the business cycle: it™s still a puzzlelchrist/research/... · 2003. 4. 10. · business cycle,...

56 Economic Perspectives

The business cycle:Its still a puzzle

Lawrence J. Christiano and Terry J. Fitzgerald

Lawrence J. Christiano is a professor of economics atNorthwestern University, a consultant to the FederalReserve Bank of Chicago, and a research associate atthe National Bureau of Economic Research. Terry J.Fitzgerald is an economist at the Federal Reserve Bankof Cleveland. The title of this article is modified fromKocherlakotas (1996) article. The findings for businesscycle analysis are similar to Kocherlakotas for theequity premium puzzle. The authors are grateful fordiscussions with Michelle Alexopoulos, Stefania Albanesi,Paul Gomme, Henry Siu, and Robert Vigfusson.

Introduction and summary

Good fiscal and monetary policy requires a clear un-derstanding of the workings of the economy, espe-cially what drives the business cyclethe periodicups and downs in economic activity. Since at leastthe late 1800s, a full swing from the start of an eco-nomic expansion to a recession and back to the startof another expansion has generally taken betweentwo and eight years. Every citizen is keenly aware ofthe state of the economy, whether it is in prosperityor recession.

Everyone is so conscious of the business cyclebecause most sectors of the economy move up anddown together.1 This phenomenon, referred to ascomovement, is a central part of the official definitionof the business cycle. The definition is set by the Na-tional Bureau of Economic Research (NBER), whichdecides when recessions begin and end. Under theNBERs definition,

... a recession is a [persistent] period of declinein total output, income, employment, and trade, usu-ally lasting from six months to a year, and marked bywidespread contractions in many sectors of theeconomy.2

Even though comovement is a defining character-istic of the business cycle, in recent decades macro-economists have tended to focus on understandingthe persistence in the ups and downs of aggregateeconomic activity. They have generally been less con-cerned with understanding the synchronized natureof this pattern across sectors. In part, the omissionreflects the conceptual difficulties inherent in think-ing about an economy with many sectors.3 Standardmodels of business cycles assume there is only onegood being produced and so they consider only oneeconomic sector. These models do not encouragethinking about the comovement of economic activityacross many sectors. Since these models were first

introduced, in the late 1970s and early 1980s, the stateof macroeconomics has advanced rapidly. The con-ceptual and computational barriers to thinking aboutmultiple sectors are quickly falling away. As a result,recent years have witnessed a renewed interest inunderstanding comovement.

We have two objectives in this article. The firstis to document business cycle comovement. Weexamine data on hours worked in a cross section ofeconomic sectors. We examine the business cyclecomponents of these data and show that the degreeof comovement is substantial. Our second objectiveis to analyze explanations for this comovement. Wefind that none is completely satisfactory. Still, thisis a growing area of research, and we are seeingsome progress.

Identifying comovement

To study comovement across sectors over thebusiness cycle, we need the following two things: ameasure of the level of economic activity in the vari-ous sectors of the economy; and a precise definitionof what we mean by the business cycle component ofthe data. Below, we address these two issues. Afterthat, we present our results, characterizing the degreeof comovement in the data.

57Federal Reserve Bank of Chicago

The dataWe measure economic activity in a given sector

by the number of hours worked in that sector. Table 1lists the sectors we consider. The hours worked mea-sure that covers the most sectors is total private hoursworked.4 This measure covers all sectors of the econ-omy, except government and agriculture. It is brokeninto hours worked in goods-producing industries andin service-producing industries. Goods-producing in-dustries are further broken into mining, manufactur-ing, and construction. Similarly, service-producing

industries are broken into five subsectors. The sub-sectors of manufacturing, durable goods and nondu-rable goods, are broken into yet smaller sectors. Thedata in the third column give an indication of the rela-tive magnitude of each subsector. In particular, anygiven row reports the average number of peopleemployed in that sector, divided by the average num-ber of people employed in the sectoral aggregate towhich that sector belongs. Thus, for example, 58 per-cent of manufacturing employment is in the durablegoods sector and 42 percent is in the nondurable

Properties of the business cycle components of hours worked

TABLE 1

Variable Relative Relative Business cyclenumber Hours worked variable magnitude volatility comovement

1 Total private 1.00 1.00 .002 Goods-producing industries .33 3.91 .993 Mining .03 5.46 .384 Construction .17 6.75 .885 Manufacturing .80 3.92 .976 Durable goods .58 6.90 .977 Lumber and wood products .06 10.18 .898 Furniture and fixtures .04 8.14 .949 Stone, clay, and glass products .05 4.98 .95

10 Primary metal industries .09 9.89 .8611 Fabricated metal products .13 7.21 .9612 Machinery, except electrical .19 11.10 .9313 Electrical and electronic equipment .15 8.75 .8814 Transportation equipment .17 7.83 .8915 Instruments and related products .08 5.03 .7616 Miscellaneous manufacturing .04 3.23 .9017 Nondurable goods .42 1.39 .9118 Food and kindred products .21 .16 .5019 Tobacco manufactures .01 1.83 .0820 Textile mill products .11 3.92 .7621 Apparel and other textile products .15 2.64 .8522 Paper and allied products .09 1.97 .8523 Printing and publishing .16 .91 .9024 Chemicals and allied products .13 1.01 .8025 Petroleum and coal products .02 2.02 .1626 Rubber and misc. plastics products .09 7.82 .8927 Leather and leather products .03 2.71 .6428 Service-producing industries .67 .25 .9329 Transpor tation and public utilities .10 .87 .9530 Wholesale trade .10 .65 .8731 Retail trade .31 .36 .8732 Finance, insurance, and real estate .10 .35 .4833 Services .38 .19 .49

Notes: The column labeled “Relative magnitude” reports an indication of the relative magnitude of each sector. Any given row reports the average number of peopleemployed in that sector divided by the average number of people employed in the sectoral aggregate to which that sector belongs, for example, 58 percent ofmanufacturing employment is in the durable goods sector and 42 percent is in the nondurable goods sector. The column labeled “Relative volatility” reports thevariance of the business cycle component of the logarithm of hours worked in the indicated row variable divided by the variance of the business cycle component ofthe logarithm of total private hours worked. The column labeled “Business cycle comovement” is calculated using the process described in note 6 of the article.

Source: Authors’ calculations from data of DRI Basic Economics database, 1964–96.


goods sector. Also, the largest goods-producing indus-try, by far, is manufacturing, which has 80 percent ofall goods-producing employees.

Next, we try to characterize how much businesscycle comovement there is across the economic sec-tors we consider. That is, if we limit ourselves to thebusiness cycle range of fluctuations in the datafluctuations that last between two and eight yearsto what extent do the data move up and downtogether?5

Business cycle component of the dataA detailed discussion of our notion of the business

cycle component of the data is in technical appendix1. Figure 1 illustrates the basic idea behind what wedo. The choppy line in panel A of figure 1 displaystotal private hours worked. The reported data are thelogarithm of the raw data. The advantage of using thelogarithm of the data in this way is that the resultingmovements correspond to percent changes in the

underlying raw data. The deviations between theactual data and the trend line in panel A of figure 1are graphed in panel B. Those deviations contain therapidly varying, erratic component, inherited from thechoppy portion of the data that is evident in panel A.The smooth curve in panel B is our measure of thebusiness cycle component of the total private hoursworked data. Specifically, that measure excludes boththe trend part of the data and the rapidly varying,erratic component. It includes only the component ofthe data that contains fluctuations in the range of twoto eight years. According to our approach, the econo-my is in recession when our business cycle measure isnegative and in prosperity when it is positive.

Figure 1 also compares our measure of the busi-ness cycle with the one produced by the NBER. Thestart of each shaded area indicates the date when,according to the NBER, the economy reached a busi-ness cycle peak. The end of each shaded area indicates

a business cycle trough. Note how totalprivate hours worked fall from peak totrough and then generally grow fromtrough to peak. An obvious difference inthe two business cycle measures is thatours is a continuous variable, while theNBERs takes the form of peak and troughdates. As a result, our measure not onlyindicates when a recession occurs, butalso the intensity of the recession. Apartfrom these differences, however, the twomeasures appear reasonably consistent.For example, note that near the troughof every NBER recession, our measureof the business cycle is always negative.But the two measures do not alwaysagree. According to our measure, theeconomy was in recession in 1967 and in1987, while the NBER did not declare arecession during those periods. In part,this is because there must be severalmonths negative employment growthbefore the NBER declares a recession.However, our procedure only requires atemporary slowdown.

Figure 1 provides informal evidencein support of the facts we wish to docu-ment. As noted in the introduction, theNBER must see a broad-based declinebefore declaring a recession. Thus, theNBER dates in figure 1 indicate periodswhen many economic sectors showedweakness. Since these dates roughly co-incide with periods of weakness in total

Total hours worked and its trend

logarithmA. Total hours worked

FIGURE 1

logarithmB. Total hours worked—High-pass and business cycle components

TrendData

Businesscycle

component

High-pass

Note: Shaded areas indicate recessions as determined by theNational Bureau of Economic Research.


