Updated Evidence on Structural Changes in Variance

The Great Moderation: Updated Evidence with Joint Tests for Multiple Structural Changes in Variance and Persistence 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 the variance of the error term of a linear regression model provided by Perron, Yamamoto, and Zhou (2020). We apply it to U.S. real GDP and its major components for the period 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 con rm evidence about the Great Moderation, i.e., a structural 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), af-ter 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 nd little 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 “ good luck”hypothesis as a source of the Great Moderation, which continues even after the Great Recession. y Department of Economics, Hitotsubashi University, 2-1 Naka, Kunitachi, Tokyo Japan, 186-8601 1 Introduction One of the striking empirical facts in the U.S. economy is the presence of a sharp decline of macroeconomic volatility in the mid-80s, the so-called Great Moderation. Blanchard and Simon (2001) documented that the sample standard deviation of quarterly real GDP growth rates, i.e., the logarithmic …rst-di¤erences, declined by half after the mid-80s. More formal statistical results were provided by Kim and Nelson (1999), who allowed a one-time structural change in the mean and the variance at the same (unknown) time in a Markov-switching model. They applied it to the real GDP growth rates and showed evidence of a decline in the mean 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., …tting some conditional mean model and then testing for a structural break in the mean of the absolute value of the residuals. 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 of the 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 and subsequently applied the same test to the mean of the absolute residuals. They investigated 168 U.S. macroeconomic variables, detrended mostly via logarithmic …rst-di¤erences (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 unconditional variance 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 speci…cation, the former is represented by a structural change in the AR coe¢ cients 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 hypothesis of “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, Sensier and van Dijk, 2004, Sims and Zha, 2006), although the conclusions of these studies are often ambivalent. Another hypothesis pertains to production smoothing achieved by better inventory management that occurred in the durable manufacturing industries (Ramey and Vine, 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 constant coe¢ cients 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 to the …nancial crisis of 2007-08, i.e., the Great Recession. Clark (2009) documented a signi…cant rise in macroeconomic volatility and claimed that it overturned the Great Moderation. Blanchard (2014) suggested that a benign view of economic ‡uctuation until 2008 needed to be reassessed. Stock and Watson (2017) argued that the low-volatility period lasted through 2007 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 even after 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 structural change tests proposed by Perron, Yamamoto and Zhou (2020); 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 of changes in regression coe¢ cients and variance of the errors; (b) tests for some unknown number of changes within some pre-speci…ed maximum; (c) tests for changes in variance (coe¢ cients) allowing for a given number of changes in the coe¢ cients (variance); (d) sequential procedures to estimate the number of changes in the coe¢ cients and in the variance. This framework has clear bene…ts to detect multiple structural changes in the coe¢ cients 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 induces a power loss and testing for variance breaks ignoring mean breaks yields tests with 1 They complemented their results using test similar to ours but allowing for a maximum of one change in each of the parameters of the conditional mean and the variance of the errors under the null hypothesis. 