Journal of Monetary Economics 33 (1994) 2333254. North-Holland

Information, forecasts, and measurement of the business cycle*

George Evans

Lucrezia Reichlin

Received November 1992, final version received January 1994

The Beveridge-Nelson (BN) technique provides a forecast-based method of decomposing a variable, such as output, into trend and cycle when the variable is integrated of order one. I(1). This paper considers the multivariate generalization of the BN decomposition when the information set includes other I(l) and/or stationary variables. We show how the relative importance of the cyclical component depends on the size of the information set, and is necessarily higher with multivariate BN decompositions. The results are illustrated using post-WWII United States data. An explanation is also provided for the empirical finding of a positive association of the multivariate BN cycle with output growth.

Key wortls: Trend; Cycle; Forecast; Information

JEL classification: C32; E32

1. Introduction

One of the basic issues of empirical macroeconomics is how to decompose real output into trend and cycle. As is now well known, when the stochastic process followed by a variable contains a unit autoregressive root, the tradi- tional method of detrending by regression on a time trend can generate spurious cycles. For the empirically important case in which a unit root is present, there is a wide range of competing models in the literature, e.g., Beveridge and Nelson (1981) Harvey (1983, Watson (1986), Lippi and Reichlin (1994a).

Corresponde~tce to: Lucrezia Reichlin, Graduate School of Business, 510 Uris Hall, Columbia University, New York, NY 10027, USA.

*We would like to thank Marco Lippt. Charles Nelson, and an anonymous referee for their comments.

0304.3932,‘94:$07.00 ,(‘: 1994 Elsevier Science B.V. All rights reserved

The differences in method depend on the dynamic specification of the non- stationary trend and on additional assumptions required to identify the de- composition. Unobserved components models impose statistical identifying assumptions on the correlation between the stationary cycle and the trend, and the results are sensitive to these a priori assumptions [on this point, see Quah (1992) and previous versions and Lippi and Reichlin (1991) and previous versions]. Equally controversial are decompositions based on structural eco- nomic models, since the required identifying assumptions then depend on one’s view of the correct model of economic growth and fluctuations.

In this paper we follow the alternative method, introduced by Beveridge and Nelson (1981), of defining the trend component of a variable in terms of its long-run forecast. In this decomposition, the components are driven by the same shock or vector of shocks. However, no a tfriuri assumption on the correlation between trend and cycle is required.’ Similarly, a forecast-based definition of the trend obviates the need for additional economic identifying assumptions.

An advantage of the Beveridge-Nelson approach is that the cyclical compo- nent of output has an attractive economic interpretation in terms of the total additional output growth, beyond normal rates, one would forecast as output returns to trend. One can thus interpret C, as the gap between current and ‘normal’ output levels, and the size of this gap is clearly crucial for policy making.

Beveridge and Nelson examined the univariate case, while multivariate ver- sions of this procedure were constructed in Evans (1989a, b), Cochrane (1990), and King et al. (1991). Empirical estimates in the univariate case attribute most of the variance to the trend component. In this paper we show that the multivariate BeveridgeeNelson decomposition depends theoretically on the information set employed and that the size of the information set is paramount to the relative importance of trend and cycle.

The first part of the paper presents some formal results on the properties of a multivariate forecast-based decomposition. Our formulation allows for the inclusion of both variables which are integrated of order one [1(l)] and stationary variables. We show that the univariate BeveridgeeNelson decomposi- tion implies a stnaller cycle-trend variance ratio than the multivariate decompo- sition, and that enlarging the information set used to forecast output growth increases a lower bound for this ratio. intuitively, in multivariate models output growth can be better forecasted, and hence more of output fluctuations is ascribed to the cyclical component. We illustrate these results empirically for the post-WWII United States using a five-variable information set.

‘Consider the following argument. Writing Jo = ‘r, + C,, where T = trend and C = cycle, we have E(J,+, 1 H,) = E(C,+, 1 ff,) + E(T,+, 1 H,) for the time r information set H,. Ask gets large. stationar- ity and zero mean of C, imply E(J, +k ( M,) = E( T,,, ( H,). If T, is a random walk with unforocastable innovations and drift d we arrive at E(T, 1 H,) -= E(Y,+~ 1 H,) - dk for large k. regardless of the correlation between trend and cycle.

