NBER WORKING PAPER SERIESMEASURING BUSINESS CYCLES BY SAVING FOR A RAINY DAY
Mario J. Crucini
Mototsugu Shintani
Working Paper 16075
http://www.nber.org/papers/w16075

NATIONAL BUREAU OF ECONOMIC RESEARCH
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Cambridge, MA 02138
June 2010

We gratefully acknowlege the financial support of the National Science Foundation. The views expressedherein are those of the authors and do not necessarily reflect the views of the National Bureau of Economic
Research.
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© 2010 by Mario J. Crucini and Mototsugu Shintani. All rights reserved. Short sections of text, notto exceed two paragraphs, may be quoted without explicit permission provided that full credit, including© notice, is given to the source.

Measuring Business Cycles by Saving for a Rainy Day
Mario J. Crucini and Mototsugu Shintani
NBER Working Paper No. 16075
June 2010
JEL No. E3
ABSTRACT
We propose a simple saving-based measure of the cyclical component in GDP. The measure is motivatedby the prediction that the represenative consumer changes savings in response to temporary deviationsof income from its stochastic trend, while satisfying a present-value budget constraint. To evaluate
our procedure, we employ the bivariate error correction model of Cochrane (1994) to the member
countries of the G-7 and Australia. Our estimates reveal, that to a close approximation, the stochastic
trend component of GDP is consumption and the transitory component is the error correction term,
which justifies the use of our saving-based measure.

Mario J. Crucini
Department of Economics
Vanderbilt University
Box 1819 Station B
Nashville, TN 37235-1819
and NBER
mario.j.crucini@vanderbilt.edu
Mototsugu Shintani
Department of Economics
Vanderbilt University
Box 1819 Station B
Nashville, TN 37235-1819
mototsugu.shintani@vanderbilt.edu

1. Introduction
Applied macroeconomic analysis inevitably begins with a decomposition of aggregate
output into a business cycle component and a growth component. Most researchers rely on
statistical …lters to achieve this decomposition and rationalize a particular parametric choiceof the …lter (or computer algorithm) based upon preconceptions of what a business cycle
should look like. A common view of what a business cycle component should look like is thatit replicate the National Bureau of Economic Research Business Cycle Dating Committee’s
choices of business cycle peaks and troughs. While we do not deny the utility of this approach,there is a certain arbitrariness in the metric used to decide if a particular business cycle is
too short or too long, particularly when the broader objective of the decomposition is to
speak to both the growth and business cycle literatures. One alternative approach is to
fully specify and estimate a particular DSGE model. A second alternative is to rely on
actual consumption responses to infer a representative agent’s view of trend and cyclicalcomponents, imposing minimal economic restrictions common to a wide class of models.
While the DSGE approach mitigates the obvious ‘measurement without theory’concern ofcritics of business cycle …lters, it is subject to the issue of model misspeci…cation which could
make the decomposition into trend and cycle totally invalid. Consequently, we opt for thelatter course of action.
In this paper, we conduct trend-cycle decompositions of aggregate output using one of
the most basic conditions imposed in almost all macroeconomic models –the representative
consumer’s long-run budget constraint. In the presence of a stochastic trend, the long-run
budget constraint requires cointegration of consumption and total income. However, consumption is less volatile and contains better information about the long-run trend than does
income alone. This motivates a multivariate approach to trend-cycle decompositions using
consumption as a proxy for the trend component, which implies that the cyclical componentof output is essentially equal to national saving. Unlike other economic based decompositions available in the literature, our procedure is extremely simple to apply, embodies an2

economic restriction found in virtually every macroeconomic model and is unlikely to involve
misspeci…cation.
The idea of using consumption as a covariate in trend-cycle decompositions of output is
not new. Motivated by the permanent income model, Cochrane (1994) considers a bivariate
error correction model (ECM) of GNP and consumption with a log consumption/GNP ratio
as the error correction term to capture the cointegration of these two economic variables. He
then conducts a multivariate Beveridge-Nelson (BN) decomposition using an ECM estimatedusing U.S. quarterly data. Cochrane’s ECM/BN approach is extended to include asset wealthas an additional covariate by Lettau and Ludvigson (2004).1 Instead of using the ECM/BN
approach, Morley (2007) considers a bivariate unobserved components (UC) model of GDP
and consumption while imposing a cointegrating restriction for the purpose of estimating
the stochastic trend component of the output.2 In these studies, the short-run dynamics ofthe model are estimated parametrically. In contrast, our saving-based measure of businesscycles does not require a parametric speci…cation of short-run business cycle dynamics.
The nonparametric nature of our approach has similarities to the Hodrick-Prescott (HP)…lter, the most common approach to business cycle decompositions presently in use. Whenusing consumption as a proxy for the stochastic trend, as emphasized in this paper, it isnatural to smooth out the transitory component from the raw consumption series.

Oneof our proposed saving measures applies the HP …lter to obtain a smoothed consumption
series before subtracting it from output to estimate the cyclical component. This measure
may be viewed as a multivariate HP …ltering method which imposes a long-run cointegrating
relationship between consumption and output. The estimated cycle based on this procedure,
however, turns out to di¤er greatly from univariate HP …lter applied to output alone, as donein the business cycle literature.
Since our approach is most closely related to Cochrane’s study, we revisit his ECM/BN
1
While not in ECM format, Rotemberg and Woodford (1996) and Ravn (1997) also use the BN decomposition applied to a three variable system of log output, log consumption/output ratio and log hours
worked.
2
For a comparison of UC models and BN decompositions, see Morley, Nelson and Zivot (2003).

3

method in detail both theoretically and empirically. The results of the ECM/BN decomposition serve as a benchmark to evaluate the performance of our simpli…ed measure. Importantly, our saving-based measure nests the ECM/BN cycle with the cointegrating restriction
in many cases. As noted by Cochrane (1994), in the special case in which consumption is anexact random walk: “the Beveridge-Nelson trend would exactly equal consumption less themean log GNP/consumption ratio” (page 252). In such a case, the saving measure would
work without the need to smooth consumption at all. Even in the presence of transitory
variation in consumption, we show that the ECM/BN estimate of the business cycle is extremely well approximated by saving when raw consumption is replaced by a moving averageof current and past consumption.
Cochrane’s analysis of U.S. output and consumption is extended to include the G-7 plus
Australia. To evaluate the plausibility of the estimated model, impulse responses and variance decomposition analysis of permanent and transitory shocks in the ECM are reported.
The estimated impulse response functions are found to be qualitatively consistent with thesimple permanent income model and even better approximations to the predictions of the
one-sector DSGE workhorse model of macroeconomics. To identify each shock, Cochrane’s
ECM is slightly modi…ed by imposing a zero restriction on the error correction term in
consumption equation.

