HIAS-E-90

The Great Moderation: Updated Evidence with Joint Tests forMultiple Structural Changes in Variance and Persistence
Pierre Perron(a) and Yohei Yamamoto(b)(a) Department of Economics, Boston University, 270 Bay State Rd., Boston, MA, 02215(b) Department of Economics, Hitotsubashi University, 2-1 Naka, Kunitachi, Tokyo Japan, 186-8601

September, 2019

Hitotsubashi Institute for Advanced Study, Hitotsubashi University
2-1, Naka, Kunitachi, Tokyo 186-8601, Japan
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The Great Moderation: Updated Evidence with Joint
Tests for Multiple Structural Changes in Variance andPersistence
Pierre Perron!

Yohei Yamamotoy

Boston University

Hitotsubashi University

September 16, 2019

Abstract
We assess the empirical evidence about the Great Moderation using a comprehensive framework to test for multiple structural changes in the coe¢cients and in thevariance of the error term of a linear regression model provided by Perron, Yamamoto,and Zhou (2019). We apply it to U.S. real GDP and its major components for theperiod 1960:1 to 2018:4. A notable feature of our approach is that we adopt an unobserved component model, allowing for two breaks in the trend function in 1973:1
and 2008:1, in order to obtain a stationary or cyclical component modelled as an autoregressive process. First, we conÖrm evidence about the Great Moderation, i.e., astructural change in variance of the errors in the mid-80s for the various series. Second,
additional breaks in variance are found in 1970:3 for GDP and production (goods), after which the sample standard deviation increased by three times. Hence, a part of
the Great Moderation can be viewed as a reversion to the pre-70s level of volatility.
Third, the evidence about systematic changes in the sum of the autoregressive coe¢cients (a measure of persistence) is weak over the whole sample period. Finally, we Öndlittle evidence of structural changes occurring in both the variance and the coe¢cients
following the Great Recession (2007-2008). These results support views emphasizing
the ìgood luckî hypothesis as a source of the Great Moderation, which continues evenafter the Great Recession.
JEL ClassiÖcation Numbers: C22, C32
Keywords: the Great Moderation; the Great Recession; multiple structural changes;
joint tests; structural change; trend-cycle decomposition.

!

Department of Economics, Boston University, 270 Bay State Rd., Boston, MA, 02215 (perron@bu.edu).
Department of Economics, Hitotsubashi University, 2-1 Naka, Kunitachi, Tokyo Japan, 186-8601(yohei.yamamoto@econ.hit-u.ac.jp).
y1Introduction

One of the striking empirical facts in the U.S. economy is the presence of a sharp declineof macroeconomic volatility in the mid-80s, the so-called Great Moderation. Blanchard andSimon (2001) documented that the sample standard deviation of quarterly real GDP growth
rates, i.e., the logarithmic ﬁrst-diﬀerences, declined by half after the mid-80s. More formal
statistical results were provided by Kim and Nelson (1999), who allowed a one-time structuralchange in the mean and the variance at the same (unknown) time in a Markov-switchingmodel. They applied it to the real GDP growth rates and showed evidence of a decline in themean gaps between booms and recessions starting in 1984:1. McConnell and Perez-Quiros(2000) used a two step testing procedure for real GDP growth, i.e., ﬁtting some conditionalmean model and then testing for a structural break in the mean of the absolute value of theresiduals. Using the single structural break tests proposed by Andrews (1993) and Andrews
and Ploberger (1994), they claimed evidence of a structural break in 1984:1 in the variance ofthe errors. Stock and Watson (2002) used a single structural change test for a break at some
unknown date applied to the conditional mean of an autoregressive model (AR) of order 4 andsubsequently applied the same test to the mean of the absolute residuals. They investigated168 U.S. macroeconomic variables, detrended mostly via logarithmic ﬁrst-diﬀerences (growth
rates), and found that aggregate output as well as employment, consumption and sectorial
output showed structural changes in the error variance in the mid-80s.
One issue which has important policy implications is that a decline in the unconditionalvariance of an observed series can be ascribed partly to a decline in the propagation dynamics(the persistence of the shocks) and partly to that of the variance of the shocks themselves.
If one adopts an AR model speciﬁcation, the former is represented by a structural change
in the AR coeﬃcients and the latter pertains to a change in the variance of errors. In
the literature, the former type of change is usually viewed as supporting the hypothesisof “improved monetary policy” as a source of the Great Moderation (Clarida, Gali and
Gertler, 2000, Cogley and Sargent, 2001, Bernanke, 2004, Herrera and Pesavento, 2005,
Gali and Gambetti, 2009), while the latter type of change is perceived as emphasizing the“good luck” hypothesis (Stock and Watson, 2002, Ahmed, Levin and Wilson, 2004, Sensierand van Dijk, 2004, Sims and Zha, 2006), although the conclusions of these studies areoften ambivalent. Another hypothesis pertains to production smoothing achieved by better
inventory management that occurred in the durable manufacturing industries (Ramey andVine, 2006, Herrera and Pesavento, 2005, Stiroh, 2009, Morley and Singh, 2016). Summers

