Spectral Covariance Instability Test and Macroeconomic. Volatility Dynamics

Transcription

1 Spectral Covariance Instability Test and Macroeconomic Volatility Dynamics Roberto Pancrazi Toulouse School of Economics Toulouse, France First Draft: 29 October 2009 This Draft: 25 July 2011 Abstract Macroeconomists mainly study business cycle uctuations (6-32 quarters) to analyze the economic cycle. In this paper I challenge this approach by showing two important results. First, the Medium-Frequency component (32-80 quarters), which captures the largest portion of the volatility of U.S. linearly detrended output, evolved di erently from the Business Cycle. Second, the dynamics of sub-cycles within the Business Cycle are qualitatively di erent. I document these ndings by analyzing a large set of macroeconomic variables. I then de ne a frequency domain structural break test which formally assesses the presence of a break in their variance. I derive its asymptotic and small sample properties, and I apply it to U.S. macroeconomic variables to provide statistical evidence regarding di erent evolutions of the cycles. JEL Classi cation: C32, C40, E32 Keywords: business cycle, frequency-domain, Great Moderation, structural break test. 1

2 1 Introduction The investigation of economic cycles plays a central role in macroeconomics, since the movements and co-movements of the variables along their trend individuate booms and recessions. As a consequence, the identi cation of the cyclical component of macroeconomic variables has drawn attention in the discipline, both from a theoretical and an empirical point of view 1. Burns and Mitchell (1946) are among the pioneers of business cycle analysis, de ning the business cycle as those uctuations that occur in the economy with periodicity up to ten years. Baxter and King (1999) formalized this de nition and provided tools to isolate the business cycle component of a time series from its trend. In the vast literature on the cyclical properties of the macroeconomic variables, the abovementioned de nition of the business cycle provided by Burns and Mitchell (1946) is the conventionally adopted measure of the economic cycle. However, in this paper I show that a more accurate analysis of the dynamics of the cyclical components of macroeconomics series leads to interesting and novel ndings about the evolution of the economic cycle in the last decades. In particular, I de ne six di erent cycles: the High-Frequency component ( uctuations between 2 and 6 quarters),the Business Cycle ( uctuations between 6 and 32 quarters, consistently with the literature), and the Medium-Frequency component ( uctuations between 32 and 80 quarters) 2. Moreover, I split the Business Cycle in two additional components: the Higher-Business Cycle component ( uctuations between 6 and 16 quarters), and the Lower-Business Cycle component ( uctuation between 16 and 32 quarters). Finally, I de ne the Medium Cycle ( uctuations between 6 and 80 quarters) as the combination of the Business Cycle and the Medium-Frequency components. Two are the main contributions of this paper. The rst one is empirical and it sheds some lights on the behavior of the U.S. economic cycles in the post-war period at Medium-Frequencies, and at sub-sets of the Business Cycle frequencies. As it will come clear later, these cycles exhibit interesting dynamics that are ignored by the dominant focus in macroeconomics just on business cycle frequencies. The second one is theoritical, since I de ne a battery of frequency domain 1 The literature on this topic is extensive, beginning early in the 20th century (see Schumpeter (1927, 1939), Kuznets (1940, 1961), Friedman and Schwartz (1963), Lucas (1977), and Hodrick and Prescott (1997)) 2 A recent papers by Comin and Gertler (2006) examines the features of a more broadly de ned cycle, including uctuations up to 50 years long. 2

3 structural break tests (the Spectral Covariance Instability Tests), which are able to detect breaks in the second moments of any vector of time-series at di erent frequencies. In the empirical section of the paper, I show that an analysis of the U.S. economic cycles that departs from the conventional de nition of the business-cycle revelas interesting and novel ndings. In particular, I point out that the Medium-Frequency component is relevant for explaining the cyclical behavior of U.S. macroeconomic variable. In fact, the Medium-Frequency component accounts for a larger fraction of the total uctuations of linearly detrended U.S. output (37 percent) with respect to the Business-Cycle component (25 percent). Therefore, ignoring the Medium-Frequencies component results in a relevant loss of information about the overall behavior of the economic cycle. I also nd that the Medium-Frequency components of U.S. macroeconomic variables in the post-war period evolved di erently with respect to their Business Cycle component. In particular, since the early 1980s, the Business Cycle volatility of macroeconomic variables has sharply declined 3. However, two results emerge from my analysis: rst, the Medium-Cycle component of the majority of the macroeconomic variables does not display a similar decline in volatility after the early 1980s, and second, within the Business Cycle, only the Higher-Business Cycle component experienced a signi cant drop in its volatility during the last decades, whereas the volatility of the Lower-Business Cycle displays an inverse-u shaped pattern. I show that these ndings are common to the majority of U.S. macroeconomic series from NIPA. The empirical analysis shows that the macroeconomic volatility dynamics are qualitatively di erent when comparing the Business Cycle with the Medium Cycle, and the Higher-Business Cycle with the Lower-Business Cycle. After documenting these empirical facts, I also propose an econometric tool which allows to formally test the presence of signi cant changes in the second moments of macroeconomic variables at di erent frequencies. Speci cally, I introduce and de ne the Integrated Cospectrum, which computes the variance and the covariance of a set of variables at any given interval of frequencies, without relying on the choice of a bandwidth. Since one of the goals of this paper is to study whether second moments of the variables at di erent cycles has signi cantly changed throughout 3 The term Great Moderation was introduced to de ne this well-known and amply documented phenomenon. Kim and Nelson (1999), McConnell and Perez-Quiros (2000), Blanchard and Simon (2001), and Stock and Watson (2002), among others, have contributed to this literature. 3