As it turns out, a simple saving-based measure tracks the moreelaborate estimated ECM/BN cycle very closely in every country. Surprisingly, the role ofshort-run dynamics in the ECMs are typically negligible. Simply put: a linearly detrended
savings (rate) alone captures most of the transitory variation in output. Thus Cochrane’s
conjecture on the usefulness of consumption as a proxy for the BN trend seems valid not
only for the U.S. but also for other industrialized countries.3
Our approach to trend-cycle decompositions is simple and gives a di¤erent cyclical component than existing approaches. The implications of this are discussed in a later sectionof the paper where we focus on the international business cycle comovement puzzle and the3Work in progress demonstrates the applicability of this decomposition to 160 or so countries over thelast 30 years.

4Great Moderation debate.
Throughout the paper, we use the notation yt = 100

ln Yt and ct = 100

ln Ct , where

Yt is Gross Domestic Product and Ct is the aggregate private consumption.

2. The trend-cycle decomposition of GDP
2.1. A saving-based measure of the business cycle
Consider the additive (in logs) trend-cycle decomposition of output yt ,yt = ytg + ytc ,where ytg is the ‘growth’ or ‘trend’ component and ytc is the ‘cyclical’ or ‘transitory’ component. Instead of relying on researcher’s preconceptions about business cycle frequencies,
the idea is to obtain the cyclical component such that a representative agent would view
it as a transitory deviation from the stochastic trend in output. One approach to achieving such a decomposition is to specify a representative agent model and estimate its deepparameters. Here, instead, the focus is on the present value budget constraint faced by
the representative consumer. The long-run budget constraint identity implies that income
and consumption share a common long-run trend. In other words, income and consumption
are cotrending. Since almost every modern macroeconomic model embodies this sensible
economic restriction, it is useful to impose it at the outset.
Simple characterizations of the permanent income model rely on restrictive assumptions
such as a constant interest rate, exogenous endowment and quadratic utility. However, as an
approximation, it is often a convenient benchmark model to understand the core implicationsof more general models. For example, under the assumption of a stochastic trend in income,a closed form solution from the rational expectation-permanent income model implies that
consumption and income are cointegrated. Campbell (1987) provides supporting evidence
for this prediction using the U.S. quarterly data.4 He further points out that the same closed
4This implication of cointegration is also examined using select international data by Campbell andClarida (1987) for Canada and U.K., MacDonald and Kearney (1990) for Australia, and Shintani (1994) for5form solution of the permanent income model implies that saving, de…ned as the di¤erence
between total disposable income and consumption, should be high when income is expected
to fall.

Essentially this is a logarithmic version of the familiar ‘saving for a rainy day’

implication of the permanent income model found in Campbell and Deaton (1989).
If aggregate output is used in place of total disposable income, the simple permanent
income model suggests that the common stochastic trend is permanent income, ytg = ct , andthe transitory component is savings, ytc = st = yt

ct . In practice, consumption does notnecessarily follow a pure random walk process as suggested by the model. The point, however,is that consumption is less volatile than output and therefore contains better information
about the trend as a result of thoughtful consumption decisions made by rational consumers.
Thus, consumption is expected to be a good proxy for the trend after removing its transitory
component, sometimes referred to as ‘transitory consumption.’ This motivates the following
simple measure of the cyclical component in GDP based on a modi…ed aggregate saving:cgt

ytc = yt

(2.1)

t;

where cgt is the estimated trend based on smoothed consumption. If cgt = ct and if there isno deterministic trend component ( = 0 and= 0), the measure reduces to just aggregate

saving itself, st = yt ct . For the consumption trend, one may simply use the moving average
P 1of lagged consumption as, cgt = (1=M ) M
j=0 ct j where M is the moving average length.
Typically, the HP …lter is applied to output and consumption by minimizing
XT(ytc )2 +XT

XT(cct )2 +

XT

t=1

t=1

g
[(yt+1

ytg )

(ytg

ytg 1 )]2 :(2.2)[(cgt+1

cgt )

(cgt

cgt 1 )]2

(2.3)

and
t=1

separately, wheret=1

is a smoothing parameter chosen by the researcher and ztc = ztztg ,z = y; c. Obviously, by separately …ltering income and consumption, their is no guaranteethat income and consumption obey a cointegrating restriction.
Japan.

6

One option is to employ a constrained multivariate HP …lter which minimizes
XT

XTg(ytc )2 +[(yt+1
ytg )
t=1
t=1
nXT
XT
+! 1
(cct )2 +[(cgt+1
t=1
t=1
XT
+! 2
(cgt ytg )2(ytg

ytg 1 )]2

cgt )

(cgt

cgt 1 )]2

o

t=1

(2.4)

where ! 1 and ! 2 are weights controlling the relative importance of the two variables as wellas the long-run restriction involving the growth components of consumption and income.
The approach developed in this paper does not require the researcher to choose ! 1 and ! 2 .
The HP trend component of consumption which minimizes (2.3), denoted, cgt , is used as themeasure of smoothed consumption. Thus simply setting cgt = cgt and substituting this into
(2.1) de…nes the HP-version of our saving-based business cycle measure.
If yt and ct are cointegrated, yt and cgt are also cointegrated. In the presence of a linear
deterministic trend, the cointegrating vector that eliminates a common stochastic trend may
or may not eliminate a common deterministic trend at the same time. Using the terminology
of Ogaki and Park (1997), two variables are deterministically cointegrated if the cointegrating
relationship also eliminates the deterministic trend. The long-run budget constraint is moreclosely related to this notion of deterministic cointegration, suggesting= 0 in the de…nitionof the saving-based measure of the cycle. However, leaving the room for the possibility of
6= 0 may be useful in some cases as it can proxy for missing additional covariates (e.g.,
an asset variable), a the gradual shift in the preference for the precautionary saving, or thepresence of measurement errors. In applications, the residual from the regression of yt

cgt

on a linear trend allows for such possibilities.
2.2. A bivariate Beveridge-Nelson decomposition of Cochrane’s (1994) error correction model
Cochrane (1994) extracts the information on the trend in GNP from consumption using abivariate error correction model (ECM) of GNP and consumption combined with a multivariate Beveridge-Nelson (BN) decomposition.
7