1

(2005) and Benati (2008) documented international evidence about the Great Moderation.
Also of interest is the fact that changes in the variance of structural shocks in a constantcoeﬃcients setting can be used as a tool to identify simultaneous equation systems (Rigobon,
2003, Lanne and Lütkepohl, 2008).
Of more recent interest is a debate as to whether the Great Moderation ended due tothe ﬁnancial crisis of 2007-08, i.e., the Great Recession. Clark (2009) documented a signiﬁcant rise in macroeconomic volatility and claimed that it overturned the Great Moderation.
Blanchard (2014) suggested that a benign view of economic ﬂuctuation until 2008 needed tobe reassessed. Stock and Watson (2017) argued that the low-volatility period lasted through2007 after which the economy seems to have returned to a higher volatility regime. Taylor(2011) pointed out that macroeconomic policies were deviating from the standard rules, arguing that such deviations ended the Great Moderation and fostered the Great Recession.
On the other hand, Gadea et al. (2018) found that output volatility remained subdued evenafter the Great Recession, applying a two step testing procedure for structural changes in
the conditional mean and variance similar to that of McConnell and Perez-Quiros (2000) for
GDP growth from 1953:1 to 2014:3.1 Gadea et al. (2019) used historical data of U.S. GDP
growth from 1875:1 to 2014:2 and reached a similar conclusion.
In this paper, we investigate the aforementioned problems using multiple structuralchange tests proposed by Perron, Yamamoto and Zhou (2019); henceforth PYZ. They provided a comprehensive treatment to jointly test for multiple structural changes in the unconditional variance of the errors and the parameters of the conditional mean in a linear
regression model. They proposed quasi likelihood ratio tests building on the work of Qu
and Perron (2007). In particular, their framework provides (a) tests for given number ofchanges in regression coeﬃcients and variance of the errors; (b) tests for some unknown
number of changes within some pre-speciﬁed maximum; (c) tests for changes in variance (coeﬃcients) allowing for a given number of changes in the coeﬃcients (variance); (d) sequential
procedures to estimate the number of changes in the coeﬃcients and in the variance. This
framework has clear beneﬁts to detect multiple structural changes in the coeﬃcients and in
the variance, compared to the two-step testing procedure commonly used in the literature(McConnell and Perez-Quiros, 2000, Stock and Watson, 2002, Herrera and Pesavento, 2005,
Gadea et al., 2018). This is so because testing for mean breaks ignoring variance breaks
1
They also complemented their results using the same method as ours but allowing only for a maximumof one change in each of the parameters of the conditional mean and the variance of the errors under thenull hypothesis.

2induces a power loss and testing for variance breaks ignoring mean breaks yields tests with
important size distortions. See Perron and Yamamoto (2019) for extensive simulation results.
Almost all existing studies use the logarithmic ﬁrst-diﬀerences of the original series.
However, this is a ﬁlter that aﬀects the spectrum of the series in diﬀerent ways at diﬀerent
frequencies and, hence, may downplay low frequency variabilities. We instead use a trendcycle decomposition based on an unobserved component model allowing two breaks in theslope of the trend component in 1973:1 and 2008:1. This is in the spirit of Perron andWada (2009) who showed that, for U.S. real GDP data from 1947:1 to 1998:2, allowing
for a break in the trend function in 1973:1 yields a cyclical component that captures the
NBER business cycle chronologies of recessions and expansions very well. Our framework
is a simple extension of their methods by allowing an additional break in 2008:1. This also
reduces the risk that the structural change tests are biased due to small shifts in the meanof the ﬁrst-diﬀerenced series, which allows us to focus more precisely on the issue of changes
in the variance of the shocks versus changes in the persistence of the series. It also allows
avoiding problems of inference when the ﬁrst-diﬀerenced series have a noise component thatis over-diﬀerenced. As in Perron and Wada (2009), our resulting cyclical component captures
the NBER business cycle chronologies of recisions and expansions very well.
The rest of the paper is structured as follows. Section 2 outlines the econometric methods
used and the multiple structural change tests proposed by PYZ. Section 3 describes the data
used and the detrending method applied to have a stationary component. Section 4 presents
the results and discusses their relevance. Section 5 provides brief concluding comments.
2

Econometric Methods

We consider an autoregressive model in which p breaks are allowed in the coeﬃcients of theconditional mean and the variance of the errors:fw = Wm−1+ 1> ===> Wmf >|w = {w  + }w  m + hw >(1)

for m = 1> ===> p + 1. This is a partial structural change model in which the coeﬃcients associated with the regressors }w (a vector of dimension t) change at the break dates (W1f > ===> Wpf ),f= W . The q breaks inwhich are treated as unknown, with the convention W0f = 0 and Wp+1yy
the variance of errors hw occur at the unknown dates (W1 > ===> Wq ). Accordingly, hw has zeroy
+ 1 ≤ w ≤ Wly with l = 1> ===> q + 1, again with W0y = 0 andmean and variance  2l for Wl−1y
= W . The breaks in the variance and in the coeﬃcients may happen at diﬀerent times
Wq+1
or may overlap partly or completely.
3

We consider three sets of testing problems (TPs). The ﬁrst pertains to testing for given
number of changes in coeﬃcients and variance: TP-1: K0 : {p = q = 0} versus K1 : {p = 0,q = qd }; TP-2: K0 : {p = pd > q = 0} versus K1 : {p = pd , q = qd }; TP-3: K0 : {p =0> q = qd } versus K1 : {p = pd , q = qd }; TP-4: K0 : {p = q = 0} versus K1 : {p = pd ,q = qd }, where pd and qd are some positive numbers selected a priori. In the second set,
we consider testing for an unknown number of changes within some pre-speciﬁed maximum:
TP-5: K0 : {p = q = 0} versus K1 : {p = 0, 1 ≤ q ≤ Q}; TP-6: K0 : {p = pd > q = 0}versus K1 : {p = pd , 1 ≤ q ≤ Q}; TP-7: K0 : {p = 0> q = qd } versus K1 : {1 ≤p ≤ P, q = qd }; TP-8: K0 : {p = q = 0} versus K1 : {1 ≤ p ≤ P, 1 ≤ q ≤ Q}.
Finally, sequential procedures to estimate the number of changes are considered via TP-9:
K0 : {p = pd > q = qd } versus K1 : {p = pd + 1, q = qd }; TP-10: K0 : {p = pd > q = qd }versus K1 : {p = pd , q = qd + 1}. These are useful to assess the adequacy of a model witha particular number of breaks by looking at whether including one more break is warranted.
Building on the work of Qu and Perron (2007), PYZ provided a quasi-likelihood ratio
supremum-type test for each testing problem. Denote the Gaussian log-likelihood of thelinear regression model by oW under {p = 0, q = 0}, oW (W1y > ===> Wqy ) under {p = 0, q 6= 0},oW (W1f > ===> Wpf ) under {p 6= 0> q = 0}, and oW (W1f > ===> Wpf ; W1y > ===> Wqy ) under {p 6= 0, q 6= 0} (seePYZ for the exact expressions). To obtain the break dates, we maximize the log-likelihood
function using a trimming [%W ] for the admissible candidate break dates, where % is somesmall positive constant fraction. Note that we denote the estimates of the break dates incoeﬃcients and variance by a “∼” when they are obtained jointly and by a “ˆ” when they
are obtained separately. Then, the test statistics for TP-1 to TP-4 are:VxsOU1>W (q) = 2[oW (Ŵ1y > ===> Ŵqy ) − oW ]>(TP-2) VxsOU2>W (q|p) = 2[oW (We1f > ===> Wepf ; We1y > ===> Weqy ) − oW (Ŵ1f > ===> Ŵpf )]>(TP-3) VxsOU3>W (p|q) = 2[oW (We1f > ===> Wepf ; We1y > ===> Weqy ) − oW (Ŵ1y > ===> Ŵqy )]>(TP-1)(TP-4) VxsOU4>W (p> q) = 2[oW (We1f > ===> Wepf ; We1y > ===> Weqy ) − oW ]>and, for TP-5 to TP-8, the corresponding UDmax type tests of Bai and Perron (1998) are:(TP-5)