4 the sample, I then de ne a structural break test in the frequency domain, the Spectral Covariance Instability test. Although the literature on structural break tests is large 4, their application to the frequency domain is one of the novelties introduced in this paper. I present three types of tests, the Spectral Average Wald test, the Spectral Exponential test, and the Spectral Nyblom test. After deriving the asymptotic proprieties of these tests, I analyze their small sample proprieties using Monte Carlo simulations. In addition, I compare the properties of the proposed frequency domain approach, with a time domain GMM-based approach, which is a natural alternative of calculating variances and covariances at any interval of frequencies. I show that the frequency domain approach I propose performs signi cantly better in small samples, because it avoids needs to de ne a bandwidth. Intuitively, den Haan and Levin (1996), and Kiefer, Vogelsang, and Bunzel (2000) have already pointed out the unsatisfactory small sample proprieties of GMM estimators, related in particular to the choice of bandwidth, since there are no guidelines for the optimal choice in small samples. The choice of the bandwidth implicitly implies a trade-o between the bias of the estimator and its variance. Therefore, the choice of the bandwidth in a small sample is not a trivial concern in practice. The Spectral Covariance Instability tests presented in this paper do not su er from the same problem, since, as mentioned above, they do not require any choice of a bandwidth. In fact, the Integrated Cospectrum is estimated as the integral of the sample periodogram and the integration procedure along the frequencies works directly as a smoothing function. Therefore, since the integration does not require the speci cation of any bandwidth parameter, the Integrated Cospectrum does not su er from any trade-o between its bias and its variance. As a consequence, the small sample properties of the Spectral Covariance Instability tests are improved with respect to its GMM-based time-domain counterpart, as showed in the paper. Finally, I apply the Spectral Covariance Instability tests to a sub-set of the main U.S. macroeconomic variables, namely output, consumption, and investment, the disaggregated components of consumption, and the disaggregated components of investment. Consistently with the Great Moderation literature, the tests detect a break in the variance of the three variables when only the Business Cycle frequencies are considered. However, when Medium-Frequencies are also included in the cyclical analysis, the tests suggest absence of any moderation. In addition, when we test 4 See Perron (2005) for a review 4

5 for a break in the second moments of the two components of the Business Cycle, the tests detect a break only for the Higher-Business Cycle component. This nding, supported by the descriptive evidence mentioned above, reveals that the decline in the variance of macroeconomic variables is present only in the higher portion of the spectrum, and it is at least mitigated, if not absent, when a broader measure of the cycle is adopted. The structure of the paper is as follows. In Section 2, I document the properties of the six di erent cyclical components of U.S. macroeconomic variables. In Section 3, I de ne the Spectral Covariance Instability tests and assess their asymptotic properties. In Section 4, I analyze their small sample proprieties using Monte Carlo simulations, with particular emphasis on the advantage of this frequency domain approach with respect to the time domain GMM-based approach. In Section 5, I apply the Spectral Covariance Instability tests to consumption, output and investment, and their disaggregated components. Section 7 concludes with some nal remarks. 2 Cyclical Macro-Volatility Dynamics The cyclical behavior of economic variables has been one of the primary interests in macroeconomics since the early stages of the discipline. During most of the last century, research was devoted to the empirical characterization of the economic cycle. Burns and Mitchell (1946) de ned the business cycles as follows: A cycle consists of expansions occurring at about the same time in many economic activities, followed by similarly general recessions, contractions, and revivals which merge into the expansion phase of the next cycle; this sequence of changes is recurrent but not periodic; in duration business cycles vary from more than one year to ten or twelve years. This de nition was formalized by Baxter and King (1999), which identify the business cycle as those cycles with periodicity from 6 to 32 quarters. This de nition is the commonly adopted measure in the cyclical analysis of macroeconomic variables. However, the adoption of this de nition might hide interesting features of the economic cycle: rst, the role of the uctuations with 5

