Sectoral Allocation, Risk Efficiency and the Great Moderation

Finance and Economics Discussion Series Divisions of Research & Statistics and Monetary Affairs Federal Reserve Board, Washington, D.C. Sectoral Allocation, Risk Efficiency and the Great Moderation Manjola Tase 2013-73 NOTE: Staff working papers in the Finance and Economics Discussion Series (FEDS) are preliminary materials circulated to stimulate discussion and critical comment. The analysis and conclusions set forth are those of the authors and do not indicate concurrence by other members of the research staff or the Board of Governors. References in publications to the Finance and Economics Discussion Series (other than acknowledgement) should be cleared with the author(s) to protect the tentative character of these papers.

Sectoral Allocation, Risk Efficiency and the Great Moderation Manjola Tase ∗ October 22, 2013 Abstract ThispaperarguesthatthedeclineinU.S.realGDPgrowthvolatility afterthemid1980swasanoutcomeofmoreriskefficientandmorediversifiedsectoralallocations. Usingaportfolioapproach,Idistinguishbetween the two determinants of GDP growth volatility: sectoral covariances and sectoral allocations. I use the sectoral growth and covariances to compute the growth-volatility frontier of the economy. I define the efficiency of the actual sectoral allocation as the distance of the economy from the frontier,measuredinthe(volatility,growth)space. Therearethreemain findings. 1) The frontier has shifted due to a lower sectoral growth rate and a higher sectoral variance. 2) The distance of the economy from the frontier has decreased. The efficiency over the period increased by 1.4 percentage points. This increase occurred along the volatility dimension and it is interpreted as the decline in the growth volatility in the economy, if there were no changes in the sectoral covariances. This efficiency improvement is comparable to the 1.5 percentage points decline in GDP growth volatility in the data after the mid 1980s. 3) The U.S. economy becamemorediversifiedacrosssectorsaftertheearly1980s,shiftingaway frommanufacturingandagriculturetowardsservices. Theincreaseinthe share of Finance and Insurance coupled with the doubling of the growth volatility in this sector, might have contributed to the recent increase in GDP growth volatility. ∗BoardofGovernorsoftheFederalReserveSystem,DivisionofMonetaryAffairs,Mailstop 76,20thStreetandConstitutionAvenueN.W.,Washington,DC20551. Phone: 202-475-6385. E-mail: manjola.tase@frb.gov. IamgratefultoGeorgeHallforinvaluableguidance. Ithank CatherineMann,BlakeLeBaron,JensHilscher,ScottSchuhandFedericoDiezfornumerous helpful comments. I also thank the seminar participants at Brandeis University, the Federal Reserve Bank of Boston and the Board of Governors of the Federal Reserve System. This paperisoneofthechaptersofmydissertation. PartofthispaperwaswrittenwhenIwasa CSWEPfellowattheFederalReserveBankofBoston. Disclaimer: Staffworkingpapersin theFinanceandEconomicsDiscussionSeries(FEDS)arepreliminarymaterialscirculatedto stimulate discussion and critical comment. The analysis and conclusions set forth are those of the author and do not indicate concurrence by other members of the research staff or theBoardofGovernors. ReferencesinpublicationstotheFinanceandEconomicsDiscussion Series(otherthanacknowledgement)shouldbeclearedwiththeauthortoprotectthetentative characterofthesepapers. 1

1 Introduction The decline in U.S. real GDP growth volatility after the early 1980s is well documentedintheliteratureanditiscoinedastheGreatModerationbyStock and Watson (2003).1 The causes of the Great Moderation mentioned in the literaturemostlyinclude: bettermonetarypolicy,betterinventorymanagement and good luck. In this paper, I use U.S. data for 22 sectors for the period 1947-2010 and I argue that the aggregate growth volatility has declined because of shifts in the production activity across different sectors. By taking a take a portfolio approach, Idistinguishbetweentheeffectofchangesinthesectoralcovariances andsectoralcompositioninthethedeclineinaggregatevolatility. Representing the economy as a portfolio of n sectors, the GDP growth volatility is given by: σ2 =ω(cid:48)Σ ω (1) t,GDP t t t where ω is an n×1 vector of sectoral shares at time t, and Σ is the n×n t covariance matrix of sectoral growth rates. Similarly, the GDP growth is the weightedaverageofthesectoralgrowthrates. Asshowninequation(1),thetwo determinantsofGDPgrowthvolatilityarethecovariancesofthesectoralgrowth ratesandthesectoralallocations. AdeclineintheGDPgrowthvolatilitywould be the outcome of a decrease in the sectoral growth volatility or correlation across sectors. Furthermore, a shifting away from the more volatile sectors or a higher diversification across sectors would also lead to a decline in volatility. To determine the effect of covariance and sectoral allocations, I present the growth and volatility in the economy in an efficient frontier framework. The growth-volatility efficient frontier is determined by the sectoral growth and covariances. The frontier is plotted in the GDP (volatility, growth) space and every point on the frontier represents a portfolio of sectors. The sectoral allocations on the frontier and the corresponding aggregate growth and volatility represent the maximum level of efficiency that the economy can achieve. An increase in the sectoral covariances or a decrease in the sectoral growth rates will shift the frontier and shrink the growth-volatility opportunity set. I define the shifts in the frontier due to changes in the sectoral covariances, as changes in the risk opportunity set. Thelocationoftheeconomyrelativetothefrontierisdeterminedbytheactualsectoralallocation. Idefinetheoptimalallocationastheonethatminimizes the distance of the economy from the frontier, where the distance is measured in the (volatility, growth) space. I measure the efficiency of the economy as the distance between the economy and the optimal allocation on the frontier. The closer the economy to the frontier, the higher the efficiency. The distance along the volatility dimension represents risk efficiency. Using this setup, I can distinguish between the two sources for the decline in GDP volatility: 1) decrease in covariances represented by an expansion of 1The decline in GDP growth volatility has been documented by Kim and Nelson (1999), McConnellandPerez-Quiros(2000)andBlanchardandSimon(2001). 2

the risk opportunity set and 2) changes in sectoral allocation represented by an increase in efficiency. The set of 22 sectors used in this paper fully represents the U.S. economy. I use a one-sided 25-year rolling window to estimate the average sectoral growth rate and the sectoral covariances and to compute the frontier for each 25-year period between 1948 and 2010. I find that: 1. The frontier has shifted down and the growth volatility opportunity set hasshrunk. Atleast2/3oftheshrinkageisduetoalowersectoralgrowth, reflectingtheproductivityslowdownofthe1970s. TherestisduetoadoublingofthegrowthvolatilityinAgriculture(1980),Information(1997)and Finance and Insurance (1998). There is no change in the growth volatility of the other sectors or in the correlation across sectors. This implies that the Great Moderation was not the outcome of a decline in sectoral covariances. As the GDP growth volatility is determined by the sectoral covariancesandsectoralallocations,thissuggeststhatthedeclineinGDP growth volatility was the outcome of changes in the sectoral allocations. 2. The distance of the economy from the frontier has decreased. The efficiency is estimated to have increased by 1.4 percentage points and it is interpreted as the decline in the growth volatility in the economy, if there were no changes in the sectoral covariances. This efficiency improvement is along the volatility dimension and is comparable to the 1.5 percentage points decline in GDP growth volatility in the data after the mid 1980s. 3. Theeconomybecamemorediversifiedaftertheearly1980s. Theeconomic activityhasbecomemoreequallyspreadacrosssectors,shiftingawayfrom Agriculture and Manufacturing towards services. Notably, the share of Finance and Insurance has increased steadily from 2.4% of GDP in 1947 to 8.4% in 2010. It became the third largest sector in the last decade, which coupled with the doubling of the growth volatility in this sector, contributed to the recent increase in GDP growth volatility. To conclude, as a result of changes in the sectoral allocation, the economy has moved closer to the growth-volatility frontier. The Great Moderation was the outcome of more risk efficient and more diversified sectoral allocations. There is a growing literature on the Great Moderation. The conclusion in thispaperiscloselyrelatedtoCarvalhoandGabaix(2010). Theydefinefundamental volatility as a weighted average of the variance of the total factor productivityofeachsector,wherethesectoralvarianceisconstantovertime. They derive GDP growth volatility to be proportional to the fundamental volatility. They conclude that the changes in fundamental volatility, therefore changes in GDP volatility, would come only from changes in the sectoral composition, corresponding to a more diversified economy. Sectoral diversification captures one dimension of the changes in the sectoral allocation. This paper further contributes to the literature by introducing 3