XGpd{1>W (Q) =(TP-6) XGpd{2>W (Q|p) =(TP-7) X Gpd{3>W (P|q) =(TP-8) XGpd{4>W (P> Q) =max q−1 VxsOU1>W (q)>1≤q≤Q

max q−1 VxsOU2>W (q|p)>1≤q≤Q

max p−1 VxsOU3>W (p|q)>1≤p≤P

max max (p + q)−1 VxsOU4>W (p> q)>1≤p≤P 1≤q≤Q

4

PYZ showed that, for all tests except for TP-4 and TP-8, one can use the critical values
provided by Bai and Perron (1998), while those of the tests for TP-4 and TP-8 are provided
in PYZ with P and Q up to 2. Here, we provide additional critical values for P andQ up to 3. Those for the VxsOU4>W and X Gpd{4>W tests of size 1%, 2.5%, 5% and 10%
are presented in Table 1 for t between 1 and 5 with the truncation parameter % = 0=10.
Results are also provided for (P> Q) = (3> 2) and (2> 3) with % = 0=15. The critical values
are obtained via simulations of the limit distributions obtained in PYZ with the Wiener
processes approximated by the partial sums of l=l=g= standard normal random variables with1,000 discrete steps and 10,000 replications.
TP-9 and TP-10 are used to select the number of breaks in coeﬃcients (intercept andautoregressive parameters) and in variance. When selecting the number of coeﬃcient breaks,
we consider TP-9 and the test VxsOU9>W (p + 1|p> Q) is applied (see PYZ for the exact
expressions), starting with K0 : p = 0 and K1 : p = 1, where Q = 3. Upon a rejection,
we proceed to K0 : p = 1 and K1 : p = 2, and so on until the test stops rejecting.
Similarly, to select the number of breaks in the variance, TP-10 is considered and the test
VxsOU10>W (q + 1|P> q) with P = 3 is used for q = 0> 1> === until a non-rejection occurs.
3

Data and detrending method

We use seasonally-adjusted quarterly US real GDP and its major components: consumption,
investment, exports, imports and government spending, covering the period 1947:1 to 2018:4.
We also use the components of real GDP by major products: production (goods), production(services) and production (structures). A logarithmic transformation is applied to the raw
data and then multiplied by 100. To eliminate the trend component we use the following
unobserved component model (Clark, 1987, Morley, Nelson and Zivot, 2003, Perron and
Wada, 2009):|w =  w + fw ;  w = w +  w−1 +  w ; !(O)fw = %w
where  w is the trend component, fw is the cyclical component and !(O) is either an AR(1)or AR(2) lag-polynomial, i.e., !(O) = 1 − !1 O − !2 O2 , with O the usual lag operator. The
AR(2) speciﬁcation is adopted for all series except consumption, investment and production(services), for which the AR(1) speciﬁcation is preferred. The shocks are speciﬁed by⎤⎞⎡ ⎤⎛⎡ ⎤ ⎡2
0  %⎦⎠ >⎣ w ⎦ ∼ l=l=g=Q ⎝⎣ ⎦ > ⎣ %w
 %  2%
05where  % = 0 is assumed; specifying a non-zero value leads to similar results. We follow
Perron and Wada (2009) who showed that, using the US real GDP from 1947:1-1998:2, if
a change in the slope of the trend is accounted for in 1973:1, the method yields a cyclicalcomponent which accords very well with the NBER chronology of recessions and expansions
and it leaves little variation, if any, to the trend component. Since our sample period includes
the Great Recession, we allow for an additional break in 2008:1 so thatw =  + g1 L(w A 1973:1) + g2 L(w A 2008:1)=
The model is estimated via the standard Kalman ﬁlter algorithm. Figure 1 presents graphsof a) the original series and the ﬁtted trend (left panels) and b) the deviations of the original
series from trend, or cyclical components (right panels). The parameter estimates 2 are reported in Table 2. What is important is that for all series, except “Government spending”,
the estimate of  2 is 0. Hence, all the source of variability is contained in the cyclical component, which is indeed nearly identical to the residuals from a piecewise linearly detrended
series with changes in slope in 1973:1 and 2008:1. Note that Luo and Startz (2014) allowed
a trend break in 2006:1 and obtained a characterization similar to that of Perron and Wada
(2009). Our results are robust to alternative choices for the second break date; e.g., 2006:1
or 2008:1, because it only aﬀects the timing when the cyclical component is perceived as
becoming above trend. Both estimates of g1 and g2 are negative. The ﬁrst relates to thewell document productivity slowdown that occurred in 1973. The second decrease in slope
in 2008:1 is more contentious but of no consequence for our analysis. Without it, the cyclicalcomponent is basically ﬂat suggesting the absence of a recovery or expansion, contrary towhat is widely perceived to have happened. With the decrease in slope, the cyclical component shows a clear upward trend that is consistent with an expansion. We prefer the lattercharacterization. It indicates that an expansion has indeed occurred but with respect to alower long-term trend, i.e., a very slow recovery as the last 10 years have been commonly
characterized. In any event, since the variance of the noise around the trend,  2 , is null,none of our results about the variability and persistence of the cyclical component is aﬀectedby this modeling choice. It is basically a matter of labeling the post-2008 period as a slow
“below trend expansion” or an “above smaller trend expansion”.
Using our approach to obtain a stationary component has several advantages over thecommonly used application of using the logarithmic ﬁrst-diﬀerences, i.e., the growth rates.
2

The standard errors reported are the diagonal elements of the negative inverse of the empirical Hessianof the log likelihood function obtained from the MATLAB version R2019a solver (fminunc function in theiroptimization toolbox).

