Stochastic Trends and Economic Fluctuations

By ROBERT G. KING, CHARLES I. PLOSSER, JAMES H. STOCK, AND MARK W. WATSON*

Are business cycles mainly the result of permanent shocks to ? This paper uses a long-run restriction implied by a large class of real-business-cycle models -identifying permanent productivity shocks as shocks to the common stochastic trend in output, , and investment-to provide new evidence on this question. Econometric tests indicate that this common-stochas- tic-trend/ cointegration implication is consistent with postwar U.S. data. How- ever, in systems with nominal variables, the estimates of this common stochastic trend indicate that permanent productivity shocks typically explain less than half of the business-cycle variability in output, consumption, and investment. (JEL E32, C32)

A central, surprising, and controversial tent with the presence of a common resultof some currentresearch on real busi- stochastic productivitytrend. Such a trend ness cycles is the claim that a common is capable of explainingimportant compo- stochastic trend-the cumulative effect of nents of fluctuationsin consumption,invest- permanent shocks to productivity-under- ment, and output in a three-variablere- lies the bulk of economic fluctuations. If duced-formsystem. However, the common confirmed, this finding would imply that trend's explanatorypower drops off sharply many other forces have been relatively when measures of money, the level, unimportantover historicalbusiness cycles, and the nominal interest rate are added to including the monetary and fiscal policy the system.The key implicationof the stan- shocks stressed in traditional macroeco- dard real-business-cyclemodel, that perma- nomic analysis. This paper shows that the nent productivityshocks are the dominant hypothesisof a common stochastic produc- source of economic fluctuations,is not sup- tivity trend has a set of econometricimpli- ported by these data. Moreover,our empiri- cations that allows us to test for its pres- cal results cast doubt on other explanations ence, measure its importance, and extract of the business cycle: estimates of perma- estimates of its realized value. Applying nent nominalshocks, which are constrained these procedures to consumption, invest- to be neutral in the long run, explain little ment, and output for the postwar United real activity. States,we find resultsthat both supportand Our econometricmethodology can deter- contradict this claim in the real-business- mine the importanceof productivityshocks cycle literature. The U.S. data are consis- within a wide class of real-business-cycle (RBC) models with permanentproductivity disturbances.To explain why this is so, we *University of Rochester, University of Rochester, begin by discussing three features of the University of California-Berkeley, and Northwestern researchon which our analysisbuilds. First, University, respectively. The authors thank , C. W. J. Granger, Robert Hall, Gary there is a long tradition of empirical sup- Hansen, Thomas Sargent, James Wilcox, and two ref- port for balanced growth in which output, erees for helpful discussions and comments and thank investment, and consumption all display Craig Burnside and Gustavo Gonzaga for valuable positive trend growthbut the consumption: research assistance. This research was supported in output and investment:output"great ratios" part by the National Science Foundation, the Sloan Foundation, and the John M. Olin Foundation at the do not (see e.g., Robert Kosobud and University of Rochester. LawrenceKlein, 1961).Second, in large part