Our multivariate estimates of cycle and trend differ from the univariate estimates also in another important respect: in contrast with the univariate results of Beveridge and Nelson, multivariate estimates of the cyclical compo- nents are positively associated with NBER reference cycles. We show that these contrasting results arise from estimation issues since theoretically the covariance between output growth and the level of the cycles is equal to the negative sum of the autocovariances of output growth. Our estimated multivariate models imply that the positive autocovariances at short lags are outweighed by negative autocovariances at longer lags, and we provide an economic interpretation of this result.

2. Framework and definitions

2. I. Multivuriute uncl uniaariate Beoeridge-Nelson decompositions

Consider a n x I stationary vector W, whose first ~1~ elements, AXI,, are first differences of I( 1) variables, and the remaining n - n, elements are levels of variables integrated of order zero [I(O)], X2,. Denote the first element of W,, which we will interpret as the first difference of the logarithm of real GNP, by

AY,. W, admits the MA representation:

w, = D + A(L)&, (1)

where v, is n-dimension white noise with Eo, = 0 and Ev,v; = Q, Q positive definite. We assume A(L) = c,7Z0 A,Lj is a rational function of L and:

(i) A, = I. (ii) The poles of A(L) are outside the unit circle. (iii) The roots of det A(L) are on or outside the unit circle. (iv) Al( 1) # 0, where AI(L) is the matrix formed from the first n, rows of

A(L).

Assumption (ii) implies the stationarity of W, and assumption (iii) implies:

c‘, = W, - Proj[W,I W,_,, W,_, ,... 1,

so that L’, is the innovation and representation (1) is fundamental for W,. Adding assumption (i) guarantees the uniqueness of representation (1); this unique representation is also called the Wold representation. Assumption (iv) implies the presence of at least one stochastic trend.

Let us partition W, and A(L) conformably:

w, = A(L) =

We have the following decomposition into a stationary and a nonstationary component:

where

i-?(L) = f iiijL’, AC?, = - f Al,, j=O m>j

and Al( 1) is of rank less or equal than n, (if XI has r cointegrating relationships, then the rank of Al( I) is nl - r). Representation (2) is a multivariate generafi- zation of the Beveridge and Nelson (BN) decomposition for the vector Wt. The sum of the first two terms on the right-hand side is interpreted as the first difference in the trend component, while the third term is interpreted as the first difference in the cyclical component.

Estimation of A(L) is performed using the autoregressive representation. If the variables in Xl are not cointegrated, then A(L) can be inverted and the system AXl, X2 estimated as a standard VAR. If components in XI are cointegrated, then we require an extension of the standard error correction representation to include I(O) variables. This issue is discussed in appendix 1.

In the remainder of this section we summarize the properties of the trend and cyclical component of output in this muttivariate context. Let us now denote the first row of A(L) as

The first difference of the log of real output satisfies

Ay, = d + r(L)u,,

from which we obtain

(3)

yt+k = 3’(_1 + (k + l)cl + r(L)&+ + ... + r(L)C.I+k*

237

Thus a~~+~/&, = (1 + rl + .I. + rk), and hence

lim aJ~l+kpL~l = r(l)ut. k-x

If we define the trend component T, of y, by

T, = lim (E,y,+k - dk): k-x

then

Ai”, = d + h(E,y,+k - Et_lJif+k) = d $ r(l)L+, k-x

so that the trend is a random walk with drift d. This corresponds to the first two terms of the first row of (2). The cyclical component is then defined as C,= Y,- T,.

Since dq’, = AT, + AC,, we can decompose Ay, as

Ay, = Ii + r(l)a, + (1 - L)F(L)v,, (4)

AT, = d + r(l)u,, (5)

AC, = (1 - L)F(L)v,, (6)

where F(L) is defined by (1 - L)?(L) = r(L) - r(l), so that the cyclical com- ponent is given by C, = F(L)o,, corresponding to the last term in the first row of (2).