2 important size distortions. See Perron and Yamamoto (2019) for extensive simulation results. Almost all existing studies use the logarithmic …rst-di¤erences of the original series. However, this is a …lter that a¤ects the spectrum of the series in di¤erent ways at di¤erent frequencies and, hence, may downplay low frequency variabilities. We instead use a trendcycle decomposition based on an unobserved component model allowing breaks in the slope of the trend component. This is in the spirit of Perron and Wada (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. This also reduces the risk that the structural change tests are biased due to small shifts in the mean of the …rst-di¤erenced 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 …rst-di¤erenced series have a noise component that is over-di¤erenced. Our framework extends Perron and Wada (2009) by allowing two trend breaks in 1973:1 and 2008:1 for the real GDP and its components. This is because (a) not all series need to have breaks at the same date when estimated individually; (b) it is di¢ cult to estimate globally two break dates jointly for all series used; (c) all prior work point to the importance of the productivity slowdown after 1973, and 2008 has been advanced for a further slowdown, hence they are natural candidates to include; (d) by including them, they are not imposed, simply allowed; (e) if spurious breaks were detected they would show up in the subsequent structural change tests, but none were detected. As in Perron and Wada (2009), our resulting cyclical component of real GDP captures the NBER business cycle chronologies of recessions and expansions very well. A caveat is that if we use the same AR(4) speci…cation as Stock and Watson (2002), the cyclical component is unstable. Hence, we follow the existing literature and use a smaller model. However, the cyclical components are nearly identical to the linear detrended series after the two breaks are accounted for. This means that our speci…cations are inconsequential to the main analysis. In any event, our approach provides a reasonable alternative to achieve a stationary transformation and therefore also provides a useful robustness check. 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. 3 2 Econometric Methods We consider an autoregressive model in which m breaks are allowed in the coe¢ cients of the conditional mean and n breaks in the variance of the errors: yt = xt + zt j t = Tjc + et ; 1 + 1; :::; Tjc ; (1) for j = 1; :::; m + 1. This is a partial structural change model in which the coe¢ cients associated with the regressors zt (a vector of dimension q) change at the break dates (T1c ; :::; Tmc ), c = T . The n breaks in which are treated as unknown, with the convention T0c = 0 and Tm+1 v v the variance of errors et occur at the unknown dates (T1 ; :::; Tn ). Accordingly, et has zero mean and variance 2 i for Tiv 1 + 1 t Tiv with i = 1; :::; n + 1, again with T0v = 0 and v Tn+1 = T . The breaks in the variance and in the coe¢ cients may happen at di¤erent times or may overlap partly or completely. We consider three sets of testing problems (TPs). The …rst pertains to testing for given number of changes in coe¢ cients and variance: TP-1: H0 : fm = n = 0g versus H1 : fm = 0, n = na g; TP-2: H0 : fm = ma ; n = 0g versus H1 : fm = ma , n = na g; TP-3: H0 : fm = 0; n = na g versus H1 : fm = ma , n = na g; TP-4: H0 : fm = n = 0g versus H1 : fm = ma , n = na g, where ma and na are some positive numbers selected a priori. In the second set, we consider testing for an unknown number of changes within some pre-speci…ed maximum: TP-5: H0 : fm = n = 0g versus H1 : fm = 0, 1 n N g; TP-6: H0 : fm = ma ; n = 0g versus H1 : fm = ma , 1 n N g; TP-7: H0 : fm = 0; n = na g versus H1 : f1 m M , n = na g; TP-8: H0 : fm = n = 0g versus H1 : f1 m M, 1 n N g. Finally, sequential procedures to estimate the number of changes are considered via TP-9: H0 : fm = ma ; n = na g versus H1 : fm = ma + 1, n = na g; TP-10: H0 : fm = ma ; n = na g versus H1 : fm = ma , n = na + 1g. These are useful to assess the adequacy of a model with a 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 the linear regression model by lT under fm = 0, n = 0g, lT (T1v ; :::; Tnv ) under fm = 0, n 6= 0g, lT (T1c ; :::; Tmc ) under fm 6= 0; n = 0g, and lT (T1c ; :::; Tmc ; T1v ; :::; Tnv ) under fm 6= 0, n 6= 0g (see PYZ for the exact expressions). To obtain the break dates, we maximize the log-likelihood function using a trimming ["T ] for the admissible candidate break dates, where " is some small positive constant fraction. Note that we denote the estimates of the break dates in coe¢ cients and variance by a “ ” when they are obtained jointly and by a “^” when they 4 are obtained separately. Then, the test statistics for TP-1 to TP-4 are: SupLR1;T (n) = 2[lT (T^1v ; :::; T^nv ) lT ]; (TP-2) SupLR2;T (njm) = 2[lT (Te1c ; :::; Temc ; Te1v ; :::; Tenv ) (TP-3) SupLR3;T (mjn) = 2[lT (Te1c ; :::; Temc ; Te1v ; :::; Tenv ) (TP-1) (TP-4) SupLR4;T (m; n) = 2[lT (Te1c ; :::; Temc ; Te1v ; :::; Tenv ) lT (T^1c ; :::; T^mc )]; lT (T^1v ; :::; T^nv )]; lT ]; and, for TP-5 to TP-8, the corresponding U D max type tests of Bai and Perron (1998) are: (TP-5) U Dmax1;T (N ) = max n 1 SupLR1;T (n); 1 n N max n 1 SupLR2;T (njm); (TP-6) U Dmax2;T (N jm) = 1 n N (TP-7) U Dmax3;T (M jn) = 1 m M (TP-8) U Dmax4;T (M; N ) = max m 1 SupLR3;T (mjn); max max (m + n) 1 SupLR4;T (m; n); 1 m M1 n N 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 M and N up to 2. Here, we provide additional critical values for M and N up to 3. Those for the SupLR4;T and U Dmax4;T tests of size 1%, 2.5%, 5% and 10% are presented in Table 1 for q between 1 and 5 with the truncation parameter " = 0:10. Results are also provided for (M; N ) = (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 i:i:d: standard normal random variables with 1,000 discrete steps and 10,000 replications. TP-9 and TP-10 are used to select the number of breaks in coe¢ cients (intercept and autoregressive parameters) and in variance. When selecting the number of coe¢ cient breaks, we consider TP-9 and the test SupLR9;T (m + 1jm; N ) is applied (see PYZ for the exact expressions), starting with H0 : m = 0 and H1 : m = 1, where N = 3. Upon a rejection, we proceed to H0 : m = 1 and H1 : m = 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 SupLR10;T (n + 1jM; n) with M = 3 is used for n = 0; 1; ::: until a non-rejection occurs. 3 Data and detrending method We use seasonally-adjusted quarterly US real GDP and its major demand components: consumption, investment, exports, imports and government spending, covering the period 5 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): yt = t + ct ; t = t + t 1 + t; (L)ct = "t where t is the trend component, ct is the cyclical component and (L) is either an AR(2) 2 or AR(1) lag-polynomial, i.e., (L) = 1 1L 2 L , with L the usual lag operator. The AR(2) speci…cation is adopted for all series except consumption, investment and production (services), for which the AR(1) speci…cation is preferred. A caveat is that if we use the same AR(4) speci…cation as Stock and Watson (2002), the cyclical component is unstable. Hence, we follow the existing literature and use a smaller model. The shocks are speci…ed by 31 2 3 02 3 2 2 0 " 5A ; 4 t 5 i:i:d:N @4 5 ; 4 2 "t 0 " " 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 cyclical component 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 t = + d1 I(t > 1973:1) + d2 I(t > 2008:1): We apply this speci…cation to all the GDP component series because (a) not all series need to have breaks at the same date when estimated individually; (b) it is di¢ cult to estimate globally two break dates jointly for all series used; (c) all prior work point to the importance of the productivity slowdown after 1973, and 2008 has been advanced for a further slowdown, hence they are natural candidates to include; (d) by including them, they are not imposed, simply allowed; (e) if spurious breaks were detected they would show up in the subsequent structural change tests, but none were detected. The model is estimated via the standard Kalman …lter algorithm. Figure 1 presents graphs of a) the original series and the …tted trend (left panels) and b) the deviations of the original series from trend, or cyclical components 6 (right panels). As in Perron and Wada (2009), our resulting cyclical component of real GDP captures the NBER business cycle chronologies of recessions and expansions very well. The parameter estimates2 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. This means that our AR(2) or AR(1) speci…cations of the cyclical component are inconsequential to the structural change analysis. 