A Markov-Switching Model of Business Cycle Dynamics with a Post-Recession Bounce-Back Effect

Transcription

1 A Markov-Swiching Model of Business Cycle Dynamics wih a Pos-Recession Bounce-Back Effec Chang-Jin Kim Korea Universiy James Morley Washingon Universiy in S. Louis Jeremy Piger Federal Reserve Bank of S. Louis Preliminary Draf 2/27/02 Absrac: This paper presens a nonlinear model of U.S. GDP growh dynamics ha allows for a pos-recession bounce-back effec in he level of GDP. While a number of sudies have aemped o capure such an effec using ad hoc recession-based dummy variable mehods, we endogenously esimae his business cycle asymmery using an exended version of Hamilon's (1989) Markov-swiching model. Like Hamilon, we find model regimes ha correspond closely o NBER-daed recession and expansions. We also find a large bounce-back effec ha, according o our Mone Carlo analysis, is saisically significan and implies a relaively small permanen effec of recessions. We would like o hank Mrinalini Lhila for providing research assisance. Responsibiliy for all errors is our own. Morley acknowledges suppor from he Weidenbaum Cener on he Economy, Governmen, and Public Policy. The views expressed in his paper should no be inerpreed as hose of he Weidenbaum Cener, he Federal Reserve Bank of S. Louis, or he Federal Reserve Sysem.

2 1. Inroducion In his seminal paper, Hamilon (1989) capures asymmery in U.S. business cycles using an endogenous regime-swiching model of real oupu. His model porrays he shor, violen naure of recessions relaive o expansions. However, oher sudies have emphasized anoher disincive feaure of U.S. business cycles ha is no capured by Hamilon s model: oupu growh ends o be relaively srong following recessions. This feaure has radiionally been modeled in a somewha ad hoc way by allowing growh dynamics o change in he quarers immediaely afer a decline in oupu below is hisorical maximum. In his paper, however, we show ha Hamilon s model can be exended in a very simple way o allow for a pos-recession bounce-back effec while mainaining endogenous esimaion of he underlying recessionary shocks. Our model provides a simple es of he bounce-back effec and produces a sraighforward measure of he long-run effecs of recessions on he level of oupu. We find ha a posrecession bounce-back has been an imporan feaure of U.S. business cycle dynamics and ha he permanen effecs of recessions are subsanially less han suggesed by Hamilon s original model. 2. Background The idea of inherenly differen dynamics in expansions and recessions has a long hisory in business cycle analysis, daing back a leas o Michell (1927) and Keynes (1936). Recen advances in economerics have allowed his idea o be formally modeled and esed. Hamilon (1989) capures asymmeric dynamics using a Markov- 1

3 swiching model ha esimaes wo regimes in U.S. GNP growh behaviour. Noably, even hough he iming of he regimes is endogenously esimaed, he finds ha he regimes correspond closely o NBER-daed recessions and expansions. While he saisical significance of he Markov-swiching behaviour in oupu is clouded by nonsandard es condiions (see Hansen, 1992, and Garcia, 1998), one implicaion of Hamilon s esimaes is clear: recessions have large permanen effecs on he level of oupu. By one measure discussed in his paper and employed here, he expeced level of oupu is permanenly lowered by as much as 4.5% as a resul of a ransiion ino recession. However, one reason his esimae may be so large is ha Hamilon s original model is unable o capure he high growh recovery phase ypical of pos-recession dynamics. We consider his possibiliy in his paper. One approach o modeling he high growh recovery phase is o add a hird regime o Hamilon s model, as in Sichel (1994). However, here is much evidence ha a recovery is no independen of he preceding recession, as would be implied by a hree regime model, bu raher he magniude of he bounce-back is closely relaed o he severiy of he recession (see Friedman, 1964, 1993, and Wynne and Balke, 1992, 1996). Kim and Nelson (1999a) allow for his ype of business cycle asymmery by modeling regime swiching in he cyclical componen of oupu only. While his relaes he bounce-back o he severiy of a recession, i consrains he effecs of recessionary shocks o be compleely ransiory, a priori. Thus, we canno use his approach o examine he permanen effecs of recessions on he level of oupu. Kim and Murray (in press) combine he Hamilon (1989) and Kim and Nelson (1999a) approaches in a mulivariae model wih regime swiching in boh he rend and cyclical componen of 2

