Graduate Macro Theory II: Stylized Business Cycle Facts and the Quantitative Performance of the RBC Model

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1 Graduate Macro Theory II: Stylized Busiess Cycle Facts ad the Quatitative Performace of the RBC Model Eric Sims Uiversity of Notre Dame Sprig Itroductio This ote describes some basic facts about US busiess cycles ad the examies how well the basic RBC model ca fit those facts. 2 Stylized US Busiess Cycle Facts The covetio withi the literature is to look at HP detreded data to examie the busiess cycle. We focus o deviatios from the smooth HP tred. There are, of course, alterative ways i which oe ca look at the cyclical compoet of the data (e.g. first differeces, the Bad Pass filter, liear detredig, etc.). Busiess cycle momets focus primarily o a variety of secod momets. I particular, the stadard deviatio of a HP filtered series is referred to as its volatility. We are also iterested i lookig at a series cyclicality, which is defied as its cotemporaeous correlatio with GDP. We call the first order autocorrelatio of a series a measure of its persistece, ad we also look at how strogly a series is correlated with output led or lagged a umber of periods, so as to say somethig about which series are laggig idicators ad which series are leadig idicators. The series we are most iterested i lookig at are the same edogeous variables that come out of a simple real busiess cycle model output, cosumptio, ivestmet, total hours worked, the real wage, ad the real iterest rate. I additio, we will look at average labor productivity (the ratio of output to total hours worked), the price level, ad total factor productivity (TFP). The price level is ot i the model as we have thus far specified it, but ca easily by added. TFP is the empirical couterpart of the drivig force a t i the model. We measure it as output mius share weighted iputs: l â t = l y t α l k t (1 α) l t (1) 1

2 Costructig this series requires a empirical measure of the capital stock. I practice, this is hard to measure ad most existig capital stock series are oly available at a aual frequecy. Typically the way i which people measure the capital stock is by usig the perpetual ivetory method. This method essetially takes data o ivestmet, a iitial capital stock, ad a estimate of the rate of depreciatio δ ad costructs a series usig the accumulatio equatio for capital: k t+1 = I t + (1 δ)k t. All series (with the exceptio of TFP, the price level, ad the iterest rate) are expressed i per capita terms after dividig by the civilia o-istitutioalized populatio aged 16 ad over. All series are also i real terms (except the price level) ad i logs (with the exceptio of the iterest rate). The measure of GDP is the real GDP from the BEA accouts. The measure of cosumptio is the sum of o-durable ad services cosumptio, also from the BEA accouts. Ivestmet is measured as total private fixed ivestmet plus cosumptio expeditures o durable goods (durable goods should be thought of as ivestmet because they provide beefits i the future, just like ew physical capital does). Cosumptio ad ivestmet data are from the BEA. Total hours is measured as total hours i the o-farm busiess sector, available from the BLS. Productivity is output per hour i the o-farm busiess sector, also from the BLS. Wages are measured as real compesatio per hour i the o-farm busiess sector, from the BLS. The price level is measured as the implicit price deflator for GDP, from the BEA. The omial iterest rate is measured as the three moth Treasury Bill rate. The real iterest rate is approximately r t = i t E t π t+1. I costruct a measure of the ex-post real iterest rate usig the measured T-bill rate at time t ad actual iflatio from t to t + 1, where iflatio is measured from the GDP deflator. This is ex-post because actual will ot equal expected iflatio i geeral. All series are HP filtered. The data are from 1948 quarter 1 to 2010 quarter 3. The selected momets are show below. Series Std. Dev. Rel. Std. Dev. Corr w/ y t Autocorr Corr w/ y t 4 Corr w/ y t+4 Output Cosumptio Ivestmet Hours Productivity Wage Iterest Rate Price Level TFP The first colum with umbers gives the stadard deviatio; the secod colum gives the stadard deviatio relative to the stadard deviatio of output. This ca be iterpreted as a 2