The speci…cation of the ECM employed here is,
ct =

c1 +

c2 t +

c1

yt =

y1 +

y2 t +y (ct 1

+ y1 ct 1 +y2

ct 1 +c2

ct 2 +

c3

yt 1 +

c4

yt 2 + "ct(2.5)

yt 1 )

ct 2 +y3

yt 1 +y4

yt 2 + "yt(2.6)

where "ct and "yt are non-orthogonalized shocks. The …xed lag length of 2 follows Cochrane.
Note that equation (2.5) involves two modi…cations of Cochrane’s original ECM: includingtime trends in both equations and omitting the error correction term, ct 1

yt 1 , fromthe consumption growth equation. The trend terms are included to re‡ect the eliminationof any deterministic trend from our saving-based cyclical component and the absence oferror correction term in consumption growth equation is related to shock identi…cation as
explained below.
Consider the role of the loading coe¢ cienty . A characteristic equation of a cointegrated

bivariate system should have one unit root and the other root outside the unit circle. Thus,in the absence of an error correction term in (2.5), cointegration of consumption and income
requiresy2 (0; 2) in (2.6). For this reason, the estimate offor the long-run budget constraint. Ifyy

has a direct implication= 0, both consumption and income are di¤er-ence stationary and they are not cointegrated.5 Since the t-statistic follows a nonstandard
distribution under the null hypothesisy= 0, Cochrane employs a bootstrap method toclaim that the coe¢ cient is signi…cantly di¤erent from zero in his analysis. As long as thecointegrating restrictiony 2 (0; 2) is satis…ed, however, the ordinary least squares (OLS)

estimator remains asymptotically normal.
To calculate the impulse response functions of this ECM, Cochrane employs the conventional recursive orthogonalization of shocks with the order consumption and then income.
The identi…ed shocks can be interpreted as the permanent and transitory shocks since theorthogonalized shock in (2.6) which has zero (contemporaneous) impact on consumption cannot be a permanent shock, according to the permanent income model. To be more speci…c,5

Ify < 0 ory > 2, the system is an explosive process.

8the permanent shock, denoted byP
t , and the transitory shock, denoted byTt , are identi…ed

as
P
t
T
t

=R 1

"ct
"yt

where R is the lower triangular matrix which satis…es RR0 = E( t 0t ) =
where

0
t

= ["ct "yt ] and0t= [ PtTt ].

and E( t 0t ) = I

In addition to the permanent response toPt , thepresence of cointegration (i.e., the error correction term) restricts the impulse responses ofconsumption and income to converge to a common level in the long-run in response to either
shock, consistent with the long-run budget constraint.6
To get a sense of how the estimated impulse responses of the ECM compare to those arising from a workhorse macroeconomic model, Figure 1 shows theoretical impulse responses
of income and consumption to permanent and persistent productivity shocks using the onesector neoclassical model calibrated to the U.S. economy by King, Plosser and Rebelo (1988).
The permanent shock scenario is the impulse response of the model following a 1% innovation to productivity under the assumption productivity follows a pure random walk. The
transitory shock scenario assumes a …rst-order autoregressive process for productivity withpersistence set at 0.9. By comparing the top panel of Figure 1 of Cochrane (1994, p. 245)with our Figure 1, we see that Cochrane’s estimated impulse responses of GNP and consumption to permanent (consumption) and transitory (GNP) shocks match up surprisingly
well with those of the calibrated model.
One notable di¤erence is a small permanent e¤ect of what Cochrane identi…es as a transitory (GNP) shock, whereas the persistent, but transitory productivity shock has no long-run
impact of either consumption or output in our estimated system or in the basic neoclassicalmodel. The di¤erence is due to the fact that Cochrane includes the error correction termin both equations of the ECM. Letc be the parameter on the error-correction-term in

consumption growth equation in Cochrane’s speci…cation. Then, the long-run responses to
6

See King, Plosser, Stock and Watson (1991) on this point.

9the identi…ed transitory shock are:@Et (yt+h )@Et (ct+h )
= lim
=
T
h!1
h!1
@ t
@ Tt

c

lim

From Table 1 of Cochrane (1994, p. 243) bc =y:c

0:02 and by = 0:08, and this formula gives along-run impulse response of 0:2 (= 0:02=0:1) consistent with Figure 1 of Cochrane’s paper.
As long as

y 6= 0, the transitory shock,e¤ect if and only ifTt , identi…ed by a recursive scheme has zero long-run

c = 0, as imposed in equation 2.4.

An alternative is to directly impose

a zero long-run impulse response assumption as Blanchard and Quah (1989) do. However,a simple application of Blanchard-Quah method cannot incorporate the contemporaneous
impact of the permanent shock implied by the permanent income model.7
For this reason, the zero restriction on the loading coe¢ cientcin the consumption

growth equation is imposed when estimating the ECM in what follows.

Due to the factthat the two equations no longer have common regressors, OLS becomes ine¢ cient since
the equivalence of OLS and generalized least squares (GLS) no longer holds. To achievee¢ ciency, we employ a restricted multivariate GLS method. The GLS estimator is asymptotically normal as long as the cointegrating restrictiony 2 (0; 2) is satis…ed as shown in

the Statistical Appendix.
Having established the appropriateness of the ECM and the estimation method, the remaining technical detail is the BN decomposition in the ECM context. The multivariate
BN decomposition in a cointegrated system was …rst proposed by Stock and Watson (1988)
and has been used in many applied studies, including Cochrane (1994), Evans and Reichlin(1994), and Lettau and Ludvigson (2004). As in the case of a univariate BN decomposition,
both the trend and cycle components are generated from a common vector error component.
The trend component follows a (multivariate) random walk process, while the cyclical component is serially correlated. This feature contrasts to an alternative decomposition basedon the ECM, often referred to as Granger-Gonzalo decomposition (Gonzalo and Granger,
7
Cochrane also argues that the conventional identi…cation scheme has an advantage over Blanchard andQuah scheme as the former does not require an estimate of the spectral density at frequency zero.

10

1995), where the trend and cycle components are orthogonal, but the trend component isgenerated from serially correlated errors.8
What is the relationship between the BN decomposition (ytg and ytc ) and shocks identi…edby the recursive scheme ( Pt andT
t )?

When there is an error correction term in (2.5),

the bivariate cointegrated system generally implies that the random-walk trend component
ytg is generated by a linear combination of current

P
t

and

T
t .

However, if we impose

c = 0, the long-run impulse response to identi…ed shocks becomes lower triangular, andthus the random-walk trend component ytg is generated only from the permanent shockPt .

In contrast, the cyclical component ytc consists of current and past values of both type ofshocks,

P
t and

T
t .

What does this imply about the relationship between our saving-based business cycle
measure (2.1) and the ECM/BN cycle?9 Cochrane points out that, if consumption follows apure random walk, as predicted by the simple permanent income model, the ECM/BN trend
becomes (log) consumption less the mean of savings, ytg = ct st . Thus the ECM/BN cycle issimply demeaned savings, ytc = yt

ytg = stst . In the presence of a deterministic trend, thecyclical component becomes detrended savings, ytc = stt, which corresponds to (2.1)

with a choice of cgt = ct . What happens if consumption growth is serially correlated? Such an
extension can be considered by imposing

c3 =c4 = 0 in (2.5).