819 820 THE AMERICAN ECONOMIC REVIEW SEPTEMBER 1991 because of this ratio stability, most RBC ond, in a three-variablemodel incorporating models are one-sectormodels which restrict output, consumption, and investment, the preferences and production possibilities so balanced-growthshock explains 60-75 per- that "balanced growth" occurs asymptoti- cent of the variationof output at business- cally when there is a constant rate of cycle horizons (4-20 quarters). Moreover, technological progress. Third, these RBC the estimatedresponse of the real variables models imply that permanentshifts in pro- to the balanced-growthshock is similar to ductivity will induce (i) long-run equipro- the dynamic multipliersimplied by simple portionate shifts in the paths of output, RBC models driven by random-walkpro- consumption, and investment and (ii) dy- ductivity.Third, these results change signif- namic adjustmentswith differential move- icantly when nominal variables are added ments in consumption,investment, and out- to the empirical model. When money, put. , and interest rates are added, the The econometric procedures developed balanced-growthshock explains less of the here use the models' long-run balanced- fluctuationsin output, from 35 percent to 44 growthimplication to isolate the permanent percent depending on the particularspeci- shocks in productivityand then to trace out fication used. Permanent nominal shocks, the short-runeffects of these shocks. These identified by imposing long-run neutrality econometricprocedures rely on the fact that for output, explainlittle of the variabilityin balanced growth under uncertaintyimplies the real variables. Much of the short-run that consumption, investment, and output variabilityin output and investmentis asso- are cointegrated in the sense of Robert ciated with a shock that has a persistent Engle and Clive Granger (1987). In turn, effect on real interest rates. These results this means that a cointegrated vector au- suggest that models that rely solely on per- toregression(VAR) nests log-linearapprox- manent productivity or long-run neutral imations of all RBC models that generate nominalshocks are not capableof capturing long-run balanced growth. Our empirical importantfeatures of the postwar U.S. ex- analysis is based on such a cointegrated perience. VAR (or vector error-correctionmodel), The paper is organizedas follows.Section which is otherwise unrestrictedby prefer- I provides theoretical backgroundand re- ences or technology.Thus, our conclusions views recent work on real business cycles. can be interpretedas casting doubt on the Section II outlines the empiricalmodel and strong claims emergingfrom an entire class discusses identification.Sections III and IV of real-business-cyclemodels. present the empirical results. Our conclu- The empirical analysis is structured sions are presented in Section V. around three questions. First, what are the cointegration properties of postwar U.S. I. Growthand Fluctuations:Theoretical data, and are these properties consistent Background with the predictions of balanced growth? Second, how much of the cyclicalvariation To fix some ideas and notation, this sec- in the data can be attributedto innovations tion outlines a simple real-business-cycle in the common stochastic trends? Third, a model with permanent productivity shocks. naturalalternative to RBC models is one in The model is of the general class put for- which nominal variablesplay an important ward by Fynn Kydland and Edward Prescott role. Do innovationsassociated with nomi- (1982) and is detailed in King et al. (1988). nal variables explain important cyclical Output, Y, is produced via a constant- movementsin the real variables? returns-to-scale Cobb-Douglas production The empirical results provide robust an- function: swers to these questions. First, cointegra- tion tests and estimated cointegratingvec- wheK) ihYt =cAtal stokaNd e tors indicate that the data are consistent with the balanced-growthhypothesis. Sec- where Kt is the capital stock and Nt repre- VOL. 81 NO. 4 KING ETAL.: STOCHASTIC TRENDS 821 sents labor input. Total factor productivity, [I(O) or "stationary"]. In Engle and At, follows a logarithmicrandom walk: Granger's(1987) terminology,the two lin- early independent cointegratingvectors, a1 (2) log(At) = /LA +log(At_1) + et = (-11,0)' and a 2 = (- 1,0, 1) isolate sta- tionary linear combinations of Xt corre- where the innovations,{ft}, are independent spondingto the logarithmsof the balanced- and identicallydistributed with a mean of 0 growthgreat ratios. and a variance of o-2. The parameter LtA In this basic one-sector model and vari- represents the average rate of growth in ants of it, the precise dynamic adjustment productivity;et representsdeviations of ac- process to a permanent productivityshock tual growthfrom this average. depends on the details of preferences and Within the basic neoclassicalmodel with technology. For example, recent RBC re- deterministic trends, it is familiar (from search has studied alterationsin the invest- [1970]) that per capita con- ment technology(time-to-build, adjustment sumption, investment, and output all grow costs, and inventories),the productiontech- at the rate jLA /0 in steady state.1The com- nology (variable , labor mon deterministic trend implies that the indivisibilities,and employmentadjustment great ratios of investmentand consumption costs), preferences (nonseparabilities in to output are constantalong the steady-state leisure and durable consumption goods), growth path. When uncertainty is added, and serial correlation in the productivity realizations of et change the forecast of growth process. Two general properties trend productivityequally at all future dates: emerge from these investigations.First, the Et log(At+s) = Et1(At+s) + et. A positive productivityshock sets off transitionaldy- productivityshock raises the expected long- namics, as capital is accumulatedand the run growth path: there is a common economy moves toward a new steady state. stochastic trend in the logarithmsof con- During this transition,work effort and the sumption, investment, and output. The great ratios change temporarily. Second, stochastictrend is log(At)/O, and its growth there is a common stochastic trend in con- rate is (kA + et)/0, the analogue of the sumption, investment, and output arising deterministicmodel's common growth-rate from productivitygrowth.2 These two prop- restriction, LAk/0. With common stochastic erties motivate the econometrictheory and trends, the great ratios Ct / Yt and It / Yt empirical research described in the next become stationarystochastic processes. sections. These theoretical results have a natural In systemsthat incorporateboth real and interpretationin terms of cointegration.Let nominal variables, additional cointegrating Xt be a vector of the logarithmsof output, consumption,and investmentat date t, de- noted by yt, ct, and it. Each componentof 2As one exampleof how an extension of the basic Xt is integratedof order one [1(1),or loosely model preservesthe stochastic-trendimplication, con- speaking, "nonstationary"]because of the sider the time-to-buildinvestment technology of Kyd- random-walknature land and Prescott (1982). All of the stages of invest- of productivity;yet, the ment in their model inherit the common stochastic balanced growth implication of the theory trend. Similarconclusions hold for the other examples implies that the differencebetween any two in the text. There are two importantcategories of RBC elements of Xt is integrated of order zero models that need not displaya single commonstochas- tic trendwhen there are permanentproductivity shocks. Multisector models can have separate productivity trends in each sector, as in John Long and Plosser (1983). Models of stochasticendogenous growth such 'This follows directlyfrom the economy'scommod- as those constructedby King and Sergio Rebelo (1988) ity resource constraint (Y, = C, + I,), its investment generate a stochastictrend in the level of productivity technology (Kt+1 = [1- JKt + It, with 8 being the when shocks are stationary;with endogenousgrowth, rate of depreciation),and the fact that the economy's permanentchanges in taxes or in the level of exoge- allocationof time between work and leisure must be nous productivitylead to permanentchanges in the constantin steadystate. growthrates. 822 THE AMERICAN ECONOMIC REV7EW SEPTEMBER 1991 relations may plausiblyarise. Two are rele- serially uncorrelatedwith a mean of zero vant for our empiricalanalysis. The first is and covariancematrix ?. Equation(5) is a the money-demandrelation: reduced-formrelation and, except for pur- poses of forecasting, is of little inherent (3) mt - Pt= 3yYt -t3RRt + vt interest. What is of interest is the set of structuralrelations that leads to (5) and the where mt - Pt is the logarithm of real bal- primarypurposes of this section are to dis- ances, Rt is the nominal interest rate, and cuss (i) how the balanced-growthand other vt is the money-demanddisturbance. The cointegrationrestrictions outlined in the last second is the conventionalFisher relation: section restrict this set of structural rela- tions and (ii) how these restrictionscan be (4) Rt =rt + Et Apt+ exploited to draw inferences about struc- tural relationsfrom consistent estimatorsof where rt is the ex ante real rate of interest, C(L) and I. Pt is the logarithmof the price level, and To be specific,consider a structuralmodel Et Apt,, denotes the expected rate of in- of the form flationbetween t and t + 1. If real balances, output, and interest rates are I(1), while the (6) AXt = ,u + r(L)>t1 money-demanddisturbance in (3) is I(O), then real balances, output, and nominal in- where qt is a n X 1 vector of serially uncor- terest rates are cointegrated.If the real rate related structuraldisturbances with a mean is I(O)and the inflationrate is 1(1), then (4) of zero and a covariance matrix St. The implies that nominal interest rates and in- reduced form of (6) will be of the form (5) flation are cointegrated.The empiricalanal- with Et = rolt and C(L)=F(L)Fo-1*3 ysis investigates these cointegrating rela- The identification problem can now be tions and isolates the common stochastic stated as follows:under what conditionsis it trends that they imply. possible to deduce the structural distur- bances at and matrix of lag polynomials II. EconometricMethodology F(L) from the reduced-form errors Et and matrixof lag polynomialsC(L)? In the clas- This section provides an overviewof the sical literature on simultaneous-equation econometrictechniques used to answer the models, the identificationproblem is solved questions posed in the Introduction.The by postulating that certain blocks of r(L) first question, concerning the integration are zero, so that some of the X's are exoge- and cointegration properties of the data, nous or predetermined.In linear rational- can be addressedusing techniquesthat are expectations models, the identification now familiar.This section therefore focuses problem is solved by imposing cross-equa- on the specificationof an econometricmodel tion restrictionson the various elements of in which the trends and their impulse re- F(L), as described,for example in Kenneth sponse functionscan be identified and esti- Wallis (1980) and Lars Hansen and Thomas mated. Sargent(1980). The literatureon vector au- Let Xt denote an n Xl vector of time toregressive models addresses the identifi- series. The individualseries are assumedto cation problem by imposing restrictionson be I(1) (so that they must be differenced the covariancematrix I and the matrixof before they are stationary)and to have the structuralimpact multipliers,ro. For exam- Wold representation: ple, in his classic paper on vector autore- gressions,Christopher Sims (1980) assumes (5) AXt = ,u +C(L)Et that 1', is diagonal and that ro is lower where Et is the vector of one-step-ahead linear forecast errors in Xt given informa- 3Thisassumes that the structuraldisturbances lie in tion on lagged values of Xt. The Eis are the space spannedby currentand laggedvalues of Xt. VOL. 81 NO. 4 KING ETAL.: STOCHASTIC TRENDS 823 triangular,assumptions analogous to a Wold impulse responses given by the first column causal chain; and Watson of r(L) are the partialderivatives of AXt+k (1986) modify Sims's original procedureby with respect to .71. The second restriction imposing restrictions on Fo analogous to specifieswhat is being held constantin com- those appearing in static simultaneous- puting these partial derivatives. equation models. Another way to motivatethese identifying In this paper, identification is achieved restrictionsis to rewritethe model in terms through two sets of restrictions.First, the of the stationary variables Zt = (A yt, cointegrationrestrictions impose constraints Ct- yt, it - yt)'. The productivity shock, '7ql, on the matrix of long-run multipliers F(1) has a long-runeffect on yt but no long-run (E=?=0 F) in (6). This identifiesthe perma- effect on the stationaryratios, ct - yt and nent components. Second, the innovations t- y. Thus, '71 can be identified as the in the permanentcomponents are assumed innovation in the long-run component of to be uncorrelatedwith the innovationsto the first element of Zt. The other two dis- the remainingtransitory components. This turbances, Y7'2and 'q3, have temporaryef- identifies the dynamicresponse of the eco- fects on yt and the ratios. nomic variables to the permanent innova- Blanchardand Danny Quah (1989) used tions. A concise algebraic summaryof the a special case of this identificationscheme identification scheme is given in the Ap- to analyze Zt = (A yt, ut)', where ut was the pendix, with a more extensive discussionin unemploymentrate, which they assumed to King et al. (1987); here, we outline the be I(O). Their disturbances were 7' a major ideas and relate our procedure to "supply shock," and t, a "demand shock." other recent work. These shocks were restrictedto be uncorre- Consider the three-variablemodel with lated, and only the supply shock, 71, was t= (YV,cO, it)'. Because there are two coin- allowed to have a long-run effect on yt. tegratingvectors, there is only one perma- Thus, except for the fact that their systemis nent innovation,the balanced-growthinno- bivariateand ours is trivariate,the identify- vation '7q.This shock correspondsto (t in ing restrictionsare identical. Indeed, if we the neoclassical model of Section I. The eliminate one of the ratios, so that our other two shocks, '7r2 and '7q',have only model is bivariate, our identifying restric- transitoryeffects on Xt. Thus, the first iden- tions are formallyequivalent to those used tification restriction (the balanced-growth by Blanchardand Quah. cointegratingvectors) implies that the ma- This equivalencehighlights what we con- trix of long-runmultipliers is sider to be two practicaladvantages of the empiricalspecification employed here. First, 1 0 0 work in the tradition of (7) r(1)= 1 0 0 (1957) links consumptionto permanent in- 1 0 0J come. This suggests that the emphasis on real flow variables, rather than on unem- where the values of the coefficients in the ployment and output as in Blanchard and first column of F(1) are normalizedto 1 to Quah, arguably will result in better esti- fix the scale of 1'q.Equation (7) serves to mates of the trend components of output identify the balanced-growthshock as the and the parametersof the structuralmodel.4 common long-run component in Xt, since Second, our application is to multivariate the innovationin the long-runforecast of Xt systems rather than bivariatesystems. This is (1 1 1)'"q71= C(1)Et, which can be calcu- lated directly from the reduced form (5). The second restriction,that r71is uncorre- lated with 'tr and '7t3,is used to determine 4This notion, that consumption might provide a the dynamiceffect of 1'r on that to good measure of permanent income, has been recently Xt, is, exploited by John Cochrane and Argia Sbordone (1988), identify the first column of r(L). The rea- Andrew Harvey and James Stock (1988), Eugene Fama son this assumptionis needed is clear: the (1990), and Cochrane (1990). 824 THE AMERICAN ECONOMIC REVIEW SEPTEMBER 1991 has two advantages:first, more macroeco- matrixwith no unknownparameters (analo- nomic variables are used to estimate the gous to the vector of l's in the k = 1 model) commontrends, and second, by allowingfor and H is a k x k lower triangularmatrix. a wider range of shocks, a richer set of As will become clear in the next section, A alternativemodels is considered. can be chosen in a way that associateseach To introduce nominal shocks, the three- shock with a familiareconomic mechanism: variable real model is augmented by real the first disturbanceis interpretedas a bal- balances, nominal interst rates, and infla- anced-growthshock, the second is a long-run tion. The resulting six-variablemodel has neutral shock and the third is a three common stochastic trends, and this permanentreal-interest-rate shock. Finally, makes identification more complicated, the constrainedreduced form is estimated since the individualpermanent innovations as a VAR with error-correctionterms [i.e., must be sorted out. We use a version of a vector error-correctionmodel (VECM)]. Sims's(1980) procedurefor this purpose. The general identificationproblem can be III. EmpiricalResults describedas follows. Suppose that there are k commonstochastic trends driving the n x 1 A. The Data vector Xt. Partitionthe vector of structural disturbances it into two components, t = The data are quarterlyU.S. observations (ta 1 2"),a where 'vl contains the distur- on real aggregate national income account bances that have permanent effects on the flow variables, the money supply, inflation, components of Xt and where i2 contains and a short-term interest rate. The three disturbancesthat have only temporaryef- aggregate real flow variables are the loga- fects. (In our six-variablemodel, k = 3 and rithms of per capita real consumptionex- it is a 3 x 1 vector containingthe balanced- penditures(c), per capita gross private do- growth shock, a long-run neutral inflation mestic fixed investment (i), and per capita shock, and a real-interest-rateshock.) Parti- "private" gross national product (y), de- tion 7(1) conformablywith nt as (1) = fined as total gross national product less [A0], whereA is the n x k matrixof long-run real total government purchases of goods multipliers for il and 0 is an n X(n - k) and services. The measure of the money matrix of zeros correspondingto the long- supply used is M2 (currency, demand de- run multipliers for i2. The matrix of posits, and savings deposits, per capita in long-run multipliers is determined by the logarithms; m). The price level is measured conditionthat its columnsare orthogonalto by the implicit price deflator for our mea- the cointegratingvectors, and Ail repre- sure of private GNP (in logarithms;p) and sents the innovationsin the long-run com- the interest rate (R) is the three-month ponents of Xt. U.S. Treasury bill rate. Regressions were Identificationof the individualelements run over 1949:1-1988:4for statisticalproce- of il becomesmore complicated when there dures that involvedonly real flows. To avoid is more than one permanent innovation, observations that occurred during periods because the unique influence of each per- of price controls, the Korean War, and the manent component needs to be isolated. Treasury-Fed accord, the regressionswere Formally, while the cointegration restric- run over 1954:1-1988:4when money, inter- tions identify the permanent innovations est rates, or priceswere involved.Data prior Adl , they fail to identify ill because All = to 1949:1(respectively, 1954:1) were used as (APXP- li) for any nonsingular matrix P. initial observationsin regressionsthat con- The followingrestrictions are used to iden- tain lags.5 tify the model. First, as in the model with k = 1, we assume that il and i2 are uncor- related. Second, the permanentshocks, All data were obtained from Citibase. Using the lt, Citibasemnemonics for the series, the precise defini- are assumed to be mutually uncorrelated. tions of the variablesare GC82 (consumption),GIF82 Third, A is assumed to be lower triangular, (investment),and (GNP82- GGE82)(real privateout- which permitswriting A = AIH,where A is a put). The Citibase M2 series (FM2) was used for VOL. 81 NO. 4 KING ETAL.: STOCHASTIC TRENDS 825