ameasureoftheefficiencyofthesectoralallocations. Giventhesectoralcovariances,thismeasureconvertsthecomplexityofthechangesinsectoralallocations into changes in efficiency. OtherexplanationsoftheGreatModerationinclude: bettermonetarypolicy (Clarida, Gali and Gertler 2000, Cecchetti, Flores-Lagues and Krause 2006), and better inventory management (Kahn, McConnell, and Perez-Quiros 2002, McCarthyandZakrajˇsek2007,IrvineandSchuh2005). Thispaperlooksatthe aggregateoutputfromtheproductionapproachusingthevalueaddedbysector and the role of inventory management is not explicit.2 However, the results in this paper do not reject the inventory hypothesis. They rather imply that the sectoral shift from manufacturing to services, especially finance, has decreased the contribution of a volatile inventory holding sector to aggregate volatility, hence lowering aggregate volatility. Finally some have looked at the technology shocks. Gal´ı and Gambetti (2009) show that the Great Moderation can be explained by the the change in the contribution of technology and non technology shocks. The findings in this paper do not explicitly exclude these factors, rather acknowledge that theirimpactmightbechanneledthroughthechangesintheeconomicstructure. Using counterfactual analysis, I show that the change in sectoral allocations is sufficient to explain the decline in GDP growth volatility. This paper weakens the case for the good luck hypothesis, as in smaller shocks, presented by Stock and Watson (2003). In a broader perspective, this paper relates to the literature that uses sectoraldata, anapproachthathasgainedprominenceinexplainingcross-country differences in the level of production diversification and aggregate volatility. In cross-country studies, Imbs and Wacziarg (2003) find a U-shaped pattern of thesectoraldiversificationalongdifferentstagesofdevelopmentandKorenand Tenreyro (2007) find that sectoral diversification can explain the variation in aggregate volatility across countries. Thispaperisorganizedasfollows. Section2presentsthesectoralallocation in a growth-volatility efficient frontier framework. Section 3 discusses the data. TheresultsarepresentedinSection4. Section5continueswithsomerobustness checks and extensions and Section 6 concludes. 2 Sectoral Allocation in an Efficient Frontier Framework In this section, I present a model of sectoral allocation in a growth-volatility efficient frontier framework. The efficient frontier is computed using the sectoral growth rates and sectoral covariances, and it provides the most risk efficient allocations that the economy can achieve. Given the sectoral growth rates and 2Value added by sector is defined as the difference between the sales and the value of intermediate inputs. The components of value added are: compensation of employees, taxes onproductionminussubsidies,andgrossoperatingsurplus. 4

covariances, the observed growth and volatility in the economy will be determinedbytheobservedsectoralshares. Theclosertheeconomyistothisefficient frontier, the more risk efficient the economy is. This set up allows for the identification of the two sources of changes in GDP volatility: 1) changes in the covariances, represented by an expansion of the growth-volatility opportunity set, and 2) changes in sectoral allocation resulting in changes in efficiency. Efficient frontier: Given the sectoral growth rates and covariances, what is the sectoral allocation that yields the lowest GDP growth variance for a given value of GDP growth? This set of sectoral allocations and the corresponding GDP growth and volatility represent the efficient frontier. The inputs in the computation of the efficient frontier are the vector of the expectedsectoralgrowthratesandthecovariancematrixofthesectoralgrowth rates. Atanytimet,giventhevectorofsectoralgrowthratesg ,thecovariance t matrix of the sectoral growth rates Σ and the vector of sectoral allocations ω , t t the GDP growth rate is given by: g =ω(cid:48)g (2) GDP,t t t the expected GDP growth rate is given by: E(g )=ω(cid:48)E (g) (3) GDP,t t t and the variance of the GDP growth rate is given by: σ2 =ω(cid:48)Σ ω (4) GDP,t t t t where Σ = E [(g −E (g)(g −E (g))(cid:48)] is an n×n covariance matrix of t t t t t t sectoral growth rates. A sectoral allocation is efficient, if for a given GDP growth rate, µ, it yields thelowestGDPgrowthvolatility. Theefficientallocation,ω˜,solvesthefollowing optimization problem: ω˜ =argmin {ω(cid:48)Σω, s.t. (ω(cid:48)E(g)=µ, ω(cid:48)1=1, ω ≤1, ω ≥0)} (5) ω Repeating(5)foreverypossiblevalueofGDPgrowth,Icomputetheefficient frontier as a set of efficient sectoral allocations, ω˜, the corresponding GDP growth,E(g )=ω˜(cid:48)E(g)andvolatility,σ2 =ω˜(cid:48)Σω˜.3 Theefficientfrontier GDP GDP isplottedinthe(volatility,growth)space. Ahighersectoralgrowthrate,alower sectoral variance or a smaller correlation across sectors will shift the efficient frontier, expanding the growth-volatility opportunity set. The shifts in the frontier, due to changes in the sectoral covariances, represent changes in the “risk opportunity”. Preferences: The economy likes GDP growth and dislikes GDP volatility. Optimal allocation: Eachofthesectoralallocationsalongthefrontierisrisk efficient. As the economy likes GDP growth and dislikes GDP volatility, the optimal risk efficient allocation is given by the portfolio on the frontier that 3The range of values for µ is determined by the range of the sectoral growth rates: µ = [min(g),max(g)]. 5