3 measure of relative volatility. We observe that cosumptio is sigificatly smoother tha output; i cotrast ivestmet is more tha 2.5 times more volatile tha output. Hours are about as a volatile as output. Productivity, the real iterest rate, ad real wages are sigificatly smoother tha output. TFP is about two-thirds as volatile as output. The ext colum gives the cotemporaeous correlatio with output. We observe that cosumptio, ivestmet, hours, ad TFP are all strogly correlated with output (ad ideed also with each other). Productivity is somewhat more weakly correlated with output; ad ideed this calculatios masks a large sig shift i the correlatios from strogly positive prior to the mid- 1980s to weakly egative thereafter. Real wages are weakly correlated with GDP. The real iterest rate is essetially acyclical, with a very slightly egative cotemporaeous correlatio (that correlatio will typically go egative if I use ex-ate expectatios of iflatio). The price level is mildly coutercyclical. We see that almost all series are strogly persistet i the sese of havig a large first order autocorrelatio coefficiet. The least persistet series is the real iterest rate, but this autocorrelatio is still We observe that hours are a laggig idicator i the sese that the correlatio with output lagged oe year is quite positive. The real iterest rate is egatively correlated with output led four quarters. 3 The Basic RBC Model ad Calibratio The basic RBC model ca be characterized by the first order coditios of the decetralized model, as described i class. These first order coditios are: u (c t ) = βe t ( u (c t+1 )(a t+1 f k (k t+1, t+1 ) + (1 δ) ) (2) v (1 t ) = u (c t )a t f (k t, t ) (3) k t+1 = a t f(k t, t ) c t + (1 δ)k t (4) l a t = ρ l a t 1 + ε t (5) y t = a t f(k t, t ) (6) y t = c t + I t (7) u (c t ) = βe t u (c t+1 )(1 + r t+1 ) (8) w t = a t f (k t, t ) (9) R t = a t f k (k t, t ) (10) I use the fuctioal form assumptios that u(c t ) = l c t, v(1 t ) = θ l(1 t ), ad f(k t, t ) = kt α 1 α t. These first order coditios ca the be re-writte imposig these fuctio forms to get: ( 1 1 ( = βe t αat+1 kt+1 α 1 c t c 1 α t+1 + (1 δ))) (11) t+1 3

4 θ = 1 (1 α)a t kt α α t 1 t c t (12) k t+1 = a t kt α t 1 α c t + (1 δ)k t (13) l a t = ρ l a t 1 + ε t (14) y t = a t k α t 1 α t (15) y t = c t + I t (16) 1 1 = βe t (1 + r t+1 ) c t c t+1 (17) w t = (1 α)a t k α t α t (18) R t = αa t kt α 1 t 1 α (19) We ca solve for the steady state of the model as we have before. I the o-stochastic steady state we set a t equal to its mea, which is 1 (0 i the log). Let variables without time subscripts deote steady states. I macroecoomics calibratio is a particular way to choose parameter values. The gist of the calibratio approach is to pick parameters so that the steady state of the model match certai log ru (i.e. average) features of the data. Oe of the ovelties of the RBC approach (ad with it calibratio) was that it took a model that was (a) desiged to explai the log ru ad (b) picked parameters desiged to explai the log ru but the (c) used that model ad those parameters to explai the short ru. Let s begi by comig up with a value for β. Go to (17), the Euler equatio for risk free bods. Evaluated i steady state, it implies: β = r I the data the average real iterest rate o riskless debt ca be measures as r = i π, where i is a safe omial iterest rate ad π is the iflatio rate. This implies a average real iterest rate of somethig o the order of two percet (at a aual frequecy), depedig. We ca use that to back out the required β to make the above hold. This implies a value of β = I m goig to roud that dow ad set β = Give the fuctio form assumptios, (18)-(19) yield: (20) w + Rk = y I other words, total output/icome is the sum of paymets to factors. Usig the expressio for the real wage, we get: w = (1 α)y w y = (1 α) (21) I other words, we ca measure α by lookig at the fractio of total icome that gets paid out i the form of wages. Doig this yields a value of α = 0.33 or so. Next, look at the accumulatio equatio for capital. This reveals that, i steady state: 4