The Statistical Appendix

shows that the ECM/BN cycle once again corresponds to (2.1) provided consumption issmoothed according to:
cgt =

2
X

(2.7)

wi ct ii=0where w0 = 1 (1
c1 + c2 ), w1 = 1 (c1
c1 + c2 ), and w2 = 1 (c2
c1 + c2 ). It is important to note thatthe moving average weights, wi ’s, depend only on the coe¢ cients in (2.5), not on those in(2.6). It is straightforward to obtain a similar result for the more generalized case beyond
8

See also Gonzalo and Ng (2001) for identi…cation of shocks combined with Granger-Gonzalo decompos-

tion, and Levtchenkova, Pagan and Robertson (1998) for a more comprehensive discussion.
9
See Appendix A.2 for our de…nition of the BN cycle in a multivariate context. We adopt the opposite
sign convention for the transitory component, making it procyclical.

11

two lags in the ECM by adding more lags in the moving average of (2.7). In summary, evenin the presence of transitory consumption variation, the ECM/BN cycle takes the form of(2.1) with the moving average weights for consumption determined by the parameters which
capture the short-run dynamics of consumption growth.

3. Estimation of ECMs in the G-7 and Australia
The bivariate error-correction models of (2.5) and (2.6) are estimated on a country-bycountry basis. Quarterly series of GDP and total consumption are used for Yt and Ct ,respectively. Both were obtained from the OECD database. The countries are: Australia,
Canada, France, Italy, Germany, Japan, the United Kingdom, and the United States. The
starting dates of the samples vary, all end in the …rst quarter of 2005 (see Table 1).
Table 1 reports the estimation results obtained using a restricted multivariate GLS
method. The coe¢ cienty on the error correction term is positive in all cases supporting thesaving for a rainy day implication of Campbell (1987): when the representative consumer
increases her consumption above income (dissaving) it foretells an increase in income growth.
Furthermore, all the point estimates fall in the range of cointegrating restrictiony 2 (0; 2)mostly with a tight con…dence interval (exceptions are Italy and the United Kingdom). Thus,
the results are consistent with long-run budget constraints.
In contrast, the short-run dynamics are imprecisely estimated: the coe¢ cients on lagged
consumption and income growth are statistically signi…cant in only 18 of 56 cases. Fourteenof these statistically signi…cant coe¢ cients are accounted for by only three countries: Japan(6), Canada (4) and Italy (4). There is no tendency for lagged growth rates to be moresigni…cant in the consumption equation than in the income equation. In this sense, thecointegrating relationship is the robust feature of the empirical model.
3.1. Decomposition of variance
Table 2 reports variance decompositions at forecast horizons of one quarter, one year andin…nity. As the Table shows, consumption changes are largely unpredictable, consistent with12

a random walk. As a consequence, virtually all of the variance in consumption growth gets
attributed to the permanent shock. In sharp contrast, the variance of output growth is split
almost exactly 50-50 between permanent and transitory shocks for the United States andthis is robust across forecast horizons. Cochrane (1994) attributed 85% of the 1-quarterahead forecast to the transitory component compared to only about 60% here. Most of thisdi¤erence is likely due to the fact that his bivariate speci…cation also included an errorcorrection term in (2.5), thereby allowing transitory shocks to alter the long-run level ofconsumption and income and elevating their importance in the variance decomposition of
output. Germany and the United Kingdom are similar to the U.S. with about 60% of output
variation attributed to the transitory shock. Japan is an outlier with a small fraction ofvariance attributed to the transitory shock (40%). What is interesting about the remaining
countries –Australia, Canada, France and Italy – is that transitory shocks are even more
important than is true of the U.S.. Thus the international evidence against the pure random
walk model of output growth seems even more compelling in other countries than it is for
the United States.
3.2. Impulse responses
The estimated impulse responses, are broadly consistent with those produced using the modelof King, Plosser and Rebelo (1988) presented earlier in Figure 1. The half-life of the income
response to a transitory shock is 7 quarters according to the quantitative theory. The averagehalf-life across countries using the estimated impulse responses (Figure 2) is very close tothis, at 8 quarters. The path back to the steady-state is predicted to be monotonic in thebenchmark theory, typical of our estimates. Consumption displays a hump-shape response
rising for a number of quarters before returning monotonically back to the steady-state level.
According to the theory, the hump-shaped response re‡ects the interaction of wealth andsubstitution e¤ects. The wealth e¤ect of a transitory shock is small as is evident from the factthat the theoretical impulse never exceeds 0.25 percent above the steady-state in response
to a persistent unit (1 percent) productivity shock. The initially high real interest rate,13

generated by higher productivity creates an upward sloping pro…le in the initial few quarters
due to an intertemporal substitution e¤ect.
Turning to the case of a pure random walk productivity shock (i.e. a permanent increasein productivity of 1%), the basic neoclassical model predicts a monotonically rising pro…le ofboth consumption and income. Both variables increase toward new higher steady-state levels,with consumption everywhere below income. The reason for this is that in the new higher
steady-state, the capital is higher than initially to re-equate the higher marginal productof capital (due to higher productivity) with the unchanged steady-state real interest rate.
To accomplish this, investment must rise above the steady-state by more in the short-run
than the long-run; this accounts for the gap in consumption and income along the transition
path. In an open economy the tension between consumption and investment in the income
identity is relaxed and as we shall see in the empirical results, consumption tends to track
income more closely in the permanent shock case in open economies. Interestingly the largest
countries look more like the closed economy simulation, consistent with the notion that thetension between consumption and investment again arises at the global level.
Figure 2 shows the impulse responses are quite similar across countries. Most countries display monotonic transitions of GDP back to the steady-state following a transitory
disturbance.

An exception is Japan, where output keeps rising for three quarters before

adjustment back toward the steady-state begins. The half-life of the response of output inthe neoclassical model is 8 quarters which is within one quarter of the empirical estimatesfor Australia, Canada, France and the United States. Germany has a much lower empirical
half-life of 4 quarters while Japan, Italy and the United Kingdom have half-lives of 2 years
or more. A signi…cant fraction of the higher half-life for Japan is due to the hump-shaped
output response; it takes 11 quarters for the impulse to return to its level at impact.
Turning to consumption, recall the impact response to a transitory shock is restricted to
be zero in the empirical model. Some delay is evident in the maximal estimated consumption
response, but not as much as the theory implies. The consumption pro…les also display

14

di¤erent sensitivity to the same magnitude disturbance across countries.
Permanent shocks carry the economy to a new higher level of consumption and income
while restoring the ratio of these two key macroeconomic variables in both the theory andthe estimated model. Most of the empirical impulses indicate rapid adjustment toward the
new higher steady state with half of the adjustment achieved in the …rst few quarters. Only
Australia, Italy and the United Kingdom take more than one year to account for one-half ofthe distance between the impact response and the new higher steady-state level of output.
The response of the calibrated neoclassical model is considerably slower.
The simplest version of the permanent income hypothesis predicts that consumption rises
in exact proportion to income if the shock is exogenous and permanent. This is what gives
rise to the classic prediction that a permanent income shock has no implication for a nation’s
current account. Here, however, a permanent shock generates a predictable rise in income.
In the neoclassical theory, this is because a permanent rise in productivity leads to a processof capital accumulation to a new higher steady-state, which takes real time to accomplish.
Consumption, therefore is expected to rise above income initially based on permanent income
logic, even in the permanent productivity shock case. Empirically, however, income moves
almost immediately to its long-run level and therefore consumption and income move quite
closely together, in response to the permanent shock. This contrasts strongly with thetransitory shock case where most of the movements are in income, while consumption barely
adjusts.