is closest to the economy. The distance of the economy from the frontier is measured in the (volatility, growth) space. The optimal sectoral allocation, ω˜ at time t, is given by: optimal ω˜ =argmin {D (ω˜ )} (6) optimal,t ω˜i t i where D (ω˜) denotes the distance of the economy from the sectoral allocation i,t ω˜ . i I compute the distance of the economy from each of the sectoral allocations on the frontier as: D (ω˜)=[(g −g )2+(σ −σ )2]1/2 (7) i,t frontier,i,t GDP,t frontier,i,t GDP,t where, t is time, i indexes portfolios (sectoral allocations) on the efficient frontier, D is the distance of the economy from the portfolio i on the frontier i,t fortimet. Thegrowthrateandthevariancefortheith portfolioonthefrontier is given by g and σ , respectively, which represent the GDP frontier,i,t frontier,i,t growthandvarianceifthesectoralallocationsintheeconomywerethesameas the sectoral allocations of the ith portfolio on the frontier. The observed GDP growth and volatility are denoted by g and σ . Because the efficient GDP,t GDP,t frontier is concave and upward slopping, at ω˜ , the GDP growth will be optimal,t nolessthantheobservedGDPgrowthandtheGDPvolatilitywillbenohigher than the observed GDP volatility. The distance of the economy from the frontier, D, is the distance from the optimal allocation on the frontier, as in: D =D(ω˜ ) (8) t optimal,t Efficiency: I define the efficiency of the economy as the distance of the economy from the optimal allocation, as in equation (8), where the distance is measured in the (growth, volatility) space. I define the risk efficiency as the distance attributed to the volatility dimension. In this efficient frontier frameworkofsectoralallocation,thetwosourcesforachangeinGDPvolatility, changes in the sectoral covariances and sectoral allocations, are converted into changes in the risk opportunity set, and changes in the risk efficiency of the economy. 3 Data The sectoral data is available from the Bureau for Economic Analysis. The data is available at an annual frequency, for the period 1947-2010 for 22 broad sectors of the economy. The list of sectors is given in Table 1 and they fully represent the economy. These sectors correspond to the 2-digit level of the 2002 North American Industry Classification System (NAICS). The sectoral data for the period 1947-1987 is available only at the 2 digit level. The more disaggregated data (at a 3-digit level) is available starting from 1987. The data includes two variables: chain-type quantity index for value added (vaqi) and 6

value added as a percentage of GDP (vapct). The sectoral growth rate is given by g = ∆log(vaqi ). Let ω ≡ vapct , then (cid:80) ω = 1, g ≡ ω(cid:48)g, and i,t i,t i,t i,t i GDP σ2 ≡ ω(cid:48)Σω. Whelan (2000) shows that in the case of chain-type indices, GDP g as given in the National Income and Product Account (NIPA), would GDP satisfy (g ≡ ω(cid:48)g), if (ω(cid:48)g) represented contributions. As vapct is defined GDP as a nominal share, its product with the sectoral growth rate represents growth contribitions. AsshowninFigure 1,theGDPgrowthrateseriesfromtheNIPA and the GDP growth series using the sectors’ growth rate and nominal shares match. The existence, timing and magnitude of the Great Moderation using annual data. ThereisconsensusintheliteratureonthestartdateoftheGreatModeration as the first quarter of 1984. This break date is estimated using quarterly data for GDP growth. Since the sectoral data is available at an annual frequency, I test for the existence and the timing of the break in the GDP growth volatilityusingannualdata. FollowingMcConnellandPerez-Quiros(2000),and Stock and Watson (2002), I compute the instantaneous volatility as, (cid:114) π σ = |(cid:15) | (9) t 2 t where (cid:15) is the estimated error term from the following AR(1) model of real t GDP growth rates: ∆y = α+β∆y +(cid:15) , where y is the log of real GDP. t t−1 t t Figure 2 plots the HP trend of the instantaneous volatility. I test the null hypothesis of no breaks, γ = 0, against the alternative of a 1 single break, in the following regression: σ =γ +γ D +ξ (10) t 0 1 t t where D is a dummy variable assuming a value of 1 for time t ≥ τ, given t an estimated break date τ ∈ [T ,T (cid:101), where T and T are defined using 15% 1 2 1 2 trimming.4 Using Bai and Perron’s (1998) Sup-F statistics and Perron and Qu (2006), I find support for a break in the GDP volatility.5 The estimated break date is 1984. The estimated values for GDP volatility before and after 1984 are 2.9and1.4respectively,whichimplyahalvingoftheGDPvolatilityafter1984. These results confirm 1984 as the estimated break date in the GDP growth volatilityandconfirmtheestimatedmagnitudeofthedeclineinGDPvolatility. Variance decomposition. Asaninitialdiagnosis,Table 2showsthevariance decomposition of the GDP growth rate. The decomposition is based on the contribution of the sector’s growth to GDP growth, which is defined as the productofthesectoralgrowthratewiththesectoralshare(ω g ). Bydefinition, i i the GDP growth volatility is the sum of the variance terms and the covariance terms of these contributions. 4Inthecaseofa15%trimming,T1=0.15N andT1=0.85N,whereNisthetotalnumber ofobservations. 5Following Perron and Qu (2006), I compute the Sup-F statistics, the critical value and theestimatedbreakdate. TheSup-Fstatisticsis12.614andthe5%criticalvalueis8.592. I thankPerronandQuforsharingtheircode. Thecodeisavailableathttp://people.bu.edu/ perron/code.html. 7

The variance decomposition of the contributions shows that the decline in the sum of the variance and covariance terms is comparable to the decline in the variance of the GDP growth rate. Similar to Irvine and Schuh (2005), the decline in the covariance of the contributions accounts for about 70 % of the decline of the aggregate volatility. Carvalho and Gabaix (2010) show that because of input linkages, there is comovement across sectors. The changes in the off-diagonal terms in the covariance matrix would simply reflect changes in theirmeasureoffundamentalvolatility,whichisdefinedastheweightedaverage of the sectoral volatilities. They also show that, while all the shocks in their modelareidiosyncratic, evensmallmeasurementerrorscancreatecomovement in total factor productivity. 4 Results Iuseaone-sided25-yearrollingwindowtocomputethefrontierforeach25-year periodbetween1948and2010,wherethefirstperiodstartsin1948andthelast periodsendsin2010. Figure3plotstheefficientfrontierforeach25-yearperiod between 1948 and 2010. The first frontier is constructed for 1972 and it uses dataonthesectoralgrowthratefortheperiod1948-1972. Thelastfrontier,the frontier for 2010, is constructed using data for the period 1986-2010. The color of the plots changes gradually, from dark blue to light green, where the dark blue corresponds to the earlier periods and the light green to the later ones. I observe that: 1. The efficient frontier has shifted down. This implies that over the period 1948-2010, if there were no changes in the sectoral allocation, the GDP volatilitywouldhaveincreasedandtheGDPgrowthwouldhavedecreased. Most of the shifts are along the growth dimension, suggesting that the shiftsshouldbemainlyduetoalowersectoralgrowthrate. Thecurvature of the frontiers is relatively unchanged, implying that there is no change in the correlation across sectors. 2. The distance of the economy from the frontier has decreased. The decline inthesectoralgrowthrates,capturedbythegrowthdimensionoftheshifts inthefrontier,ishigherthandeclineintheGDPgrowthinthedata. Also the decline in the distance is mostly along the volatility dimension. That is, the economy became more risk efficient. These observations are elaborated in the following subsections. 4.1 Efficient Frontier and Growth-Volatility Opportunity Set To get a time series of the shifting down of the frontier, I compute the distance of the each frontier from the first frontier. The distance is measured in the (volatility, growth) space. Note that each frontier is computed by constructing 8