6

First, ﬁrst-diﬀerencing is a ﬁlter that aﬀects the spectrum of the series in diﬀerent ways
at diﬀerent frequencies. It may downplay changes in variability at low frequency. Second,
when applying the structural change tests, the trend breaks may be diﬃcult to capture asﬁrst-diﬀerencing translates them into small changes in mean. Testing for such changes inmean becomes problematic because the ﬁrst-diﬀerences involves a resulting series with anover-diﬀerenced noise. Third, the unobserved component model shows that the resulting
noise is at least mean stationary, so that it will not bias the tests and allow us to focus
on changes in persistence and variance. Fourth, the series looks much more like a cyclicalcomponent which accords well with the NBER chronology of expansions and recessions.
Note that for “Government spending” one should also consider possible changes in thevariance of w since the estimate of  2 is non-zero. We obtained an estimate of w as theﬁrst-diﬀerences of the smoothed estimates of  w from which we subtracted the estimate ofw . A plot of this estimate is presented in Figure 2 along with the estimate of the cyclicalcomponent fw . It is evident that the magnitude of the former is negligible compared to
the latter. Hence, any issue of changes in the variance of  w will be negligible compared tochanges occurring in fw . Hence, we shall only consider fw for “Government spending”, as we
do for the other series, without loss of substance.
4

Results of multiple structural change tests

We now present the main results; those for the structural change tests in Section 4.1 andthe estimates of the break dates and parameter estimates in Section 4.2. Throughout, weconsider a simple autoregressive model of order 4, AR(4), with a ﬁtted intercept. Hence,in terms of model (1), {w is null and }w = [1> |w−1 > |w−2 > |w−3 > |w−4 ]. We include a constant as
parts of the regressors whose coeﬃcients are subject to change in order to assess whether
any change in mean is left in the cyclical components when using our detrending procedure
with two breaks in 1973:1 and 2008:1. It also serves as a diagnostic check to assess theadequacy of allowing breaks at these dates. We also use an AR(4) speciﬁcation even though
our unobserved components model speciﬁes a cyclical component with either an AR(2) orAR(1) structure. This is done to ensure that any remaining correlation is taken into account.
It involves no loss in generality or precision as the AR(4) model is encompassing and that
we mainly focus on the sum of the autoregressive coeﬃcients as a measure of persistence.
We start the sample from 1960:1, consistent with Stock and Watson (2002) although thedata are available from 1947, because including the 1950s makes the results sensitive to thestarting date, possibly because of the inclusion of the Korean War period.
7

4.1

Presence of structural changes

Our ﬁrst results, presented in Table 3, pertain to testing if there is evidence for at least somebreaks by using the VxsOU4>W test with pd and qd ranging from 1 to 3 and the XG max4>W
test with P = Q = 3 (using, here and throughout all results, the truncation parameter% = 0=10). These tests are complementary. The VxsOU4>W is directed at having maximal
power for the pre-speciﬁed number of break, though they still have power when the numberof breaks is misspeciﬁed, whether in the variance or in the coeﬃcients. The XG max4>W aims
at testing for any numbers of breaks within the pre-speciﬁed upper bounds. As in Bai andPerron (1998, 2003), these are very useful because the power is usually as good as the testwith the correct number of breaks. They are, however, less informative as one can only
conclude that at least one break in either the coeﬃcients or the variance has occurred. Bothtests are a good starting point to assess whether breaks are present. As shown in Table
3, we ﬁnd evidence for at least some breaks in consumption, exports, imports, government
spending and production (services) as the XG max4>W test rejects at the 5% level for theseseries. We ﬁnd less clear evidence for GDP, investment, production (goods) and production(structures) as some of the VxsOU4>W tests reject but not the XG max4>W test at the 10%
level. For series with a clear rejection, it is useful to compute a wide range of tests to decide
which model to select. For series with a weaker or no rejection, it may be the case that the
tests lack power if only changes in the variance or in the coeﬃcients occur. Hence, we also
use other tests to narrow down the types of breaks present.
Table 4 reports results when testing for multiple structural changes in the variance conditional on changes in the coeﬃcients occurring (including zero change). We allow for up to
3 changes in the coeﬃcients (i.e., pd = 0, 1, 2 or 3) when testing for such variance breaks.
These tests have the right size for all values of pd but, of course, are more powerful when
the correct number of breaks is used. The tests used are then the VxsOU1>W test for qd = 0
and the VxsOU2>W test for qd = 1> 2 and 3. For an unspeciﬁed number of changes in coefﬁcients within a maximal value Q = 3, the XG max2>W test is used to test for pd changes
in coeﬃcients with pd = 0> 1> 2> 3. The results show that the latter test rejects the null
hypothesis of no variance break for a broad set of variables; in particular, for consumption,
exports, imports, government spending and production (services). These are consistent withthe results obtained with the XG max4>W tests. In addition, rejections also occur for GDP
and production (goods), and to a lesser extent for production (structures) and for investment(higher p-values and/or results more sensitive to the choice of pd ). Overall, the results show
widespread evidence of structural changes in the variance of errors.
8