Because the national product measure is A...... not the standardone, we have graphedthe logarithmof the real variables(y, c, i, and m - p) in Figure 1A. These plots show the + _y familiar growth and cyclical characteristics of the data. Output, consumption,and in- vestment display strong upwardtrends. In- vestment is the most volatile component, 0 followed by output and then consumption. Real balances (m - p) also display an up- ward trend. Figure 1B plots the logarithm IN rn-p of the consumption:outputratio (c - y) and

0 the logarithmof the investment:output ra- 51 54 57 60 63 66 69 72 75 78 81 84 87 90 tio (i - y). Over the postwar period, these yeor ratios displaythe stabilityreported by prior it is researchers; easy to view them as fluc- B. tuating around a constant mean. This sug- ...... gests that the growth evident in Figure 1 0 occurs in a manner that is "balanced"be- tween investmentand consumption. i-y B. Integration and Cointegration Properties of the Data

Univariate analysis of these six variables a0 indicates that the real flow variables, y, c, and i can be characterizedas 1(1) processes with positive drift, and that R, price infla- ? 51 54 57 60 63 66 69 72 75 78 81 84 87 90 tion (Ap), and nominalmoney growth(Am) yeo can be characterizedas 1(1) processes with- out drift. These results are consistent with FIGURE 1. A) LOGARITHMS OF PRIVATE OUTPUT the large literatureon the "unit-root"prop- (y), CONSUMPTION (C), INVESTMENT (i), AND REAL MONEY BALANCES (M -P); erties of U.S. macroeconomictime-series.6 B) LOGARITHMS OF THE CONSUMPTION:OUTPUT (C - Y) AND INVESTMENT:OUTPUT (i - y) RATIOS Note: To facilitate graphing,constants were added to the logarithmsof the variables.The horizontallines in part B are the means of the (constant-adjusted)vari- 1959:1-1988:4;the earlier M2 data were formed by ables. splicingthe M2 series reportedin Bankingand Mone- tary Statistics, 1941-1970 (Board of Governors of the Federal Reserve System,1976) to the Citibasedata in January 1959. The monthly data were averaged to The obtain the quarterlyobservations. The price deflator balanced-growthconditions also ap- was obtainedas the ratio of nominalprivate GNP (the pear to be consistentwith the data: we can differencebetween Citibaseseries GNP and GGE) to reject the presence of unit-rootcomponents real privateGNP (the differencebetween Citibasese- in the great ratios. Augmented Dickey- ries GNP82and GGE82).The interestrate is FYGM3. Fuller t statistics (-v,with five lags; see It is measuredas an annualpercentage (a typicalvalue is 10.0 percent). Price inflationwas also measuredas David Dickey and Wayne Fuller, 1979) test- an annual percentage[400ln(P,1/P,_1)]. Output, con- ing for a unit autoregressiveroot in c - y sumption,investment, and money are determinedon a and i - y have values of - 4.21 and - 3.99, per capita basis using total civilian noninstitutional respectively;both are significantat the 1- population(P16). percentlevel, that 6Because the techniquesand results are now famil- suggesting (c, y) and (i, y) iar, they are omitted here; interested readers are re- are cointegrated.The log of real balances, ferred to an earlier version of this paper (King et al., m - p, appears to be an 1(1) process with 1987)for details. drift, even though both m and p can be 826 THE AMERICAN ECONOMIC REVIEW SEPTEMBER 1991

TABLE 1-COINTEGRATION STATISTICS: THREE-VARIABLE MODEL, (y, c, i), 1949:1-1988:4

A. Results from UnrestrictedLevels VectorAutoregressions: Largest Eigenvalues of Estimated Companion Matrix VAR(6) with constant VAR(6) with constant and trend Real Imaginary Modulus Real Imaginary Modulus

1.00 0.00 1.00 0.97 0.00 0.97 0.83 0.19 0.85 0.83 0.18 0.85 0.83 -0.19 0.85 0.83 -0.18 0.85 - 0.62 0.46 0.77 - 0.62 0.46 0.77 -0.62 - 0.46 0.77 - 0.62 - 0.46 0.77 0.50 0.50 0.71 0.49 0.49 0.69 Log likelihood: 2,199.66 2,200.95 B. Multivariate Unit-Root Statistics Statistic Value P value Null/alternative

JT(0) 35.4 0.13 3 unit roots/at most 2 unit roots qf(3, 2) - 29.4 0.21 3 unit roots/at most 2 unit roots qf(3, 1) - 29.4 < 0.01 3 unit roots/at most 1 unit root