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 a¤ects the timing when the cyclical component is perceived as becoming above trend. Both estimates of d1 and d2 are negative. The …rst relates to the well 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 cyclical component is basically ‡at suggesting the absence of a recovery or expansion, contrary to what 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 latter characterization. It indicates that an expansion has indeed occurred but with respect to a lower 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 a¤ected by 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 the commonly used application of using the logarithmic …rst-di¤erences, i.e., the growth rates. First, …rst-di¤erencing is a …lter that a¤ects the spectrum of the series in di¤erent ways at di¤erent frequencies. It may downplay changes in variability at low frequency. Second, when applying the structural change tests, the trend breaks may be di¢ cult to capture as …rst-di¤erencing translates them into small changes in mean. Testing for such changes in mean becomes problematic because the …rst-di¤erences involves a resulting series with an over-di¤erenced noise. Third, the unobserved component model shows that the resulting 2 The standard errors reported are the diagonal elements of the negative inverse of the empirical Hessian of the log likelihood function obtained from the MATLAB version R2019a solver (fminunc function in their optimization toolbox). 7 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 cyclical component which accords well with the NBER chronology of expansions and recessions. Note that for “Government spending” one should also consider possible changes in the variance of t since the estimate of 2 is non-zero. We obtained an estimate of t as the …rst-di¤erences of the smoothed estimates of t from which we subtracted the estimate of t . A plot of this estimate is presented in Figure 2 along with the estimate of the cyclical component ct . It is evident that the magnitude of the former is negligible compared to the latter. Hence, any issue of changes in the variance of t will be negligible compared to changes occurring in ct . Hence, we shall only consider ct 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 and the estimates of the break dates and parameter estimates in Section 4.2. Throughout, we consider a simple autoregressive model of order 4, AR(4), with a …tted intercept. Hence, in terms of model (1), xt is null and zt = [1; yt 1 ; yt 2 ; yt 3 ; yt 4 ]. We include a constant as parts of the regressors whose coe¢ cients 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 the adequacy of allowing breaks at these dates. We also use an AR(4) speci…cation even though our unobserved components model speci…es a cyclical component with either an AR(2) or AR(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 coe¢ cients as a measure of persistence. We start the sample from 1960:1, consistent with Stock and Watson (2002) although the data are available from 1947, because including the 1950s makes the results sensitive to the starting date, possibly because of the inclusion of the Korean War period. 4.1 Presence of structural changes Our …rst results, presented in Table 3, pertain to testing if there is evidence for at least some breaks by using the SupLR4;T test with ma and na ranging from 1 to 3 and the U D max4;T test with M = N = 3 (using, here and throughout all results, the truncation parameter 8 " = 0:10). These tests are complementary. The SupLR4;T test is directed at having maximal power for the pre-speci…ed number of breaks, though they still have power when the number of breaks is misspeci…ed, whether in the variance or in the coe¢ cients. The U D max4;T test aims at testing for any numbers of breaks within the pre-speci…ed upper bounds. As in Bai and Perron (1998, 2003), these are very useful because the power is usually as good as the test with the correct number of breaks. They are, however, less informative as one can only conclude that at least one break in either the coe¢ cients or the variance has occurred. Both tests are a good starting point to assess whether breaks are present. As shown in Table 3, we …nd evidence for at least some breaks in consumption, exports, imports, government spending and production (services) as the U D max4;T test rejects at the 5% level for these series. We …nd less clear evidence for GDP, investment, production (goods) and production (structures) as some of the SupLR4;T