→


private hours worked, this is consistent with the viewthat most sectors move up and down together, atleast in the two- to eight-year frequency range. Westress, however, that the NBERs dating proceduresare informal. Our objective in this section is to providea formal, quantitative assessment of the degree ofcomovement among economic sectors.

We computed a business cycle component foreach of the 33 series listed in table 1. As we anticipated,we find that the business cycle components in mostof the series move together closely. This is true, de-spite a striking lack of uniformity in other respects.For example, note how different the trends in figure 2are. The first two columns report data for the goods-producing industries and its major components. Thesecond two columns report the analogous data forthe service-producing industries. Generally, trend em-ployment is down in the goods-producing industriesand up in the service-producing industries. The lev-els of volatility in the business cycle components ofthe various series are also very different. The fourthcolumn of table 1 reports the variance of the businesscycle component of a variable, divided by the varianceof aggregate hours worked. The relative variance ofhours worked in goods-producing industries is typi-cally quite high, substantially above 2, and it is quitelow for service-producing industries. That goods-producing industries are volatile relative to the service-producing industries is well known.

Measuring business cycle comovement

Despite the very substantial differences in thetrends of the data series shown in figure 2, their move-ments over the business cycle are quite similar. Figure 3illustrates the business cycle components of the samevariables used in figure 2. In each case, we computedthe business cycle component using exactly the samemethod underlying the calculations in panel B of fig-ure 1. Each graph contains the business cycle compo-nent of the variable indicated and the business cyclecomponent for total private hours. This was takendirectly from panel B of figure 1.

In most of the series in figure 3, the data moveup and down closely with the business cycle compo-nent of total hours worked. There are some exceptions.For example, the business cycle movements in miningbear little resemblance to the business cycle move-ments in total hours worked. At the same time, miningrepresents a very small part of the private economyand employs only 3 percent of workers in the goods-producing industry. Another exception is the finance,insurance, and real estate (FIRE) industry, whosebusiness cycle component exhibits reasonably high

comovement with aggregate employment until the1980s, after which this relationship breaks down.

To measure the degree of business cycle comove-ment between a given series and total hours worked,we use a statistic that is like the square of the correla-tion between the business cycle components in thetwo variables. Our statistic measures the fraction ofthe variance in the series that can be accounted forby the total hours worked data.6 If this number is, say,98 percent, this means that 98 percent of the businesscycle variance in the variable can be accounted for bythe business cycle in aggregate hours worked. Theseresults are reported in the fifth column of table 1. Asexpected, the results indicate that this measure ofcomovement is relatively low, in the sense of beingbelow 0.50, for the mining, FIRE, and services sec-tors. Overall, however, the degree of comovement bythis measure is high.

Going one step further in the level of disaggrega-tion, we can get an idea about the comovement in thecomponents of durable and nondurable manufactur-ing. Figure 4, panel A displays the business cyclemovements in the components of durable manufac-turing sectors. Panel B does the same for nondurablemanufacturing. In each case, the data series graphedat the top of the figure is the business cycle compo-nent of total hours worked. The series are presentedso as to allow one to focus exclusively on the degreeof comovement between them. Thus, we added a con-stant to each series to spread them out across thefigure and divided each series by its sample standarddeviation, so that the standard deviation of thereported data is unity in each case.7 The number tothe right of each line in the figure identifies the dataseries. Figure 4 also displays the NBER peak andtrough dates as a convenient benchmark.

Figure 4, panel A shows that the comovementamong sectors in durable manufacturing is very high.With only one minor exception, the variables moveclosely with each other and with aggregate employ-ment. The exception is that instruments and relatedproducts, series 15, does not move closely with theother variables during 1987, when the other businesscycle components are signaling a recession. However,overall the degree of comovement is strikingly high.Figure 4, panel B shows that the business cycle co-movement in the nondurable goods manufacturingindustries is lower than in the durable goods sector.Two variables that do not comove closely with theothers at business cycle frequencies are tobacco man-ufactures, series 19, and petroleum and coal products,series 25. Both these variables are rising in the firstand last NBER recession periods in our data. The

60

Eco

nom

ic Persp

ectives

FIGURE 2

Hours worked in various sectors: Data and trends

logarithmA. Goods-producing industries

logarithmD. Mining

logarithmG. Service-producing industries

logarithmJ. Retail trade

logarithmB. Manufacturing, durable goods

logarithmE. Construction

logarithmH. Transportation and utilities

logarithmK. Finance, insurance, and real estate

logarithmC. Manufacturing, nondurable goods

logarithmF. Wholesale trade

logarithmI. Services

Trend

Data


→

61

Federal R

eserve Ban

k o

f Chicag

o

FIGURE 3

Business cycle component comparison: Total hours worked versus hours worked in various sectors

logarithmA. Goods-producing industries

logarithmD. Mining

logarithmG. Service-producing industries

logarithmJ. Retail trade

logarithmB. Manufacturing, durable goods

logarithmE. Construction

logarithmH. Transportation and utilities

logarithmK. Finance, insurance, and real estate

logarithmC. Manufacturing, nondurable goods

logarithmF. Wholesale trade

logarithmI. Services

Total

Note: The information displayed in the “total hours worked” line is “business cycle component,” taken from figure 1, panel B.


Subsector


FIGURE 4

Business cycle component of total hours worked and hours workedin various manufacturing subsectors

A. Two-digit manufacturing durableand total hours worked

B. Two-digit manufacturing nondurableand total hours worked

1

16

15

14

13

12

11

10

9

8

7

1

27

2625

24

232221

2019

18

Notes: The number on the right indicates the variable sector from table 1 that is being highlighted. Panel A shows:1—total private hours; 7—lumber and wood products; 8—furniture and fixtures; 9—stone, clay, and glass products;10—primary metal industries; 11—fabricated metal products; 12—machinery, except electrical; 13—electrical and electronicequipment; 14—transportation equipment; 15—instruments and related products; and 16—miscellaneous manufacturing.Panel B shows: 1—total private hours; 18—food and kindred products; 19—tobacco manufactures; 20—textile mill products;21—apparel and other textile products; 22—paper and allied products; 23—printing and publishing; 24—chemicals and alliedproducts; 25—petroleum and coal products; 26—rubber and miscellaneous plastics products; and 27—leather and leather products.

Each variable has been scaled by its sample standard deviation and a constant has been added in order to spread out the data in the panels. Shaded areas indicate recessions as determined by the National Bureau of Economic Research.


comovement statistic for these variables reported intable 1 is very low, 0.08 for tobacco manufacturesand 0.16 for petroleum and coal products. The othervariables in nondurable manufacturing display stron-ger comovement, with comovement statistics of 0.50or higher.

Up to now, the statistics we have used to charac-terize comovement emphasize association with aggre-gate hours worked. This nicely complements thevisual evidence in the graphs. However, there is apitfall to relying exclusively on statistics like this tocharacterize comovement. A simple example illustratesthe point. Suppose there is a variable, yt, which is thesum of two other variables, y1t and y2t:

yt = y1t + y2t.

Suppose further that y1t and y2t are uncorrelated. Noone would say there is comovement between thesevariables. Still, each variable is strongly correlatedwith the aggregate. To see this, take the simple casewhere the variance of y1t and y2t is σ

2. Then, the corre-lation between yit and yt is 0.71, for i = 1,2, despite the

absence of comovement between the variables.8 Thisexample exaggerates the point somewhat, since resultsare less severe when there are more than two sectors.9

Still, this pitfall is of some concern.With this in mind, we consider the correlation

between the business cycle components of all thevariables. A difficulty with this is that there are manysuch correlations. For example, with three variables,there are three possible correlations, with four thereare six, with five there are ten, and with n there aren(n 1)/2. So, with n = 33, there are 528 possible cor-relations. It is a challenge to organize and presentthis many correlations in a coherent way. We presentthe mean and histogram of the correlations for differentsubsectors in figure 5. The histograms display, on thevertical axis, the fraction of correlations lying within agiven interval, whose midpoint is indicated on thehorizontal axis.10

Figure 5, panel A displays the correlations for thefinest levels of aggregation for which we have data.This means hours worked in mining, construction, the20 components of manufacturing, and the five compo-nents of the service-producing industries. Thus, we


have 27 data series, with 351 correlations between them.Figure 5, panel A indicates that the mean of these corre-lations is 0.55. When we eliminate the three data seriesthat we already know do not display strong businesscycle comovement, the mean rises to 0.68. The histo-gram shows that there is a substantial fraction of highcorrelations in these data. We infer that the data areconsistent with the impression from the precedingstatistics that there is considerable evidence of comove-ment. Figure 5, panel B presents the results for themanufacturing durable sector. Consistent with ourprevious findings, the degree of comovement in thissector is very high, with a mean correlation of 0.82.Figure 5, panels C and D show the results for thenondurable manufacturing sector and the service-producing sectors, respectively. Again, the results areconsistent with the notion that there is less comove-ment in these sectors than in manufacturing durables.Still, the degree of comovement is substantial, withmean correlations in excess of 0.6 if we consider all sec-tors except tobacco and petroleum and coal products.

In conclusion, we find that there is substantialbusiness cycle comovement in the data. Only two rel-atively small sectorstobacco manufactures and pe-troleum and coal productsexhibit little tendency tomove up and down with general business conditionsover the business cycle.