5 I k = δ (22) Give data o ivestmet ad capital, we ca thus measure δ as the average ratio of ivestmet to capital. This comes out to be about δ = 0.025, or about 10 percet at a aual frequecy. Next go to (11) to solve for the steady state capital-labor ratio: ( k = α 1 β (1 δ) ) 1 1 α Usig the umbers for α, β, ad δ, this implies that the capital-labor ratio should be about 28. labor: Now solve the accumulatio equatio for steady state cosumptio per worker: c = ( ) k α δ k Now solve the first order coditio for labor supply for aother expressio for cosumptio to (23) (24) Equate (21) ad (22): c = 1 θ 1 (1 α) ( ) k α (25) 1 θ Solve for θ, takig as give: 1 (1 α) θ = ( ) k α = 1 ( k (1 α) ( k ) α δ k ( ) k α δ k I the data, people work about oe-third of their time edowmet (this may be a bit of a overstatemet, but it s simple eough). So set = 0.33 ad solve for θ. I get That is all the parameters of the model that do ot gover the stochastic process for a t, which we tur to ext. ) α (26) (27) 4 How Well Ca the Model Fit the Data? We eed a systematic way to calibrate the parameters goverig the process for TFP. I begi by liearly detredig my measure of log TFP. I eed to liearly detred because it is (implicitly) assumed i the model that a liear, determiistic tred drives the observed treds i the actual data (though we igore the small variatios to the first order coditios that are ecessary to be cosistet with that). As such, I regress my empirical measure of TFP o a costat ad a liear time tred: l â t = φ 0 + φ 1 t t + u t (28) 5

6 I get the followig estimates: φ 0 = ad φ 1 = This basically ca be iterpreted that TFP grows, o average, at about 0.012, or 1.2 percet, per year. The I take the measured residual, û t, which ca be iterpreted as the detreded TFP series, ad estimate a AR(1) process: û t = ρû t 1 + e t (29) I get the followig estimates: ρ = ad the stadard deviatio of the residual of I use these parameters to solve the model. Below is a table of momets from the model: Series Std. Dev. Rel. Std. Dev. Corr w/ y t Autocorr Corr w/ y t 4 Corr w/ y t+4 Output Cosumptio Ivestmet Hours Productivity Wage Iterest Rate TFP Let s take a momet to compare the umbers i this table (from model-geerated data) to the umbers from the US data. Let s begi by focusig o areas where the model does well. We see that the model does a pretty good job at matchig the volatilities of output, cosumptio, ad ivestmet i particular, cosumptio is sigificatly smoother tha output, ad ivestmet is sigificatly more volatile tha output. The model does a good job of matchig the volatilities of labor productivity ad TFP. The model also does a good job at matchig the ow autocorrelatios the series are all persistet with first order autocorrelatio coefficiets typically i the eighborhood of Lastly, the model captures the fact that most quatity series (cosumptio, ivestmet, hours, productivity, ad TFP) are quite procyclical (high cotemporaeous correlatios with output), though these correlatios are too high i the model relative to the data. Now let s move o to where the model does less well. Whereas it does pretty well with quatity momets (see above), it does much less well with prices. I particular, the model does ot geerate eough volatility of iterest rates (relative stadard deviatio of 0.06 i model vs i the data). Further, it geerates wages ad real iterest rates that are far too procyclical relative to the data. I the data, wages are very modestly procyclical ad real iterest rates are acyclical or coutercyclical, depedig o how you measure them. There is some evidece that aggregate wage date uderstates the procyclicality of wages due to a compositio bias (Solo, Barsky, ad Parker, 1994), so the wage cyclicality ca potetially be recociled. It s much harder to deal with the iterest rate cyclicality. The model does ot do great at the dyamic correlatios. Oe particular area of failure is the fact that real iterest rates positively lead output i the model, whereas they 6