N portfolios which are equally spaced in the range of the growth rates for that frontier. For each frontier, I calculate the distance between the corresponding i portfolios, where i=1 to N, as: D =[(g −g )2+(σ −σ )2]1/2(11) t,i,1 frontier,i,t frontier,i,1 frontier,i,t frontier,i,1 The distance of each frontier from the first one is given by the average distance between the corresponding portfolios as in equation ( 12). D =mean {D } (12) t,1 i t,i,1 Figure 4 plots the distance of each frontier from the first frontier. ThedistanceisinterpretedastheexpectedincreaseinGDPvolatilityorthe expected decline in GDP growth in that period relative to the first one. For example,whencomparingthelastfrontier(period1986-2010)withthefirstone (period 1948-1972), a distance of 2.418 can be interpreted as: if there were no changeintheefficiencyoftheeconomy, thegrowthGDPvolatilitywouldbeup to 2.418 percentage points higher or the average growth rate would be up to 2.418 percentage points lower. The shifting down of the frontier can be the outcome of a higher sectoral variance,alowercorrelationacrosssectorsoralowersectoralgrowth. Icompute the contribution of the changes in sectoral growth, volatility and correlation, by computing counterfactual frontiers where I allow for only one of these three variablestochange. Todistinguishbetweentheeffectofachangeinthesectoral growth and sectoral covariance, I compute a counterfactual frontier, using the sectoralgrowthfromthefirstperiod,1948-1972,andthecovariancematrixfrom the period (1948+n) to (1948+n+25), where n takes values from 1 to 38. For example the counterfactual frontier for the last period, 1986-2010, is computed using the vector of sectoral growth rate from the period 1948-1972 and the covariance matrix from the period 1986-2010. I further decompose the change in the covariance matrix into a change in the sectoral variance and correlation across sectors. I compute a counterfactual covariance matrix as σ2,n =ρ(1948+n)to(1948+n+25)σ1948−1972σ1948−1972 (13) i,j,counterfactual i,j i j where, n = 1, 2,...38, i and j denoted sectors and the superscripts denote the corresponding periods. For example the counterfactual frontier for the last period, 1986-2010, is computed using the vector of sectoral growth rate from the period 1948-1972 and the covariance matrix as σ2,1986−2010 =ρ1986−2010σ1948−1972σ1948−1972 (14) i,j,counterfactual i,j i j Figure 5 plots a time series decomposition of the frontier shifts due to changes in the sectoral growth rate, variance and correlation. The vertical axis measuresthedistanceofeachfrontierfromthefirstfrontier. Anon-zeroslope represents a shift in the frontier for the corresponding period. For example, the frontiers computed using data until the mid 80s, have shifted due to changes 9

in the sectoral growth rates, variances and correlation. During this period, while the growth effect is larger, the magnitude of the shifts is comparable acrossthethreefactors. However,thevarianceandthecorrelationeffectforthe latter frontiers is either unchanged or smaller, as the variance effect (the red line) is almost flat and the correlation effect (the blue line) is flat and slightly downward sloping in the latest period. The illustration of the decomposition of these effects for the latest frontier is shown in Figure 6. The latest frontier is computed using the data for the period 1986-2010 and as such it almost fully represents the period of the Great Moderation. It is evident that the frontier has shifted down because of a lower sectoral growth rate, and a higher sectoral variance. A lower sectoral growth accounts for 2/3 of the shift in the frontier, representing the productivity slowdown in the 1970s.6 I test for breaks in growth volatility for each of the sectors. The Sup-F statistics, estimated break date and volatilities are given in Table 3. I find support for no change in the sectoral volatility for 19 sectors and an increase in sectoral volatility in 3 sectors: Agriculture, Information and Finance and Insurance. Figure 7 plots the sectoral volatility for these three sectors. As the sectoralvariancehasbeenhigherandtherehasbeennochangeinthecorrelation across sectors, we expect to observe a higher GDP volatility. Instead, the GDP volatility has decreased, implying that the decline in the GDP volatility was due to changes in the sectoral allocations. 4.2 Efficiency Figure 3 suggested that the economy has been getting closer to the frontier. I defined the efficiency of the economy as the distance of the economy from the frontier. Figure 8 shows that the distance of the economy from the frontier has been decreasing over time, i.e. the economy has become more efficient. The distance of the economy from the frontier was 2.32 in the period 1948-1972 and 0.93 in the period 1986-2010. The change in the efficiency of the economy from the period 1948-1972 to the period 1986-2010, given by the difference in distances, equals 1.39. Interpreted in terms of volatility, this implies that, if there were no change in the covariance matrix, the GDP growth volatility in the period 1986-2010 would be 1.39 percentage points lower than in the period 1948-1972. This improvement in efficiency is comparable to the 1.5 percentage points decline in the GDP growth volatility after 1984. To illustrate the effect of the increased efficiency of the economy in the decline of GDP volatility, I compute the counterfactual values for growth and volatility in the period 1986- 2010, if there were no changes in the efficiency of the economy. Isettheefficiencyattheperiod1948-1972. Inthe(volatility,growth)space, I use the vector between the optimal portfolio and the economy in the period 1986-2010 and the distance of the economy from the frontier in the period 1948-1972 to locate the economy under this “what if” scenario. The results 6The shift due to the change in the average sectoral growth rate is 2.2126 and the shift duetothechangeinthecovariancematrixis0.9686. 10

are plotted in Figure 9. If there were no change in the efficiency, the GDP volatility in the period 1986-2010, would be 2.82%, which is almost the same as theGDPvolatilityinthedataduringtheperiod1948-1972. Thedistanceofthe economy from the frontier has two dimentions: growth and volatility. As such, theincreasedefficiencycanbeattributedtoadecreaseinthedistancealongthe growth or the volatility dimension. A decrease in the distance along the growth (volatility) dimension implies a declineinthedifferencebetweenthegrowth(volatility)observedinthedataand thegrowth(volatility)attheoptimalallocation. Figure10plotsthegrowthrate and volatility at the optimal sectoral allocation, the GDP growth and volatility observed in the data, and the difference between the two. Figure 10 shows that thedifferencebetweenthegrowthvolatilityinthedataandthevolatilityatthe optimal allocation has been decreasing over time. The increase in the efficiency occured along the volatility dimension, an increase in risk efficiency. 4.3 Sectoral Allocation In the previous section, I showed that since the sectoral covariances have either increased or not changes, the Great Moderation was the outcome of changes in the sectoral allocations. Furthermore, the risk efficiency, as measured by the difference between the GDP volatility in the data vs.the model, has increased. Thesefindingsimplythatthesectoralallocationsweremoreriskefficient. From Finance, a lower volatility can be a benefit of diversification. To illustrate, I present a measure of sectoral diversification, which refers to the dispersion of the economic activity across sectors. Figure 11 plots, the Herfindal Index, the most commonly used index of sectoral concentration.7 The Herfindal Index, (HI), is defined as, (cid:88) HI = ω2 (15) t i,t whereω isthesectorsharetoGDP,denotedbythevariablevapctinthedata. i,t A decrease in the Herfindal index corresponds to an increase in diversification, implying that the economic activity is more equally spread across sectors. The U.S. economy was more diversified during the Great Moderation. Figure 12 showsthatthereisadeclineinthesectorsharefortheManufacturingofDurable and Nondurables goods, and Agriculture and an increase in services, particularly Finance and Insurance, Health, Professional and Administration services. AsshowninTable 3, thegrowthvolatilityinFinanceandInsuranceandInformation doubled after 1998/1999, contributing to the increase in the volatility in the recent years. Figure11suggeststhattherearetwobreaks(threeregimes)intheHIseries. I find strong evidence for the presence of two breaks.8 The estimated break 7Other measures of sectoral concentration, such as the Coefficient of Variation and the Max-Minspreadofsectoralsharesshowthesamepattern. Thecorrelation(HI,coefficientof variation)is0.9997andthecorrelation(HI,Max-Minspread)is0.945. 8TheSup-Fstatisticsis129.229andthe1%criticalvalueis9.801. 11