6 periodicity higher than 32 quarters, and second, the di erent dynamics of the sub-cycles within the 6-32 quarters periodicity. One of the goal of this paper is to shade light on these features, mostly ignored by macroeconomists. In detail, consider the conventional de nition of business-cycle: all uctuations with periodicity larger than 32 quarters are included into the trend, and therefore excluded from the investigation of the cyclical properties of the economic series. Figure 1 displays the log-level of real per-capita U.S. Gross Domestic Product (solid line) and the trend identi ed by eliminating the uctuations with periodicity up to 32 quarters (dashed line). The cyclical component is de ned as the di erence between the level and the trend. However, notice that the trend generated by this procedure displays evident waves that a ect the medium-run behavior of the series. This source of uctuations has been ignored in the conventional business cycle analysis 5. In the rest of paper I will show that medium-run cycles capture a large portion of the uctuations of macroeconomic variables, and therefore, might be relevant to analyze their behavior. In addition, the conventional de nition of business cycle gathers all the frequencies between 6 and 32 quarters, and, as a consequence, it is silent about the possibility of a di erent evolution of cycles within that range of frequencies. In the empirical analysis of this paper I will show that cycles identi ed within the business cycle frequencies evolved with qualitatively di erent dynamics. In this paper I study the properties of several cycles de ned in a ner set of frequencies than the conventional de nition of business cycle. This approach allows to understand the role of a larger set of intervals on frequencies on capturing the dynamics of macroeconomic variables. The idea of a presence of several economic cycles was introduced at the beginning of the 20th century; Schumpeter (1954) decomposes a stationary series in four di erent waves, named after the economist that rst introduced them, i.e. the Kitchin inventory cycle (2-5 years) the Juglar xed investment cycle (7-11 years) the Kuznets infrastructural investment cycle (10-20 years) 5 Comin and Gertler (2006) present a similar gure, plotting the Medium Term Cycle (whose maximum periodicity is 40 years) for the per-capita non-farm business output; the following ndings are consistent with their analysis. 6

7 the Kondratie wave or cycle (45-60 years). Therefore, the economic series is thought as a combination of these four components. The purpose of this paper is to apply an idea similar to Schumpeter s (1954). Whereas the conventional analysis of the cycle is based on the business cycle, which approximately includes the Kitchin inventory cycle and the Juglar xed investment cycles, I also consider the role of the Kuznets infrastructural investment cycle, which corresponds to those uctuations up to 20 years. Therefore, I ask whether a measure of the cycle de ned more broadly, including uctuations with periodicity up to twenty years, provides additional and relevant information about the cyclical behavior of macroeconomic variables. In order to clearly isolate the contribution of the di erent cyclical components, I will refer to the High-Frequency component as uctuations included between 2 and 6 quarters, to the Business Cycle component as the uctuations included between 6 and 32 quarters, and to the Medium- Frequency component as the uctuations included between 32 and 80 quarters. Moreover, I will split the Business Cycle in two sub-components: the Higher Business Cycle component, arbitrarily de ned to include uctuations between 6 and 16 quarters, and the Lower Business Cycle component, which includes uctuations between 16 and 32 quarters. A formal de nition of the cyclical components is provided as follows: De nition 1 Given a time series x t ; the High-Frequency component (HF), x HF t ; corresponds to the cyclical component of x t with periodicity between 2 and 6 quarters. In the frequency domain, these uctuations belong to the interval ; 3 for quarterly data. De nition 2 Given a time series x t ; the Higher Business Cycle component (HBC), x HBC t ; corresponds to the cyclical component of x t with periodicity between 6 and 16 quarters. In the frequency domain, these uctuations belong to the interval 3 ; 8 for quarterly data. De nition 3 Given a time series x t ; the Lower Business Cycle component (LBC), x LBC t ; corresponds to the cyclical component of x t with periodicity between 16 and 32 quarters. In the frequency domain, these uctuations belong to the interval 8 ; 16 for quarterly data. 7

8 De nition 4 Given a time series x t ; the Medium-Frequency component (MF), x MF t ; corresponds to the cyclical component of x t with periodicity between 32 and 80 quarters. In the frequency domain, these uctuations belong to the interval 16 ; 40 for quarterly data. In the paper I will describe the dynamic evolution of these four cycles. However, a large part of the analysis will be devoted to understand the implication of considering the medium-frequency together with the business-cycle frequencies. To achieve this goal I de ne two additional broader cycles, the Business-Cycle, as conventionally de ned, and the Medium-Cycle, which includes also the Medium-Frequency component, i.e.: De nition 5 Given a time series x t ; the Business Cycle, x BC t ; corresponds to the cyclical component of x t with periodicity between 6 and 32 quarters. In the frequency domain, these uctuations belong to the interval 3 ; 32 for quarterly data. Therefore, the Business Cycle is the sum of the Higher Business Cycle component and the Lower Business Cycle component: x BC t = x HBC t + x LBC t : De nition 6 Given a time series x t ; the Medium Cycle component, x MC t ; corresponds to the cyclical component of x t with periodicity between 6 and 80 quarters. In the frequency domain, these uctuations belong to the interval 3 ; 40 for quarterly data. Therefore, the High-to-Medium frequency component is the sum of the Higher Business Cycle component, the Lower Business Cycle component, and the Medium-Frequency: x MC t = x HBC t + x LBC t + x MF t : 2.1 Cyclical Components of Output To study the behavior of the di erent cycles de ned above, in Figure 2a I plot the four cyclical components of output. Output is de ned as the per-capita quarterly real GDP series, in the period 1947:1-2007:4 6. The solid line displays the High-Frequency component, the dotted line displays 6 Source: US Bureau of Economic Analysis (BEA) 8