C, can be interpreted by noting that

C, = y1 - lim (E,Y~+~ - dk) = lim dk - 5 E,Ayt+i , k-x> k-x i=l

or

Cl = - f (E,A.Y,+i - d), (7) i=l

which is the negative of the sum of expected future output growth, given information at t, in excess of the unconditional mean rate of growth.

Eqs. (4), (5), and (6) give us the multivariate generalization of the univariate BN decomposition for y,. The latter case corresponds to k = I and the representation

Ayt = d + u(L)u,, (8)

238 G. Erat7s and L. Reich/in, Forecusi-bused output &compo.vition

where U, is a scalar white noise process with variance CJ,’ and a(L) = x1?=, aj L’ is a rational function in L in which a(O) = 1 and the usual assumptions apply. The univariate BN decomposition will then be

Ay, = d + a(l)u, + (1

where

As, = d + a(l)u,,

AC, = (1 - L);(L)&.

(10)

(11)

2.2. Trendcy and permanent-transitory decompositions

In contrast with the BN definition of the trend as the long-term forecast of the level of the variable of interest, several authors have taken a more structural approach. In particular, Blanchard and Quah (1989) and King et al. (1992) have suggested using restrictions on the matrix of the long-run multipliers in order to identify permanent versus transitory shocks. The permanent component is then defined as the one resulting from the dynamic effect of the permanent shocks.

In order to clarify the difference between this approach and the BN one, we consider a bivariate example. In the representation (1) assume that U: is a 2 x 1 vector composed of two I(1) variables. The Wold representation is

w, = A(L)&, (12)

where

w, =

Identification of the permanent and structural shocks on output is achieved by imposing two u priori requirements. First, the structural shocks are assumed orthonormal, i.e., we pick a matrix S such that

II, = s-l c,, Eu,u; = 1.

We can thus write

w, = B(L)&, (13)

239

where

B(L) = A(L)S.

Second, S is selected so that

This implies that only the second component of c’, has a permanent effect on output. The permanent component of output for this example is then defined as

Clearly, isolation of the permanent component (14) depends on the two a priori identifying assumptions imposed.

The BN trend of output, on the other hand, is defined in terms of forecasts E,dytck (which depend on the long-run dynamics of all shocks). It is therefore independent of the identification assumptions. This can be seen by noting that, for any S,

A(l)& = A(I)SS-‘z?, = B(l)&.

In particular, for the above identification we have

Finally we note that if J+ and _xt are cointegrated, b,,(l) = 0 implies b,,(l) = 0, and we have

The permanent component and the BN trend of output are defined as above, but now .q has only one permanent shock.’

ZBlanchard and Quah (1989) used the same identification scheme, but considered the case of one stationary and one nonstationary variable. King et al. (1992) considered the cointegration case for a tri-variate and a six-variate system. For the case of more than two variables, some additional identifi~tion assumptions are needed.

240 G. Emms and L. Reichlin. Forrcust-bawd output decomposition

3. Some formal results

3.1. Size qf the information set und cycle-trend variance ratios

Although var(A T,) is invariant to the choice of the information set, var(AC,) is not. The latter depends on the covariance between Ay, and AT, as well as their variances. We now show that the cycle-trend variance ratio R =

var(AC,)/var(AT,) is smaller in the univariate case than in the multivariate case and, in general, by increasing the number of variables used to forecast income growth, its lower bound increases.

Let us consider first representation (8) and the univariate decomposition (9). The variance of the trend component (in first differences) is

vards, = Thai. (16)

The variance of the cyclical component (in first differences) is

vardc, = vardy, + vardz, - 2cov(Ay,, As,). (17)

Consider now the multivariate decomposition (4). The variance of the first difference of the trend is equal to the spectral density of Ay, at frequency zero:

varAT, = r(l)Qr(l)‘, (18)

where Q = E(va’). We now note that var(Ar,) = var(AT,). This follows from the fact that in decompositions where the trend is a random walk and the cycle is stationary, the variance of the trend is equal to the spectral density of the process at frequency zero [Lippi and Reichlin (1991)], i.e.,

var(Ar,) = var(AT,) = y.,,.(O), (19)

where y.,j(o) is the spectral density of Ay, frequency o. The variance of the cycle is

varAC, = varAq>, + varAT, ~ 2cov(Ay,, AT,),

with

cov(Ay,, AT,) = r,Qr(l)‘,

(20)

where r0 = (1 0 ... 0). From eqs. (17) and (20) it is clear that, unless the covariance between the trend

and output is the same, the variance of the cycle will be different in the univariate and the multivariate cases.