4 oupu. While his approach is capable of providing a measure of he permanen effecs of recessions, i comes a he price of considerable added complexiy and he need for srong idenificaion assumpions. A relaed lieraure models he bounce-back effec using nonlinear ARMA processes in which dynamics change when an observed indicaor variable exceeds a given hreshold. In an imporan paper, Beaudry and Koop (1993) augmen a sandard ARMA model of oupu growh wih a curren-deph-of-recession dummy variable ha measures he disance oupu has fallen below is hisorical maximum. They find ha his addiional variable is highly significan using a sandard -es and ha ypical recessions have no significan permanen effec on he level of GDP. However, Hess and Iwaa (1997) argue ha he dummy variable is nonsaionary and he -es oversaes he significance of he bounce-back effec. The Beaudry and Koop model has been exended and modified by several auhors, mos noably Pesaran and Poer (1997) who endogenize he hreshold. Our approach in his paper is o direcly augmen Hamilon s original model wih a new erm ha is able o capure he lengh and severiy of a recession. In his way, our model is like Beaudry and Koop s (1993). However, unlike he curren-deph-ofrecession variable used in heir paper, our bounce-back erm is direcly relaed o he underlying recessionary regimes and is, herefore, endogenously esimaed. I is also saionary by consrucion and so does no suffer from he Hess and Iwaa (1997) criique. Meanwhile, our model places no consrains a priori on he permanen effecs of a ypical recession and, like Hamilon s original model, yields a sraighforward measure of his effec. 3

5 3. Model Our exended version of Hamilon s model, augmened o allow for a posrecession bounce-back effec, is given as follows: m 2 φ ( L) y µ µ S λ S 0 1 j = ε, ε i. i. d. N(0, σ ) j= 1, where he lag operaor φ (L) is p-h order wih roos ouside he uni circle, y is he firs difference of log U.S. real GDP, and S is an unobserved Markov-swiching sae variable ha akes on discree values of 0 or 1 according o ransiion probabiliies Pr[ S = 0 S 1 = 0] q and Pr[ S = 1 S 1 = 1] p. We normalize he saes by = = resricing µ 0. Tha is, S = 1 corresponds o a lower growh regime or, if 1 < µ + µ 0, a conracionary regime. 0 1 < The innovaion in our model is he summaion erm, which for fuure convenience we denoe S ( m) m S j j= 1. This erm is he only addiion o Hamilon s model, as his model obains if λ = 0. The erm reflecs he lengh and severiy of he mos recen lower growh or conracionary regime. In pracice, we se m = 6, which is equal o he lengh of he longes poswar U.S. recessions ( and ). In erms of our model, a bounce-back effec occurs if λ > 0. Figure 1 shows his effec by simulaing sylized versions of our model and Hamilon s original model. 4

6 For boh models, we se he underlying growh rae parameers o be µ = 0 1 and µ 2 1 =. For our model, we se he bounce-back coefficien o be λ = For Hamilon s model, λ = 0. We ignore he auoregressive parameers since for he simulaion we assume ha here are no regular shocks (i.e., ε = 0 for all ). In he boom of he figure, he hick line represens a hypoheical ime pah for he sae variable S. The shif in S from 0 o 1 represens a movemen of he economy ino a conracionary regime for l = 4 quarers, denoed by he shading. As he regime his in period 0 and persiss unil period 4, oupu falls boh for our model and for Hamilon s model. Meanwhile, he summaion erm S ( m ) increases up o he min{ m, l}, which is l = 4 in his case. The S ( m ) erm behaves in a similar fashion as he curren-deph-ofrecession variable in Beaudry and Koop (1993). However, again, i is no an ad hoc dummy variable, bu is endogenously deermined by he underlying saes. For our model, he effec of he S ( m ) erm begins o offse he effec of he S erm as he recession persiss, and oupu levels off. Afer S reurns o 0 and he economy moves back ino expansion, he S ( m ) erm reaches is maximum, and he level of oupu rises dramaically due o λ > 0. This bounce-back in he level of oupu coninues as he expansion persiss, bu is effec diminishes as he S ( m ) erm evenually falls back o is minimum of 0. By conras, for Hamilon s model wih λ = 0, oupu rises from is rough a is regular expansionary growh rae only, implying a much larger permanen effec of he recession on he level of oupu. Esimaion of our model is a sraighforward applicaion of Hamilon s (1989) 5