dates are the 1970 and 1980, where the period 1970-1980 corresponds to the decline in the sectoral concentration and the period 1980 onwards corresponds to the period of a higher sectoral diversification. While the estimated date for the second break in the HI, 1980, occurs before the estimated break date in the GDPvolatility,1984,the90%confidenceintervalforthesecondbreakinHIfalls within the 90% confidence interval for a break in the GDP growth volatility.9 TofurtherillustratetheeffectofchangesinthesectoralallocationinthedeclineofGDPgrowthvolatility, IconstructacounterfactualGDPgrowthseries, using the sectoral growth rates inthe data and setting thesectoralcomposition as in 1948: (cid:88) g = g ω (16) GDP,t,counterfactual i,t i,1948 I compute the instantaneous volatility of g . Figure 13 GDP,t,counterfactual plots the GDP volatility in the data and the GDP volatility for the counterfactual growth series. The only difference between these two series comes from the changes in the sectoralallocationsovertime. Figure 13showsthattheeffectofthechangesin the sectoral allocation occurs in the mid 1980s. I cannot reject the hypothesis of no break in the counterfactual GDP growth series. The change in sectoral allocations is sufficient to explain the decline in GDP growth volatility. 5 Robustness Checks and Extensions 5.1 Robustness Checks In this section, I test for the robustness of results using a less disaggregated classification and a more rigid frontier, where the sectoral shares are allowed to move within certain bounds. Sectoral Classification. The selection of the 22 sectors, corresponding to the 2-digit NAICS, was dictated by the data availability. To test the robustness of the results with respect to the sectoral classification, I use 15 sectors corresponding to the sector level classification in the Input-Output Tables. The list of sectors is given in Table 4. I cannot test the robustness of the results for a less disaggregated classification as the data at a 3-digit level data is availabe starting from 1987, leaving out the period before the Great Moderation. Bounds on Sectoral Shares. When computing the efficient frontier, the sectoral shares were just constrained to be between 0 and 1 and to sum up to 1, allowing for a very feasible structure of the economy. To introduce rigidities, I impose further constraints on the sectoral shares. I set the lower (L) and the upper bounds (U) as: ω =max{0,[min(vapct )−3stdev(vapct )]} (17) L,i i i ω =min{1,[max(vapct )+3stdev(vapct )]} (18) U,i i i 9The90%confidenceinterval: GDPvolatility=[1974, 1996].HI=[1979, 1987]. 12

Furthermore, I define 15 groups as in the sector level classification of the Input-OutputTable. Iimposeboundsontherelativeshareofgroupsasfollows: (cid:18) (cid:19) (cid:26) (cid:20) (cid:18) (cid:19) (cid:18) (cid:19)(cid:21)(cid:27) ω vapct vapct groupi =max 0, min groupi −3stdev ( groupi (19) ω vapct vapct groupj L groupj groupj (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19) ω vapct vapct groupi =max groupi +3stdev groupi (20) ω vapct vapct groupj U groupj groupj The allocations along the efficient frontier are computed as in equation 21. ω˜ =argmin {ω(cid:48)Σω, s.t. (ω(cid:48)E(g)=µ,ω(cid:48)1=1, ω ≤1, ω ω ≤ω ≤ω L U (21) (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19) ω ω ω groupi ≤ groupi ≤ groupi )} ω ω ω groupj L groupj groupj U Table 5 shows the correlation coefficient for the measures of frontier shifts, efficiency and risk efficiency produced by these two specifications with the 22sector baseline. I find that the results are robust to the less disaggregated classification and robust to the constraints on the sectoral shares: the growthvolatility opportunity set has shrunk, the economy has become more efficient and the improvement in efficiency is along the volatility dimension. Table 6showsthecounterfacturalGDPvolatilityfortheperiod1986-2010if therewerenochangesinefficiency. TheGDPvolatilityfortheperiod1948-1972 was 2.62 %. If the efficiency in 1986-2010 were the same as in the 1948-1972, the counterfactual GDP volatility produced by the 22-sector, 15-sector and 22sectorwithsectoralshareboundswouldbe2.41%,2.64%and2.82%respectively. These results confirm that the decline the GDP volatility was the outcome of more risk efficient sectoral allocations. The Herfindal Index, both in the 22-sector and 15-sector case, shows an increase in sectoral diversification, implying that the more diversified sectoral allocations were more risk efficient.10 5.2 Extensions Preferences. Thefrontierisdeterminedbythevectorofthesectoralgrowthrate and the covariance matrix of the sectoral growth rates. Given the frontier, the actualsectoralallocationsdeterminethelocationoftheeconomyrelativetothe frontier. The preferences in terms of growth and volatility are represented in the distance of the economy from the frontier as in equation ( 7). The optimal allocation then corresponds to efficient allocation that minimizes the distance of the economy from the frontier in the (growth, volatility) space. I use the ratio of growth-to-volatility at the optimal allocation to capture the “revealed” preferences. Figure 14plotstheratioofgrowth-to-volatilityinthedataandthe 10ThecorrelationcoefficientfortheHerfindalIndexinthe22-sectorand15-sectorclassificationis0.986. 13

optimal allocation on the frontier computed for three specifications: 22-sector and 15-sector classsification and the 22-sector with bounds on sectoral shares. The growth-to-volatility ratio in the economy is stable at about 2. Across the three specifications, the growth-to-volatility ratio of the optimal allocation is higher and volatile in the period before the Great Moderation. During the Great Moderation, the ratio is in the range of 2 to 4, which is also the range usually used for the coefficient of risk aversion in the the mean-variance utility function. Sectoral Volatility and Trade Balance. While the trade volume has been increasing since the 1950s, the Great Moderation corresponds to a period of persistent trade deficits, as plotted in Figure 15. Figure 16 plots the trade balance and growth volatility for each sector. Figure 16 shows that the sectors with a trade deficit (Durables, Mining and Non durables) are among the more volatile sectors. ThissuggestsanotherpointofviewoftheGreatModeration,asanoutcome of“exportingvolatility”byincurringatradedeficitinthemorevolatilesectors. Itisnotplausibletosuggestthattheincreasedtradeopennessledtoanincrease in growth volatility in these sectors, as I did not find support for a change in volatility in these sectors. 6 Concluding Remarks Modelingtheeconomyinthe(growth,volatility)space,Icanpresentthesectoral allocationinagrowth-volatilityefficientfrontieranddiscussthecontributionof changes in sectoral covariances and sectoral allocations in explaining the Great Moderation. Iconvertthecomplexityofchangesinthesectoralallocationsinto a measure of efficiency of the economy. I measure the efficiency as the distance of the economy from the growth-volatility efficient frontier. While the frontier has shifted down, shrinking the growth-volatility opportunity set, the efficiency has increased. I conclude that the decline in GDP growth volatility after 1984, up until the last recession, can be sufficiently explalined by more efficient and more diversified sectoral allocations. Inaddition,sectoralshiftshavebeenproposedasanexplanationfortheslow increase in employement during the recoveries since the early 1990s. Risssman (2009) shows the structural changes were more prounounced in Finance, Insurance and Real Estate sector. These structural changes would contribute to the increaseinthegrowthvolatilityinFinanceandInsurance,whichcombinedwith the increase in the share of these sectors contributed to the recent increase in aggregate growth volatility. Furthermore, the Great Moderation also concurs with a period of persistent trade deficits. These deficits were in the more volatile sectors. These facts suggest that the impact of the globalization on growth volatility is affected by the sectoral composition of the trade balance. 14