G. Eoans and L. Reichlin, Forecast-hasrd output decomposition 241

Proposition 1. (i) The cycle-trend variance ratio derivedfrom the univariate BN

decomposition is less than or equal to the cycle-trend variance ratio derived from the multivariate BN decomposition. (ii) By expanding the information set used to forecast income growth, we increase a lower bound of the cycle-trend variance ratio derived from the BN decomposition.

Proaf Since Evv’ = Sz is positive definite, there exists a nonsingular matrix P such that Q = PP’. Define

s(1) = r(l)P and sO = rOP.

Note that s,, is just the first row of P. We can then write

cov(dyt, AT,) = sOs(l)‘.

The variance of dy, conditional on the history of output growth alone is

var(dy,/I,_,)= a,Z,

where I,_ 1 is the information set containing the past values of dy,. The variance conditional on the larger information set H,_ r, containing the history of the k variables in W, _ , , is

var(dy,IH,_l) = sosb.

var(dyt/l,-,)2 var(dy,IH,-,).

We first establish the lower bound in the multivariate case. From Cauchy- Schwartz we have

(slJs(1)‘)2 I (s()sb)(s(l)s(l)‘).

Noting from (1X) that vardT, = s(l)s(l)‘, we have

cov(dy,, AT,) I Jvar(dy, 1 H,_,)var(dT,).

We thus obtain the lower bound:

var(dC,) > var(dy,) + 1 _ 2 var(~AHt-I) var(d T,) - var(d T,) J var(dT,)

Since var(dy, j H,- 1) I var(dy, 1 *I,_ 1) for any information subset J,_ , c H,_ , , we have established (ii).

Next, from the univariate representation:

cov(4yt, dr,) = a(l)o,2 = J_ J_.

Thus

var(dc,) var(4y,) + f _ 2 var(4ytlf,-r)

var(4 r,) = var(4 r,) J var(dT,) ’

from which (i) follows. Q.E.D.

Remark 1: If 4yt is Granger-caused by any of the other variables in ul,, then var(dy, 1 I,_ 1) > var(dy, III-r) and result (i) holds with strict inequality.

Rernmk 2: All through the discussion we have limited the analysis to funda- mental representations for ul,. This is justified because we are interested in a forecasting interpretation of the trend from which we derive our definition of the cycle as the output gap. However, one should be aware that structural interpretations of the error terms are not warranted.3 Moreover, to restrict the analysis to fundamental representations of W, does not imply that the repre- sentation of the multivariate cyclical component is itself fundamental. On this point, see Sargent (1987, ch. XI, exercise no. 25).

We conclude this section with a simple example which illustrates the im- portance of the information set. Consider the case of a univariate IMA(1, 1) process with positive coefficient:

dy, = (I + aL)u,, (21)

where u1 is i.i.d. white noise with mean zero and variance ~z and where a > 0. This case was considered explicitely by Beveridge and Nelson (1981, p. I .57).4 Notice that var(dc,) < var(dt,) since

vardc, - vardr, = [(u’ - 1) - Za] 0,” < 0,

with a < 1 by the invertibility condition.

3For a more general analysis of identification of ftmdamental and nonfundamental representa- tions in multivariate models, see Lippi and Reichlin (1993, 1994b).

4A related discussion can be found in Nelson and Plosser (1982, pp. 154ff.)

G. Ecans and L. Rrichlitl, Forecast-based ourput decomposition 243

Consider now a multivariate IMA(l, 1) process in which output is generated by two orthogonal shocks:

AY, = 6, + U&,-l + Bvr-1, (22)

where E, and qr are i.i.d. white noises with variance CJ~ and g,’ respectively, EE,~, = 0, and 1 xl, 181 < 1. Assume that both E, and qt are in the time t informa- tion set available for the multivariate BN decomposition. We have

?i’l = cov(Ay,, Ay,_1) = cm~,

and the multivariate BN decomposition satisfies

var(d C,) - var(d T1) = (x2 - 1 - 2c()a,2 + fl’ D,‘.