9 the Higher Business Cycle component, the star line displays the Lower Business Cycle component, and the dashed line displays the Medium-Frequency component. The series are computed using a bandpass lter, as implemented by Christiano, Fitzgerald (2003). Recall that this procedure eliminate the long-run trend of the series. Therefore, the cycles display the deviation from the trend. First, note that the magnitude of the Medium-Frequency component is large if compared to the other components, as visualized by the peak-to-through distances. Also, within the business cycle frequencies, the Lower-Business Cycle component is larger than the Higer- Business Cycle component especially after the post Finally, we observe the reduction of the volatility of the High-Frequency and Higer- Business Cycle components after the mid-1980s. This reduction in variance seems milder or even absent for the Lower-Business Cycle and the Medium-Frequency component. In order to explore the contribution of di erent cycles to capturing the overall volatility of detrended output, in Table 1 I report the standard deviation of the six di erent components and the total standard deviation of the linearly detrended output, both in levels and percentage. The High-Frequency component captures a small part of the overall variance of detrended output (less than 2 percent). Also, the volatility of the Medium-Frequency component is larger than the one of the Business Cycle component: the former accounts for 38 percent of the total variability of detrended output, whereas the latter accounts for only the 23 percent of the total variance of detrended output. Therefore, focusing on the Medium-Cycle would allow to study a much larger portion of the overall variance of detrended output (61 percent) than just considering the Business Cycle (23 percent). Finally, the lower part of the business cycle includes a larger portion of variance than the higher part of the business cycle (15 percent and 9 percent respectively). To study what is the contribution added by including the Medium Frequencies in the cyclical analysis, in Figure 2b I plot the commonly used Business Cycle component (solid line) of output and its Medium Cycle component (dashed line), which is, as de ned above, the combination of the Business Cycle and the Medium-Frequency component. We observe that the booms and the recessions identi ed by the Medium Cycle are ampli ed with respect to those identi ed by the 9

10 Business Cycle. For example, the substantial upward movement of output during the 1960s caused a growth of roughly 10 percent in the Medium Cycle, versus a 4 percent growth in the Business Cycle. Similarly, the decline of output at the end of the 1970s and at the beginning of the 1980s was almost 10 percent in the Medium Cycle and only 6 percent in the Business Cycle. Moreover, Figure 3 suggests that the correlation between the Business Cycle and the Medium Cycle declined in the last part of the sample; whereas until the mid-1980s the two series have a similar pattern, in the last two decades the two cycles diverge. To show this fact, I divide the sample into two sub-samples: the rst sub-sample includes observations in the period 1947:1-1983:4; the second sub-sample includes observation in the period 1984:1-2007:4. The sample correlation between the Business Cycle and the Medium Cycle is 0.70 in the rst sub-sample, and 0.40 in the second sub-sample. This divergence can be attributed to the larger magnitude of the Medium Frequency component in the second sub-sample. To observe the evolution of the variance of the cycles, in Table 2, I compute the standard deviations of the six components in both sub-samples. Although the total standard deviation of the linearly detrended output has declined from 3.7 to 2.3 in the second sub-sample, this reduction might be located especially at higher frequencies. In fact, the decline of the variance in the sub-samples is uneven across frequencies: the High-Frequencies and the Higher-Business Cycle experienced a more than 80 percent decline in their variance in the second subsample, the Lower- Business Cycle variance dropped by about 65 percent and the Medium Cycle variance declined only by 22 percent. As a natural consequence of this uneven changed in the variances across components, the relative contribution of each cycle on the total output variance changed as well. In fact, whereas in the pre-1984 period, the Business Cycle component accounts for about 25 percent of the total volatility of output, this value declines to 16 percent in the post-1984 period. On the other hand, the relative contribution of the Medium-Frequency component doubled in the second sub-sample, from 31 percent to 62 percent. As a consequence, in the second sub-sample the behavior of the Medium Cycle component is mostly driven by the Medium Frequency component. Also, within the business cycle frequencies, the Higher and Lower Business cycle components experienced a reduction of the volatility at di erent magnitude, since the Lower-Business Cycle 10

11 variance declined only about 67 percent. As a result, in the second subsample the Business Cycle dynamics are driven mainly but its the Lower-Business Cycle component. This descriptive evidence suggests that the Medium-Frequency component has become more relevant in the last part of the sample, thus implying a divergence between the Business Cycle and the Medium Cycle. In other words, if a researcher who studies the uctuations of the economy focuses just on the Business Cycle, she would not take into account the larger amount of uctuations now than in Obviously, this date is arbitrary, but the example above suggests the importance of exploring the contribution of the Medium Frequencies on capturing the uctuations of the economy. 2.2 The Business Cycle and Medium Cycle Volatility Ratio In the previous section, I showed that the correlation between the Business Cycle component and the Medium Cycle component has declined in the last twenty years. This implies that in this period the Medium-Frequency component has increased its relative weight with respect to the Business Cycle component. Intuitively, if there were no uctuations with periodicity between 32 and 80 quarters, the Medium Cycle component would exactly coincide with the Business Cycle component; this obviously means that their correlation would be one. On the other hand, the more uctuations belong to the Medium-Frequency cycle, the more the Business Cycle component and the Medium Cycle component diverge. The intuition presented above is formally supported by Theorem 7, which provides a useful representation of this correlation in terms of the variances of the ltered process: Theorem 7 Let I 1 = [! 1 L ;!1 H ] and I 2 = [! 2 L ;!2 H ]be two disjoint set of frequencies, and let I 3 = I 1 [ I 2 : Let x 1 t ; x 2 t and x 3 t be the ltered series obtained by the same process y t, isolating respectively the frequencies in I 1, I 2, and I 3 : Then, the correlation between x 1 t and x 3 t is equal to the ratio of their standard deviation, i.e. x 1 t ; x 3 t = sv ar (x 1 t ) V ar (x 3 t ) : (1) (For the proof see Appendix B). 11