Explaining business cycle comovement

What is it that at times pulls most sectors of theeconomy up, and at other times pushes them downagain? This is one of the central questions in busi-ness cycle analysis. Although economists have de-veloped a number of possible explanations, thephenomenon remains a puzzle.

In a classic article devoted to this puzzle, RobertE. Lucas, Jr., conjectures that the resolution must liein some sort of shock that hits all sectors of the econ-omy, a so-called aggregate shock (Lucas, 1981).Many economists today would probably agree withthis conjecture. That is why, in practice, the search forthe ultimate cause of business cycles often focuses

FIGURE 5

frequency

Distribution of correlations between business cycle components of hours worked in various sectors

A. All sectors and all sectors without mining (3), tobacco manufactures (19),and petroleum and coal products (25)

–0.4 –0.2 0.2 0.40.0 0.6 0.8 1.0

0.0 0.5 1.0 1.00.0 0.5 1.00.0 0.5

correlation

frequencyB. Two-digit durable manufacturing

frequencyC. Two-digit nondurable manufacturing

frequencyD. Service-producing

correlation correlation correlation

All sectors (0.55)

All sectors minus three (0.68)

All sectors (0.82) All sectors (0.46)

All sectors minus (19)and (25) (0.66)

All sectors (0.63)

Notes: Whole numbers in parentheses represent the variable sector designated in table 1. Decimals in parentheses representmean over indicated set of correlations.



on identifying an aggregate shock. However, re-search conducted since Lucas published his articlesuggests that identifying the cause of business cyclesmay not be so simple.

First, even if we do manage to identify a shockthat clearly affects the whole economy, it does notnecessarily follow that shock is responsible for thebusiness cycle. A shock might well be experienced byall sectors of the economy, but they need not all re-spond in the same way. The business cycle shock, ifindeed there is only one, seems to lead to a synchro-nized response across sectors. Second, we now knowthat the search for a single aggregate shock may itselfbe off base. Following the work of Long and Plosser(1983), we know that, at least in theory, disturbancesto individual industries, even if they are uncorrelatedacross industries, could result in comovement.

Currently, there is no consensus among econo-mists as to what causes business cycles and, in partic-ular, their key feature, comovement. At the same time,researchers are exploring a large range of possibilities.Next, we provide a selective overview of this research.

A natural starting point is what is perhaps themost thoroughly developed theory of business cycles,the real business cycle theory associated with Kydlandand Prescott (1982), Long and Plosser (1983), andPrescott (1986).11 We focus specifically on the stan-dard real business cycle model, developed in Hansen(1985) and analyzed in Prescott (1986). Although thatmodel posits an aggregate shock, it is inconsistentwith business cycle comovement. We then exploretwo sets of modifications to this model. The first canbe viewed as natural extensions of the model. Thesecond depart more significantly from the modelsassumptions.

Standard real business cycle theory12

A key component of real business cycle theoryis a production technology. This is a relationship thatspecifies how much output a firm can obtain from agiven amount of capital and labor resources. Thistechnology is subject to shocks. Sometimes a goodshock occurs and more output can be produced for agiven level of inputs. In this case, we say the technol-ogy has been shifted up. A good technology shockmight reflect the implementation of a more efficientway to organize the work force, the acquisition ofmore efficient manufacturing equipment, or perhapsthe discovery of a way to alter the firms product sothat it better meets customers needs. At other times,a bad technology shock can shift a production tech-nology down. A bad shock might reflect bad weather,a labor dispute, an accident in the workplace, a machine

breakdown, or a government policy that encouragesan inefficient way of organizing production. Accord-ing to real business cycle theory, business cycleexpansions reflect that shocks affecting firms aremostly on the positive side, while recessions reflectperiods when most firms shocks are on the negativeside. Standard formulations abstract from the differ-ences between firms and simply assume they all havethe same production technology and are affected bythe same shock. Thus, real business cycle theoryproposes that the aggregate shock to which Lucasrefers is a productivity shock.13

The standard real business cycle model not onlyassumes that all firms are affected by the same produc-tivity shock, but also that there is just one type ofgood produced (and, therefore, one industry sector)in the economy. At least at first glance, this modeldoes not seem useful for examining business cyclecomovement among many sectors. However, it hasrecently been pointed out that this impression is mis-leading.14 In fact, one can use the model to examinebusiness cycle comovement. When we do so, we findthat its implications are strongly counterfactual. Thestandard real business cycle model is at variance withthe observation of business cycle comovement, despitethe fact that it views the economy as being driven bya single aggregate shock. Understanding why it isincompatible with comovement is useful for gaininginsight into the various lines of inquiry researchershave pursued.

The standard real business cycle model imaginesthat households interact with firms in competitive mar-kets, in which they supply labor and physical capitaland demand goods for consumption and to add totheir stock of capital. Although there is only one typeof production technology in this model, we can rein-terpret the model to suggest that one type of firm pro-duces goods for consumption (the consumption goodsindustry) and another type produces new investmentgoods for maintaining or adding to the stock of capi-tal (the investment goods industry).

When a positive productivity shock hits, so thatthe real business cycle model shifts into a boom, theoutput of both consumption and investment goodsindustries increases. However, there is a relativelylarger increase in the output of investment goods.This reflects a combination of two features of the mod-el. First, a positive technology shock increases the ex-pected return to investment, raising the opportunitycost of applying resources to the consumption sector.Second, the model assumes that households prefernot to increase consumption substantially duringbooms but to smooth consumption increases over a


longer time horizon. The increase in the demand forinvestment goods relative to consumption goods thatoccurs in a boom implies, in the standard model, thatcapital and labor resources are shifted out of the pro-duction of consumption goods and into the produc-tion of investment goods. The model does predict asmall rise in consumption in a boom. However, thisrise is driven by the favorable technology shock, whichis not fully offset by the flow of productive resourcesout of that sector. Thus, the model implies that hoursworked in the consumption sector are countercyclical,in contrast with our empirical findings in the previoussection. This is a feature of the model, despite its im-plication that total hours worked rise in a boom. Thatis, the additional hours of work all flow into the invest-ment good sector. The standard real business cyclemodel also implies that investment in capital for usein the consumption sector is countercyclical. This too,is counterfactual, according to the results reported inHuffman and Wynne (1998).

So, this model is strongly at variance with comove-ment. Why is this so? The result may seem especiallysurprising to those who expect an aggregate shock toall sectors of the economy to produce comovement.

Intuitively, there are two related ways to under-stand the models implication that inputs are allocatedaway from the sector that produces consumptiongoods during a boom. One is that the model overstatesthe value of leisure at that time. This inflates the costof allocating labor resources to the consumption sec-tor then. The other is that the model understates thevalue of the output of the sector producing consump-tion goods in a boom. This undercuts the incentiveto allocate resources to that sector then.15

Natural extensions of the standard theory

Among the various extensions to the model thateconomists have pursued,16 we focus on approachesthat stress 1) factors that prevent the rise in the mar-ginal utility of leisure in a boom and 2) factors thatprevent the decline in the value of the output of theconsumption sector in a boom. As in the discussionabove, the work we survey here assumes two marketsectors.17

Value of leisureOne factor that can slow the decline of the mar-

ginal utility of leisure when the economy moves intoa boom was explored in Benhabib, Rogerson, andWright (1991) and Einarsson and Marquis (1997). Eachof these papers points out that if there is a third useof time, in addition to leisure and time spent workingin the market, and if that use of time declines during a

boom, the marginal utility of leisure need not increaseas market effort increases.18

Benhabib, Rogerson, and Wright (1991) suggestthat the third use of time may be working in the home.For example, the amount of leisure time enjoyed by ahomemaker may not change significantly if the home-making job is exchanged for a market job. Consider-ations like this lead Benhabib, Rogerson, and Wrightto argue that work time can be reallocated from thehome to the market during a boom without substan-tially raising the marginal utility of leisure.19

Einarsson and Marquis (1997) suggest that thethird use of time may be time spent accumulatinghuman capital, such as going to school. In principle,this is an appealing idea, since it is known that timespent in educational pursuits goes down in businesscycle expansions. Their work is primarily theoretical,however. A crucial issue one would have to addressin pursuing this explanation at the empirical level iswhether the time spent on education is sufficientlycountercyclical, in a quantitative sense, to have a sub-stantial effect in a suitably modified real businesscycle model. In assessing this, one would have toconfront a substantial measurement problem. In par-ticular, time spent in educational institutions is onlypart of the time spent in education. Some of that timeis applied in the workplace, by diverting workers fromdirect production. Our understanding is that there donot exist reliable measures of this use of time.