References [1] Bai, Jushan, and Pierre Perron. 1998. Estimating and testing linear models with multiple structural changes. Econometrica 66(1), 47-78. [2] Blanchard,Olivier,andJohnSimon.2001.ThelongandlargedeclineinU.S. output volatility. Brookings Papers on Economic Activity 2001(1), 165-71. [3] Cavallo,Eduardo.2007.Opennesstotradeandoutputvolatility: areassessment. Inter-American Development Bank Working Paper 604. [4] Carvalho, VascoM., andXavierGabaix.2010.Thegreatdiversificationand its undoing. National Bureau for Economic Research Working Paper16424. [5] Cecchetti, Stephen G., Alfonso Flores-Lagunes, and Stefan Krause. 2006. Has monetary policy become more efficient? A cross-country analysis. The Economic Journal 116: 408-433. [6] Clarida, Richard, Jordi Gal´ı, and Mark Getler. 2000. Monetary policy rules and macroeconomic stability: evidence and some theory. Quarterly Journal of Economics 115(1), 147-180. [7] Gal´ı, Jordi and Luca Gambetti. 2009. On the sources of the great moderation. American Economic Journal: Macroeconomics 1(1), 26-57. [8] Imbs,Jean.2007.Growthandvolatility.JournalofMonetaryEconomics54, 1848-1862. [9] Imbs, Jean and Romain Wacziarg. 2003. Stages of diversification. American Economic Review 93(1), 63-86. [10] Irvine, Owen and Scott Schuh. 2005. The Roles of comovement and inventory investment in the reduction of output volatility. Federal Reserve Bank of Boston 05(9). [11] Kahn,James,MargaretM.McConnell,andGabrielPerez-Quiros.2002.On the causes of the increased stability of the U.S. economy. Federal Reserve Bank of New York Economic Policy Review 8(1), 183 - 202. [12] Kim, Chang-Jin, and Charles R. Nelson. 1999. Has the U.S. economy becomemorestable? ABayesianapproachbasedonaMarkov-switchingmodel ofthebusinesscycle.TheReviewofEconomicsandStatistics81(4),608-616. [13] Koren, Mikl´os and Silvana Tenreyro. 2007. Volatility and development. Quarterly Journal of Economics, 122(1), 243-87. [14] McCarthy, Jonathan and Egon Zakrajˇsek. 2007. Inventory dynamics and businesscycles: Whathaschanged? JournalofMoney,Credit,andBanking 39(2-3), 591-613. 15

[15] McConnell, MargaretandGabrielPerez-Quiros.2000.Outputfluctuations in the United States: What has changed since the early 1980’s. American Economic Review 90(5), 1464-76. [16] Perron, Pierre and Zhongjun Qu. 2006. Estimating restricted structural change models. Journal of Econometrics 134(2), 373-399. [17] Rissman,EllenR.2009.Employmentgrowth: cyclicalmovementsorstructural change? Federal Reserve Bank of Chicago Economic Perspectives 33(4), 44-57. [18] Stock, James and Mark Watson. 2003. Has the business cycle changed? Evidence and explanations. FRB Kansas City symposium, Jackson Hole, Wyoming, August 28-30, 2003. [19] Whelan, Karl. 2000. A guide to the use of chain aggregated NIPA data. BoardofGovernorsoftheFederalReserveSystemWorkingPaper2000(35). 16

Abbreviation Sector 2002NAICSCode Agri Agriculture,Forestry,FishingandHunting 11 Mining Mining 21 Utilities Utilities 22 Construction Construction 23 Durables Durablegoods 33,321,327 Nondurables Nondurablegoods 31,32(except321&327) Wholesale Wholesaletrade 42 Retail Retailtrade 44,45 Transp TransportationandWarehousing 48,49(except491) Info Information 51 FinIns FinanceandInsurance 52 Real Realestate,Rental,Leasing 53 ProfScien Professional,ScientificandTechnicalServices 54 Manage ManagementofCompaniesandEnterprises 55 Admin AdministrativeandWasteManagementServices 56 Education Educationservices 61 Health HealthcareandSocialassistance 62 ArtsEntRec Arts,EntertainmentandRecreation 71 AccomFood AccomodationandFoodservices 72 Other OtherServices,exceptGovernment 81 Fed FederalGovernment na StateLocal StateandLocalGovernment na Table1: List of Sectors. Theselectionofthesectorsisdictatedbythedataavailability,as toincludeobservationsbeforeandaftertheGreatModeration. Thedatafortheperiod1947- 1987isavailableonlyatthe2-digitlevel,representing22sectors. Themoredisaggregateddata isavailableonlystartingfrom1987. Idoarobustnesstestusingthesector-levelclassification oftheInput-Outputtable,representing15sectors,andIfindthattheresultsarerobusttoa lessdisaggregateddata. 17

Early(1948-1983) Late(1984-2010) Late/Early VarianceRealGDPgrowth 7.32 3.07 0.42 Sumofthevarianceterms 2.05 0.77 0.38 Sumofthecovarianceterms 5.27 2.30 0.44 Table2: Variance Decomposition. Sector’scontributiontoGDPgrowthisdefinedasthe sector’s growth rate weighted by the sector’s share to GDP (ωigi). The sum of the variance termsis((cid:80)ω i 2σ i 2)andthesumofthecovariancetermsis(2(cid:80)ωiωjρi,jσiσj). 18

Sector Sup-F Date Before After Agriculture,Forestry,FishingandHunting 8.934** 1980 3.8 9.5 Mining 6.341 Utilities 4.279 Construction 2.885 Durablegoods 4.555 Nondurablegoods 2.395 Wholesaletrade 3.579 Retailtrade 1.57 TransportationandWarehousing 6.327 Information 9.056** 1997 2.4 5.4 FinanceandInsurance 13.785*** 1998 2.5 5.3 Realestate,Rental,Leasing 3.662 Professional,ScientificandTechnicalServices 1.87 ManagementofCompaniesandEnterprises 5.178 AdministrativeandWasteManagementServices 4.178 Educationservices 2.24 HealthcareandSocialassistance 5.759 Arts,EntertainmentandRecreation 6.984 AccomodationandFoodservices 1.984 OtherServices,exceptGovernment 2.36 FederalGovernment 2.043 StateandLocalGovernment 4.004 Table3: BreaksinSectoralVolatility. Ifindsupportfornochangeinthesectoralvolatility for19sectorsandanincreaseinsectoralvolatilityin3sectors: Agriculture,Informationand Finance and Insurance. For these three sectors, Date, Before and After show the estimated break date, and the estimated volatility before and after the break date. The volatility in thesethreesectorshasdoubled. TheseresultsshowthatthedeclineinGDPgrowthvolatility couldhavenotbeentheoutcomeofalowersectoralvariance. Levelsofsignificance: 1%(***), 5%(**). 19

sedoc SCIAN 2002 sedoC O-I rotceS 11 11 gnitnuhdna,gnihsfi,yrtserof,erutlucirgA 12 12 gniniM 22 22 seitilitU 32 32 noitcurtsnoC 33,23,13 G13 gnirutcafunaM 24 24 edartelaselohW 54,44 TR44 edartliateR )194tpecxe(94,84 WT84 gnisuoherawdnanoitatropsnarT 15 15 noitamrofnI 35,25 ERIF gnisaeldna,latner,etatselaer,ecnarusni,ecnaniF 65,55,45 FORP secivresssenisubdnalanoisseforP 6 6 ecnatsissalaicosdna,erachtlaeh,secivreslanoitacudE 7 7 secivresdoofdna,noitadommocca,noitaercer,tnemniatretne,strA 18 18 tnemnrevogtpecxe,secivresrehtO AN G tnemnrevoG 51otsdnopserrocselbaTtuptuO-tupnIehtfonoitacfiissalclevel-rotcesehT .selbaT tuptuO-tupnI eht ni noitacfiissalC level rotceS :4elbaT .7891morfgnitratselbaliavasiatadtigid-3ehT .noitagerggafolevelsihtottsuborerastluserehT .srotces 20