12 It is worth noticing that this result holds when the frequencies in I 1, I 2, and I 3 are perfectly isolated. For simplicity of notation, henceforth I denote = x BC t ; x MC t, HF = p V ar (x HF t ); HBC = p V ar (x HBC ); LBC = p V ar (x LBC ); MF = p V ar (x MF ); BC = p V ar (x BC );and MC = p V ar (x MC t ): These seven parameters are estimated using the sample correlation and the sample standard deviation of the series, ltered at the appropriate frequencies with a bandpass lter. In order to visualize the evolution of ^ over time, I construct a rolling window statistic as follows: x HF ^ t = ^ J t j=t ; x HM k J t j=t k for t = k + 1; :::; T; where k indicates the length of the window, T is the length of the time series, and fxg t 2 t 1 represent the subset of observations of the time series x included between the periods t 1 and t 2 : Accordingly, ^ t is the value of the correlation between the Business Cycle component and the Medium Cycle component computed by considering the k observations of the series x prior to time t: Using a similar procedure, we can estimate the rolling window standard deviations of all the cycles: Figure 3a plots the rolling window statistics of output computed for a window-length of 20 years (k = 80). The rst panel shows the evolution of ^ t, the second panel shows the evolution of ^ HF t (solid line), ^ HBC t shows the evolution of ^ BC t (dotted line); ^ LBC t (solid line), ^ MC t (star line); and ^ MF (dashed line), and the third panel (dotted line): The rst panel shows that the correlation between the Business Cycle component and the Medium Cycle component has declined in the second part of the sample, from a maximum value of 0.8 to a minimum value around 0.2. As stated in the previous section, the decline of this correlation is explained by the increase in the relative importance of the Medium-Frequencies volatility with respect to the Business Cycle volatility. In fact, whereas the standard deviations of output at business cycle frequencies largely declined in the last decades (mainly due to the Lower-Business Cycle component), in the same period the standard deviation of the medium frequencies did not dropped as much. In order to study the temporal evolution of the volatility of di erent cycles, the second panel 12

13 plots their rolling windows standard deviations. Their dynamics are qualitatively di erent. The High-Frequency component, which accounts for a small portion of the variance, displays a roughly constant standard deviation until the mid-80s and then a sharp decline in particular during the 1990s. The standard deviation of the Higher-Business Cycle component, instead, declines very uniformly since the beginning of the sample. On the contrary, the Lower-Business Cycle standard deviation has an interesting inverse U-shape dynamics, since it increases during the 1970s and the 1980s and then declines in the last two decades. Finally, after an initial drop, the Medium- Frequency standard deviation rises after the mid-1980s. This evidence suggests that it is worth investigating the properties of the economic cycles at di erent frequencies, since cycles with different periodicity do not share the same time evolutions. The decline of the volatility of the macroeconomic variables after the mid-1980s has attracted attention in the recent macroeconomic literature. The term Great Moderation was in fact created to refer to the evident stabilization of the macroeconomic variables 7. However, the rolling window statistics presented above suggest that the reduction of the volatility of output is concentrated mainly at higher frequencies (High-Frequency and Higher-Business Cycle Frequencies). On the other hand, the Lower-Business Cycle standard deviation increased until the early 1990s and only after that period it declines. Finally, the Medium-Frequency volatility actually increased. The third panel, in fact, shows that accounting for the Medium-Frequency component in the economic analysis (which might be intuitive to do since as we have seen in the Table 1 the Medium- Frequency component accounts for the largest part of the overall volatility of detrended output) might undermine the presence of the Great Moderation: in fact, whereas the Business Cycle displays the evident drop in its standard deviation in the last decades, the Medium Cycle does not display evident moderation. In the next sections of the paper, I will rst show that this pattern is common to other macroeconomic variables, and then I will formally test the presence of moderation when the medium frequencies are considered. In summary, the empirical evidence described in this section suggests that an analysis of the uctuations of output conducted just at business cycle frequencies might be 7 See Stock and Watson (2002) for a survey of the literature. 13