tests reject but not the U D max4;T 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 coe¢ cients 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 coe¢ cients occurring (including zero change). We allow for up to 3 changes in the coe¢ cients (i.e., ma = 0, 1, 2 or 3) when testing for such variance breaks. These tests have the right size for all values of ma but, of course, are more powerful when the correct number of breaks is used. The tests used are then the SupLR1;T test for ma = 0 and the SupLR2;T test for ma = 1; 2 and 3. For an unspeci…ed number of changes in variance within a maximal value N = 3, the U D max2;T test is used to test given ma changes in coe¢ cients with ma = 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 with the results obtained with the U D max4;T 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 ma ). Overall, the results show widespread evidence of structural changes in the variance of errors. To select the number of structural changes in the variance, we adopt a sequential procedure using the SupLR10;T test with ma = 3 and a 10% signi…cant level when testing for na versus na + 1 breaks in variance applied sequentially for na = 0; 1; 2; 3. The number of breaks selected is the value na for which the test rejects but fails to reject for na + 1. The results are 9 presented 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 with the previous testing results. Since Table 4 suggests evidence for structural breaks in the variance 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 coe¢ cients. Table 6 provides results for the tests SupLR3;T with ma = 1; 2; 3 and U D max3;T with M = 3. In both cases, we consider na = 0; 1; 2 and 3. The case with ma = 1 and na = 0 reduces to the standard SupLR test for a change in the coe¢ cients (Andrews, 1993). As explained above, two breaks in the trend component are allowed to obtain the cyclical component. Hence, our focus here is on testing for changes in persistence. Both tests show strong rejections under any speci…ed number of breaks for exports and imports. However, for the other series, the results show weak, if any, evidence for changes in the parameters. This is con…rmed in Table 7, which presents results for the sequential SupLR9;T test to select the number of breaks in the coe¢ cients with na = 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 of the break dates for the selected number of breaks in the variance and coe¢ cients via jointly maximizing the likelihood function (Qu and Perron, 2007). We also provide point estimates of the segment-wise variance and the coe¢ cients. In particular, the sample standard deviation as a ratio over that of the …rst regime and the sum of the autoregressive coe¢ cients are reported. The con…dence 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 coe¢ cient 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 estimated in 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 for imports and 1983:4 for production (goods). Other break dates in variance are also consistent with their results such as 1992:1 for consumption. Second, however, our multiple break tests …nd additional breaks in variance in 1970:3 for GDP and production (goods), after which the sample standard deviation increased threefold. This …nding suggests a partial reversal instead of a moderation, i.e., the presence of a temporary regime with a large variance in 10 the 70s and early 80s. Hence, part of the Great Moderation can be viewed as a reversion to the pre 1970 volatility level. Third, we …nd little evidence of structural changes in the coe¢ cients once the two breaks in the slope of the trend are accounted for. Finally, we …nd no evidence for structural changes in both the variance and the coe¢ cients around the Great Recession. This result is consistent with those obtained by Gadea et al. (2018, 2019). 