To select the number of structural changes in the variance, we adopt a sequential procedure using the VxsOU10>W test with pd = 3 and a 10% signiﬁcant level when testing for qd
versus qd +1 breaks in variance applied sequentially for qd = 0> 1> 2> 3. The number of breaks
selected is the value qd for which the test rejects but fails to reject for qd + 1. The results arepresented in Table 5. We obtain two breaks in the variance for GDP and production (goods)
and one break in the other series except that no variance break is detected in production(structures). It is important to ensure that the results of the sequential test comply withthe previous testing results. Since Table 4 suggests evidence for structural breaks in thevariance for all series, though with only marginal evidence for investment and production(structures), the number of breaks selected by the sequential procedure are adequate.
We now turn to changes in the coeﬃcients. Table 6 provides results for the tests VxsOU3>W
with pd = 1> 2> 3 and XG max3>W with P = 3. In both cases, we consider qd = 0> 1> 2 and 3.
The case with pd = 1 and qd = 0 reduces to the standard SupLR test for a change in thecoeﬃcients (Andrews, 1993). As explained above, two breaks in the trend component areallowed to obtain the cyclical component. Hence, our focus here is on testing for changes
in persistence. Both tests show strong rejections under any speciﬁed number of breaks forexports and imports. However, for the other series, the results show weak, if any, evidence
for changes in the parameters. This is conﬁrmed in Table 7, which presents results for thesequential VxsOU9>W test to select the number of breaks in the coeﬃcients with qd = 3. It
suggests one break for exports and imports and no break for the other series.
4.2

Break dates and parameter estimates

We now present the estimates the break dates for the selected number of breaks in thevariance and coeﬃcients via jointly maximizing the likelihood function (Qu and Perron,2007). We also provide point estimates of the segment-wise variance and the coeﬃcients. In
particular, the sample standard deviation as a ratio over that of the ﬁrst regime and the sumof the autoregressive coeﬃcients are reported. The conﬁdence intervals for the break dates
were obtained using Theorem 3 of Qu and Perron (2007) based on the Gaussian likelihood
function with regimes segmented by the union of the variance and the coeﬃcient breaks
selected by the sequential procedure. The results in Tables 8 yield the following conclusions.
First, consistent with the Great Moderation, structural changes in the variance are estimatedin the mid-80s for GDP, investment, imports and production (goods). These break dates
closely match those reported by Stock and Watson (2002), i.e. 1983:2 for GDP, 1986:2 forimports and 1983:4 for production (goods). Other break dates in variance are also consistent9with their results such as 1992:1 for consumption. Second, however, our multiple break tests
ﬁnd additional breaks in variance in 1970:3 for GDP and production (goods), after which
the sample standard deviation increased threefold. This ﬁnding suggests a partial reversal
instead of a moderation, i.e., the presence of a temporary regime with a large variance inthe 70s and early 80s. Hence, part of the Great Moderation can be viewed as a reversion
to the pre 1970 volatility level. Third, we ﬁnd little evidence of structural changes in thecoeﬃcients once the two breaks in the slope of the trend are accounted for. Finally, we ﬁnd
no evidence for structural changes in both the variance and the coeﬃcients around the GreatRecession. This result is consistent with those obtained by Gadea et al. (2018, 2019).
5

Conclusion

In this paper, we provided additional empirical evidence about the Great Moderation using
a comprehensive framework to test for multiple structural changes in the coeﬃcients and in
the variance of error term of a linear regression model provided by Perron, Yamamoto andZhou (2019). In particular, the framework provides methods to (a) test for given numberof changes in regression coeﬃcients and variance of the errors, (b) test for some unknown
number of changes within some pre-speciﬁed maximum, (c) test for changes in variance(coeﬃcients) allowing for a given number of changes in the coeﬃcients (variance) and (d)sequential procedures to estimate the number of changes. We applied these methods toquarterly U.S. real GDP and its major components for the period 1960:1 to 2018:4. Unlike
most related studies, we adopted an unobserved component model, allowing for two breaks inthe slope of trend in 1973:1 and 2008:1. Our cyclical components are similar to the residuals
obtained from a piecewise linear regression accounting for the two trend breaks. The method
follows Perron and Wada (2009) to an extended sample period and, similar to their results,
our detrended series follows closely the NBER chronology for recessions and expansions.
We obtained the following important empirical ﬁndings. First, we conﬁrmed evidence for
the Great Moderation, i.e., a structural change in the variance of the errors in the mid-80s for
GDP, investment, imports and production (goods). Second, we found an additional breakin variance at 1970:3 for GDP and production (goods), after which the sample standard
deviation increased threefold. Hence, a part of the Great Moderation can be viewed as areversion to the pre 1970 volatility level. Third, we found little evidence of structural change
in the coeﬃcients once the two breaks in the slope of the trend are accounted for. Finally,
we found no evidence for changes in the variance nor in the coeﬃcients around the GreatRecession. Hence, the low volatility regime is still in eﬀect.
10

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13

% = 0=10% = 0=15
qd = 1 qd = 2
qd = 3
qd = 1 qd = 2
qd = 3tpd = 3 pd = 3 pd = 1 pd = 2 pd = 3 pd = 3 pd = 3 pd = 1 pd = 2
1 .90
5.79
5.74
5.77
5.74
5.43
6.21
5.72
5.09
5.00
.95
6.39
6.27
6.43
6.26
5.90
7.18
6.46
5.68
5.50
.975
6.99
6.72
6.93
6.77
6.31
8.12
7.23
6.16
5.95
.99
7.60
7.25
7.77
7.36
6.83
9.24
8.00
6.79
6.48
2 .90
7.52
7.12
6.40
6.70
6.59
7.45
6.54
5.68
5.87
.95
8.15
7.68
6.96
7.23
7.08
8.45
7.36
6.27
6.45
.975
8.73
8.16
7.54
7.77
7.48
9.45
8.02
6.84
6.94
.99
9.68
8.80
8.37
8.37
8.01
10.73
8.93
7.57
7.58
3 .90
9.04
8.33
6.94
7.52
7.56
8.53
7.30
6.20
6.66
.95
9.73
8.91
7.58
8.09
8.09
9.52
8.07
6.84
7.23
.975
10.36
9.46
8.18
8.60
8.57
10.61
8.80
7.39
7.74
.99
11.18
10.09
8.91
9.20
9.15
11.87
9.67
8.16
8.38
4 .90
10.39
9.43
7.41
8.25
8.48
9.51
7.87
6.67
7.38
.95
11.14
10.06
8.08
8.85
9.02
10.54
8.73
7.32
8.02
.975
11.79
10.56
8.72
9.39
9.52
11.61
9.47
7.92
8.60
.99
12.62
11.30
9.44
10.02
10.09
13.08
10.42
8.71
9.28
5 .90
11.74
10.44
7.88
8.93
9.29
10.45
8.53
7.18
8.10
.95
12.52
11.10
8.59
9.55
9.88
11.66
9.33
7.82
8.70
.975
13.13
11.66
9.18
10.08
10.36
12.72
10.09
8.43
9.23
.99
14.00
12.31
9.83
10.70
10.91
14.06
11.15
9.12
9.87
Note: For cases of both nd and md being less than 3, see Perron, Yamamoto, and Zhou (2019).
P = 3> Q = 2% = 0=10 % = 0=15
7.17
6.61
8.04
7.45
8.94
8.28
10.00
9.20
8.51
7.89
9.34
8.84
10.27
9.70
11.34
10.75
9.85
9.15
10.71
10.11
11.58
10.99
12.72
12.19
10.99
10.33
11.94
11.28
12.82
12.17
13.94
13.43
12.28
11.49
13.19
12.51
14.08
13.29
15.05
14.50