Log likelihood (3 unit roots): 2,183.27 Log likelihood (2 unit roots): 2,193.11 Log likelihood (1 unit root): 2,199.43 Log likelihood (0 unit roots): 2,200.95 C. Estimated Cointegrating Vectors Null hypothesis Estimates

Variable a1 a2 a, 62

C 1 0 1.OOa o.ooa i 0 1 o.ooa 1.ooa y -1 -1 -1.058 -1.004 (0.026) (0.038) V Wald test of balanced growth restrictions: 2 = 4.96 (P = 0.08)

Notes: Values in parentheses are P values (for the test statistics) or standard errors (for the estimators). The roots and likelihoods in part A correspond to an unrestricted VAR(6) in levels. The multivariate unit-root tests in part B, which are described in detail in footnote 7, were computed from VAR(6)'s in levels. J(r) is Johansen's (1988) test of the null of r cointegrating vectors against more than r cointegrating vectors, and qf(k, m) is Stock and Watson's (1988) test of the null of k unit roots in the multivariate system against the alternative of m (m < k) unit roots, where "T" denotes linear detrending. The log likelihoods in part B are for models that include a constant and linear time trend. Part C reports the cointegrating vectors for (c, i, y), estimated by Stock and Watson's (1989) dynamic OLS (with a constant, five leads, and five lags) procedure. The t statistics formed using the standard errors have asymptotic normal distributions. The Wald statistic, which tests whether the cointegrating vectors lie in the hypothesized subspace, is computed using the dynamic OLS estimates and standard errors described in footnote 8. a Normalized. characterized as I(2) processes. (The Tr stationary behavior, but the augmented statistic for A(m - p) is significant at the Dickey-Fullert statistic (' , with five lags) 1-percent level). The best characterization is only - 1.8, which is consistentwith a unit of the real interest rate is unclear. The root in the real rate. ex post real rate, R - Ap, has a sample Tables 1-3 present a variety of statistics AR(1) coefficient equal to 0.86, suggesting calculated from multivariatesystems, start- VOL. 81 NO. 4 KING ETAL.: STOCHASTIC TRENDS 827 ing with the results for the three-variable growth in output, consumption,and invest- (y,c,i) model. Panel A of Table 1 shows ment. These balanced-growthrestrictions the largest eigenvaluesfrom the companion impose two constraintson the cointegrating matrixof an estimated VAR(6). The model vectors. In Table 1, these restrictions are with one common stochastic trend (bal- tested using a Wald statistic based on the anced growth) implies that the companion dynamicOLS point estimates and standard matrixshould have one unit eigenvalue,cor- errors;under the null hypothesis,this statis- responding to the common trend, and all tic has an asymptoticchi-square distribution the other eigenvaluesshould be less than 1 with two degrees of freedom. Although the in modulus. The point estimates are in ac- restrictionis rejected at the 10-percent(but cord with this prediction.Panel B presents not the 5-percent)level, the estimatedcoin- formal tests for cointegrationusing proce- tegrating vector is broadly consistent with dures developed by Soren Johansen (1988) the balanced-growthprediction.8 and Stock and Watson (1988). Both proce- Table 2 explorestwo sets of cointegrating dures take as their null hypothesisthat the relationssuggested by the nonstationarityof data are integratedbut not cointegrated,so the nominal and real interest rates. Table that there are three unit roots in the com- 2A reports an estimated cointegratingrela- panion matrix.The first two rows of panel B tion between real balances, output, and test this against an alternative of at most nominalinterest rates.9The estimate of the two unit roots, while the third row tests the long-run income elasticity is close to unity null against the sharper alternative of at (although statistically significantly larger most one unit root. The results are consis- than 1); the estimated interest rate tent with the one-unit-root (one common semielasticityis small but statisticallysignif- trend) specification.7 icantlyless than zero. The final panel in Table 1 presents maxi- mum-likelihoodestimates of the cointegrat- ing vectors, conditionalon the presence of one unit root in the VAR, computed using 8The cointegratingvectors reportedin Tables 1-3, the dynamic ordinary least-squares (OLS) and the estimated cointegratingvectors used as the procedureof Stock and Watson (1989). The basis of the VAR analysis in Tables 4-6, were esti- point estimates are close to (1,0, - 1) and mated using the Stock and Watson (1989) dynamic (0, 1, -1), the values that imply balanced OLS procedure,which is asymptoticallyequivalent to the Gaussianmaximum-likelihood procedure for a tri- angularerror-correction system. If there are r cointe- gratingvectors, then there are r regressionequations; 7The multivariateunit-root tests in Tables 1-3 are each equationhas n - r regressorsin levels (wheren is based on the J,L(r), J(r), q,(k, m), and q'(k, m) statis- the numberof variables),a constant,and m leads, m tics. The A(r) statistics are Johansen's (1988) test of the lags, and the contemporaneousvalues of the differ- null of r cointegratingvectors against more than r ences of right-hand-sidelevels variables.Standard er- cointegratingvectors, and the qf(k, m) statistics are rors, calculatedusing a VAR(4) estimatorof the spec- Stock and Watson's(1988) test of the null of k unit tral density matrixof the errors in these r equations, roots in the multivariatesystem against the alternative are given in parenthesesin Tables 1-3. All results are of m (m < k) unit roots; ",u" and "TX" subscripts de- based on m = 5; to allow for the leads, the dynamic note the tests computedusing demeaned data and data OLS regressionsend at 1987:3(all other regressionsgo that have been both demeanedand linearlydetrended, through 1988:4).Wald statistics computed using the respectively.(Detrending is appropriateif the series estimatedcovariance matrix have the usual large-sam- have a nonzerodrift under the null.) Asymptoticcriti- ple chi-squaredistributions. cal values for qf and qf are taken from Stock and The log likelihoodsprovided in Table 1 and subse- Watson (1988). Asymptotic p values for J. and J, quent tables are providedas a guide for readersinter- differfrom those tabulatedin Johansen(1988) because ested in exploringthe shape of the likelihoodsurface. of the demeaning/detrending.It is straightforwardto It shouldbe stressedthat, becauseof the hypothesized derive and to computethe asymptoticnull distribution unit roots, the usual chi-square inference does not of J,. and J, using the results in Sims et al. (1990). We always apply to likelihood-ratiostatistics computed have done this, and the p values shown in the tables from the reportedvalues. are based on these asymptoticdistributions. All multi- 9See Robert Lucas (1988), Dennis Hoffman and variate tests were computed using six lagged levels, Robert Rasche (1989), and BenjaminFriedman and which these proceduresparameterize as five lags in KennethKuttner (1990) for recent empiricalinvestiga- first differencesand a single lagged level. tions of the stabilityof long-runmoney demand. 828 THE AMERICAN ECONOMIC REVIEW SEPTEMBER 1991

TABLE2-ESTIMATED COINTEGRATING VECTORS

A. Money Demand, 1954:1-1988:4

(i) m -p = 1.197 y -0.013 R q'(3, 2) =-20.6 JT(0)= 42.6 (0.062) (0.004) (0.54) (0.03)

Wald test of velocity restriction (f3y = 1 and JPR = 0): X12]= 12.7 (P<0.01)

B. Real Ratios and Real Interest, 1954:1-1988:4 (ii) c - y = 0.0033 (R - Ap) q'(2, 1) = -70.0 J(0) = 19.3 (0.0022) (< 0.01) (0.17) q,f(2,1) = -62.2 JIU (0) = 15.6 (< 0.01) (0.11)

(iii) i y = 0.0028 (R -Ap) qf(2, 1) = 73.8 JT(O)= 26.9 (0.0050) (< 0.01) (0.02) qAf(2,1) =-65.1 J, (0) = 24.8 (< 0.01) (0.01) Notes: Values in parentheses are P values (for the test statistics) or standard errors (for the estimators). The cointegrating vectors (i)-(iii) were estimated by dynamic OLS (with five leads and five lags), including a constant in the regression, equation-by-equa- tion. The qf and J tests are computed using demeaned data (see footnote 7). The Wald statistic, which tests whether the cointegrating vectors lie in the hypothesized subspace, is computed using the dynamic OLS estimates and standard errors described in footnote 8.