14 misleading and incomplete on explaining the macroeconomic volatility dynamics, since the Medium Cycle has captured an increasing share of the total volatility of output during the last twenty years and since we observe a heterogenous behavior of sub-cycles within the Business Cycle. 2.3 Cyclical Components of Disaggregated Data In the previous section I pointed out that the four di erent cycles presented in this paper (HF, HBC, LBC, and MF) show qualitatively di erent evolutions of their variance, and that the relative contribution of the Medium-Frequency component to the total volatility of detrended output in the last twenty- ve years has increased. Similar descriptive evidence is found in the disaggregated components of GDP, as I show in this section. The data set is composed of the quarterly real percapita NIPA series and covers the period 1947:1-2007:4. The list of the series and their identi cation number can be found in Appendix A. From Figure 3b to Figure 3w I plot the rolling window statistics for all the NIPA series. As in the previous section, the rst panel shows the evolution of ^ t ; the second panel shows the evolution of ^ HF t (solid line), ^ HBC t panel shows the evolution of ^ BC t (dotted line); ^ LBC t (star line); and ^ MF (dashed line), and the third (solid line), ^ MC t (dashed line). Some ngings are common to the majority of the variables: rst, all the series, although to a di erent degree, display a decline in the variance of the High-Frequency component and of the Higher-Business Cycle component in the last part of the sample; second, the Lower-Business Cycle component shows an inverse u- shaped evolution, where its variance declines only after the 1990s; nally, the Medium-Frequency volatility has increased. As a consequence, the Medium-Cycle volatility has not declined as much as the Business Cycle volatility during the post mid-80s, and, also, the correlation between the High-Frequency and the High-to-Medium-Frequency cycles of many series had drop from values around 0.7 to values around 0.2 in the last part of the sample. These observations motivate one of the goal of this paper, which is to study whether the cyclical economic dynamics are robust to the di erent de nitions of the cyclical components. 14

15 2.3.1 Consumption Figure 3b shows that the correlation between the Business Cycle and Medium-Frequency components of the Personal Consumption Expenditure series has dropped from its maximum of 0.7 in the early-80s to 0.2 in the last years. This decline is due to the decrease of the business cycle variance and to the increase of the Medium-Frequency volatility. The analysis of the component of Consumption (Durable, Non-Durable, and Services, respectively, in Figure 3c, Figure 3d, and Figure 3e), shows that the standard deviation of the Medium-Frequency has risen especially for Durables and Services. In particular, in the last two decades the Durable series, the most volatile component of consumption, displays an evident divergence between the declining standard deviation of the Business Cycle, and the rising standard deviation of the Medium-Frequencies. As a result, the correlation ^ t constantly drops in the Durables series. The Non-Durable series have similar features: the decline of the Business Cycle volatility is mainly driven by the Higher-Business Cycle component, since the Lower-Business cycle variance does not decline in the last two decades, and the increase of the Medium-Frequency volatility is once again evident. Finally, the Service series displays a substantial increase in the Medium-Frequency variance, but it does not share the same decline of the variance of the other components. As a result, the Great Moderation does not appear as strong for the Service series and the correlation ^ t remains quite steady Investment The role of the Medium Frequency component is particularly evident in the Investment series, the most volatile component of GDP. As Figure 3f and Figure 3g show, the Gross Private Domestic Investment and the Fixed Investment series display a sharp decline of the correlation ^ t after 1990, from 0.9 to almost 0.4. This drop is caused by both the decline of the Business Cycle volatility and the increase of the Medium-Frequency volatilities in the last twenty years. An additional implication of this fact is that in the last observations of the sample the standard deviation of the Medium-Frequency component is much larger than the standard deviation of the Business-Cycle component. Moreover, as for GDP, the High-Frequency component and the Higher-Business Cycle component display a sharp decline of their standard deviation, the Lower-Business Cycle displays 15

16 a inverse-u shaped standard deviation evolution. I also analyze the behavior of the di erent components of investment, i.e. the Non-Residential (Figure 3h) and Residential (Figure 3k) investments. Several important implications can be inferred from comparing the two components. First, the Business Cycle volatility has declined in both Residential and Nonresidential investment in the last two decades. On the other hand, the standard deviation of the Medium-Frequency component has a di erent pattern in the two variables. In fact, in Nonresidential investment, the standard deviation ^ MF t rises sharply after 1990 until the end of the sample, whereas it has increased in Residential Investment since the early 80s, and it slightly decreases at the end of the sample. As a consequence, in Residential Investment the volatility of the Medium Cycle follows the increasing Medium-Frequency volatility and in the Nonresidential it is driven by the decreasing Business Cycle volatility. For this reason, although Residential Investment is more volatile than Non-Residential in most of the sample, this spread has shrunk in the last decades Imports, Exports, and Government Spending Since Imports, Exports and Government Spending together account for a smaller fraction of output than Consumption and Investment, their impact on the cyclical behavior of output is limited. However, Figure 3l and Figure 3o show that Exports and Imports, respectively, share the same features as most of the series analyzed above, i.e. the declining Business Cycle volatility (with di erent dynamics when considering its higher and lower part) and the rising Medium Frequency volatility in the last part of the sample. In contrast, the Government Spending series, Figure 3r, displays a large volatility only in the beginning of the sample, mainly because of Korean war military spending. Afterwards the rolling window standard deviations do not show relevant movements. 3 Spectral Covariance Instability (SCI) Tests The empirical evidence shows that the Medium-Frequency component contains a large part of the information about the cyclical behavior of macroeconomic series, since it captures a large portion of 16