Table 1: Asympotic critical values of the upper bound of the sup OU4>W test¢¡)(the entries are quantiles { such that S (qd + pd )−1 sup OU4 ≤ {X G max OU4
P = 2> Q = 3% = 0=10 % = 0=15
7.17
6.62
8.04
7.45
8.94
8.28
10.00
9.20
8.48
7.87
9.31
8.83
10.25
9.69
11.34
10.75
9.74
9.08
10.65
10.08
11.54
10.95
12.69
12.19
10.86
10.25
11.81
11.21
12.73
12.14
13.85
13.41
12.06
11.35
13.04
12.45
13.97
13.26
14.95
14.48

P = 3> Q = 3% = 0=10
7.18
8.05
8.94
10.00
8.52
9.35
10.27
11.34
9.85
10.71
11.58
12.72
10.99
11.94
12.82
13.94
12.28
13.19
14.08
15.05

Table 2. Parameter estimates of the unobserved component model

%

g1
g2
!1
!2
oW
GDP
0.00
0.84
0.96
-0.21
-0.58
1.30
-0.36
369.52(0.02) (0.03) (0.04) (0.07) (0.20) (1.17)
Consumption
0.00
0.77
0.97
-0.16
-0.51
0.96
346.39(0.02) (0.03) (0.05) (0.08) (0.02)
Investment
0.00
4.65
1.11
-0.16
-0.53
0.91
862.97(0.00) (0.11) (0.16) (0.28) (0.02)
Exports
0.00
3.12
1.52
-0.03
-0.83
0.76
0.18
614.62(0.00) (0.19) (0.23) (0.27) (1.08) (1.60)
Imports
0.00
3.64
1.51
0.04
-0.87
1.00
-0.09
792.26(0.00) (0.09) (0.13) (0.24) (0.36) (1.94)
Government Spending
0.76
0.96
0.96
-0.53
-0.87
1.71
-0.73
497.40(0.073) (0.05) (0.13) (0.19) (0.31) (0.39) (2.07)
Production (Goods)
0.00
1.75
0.94
0.00
-0.14
1.08
-0.20
581.60(0.01) (0.03) (0.05) (0.10) (0.18) (0.93)
Production (Services)
0.00
0.68
0.99
-0.28
-0.67
0.97
309.47(0.02) (0.04) (0.05) (0.08) (0.02)
Production (Structures)
0.00
2.33
1.00
-0.72
-1.19
1.35
-0.38
663.40(0.00) (0.13) (0.20) (0.31) (0.54) (3.37)
Note: Standard errors in parentheses. When   = 0, they are not well-deﬁned and thus unreported.

coeﬃcients (pd )variance (qd )VxsOU4>W (p = pd > q = qd )
1
1
1
2
2
2
1
2
3
1
2
3
∗
∗∗
GDP
10.48
9.56
9.28
10.00
9.26
9.54∗
Consumption
13.28∗∗
9.51∗
7.91∗
13.12∗∗
10.61∗
8.71
Investment
10.61
8.26
7.71
11.60
11.91∗∗
9.83∗∗
∗∗∗
∗∗∗
∗∗∗
∗∗∗
∗∗∗
Exports
31.73
21.92
25.27
27.11
26.44
23.08∗∗∗
∗∗∗
∗∗∗
∗∗∗
∗∗∗
∗∗∗
Imports
18.53
12.41
11.78
17.62
12.51
11.29∗∗∗
∗∗∗
∗∗∗
∗∗∗
∗∗∗
∗∗∗
Government Spending
15.21
13.66
10.56
18.75
14.54
11.89∗∗∗∗∗
Production (Goods)
10.66
9.74
8.29
8.33
8.09
7.35
Production (Services)
15.86∗∗∗ 12.39∗∗∗ 9.47∗∗
12.38∗
11.08∗∗
9.43∗∗∗∗∗
Production (Structures) 10.23
9.71
10.10
9.31
9.98
9.67∗∗∗ ∗∗ ∗∗∗
Note: , ,indicate signiﬁcance at the 10%, 5%, and 1% levels, respectively.
3
1
8.96
13.40∗∗
11.52
29.43∗∗∗
13.74∗∗
17.41∗∗∗
7.65
12.11∗
8.86

3
2
9.09
11.26∗∗
11.21∗∗
24.77∗∗∗
11.42∗∗
14.51∗∗∗
7.39
11.51∗∗
11.57∗∗

3
3
9.03
9.48∗
9.51∗
21.15∗∗∗
15.67∗∗∗
12.47∗∗∗
7.60
10.04∗∗
10.19∗∗

Table 3. Tests for multiple structural changes in the variance and the coeﬃcients
X G max4>W
P =3
Q =3
10.48
13.40∗∗
11.91
31.73∗∗∗
18.53∗∗∗
18.75∗∗∗
10.66
15.86∗∗∗
11.57

Table 4. Tests for multiple structural changes in the variance of the errors

variance (qd )
coeﬃcients (pd )GDP
Consumption
Investment
Exports
Imports
Government Spending
Production (Goods)
Production (Services)
Production (Structures)