The second issue examined in Table 2 is long-run money-demand relation. The the possibilitythat the consumption:output qf(6,3) statistic reported at the bottom of and investment:outputratios might exhibit Table 3A provides some evidence for the permanentshifts resultingfrom permanent three-trend specification, rejecting six unit shifts in real rates. Estimatedbivariate coin- roots againstthree with a P value of 0.11. tegrating relations (c - y) = 0 (R - Ap) and The Wald tests in Table 3B investigate (i- y) = 02(R- Ap) are shown in Table various hypotheses about the cointegrating 2B. As predicted by the long-runtheory of vectors, under the maintained hypothesis the growth model, for example, a higher that the number of cointegrating vectors real interest rate lowers the share of prod- is correctly specified. The first hypothesis uct going into investment and, symmetri- (Table 3B, row 1) is that the cointegrating cally, raises the share of consumption.How- vectors are the balanced-growth and ever, the long-run effects are imprecisely money-demandcointegrating vectors. This estimated and small: a permanentincrease hypothesisis rejectedat the 10-percentlevel in the annual real rate of one percentage but not at the 5-percent level. Despite this point is associated with an increase in the formal rejection at the 10-percentlevel, we consumption:outputratio of 0.3 percentage interpret the balanced-growth/money- points. demandcointegrating restrictions as provid- The cointegrationproperties of the six- ing a good qualitative description of the variable system (y, c, i, m - p, R, Ap) are in- cointegratingvectors for the system. The vestigatedin Table 3. The theoreticalanaly- remaininglines of Table 3B investigateal- sis suggests three stochastic trends in the ternativecointegration restrictions. There is system: a balanced-growthtrend, an infla- strong evidence against a fourth cointegrat- tion/money-growth trend and, possibly, a ing vector implyingstationary real rates (line real-interest-ratestochastic trend. Equiva- 2) and againstthe stationaryvelocity model, lently, three cointegratingrelations should even permittingcointegration between the be present in the system:two (interest-rate- great ratios and real rates (line 4). The adjusted) balanced-growthrelations and a evidence is weakest against the hypothesis VOL.81 NO.4 KING ETAL.: STOCHASTIC TRENDS 829

TABLE 3-COINTEGRATION STATISTICS:SIX-VARIABLE MODEL (y, c, i, m - p, R, Ap), 1954:1-1988:4

A. Estimated Cointegrating Vectors Variable a1 a2 a3

C 1.00a o.ooa o.ooa

i 0oooa 1.ooa o.ooa

m-p 0.00a 0.ooa 1.ooa y -1.118 -1.120 -1.152 (0.050) (0.083) (0.063) R 0.004 0.002 0.009 (0.003) (0.005) (0.004) Ap 0.004 0.006 0.002 (0.003) (0.004) (0.003)

qf(6,3)= -27.5 (P value = 0.11) Log likelihood = 2,826.54 B. Tests of Restrictions on Cointegrating Vectors Null hypothesis d.f. Waldtest

(c - y), (i - y), m -p - 3yy +83RR 7 12.6(0.08) (c - y), (i - y), m - p - ,lyY +PORR, R - Ap 6 40.1 ( < 0.01) (c - y)-,41(R - Ap), (i - y)- 42(R - Ap), 5 7.6 (0.18) m-p-0 + PRR (c - y)-4 1R - Ap), (i- y)-4'2(R - Ap), m - p- y 7 42.0 ( < 0.01) Notes: Values in parenthesesare P values (for the test statistics)or standarderrors (for the estimators).Part A reports dynamicOLS estimates for the three-equation system.Part B reportstests of whether the cointegratingvectors fall in the hypothe- sized subspace,conditional on the numberof cointegratingvectors. The Wald statis- tics, whichtest whetherthe cointegratingvectors lie in the hypothesizedsubspace, are computedusing the dynamicOLS estimatesand standarderrors described in footnote 8. a Normalized. that the great ratios and real rates are coin- time-series.There is some evidence that the tegrated,combined with the money-demand sharesof consumptionand investmentmove cointegratingvector (line 3).10 with permanentshifts in the real rate. Yet, Taken together, these results suggest that this effect is negligiblysmall in the long run, a money-demandcointegrating relation is and the hypothesis of "balanced growth" consistentwith the observedbehavior of the also appearsto be generallyconsistent with the data.

10To check robustness, the cointegrating vectors in C. A Three-VariableSystem of Real Tables 1, 2, 3, and 6 were also estimated using Jo- hansen's (1988) maximum-likelihood estimate (MLE) Flow Variables for a vector error-correction model with five lagged differences, one lagged level, and a constant. The The results for the three-variablesystem Johansen MLE point estimates (available from the are based on an estimated VECM using authors upon request) are similar to the dynamic OLS eight lags of the first differencesof y, c, and point estimates. For example, the Johansen MLE of the money-demand equation is m - p = 1.134y - i with an intercept and the two theoretical 0.0093R [cf. model (i) in Table 2]. error-correction terms, c - y and i - y. The 830 THE AMERICAN ECONOMIC REVIEW SEPTEMBER 1991

A. o i . .,, -,,,,,,,,,,,,,,,,,,, TABLE 4-FORECAST-ERROR VARIANCE Cn DECOMPOSITIONS: THREE-VARIABLE- REAL MODEL (y, C, i), O _ - - - - 1949:2-1988:4 OFraction______of the forecast-error variance o attributed to the real permanent shock o Horizon y c 1 3 5 7 9 11 13 15 17 19 21 23 25 log 1 0.45 0.88 0.12 (0.28) (0.21) (0.18) B0 - 4 0.58 0.89 0.31 (0.27) (0.19) (0.23)

- 8 0.68 0.83 0.40 ----______(0.22) (0.18) (0.18)

O.135791113151719212325 12 0.73 0.83 0.43 (0.19) (0.18) (0.17) 1 3 5 7 9 1 1 1 315 17 1 9 21 23 25 aog 16 0.77 0.85 0.44 (0.17) (0.16) (0.16)

Cn 20 0.79 0.87 0.46

to t 1(0.16)24 (0.15) (0.16) - -/- -______0.81 0.89 0.47 O __ (0.16) (0.13) (0.16)

o 00 1.00 1.00 1.00 I 1 3 5 7 9 1 1 13 15 17 19 21 23 25 bg Note: Based on an estimated vector error-correction model of X, = (yt, ct, it) with eight lags of AX1, one lag FIGURE 2. RESPONSES IN A THREE-VARIABLE each of the error-correction terms c - y and i - y, and MODEL TO A ONE-STANDARD-DEVIATION SHOCK a constant. Approximate standard errors, shown in IN THE REAL PERMANENT COMPONENT: parentheses, were computed by Monte Carlo simula- A) RESPONSE OF LOG OUTPUT; B) RESPONSE OF LOG tion using 500 replications. CONSUMPTION; C) RESPONSE OF LOG INVESTMENT