Value of the output of the consumption sectorSeveral papers attempt to get at comovement by

reducing the decline in the value of output in the con-sumption sector during booms. For example, Baxter(1996) adapts the standard real business cycle modelby assuming that the consumption of market goodsand the services of home durables are good substi-tutes. An example of two goods that substitute isa movie watched in a theater (a market good) and amovie watched on a home television set (a homedurable good).20

Under Baxters substitutability assumption, theappropriate measure of household consumption is notjust market consumption, but consumption of marketgoods plus the service flow on the stock of homedurables. If home durables consumption is sufficientlylarge, then a given jump in the consumption of marketgoods leads to a smaller percent drop in the marginalutility of consumption. In the extreme case where thestock of home durables is extremely large and accountsfor essentially all of consumption, then a rise in mar-ket consumption would produce essentially no dropin the marginal utility of consumption.21 Although


Baxter shows that this mechanism does indeed pro-duce comovement in employment across consump-tion and investment sectors in her model, there is asense in which the comovement is not strongenough. That is because investment in the capitalused in the two sectors is essentially uncorrelated.As noted above, the data suggest that investmentacross sectors comoves as well, in addition to outputand employment.

One can also view the home production ap-proach of Benhabib, Rogerson, and Wright (1991) asa strategy to generate comovement by reducing thedecline in the value of output in the consumptionsector during booms. In a boom, as labor is allocatedaway from home-produced goods toward the produc-tion of market goods, the marginal utility of the mar-ket good does not fall much because the market andhome goods are assumed to be highly substitutable.22

This allows the value of output in the consumptionsector to rise sufficiently so that employment in thatsector is procyclical. A shortcoming of the analysis,emphasized by Benhabib, Rogerson, and Wright(1991), is that the high substitutability between homeand market goods needed for comovement of laborinputs hurts on another dimension. It has the effectthat purchases of durables are countercyclical overthe cycle.23

Christiano and Fisher (1998) take another ap-proach. Following Boldrin, Christiano, and Fisher(1995), they modify a standard real business cyclemodel in two ways. First, they specify that it takestime before labor can shift between economic sectorsin response to a shock. The reasons for this are notmodeled explicitly, but the assumption is motivatedwith an informal reference to such factors as thesearch and training costs which inhibit real world la-bor mobility between industry sectors. This assump-tion alone is not sufficient to guarantee comovement,however. Without further changes, their model pre-dicts that resources would be reallocated out of theconsumption sector and into the investment sectoras soon as labor becomes fully mobile, which theyspecify to occur in three months time. As a result,this version of the model is still inconsistent with theevidence on business cycle comovement. Christianoand Fisher therefore introduce a second modification,by changing the specification of household prefer-ences over consumption. They assume that house-holds have a tendency to become accustomed to thelevel of consumption they have enjoyed in the recentpast. This property of preferences is known as habitpersistence. A household with habit persistence pref-erences whose consumption has recently increased

is particularly unhappy if later it must return to itsprevious level of consumption.24 Habit persistencepreferences have the implication that when consump-tion rises in a boom, the marginal value of continuingto maintain consumption at a high level is increased.Christiano and Fisher show that habit persistenceand limitations on the intersectoral mobility of laborare sufficient to produce comovement in hoursworked and investment. To our knowledge, this is theonly quantitative model in the comovement literaturewith this property.

The credibility of this result depends on thecredibility of the underlying assumptions. The as-sumption that there are limitations on the speed withwhich productive resources can be transferred acrosssectors seems uncontroversial, though the model cer-tainly takes an extreme stance. What does call for adefense is the assumption of habit persistence prefer-ences. One defense is that these preferences help toaccount for observations that otherwise seem puz-zling. For example, Boldrin, Christiano, and Fisher(1995) show that, consistent with results in Constan-tinides (1990), habit persistence and limited intersec-toral mobility can account for the magnitude of theobserved average premium in the return on equityover risk-free securities. The solution to this premiumhas eluded many researchers.25 In addition, Christianoand Fisher (1998) show that habit persistence canhelp account for the so-called inverted leading indi-cator property of interest rates, that high interestrates tend to forecast bad economic times. King andWatson (1996) document that standard models havedifficulty accounting for this observation.26

A third approach toward understanding comove-ment was recently pursued by Hornstein and Praschnik(1997). They observe that some of the output of thesector that produces consumption goods (the non-durable goods sector) is also used as intermediategoods in the production of investment goods. Forexample, both households and investment-good pro-ducing firms make use of the services of the transpor-tation sector. Hornstein and Praschnik (1997) modifya real business cycle model to accommodate this fea-ture of the economy. The modification has the effectof increasing the value of output in the consumptionsector in a boom. This increased value reflects the in-creased need for the output of the consumption goodsector during a boom for use in the investment goodsector.27 We refer to this demand channel going frominvestment sector to the nondurable goods sector asthe intermediate goods channel.

There are two shortcomings of the Hornsteinand Praschnik (1997) analysis. First, the model is not


consistent with the observed comovement in invest-ment across sectors. That is, the intersectoral linkag-es in the model are not strong enough to produce fullcomovement. Second, data on subsectors of the non-durable goods sector cast doubt on the notion thatthe intermediate good channel is the only reasonthere is comovement. We studied the subsectors ofthe nondurable goods sector and found that there isconsiderable variation in the fraction of total outputsent to the investment goods industry. But, as docu-mented in the previous section, most of these sectorsnevertheless display strong business cycle comove-ment. Figure 6 is a scatter plot of the subsectors de-gree of comovement, drawn (with one exception) fromthe fifth column of numbers in table 1, against thestrength of each sectors intersectoral linkage withthe investment sector, Ic. The variable Ic is the frac-tion of the gross output of a sector which is allocatedto intermediate goods destined directly or indirectlyfor the production of final investment goods.28 (Tech-nical appendix 3 has details of how we computedthis.) The numbers in parentheses in figure 6 indicatethe relative magnitude of the gross output of the sec-tor (gross output of the sector in 1987, divided by thesum of the gross outputs across all 17 sectors). Horn-stein and Praschniks concept of the nondurablegoods sector is broader than the one in table 1. Theyalso include agriculture; retail trade; wholesale trade;transportation, communication, and utilities; servic-es; FIRE; and mining.29

Figure 6 shows that employment in most (13 of

17) nondurable good sectors is substantially procy-clical (that is, the comovement statistic is 0.45 orhigher), even though the strength of the intermediategood channel (the magnitude of Ic) varies from almostzero in the case of food to nearly 0.25 in the case ofwholesale trade. Interestingly, although the comove-ment in mining is moderately weak in our data set, itis one of the sectors in which the intermediate goodschannel is the strongest. Conversely, the comove-ment in apparel is strong, although this sectors inter-mediate goods channel is almost nonexistent. Basedon these results, we suspect that the intermediategoods channel to the investment sector plays at bestonly a small role in accounting for comovement ofemployment in nondurable goods.30 To further ex-plore the Hornstein and Praschnik hypothesis, onewould have to construct a version of their model witha disaggregated nondurable goods sector and see ifit is consistent with comovement, in the sense of be-ing able to reproduce patterns like those in figure 6.31

Alternative approaches

Here, we summarize three other approaches thatmay ultimately lead to a satisfactory explanation ofbusiness cycle comovementstrategic complemen-tarities, information externalities, and efficiencywage theory. The first two approaches emphasize theimportance in business decisions of expectationsabout the future. They draw attention to the possibil-ity that general shifts in expectations may triggerbusiness cycle fluctuations. If so, these shifts in ex-

pectations may well constitute the ag-gregate shock to which Lucas (1981)refers. The third approach looks at effi-ciency wage theory. Although promis-ing, the ability of these three theories toquantitatively account for the comove-ment aspect of business cycles is yet tobe fully explored.

Strategic complementaritiesSuppose there are two people, A

and B. Each has to decide on a level ofwork effort: high or low. Suppose thatthe net gain to A of exerting a high levelof effort is greater if B exerts a high levelof effort and that B is in a similar posi-tion. The situation is depicted in table 2.

Table 2 has four entries, one foreach possible combination of work ef-fort. In each entry, the first number indi-cates the net gain to A, and the secondnumber indicates the gain to B. SupposeA exerts high effort. Then, if B is putting

Business cycle comovement in nondurable goods sectors

comovement

FIGURE 6

importance of intermediate goods channel to investment sector (lc )

Printing (0.02)Transportation, communication,and public utilities (0.13)

Apparel (0.02) Retailtrade (0.07)

Paper (0.02)Chemicals (0.03)

Rubber (0.02)

Textiles(0.01)Leather products (0.00)

Services (0.28)

Finance, insurance,and real estate (0.21)

Food (0.06)

Tobacco (0.01)

Agriculture (0.04)

Mining (0.02)

Petroleum (0.02)

Wholesaletrade (0.07)

Note: Plot of sector is calculated using the comovement statistic reported intable 1 and as explained in note 6. The number in parentheses following the sectorname indicates the relative size of that sector.



TABLE 2

Example of strategic complementarity

Person B

High effort Low effort

Person A High effort (5, 5) (–5, 0)Low effort (0, –5) (2, 2)

out high effort too, A receives 5. But, if B exerts loweffort, then A receives 5. The situation is the samefor B. Table 2 captures the idea that the gain to eitherperson from exerting high effort is high only whenthe other person exerts high effort. A situation likethis is said to be characterized by strategic comple-mentarity. What do we expect to happen? If the twopeople could sit down and reach an agreement, theywould clearly both choose to exert high effort. Butwhat if they have difficulty coordinating in this way?There are now two possibilities. One is that each ex-pects the other to exert low effort, in which case eachfinds it optimal to exert low effort. This would put thetwo people in the bottom right box, with a low payoffgoing to each. They would stay there until theyfound a way to communicate and reach an agreementor until something happened to alter their expectationabout the others plans. Another possibility is thateach expects the other to exert high effort, in whichcase it is in the private interest of each person toexert high effort. This situation could persist for a while,again, unless something happens to shift one personsexpectations about what the other one will do.

