrained o be economically plausible. For he Loose consrains case, he only resricion imposed is ha an equilibrium can be compued. The numbers in parenheses are he roo mean squared errors. Some saisics are no repored for he VARMA represenaion because hey are no idenifiable. I is clear from Table 1 ha he differences in resuls for he resriced sae space and he oher wo repre- esimaes. There is lile bias in he esimaes for he resriced sae space. This is especially rue when igh consrains are used during maximum likelihood esimaion. However, even in he case of modes consrains, he means of he esimaes are very close o he rue space represenaion and he VARMA, he biases are large. For example, all of he prediced responses fol- acual responses. In he case of he shocks o he labor and invesmen wedges for he unresriced sae space model, large biases are also eviden. Nex consider he RMSEs ha appear in parenheses below he means. As I remove resricions, hese errors grow large. Compare, for example, he errors of he resriced sae space represenaion wih igh consrains wih hose of he VARMA in columns 1 hrough 3. In he laer case, he size of he errors indicaes ha he impulse response predicions range from large negaives o large posiives. In oher words, he VARMA predicions are uninformaive. Similarly, he unresriced sae space has large RMSEs for all of he saisics repored in Table 1 and, like he VARMA, is herefore uninformaive abou impulse responses. To generae igh predicions, we need o impose he cross-equaion resricions and resric parameer esimaes o lie in he economically plausible range. When I allow all of he parameers o be compleely free for

10 FEDERAL RESERVE BANK OF MINNEAPOLIS QR Table 2 Variance Decomposiion of Produciviy Growh, Labor, and Invesmen Share (Means and Roo Mean Squared Errors) Wha Fracion of Variance Is Due o z? Wha Fracion of Variance Is Due o l? Wha Fracion of Variance Is Due o x? log y /l log l logx / y log y /l logl logx / y log y /l logl logx / y True Resriced SS Tigh consrains (4.1) (.9) (3.1) (7.0) (12) (8.5) (6.6) (12) (11) Modes consrains (6.9) (1.9) (5.6) (8.0) (14) (14) (8.5) (14) (17) Loose consrains (9.3) (8.2) (12) (10) (23) (22) (15) (19) (22) Unresriced SS (23) (21) (22) (24) (35) (30) (25) (30) (35) VARMA (30) (40) (35) Noes: For each model, parameers are esimaed by he mehod of maximum likelihood. This is done for 1,000 daa ses of lengh 200 periods. The esimaed parameers are used o compue he impac coefficiens repored in he able. The erm logy /l is he growh in labor produciviy, y is oupu, l is labor, and x is invesmen. SS indicaes sae space model, and VARMA indicaes vecor auoregressive moving average model of order (1,1). For he Tigh consrains case of he resriced sae space model, only,, and he sochasic processes of he exogenous shocks are esimaed. For he Modes consrains case, all parameers are esimaed, bu he parameers are consrained o be economically plausible. For he Loose consrains case, he only resricion imposed is ha an equilibrium can be compued. The numbers in parenheses are he roo mean squared errors. Some saisics are no repored for he VARMA represenaion because hey are no idenifiable. he invesmen share. The nex saisics ha I consider are variance decomposiions. For a general sae space sysem of he form S = AS + Be Y = CS wih Ee e, he populaion variances of he observables in Y are he diagonal elemens of he marix V, where V AVA BLLB and L LL. In oher words, he ii elemen of V is he variance of he ih variable in Y. The variance decomposiion summarizes he conribuion of he variances due o each of he shocks in e.v j be he conribuion of he variance of Y due o shock j. This is given by V AV A BL LB, j j j where j is a marix wih he same dimensions as and one nonzero elemen, elemen j j ha is equal o 1. In his case, he ii elemen of V j is he variance of he ih variable in Y which is due o he jh shock. Noe ha V j V j. namely, 10