If x > 0 and 0,’ is sufficiently large, we will thus have yI > 0, but var(dC,) > var(dT,). Since the univariate Wold representation of the process (22) is an IMA(l, 1) of the form (21) the univariate BN decomposition yields var(dc,) < var(dr,), while the bivariate BN decomposition based on the larger information set reverses the relative size of cycle and trend.

3.2. The autocovariance jiinction und the BN decomposition

In the previous section we saw that going from the univariate to the multivari- ate BN representation we obtain larger cycles. Here we will show how some features of the BN decomposition, univariate and multivariate, depend upon the autocorrelation function, and hence are independent of the information set. We begin with some results that obtain when dy, is positively autocorrelated at all lags.

Proposition 2. If Ayt is positively autocorrelated at all lags, its spectral density has a maximum at zero.

Proof: This follows trivially from the definition of the spectral density:

k=-z

= yo + ;‘k COS ok, k=l

where yk = cov(Ayr, Ay,_,) for k = 0, 1,2,. .

244 G. Ewns und L. Reichlin, Forecast-bused output decomposition

Corollary 1. If Ay, is positively autocorrelated at all lags, then

var(Ay,) < var(AT,).

Proof: Since the spectral density of A T, is equal to the spectral density of A y, at frequency zero, this follows from var(Ay,) = (1/2x)1 g3>(m)dW.

Corollary 2. If Ay, is positively autocorrelated at all lags, then

cov(AC,, AT,) < 0.

ProoJ: This follows from the definition:

cov(AC,, AT,) = f [var Ay, - varAT, - varAC,]

A final consequence, which is implied by the next Proposition, is that if Ayt is positively correlated at all lags, then

cov(Ayl, C,) < 0.

We formulate this proposition more generally.

Proposition 3. Let C, he the univariate or multivariate BN cycle for any informa-

tion set. Then

cov(Ay,, C,) = - f ‘Jo, i=l

where yi = cov(A y,, AY,_~). (The proof is in appendix 2.)

Proposition 3 indicates the properties Ay, must satisfy to generate cov(Ay,, C,) > 0. In section 5 we will use Proposition 3 to interpret the empirical results on the correlation between BN cycles and NBER cyclical datings.

4. Empirical results

To implement the multivariate decomposition procedure defined in section 2, we require a set of variables which are good forecasters of output growth. There are obviously many possible candidates: Evans (1989a, b) used the unemploy- ment rate, which provides information on excess capacity in the labor market. Cochrane (1990) uses the saving ratio as a measure of households’ expected future incomes. In addition to these variables, we select two indices which have

G. ELWIS und L. Reichlin, Fwecust-bused output decomposition 245

Table 1

Bivariate causality tests; dependent variable AL’; sample period 1948: l-1990:3.

Causal variable

AC0 ALE ACS u

4 lags

0.0024 0.0000 0.0056 0.0003

Significance level

8 lags

0.0025 0.0000 0.00 14 0.001 I

been constructed for the purpose of measuring and forecasting the aggregate level of economic activity: the index of leading economic indicators and the index of coincident indicators. The second index reflects the widely held view that business cycles are characterized by comovements of numerous macro- economic variables, while the first index exploits systematic timing relationships between economic variables.5

Our specific choice of variables is: real GNP (Y), the unemployment rate (U), real nondurables plus services consumption (CS), the index of leading indicators (LE), and the index of coincident indicators (CO). Y, CS, LE, and CO are all measured in log form, and U is detrended to allow for a possible increase in the natural rate of unemployment. These variables do appear useful for forecasting output growth, as indicated by the results in table 1: d Y is Granger-caused by each of the variables ACO, ALE, AC& and U.