7 egatively lead output i the data (Kig ad Watso, 1988). Fially, aother failure of the model is that it does ot geerate sufficiet volatility of hours i the data, hours are actually slightly more volatile tha output (this relative volatility has rise over time), but i the model hours are about half as volatile as output. Some of these deficiecies are easier to deal with tha others. For example, we ca employ a versio of Hase (1985) ad Rogerso (1988) idivisible labor model to get what essetially amouts to ifiitely elastic labor supply. This will work to raise the volatility of hours ad lower the cyclicality of wages. We could also add moey to the model i a way that does t chage ay of the above results, but which makes the model capable of matchig the coutercylicality of the price level. I additio, we ca add shocks to thigs other tha techology thigs like govermet spedig shocks or distortioary tax shocks (McGratta, 1994). These ca work to lower the cotemporaeous correlatios with output, which are too high i the data. At a eve deeper level, people have criticized RBC models because they do t seem particularly realistic. To geerate fluctuatios that resemble those i the US, oe eeds large, high frequecy variatio i a t. No other shock (e.g. govermet spedig, prefereces, etc.), withi the cofies of the model, ca be the mai drivig force behid the data, although, as metioed i the precedig paragraph, addig other shocks ca make the overall correlatios with output look better. The reaso is because hours ad cosumptio are highly positively correlated i the data (correlatio betwee HP filtered series of 0.78). Combie (3) with (9), ad use our baselie fuctioal form assumptios. You get: Let s log-liearize this: θ = 1 (1 α)a t kt α α t 1 t c t Agai defie γ = l θ l(1 t ) = l c t + l(1 α) + l a t + α l k t α l t ( ) 1 ñ t = c t + ã t + α k t αñ t 1 ad we ca write this as: ñ t = ( ) 1 ( ) c t + ã t + α k t γ + α If either k or a move, the it must be the case that ad c move i opposite directios. Sice shocks to k do t seem like a very plausible explaatio, we are left with shocks to a. These shocks must be the mai drivig force behid the data, otherwise cosumptio ad hours will ot be correlated strogly eough. People have said that assumig busiess cycles result from exogeous techological progress (or worse, regress) is uappealig (Summers, 1986). To geerate recessios, oce eeds a to declie. What does this eve mea, especially for a moder ecoomy? If a is movig aroud so much, why 7

8 do t we read about it i ewspapers? Rather, we typically read about output ad employmet decliig with some speculatio as to why, but that speculatio rarely if ever metios fluctuatios i techological possibilities or somethig similar. Hece, critics of the real busiess cycle model are ucomfortable with the facts that it is (a) drive by techology shocks ad that (b) these shocks must be large ad sometimes egative. Hece, much of busiess cycle research sice the 1980s has bee ivolved i modifyig the basic model to (a) allow other shocks to matter i a way that they ca t i the basic model (e.g. moetary policy) ad (b) geeratig better ad more realistic mechaisms for the model to take small shocks (as opposed to large) ad produce relatively large busiess cycles. The real busiess cycle model has a fairly weak amplificatio mechaism ad a eve weaker propagatio mechaism. Amplificatio refers to a model s ability to have output react by substatially more tha the exogeous shock i.e. to take small shocks ad produce large fluctuatios. The oly amplificatio mechaism here is labor supply, ad it is fairly weak. Propagatio refers to a model s ability to make shocks have persistet effects. The oly propagatio mechaism i the model is capital accumulatio, but this is weak as well. Hece, the time series properties of output ed up essetially lookig just like those of TFP output is about as variable ad about as persistet as the mai drivig force. Much of the rest of what we do i this class will ivolve addig amplificatio ad propagatio mechaisms ad re-toolig the model i other ways such that o-techology shocks (e.g. moey) ca have large real effects. 8

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