11 Table 3 Sandard Deviaions and Correlaions of HP-Filered Oupu, Labor, and Invesmen (Means and Roo Mean Squared Errors) Sandard Deviaions of HP-Filered Series Auocorrelaions of HP-Filered Series Cross-Correlaions of HP-Filered Series Oupu and Oupu and Labor and Oupu Labor Invesmen Oupu Labor Invesmen Labor Invesmen Invesmen True Resriced SS Tigh consrains (.086) (.13) (.36) (.008) (.011) (.011) (.013) (.012) (.011) Modes consrains (.097) (.13) (.37) (.011) (.015) (.014) (.015) (.013) (.011) Loose consrains (.098) (.13) (.38) (.012) (.017) (.016) (.015) (.013) (.011) Unresriced SS (.13) (.17) (.47) (.035) (.034) (.035) (.020) (.017) (.014) VARMA (.14) (.16) (.51) (.036) (.029) (.035) (.018) (.018) (.015) Noes: For each model, parameers are esimaed by he mehod of maximum likelihood. This is done for 1,000 daa ses of lengh 200 periods. The esimaed parameers are used o compue he second momens repored in he able. SS indicaes sae space model, and VARMA indicaes vecor auoregressive moving average model of order (1,1). For he Tigh consrains case of he resriced sae space model, only,, and he sochasic processes of he exogenous shocks are esimaed. For he Modes consrains case, all parameers are esimaed, bu he parameers are consrained o be economically plausible. For he Loose consrains case, he only resricion imposed is ha an equilibrium can be compued. The numbers in parenheses are he roo mean squared errors. S B1 M I S 0 0 I M e 1 1 Y I 0 S, = [ ] S and e +1 can be mapped o A, B, and C In Table 2, I repor he predicions of he populaion variance decomposiions. The ordering of resuls in Table 2 is he same as in Table 1, wih he mos resricive Comparing he means of he saisics wih he acual values, we again see large biases for he unresriced sae space and VARMA represenaions, especially for decomposiions of labor and invesmen shares. In erms of he RMSEs, he resuls for he unresriced sae space and VARMA represenaions again show ha he predicions are no informaive. In effec, he range of variances for he VARMA represenaion is close o everyhing in 0 o 100 percen. The hird se of saisics is very common in he real business cycle lieraure ha ypically repors saisics senaion and each se of parameer esimaes, I simulae 500 ime series for oupu, labor, and invesmen of lengh 200. In each case, he oupu and invesmen ake averages of sandard deviaions, auocorrelaions, and cross-correlaions over he 500 simulaions. This is done for each represenaion and for each of he 1,000 11

12 FEDERAL RESERVE BANK OF MINNEAPOLIS QR Table 4 Variance Decomposiion of HP-Filered Oupu, Labor, and Invesmen (Means and Roo Mean Squared Errors) Wha Fracion of Variance of HP-Filered Series Is Due o z? Wha Fracion of Variance of HP-Filered Series Is Due o l? Wha Fracion of Variance of HP-Filered Series Is Due o x? Oupu Labor Invesmen Oupu Labor Invesmen Oupu Labor Invesmen True Resriced SS Tigh consrains (3.0) (0.7) (2.2) (8.2) (12) (10) (8.6) (12) (12) Modes consrains (4.8) (1.0) (2.9) (12) (15) (16) (11) (15) (16) Loose consrains (8.5) (3.3) (6.0) (19) (23) (24) (14) (21) (22) Unresriced SS (22) (26) (25) (29) (37) (30) (27) (31) (34) VARMA (28) (37) (29) Noes: For each model, parameers are esimaed by he mehod of maximum likelihood. This is done for 1,000 daa ses of lengh 200 periods. The esimaed parameers are used o compue he variance decomposiions repored in he able. Tables do no necessarily sum o 100 percen because a wo-sided filer is applied o he ime series. SS indicaes sae space model, and VARMA indicaes vecor auoregressive moving average model of order (1,1). For he Tigh consrains case of he resriced sae space model, only,, and he sochasic processes of he exogenous shocks are esimaed. For he Modes consrains case, all parameers are esimaed, bu he parameers are consrained o be economically plausible. For he Loose consrains case, he only resricion imposed is ha an equilibrium can be compued. The numbers in parenheses are he roo mean squared errors. maximum likelihood parameer vecors. The implied saisics are repored in Table 3. Noice ha he bias and RMSEs of he predicions are small for all represenaions. For example, in all cases, he disribuion of cross-correlaions of oupu and labor has a all of he deails of a model o ge an accurae predicion for uncondiional momens. Table 2. In Table 4, I repor he variance decomposiions done in Table 2 bu is included for easy comparison wih esimaes in he business cycle lieraure. As before, he RMSEs for he unresriced sae space and he VARMA represenaions are so large ha hey are uninformaive. In he resriced sae space model, he esimaes for he echnology shock are very informaive. This is rue even for labor and invesmen, whose variaion depends lile on echnology shocks. The resriced sae space esimaes for he labor shock imply ha i conribues space esimaes for he invesmen shock are he leas informaive bu sill imply ha x has a big effec on invesmen. Conclusion In his aricle, I conduc a simple small-sample sudy. I ask how much can business cycle heoriss learn from acual ime series if hey impose very lile heory when applying heir saisical mehods. The answer is very lile. 12

13 References Anderson, Evan W., Lars P. Hansen,, and Thomas J. Sargen. Handbook of compuaional economics, ed. Hans M. Amman, David A. Ken- auoregressive moving average model o enforce saionariy. Journal of Saisical Compuaion and Simulaion demand and supply disurbances. American Economic Review- Journal of Business and Economic Saisics couning. Economerica business cycle heory? NBER Macroeconomics Annual 2004 versiy Press. space forms. Economerica An empirical invesigaion. models. easily. In, Universiy Press. Paper presened a he Sixh Inernaional Conference of he Sociey for Economic Dynamics and Conrol, Nice, France, June. 13