To compute our estimate of the cyclical component of Y, based on formula (7), we set up an appropriate cointegrated VAR (vector autoregression) which can be used to construct conditional forecasts of GNP growth. Since the variables Y, CO, LE, and CS appear to be 1(1),‘j we follow a two-step procedure in which we first test for cointegration among these variables. Using Johansen’s test, there appear to be three cointegrating relationships (parameter estimates are given in table 2) and we include the corresponding cointegration residuals in a five- variable VAR in the first differences of Y, CO, LE, and CS and in the level of U. The estimation period for the VAR is 1949.1 to 1990.3. Data are quarterly’ and four lags (of levels) were used. C, was computed from (7) with the sum truncated at k = 300 quarters.

‘For a recent discussion of the value of leading indicators in economic forecasting, see Lahiri and Moore (1991).

‘For evidence on the stationarity of the unemployment rate, see Evans (1989a,b).

‘For LE, CO, and U quarterly averages of monthly data are used.

246

Table 2

Cointegrating vectors; 4 lags; estimation period 1949: 1 - 1990:3.”

Variable Vector 1 Vector 2 Vector 3 _ --.-.

1 -.. I -I -1 CO - 0.6975 0.4641 - 0.6050 LE - 0.3369 0.3818 0.7655 CS 1.9335 0.1278 0.8925

~_ “Eigenvalues are 0.1525, 0.1172. 0.0921, and 0.0251. The chi-square statistic for the test of H,:

r I 2 vs. r = 3. where r is the number of cointegrating vectors, is ;y* = 16.14, and for the test of H,: r I 2 vs. r 2 3 it is yz = 20.39. Both are significant at the 5% level.

Table 3

Measures of cyclical component for alternative models; VAR models with 4 lags of levels; estimation period 1949: l-1990:3.”

.-____ ~- ..____ var(dC) R = var(dC)/var(d T)

_.

Model 1: dy is AR(3) 0.51s 0.23 1 Model 2: VAR in Ag, ACO, and y-C0 1.076 0.916 Model 3: VAR in do, AC’S, and JCS 0.896 1.376 Model 4: VAR in Ay and [I 0.985 2.690 Model 5: VAR in dy, ACO, ALE, ACS,

3 cointegrating residuals, and U 1.503 5.511 .____-

%tatistics given in table correspond to the period 1951: 1 to 1990:3.

In order to examine the importance of larger information sets, we compare the results with those obtained when fewer variables are included. Table 3 re- ports the sample variance of the estimated AC’, series, and the sample estimates of R = var(~C~)/var(~ T,), for each of the five models8 The smallest values are obtained for model 1 which includes only past output growth as information. All of the other models have substantially higher values of var(dC,) and R. As anticipated, model 5, which uses all the information, attributes the most import- ance to the cyclical component. Model 5 also provides the smoothest estimate of the trend.’

*Because of the instability of the estimates of C, for 1949-50 (model 5 giving particularly large negative estimates), tables 3 and 4 (and the figures) report results over 195 1: 1 to 1990: 3. The patterns shown in tables 3 and 4 are little affected by this choice.

‘Estimates of var(d r,). given by the ratio of the columns in table 3, vary substantially, although theoretically they should not. The univariate estimate of vartdr,), model 1, is particularly high. Presumably the explanations are specification and sampling errors. In model 1 an infinite-order AR process has been approximated by an AR(3). More lags could be included, but the associated sampling error would be large. A closely related point is discussed in Evans (1989, sect. 4.A).

6 n’

-f-i’ 2

50 55 60 65 70 75 80 85 90 95

Fig. 1. Estimates of C from model 1 (Cl) and model 2 (C2).

Figs. 1 and 2 present the estimates of C, obtained from models 1,2, and 5, and table 4 presents the correlations between the different measures of C, and AC,. All of the models, except the univariate model 1, exhibit the qualitative features of the NBER business cycles dates, though there are important differences in the sizes of particular booms and recessions. An interesting feature of model 5 esti- mates shown in fig. 2 are the large negative magnitudes of C, during the 1958 and 1961 troughs. Another intriguing finding (see fig. 3) is that the sample distribution of model 5 estimates is skewed, with the largest values of IC,I occurring during recessions.