1
0
15.90∗∗∗
18.19∗∗∗
8.17∗
7.52∗
12.87∗∗
22.20∗∗∗
7.61∗
17.59∗∗∗
6.76

1
1
13.75∗∗∗
17.29∗∗∗
6.53
5.98
18.14∗∗∗
18.63∗∗∗
13.65∗∗∗
18.37∗∗∗
6.93

VxsOU2>W (q = qd |p = pd )
1
1
2
2
2
3
0
1
∗∗
∗∗
∗∗∗
12.37
10.75
10.64
10.28∗∗∗
14.05∗∗∗ 15.82∗∗∗ 9.96∗∗
9.49∗∗
5.82
7.18
5.99
5.34
9.22∗∗
17.04∗∗∗ 6.27
5.46
16.00∗∗∗ 18.82∗∗∗ 8.21∗∗
10.73∗∗∗
∗∗∗
∗∗∗
∗∗∗
16.36
17.38
12.79
14.02∗∗∗
∗∗
9.13
5.89
6.49
11.63∗∗∗
∗∗∗
∗∗∗
∗∗∗
22.34
19.58
13.30
13.46∗∗∗
∗
2.88
1.09
7.72
7.91∗

2
2
10.42∗∗∗
7.69∗
7.55∗
7.81∗
9.97∗∗
9.17∗∗
8.20∗∗
16.17∗∗∗
7.38∗

2
3
9.91∗∗
9.33∗∗
7.07∗
11.06∗∗∗
11.40∗∗∗
10.99∗∗∗
6.84
15.72∗∗∗
9.65∗∗

VxsOU2>W (q = qd |p = pd )X Gpd{2>W (Q> p = pd )
3
3
3
3
Q =3
Q =3
Q =3
Q =3
0
1
2
3
0
1
2
3
GDP
8.37∗∗
8.81∗∗∗ 9.35∗∗∗
8.95∗∗∗
15.90∗∗∗ 13.76∗∗∗ 12.37∗∗
10.75∗∗
Consumption
7.46∗∗
7.20∗∗
5.44
6.42∗
18.19∗∗∗ 17.29∗∗∗ 14.05∗∗∗ 15.82∗∗∗
Investment
5.21
5.11
5.60
4.93
8.17∗
6.53
7.55
7.18
∗∗
∗∗
∗∗
Exports
5.07
8.22
7.07
7.85
7.52
8.22∗
9.22∗
17.04∗∗∗
∗
∗∗
∗∗
∗∗∗
∗∗∗
∗∗∗
∗∗∗
Imports
6.17
8.33
7.50
8.45
12.87
18.14
16.00
18.82∗∗∗
∗∗∗
∗∗∗
∗∗
∗∗
∗∗∗
∗∗∗
∗∗∗
Government Spending
9.05
9.75
6.87
7.80
22.20
18.63
16.36
17.38∗∗∗
∗∗∗
∗∗
∗∗
∗∗∗
∗
Production (Goods)
5.80
9.36
7.20
7.08
7.61
13.65
9.13
7.08
Production (Services)
9.26∗∗∗ 9.30∗∗∗ 11.49∗∗∗ 11.05∗∗∗ 17.59∗∗∗ 18.37∗∗∗ 22.34∗∗∗ 19.58∗∗∗
Production (Structures) 8.44∗∗∗ 8.18∗∗
7.00∗∗
7.50∗∗
8.44∗
8.18∗
7.38
9.65∗∗∗ ∗∗ ∗∗∗
Notes: 1. , ,indicate signiﬁcance at the 10%, 5%, and 1% levels, respectively.
2. The VxsOU2>W test and the X Gpd{2>W test reduce to the VxsOU1>W test and the X Gpd{1>W test,
respectively when pd = 0=variance (qd )
coeﬃcients (pd )Table 5. Sequential tests for structural changes in the variance of the errors
VxsOU10>W (q = qd + 1|q = qd > p = pd ) Number
variance (qd )0123of Breaks
coeﬃcients (pd )
3
3
3
3
GDP
14.04∗∗∗ 11.46∗∗ 4.52
2.72
2Consumption
22.89∗∗∗ 3.71
3.62
3.62
1
Investment
8.70∗
8.35
8.35
0.88
1
Exports
14.13∗∗∗ 7.87
7.87
7.87
1
Imports
17.03∗∗∗ 4.02
4.02
4.02
1
Government Spending
19.33∗∗∗ 3.96
4.83
1.94
1
Production (Goods)
12.79∗∗
10.87∗∗ 3.96
2.38
2
Production (Services)27.88∗∗∗ 8.30
3.69
4.57
1
Production (Structures) 6.41
11.84∗∗ 3.30
5.51
0Notes: Same as in Table 3.

Table 6. Tests for multiple structural changes in the coeﬃcients

coeﬃcients (pd )variance (qd )GDP
Consumption
Investment
Exports
Imports
Government Spending
Production (Goods)
Production (Services)
Production (Structures)

1
0
8.27
8.94
16.01
78.89∗∗∗
46.24∗∗∗
18.45∗
10.54
15.09
14.75

1
1
2.71
9.00
12.72
50.36∗∗∗
33.35∗∗∗
9.78
16.45
13.46
12.76

VxsOU3>W (p = pd |q = qd )
1
1
2
2
2
3
0
1
7.83
12.38
5.50
5.96
9.16
10.74
7.01
13.05
9.69
9.78
8.04
14.13
35.04∗∗∗ 46.60∗∗∗ 23.57∗∗∗ 38.79∗∗∗
22.92∗∗∗ 23.73∗∗∗ 19.12∗∗∗ 27.25∗∗∗
15.46
14.54
9.87
15.99∗
∗
18.05
12.61
5.41
10.27
6.82
7.13
6.01
12.08
10.05
13.99
8.16
10.66