4. Saving cycles and ECM/BN cycles
4.1. Summary statistics
This section computes our saving-based business cycle measure for each nation and compares
them with the BN cycle obtained from the restricted bivariate ECM estimates as well assome commonly used univariate business cycle measures. We denote Cochrane’s ECM/BN
cycle by ECM-BN. The simplest version of our saving-based measure, namely, the detrended15

savings with the choice of cgt = ct in (2.1) is denoted by SV. The saving-based measure
P4 1
derived using a moving average of consumption cgt = (1=4) j=0
ct j , is denote by SV-MA.

The saving-based measure with the HP-…ltered consumption obtained by minimizing (2.3)
with= 1600 is denoted by SV-HP. The univariate BN business cycle measure obtained

from the OLS estimate of a second-order autoregressive (AR) model of output growth isdenoted by BN, and the standard HP cycle obtained by minimizing (2.2) with= 1600 isdenoted by HP.
Table 3 reports the standard deviations of the cyclical measures. The notable …nding hereis that the magnitude of volatility in general is much larger for the multivariate measures thanfor the univariate measures. For the multivariate measures, ECM-BN cycle and saving-based
cycles, SV in particular, are remarkably similar to each other. For the univariate measures,the largest (across country) cyclical volatility for the BN cycle barely reaches a standard
deviation of 0.5%, the mean across countries is a mere 0.34%. The small contribution of
the BN cycle to the variance of output is the ‡ipside of the argument, by Beveridge andNelson (1981), that output is close to a random walk. The standard deviation of the HP
cycle is always intermediate between the univariate BN model and the multivariate models.
Moreover, cyclical variability of output according to the univariate BN model is trivial.
Table 4 examines the time series correlations of the various estimates of the cycle. The BN
cycle has an average (across countries) correlation of near zero with multivariate measures,such as ECM-BN and SV. The HP cycle is positively correlated with multivariate measures
for all countries, with the highest correlation being 0.72 with SV-HP. Based on the similarityof the moments of the ECM-BN cycle and the SV cycle, one might suspect that the two follow
a similar time path. Figure 3, in fact, shows that the two estimates are virtually identical.
Simply put: to a very close approximation, savings is the cycle in GDP and consumption is
the trend.
The obvious di¤erence between the HP cycle and the SV cycle (see Figure 3) is that the
SV produces business cycles signi…cantly longer in duration and greater in amplitude than

16the HP cycle (for the U.S., the two are more comparable). The average duration of thesecycles would push the upper limits of what business cycle theorists (or the NBER dating
committee, for that matter) would consider reasonable. And yet, if the goal of the exercise
is to parse the data into trend and cycle for purposes of applications to growth theory andbusiness cycle theory, it seems logical to infer the stochastic trend from permanent income
behavior of the mythical representative agent rather than relegate the task to a purely
statistical procedure.
Taking the view that non-in‡ationary or sustainable economic growth is tracked by thegrowth component, the implication of our cyclical measure is dramatically di¤erent fromthe conventional view of the output gap using the HP …lter. The di¤erences involve many
policy relevant aspects: i) the extent of variance around the growth trend, ii) the durationof cycle; and iii) the timing of turning points. The welfare implications of business cycles
is also altered given the greater persistence and volatility of the deviations from the growthpath.
4.2. The international comovement puzzle
Given the analysis above, it should not be surprising that the simple saving-based measureof business cycles has dramatic implications for some key business cycle facts. Consider
the most cited of these: the international correlation of GDP. The correlation of U.S. andforeign business cycles, is uniformly positive using the HP cycle, averaging about 0.5 acrosscountries. This is higher than the correlation of 0.02 produced in the two-sector, two-country
benchmark model of Backus, Kehoe and Kydland (1994).10 The contrast of the observed andsimulated moments has been dubbed the comovement puzzle: pointing to the observationthat models have di¢ culty producing the high international output correlations observed in
the data. Table 5 shows that the SV measure of the cycle in U.S. output has an averagecorrelation of nearly zero with its foreign counterpart. Aiming for this empirical target seems
10

Baxter and Crucini (1995) explore the role of incomplete markets in a one-sector, two-country businesscycle model and …nd signi…cantly positive output correlations arise only when productivity are near random
walks with modest international correlations in the innovations and no dynamic spillovers.

17

likely to change the relative merit of alternative international business cycle models.
4.3. The Great Moderation
Stock and Watson (2005) documented a decline in the size of common, international, shocks
in recent decades. In this sense business cycle volatility may be sensitive not only to the
choice of the method of the trend-cycle decomposition but also to the time period. Table
6 shows the standard deviations of alternative business cycle measures computed for two
subsamples. We follow Stock and Watson and divide the sample into a period that ends
in 1993 and a period starting in 1994. On average, we observe declines in all the businesscycle measures as we move from the …rst period to the second period. However, volatility
reductions are not so obvious when multivariate measures are employed in place of univariate
measures. In fact, standard deviations are actually larger in the second subsample when SV
and SV-MA cycles are employed. It could be that better economic policy smoothed out highfrequency variation while the underlying cycle perceived by consumers remained as volatile
as ever.

5. Conclusions
As emphasized by Canova (1998), alternative detrending …lters extract di¤erent types ofbusiness cycle information from the data.11 The standard approach to business cycle decompositions in macroeconomics is the univariate HP …lter. Baxter and King (1999) andthe NBER business cycle dating committee take nuanced approaches, starting with a formal
de…nition of what a business cycle should look like in terms of duration or judging businesscycle peaks and troughs using multiple indicators but not holding to an algorithm, respectively. Applying minimal theoretical restrictions embodied in traditional macroeconomic
models, such as the permanent income model, we …nd that business cycles do not conformto these conventional approaches. Consequently, many of the empirical challenges in the11

Cogley and Nason (1995) go further and show that the Hodrick-Prescott …lter can create periodicities
in the cyclical component when none exist in the original series.