A caveat to the findings of this section is that some quantitative results depend on details of specification. For a given set of information variables, estimated magnitudes of C, are somewhat sensitive to lag length and sample period and will depend on estimates, and the estimated number, of cointegrating relation- ships. However, the qualitative aspects of the estimated C, series do appear fairly robust to these estimation issues for each of the multivariate models exam- ined.” In particular, when model 5 is reestimated dropping the least significant

“‘Another potentially important empirical issue is the possibility of structural change over the period, and in particular a shift in the mean rate of growth of output.

248 G. Evans and L. Rrichlin. Forecast-based output decomposition

6,

4 I 2-

: u O

%

!zJ -2-

a 2 -4 - :: u

-6

-8

t

fl P v

50 55 60 65 70 75 80 85 90

Fig. 2. Estimates of C from model 5

Table 4

Correlations between cyclical measure for model 5 and other models; sample period 195 1: 1~ 1990: 3.

C, AC,

Model 4 0.725 0.815 Model 3 0.797 0.587 Model 2 0.434 0.578 Model 1 0.008 ~ 0.490

of the cointegrating vectors, the ratio R is little changed and the correlation with the cyclical measure of model 5 using all three cointegrating vectors is high (0.927 in levels and 0.951 in first differences).

5. Discussion

The contrasting results between univariate and multivariate BN decompo- sition are partly the result of differences in the theoretical features of the

20 -

E

5 15-

w ‘+d

f lo-

-12 -10 -8 -6 -4 -2 I

4 IL 6

Value of C

Fig. 3. Histogram of C from model 5.

decompositions and partly the result of estimation issues. First, var(nC,) is estimated to be larger than var(dT,) in models 3, 4, and 5, whereas var(dc,) is estimated to be substantially less than var(&) for the univariate model. The reversal of the sizes of the trend and the cycle are not at all surprising in terms of the results in section 3.1. Because the information set is larger and includes variables that Granger-cause output, more of output growth is forecastable and ascribed to the cyclical component. Note that the same result may be used to explain a well-known empirical finding of the aggregation literature whereby estimates of the BN cycle for output obtained from multivariate systems using sectoral data have typically larger variance than univariate estimates [e.g., Lee and Pesaran ( 1992)].

Second, the relationship to the NBER reference cycles is qualitatively different across models; NBER recessions correspond to negative values of the multivari- ate BN measures, whereas the opposite was true in the original Beveridge- Nelson results.

The explanation of the contrasting empirical univariate and multivariate results on the correlation of the leuel of C, with the NBER recessions rests on the Proposition 3 of section 3.2. Although there is no mechanical rule for NBER datings of trough and peak, their datings correspond approximately to the rule

250 G. ISam and L. Reich&z, Forecast-based output decomposiliorz

that recessions are periods of decline of GNP. Proposition 3 shows that cov(C,, dy,) is determined by the autocovariance function of dy, and is therefore in theory independent of the choice of the information set.

This indicates that the contrasting empirical results on cov(C,, dyt) depend on estimation issues. Univariate estimates of yi are positive at low lags but impre- cise at higher lags. Low-order univariate models of dy,, e.g., IMA( 1, 1) models as in Nelson and Plosser (1981), will obtain a negative covariance if all implied estimated autocorrelations are positive. Intuitively, a higher than average value of dy, then leads to above-average expected future output growth and hence

negative values of C,, as seen from (7). However, a positive association between the multivariate C,, in models 2 to 5,

and the NBER cycles can be explained if positive values of yi for low i are outweighed by negative values for large i. To see that this is indeed what is happening we present in table 5 the implied autocorrelation function for the estimated model 5. The positive autocorrelations at lags 1 and 2 quarters are outweighed by the negative autocorrelations at lags 4 through 10 quarters.

The economic intuition for the result is that, during periods of recession (declines in output), changes in the state of the economy take place which set the stage for eventual recovery. For example, during recessions the unemployment rate rises, a pattern well documented as ‘Okun’s Law’. An expected later decline in the unemployment rate would then be associated with faster than average employment and output growth. Other economic variables, such as the saving ratio or the relationship between output and the coincident and leading indi- cators may also provide information indicating higher future growth rates at a later time.

We conclude this section with a brief discussion of the role of cointegration for the empirical results. The difference in results for the univariate and the

Table 5

Correlogram of fly, for model 5.”