What does this have to do with business cyclesand comovement? Possibly a lot. There are aspectsof business decisions that exhibit strategic comple-mentarity. For example, suppose a firm is consideringreopening a plant or starting a large capital invest-ment project. Suppose the project involves a sub-stantial outlay of funds, not just to hire more workersbut also to purchase materials and supplies from oth-er firms. The higher the sales the firm expects in thefuture, the more inclined it will be to shift to a highlevel of activity in this way. However, much of afirms sales come from other firms. And those salesare greater if other firms are themselves operating at ahigh level of activity, for example, reopening plants orundertaking new capital investment projects. So, firmA has a greater incentive to shift to a high level of ac-tivity if it believes firm B plans to operate at a highlevel of activity.

What do we expect in this situation? Coordina-tion in this setup is much more difficult than in the

two person example. There are millions of firms in theeconomy and, even if it were technically feasible forsome firms to coordinate, the antitrust laws representanother barrier. In light of these considerations, wemight well expect to find results similar to those inthe two person example. Thus, if firms were pessimis-tic about prospects for future sales, they would chooseto be inactive and their pessimistic expectations wouldbe fulfilled.32 Optimistic expectations would be self-fulfilling in the same way. It is clear that in this setting,expectations have the potential to act as an aggre-gate shock driving the business cycle. Of course,that does not guarantee that they can necessarilyaccount for comovement.33 This is an important topicof research.34

Information externalitiesAnother potential source of comovement is the

way information about the state of the economy istransmitted to individual firms. Forecasts of the futurestrength of the economy are a factor in individualfirms current investment decisions. If a firm observesa series of construction projects being initiated byother firms, it may infer that those other firms haveinformation that bodes well for the general economicoutlook. When the firm combines this inference withits own information about the economic outlook, itmay decide to invest too. Other firms may follow forsimilar reasons.

These considerations are logically distinct fromthe strategic complementarities discussed above. There,a firm is interested in the actions of other firms becausethese actions have a direct impact on the firms prof-itability. Here, a firm is interested in the actions of otherfirms because of the associated information externality.The externality refers to the fact that a firms actionmay reveal information it has on something of inter-est to other firms, such as the state of the economy.It is a positive externality, unlike the more familiarexamples of externalities which tend to be negative.35

We present an example, taken from Banerjee(1992), to illustrate the sort of things that can happenwhen there are information externalities. When firmslook to what other firms are doing for guidance in de-ciding what they should do, this can lead to whatBanerjee (1992, p. 798) calls herd behavior, a situationwith everyone doing what everyone else is doing,even when their private information suggests doingsomething quite different. It hardly needs to be stat-ed that herd behavior sounds like comovement.

Here is the example. Suppose there are 100 peo-ple trying to decide between two restaurants, A andB. Each person knows very little about the two


restaurants, but thinks the odds favor A slightly. In ad-dition, each person receives a signal about the relativequality of the two restaurants. For example, one per-son may read a review of the two restaurants in atravel guide. The review is several years old, howev-er, and the signal may not be accurate. The signals re-ceived by each of the 100 persons are equally reliable.Everyone knows this, but they do not know what sig-nal the others received. Now, suppose that 99 peopleget a signal that suggests B is better than A, whileone person gets a signal that A is better than B. If allinformation were known to everyone, they would rec-ognize that the preponderance of the evidence favorsrestaurant B and all 100 people would go to B. How-ever, what actually happens is that the 99 peoplewhose signal indicates B is better ignore their signaland flock to restaurant A, following the one personwho received the signal that A is better.

This result is not due, as one might suppose, toan assumption that agents are irrational. On the con-trary, the example assumes agents are completelyrational. The result reflects that not all people maketheir decisions at the same time. Some have to befirst, and as a result, the information they have hasdisproportionate impact, since almost everyone elseis watching them. This timing assumption does notseem unrealistic. In practice, the exact timing of firmsdecisions is not completely under their control.36

The example adopts an extreme version of theassumption that the timing of a decision is out of theagents control, specifying that someone must choosefirst, then someone else must choose second afterobserving the choice of the first, and so on. The per-son choosing first happens to be the one who receivesthe signal that A is better than B. Since person 1ssuspicion that A is better is apparently confirmed bythe signal, this person rationally chooses A. The sec-ond persons signal suggests that B is better. However,person 2 knows that person 1s signal must havefavored A. Since the two signals are equally reliable,they cancel in the mind of person 2. Since person 2originally thought restaurant A was better, the ratio-nal thing for person 2 to do is to go to restaurant A.Person 3 is in precisely the same position as 2, becauseperson 3 knows that, given person 1 went to A, per-son 2 would have gone to A no matter what signalshe received. That is, person 3s observation thatperson 2 went to A provides no information at allabout the relative quality of the two restaurants. Beingin the same position as 2, person 3 also chooses Aregardless of the signal received. In this way, all 99people after the first ignore their own signal and go

to restaurant A. Although there is a lot of informationin the economy about the relative quality of the res-taurants, one person acts on a small piece of it, andeveryone else follows.

This example and others like it hold out somehope that a fully developed business cycle modelincorporating information externalities might exhibitthe synchronization of behavior across economicsectors that we observe over the business cycle. How-ever, the above example only illustrates how informa-tion externalities can lead rational people to ignoreinformation and synchronize on bad decisions. Syn-chronization of actions would have occurred anyway,even if there had been no information problem and allsignals had been known to everyone. Another con-cern with this example is how heavily dependent it isupon details of the environment. For example, theoutcome is very different if two people are required tochoose a restaurant first. In this case, the 99 peoplewho received the signal that B is better than A go toB.37,38 Despite these considerations, we believe thegrowing literature on information externalities mayeventually provide at least part of the explanation forbusiness cycle comovement.39

Efficiency wage theoryA third strategy for understanding comovement

is to make use of efficiency wage theory.

Efficiency wage theory: A sketchUnder this view of labor markets, the amount of

effort a worker makes (the workers efficiency) dependson the wage that the worker is paid. Developmenteconomists hypothesized that in economies at a veryearly stage of development, a higher wage leads togreater worker efficiency because it facilitates improve-ments in diet and health. Efficiency wage theory holdsthat a higher wage also results in greater worker effi-ciency in a modern, developed economy, but for differ-ent reasons. Because employers cannot perfectlymonitor the amount and quality of work effort expendedby their employees, there is a temptation for workersto shirk. Efficiency wage theory says that a high wagerate is an effective way to combat this temptation.The higher the wage, the more a worker has to loseif caught and fired for poor job performance.

The simplest version of this idea was articulatedby Robert Solow,40 who theorized that the firm selectsa wage rate, the efficiency wage, which maximizesworker effort per dollar paid. The firm is not willingto pay more because the resulting increase in workereffort would not be enough to warrant the extra cost.The firm is also not willing to pay less, because the


resulting fall in effort would exceed the fall in cost.41

In the Solow model, the amount of effort expendedper hour by a worker is a function only of the currentwage and, for example, does not depend on the generalstate of business conditions. As a result, the efficiencywage rate does not vary over the business cycle underSolows efficiency wage theory.

The firm also has to decide how many workers tohire. It hires workers up to the point at which the mar-ginal productivity of the last worker is just equal to theefficiency wage.42 The downward sloping marginalproductivity of labor curve in figure 7 shows how themarginal productivity of labor is lower at higher lev-els of employment. At the level of employment, L, themarginal productivity of the last worker hired is equalto the efficiency wage. Since the efficiency wage inthe Solow model is a constant, it follows that employ-ment over the business cycle is determined by the re-quirement that the marginal product of labor does notchange. The downward sloping curve in figure 7,marginal productivity of labor′, shows the marginalproductivity curve after it has been shifted up by apositive technology shock.43 If the firm kept employ-ment fixed at L when technology shifted up, marginalproductivity would rise to W′, a point far above theefficiency wage. By increasing L to L′, the firm keepsmarginal productivity unchanged despite the shift upin technology.44

A notable feature of efficiency wage theory isthat labor supply plays no role in the determinationof the wage rate. The theory assumes that there aremore workers willing to work than the firm is willing

to hire at the efficiency wage. Still, un-employed workers cannot bid the wagedown below the level of the efficiencywage. Firms are not interested in hiringworkers at such a low wage becausethey fear it would not provide workerswith enough incentive to work hard. Thequantity of unemployed people is thenumber who are willing to work at the ef-ficiency wage, minus the number thatfirms want to hire. Note how the upwardsloping labor supply curve in figure 7 isshifted to the right. At the efficiencywage, Ls, workers would like to work, butonly L are hired, so unemployment is Ls