18

business cycle literature may be di¤erent than was previously thought. Two key dimensionsof business cycle facts found to be very sensitive to …ltering relate to ongoing debates about
the extent of international comovement and the Great Moderation. While obviously central
to business cycle research, these two examples are just illustrative, there are likely otherkey relationships such as the association of nominal and real variables deserving of the same
scrutiny. Much remains to be done.

19Appendix
A1. Multivariate GLS Estimation of Restricted Vector ECMs
An unrestricted bivariate ECM with lag two (with constant and trend term),
ct
yt

c1

=

c2

+

y1

+

c

t+

y2
c1

c3

y1

y3

(ct 1

yt 1 )

y

ct 1
yt 1

+

c2

c4

y2

y4

ct 2
yt 2

+

"ct
"yt

can be written in a matrix form
X = AZ + E
where

X = [ X1 ; :::; XT ],A=

Xt = ( ct ; yt )0 , Z = [Z0 ; :::; ZT 1 ] ;
2
3
1
6 t
7
6
7
7;
c
y
Zt = 6t1
t
1
6
7
4 Xt 1
5
Xt 2
c1

c2

c

c1

c3

c2

c4

y1

y2

y

y1

y3

y2

y4

and E = ["1 ; :::; "T ]. A linear restriction c = 0 on A can be expressed by a =vec(A) = Sr+s,
where vec(A) = ( c1 ; y1 ; c2 ; y2 ; c ; y ; c1 ; y1 ; c3 ; y3 ; c2 ; y2 ; c4 ; y4 )0 ,3
2
I4 0
S = 4 0 0 5;
0 I9
r = ( c1 ;

y1 ;

c2 ;

y2 ;

y;

c1 ;

vec( X) = (Z 0

y1 ;

c3 ;

y3 ;

c2 ;

y2 ;

c4 ;

0
y4 ) and s = 0. Since

IK )vec(A) + vec(E) = (Z 0

IK )Sr + vec(E);a restricted GLS estimator is given byb = Sbb
a = vec(A)r= S S 0 (ZZ 0with its limit distributionpwhere Q =plim ZZ 0 =T .

T (b
a

1
"

)S

1

S 0 (Zhda) ! N 0; S S 0 (Q

20

1
"

1

" )S

)vec( X)

1

S0i

A2. Bivariate Beveridge-Nelson Decomposition of Restricted Vector ECMs
To simplify the derivation, we omit constant and trend term without the loss of generality.
Suppose a vector ECM
p
X
0
Xt =
Xt 1 +Bi Xt i + "ti=1

where Xt is an n 1 vector of variables, is an n r matrix representing cointegrating
vectors, is an n r loading coe¢ cients, Bi ’s are n n coe¢ cient matrices and "t is an
n 1 zero mean error vector. Its VAR(1) representation is given by
Wt = AWt 1 + ut
where Wt = [ Xt0 ; :::; Xt0 p+1 ; 0 Xt 1 ], A is an (np + r) (np + r) coe¢ cient matrix and
ut is an (np + r) 1 error vector. A multivariate version of Beveridge and Nelson (1981)
decomposition for the i-th element of Xt yields its cyclical component given bye c = lim
X
it

k!1

k
X
j=1

ct+jjt =
e0i W

lim

k!1

k
X

e0i Aj Wt = e0i A(IA) 1 Wt

j=1

and trend component given bye c = Xit + e0 A(IXitg = Xit + X
itiA) 1 Wt

where ei is a selection vector for i-th element. Note that we can simply ‡ip the sign of theec .
original BN cycle to much the de…nition of cyclical component in the main text as Xitc = X
itFor a restricted bivariate ECM with transitory consumption ( c = 0 and c3 = c4 = 0), its
VAR(1) representation is simpli…ed to
3
3 2
2
3 2
32
0
0
0
"1t
ct 1
ct
c1
c2
7
7 6 "2t
7 6 y17 6 yt 1
6 yt
y3
y2
y4
y
7
7 6
7 6
76
6
7:
7
6
7
6
6 ct 1 7 = 6 1
0
0
0
0
7+6 0
7
6
7 6
7 6 ct 2
5 4 0
5
5 4 yt 2
4 yt 1 5 4 01
000
ct 1 yt 1"1t "2t
ct yt
c1
y1
y3
c2
y2
y4
y +1

21

Thus, for the transitory component of yt , we haveyetc =

ytc =

2

c1

1

6
y1
6
6 1
6
4 0
h

= yt

0
1
0

c1

y1

=

y3

c1

0

c1 +

1

y3

y2

0
1

y1

y3

c2

0

c2

y

2

c2

ct

c2 )

0
0

3
ct
7
i6
6 yt
7
7
0 1 6
6 ct 1 7
4 yt 1 5
ct yt

c2( c1 +

7
7
7
7
5

y

y4

1

( c1 +

0
0

3 12

0

0
1

1 ( c1 + c2 )

c2

y2

y4

y2

y

0
0
c2

3

0
y4

0
0

1
0

y3

0

c2

y2

1

c1 + c2
1 ( c1 + c2 )

ct

0

c1

6 y1
6
[ 0 1 0 0 0 ]6
6 1
4 0

=

A) 1 Wt[ 0 1 0 0 0 ]A(I
2

y4

y +1

3
ct
6 yt
7
6
7
6 ct 1 76
7
4 yt 1 5
ct yt

7
7
7
7
5

ct 1 :c2 )The trend component of yt can be obtained as
ytg = yt=

ytc = yt + ct
1

1

( c1 +

c2 )

yt +

ct +

c1 +

1

c2

( c1 +c1

1( c1 +

c2 )

c2 )

ct +

ct 1 +

1c2

1( c1 +c2 )c2

ct 2 :( c1 +c2 )ct 1

Consistent with the argument of Cochrane, in the case of random walk consumption withcc1 = c2 = 0, the results above implies that the cyclical component reduces to yt = yt ct =gst (saving) and the trend component reduces to yt = ct (consumption).