Lag, i

1 0.3831 2 0.2306 3 - 0.0052 4 - 0.0988 5 - 0.1958 6 - 0.1777 7 - 0.1514 8 - 0.1146 9 - 0.0689

10 - 0.0495 11 - 0.0111

‘pi is the implied value of corr(dy,, AJ~-~) for the estimated multivariate model 5.

G. Eouns and L. Reichlin, Forecast-based ouiput decomposirion 251

multivariate cases is driven fundamentally by the difference in information used to forecast output. Cointegration is relevant only for estimation purposes. Given the information set, including cointegration residuals does not in theory add extra information. The presence of cointegration does imply that representation (1) is not invertible; however, with an infinite amount of data the moving average system (1) could be precisely estimated, with the innovation u, first obtained as the limiting projection of I+‘: = (AXI;, X21) on its lags [see, for example, Gourieroux and Monfort (1990, pp. 192- 193)]. From the known system (1) our measures C, and 7’, can then be obtained. Thus in principle cointegrating errors are not required since the information is contained in the infinite past history of W,. However, with finite data, estimation is most satisfactorily performed using the error-correction representation of the VAR incorporating cointegrating residuals.

6. Conclusions

The BeveridgeeNelson technique provides a forecast based measure of the decomposition of output into trend and cycle, y, = C, + r,, when y, has a unit root. A number of aggregate economic data series help forecast GNP growth, and thus should be included in the information set when computing the cyclical component. This is not simply an estimation issue. Although the theoretical value of var(AT,) is determined by the spectral density at zero, and hence is independent of the representation of A Y,, we have shown that var(AC,) depends theoretically on the information set available. In principle we want to use all variables which provide incremental information in forecasting Ay,. Whether additional variables beyond those included in this paper provide significant incremental forecasting value is a testable issue amenable to further empirical investigation.

Appendix 1

We can distinguish two cases (we have dropped the constant term D for simplicity):

(A) The roots of det A(L) are all outside the unit circle. In this case u, can be recovered by running a VAR on W,. The latter is an approximation to

~ w, = B(L) w, = u,: det A(L)

which can be estimated from the data.

252 G. Evans and L. Reichtin, Forecust-based outpi de~~l~n~~).sition

(B) The determinant of ,4(L) has a root on L = 1, so that there is at least one cointegration relation. Therefore,

det A(L) = (1 - L)p(L).

Furthermore, by the cointegration assumption:

rankle < n,.

Using the definition of the adjoint matrix it follows that we can write

&d(L) = [N(L) (1 - ~Kv-)I, (24)

so that

IIN(L) (1 - W(L)1 “,“it ( 1 = (1 - L)PWh **

(25)

or

N(L)XI, + C(L)XZ, = p(L)?_+. (26)

Using standard arguments, the corresponding error correction representation is

F(L)dXI, + C(L)XZ, = - 6z,_ 1 + @L(L)&, (27)

where

z,_1 = ;‘Xl,_l, N(1) = cT1’, F(L) = N(L) - N(l)L

1-L ’

and 6 is nr x r, y is Y x n,, and r is the cointegration rank. In this case, estimation should be performed on the basis of a constrained VAR which is an approxima- tion to model (27).

Appendix 2: Proof of Proposition 3

We first show the result for the univariate case:

Lly, = n(L)u,.

253

From J’i = CJ~ CT:, ajaj+i we have

The result thus follows using c, = L?(L)u,, where ii(L) = ~IFI_,&Li where

Gi = - zzzi+ 1 a,. In the multivariate case, write

Ay, = r(L)zl, = s(L)e,,

where s(L) = r(L)P and e, = P-‘u, for PP’ = $2. Then

Ay, = i: s’j’(L)ejt = i: A$‘, j=l j=l

c, = i: S’j’(L)ejt = i: cy, j= 1 j= 1

where s(j)(L) and S”‘(L) denote thejth components of s(L ly. Since Eee’ = I, we then have

f and s”(L ) respective-

and

cov(A y,, c,) = f c0v(d yjj), cjj)) . j=l

The result then follow from the univariate case.

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