only identifying assumption needed to ana- innovation eventually leads to a one-unit lyze the dynamics of the system is that the increase in y, c, and i.) In response to a permanent shock is uncorrelated with the shock that leads to a 1-percent permanent transitory shocks. The impulse response of increase in y, c, and i, output and invest- y, c, and i to an innovation of one standard ment increase by more than 1 percent in the deviation in the common trend is plotted in near term (1-2 years), while consumption Figure 2, together with one-standard- moves less. Most of the adjustment is com- deviation confidence bands." (The standard pleted within four years. The results for c deviation of the balanced-growth shock is and i are consistent with the simple theoret- 0.7 percent, and as discussed in Section II, ical model discussed in Section I, where the the system is normalized so that a one-unit capital stock rapidly increases at the short- run cost of consumption. Are these responses large enough to ex- plain a substantial fraction of the short-run 1"The standard errors for the impulse response functions and variance decompositions were approxi- variation in the data? This key question mated using 500 simulations as discussed by Thomas is addressed in Table 4, which shows the Doan and Robert Litterman (1986 p. 19-4). fraction of the forecast-error variance at- VOL. 81 NO. 4 KING ETAL.: STOCHASTIC TRENDS 831 tributed to innovations in the common link variation in the real ratios to perma- stochastic trend, at horizons of 1-24 quar- nent shifts in the real interest rate, although ters. These variance decompositions suggest the estimates reported above suggest that that innovations in the permanent compo- this effect is small. The third implies that nent appear to play a dominant role in the "money-demand" disturbances are I(0). The variation of GNP and consumption. At the estimates of p1, 02, fy, and fR are the 1-4 quarter horizon, the point estimates restricted dynamic OLS estimates given in suggest that 45-58 percent of the fluctua- Table 2. tions in private GNP can be attributed to The permanent components and their im- the permanent component. This increases pulse responses are identified by specifying to 68 percent at the two-year horizon and to a structure for the matrix of long-run multi- 81 percent at the six-year horizon. The re- pliers. In the notation of Section II, with sults for consumption are broadly similar. Xt = (y, c, i, m - p, R, p)', the particular Notably, the permanent component ex- structure adopted is: plains a much smaller fraction of the move- ments in investment: only 31 percent at the (8) one-year horizon, increasing to 47 percent at the six-year horizon. 1 0 0 This evidence suggests the existence of a 1 0 1 0 0 persistent, potentially permanent, compo- 1 0 02 I nent that shifts the composition of real out- A= 07r21 j 1 01 put between consumption and investment. 1? 13R 13R [w31 732 1 (If there were temporary components with 0 1 1 negligible effect on forecast errors after 0 1 0 three or more years, then the population counterparts of the variance ratios in Table The 6 x 3 matrix A is the matrix of long-run 4 would increase more sharply at the longer multipliers from the three permanent horizons.) Thus, the results motivate us to shocks. In the notation of Section II, the investigate the possibility of additional per- two matrices on the right-hand side of (8) manent components. are A and H[, respectively. Our interpretation of the shocks follows D. Six-VariableSystems with Nominal from the structure placed on the long-run Variables multipliers in (8). The first shock is a real- balanced-growth shock, since it leads to a Augmenting output, consumption, and in- unit long-run increase in y, c, and i. vestment by real balances, nominal interest Through the money-demand relation, it also rates, and inflation yields a six-variable sys- leads to a By increase in real balances. The tem. The results of Subsection Ill-B suggest second shock is a neutral inflation shock. It that a reasonable specification incorporates has no long-run effect on y, c, or i and has three cointegrating relations (and thus three a unit long-run effect on inflation and nomi- common trends) among the six variables. nal interest rates. Further, the unit increase We have estimated a variety of six-variable in nominal interest rates arising from this models, with different numbers of trends, shock leads to reduction of real balances of different cointegrating relations, and dif- ,R. The final permanent shock is a real- ferent orderings of the shocks. The detailed interest-rate shock. A one-unit increase in results for a benchmark model are reported this shock leads to a change of f1 and O2 in in this subsection, and the results for the c - y and i - y. There is also a one-unit other models are summarized in the next increase in nominal interest rates and a subsection. The benchmark model incorpo- decrease of fR in real balances. The coef- rates the cointegrating relations (c - y) = ficients in H are determined by the require- l(R - A p), (i - y) = 02(R - A p), and ment that the permanent innovations are - = m p Byyy- JRR. The first two relations mutually uncorrelated. In the standard VAR 832 THE AMERICAN ECONOMIC REVIEW SEPTEMBER 1991

TABLE 5-FORECAST-ERROR VARIANCE DECOMPOSITIONS: SIX-VARIABLE MODEL (8), 1954:1-1988:4

A. Fraction of the forecast-error variance attributed to balanced-growthshock Horizon y c i mr-p R Ap

1 0.00 0.02 0.11 0.79 0.14 0.30 (0.13) (0.09) (0.16) (0.23) (0.19) (0.13) 4 0.05 0.15 0.06 0.76 0.11 0.22 (0.14) (0.13) (0.11) (0.23) (0.19) (0.08) 8 0.22 0.31 0.14 0.70 0.11 0.20 (0.13) (0.18) (0.11) (0.24) (0.20) (0.07) 12 0.44 0.48 0.27 0.72 0.11 0.17 (0.14) (0.21) (0.16) (0.25) (0.20) (0.06) 16 0.54 0.59 0.32 0.74 0.11 0.16 (0.15) (0.21) (0.17) (0.24) (0.20) (0.07) 20 0.59 0.63 0.33 0.75 0.12 0.15 (0.15) (0.19) (0.17) (0.22) (0.21) (0.07) 24 0.62 0.65 0.33 0.77 0.14 0.14 (0.14) (0.17) (0.16) (0.22) (0.22) (0.08) 00 1.00 0.92 0.97 0.78 0.23 0.04

B. Fraction of the forecast-error variance attributable to inflation shock Horizon y c i mr-p R Ap

1 0.00 0.02 0.08 0.01 0.03 0.43 (0.12) (0.11) (0.16) (0.13) (0.14) (0.19) 4 0.04 0.01 0.23 0.04 0.04 0.37 (0.14) (0.10) (0.19) (0.14) (0.15) (0.13) 8 0.04 0.01 0.20 0.01 0.02 0.36 (0.12) (0.11) (0.15) (0.14) (0.16) (0.12) 12 0.03 0.01 0.12 0.01 0.02 0.45 (0.10) (0.12) (0.12) (0.14) (0.15) (0.11) 16 0.02 0.01 0.10 0.01 0.03 0.49 (0.10) (0.12) (0.11) (0.13) (0.15) (0.11) 20 0.02 0.02 0.10 0.01 0.03 0.53 (0.10) (0.12) (0.11) (0.13) (0.15) (0.12) 24 0.02 0.02 0.09 0.01 0.02 0.55 (0.10) (0.11) (0.11) (0.13) (0.14) (0.13) 00 0.00 0.02 0.00 0.00 0.01 0.96 C. Fraction of the forecast-error variance attributable to real-interest-rateshock Horizon y c i mr-p R AP 1 0.67 0.35 0.44 0.03 0.63 0.00 (0.19) (0.22) (0.20) (0.13) (0.21) (0.11) 4 0.74 0.24 0.50 0.07 0.72 0.10 (0.20) (0.18) (0.20) (0.14) (0.21) (0.08) 8 0.55 0.12 0.37 0.20 0.77 0.16 (0.16) (0.10) (0.14) (0.17) (0.22) (0.09) 12 0.39 0.11 0.36 0.21 0.78 0.14 (0.11) (0.09) (0.12) (0.18) (0.22) (0.08) 16 0.32 0.09 0.37 0.20 0.78 0.13 (0.10) (0.09) (0.12) (0.17) (0.22) (0.07) 20 0.28 0.09 0.34 0.18 0.78 0.12 (0.09) (0.08) (0.12) (0.16) (0.22) (0.07) 24 0.25 0.10 0.34 0.16 0.77 0.11 (0.08) (0.08) (0.11) (0.15) (0.22) (0.07) 00 0.00 0.06 0.03 0.21 0.77 0.00

Notes: Based on an estimated vector error-correction model of X =(y, c, i, m - p, R, Ap) with eight lags of AXI, one lag each of the error-correction terms c - y - 01(R - Ap), i - y - 02(R - Ap), and m - P - I3yy + ORR and a constant. Approximate standard errors, shown in parentheses, were computed by Monte Carlo simulation using 500 replications. VOL. 81 NO. 4 KING ETAL.: STOCHASTIC TRENDS 833

0)~~~~~~~~~~~~~~( ...... S O , , , , , , , , , , ...... lTlll

Z o eDNI A- A AA 0 ; A

1 58 62 66 70 74 78 82 86 90 1 58 62 66 70 74 78 82 86 90 58 62 66 70 74 78 82 86 90 1o -4 ^ 1o0A C A-1? 2 tsA

I J 1 / U?6< ?I __o__ _ _ o o ______0

0)0 'IO C1o

1 58 62 66 70 74 78 82 86 90 1 58 62 66 70 74 78 82 86 90 58 62 66 70 74 78 82 86 90

4)0 ~~~~~00

I~.______Cause Ivsmn 0 'A~~~~~~A 0~ 6

yeor ~~~~~~~year year ou~tput Consumption Investment

FIGURE 3. HISTORICAL FORECAST-ERROR DECOMPOSITION IN THE SIX-VARIABLE MODEL

Note: The forecasterrors are shownas percentages,on a decimalbasis.