7 filer. The only new wrinkle is ha, due o he S ( m ) erm, we need o keep rack of saes in each period, whereas Hamilon only needed esimaion deails. p+m 2 p 2 saes. See Hamilon (1989) for 4. Esimaes The daa for y are 100 imes he log of real U.S. GDP over he sample period of 1952:Q1 o 2001:Q2. Given a maximum lag order of p = 4, boh he AIC and BIC pick p = 1. Table 1 repors model esimaes for his case. The firs imporan resul o noice is ha µ + µ 0, implying ha S = 1 corresponds o a conracionary regime. The 0 1 < ransiion probabiliies also sugges ha expansions are much more persisen han conracions, much like he NBER reference cycle. Figure 2 reveals a srong correspondence beween he smoohed probabiliy of being in a conracionary regime and he NBER recession daes. This resul is paricularly noable since i has been widely repored ha Hamilon s original model does no capure NBER recession daes when applied o he longer daa sample employed here (see, for example, Kim and Nelson, 1999b, and McConnell and Perez-Quiros, 2000). The figure also displays he smoohed esimae of S ( m ). As wih he previous figure, his erm increases as he lengh of each conracion progresses, and declines soon afer he recession is over. Again, his erm and is coefficien λ deermine he size of he bounce-back effec. Our esimae of λ is posiive, corresponding o faser growh during pos-recession recoveries. The -saisic for H : λ 0 is 4.2, which is highly 0 = 6

8 significan using sandard asympoic criical values. A possible concern is wheher he sandard asympoic disribuion applies in his case. Hess and Iwaa (1997) argue ha Beaudry and Koop s (1993) curren-deph-ofrecession variable is nonsaionary. Thus, he esimae for is coefficien has a nonsandard disribuion. In our case, given finie m, he S ( m ) erm will be saionary since S is saionary. However, given he persisence of he S ( m ) erm, he small sample disribuion may be very differen o he asympoic disribuion. To examine his possibiliy, we conduc a Mone Carlo experimen. For our daa generaing process, we use Hamilon s (1989) original esimaed model for which λ = 0. We esimae our model allowing λ 0 for each simulaion and calculae -saisics for he null hypohesis H : λ 0 = 0. Table 2 repors criical values for our experimen. We consider sample sizes of T=200 and T=500. The criical values are larger han he sandard normal case, reflecing a small-sample disorion. However, he disorion ges smaller as he sample size ges larger. Meanwhile, our esimae of λ is sill significan a he 5% level using he T=200 resuls. Given a bounce-back effec, he quesion is wheher recessions have permanen effecs on he level of oupu. Hamilon (1989) provides a useful measure of he long-run effecs of recessions in he conex of a regime swiching models. He considers he expeced difference in he long-run level of oupu given a conracionary regime versus an expansionary regime in period : { [ y S 1, I ] E[ y S = 0, I ]} lim E + j = 1 + j 1, j 7

9 where I y, y,...; S, S,...}. For our model, his limi converges o 1 = { ( µ 1 + mλ) (2 q p), which given he esimaes in Table 1 is equal o 0.945, or abou a 1% permanen drop in he level of GDP, and is no saisically significan. By conras, Hamilon s esimaes imply a 4.5% permanen drop ha is saisically significan. I should be noed ha an alernaive meric is also repored in Hamilon s paper ha condiions on I raher han I 1. Insead of giving he dynamic muliplier for a shif in S, his alernaive meric calculaes he forecasable consequences of a recession for fuure oupu. For Hamilon, his number is 3%. For our model, he number is acually posiive and abou 1%, corresponding o a large prediced bounce-back. In addiion o very differen implicaions for he permanen effecs of recessions, anoher noable difference beween our resuls and Hamilon s (1989) relaes o he auoregressive dynamics propagaing he regular ε shocks. Hamilon repors hird and fourh order lags ha are large and negaive. By conras, we find ha higher order lags are small and insignifican. One possible explanaion for his difference is ha he negaive serial correlaion in Hamilon s specificaion is beer capured by he addiional S ( m ) erm han by linear auoregressive dynamics. Thus, our resuls imply very lile serial correlaion in oupu ouside of recessions and heir recoveries. 5. Conclusions In summary, we find ha poswar recessions have no significan permanen impac on U.S. real GDP. Insead, we find a significan and large bounce-back effec 8