5 Conclusion In this paper, we provided updated empirical evidence about the Great Moderation using a comprehensive framework to test for multiple structural changes in the coe¢ cients and in the variance of error term of a linear regression model provided by PYZ. In particular, the framework provides methods to (a) test for given number of changes in regression coe¢ cients and variance of the errors, (b) test for some unknown number of changes within some prespeci…ed maximum, (c) test for changes in variance (coe¢ cients) allowing for a given number of changes in the coe¢ cients (variance) and (d) sequential procedures to estimate the number of changes. We applied these methods to quarterly 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 in the 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 …ndings. First, we con…rmed 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 break in 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 a reversion to the pre 1970 volatility level. Third, we found little evidence of structural change in the coe¢ cients 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 coe¢ cients around the Great Recession. Hence, the low volatility regime is still in e¤ect. 11 References Andrews, D.W.K. (1993): “Tests for parameter instability and structural change with unknown change point,”Econometrica 61, 821-856. Andrews, D.W.K., and Ploberger, W. 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(2017): “Twenty years of time series econometrics in ten pictures,”Journal of Economic Perspectives 31(2), 59-86. Summers, P.M. (2005): “What caused the Great Moderation? Some cross-country evidence,” Economic Review QII, Federal Reserve Bank of Kansas, 5-32. Taylor, J.B. (2011): “Macroeconomic lessons from the Great Deviation,” in NBER Macroeconomics Annual 25. D. Acemoglu and M. Woodford (eds). 387-395. 14 Table 1: Asympotic critical values of the upper bound of the sup LR4,T test ¢ ¡ (the entries are quantiles x such that P (na + ma )−1 sup LR4 ≤ x ≥ α) ε = 0.10 ε = 0.15 na = 1 na = 2 na = 3 na = 1 na = 2 na = 3 q α ma = 3 ma = 3 ma = 1 ma = 2 ma = 3 ma = 3 ma = 3 ma = 1 ma = 2 1 .90 5.79 5.74 5.77 5.74 5.43 5.06 4.97 5.09 5.00 .95 6.39 6.27 6.43 6.26 5.90 5.69 5.48 5.68 5.50 .975 6.99 6.72 6.93 6.77 6.31 6.17 5.94 6.16 5.95 .99 7.60 7.25 7.77 7.36 6.83 6.76 6.57 6.79 6.48 2 .90 7.52 7.12 6.40 6.70 6.59 6.74 6.29 5.68 5.87 .95 8.15 7.68 6.96 7.23 7.08 7.46 6.86 6.27 6.45 .975 8.73 8.16 7.54 7.77 7.48 8.15 7.41 6.84 6.94 .99 9.68 8.80 8.37 8.37 8.01 8.84 8.04 7.57 7.58 3 .90 9.04 8.33 6.94 7.52 7.56 8.12 7.41 6.20 6.66 .95 9.73 8.91 7.58 8.09 8.09 8.86 8.05 6.84 7.23 .975 10.36 9.46 8.18 8.60 8.57 9.48 8.63 7.39 7.74 .99 11.18 10.09 8.91 9.20 9.15 10.25 9.33 8.16 8.38 4 .90 10.39 9.43 7.41 8.25 8.48 9.45 8.40 6.67 7.38 .95 11.14 10.06 8.08 8.85 9.02 10.27 9.03 7.32 8.02 .975 11.79 10.56 8.72 9.39 9.52 11.01 9.58 7.92 8.60 .99 12.62 11.30 9.44 10.02 10.09 11.84 10.29 8.71 9.28 5 .90 11.74 10.44 7.88 8.93 9.29 10.69 9.43 7.18 8.10 .95 12.52 11.10 8.59 9.55 9.88 11.49 10.09 7.82 8.70 .975 13.13 11.66 9.18 10.08 10.36 12.28 10.72 8.43 9.23 .99 14.00 12.31 9.83 10.70 10.91 13.19 11.38 9.12 9.87 Note: For cases of both na and ma being less than 3, see Perron, Yamamoto, and Zhou (2020). M = 3, N = 2 ε = 0.10 ε = 0.15 7.18 6.61 8.04 7.51 8.94 8.32 10.00 9.42 8.51 7.93 9.37 8.88 10.32 9.77 11.47 10.96 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 12.20 13.99 13.43 12.28 11.49 13.19 12.51 14.08 13.38 15.16 14.50 U D max LR4 M = 2, N = 3 ε = 0.10 ε = 0.15 7.18 6.62 8.04 7.51 8.94 8.32 10.00 9.42 8.48 7.93 9.37 8.88 10.32 9.77 11.47 10.96 9.74 9.09 10.66 10.08 11.54 10.95 12.69 12.19 10.88 10.25 11.85 11.21 12.81 12.20 13.99 13.41 12.07 11.35 13.06 12.45 13.99 13.38 15.16 14.50 M = 3, N = 3 ε = 0.10 7.18 8.05 8.94 10.00 8.52 9.37 10.32 11.47 9.85 10.71 11.58 12.72 10.99 11.94 12.82 13.99 12.28 13.19 14.08 15.16 Table 2. Parameter estimates of the unobserved component model d1 d2 lT " 1 2 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) 0.00 0.68 0.99 -0.28 -0.67 0.97 309.47 Production (Services) (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-de…ned and thus unreported. Table 3. Tests for multiple structural changes in the variance and the coefficients coefficients (ma ) variance (na ) 1 1 GDP 10.48 Consumption 13.28∗∗ Investment 10.61 Exports 31.73∗∗∗ Imports 18.53∗∗∗ Government Spending 15.21∗∗∗ Production (Goods) 10.66 Production (Services) 15.86∗∗∗ Production (Structures) 10.23 Note: ∗ , ∗∗ , ∗∗∗ indicate significance 1 1 2 3 ∗ 9.56 9.28∗∗ ∗ 9.51 7.91∗ 8.26 7.71 21.92∗∗∗ 25.27∗∗∗ 12.41∗∗∗ 11.78∗∗∗ 13.66∗∗∗ 10.56∗∗∗ 9.74∗ 8.29∗ ∗∗∗ 12.39 9.47∗∗ ∗ 9.71 10.10∗∗∗ at the 10%, 5%, and SupLR4,T (m = ma , n = na ) 2 2 2 1 2 3 10.00 9.26 9.54∗ ∗∗ ∗ 13.12 10.61 8.71 11.60 11.91∗∗ 9.83∗∗ ∗∗∗ ∗∗∗ 27.11 26.44 23.08∗∗∗ ∗∗∗ ∗∗∗ 17.62 12.51 11.29∗∗∗ ∗∗∗ ∗∗∗ 18.75 14.54 11.89∗∗∗ 8.33 8.09 7.35 12.38∗ 11.08∗∗ 9.43∗ 9.31 9.98 9.67∗∗ 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∗∗ U D max4,T M =3 N =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 (na ) coe¢ cients (ma ) 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 13.76 17.29 6.53 5.98 18.14 18.63 13.65 18.37 6.93 1 2 12.37 14.05 5.82 9.22 16.00 16.36 9.13 22.34 2.88 SupLR2;T (n = na jm = ma ) 1 2 2 3 0 1 10.75 10.64 10.28 15.82 9.96 9.49 7.18 5.99 5.34 17.04 6.27 5.46 18.82 8.21 10.73 17.38 12.79 14.02 5.89 6.49 11.63 19.58 13.30 13.46 -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 SupLR2;T (n = na jm = ma ) U Dmax2;T (N; m = ma ) 3 3 3 3 N =3 N =3 N =3 N =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 9.05 9.75 6.87 7.80 22.20 18.63 16.36 17.38 Government Spending Production (Goods) 5.80 9.36 7.20 7.08 7.61 13.65 9.13 7.08 9.26 9.30 11.49 11.05 17.59 18.37 22.34 19.58 Production (Services) Production (Structures) 8.44 8.18 7.00 7.50 8.44 8.18 7.38 9.65 Notes: 1. , , indicate signi…cance at the 10%, 5%, and 1% levels, respectively. 2. The SupLR2;T test and the U Dmax2;T test reduce to the SupLR1;T test and the U Dmax1;T test, respectively when ma = 0: variance (na ) coe¢ cients (ma ) Table 5. Sequential tests for structural changes in the variance of the errors SupLR10;T (n = na + 1jn = na ; m = ma ) Number variance (na ) 0 1 2 3 of Breaks coe¢ cients (ma ) 3 3 3 3 GDP 14.04 11.46 4.52 2.72 2 Consumption 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) 27.89 8.30 3.69 4.57 1 Production (Structures) 6.41 11.84 3.30 5.51 0 Notes: Same as in Table 3. Table 6. Tests for multiple structural changes in the coe¢ cients coe¢ cients (ma ) variance (na ) GDP Consumption Investment Exports Imports Government Spending Production (Goods) Production (Services) Production (Structures) 1 0 8.27 8.95 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 1 2 7.83 9.16 9.69 35.05 22.92 15.46 18.05 6.82 10.05 SupLR3;T (m = ma jn = na ) 1 2 2 3 0 1 12.38 11.01 5.96 10.75 14.02 13.05 9.78 16.08 14.13 46.60 47.14 38.79 23.73 38.24 27.25 14.54 19.73 15.99 12.61 10.83 10.27 7.13 12.02 12.08 13.99 16.32 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 SupLR3;T (m = ma jn = na ) U Dmax3;T (M; n = na ) 3 3 3 3 M =3 M =3 M =3 M =3 0 1 2 3 0 1 2 3 GDP 11.59 6.69 8.85 11.79 11.59 6.69 10.10 13.70 Consumption 11.95 14.38 15.49 14.05 14.02 14.38 15.49 14.05 Investment 14.23 13.96 15.38 12.37 16.08 14.13 18.80 14.46 Exports 35.69 39.47 33.77 28.23 78.89 50.36 35.05 46.60 Imports 29.65 25.18 21.48 19.07 46.24 33.35 22.92 23.73 17.78 14.75 14.74 14.66 19.73 15.99 15.46 14.95 Government Spending Production (Goods) 10.68 8.29 8.91 9.65 10.83 16.45 18.05 12.61 12.16 11.82 12.46 13.03 15.09 13.46 12.46 13.03 Production (Services) Production (Structures) 15.24 9.09 14.29 11.86 16.31 12.76 14.29 13.99 Notes: 1. , , indicate signi…cance at the 10%, 5%, and 1% levels, respectively. 2. The SupLR3;T test and the U Dmax3;T test reduce to the standard SupLR test and the U Dmax LR test, respectively, when na = 0 (see Andrews, 1993, and Bai and Perron, 1998). coe¢ cients (ma ) variance (na ) Table 7. Sequential tests for structural changes in the coe¢ cients SupLR9;T (m = ma + 1jm = ma ; n = na ) Number coe¢ cients (ma ) 0 1 2 3 of Breaks variance (na ) 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 25.28 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 19.19 6.73 0 Notes: Same as in Table 3. Table 8. Break dates and parameter estimates Break Dates variance GDP Consumption Investment Exports Imports Government Spending Production (Goods) Production (Services) 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) coefficients 1983:2 (1982:2, 1983:3) 1978:3 (1976:1, 1979:3) 1986:2 (1985:2, 1986:4) 1983:4 (1979:4, 1984:2) Production (Structures) Note: the 90% confidence intervals are in parentheses. Ratio of s.d. over Regime 1 Regime 2 Regime 3 3.15 0.67 Sum of AR coefficients Regime 1 Regime 2 0.93 0.30 0.95 0.41 0.90 0.27 0.70 0.96 0.14 0.91 0.97 0.28 0.99 3.01 0.32 0.96 0.86 0.97 0.96 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 -20 1960 1970 1980 1990 2000 2010 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 -30 1960 1970 1980 1990 2000 2010 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 Deviation (%) original 850 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