L. At the higher level of technology,unemployment falls to Ls L′.

Efficiency wage theory and businesscycle comovement

How might efficiency wage theory help accountfor business cycle comovement? Suppose the busi-ness cycle is driven by an aggregate, real-business-cycle-type technology shock. As we explained earlier,in the standard real business cycle model such ashock does not lead to comovement in employment.In that model, a positive shock leads to a transfer ofresourceslabor and capitalaway from the firmsproducing consumption goods and toward the firmsproducing investment goods. Now suppose the labormarket part of the real business cycle model is replacedby efficiency wage theory, which implies that firmsvary the number of workers they employ to ensurethat the marginal product of labor remains constantand equal to the efficiency wage rate. So, when a posi-tive technology shock shifts up the marginal produc-tivity of labor, employment must increase to maintainequality between the marginal product of labor andthe efficiency wage.45

We indicated earlier that a positive real-business-cycle-type shock pushes up the production functionsand the marginal labor productivity curve of all firms.According to efficiency wage theory, this results inan increase in employment by all firms, as they seekto maintain equality between marginal labor produc-tivity and the unchanging efficiency wage rate. Thisis the intuition underlying the idea that efficiencywage theory may help explain business cycle co-movement.46, 47

Have we now established that efficiency wagesare sufficient to account for comovement? Absolute-ly not. When we examine efficiency wage theory moreclosely, we discover that it need not necessarily work

Efficiency wage model

FIGURE 7

Solowefficiencywage rate

W′

L′ Ls

Marginalproductivityof labor

Marginalproductivityof labor ′

← →

Marginalproductivity, wage

Number ofworkers

Laborsupply

→

L


as just outlined. The relationship between how hard aworker works and the wage rate (the workers effortfunction) is a function of the households attitude to-ward risk, the resources it has available if the workeris caught shirking and fired, the probability of beingcaught conditional on shirking, and the precise con-sequences of being fired for shirking. The householdeffort function used in the analysis must integrate allthese factors in a logically coherent way. In addition,it must be consistent with other household decisions,such as how to split income between consumptionand saving. To build confidence in the idea that effi-ciency wage theory helps account for comovement,we must integrate all these aspects of the householdinto a coherent framework which also includes firmsand their decisions to see if it works.

To understand why it might not work, recall theSolow models assumption that worker effort is a func-tion only of the wage rate. That is what led to our con-clusion that the efficiency wage is a fixed number,unrelated to the state of the business cycle. But thelogic of the efficiency wage argument suggests thatthe Solow assumption may not be consistent with ra-tional behavior by households. According to efficiencywage theory, what motivates hard work is the fear oflosing a high-wage job. Of course, the cost of thatloss is not a function of the wage rate alone. It is alsoa function of the amount of time the worker can expectto be out of a job after being fired. This suggests thatthe horizontal line in figure 7 shifts up in a boom, whenthe duration of unemployment is low.48 However, if itshifts up high enough, the comovement result coulddisappear. This highlights the importance of integratingefficiency wage theory into a logically coherent model,before we conclude that it provides a solid foundationfor understanding business cycle comovement.

Important steps have been taken in this direc-tion, for example, Shapiro and Stiglitz (1984) and Dan-thine and Donaldson (1995).49 Recent work by Gomme(1998) and Alexopoulos (1998) makes a significant

further contribution toward understanding the impli-cations of efficiency wage theory for business cy-cles. However, this work does not focus on theimplications for business cycle comovement. We ar-gue that doing so is a good idea.50

ConclusionConclusionConclusionConclusionConclusion

A key feature of the data is that, in a frequencyrange of two to eight years, output, employment andinvestment across a broad range of sectors move upand down together. We have documented this phenom-enonbusiness cycle comovementas it pertains toemployment. Our survey of possible explanations forit is by no means exhaustive. Many other approach-esthose based on sticky prices and wages, coun-tercylical markups, and credit market frictionsalsodeserve consideration.51 Still, we have covered a widerange of models, from straightforward modificationsto standard business cycle theory to theories thatsuggest analogies between businesspeople and herdsof animals.

Many of the approaches we have surveyed arein early stages of development, while some have beendeveloped to the point where their implications havebeen quantified and compared with the data. Amongthese, only one has been shown to be consistent withthe observed strong comovement in output, employ-ment, and investment across sectors of the economythe model presented in Christiano and Fisher (1998).This model incorporates a specification of householdpreferences, habit persistence, that is not currentlystandard in the macroeconomics literature. We believethat the success this model has in generating comove-ment warrants giving this specification of preferencesfurther consideration.

Because comovement is such a central featureof business cycles and because we do not have agenerally agreed upon theory of comovement, weconclude that the business cycle is still a puzzle.

Extracting the business cycle componentof a time series

In casual discussions of economic time series, weoften think of the data as being the sum of compo-nents that have different frequencies of oscillation:the business cycle component lasting two to eightyears, components lasting shorter periods, etc. Thetheory of the spectral analysis of time series makesthis intuition rigorous. It clarifies how one can think

TECHNICAL APPENDIX 1

of data as being composed of components that fluc-tuate at different frequencies. The method we use toextract the business cycle component of economictime series builds on this theory. For this reason, webegin with a brief section which attempts to conveythe basic intuition of spectral analysis. The secondsection uses this intuition to describe and motivateour method for extracting the business cycle compo-nent of a time series.


Decomposing a time series intofrequency components

At the core of the spectral analysis of time seriesis the view that the data can be thought of as the sumof periodic functions. The purpose of this section isto explain this. We begin by reviewing the basic peri-odic function used in spectral analysis, which is com-posed of a sine and a cosine function.

Consider the following cosine function of time, t:

cos (tω), t = 0, 1, 2, ... .

A graph of this, with cos (tω) on the vertical axisand t on the horizontal axis, exhibits the oscillationsbetween 1 and 1 familiar from high school trigonom-etry. Recall too, that the period of the cosine functionis 2π. That is, cos(y) = cos(y + 2πh), for h = 1, 2, ... .Thus, after the argument of the cosine function in-creases by 2π, the function repeats itself in a periodicfashion.

What is of interest here is the period of oscilla-tion of cos(tω), expressed in units of time. This is theamount by which t must increase so that tω increasesby 2π. Thus, suppose t

1 is a given point in time. We

want to know what is the later point in time, t2 > t

1,

when the cosine function begins to repeat itself. Thisis just t

2 such that t

2ω t

1ω = 2π, or, t

2 t

1 = 2π/ω.

Thus, the period of oscillation of cos(tω), in units oftime, is 2π/ω. The parameter ω is referred to as the fre-quency of oscillation.

The function, sin(tω), behaves similarly. It fluctu-ates between 1 and 1, and has a period of oscillationof 2π/ω. Thus, the two functions have the same ampli-tude (magnitude of vertical variation) and period.However, the sine function has a different phase thancos(tω). For example, a graph of the two functions to-gether shows that one looks like the other, apart froma horizontal shift. The phase difference between thetwo functions is a measure of the magnitude of thishorizontal shift. Figure A1 displays sine and cosinefunctions for t = 0, 1, ... , 200. The period of oscilla-tion is 100, so that the frequency is ω = 2π/100. Thus,the figure displays the graphs of cos(t2π/100) andsin(t2π/100).

We can now describe the central periodic functionin spectral analysis, namely the linear combination of asine and a cosine function

1) cos( ) sin( ),a t b tω ω+

where a and b are parameters. This function obviouslyhas a period, in units of time, equal to 2π/ω. But, its

amplitude and phase depend on the values of a andb. If a and b are both very small, the resulting functionhas very small amplitude and if a and b are both verylarge it has a large amplitude. Also, as the size of a isincreased and the size of b is decreased, the phase ofthe function shifts, as more weight is allocated to thecosine and less to the sine.

It turns out that sums of periodic functions likeequation 1 look very much like actual data. Thus,suppose we have a time series of data, xt, t = 1, ..., T.To see that xt can be expressed as a sum of periodicfunctions, suppose we specify T/2 (suppose T is even)such functions, each with a different frequency ofoscillation ω. For concreteness, let ωj = 2πj/T, for j = 1,..., T/2. To distinguish the parameters associated witheach of these functions, we denote them by aj and bjfor j = 1, ..., T/2. It should not be surprising that, ingeneral, a time series, x

1, ..., xT can be written as the

sum of these T/2 functions

2 1 1 1 1

2 2 2 2

) cos( ) sin( ) ...

cos( ) sin( ),/ / / /

x a t b t

a t b tt

T T T T

= + ++ +

ω ωω ω

for t = 1, ..., T. That is, we can always find values forthe T parameters, (aj, bj; j = 1, ..., T/2), so that the Tequations, equation 2 for t = 1, ..., T, are satisfied. Tosee this, consider the following regression. Let theexplanatory variables be:

X

T T T T

T T

T T

T T

=

�

!

"

$

####

cos( ) sin( ) cos( ) sin( )



.

/ /

/ /

/ /

ω ω ω ωω ω ω ω

ω ω ω ω

1 1 2 2

1 1 2 2

1 1 2 2

2 2 2 2

L

L

M M M M

L

FIGURE A1

Trigonometric functions

t

←

→sin(.06t)

cos(.06t)

function

0 20 40 60 80 100 120 140 160 180 200

r


Let the T × 1 vector of independent variables, Y,and the T × 1 vector of regression coefficients, β, be

Y

x

x

x

a

b

a

bTT

T

=

�

!