terminology,following Sims (1980), the bal- nent associatedwith permanentshifts in the anced-growthdisturbance is ordered first, rate of inflation explains a considerable the inflationdisturbance is second, and the amount of the variation in inflation, but real-rateshock is third. little else. Fourth, the component associ- The model is estimated using a VECM ated with permanentmovements in the real with eight lags and the three error-correc- interest rate explains most of the forecast tion terms impliedby the cointegratingrela- errors in the nominal rate. It also is impor- tions. Table 5 presents the variancedecom- tant for real activity:it explains substantial positions of the forecast errors from the amountsof the output and investmentfore- benchmarkmodel. Four aspects of this table cast errors, particularlyat the short hori- are of particularinterest. First, relative to zons. the three-variablemodel, the real or "bal- Figure3 illustratesthe roles playedby the anced-growth"shock is less importantfor different shocks by plotting the forecast er- output and consumption, especially at the ror at the three-yearhorizon and the por- 1-4 quarterhorizon. At the 3-5 year hori- tion attributableto each stochastictrend for zon, however, this shock has importantex- y, c, and i. These plots highlightthe negligi- planatorypower: roughly one-half the varia- ble role of the inflation shock and the sub- tion in these forecast errors is attributable stantial role played by the balanced-growth to the first permanentcomponent. Second, shock and the real-interest-rate shock. including the additional shocks in this ex- Looking at specific episodes in this figure, panded model does not enable the first per- one finds that the balanced-growthshock manent componentto explain the short-run has particular explanatory power for the variationsin investment.Third, the compo- sustainedgrowth of the 1960's.On the other 834 THE AMERICAN ECONOMIC REVIEW SEPTEMBER 1991

C) '-_

CO -0 M 0 3 6 9 12 15 18 21 24 27 ' 3 6 9 12 15 18 21 24 27 ' 3 6 9 12 15 18 21 24 27

o 6-~~~~~~~~~~~~~~~~~~~c

C~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~C

o IIIIIIIIIIII~~II~IIIIIII *IIIIIIIIIIIIIIII~~IIIIII, I I, I II ; 3 6 9 12 15 18 21 24 27 ' 3 6 9 12 15 18 21 24 27 1 3 6 9 12 15 18 21 24 27

T 0 3 6 9 12 15 18 21 24 27 ' 3 6 9 12 15 18 21 24 27 0 3 6 9 12 15 18 21 24 27 lag lag lag Output Consuirnption Investment

FIGURE 4. RESPONSEIN THE SIX-VARIABLE MODEL TO A ONE-STANDARD-DEVIATIONSHOCK IN THE PERMANENTCOMPONENTS

hand, the real-rate shock seems particularly impact on output and consumption. Invest- important in the contraction of 1974 and ment, on the other hand, shows a large the 1981-1982 recession. positive response to this shock for the first Figure 4 shows the responses of the vari- three quarters. ables to one-standard-deviation impulses in We have already noted that the real- the balanced-growth shock, the inflation interest-rate shock plays an important role shock, and the real-interest-rate shock. The in explaining the short-run behavior of out- estimated standard deviations of these un- put and investment. The impulse response derlying shocks are, respectively, 0.7 per- functions make interpreting this shock as a cent, 0.08 percent, and 0.12 percent per permanent change in the real rate of inter- quarter. The response of output to the bal- est somewhat difficult. All three of the real anced-growth shock is negligible over the flow variables have an initial response to first few quarters, while consumption in- this shock that is strongly positive, before creases slightly and investment declines. Af- turning negative after 2-3 quarters. While ter one year, major increases in output, there may be economic models that predict consumption, and investment are present. such responses to a permanent change in While these responses are smaller than the real rate, standard ones do not. those in the three-variable model, they con- We draw four conclusions from this anal- form to how one might think a system would ysis. First, permanent innovations account respond to news about technological devel- for a substantial fraction of transitory eco- opments. The inflation shock has very little nomic fluctuations. Second, the balanced- VOL. 81 NO. 4 KING ETAL.: STOCHASTIC TRENDS 835

TABLE 6-THREE-YEAR-AHEAD FORECAST-ERROR VARIANCE DECOMPOSITIONS: SUMMARY OF RESULTS OF VARIOUS MODELS

Fraction of forecast-error Tests of restrictions variance attributed to on cointegrating vectors the permanent real shock Model Estimate period d.f. Wald test (P) Log likelihood y c i m-p R Ap

R.1 1949:2-1988:4 2 4.96 (0.08) 2,196.67 0.73 0.83 0.43 - - M.1 1954:1-1988:4 5 7.60 (0.18) 2,816.06 0.44 0.48 0.27 0.72 0.11 0.17 M.2 1954:1-1988:4 7 12.60 (0.08) 2,814.64 0.42 0.52 0.25 0.68 0.07 0.16 M.3 1954:1-1988:4 same as M.1 0.35 0.30 0.12 0.26 0.02 0.16 M.4 1954:1-1988:4 6 40.10 ( < 0.01) 2,820.48 0.37 0.40 0.15 0.56 0.01 0.18 M.5 1954:1-1988:4 4 3.04 (0.55) 2,812.52 0.42 0.47 0.23 0.64 0.06 - M.6 1954:1-1988:4 same as M.5 0.42 0.36 0.19 0.46 0.01

Model Description Model R.1: Three-variable (y, c, i) model with cointegrating relations c - y and i - y Model M.1: Six-variable (y, c, i, m - p, R, Ap) baseline model of Table 5 Model M.2: Identical to M.1, except that the coefficients 01 and 02 are set to zero in the cointegrating vectors and the A matrix (i.e., cointegration of shares and the real interest rate is dropped) Model M.3: Identical to M.1, except that the stochastic trend innovations are reordered to place the inflation shock first, the real-interest rate shock second, and the balanced-growth trend third Model M.4: A two-stochastic-trend model for (y, c, i, m - p, R, Ap), obtained by assuming that the real interest rate is stationary; the cointegrating relations are c - y, i - y, (m - p)- fJ3Yy+ I3RRand R - Ap, and A = [A1 A2], where A1 = (1 11 3y 0 0)' (balanced-growth shock) and A2 = (?00 ?PR 1 1)' (neutral inflation shock) Model M.5: A five-variable system (y, c, i, m - p, R) with cointegrating relations c - y, i - y, and (m - p)- 18yy + pRR, and A = [A3 A4], where A3 = (1 11 y 0)' (balanced-growth shock) and A4 = (0 0 0 - PR l' (neutral interest-rate shock) Model M.6: Identical to M.5, except that the ordering of stochastic trend innovations is reversed, so A = [A4 A3]

Notes: The estimation period denotes the sample used to estimate the VECM, with earlier data used for initial conditions for the lags. The Wald statistics, which test the hypothesis that the true cointegrating subspace is spanned by the hypothesized cointegrating vectors or, equivalently, that it is orthogonal to the A matrix, are computed using the dynamic OLS estimates and standard errors described in footnote 8.

growth factor retains a significant role in E. SensitivityAnalysis explaining movements at horizons greater than two years, although it has considerably It is important to explore the sensitivity of less explanatory power in the six-variable these main conclusions to changes in cointe- system than in the three-variable system. grating vectors and changes in the ordering Third, a large fraction of the short-run (0-2 of the permanent components: we do this year) variability in output and investment is by estimating a variety of five- and six-vari- explained by a factor that is related to per- able models. To save space, we focus on a sistent movements in the real rate of inter- key measure, the fraction of the variance of est. Fourth, the impulse response functions the three-year-ahead forecast error in each appear to be consistent with the interpreta- variable explained by the balanced-growth tion of the first shock as a real or balanced- permanent innovation. The results, summa- growth shock, but lead us to be uncertain rized in Table 6, suggest four conclusions. about the interpretation of the third, real- First, looking across specifications, substan- rate shock, at least within the context of tial fractions of the forecast errors in output standard macroeconomic models. and consumption are explained by the bal- 836 THE AMERICAN ECONOMIC REVIEW SEPTEMBER 1991 anced-growth innovation; the point esti- A. , mates range from one-third to two-thirds. Second, in systems includingnominal vari- ables, the fractionof the forecast-errorvari- ance of investment explained by the bal- Vt - anced-growthreal permanentcomponent is -.-l Itll 'd stt o. never large (at most 27 percent in model M.1). Third, little changes when bal- anced growth is imposed by setting 01 2 Iy and O2 equal to zero. Fourth, changingthe ordering of the shocks (e.g., putting the balanced-growth shock last in the Wold causal ordering, as in model M.3) B. tn- W | ? W\ . , , does not change the main qualitative fea- tures of the results. In short, the sensitivity analysis indicates that the principal results for the base six-variablemodel are robustto a wide variety of changes in the identifying restrictions.'2