22

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Trend and Di¤erence Stationary Time Series: Implications for Business Cycle Research.
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[12] Evans, George, and Lucrezia Reichlin, 1994. Information, Forecasts, and Measurementof the Business Cycle. Journal of Monetary Economics 33, 233-254.
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[14] Gonzalo, Jesús and Serena Ng, 2001. A Systematic Framework for Analyzing the Dynamic E¤ects of Permanent and Transitory Shocks, Journal of Economic Dynamics andControl 25, 1527-1546.
[15] Hodrick, Robert J. and Edward C. Prescott, 1997. Postwar U.S. Business Cycles: An
Empirical Investigation. Journal of Money, Credit and Banking 29(1), pages 1-16, February.
[16] King, Robert G., Charles I. Plosser and Sergio Rebelo, 1988. Production, Growth andBusiness Cycles I: The Basic Neoclassical Model. Journal of Monetary Economics 21,
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[17] King, Robert G., Charles I. Plosser, James H. Stock and Mark W. Watson, 1991,
Stochastic Trends and Economic Fluctuations. American Economic Review 81, 819840.
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24

[19] Levtchenkova, S., A. R. Pagan and J. C. Robertson, 1998. Shocking Stories. Journal of
Economic Surveys 12(5), 507-532.
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Income. Journal of Money, Credit and Banking 39(2-3), 615-638.
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[26] Shintani, Mototsugu, 1994. Cointegration and Tests of the Permanent Income Hypothesis: Japanese Evidence with International Comparisons. Journal of the Japanese andInternational Economies 8, 144-172.
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25

Table 1. –Bivariate Vector ECM Estimates
Country
Australiact
1959:3 to 2005:1
T = 183yt

const.

trend

0:81(5:24)
4:82(2:44)

0:00(0:78)
0:00(0:70)

0:90(4:72)
6:02(3:23)

ct 1

yt 1

ct 1

ct 2

yt 1

yt 2

0:00
( )
0:08
(1:91)

0:04(0:48)
0:09(0:70)

0:01(0:13)
0:06(0:47)

0:09(1:62)
0:06(0:67)

0:05(0:94)
0:01(0:15)

0:00(1:67)
0:00(0:19)

0:00
( )
0:10
(2:84)

0:19(2:17)
0:15(1:66)

0:03(0:30)
0:19(2:15)

0:29(3:21)
0:18(1:97)

0:05(0:54)
0:05(0:55)

0:67(3:84)
3:18(2:32)

0:00(1:60)
0:00(0:32)

0:00
( )
0:05
(2:05)

0:13(1:21)
0:11(1:45)

0:22(2:17)
0:15(2:07)

0:02(0:16)
0:12(1:07)

0:09(0:64)
0:13(1:35)

1:19(5:17)
9:04(3:87)

0:01(2:71)
0:00(2:11)

0:00
( )
0:15
(3:58)

0:33(3:13)
0:14(1:46)

0:14(1:36)
0:05(0:55)

0:04(0:35)
0:07(0:61)

0:14(1:24)
0:13(1:24)

Canadact
1961:1 to 2005:1
T = 177yt

Francect
1970:1 to 2005:1
T = 141ytGermanyct
1970:1 to 2005:1
T = 141yt

Notes: The regressions are of the form:
ct =

c1 +

c2 t +

c1

yt =

y1 +

y2 t +y (ct 1

+ y1 ct 1 +where ct = 100

y2

ln Ct and yt = 100

ct 1 +c2

ct 2 +

c3

yt 1 +

c4

yt 2 + "ct

yt 1 )

ct 2 +y3

yt 1 +y4

yt 2 + "yt

ln Yt . Restricted multivariate generalized least squares

estimates.

26

Table 1 (continued). –Bivariate Vector ECM Estimates
CountryItalyct
1970:1 to 2005:1
T = 141yt

const.

trend

0:40(3:30)
7:83(3:24)

0:00(1:98)
0:01(3:09)

2:51(7:72)
4:57(4:21)

ct 1

yt 1

ct 1

ct 2

yt 1

yt 2

0:00
( )
0:13
(3:16)

0:42(4:56)
0:31(2:20)

0:16(1:75)
0:15(1:09)

0:14(2:27)
0:30(3:17)

0:08(1:34)
0:05(0:54)

0:01(6:48)
0:01(2:80)

0:00
( )
0:05
(2:45)

0:48(4:42)
0:26(2:56)

0:20(1:80)
0:10(0:98)

0:26(2:28)
0:22(2:06)

0:32(2:77)
0:32(2:96)

0:55(3:09)
2:27(2:24)

0:00(0:82)
0:00(1:32)

0:00
( )
0:03
(1:70)

0:08(0:86)
0:11(1:31)

0:10(1:06)
0:11(1:30)

0:01(0:12)
0:14(1:58)

0:01(0:06)
0:02(0:20)

0:60(4:71)
5:15(3:20)

0:00(0:17)
0:01(2:77)

0:00
( )
0:10
(3:06)

0:12(1:30)
0:44(3:76)

0:10(1:03)
0:20(1:62)

0:10(1:41)
0:02(0:23)

0:01(0:15)
0:00(0:05)

Japan
ct
1955:2 to 2005:1
T = 200

yt

United Kingdom
ct
1955:1 to 2005:1
T = 201yt

United Statesct
1955:1 to 2005:1
T = 201yt

27

Table 2. –Consumption and Income Variance Decompositions
Variance of:Country
Australia

CanadaFranceGermany

Italy

Japan

UnitedKingdom
UnitedStatesct
Accounted for by
Permanent Transitory
Horizonshocksshocks
1
100:0
0:0
4
98:3
1:7
1
98:2
1:8
1
100:0
0:0
4
94:9
5:1
1
94:6
5:4
1
100:0
0:0
4
99:7
0:3
1
99:7
0:3
1
100:0
0:0
4
99:1
0:9
1
99:1
0:9
1
100:0
0:0
4
97:7
2:3
1
97:6
2:4
1
100:0
0:0
4
96:0
4:0
1
95:7
4:3
1
100:0
0:0
4
100:0
0:0
1
100:0
0:0
1
100:0
0:0
4
99:1
0:9
1
99:0
1:0

Note: See Table 1 for the sample period.

28

yt
Accounted for by
Permanent Transitory
shocksshocks
22:5
77:5
22:6
77:4
22:3
77:7
29:5
70:5
38:1
61:9
37:2
62:8
37:3
62:7
47:0
53:0
46:8
53:2
38:3
61:7
38:4
61:6
37:5
62:5
18:1
81:9
35:4
64:6
36:1
63:9
58:4
41:6
58:4
41:6
57:3
42:7
38:9
61:1
39:2
60:8
39:1
60:9
41:2
58:8
51:5
48:5
50:8
49:2

Table 3. –Standard Deviations of Alternative Business Cycle Measures
Multivariate measures
Cochrane’s
Saving-based
measuremeasures
ECM-BNSV SV-MA SV-HP
Australia
2.49
2.21
2.50
2.31
Canada
2.15
1.57
1.77
1.85
France
1.49
1.38
1.48
1.49
Germany
1.73
1.58
1.59
1.77
Italy
2.06
1.53
1.76
1.69
Japan
3.22
2.15
2.34
2.54
United Kingdom
3.30
3.31
3.30
3.37
United States
1.69
1.40
1.60
1.94
Averages
2.27
1.89
2.04
2.12
Note: See Table 1 for the sample period.