"

$

####=

�

!

"

$

######

1

2

1

1

2

2

MM, .

/

/

β

Then, the regression is

Y = Xβ + u.

Note, however, that since the number of explanatoryvariables is T, the error term is exactly zero, and β iscomputed from (X′ X)1 X′ Y = X1Y.1 Thus a time se-ries of length T can be expressed exactly as the sumof T/2 simple processes like equation 1, each having adifferent frequency of oscillation.

Unfortunately, taken literally, equation 2 is not avery sensible way to think of the data. With T obser-vations, one has only to compute the ais and the bisand then the T + 1st observation can be predicted ex-actly. No one believes that there is any way to use Tobservations on any economic data series and predictthe next observation exactly. Imagine, for example,that you could do this with the Dow Jones IndustrialAverage. If you could, then after one minute of read-ing this appendix, you would have the informationneeded to go out and become fabulously wealthy.

Of course, one could instead suppose that thedata are a realization from an expression like equation2, in which the number of periodic functions exceedsthe number of observations by, say, 10. In this case,there is no longer the implication that one can per-fectly predict next periods value of xt. However, thereis the implication that after 20 more periods, the dataseries will then become perfectly predictable. No onewould think this is a sensible way to view economicdata either. The theory of spectral analysis assumes,sensibly, that no matter how many observations on xtone accumulates, the data never become perfectlypredictable. That is, it in effect assumes that the num-ber of periodic functions in equation 2 is infinitelylarge by comparison with the size of the available dataset. When this is so, equation 2 is written in the formof an integral, as follows:2

3 0) [ ( )cos( ) ( )sin( )] ,x a t b t dt = I +ω ω ω ω ωπ

where a(ω) and b(ω) are functions of ω. In view ofthese observations, it is perhaps not surprising that

any covariance stationary time series process, xt, canbe expressed in the form of equation 3 (Koopmans,1974).

Extracting the business cycle componentEquation 3 allows us to make precise the notion

of extracting the business cycle component of xt. Thatrepresentation views the time series process, xt, asthe sum of components with periods of oscillation2π/ω for ω lying in the interval 0 to π. In monthlydata, the business cycle corresponds to componentswith period greater than 24 months and less than96 months. In terms of frequencies of oscillation,this corresponds to ω belonging in the intervalω π ω π= =2 96 2 24/ / .to Thus, we seek thebusiness cycle component of xt, yt, such that

4) [ ( )cos( ) ( ) sin( )] .y a t b t dt = I +ω ω ω ω ωωω

It is well known that yt can be computed as a par-ticular centered moving average of observations onthe observed data, xt

5) yt = B0xt + B1(xt1 + xt+1 ) + B2(xt2 + xt+2 ) + ... ,

where

61 2

0

)

sin( ) sin( ), , , ...

.

B Bj j

jj

B

j j= =− =

=−

−ω ω

πω ω

π

There is a major practical stumbling block to usingequation 5 for extracting the business cycle compo-nent of xt. It requires an infinite amount of data! Somesort of approximation is needed, if one is to estimateyt given only the available data, x1, ..., xT.

An extensive analysis of this problem appears inChristiano and Fitzgerald (1998), which also providesa review of the related literature. We provide only thebriefest review of that discussion here, just enoughto enable us to describe exactly how we isolated thebusiness cycle component of the data.

We denote our approximation of yt by ŷt. Here, wefocus on the approximations of the following form:

7 0 1 1 1) $$ $ ( )

$ ( ).

y B x B x x

B x x

t t t t

K t K t K

= + + +

+ +− +

− +

K

That is, we approximate yt by a finite ordered, cen-tered, symmetric moving average. But, how should wechoose the weights? The natural way is to choose


them so that ŷt is as close to yt as possible, that is, sothat they solve

80 1

2) min ( $ ) .$ , , ,...,Bi i K

t tE y y=

−

The solution to this problem is a function of thedetails of the time series representation of xt. For ex-ample, if we suppose that xt is a random walk, that is,xt xt1 is a process that is uncorrelated over time,then the solution is:3

9 0 1

12 0 1 2 1

) $ , , ...,

$ ... .

B B j K

B B B B B

j j

K K

= = −

= − + + + + −

Suppose the data at hand are x1, ..., xT, so that

the objects of interest are, y1, ..., yT. We computed ŷ36,

..., ŷT36 as follows. For ŷ36 we applied equation 7 withK = 35, for ŷ

37 we applied equation 7 with K = 36, and

so on. For each ŷt that we computed, we used thelargest value of K possible. Christiano and Fitzgerald(1998) argue that this procedure for estimating yt workswell in terms of optimizing equation 8, even if the true-time series representation of xt is not a random walk.They show that an even better approach uses anasymmetric set of weights, so that the estimate of ŷtfor each t uses all the available data on xt.

1Note that sin(tωT/2

) = 0 for all integers t. Since the right col-umn of X is zero in this case, X is singular and so cannot be in-verted. In practice, the last column of X is replaced by acolumn of ones, to accommodate a non-zero sample mean inx

t. Under these conditions, the columns of X are orthogonal, so

that X1Y is trivial to compute. In particular, for j = 1, ..., T/2 1:

a t x

b t x

j T j tt

T

j T j tt

T

= ∑

= ∑

=

=

2

1

2

1

cos( ) ,

sin( ) .

ω

ω

Also,

a t x T

b x T

T T tt

T

T tt

T

/ /

/

cos( ) /

/ .

2 21

21

= ∑�

!

"

$#

= ∑�

!

"

$#

=

=

ω

2To gain further intuition into the relationship between equa-tions 2 and 3, it is useful to recall the simplest definition ofan integral, the Riemann integral. Thus, let f(y) be a function,with domain y y y≤ ≤ . Let y

i, j = 1, ..., M be a set of numbers

that divide the domain into M equally spaced parts. That is,y y y y y yM M M M M1 2 1 1= + = + = +−∆ ∆ ∆, , , ,K

where ∆ M y y M= −( ) / . Note that y yM = . The integral of fover its domain is written,

f y dyyy ( ) .I

This is approximated by the sum of the areas of theM f(y

j) by ∆

M rectangles:

f y j Mj

M( ) .∆

=∑

1

The Riemann interpretation of the integral is that it isthe limit of the above sums, as M → ∞. The relationship be-tween the above finite sum and the integral resembles thatbetween equations 2 and 3 if we adopt y

j = ω

j = 2πj/T, ∆

M = 2π/T,

M = T/2, f(yj) = a(ω

j) cos(ω

jt) + b(ω

j) sin(ω

jt), a(ω

j) = a

jT/ 2π,

b(ωj) = b

jT/2π.

3Actually, the theory as we summarized it here technically doesnot accommodate nonstationary processes like random walks.Christiano and Fitzgerald (1998) discuss standard ways of ex-tending the theory to this case. Also, optimizing the meansquare error criterion, equation 7, requires a constant term inequation 8. See Christiano and Fitzgerald (1998) for more de-tails.


Comovement and the elasticityof substitution

The standard real business cycle model assumes thatthe elasticity of substitution between capital and laborin production is unity. In the text, we discussed aresult due to Benhabib, Rogerson, and Wright (1991):With this kind of production function and with utilityfunctions consistent with balanced growth, comove-ment in employment is impossible (see note 16). Here,we show that comovement is a technical possibilitywhen the elasticity of substitution is different fromunity. However, we find that comovement does not

occur for plausible parameter values. These resultssuggest that attempts to account for comovement byadjusting the elasticity of substitution in productionin a standard real business cycle model are unlikelyto be successful.

We begin by describing a version of the standardreal business cycle model. We assume that house-holds have identical preferences of the following form:

EC L

t

t t t

00

11

1β

σ

ψ σ

=

∞−

∑−

−

( ),


where σ, ψ > 0 satisfy the various conditions requiredfor utility to be strictly concave. Also, Ct > 0 denotesper capita consumption, and Lt denotes per capitahours worked. We require 0 ≤ Lt ≤ 1. The resourceconstraint is

c k k k L zt t t t t t+ − − ≤ − +�!

"$##+

− − −

1

1 1 1

1 1( ) ( ) .δ α αν

νν

ν

νν

Here, 0 < δ < 1 is the rate of depreciation on capi-tal and 0 < α < 1 is a parameter. The elasticity of substi-tution between capital and labor is ν > 0. Also,

log( ) log( ) , ,z zt t t= + < 0 denotesthe beginning of period t stock of capital, which is agiven quantity at time t.

As noted in the body of the article, it is possibleto interpret this as a two sector model: one in whichconsumption goods, ct, and investment goods, kt+1 (1 δ) kt, are produced in different sectors by differ-ent firms. It is assumed that both sectors use the sameproduction function, the one stated above, and thatcapital and labor can move freely between the twosectors, subject only to the obvious constraint thatthe sum of capital and labor in the two sectors equalskt and Lt, respectively. Thus, letting Lct, kct and Lit,, kitdenote the amount of labor and capital, respectively,used in the consumption and investment sectors,we require

Lct + Lit = Lt, kct + kit = kt.