2
2
10.10
12.94
18.80∗∗
33.20∗∗∗
21.22∗∗∗
14.52
10.36
11.83
11.21

2
3
13.70
11.14
14.46
29.04∗∗∗
19.06∗∗∗
14.95
9.86
12.28
10.69

VxsOU3>W (p = pd |q = qd )X Gpd{3>W (P> q = qd )
3
3
3
3
P =3
P =3
P =3
P =3
0
1
2
3
0
1
2
3
GDP
3.86
6.69
8.85
11.79
8.27
6.69
10.10
13.70
Consumption
3.98
14.38∗
15.49∗∗
14.05
8.94
14.38
15.49
14.05
Investment
4.74
13.96
15.38∗∗
12.37
16.01
14.13
18.80∗∗
14.46
Exports
11.90 39.47∗∗∗ 33.77∗∗∗ 28.23∗∗∗ 78.89∗∗∗ 50.36∗∗∗ 35.05∗∗∗ 46.60∗∗∗
Imports
9.88
25.18∗∗∗ 21.48∗∗∗ 19.07∗∗∗ 46.24∗∗∗ 33.35∗∗∗ 22.92∗∗
23.73∗∗∗∗∗∗∗
Government Spending
5.92
14.75
14.74
14.66
18.45
15.99
15.46
14.95
Production (Goods)
3.56
8.29
8.91
9.65
10.54
16.45
18.05∗
12.61
Production (Services)
4.05
11.82
12.46
13.03
15.09
13.46
12.46
13.03
Production (Structures) 5.08
9.09
14.29∗
11.86
14.75
12.76
14.29
13.99
∗ ∗∗ ∗∗∗
Notes: 1. , ,indicate signiﬁcance at the 10%, 5%, and 1% levels, respectively.
2. The VxsOU3>W test and the X Gpd{3>W test reduce to the standard VxsOU test and the X Gpd{
LR test, respectively, when qd = 0 (see Andrews, 1993, and Bai and Perron, 1998).
coeﬃcients (pd )variance (qd )Table 7. Sequential tests for structural changes in the coeﬃcients
VxsOU9>W (p = pd + 1|p = pd > q = qd ) Number
coeﬃcients (pd )0123of Breaks
variance (qd )
3
3
3
3
GDP
14.08
6.61
11.48
12.47
0
Consumption
10.76
20.96∗∗
15.29
11.05
0
Investment
13.80
25.29∗∗∗
11.92
11.84
0
Exports
45.45∗∗∗
11.54
37.00∗∗∗
11.20
1
Imports
31.94∗∗∗
18.04
16.54
19.39
1
Government Spending
16.40
17.31
17.31
16.44
0
Production (Goods)
12.58
8.39
9.13
10.02
0
Production (Services)
8.15
13.98
12.21
10.84
0
Production (Structures)
11.34
12.00
19.18
6.73
0
Notes: Same as in Table 3.

1970:3(1962:1, 1975:2)
1992:1(1991:2, 1993:1)
1984:1(1977:1, 1986:1)
1978:3(1976:1, 1979:3)
1986:2(1985:2, 1986:4)
2002:1(2001:3, 2002:4)
1970:3(1964:2, 1980:4)
2000:3(2000:1, 2001:3)
1983:4(1979:4, 1984:2)

1983:2(1982:2, 1983:3)

Production (Structures)Note: the 90% conﬁdence intervals are in parentheses.

Production (Services)

Production (Goods)

Government Spending

Imports

Exports

Investment

Consumption

GDP

Break Dates
variance

1978:3(1976:1, 1979:3)
1986:2(1985:2, 1986:4)

coeﬃcients

0.32

3.01

0.28

0.14

0.27

0.41

0.30

0.96

Ratio of s.d.
over Regime 1
Regime 2 Regime 3
3.15
0.67

Table 8. Break dates and parameter estimates

0.96

0.97

0.86

0.99

0.91

0.70

0.90

0.95

0.97

0.96

Sum of ARcoeﬃcients
Regime 1 Regime 2
0.93

Figure 1. Trend and cycle decomposition of U.S. real GDP and its components.
GDP
1000

8
original
trend

950

Deviation (%)

6
4
2

900

0
-2

850

-4
-6

800

-8
1960

1970

1980

1990

2000

2010

1960

1970

1980

1990

2000

2010

1990

2000

2010

1990

2000

2010

Consumption
950

6

Deviation (%)

4
900
2
850

0
-2

800
-4
750
1960

1970

1980

1990

2000

2010

-6
1960

1970

1980

Investment
850

30

800

20

Deviation (%)

10

750

0
700
-10
650

-20

600

-30
-40

550
1960

1970

1980

1990

2000

2010

1960

1970

1980

Figure 1. Trend and cycle decomposition of U.S. real GDP and its components (continued).
Exports
800

20
original

750

Deviation (%)

15

trend

10

700

5

650

0
600

-5

550

-10

500

-15

450
1960

1970

1980

1990

2000

2010

-20
1960

1970

1980

1990

2000

2010

1990

2000

2010

Imports
850

30

Deviation (%)

800

20

750
10

700
650

0

600

-10

550
-20

500
450
1960

1970

1980

1990

2000

2010

-30
1960

1970

1980

Government spending
850

15

Deviation (%)
10
800
5
750

0
-5

700
-10
650

-15
1960

1970

1980

1990

2000

2010

1960

1970

1980

1990

2000

2010

Figure 1. Trend and cycle decomposition of U.S. real GDP and its components (continued).
Production (goods)
900

15
original

850

Deviation (%)

10

trend

5
800
0
750
-5
700

-10

650

-15
1960

1970

1980

1990

2000

2010

1960

1970

1980

1990

2000

2010

1990

2000

2010

1990

2000

2010

Production (services)950

4

Deviation (%)

3
900

2
1

850

0
-1

800

-2
-3

750
1960

1970

1980

1990

2000

2010

-4
1960

1970

1980

Production (structures)
750

30

Deviation (%)
20
700

10
0

650

-10
-20

600
1960

1970

1980

1990

2000

2010

-30
1960

1970

1980

Figure 2. Cyclical component and the growth rate of the stochastic trend (! t ) for
Government Spending
15(%)

cycle component
growth rate of the trend

10

5

0

-5

-10

-15
1960

1970

1980

1990

2000

2010