29

Univariate
measures
BN HP
0.09 1.44
0.34 1.39
0.44 0.91
0.07 1.34
0.57 1.43
0.52 1.50
0.10 1.45
0.55 1.55
0.34 1.38

Table 4. –Correlation of Alternative Business Cycle Measures
Multivariate measures
Cochrane’s
Saving-based
measuremeasures
ECM-BNSV SV-MA SV-HP(1) Correlation with ECM-BN
Australia
1.00
1.00
0.94
0.87
Canada
1.00
0.99
0.85
0.75
France
1.00
0.99
0.83
0.76
Germany
1.00
0.98
0.91
0.80
Italy
1.00
0.83
0.50
0.43
Japan
1.00
0.99
0.92
0.88
United Kingdom
1.00
1.00
0.95
0.88
United States
1.00
0.97
0.75
0.72
Averages
1.00
0.97
0.83
0.76
(2) Correlation with SV
Australia
–
1.00
0.95
0.89
Canada
–
1.00
0.88
0.77
France
–
1.00
0.90
0.81
Germany
–
1.00
0.85
0.77
Italy
–
1.00
0.90
0.66
Japan
–
1.00
0.92
0.87
United Kingdom
–
1.00
0.95
0.89
United States
–
1.00
0.88
0.79
Averages
–
1.00
0.90
0.81
(3) Correlation with HP
Australia
–
–
0.58
0.74
Canada
–
–
0.38
0.77
France
–
–
0.29
0.66
Germany
–
–
0.49
0.79
Italy
–
–
0.36
0.79
Japan
–
–
0.51
0.72
United Kingdom
–
–
0.18
0.43
United States
–
–
0.50
0.86
Averages
–
–
0.41
0.72
Note: See Table 1 for the sample period.

30

Univariate
measures
BN HP
0.27
-0.05
-0.03
-0.18
0.14
-0.09
0.03
0.06
0.02

0.49
0.26
0.23
0.42
0.07
0.39
0.08
0.39
0.29

0.34
-0.18
-0.16
-0.20
-0.24
-0.14
0.01
-0.16
-0.09

0.51
0.30
0.25
0.40
0.22
0.39
0.08
0.45
0.33

0.39
-0.31
-0.34
-0.36
-0.30
-0.38
-0.34
-0.34
-0.25

1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00

Table 5. –International Business Cycle Comovement with the U.S.
Multivariate measures
Cochrane’s
Saving-based
measuremeasures
ECM-BNSV SV-MA SV-HP
Australia
0.24
0.27
0.31
0.40
Canada
0.01
0.00
0.06
0.45
France
0.25
0.24
0.20
0.23
Germany
-0.05
-0.06
-0.09
0.15
Italy
-0.17
-0.15
-0.04
0.19
Japan
-0.04
-0.04
-0.07
0.15
United Kingdom
-0.10
-0.11
-0.04
0.20
United States
1.00
1.00
1.00
1.00
Averages
0.02
0.02
0.05
0.25
Note: Sample period is 1970:Q4 to 2005:Q1.

31

Univariate
measures
BN HP-0.33 0.43
0.57 0.72
0.29 0.36
0.17 0.43
0.15 0.33
0.30 0.40
0.34 0.65
1.00 1.00
0.21 0.47

Table 6. –Standard Deviations of Alternative Business Cycle Measures: Subsamples
Multivariate measures
Cochrane’s
Saving-based
measuremeasures
ECM-BNSV SV-MA SV-HP(1) 1970Q4 - 2005Q1
Australia
2.57
2.27
2.55
2.24
Canada
1.96
1.45
1.78
1.81
France
1.48
1.37
1.48
1.50
Germany
1.68
1.54
1.59
1.75
Italy
2.05
1.52
1.76
1.68
Japan
2.70
1.79
1.84
2.11
United Kingdom
3.05
3.11
3.11
3.18
United States
1.68
1.31
1.29
1.87
Averages
2.15
1.80
1.93
2.02
(2) 1970Q4 - 1983Q4
Australia
3.36
2.97
3.28
2.77
Canada
2.00
1.47
1.73
1.71
France
1.61
1.44
1.53
1.49
Germany
2.12
1.87
2.01
2.20
Italy
1.86
1.34
1.81
2.08
Japan
2.66
1.78
2.08
2.23
United Kingdom
2.05
2.05
2.21
2.29
United States
2.00
1.42
1.14
2.41
Averages
2.21
1.79
1.97
2.15
(3) 1984Q1 - 2005Q1
Australia
1.63
1.44
1.68
1.65
Canada
1.95
1.45
1.80
1.87
France
1.40
1.34
1.45
1.49
Germany
1.34
1.28
1.25
1.41
Italy
2.13
1.61
1.71
1.35
Japan
2.02
1.38
1.46
1.63
United Kingdom
2.02
2.07
2.33
1.96
United States
1.41
1.19
1.30
1.33
Averages
1.74
1.47
1.62
1.59

32

Univariate
measures
BN HP
0.08
0.34
0.44
0.06
0.58
0.45
0.10
0.51
0.32

1.34
1.50
0.91
1.34
1.43
1.32
1.50
1.61
1.37

0.11
0.43
0.51
0.07
0.78
0.50
0.14
0.72
0.41

1.68
1.75
0.97
1.60
2.05
1.57
1.96
2.29
1.74

0.05
0.28
0.40
0.06
0.40
0.37
0.06
0.31
0.24

1.08
1.31
0.87
1.16
0.87
1.16
1.13
0.96
1.07

Figure 1. Impulse responses from King, Plosser and Rebelo Model
0.8

0.7

percentage deviation

0.6

0.5
Consumption response to a permanent shock
Output response to a permanent shock
Consumption response to a transitory shock
Output rseponse to a transitory shock

0.4

0.3

0.2

0.1

0
0

5

10

15

20
25
quarters after the shock

30

35

40

Figure 2. Impulse Response
Functions ECM/BM Model
Australia

Canada

1

1

0.5

0.5

0
0

10

20

30

0
0

40

10

20

30

40

30

40

30

40

30

40

Germany

France
1

1

0.5

0.5

0

0

0

10

20

30

40

Italy
1.5

c-permy-permc-trans
y-trans

0

Japan

1

0.5

0.5
10

20

30

40

20

1.5

1

0
0

10

0
0

10

United Kingdom

20
United States

1.5

1.5

1

1

0.5
0.5
0
0

10

20

30

40

0
0

10

20

Figure 3. Business cycle estimates for select countries
CanadaAustralia
8

10

6
5

4
2

0

0
-5

-2
-4

-10
-6
-15
1960

1970

1980

1990

2000

-8
1960

1970

1980

1990

2000

1990

2000

BN-ECMHP
SAVING RATE
United Kingdom

United States

15

5

10
5
0
0
-5
-10
1960

1970

1980

1990

2000

-5
1960

1970

1980