Correlationcoefficientfor FrontierShifts Efficiency RiskEfficiency 22sectors 22sectors 22sectors 22sectors 1 1 1 15sectors 0.979 0.995 0.994 22sectors&sectoralsharesbounds 0.965 0.974 0.866 Table5: ComparisonofDifferentSpecifications. “Frontiershifts”measuresthedistance of each frontier from the first one. “Efficiency” is measured by the distance of the economy fromtheefficientallocation,and”RiskEfficiency”correspondstothevolatilitydimensionof “Efficiency”. I test the robustness of the results to the level of dissagregation (“22 sectors” versus “15 sectors”) and variability of sectoral shares (“22 sectors” versus “22 sectors & bounds”). In “22 sectors & bounds”, the sector shares and the ratio of sector shares are bounded to [min−3stdev,max+3stdev] of the observed values. The results are robust to these changes. The growth-volatility opportunities set has been shrinking and the risk efficiencyhasbeenincreasing. 21

GDPgrowthvolatility data(1948-1972) 2.62 data(1986-2010) 1.68 counterfactual(1986-2010)ifefficiencyasin(1948-2010) —22sectors 2.41 —15sectors 2.64 —22sectors&sectoralsharesbounds 2.82 Table 6: GDP volatility and Efficiency. The counterfactual GDP growth volatility is computed keeping the efficiency as in (1948-1972). Efficiency is measured by the distance of the economy from the frontier. The counterfactual GDP growth volatility in the three specifications is almost the same as the volatility in the data in (1948-1972), reconfirming thatthedeclineinGDPvolatilitywastheoutcomeofmoreriskefficientsectoralallocations. “22sectors”isthebaseline,“15sectors”correspondstoamoreaggregateddatatsetand“22 sectors & bounds” restricts the variability of the sector shares and the ratio of sector shares to[min−3stdev,max+3stdev]. 22

10 8 6 4 2 0 −2 −4 1940 1950 1960 1970 1980 1990 2000 2010 )tnecrep ni( etar htworg GDP growth (NIPA) GDP growth (sectoral data) Figure1: TheGDPgrowthratefromNationalIncomeandProductAccountsand as a weighted sum of the sectoral growth rates. 23

) % ni ( ytilitalov htworg PDG 4 3 2 1 1940 1950 1960 1970 1980 1990 2000 2010 year Figure 2: HP Trend of Instantaneous GDP Growth Volatility. The instantaneous (cid:113) GDP volatility is defined as σt = π 2 |(cid:15)t|, where (cid:15)t is the error term from the AR(1) model ofGDPgrowth. Intheliterature,theestimatedstartdateofthedeclineintheGDPgrowth volatilityis1984:1. Theestimatedbreakdate,usingannualdata,is1984. 24

8 7 6 5 4 3 2 1 0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 standard deviation of the growth rate (in%) )% ni( etar htworg egareva frontier shifting down distance of the economy from the frontier decreasing Figure 3: The Efficient Frontier and the Economy Over Time. Each line shows the efficientfrontierforevery25-yearrollingperiodbetween1948and2010. Thecirclesshowthe GDPgrowthandvolatilityobservedinthedataforevery25-yearrollingperiod. Thereare39 linesand39circlesinthisFigure. Thedistanceoftheeconomyfromthefrontiercorresponds todistanceofthecirclefromthelinewiththesamecolor. Thefontierhasbeenshiftingdown totheright,implyingthattheopportunitysethasshrunk. Thedistanceoftheeconomyfrom thefrontierhasdecreased,implyingthattheefficiencyoftheeconomyhasincreased. 25

2.5 2 1.5 1 0.5 0 1970 1975 1980 1985 1990 1995 2000 2005 2010 year )stniop egatnecrep ni( eno tsrif eht morf reitnorf hcae fo ecnatsid Figure4: DistanceofeachFrontierfromtheFirstFrontier. Thedistanceisinterpreted as the expected increase in GDP volatility or the expected decline in GDP growth in the corresponding period relative to the first one, if there were no change in efficiency. For example, the distance of the last frontier (1986-2010) from the first frontier (1948-1872) is 2.418. Iftherewerenochangeinthedistanceoftheeconomyfromthefrontier,i.e. nochange intheefficiency,theGDPgrowthvolatilityinthelastperiodwouldbeupto2.418percentage pointhigherthaninthefirstperiod. 26

2.5 2 1.5 1 0.5 0 1970 1975 1980 1985 1990 1995 2000 2005 2010 year )stniop egatnecrep ni( eno tsrif eht morf reitnorf hcae fo ecnatsid correlation effect variance effect growth effect Figure5: Time Series of the Effect of Sectoral Growth, Variance and Correlation in Shifting the Frontier.The vertical axis measures the distance of each frontier from the firstfrontier. Anon-zerosloperepresentsashiftinthefrontierforthecorrespondingperiod. 27

8 7 6 5 4 3 2 1 0 0 1 2 3 4 5 6 standard deviation of growth rate (in %) )% ni( etar htworg egareva frontier (1948−1972) variance effect correlation effect growth effect frontier (1986−2010) growth 1948−1972, var 1948−1972, corr 1948−1972 growth 1986−2010, var 1986−2010, corr 1986−2010 growth 1948−1972, var 1986−2010, corr 1986−2010 growth 1948−1972, var 1948−1972, corr 1986−2010 Figure 6: Decomposing the Effect of Sectoral Growth, Variance and Correlation. The frontier for the first period (1948-19572) is given by the solid line and the frontier for the last period (1986-2010) is given by the dash-dot line. The dash line and the dotted line representcounterfactualfrontiers. Thesolidlineandthedottedlinearecomputedusingthe samevectorofsectoralgrowthrateandthesamesectoralvariance. Thedistancebetweenthe solidlineandthedottedlineshowstheeffectofthechangeinthecorrelationacrosssectors. The dotted line and the dash line are computed using the same vector of sectoral growth rate and the same correlation across sectors. The distance between the dotted line and the dash line shows the effect of a change in the variance. The dash line and the dash-dot line arecomputedusingthesamecovariancematrix. Thedistancebetweenthedashlineandthe dash-dotlineshowstheeffectofachangeinthegrowthrate. Alowersectoralgrowthanda highersectoralvarianceaccountfor2/3and1/3oftheshiftinthefrontier,respectively. 28