17 their total volatility. Moreover, the Medium-Frequency components do not share similar properties with the other components. For example, whereas there was a reduction of the Business Cycle volatility after the early 1980s, there was not a similar decline of the Medium-Frequency volatility in the same time period. In addition, even within the Business Cycle, its Higher-Frequency component behaves qualitatively di erent than the Lower-Frequency component. Therefore, it is natural to ask whether the properties of the economic cycle are robust to a di erent de nition of the cyclical component. In particular, one could wonder whether the decline of the volatility after the mid- 1980s is a signi cant phenomenon across di erent intervals of frequencies. For this purpose, I introduce the Spectral Covariance Instability (SCI) test, an useful tool to test whether a multivariate process has experienced a structural break in its variance or covariance at any interval of frequencies of interest. The frequency domain approach for a structural break test is a noverly I introduced in this paper. In the next section I will show that the frequencydomain approach has signi cant advantages with respect to the time-domain approach in terms of statistcal inference. The basic concept at the base of this test is the Integrated Cospectrum, which computes the variance and covariance attributable to any interval of frequencies The Integrated Cospectrum Let y t be a (N x 1) multivariate linear stationary process whose in nite moving average, MA (1), representation is: y t = + (L) " t ; (2) 8 Ahmed, Levin, and Wilson (2004) used the Integrated Spectrum, the univariate version of the Integrated Cospectrum in their analysis. 17

18 where L is the lag operator, is the mean vector, and (L) = P 1 k=0 kl k with f kg 1 k=0 absolutely summable. The (N x 1) vector " t is a vector of white noise, i.e.: E (" t ) = 0 8 >< for t = E (" t " 0 ) = >: 0 otherwise. The k th autocovariance matrix of y is given by: E (y t ) (y t k ) 0 = (k) : (3) and the autocovariance-generating function of y is: G Y (z) = 1X k= 1 (k) z k ; (4) with (k) 1 k= 1 absolutely summable and with z being a complex scalar. Then, the population spectrum of the vector y is given by s Y (!) = (2) 1 G Y e i! = (2) 1 1 X k= 1 (k) e i!k ; (5) where i = p 1 and! is a real scalar. Thus, s Y (!), known as the cross-spectrum, is a (N x N) matrix in which the diagonal elements are real and the o -diagonal elements are complex. The cross-spectrum can be written in terms of its real and imaginary components, i.e. s (!) = c (!) + iq (!) : (6) where c (!) and q (!) are known as the cospectrum and the quadrature of y t : First, it is useful to recall some results of the frequency domain analysis in a univariate framework. Let x t be a linear univariate stationary process with nite variance and with an MA (1) 18

19 representation, x t = x + (L) " t ; (7) with mean x and whose spectrum is denoted by s x (!) : Then, the integral between - and of the spectrum of x t is equal to its variance, that is: Z s x (!) = E (x t x ) 2 : For a given frequency range [! 1 ;! 2 ], with 0! 1 <! 2, the variance attributable at that interval of frequencies can be computed with the Integrated Spectrum, as in the following de nition: De nition 8 Given a univariate process x t as in (7), the Integrated Spectrum H(! 1 ;! 2 ) for the interval of frequencies [! 1 ;! 2 ], with 0! 1! 2 ; is the integral of s x (!)between! 1 and! 2 ; i.e. Z!2 H(! 1 ;! 2 ) = 2 s x (!) d!: 9 (8)! 1 and it corresponds to the variance of the processes x t due to cycles identi ed by the interval of frequencies [! 1 ;! 2 ] : I now extend these results in the multivariate case; in fact, the integral of the cross-spectrum corresponds to the variance-covariance matrix of the multivariate process, i.e.: Z s (!) d! = E (y t ) (y t ) 0 : (9) However, since q (!) = q (!) ; the variance-covariance matrix can be computed from the area below the Cospectrum: Z c (!) d! = E (y t ) (y t ) 0 ; (10) It is then straightforward to extend the concept of the Integrated Spectrum to de ne the Integrated Cospectrum. 9 Note that s x (!) = s x (!) : 19

20 De nition 9 Given a multivariate process y t as in (2), the Integrated Cospectrum for the interval of frequencies [! 1 ;! 2 ], with 0! 1! 2 ; is the integral of c (!) between! 1 and! 2 ; i.e. Z!2 H(! 1 ;! 2 ) = 2 c (!) d!: (11)! 1 and it corresponds to the variance-covariance matrix of the processes y t due to cycles identi ed by the interval of frequencies [! 1 ;! 2 ] : To provide a consistent estimate for the Integrated Cospectrum, I generalize Priestley (1982) s univariate approach to a multivariate process. De ne the multivariate sample periodogram for a sample size T process y t as ^I (!) = 1 2 XT 1 j=1 T ^(j) e i!j = 1 2 XT 1 j=1 T ^(j) cos(!j); (12) where ^(J) represents the sample autocovariance given by: ^(j) = 1 T T j X (y t y) (y t j y) 0 : (13) i=1 A consistent estimate of the Integrated Cospectrum H(! 1 ;! 2 ) is given by Z!2 ^H(! 1 ;! 2 ) = 2 Z!2 1 = 2! 1 2 = 1 " = 1! 1 ^I () d (14) XT 1 j=1 T XT 1 j=1 T ^(j) Z!2 T 1 ^(0) + ^(j) cos(j)d cos(j)d! 1 # X ^( sin j) + ^(j) (!2 j) sin (! 1 j) ; (15) j j=1 where the second equality comes from (12), the third equality comes from switching summation and integral, and the fourth equality comes from R cos(j)d = sin(j) j : To prove the asymptotic proprieties of ^H(! 1 ;! 2 ); it is convenient to recall some general results 20