As mentioned in note 16, the marginal productof labor in the sector producing the consumptiongood equals households marginal rate of substitutionbetween consumption and leisure.

α ψν ν

νC

Lz

C

L LC c i

�� = − −

−1

1

1( ).

We drop the time subscripts to simplify the nota-tion. Rearranging this equation, we obtain

αψ

ν νν�

�� − − =

−C

zL L Lc i c

1

1( ) .

Note first that when ν = 1, we reproduce the resultin note 16, which indicates that Lc and Li cannot movein the same direction. When ν ≠ 1, this reasoning nolonger holds. We can see intuitively that employment

in the two sectors might move together with ν > 1. Inparticular, consider the case ν = 1. In this case, wehave found in many numerical examples that C/z fallswith a rise in z due to a positive shock in ε. Continuitysuggests that this also happens when ν is a little aboveunity. We conclude that if the resulting rise in (C/z)1ν

is sufficiently large, then it is possible for both Lc andLi to increase in response to a positive shock in ε (seenote 16 for the sort of reasoning used here).

We approximated the solution to this model us-ing the undetermined coefficient method in Christiano(1991). We assigned parameter values in the followingway. For our baseline parameterization, we set β = 0.99,δ = 0.025, ρ = 0.95, and σ = 1. We chose ψ and α toguarantee that, in the models steady state, laborsshare of income is 0.64 and steady state hoursworked is 0.30. An empirical defense for the choice ofvalues for β, δ, ρ , labors share, and steady state hoursworked may be found in the real business cycle litera-ture. For the calculations reported below, we set σ =0.01, and drew 1,000 observations on εt from a normaldistribution with mean zero and standard deviation, σε.

In our first experiment, we considered values of νon a grid between 0.7 and 20. For each value of ν,1,000 observations on Lct and Lit, and

Ic,t = kc,t+1 (1 δ)kc,t, Ii,t = ki,t+1 (1 δ) ki,t

were generated using our approximation to themodels solution. The 1,000 observations were thenused to compute the correlations, rlic between Lct andLit and the correlations, ρic

i , between Ic,t and Ii,t. Amodel exhibits comovement in employment and in-vestment if both ρic

i , ρicl > 0. We found ρic

i , ρicl < 0 for

each value of n using the benchmark parameterization.We repeated these calculations several times,

each time perturbing one, and only one, of the param-eters in the benchmark parameterization. We consid-ered the following alternatives: σ = 5; ρ = 0.99; ρ = 0.0;steady state hours equal to 0.1; steady state laborsshare equal to 0.3; δ = 0.05, δ = 0.01; and β = 0.97,β = 0.995. The perturbations in σ, ρ, β, and δ did notproduce a parameterization exhibiting comovement.The reduction in labors share resulted in comovementin employment, but not investment, for values of νbetween about 3 and 13. Lowering steady state hoursto 0.10 also resulted in comovement in employmentbut not investment. Hence, we conclude that alteringthe elasticity of factor substitution in productiondoes not improve the standard real business cyclemodels ability to reproduce full comovement for rea-sonable parameter values.



Analysis of the input–output tables

Our analysis of the inputoutput tables is based onthe 1987 benchmark, 95 variable inputoutput tablefor the U.S. economy. Our main objective here is todefine the fraction of a sectors final output which isused directly or indirectly in the production of finalinvestment goods. Let Y denote the vector of grossoutputs for the production sectors of the economy.Let A = [aij] be the matrix of inputoutput coefficients.That is, aij is the quantity of the ith industrys outputused to produce one unit of the jth industrys output.Let If, C, G, O denote the vectors of gross private fixedinvestment, personal consumption expenditures, gov-ernment (federal, state, and local) purchases, andother for each sector. Here, other is essentially ex-ports minus imports. Total output, Y, is broken downinto a part allocated to intermediate inputs, AY, and apart allocated to final output, If + C + G + O as follows:

AY + I f + C + G + O = Y.

Solving this for Y, we get

Y = YI f +YC + YG + YO,Yi = [ I A]

1 i, i = If, C, G, O.

For convenience, we report Yi , i = If, C, G, O for

the 95 sectors of the U.S. economy which are includ-ed in the inputoutput table underlying the analysisreported in figure 6. Table A1 reports results for the17 sectors of the nondurable goods industry, as de-fined in the Hornstein and Praschnik (1997) analysis.That table reports the inputoutput table industrynumbers that make up the industries whose name isin the middle column. Table A2 reports the numbersfor the other sectors. The sum of the numbers in arow must be unity.

Results for consumption

TABLE A1

I–O industry number I–O industry title YIf Y

CYG

YO

1+2+3+4 Agriculture, forestry, and fisheries 0.060 0.894 0.052 –0.0065+6+7+8+9+10 Mining 0.207 0.893 0.181 –0.28214 Food and kindred products 0.018 0.962 0.041 –0.02115 Tobacco products 0.000 0.914 0.000 0.08616+17 Textile mill products 0.185 0.995 0.072 –0.25218+19 Apparel and other textile products 0.037 1.284 0.041 –0.36224+25 Paper and allied products 0.103 0.833 0.112 –0.04726A+26B Printing and publishing 0.058 0.795 0.121 0.02627A+27B+28 Chemicals and allied products 0.180 0.698 0.138 –0.01631 Petroleum refining and related products 0.105 0.782 0.145 –0.03232 Rubber and miscellaneous plastics products 0.246 0.761 0.127 –0.13433+34 Footwear, leather, and leather products 0.031 2.154 0.037 –1.22265A+...+68C Transportation, communications, and utilities 0.107 0.740 0.123 0.02969A Wholesale trade 0.232 0.589 0.098 0.08269B Retail trade 0.066 0.919 0.015 0.00070A+70B+71A+71B Finance, insurance, and real estate 0.061 0.877 0.043 0.02072A+...+77B Services 0.076 0.879 0.044 0.002

Notes: Yi measures amount of gross output of industry in indicated row sent directly or indirectly to industry i, where i = If, C, G, O.

Row numbers are scaled so they sum to unity.

Source: Authors’ calculations based on data from U.S. Department of Commerce, Bureau of Economic Analysis, 1992,Survey of Current Business, Volume 72, Number 4, April.


Results for nonconsumption

TABLE A2

I–O industrynumber I–O industry title Y

If Y

CYG

YO

11 New construction 0.805 0.000 0.195 0.00012 Maintenance and repair construction 0.180 0.574 0.243 0.00213 Ordnance and accessories 0.011 0.051 0.838 0.10020+21 Lumber and wood products 0.542 0.340 0.169 –0.05122+23 Furniture and fixtures 0.477 0.585 0.067 –0.12829A Drugs 0.018 0.963 0.125 –0.10729B Cleaning and toilet preparations 0.018 0.949 0.033 0.00030 Paints and allied products 0.422 0.445 0.168 –0.03635 Glass and glass products 0.202 0.798 0.116 –0.11636 Stone and clay products 0.575 0.314 0.206 –0.09537 Primary iron and steel manufacturing 0.515 0.501 0.207 –0.22338 Primary nonferrous metals manufacturing 0.400 0.485 0.247 –0.13239 Metal containers 0.057 0.891 0.064 –0.01240 Heating, plumbing, and fabricated structural metal products 0.604 0.201 0.198 –0.00241 Screw machine products and stampings 0.358 0.641 0.132 –0.13142 Other fabricated metal products 0.377 0.573 0.175 –0.12443 Engines and turbines 0.377 0.362 0.246 0.01544+45 Farm, construction, and mining machinery 0.731 0.151 0.097 0.02146 Materials handling machinery and equipment 0.876 0.134 0.105 –0.11547 Metalworking machinery and equipment 0.779 0.261 0.108 –0.14748 Special industry machinery and equipment 0.962 0.154 0.028 –0.14549 General industrial machinery and equipment 0.729 0.305 0.130 –0.16450 Miscellaneous machinery, except electrical 0.309 0.438 0.258 –0.00651 Computer and office equipment 0.786 0.148 0.156 –0.09052 Service industry machinery 0.636 0.289 0.120 –0.04553 Electrical industrial equipment and apparatus 0.639 0.308 0.168 –0.11454 Household appliances 0.242 0.842 0.045 –0.12955 Electric lighting and wiring equipment 0.471 0.447 0.185 –0.10456 Audio, video, and communication equipment 0.626 0.564 0.206 –0.39657 Electronic components and accessories 0.338 0.437 0.322 –0.09758 Miscellaneous electrical machinery and supplies 0.321 0.687 0.148 –0.15659A Motor vehicles (passenger cars and trucks) 0.478 0.776 0.051 –0.30459B Truck and bus bodies, trailers, and motor vehicles parts 0.437 0.746 0.080 –0.26360 Aircraft and parts 0.134 0.049 0.546 0.27061 Other transportation equipment 0.145 0.543 0.336 –0.02462 Scientific and controlling instruments 0.442 0.166 0.372 0.02063 Ophthalmic and photographic equipment 0.347 0.590 0.228 –0.16564 Miscellaneous manufacturing 0.175 1.121 0.071 –0.36878 Federal gover

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