02 51 01 5 0 Agriculture Finance and Insurance Information 1940 1960 1980 2000 1940 1960 1980 2000 1940 1960 1980 2000 )% ni( ytilitalov htworg year Figure7: SectoralGrowthVolatility. Thefigureplotsthegrowthvolatilityforthesectors thathaveastatisticallysignificantbreakinvolatility. 29

2.6 2.4 2.2 2 1.8 1.6 1.4 1.2 1 0.8 1970 1975 1980 1985 1990 1995 2000 2005 2010 year )stniop egatnecrep ni( reitnorf eht morf ymonoce eht fo ecnatsid Figure 8: Distance of the Economy from the Frontier. The efficiency is measured by the distance of the economy from the frontier. The distance of the economy from the frontierismeasuredastheminimumdistancefromeachoftheportfoliosfromonthefrontier, where the distance is measured in the (growth, volatility) space: min[Di,t], where Di,t = [(g frontier,i,t −gGDP,t)2+(σ frontier,i,t −σGDP,t)2]1/2. Theclosertheeconomytothefrontier the higher the efficiency. A change in the efficiency over time is interpreted as the expected increase in GDP growth, or decline in GDP volatility. For example, the efficiency between 1986-2010and1948-1972increaseby1.39(thedistanceoftheeconomyfromthefrontierwas 0.93in1986-2010and2.32in1948-1972). Interpretedalongthevolatilitydimension,ifthere were no change in the covariance matrix, the GDP volatility in the period 1986-2010 would be1.39percentagepointslowerthanintheperiod1948-1972. 30

8 7 6 5 4 3 2 1 0 0 1 2 3 4 5 6 standard deviation of growth rate (in %) )% ni( etar htworg egareva frontier (1948−1972) frontier (1986−2010) econ (optimal, 1948−1972) econ (data, 1948−1972) econ (optimal, 1986−2010) econ (data, 1986−2010) econ (counterfactual, 1986−2010) Figure9: EfficiencyandtheDeclineinGrowthVolatility. Theefficiencyoftheeconomy ismeasuredbythedistanceoftheeconomyfromtheoptimalallocationonthefrontier. The distanceoftheeconomyfromthefrontierisgivenbythedistancebetweentheecon(data)and econ(optimal),whereecon(optimal)istheoptimalallocationonthefrontier. Astheeconomy likes GDP growth and dislikes GDP growth volatility, the optimal allocation is determined by the portfolio on the frontier closest to the corresponding econ(data), where the distance ismeasuredinthe(growth, volatility)space. Thedistanceoftheeconomyfromthefrontier has decreased, implying that the efficiency has increased. To illustrated the effect of the increased efficiency in the decline of GDP volatility, I compute a counterfactual economy: econ(counterfactual, 1986-2010) is computed such that the distance of econ(counterfactual, 1985-2010) from frontier(1986-2010) is the same as the distance of the economy from the frontier (1948-1972). If there were no changes in efficiency, there would be no decline in the growthvolatility: theGDPvolatilityintheperiod1986-2010,wouldbe2.82%,whichisalmost thesameasthevolatilityduringtheperiod1948-1972. 31

5 4.5 4 3.5 3 2.5 2 1.5 1 0.5 0 1970 1980 1990 2000 2010 year )% ni( Growth Dimension Volatility Dimension 3 2 1 0 −1 −2 −3 1970 1980 1990 2000 2010 year )% ni( econ (data) econ (frontier) econ (frontier) econ (data) difference difference Figure10: RiskEfficiency. Thisfigureplotstheimprovementsinefficiencyalongthegrowth andvolatilitydimension. Thedistanceoftheeconomyfromthefrontierhasbeendecreasing along the volatility dimension, implying an improvement in the risk efficiency. Efficiency is measured by the distance of the economy from the frontier in the (growth, volatility) space. Theoptimalallocationisdefinedastheallocationonfrontierclosesttotheeconomy. 32

0.082 0.08 0.078 0.076 0.074 0.072 0.07 0.068 0.066 0.064 1940 1950 1960 1970 1980 1990 2000 2010 xednI ladnifreH year Figure 11: Sectoral Concentration in the Economy. The sectoral concentration is measured bytheHerfindalIndex,as(cid:80)ω i 2,whereωi isthesectorsharetoGDP.TheHerfindalIndexis calculated using the annual data for 22 broad sectors in the economy. The list of sectors is giveninTable 1. Theindextakesvaluesfrom1/n(equalsectoralshares)to1(asinglesector economy), wheren=numberofsectors. AdecreaseintheHerfindalIndexcorrespondstoan increasedindiversification. TheeconomywasmorediversifiedduringtheGreatModeration. 33

5101 5 0 5101 5 0 5101 5 0 5101 5 0 5101 5 0 AccomFood Admin Agri ArtsEntRec Construction Durables Education Fed FinIns Health Info Manage Mining Nondurables Other ProfScien Real Retail StateLocal Transp 1940 1960 1980 2000 20201940 1960 1980 2000 20201940 1960 1980 2000 2020 Utilities Wholesale 1940 1960 1980 2000 20201940 1960 1980 2000 2020 serahS rotceS year Figure 12: Sector Shares. The economy has been shifting away from Agriculture and Manufacturingandmovingtowardservices. 34

) % ni ( 4 3 2 1 1940 1950 1960 1970 1980 1990 2000 2010 year Growth Volatility Growth Volatility (if no changes in the sectoral allocations) Figure 13: Sectoral Allocation and GDP Volatility. The dashed line plots the GDP growth volatility in the data. The solid line plots the growth volatility of a counterfactual GDP series, where the sectoral allocations are time invariant, and the sectoral growth rates vary as in the data. I cannot reject the hypothesis of no break in the counterfactual GDP growth series, providing further support that the decline in GDP growth volatility was the outcomeofchangesinthesectoralallocations. 35

12 10 econ (frontier, 22 sectors) 8 6 econ (frontier, 15 sectors) 4 econ (frontier, 22 sectors & bounds) 2 econ (data) 0 1970 1975 1980 1985 1990 1995 2000 2005 2010 Figure 14: Preferences: Growth-to-Volatility Ratio. It is assumed that the economy likesgrowthanddislikesvolatility. Theoptimalallocationisdefinedastheallocationonthe frontier that minimizes the distance of the economy from the frontier. Growth-to-Volatility Ratio produced by these allocations shows the same pattern across the three specifications. TheratioislowerandstableduringtheGreatModeration. Ittakesvaluesfrom2to4,which is also the range usually used for the coefficient of risk aversion in the the mean-variance utilityfunction. 36

Trade Balance/GDP Trade Volume/GDP )% ni( PDG/emuloV edarT 03 52 02 51 01 5 4 2 0 2− 4− 6− )% ni( PDG/ecnalaB edarT 1940 1950 1960 1970 1980 1990 2000 2010 year Figure15: TradeBalanceandTradeDeficit. Thetradevolumehasbeenincreasingsince the1950s. TradedeficitshavebeenpersistentduringtheGreatModeration. 37

StateLocal Real Health GDP Education ProfScien Other FinIns Info AccomFood Wholesale Retail ArtsEntRec Admin Manage Fed Nondurables Transp Utilities Construction Mining Durables Agri −4 −2 0 2 4 6 8 10 (in %) Growth Volatility Trade Balance/GDP Figure16: GrowthVolatilityandTradeBalance. Themorevolatilesectorshavealarger tradedeficit. Thetradedataarefromthe1998-2009fromtheInput-OutputUseTables. 38