10 during he recovery phase of he business cycle. Meanwhile, here appears o be lile serial correlaion in oupu growh during he regular expansion phase of he business cycle. A virue of our model is is simpliciy. In paricular, i is able o capure a defining feaure of he business cycle wih only a small modificaion o Hamilon s original Markov-swiching model of nonlinear dynamics. Again, he modificaion is he addiion of a erm ha reflecs he lengh and severiy of he mos recen recession. In his way, our model is reminiscen of Beaudry and Koop s (1993) model, which also implies small permanen effecs of recessions. However, i should be emphasized ha our model is able o capure he bounce-back effec using an endogenously esimaed sae variable. Meanwhile, he simpliciy of our model suggess ha exensions, such as mulivariae analysis o capure he co-movemen feaure of business cycles or allowing for imevarying ransiion probabiliies, should be relaively easy o implemen. We leave hese exensions o fuure research. 9

11 References Beaudry, P. and G. Koop, 1993, Do recessions permanenly change oupu?, Journal of Moneary Economics 31, Friedman, M., 1964, Moneary Sudies of he Naional Bureau, he Naional Bureau eners is 45 h Year, 44 h Annual Repor, 7-25 (NBER, New York); Reprined in Friedman, M., 1969, The opimum quaniy of money and oher essays (Aldine, Chicago). Friedman, M. 1993, The plucking model of business flucuaions revisied, Economic Inquiry 31, Garcia, R., 1998, Asympoic null disribuion of he likelihood raio es in Markov swiching models, Inernaional Economic Review 39, Hamilon, J.D., 1989, A new approach o he economic analysis of nonsaionary ime series and he business cycle, Economerica 57, Hansen, B.E., 1992, The likelihood raio es under nonsandard condiions: esing he Markov swiching model of GNP, Journal of Applied Economerics 7, S61-S82. Hess, G.D. and S. Iwaa, 1997, Asymmeric persisence in GDP? A deeper look a deph, Journal of Moneary Economics 40, Keynes, J.M., 1936, The general heory of employmen, ineres, and money (Macmillan, London). Kim, C.-J. and C.J. Murray, Permanen and ransiory componens of recessions, Empirical Economics in press. Kim, C.-J. and C.R. Nelson, 1999a, Friedman s plucking model of business flucuaions: 10

12 Tess and esimaes of permanen and ransiory componens, Journal of Money, Credi and Banking 31, Kim, C.-J. and C.R. Nelson, 1999b, Has he U.S. economy become more sable? A Bayesian approach based on a Markov-swiching model of he business cycle, Review of Economics and Saisics 81, McConnell, M.M. and G. Perez-Quiros, 2000, Oupu flucuaions in he Unied Saes: Wha has changed since he early 1980s? American Economic Review, 90, Michell, W.A., 1927, Business cycles: The problem and is seing (NBER, New York). Pesaran, M.H. and S. M. Poer, A floor and ceiling model of U.S. oupu, Journal of Economic Dynamics and Conrol, 21, Sichel, D. E., 1994, Invenories and he hree phases of he business cycle, Journal of Business and Economic Saisics 12, Wynne, M.A. and N.S. Balke, 1992, Are deep recessions followed by srong recoveries?, Economics Leers 39, Wynne, M.A. and N.S. Balke, 1996, Are deep recessions followed by srong recoveries? Resuls for he G-7 counries, Applied Economics 28,

13 Table 1 Maximum Likelihood Esimaes Parameer Esimae Sandard Error µ µ λ q p σ φ µ 0 + µ ( µ 1 + mλ) (2 q p)

14 Table 2 Mone Carlo Resuls Criical Values p-value T=200 T=500 N(0,1)

15 y Hamilon wih bounce-back S ( m ) Hamilon 1 S Time Fig. 1 The Bounce-Back Effec (Simulaed recession is shaded) 14

16 Pr[ S y1,..., yt ] Fig. 2 Smoohed Inferences for E[ S ( m) y1,..., yt ] S and S ( m ) (NBER recession daes are shaded) 15