I50 54 58 62 66 70 74 78 82 86 90 IV. Analysisof TrendComponents of yeor Private GNP FIGURE 5. BALANCED-GROWTH SHOCK FROM THE In the neoclassical growth frameworkof SIX-VARIABLE BENCHMARK MODEL (SOLID LINE) Section I, the commonlong-run movements AND A) HALL'S SOLOw RESIDUAL OR B) PRESCOTr'S SOLOw RESIDUAL (DASHED LINES) in aggregatevariables arise from changes in productivity.Is there any evidence that pro- ductivitymovements are related to innova- tions in the balanced-growthcomponent of GNP? We investigate this by comparing The time path of the Solow residual and these estimated innovations to a popular the change in the balanced-growth-trend estimate of the change in total factor pro- component of private GNP from the six- ductivity in the economy, Solow's (1957) variablemodel are plotted in Figure 5A for residual. If the economy can be described Hall's measure and in Figure 5B for by a Cobb-Douglasproduction function-as Prescott's measure. The graphs suggest a in the theoretical model of Section I-the very modest relation between Hall's Solow Solow residual has the convenientinterpre- residualand our estimatedbalanced-growth tation of being exactly(t in (2). We use two shock(the correlationis 0.19) and a stronger measures of this productivity residual: relationshipbetween Prescott'sSolow resid- Robert Hall's (1988 table 1) for total manu- ual and the estimatedshock (the correlation facturing and that produced by Prescott is 0.48). (1986).13 The Solow residual is an imperfectmea- sure of technical change. For example, Prescott (1986) points to errors in measur- 12 Additional sensitivity analyses were performed: ing the variables used in its construction, substitutingshort-term private and long-term public and Hall (1988) suggests that this measure interest rates for the short-termpublic rate, dropping of productivitymisrepresents true techno- interest rates entirely, and changing the number of logical progress in noncompetitiveenviron- lags. The results, available from the authors on re- ments where exceeds cost. quest, are consistentwith the summaryconclusions in price marginal this and subsequentsections. Nonetheless, these results suggestsome link 13BecauseHall's series is annual, Prescott's quar- between the real permanent shocks from terly series was aggregatedto an annuallevel. the model and the two measures of the VOL. 81 NO. 4 KING ETAL.: STOCHASTIC TRENDS 837

proaches used to construct the two trend estimates, they are broadly similar. The three major differences between the two series are the prolonged growth of the 1960's, the 1974 contraction, and the slow- an~~~~~ / down of the late 1970's.

- /~7 V. Conclusion

LCj In this paper, we have analyzed the 1 56 60 64 68 72 76 80 84 stochastic trend properties of postwar U.S. year macroeconomic data to evaluate the empiri- FIGURE 6. ESTIMATES OF ANNUAL TREND cal relevance of standard RBC models with OUTPUT: DENISON'S (1985 TABLE 2-2) ESTIMATE permanent productivity shocks. Several as- OF REAL POTENTIAL GNP PER CAPITA (SOLID pects of these results are consistent with the LINE) AND THE PERMANENT COMPONENT OF Y central proposition of most real-business- FROM THE SIX-VARIABLE BENCHMARK MODEL (DASHED LINE) cycle models. Real per capita output, con- sumption, and investment (as well as real balances and interest rates) appear to share common stochastic trends. The cointegrat- Solow residual. These comparisons thus lend ing relations among the real flow variables some credence to the interpretation in Sec- are consistent with balanced growth; in ad- tion III of the permanent real shocks as dition, money, prices, output, and interest measuring economy-wide shifts in produc- rates are consistent with a long-run money- tivity. demand cointegrating relation. In a three- The focus so far has been to use the variable-real model, innovations in the bal- empirical model to evaluate a class of anced-growth component account for more real-business-cycle models. However, the than two-thirds of the unpredictable varia- empirical model also provides a solution to tion in output over the 2-5 year horizon. a classic problem in descriptive economic Yet much evidence is at odds with one- statistics: how to decompose an economic sector RBC models in which permanent time-series into a "trend" and a "cyclical" productivity changes play a major role. Even component. A natural definition of the trend in the three-variable model, the balanced- is the long-run forecast of the variables (see growth innovation accounts for less than Harvey, 1989 Ch. 6), and some simple alge- two-fifths of the movements of investment bra (see the Appendix) shows that changes at horizons up to six years. The explanatory in this trend are just linear combinations power of the balanced-growth innovation of the permanent innovations. Thus, the for output is reduced to less than 45 percent empirical model can be used to form a by introducing nominal variables. Moreover, multivariate generalization of the trend- the explanatory power arises mainly from cycle decomposition proposed by Stephen some specific episodes, notably the sus- Beveridge and Charles Nelson (1981). tained growth of the 1960's. The balanced- The implied trend component of output growth innovation sheds little light on other computed using the six-variable model is important episodes, such as 1974-1975 and shown in Figure 6 along with Edward Deni- 1981-1982. Thus, we are led to conclude son's (1985) estimate of real potential GNP per capita.14 Despite the very different ap-

relationship, by adjusting for capacity utilization and by making other adjustments such as for labor disputes, 14Denison computed his measure of potential out- the weather, and the size of the armed forces (Deni- put by adjusting actual output using an Okun's-law son, 1985 tables 2-4). 838 THE AMERICAN ECONOMIC REVIEW SEPTEMBER 1991 that the U.S. data are not consistent with Equations (Al) and (A2) are the defini- the view that a single real permanent shock tions of the reduced form and structural is the dominant source of business-cycle models given in (5) and (6) in the text. fluctuations. Assumption (A3) says that the structural What are the omitted sources of the busi- disturbances are in the space spanned by ness cycle? From a monetarist perspective, current and lagged values of Xt and that it is surprising that such a small role is there are no singularities in the structural played by the inflation shock. Accelerations model. Assumption (A4) is discussed in the and decelerations in money growth and in- text explicitly for the six-variable model. It flation, which are assumed to have no long- also applies to the three-variable model by run effect on real flow variables and real defining A to be a vector of l's. In Assump- interest rates, explain a trivial fraction of tion (A4), the diagonal elements of HI are the variability in output and consumption normalized to unity without loss of general- and a small fraction of the variability in ity, since the variances of ilt, in (AS), are investment. The results point toward an ad- unrestricted. ditional permanent (or at least highly per- The permanent innovations, it, can be sistent) component associated with real determined from the reduced form, (Al), as interest rates which has large effects on follows. From (Al)-(A3), C(L)= r(L)r&1- investment. so that C(l) = r(l)ro-1 Let D be any solu- tion of C() = AD [for example, D= (A'A)-YA'C(l)]. Thus, ADEt = AHIItl and, APPENDIX since Enlq'l' =I i, DI D'=H in,'. Let II* be the unique lower triangular square This appendix presents a discussion of root of DID', and let H and be the identification and estimation. The defini- unique solutions to fl1T2 = *, where H tions in the text are: and I, i satisfy (A4) and (A5). Then, A = AH, and the first k rows of r&- are given (Al) reducedform: AX,=ti+C(L)E, by G = H -1D. Since D is unique up to pre- multiplication by a nonsingular matrix, G is = (A2) structural model: AX, u + r(L) q,. unique. Finally, nt = rJ'E t implies that Mt = GE. The identifying restrictions are: The dynamic multipliers associated with Mt1are given by the first k columns of r(L). (A3) rO t = t These can be calculated as follows. First, write ro = (HJ), where H has dimensions (A4) r(l) =[A o] n x k and J has dimensions n x(n - k). Since r(L) = C(L)ro, the first k columnsof where rO-1 exists and where A is a known r(L) are given by C(L)H. Finally, Et =roNt n x k matrix with full column rank, H is implies 1[r6=Lr'-T, so that from (AS) a k x k lower triangular matrix with full H'= I -1'G?; thus, the dynamic multipliers rank and l's on the diagonal, and 0 is a for i t are C(L).G'GP . n x(n - k) matrix of O's. The covariance Both the structural and reduced form lead matrix of the structural disturbances is as- naturally to the multivariate version of the sumed to be Beveridge and Nelson (1981) decomposition used to estimate trend output, which is plot- ted in Figure 6. The structural form can be (A5) E(lq,lq)=[? 2 In77 expressed as Xt= X + ,ut +E=jLr(L)Qq or, setting XO = O, Xt = lt + r(l)Et=q1 + where 5:, is partitioned conformably with r*(L),t, where r* =-E.1 1ri. Let Tt= mt =(lt it ) , where nt is a k x 1 vector, E1=jil; then, this becomes Xt = Xtp+XS, 2 is a (n - k) x 1 vector, and where I is where Xs = r*(L),t is the stationarycom- diagonal. ponent of Xt and XP = t + r(l)Eo1o = VOL. 81 NO. 4 KING ETAL.: STOCHASTIC TRENDS 839

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