21 about weighted integrals of the periodogram, as in the following Lemma 10 : Lemma 10 Let 1 (!) and 2 (!), be two real valued functions de ned in! ; each of which has at most a nite number of discontinuities and is both absolutely integrable and square integrable, i.e. for i=1,2, Z j i (!)j d! < 1 and Z 2 i (!) d! < 1: Let x t be a general univariate linear process as in (7) with " t normally distributed 11, and whose spectrum is s x (!). Let, for i=1,2, ^ i = i = Z Z i (!) ^I (!) d! (16) i (!) s x (!) d!; (17) with ^I (!) being the sample periodogram of x t. Then: 1. lim ^i E = i ; i = 1; 2: T!1 2. lim T cov ^ ^1 2 T!1 Z = 4 1 (!) 2 (!) s 2 x (!) d! with 2 (!) = 1 2 [ 2 (!) + 2 (!)] : In particular, when 1 (!) = 2 (!) = (!) ; we have: 10 For the proof, see Priestley (1982) pp The normality assumption of the errors, although not crucial for results 1-2-3, is convenient to obtain simpler expressions for the asymptotic variance and covariance for the following estimators. 21

22 3. ^ lim T var T!1 Z = 4 (!) (!) s 2 x (!) d! with 1 (!) = 12 [ (!) + (!)] 2 The following Theorem shows the asymptotic proprieties of the element of the Integrated Cospectrum: Theorem 11 Let y t be a multivariate linear process as in (2), where " t is a multivariate normal. Then, the (m; n)-th element, m = 1; ::N, n = 1; ::N of the sample Integrated Cospectrum in (15) has the following proprieties: 1. asymptotic unbiasedness: lim E ^H m;n (! 1 ;! 2 ) = H m;n (! 1 ;! 2 ) T!1 2. consistency: ^H m;n (! 1 ;! 2 ) p!h m;n (! 1 ;! 2 ) 3. asymptotic normality: p h i d! T ^H m;n (! 1 ;! 2 ) H m;n (! 1 ;! 2 ) N 0; m;n(! 1 ;! 2 ) ; with Z!2 m;n (! 1 ;! 2 )= 8! 1 c 2 m;n(!)d!: (18) (For the proof see Appendix B). In order to derive any inference result on the Integrated Cospectrum, we need an estimate of its asymptotic variance-covariance matrix in (18). An obvious procedure is to substitute the cospectrum c (!) with the multivariate periodogram in (12). Moreover, I approximate the integral 22

23 in (18) with a discrete sum, dividing the interval [! 1 ;! 2 ] into q segments of length! =! 2! 1 q ; with q! 1 as T! 1. The following Theorem assures the consistency of the estimate of m;n (! 1 ;! 2 ): Theorem 12 Let y t be a multivariate linear process as in (2), where " t is a multivariate normal. Also, let 0! 1! 2 ; and de ne! =! 2! 1 ; with q! 1 as T! 1: Then, a consistent q estimate of m;n (! 1 ;! 2 ), m = 1; ::N, n = 1; ::N is given by ^ m;n = 4 " qx 1 2 i=1 XT 1 j=1 T ^(j) m;ne i!j # 2!: (19) (For the proof see Appendix B). Hence, the approximated distribution of ^H m;n (! 1 ;! 2 ) is given by: ^H m;n (! 1 ;! 2 ) N H m;n (! 1 ;! 2 ); ^! m;n (! 1 ;! 2 ) : (20) T 3.2 Spectral Covariance Instability (SCI) Test Once the asymptotic distribution of the Integrated Cospectrum is derived, I can use its estimate for testing purposes. In particular, in what follows I derive a structural break test for the elements of the Integrated Cospectrum. In particular, let y m;t and y n;t be two components of the process y t : An estimate of the covariance between y m;t and y n;t at the frequencies in [! 1 ;! 2 ] is given by ^H m;n (! 1 ;! 2 ): Even though in this section I focus on the covariance between two series (m 6= n), the same procedure can be applied when m = n, that is when the variance of y m;t is the object of the analysis Suppose one wants to test whether the population covariance attributable to the frequencies in [! 1 ;! 2 ] ; H m;n (! 1 ;! 2 ) ; has experienced a one-time structural change in a known period : Let = [T c] ; with [] being the integer part operator and c 2 (0; 1) : i.e. the null and the alternative hypothesis and that de ne the test are: 23

24 H 0 : H (1;c) m;n (! 1 ;! 2 ) = H (2;c) m;n (! 1 ;! 2 ) with c known vs H 1 : H (1;c) m;n (! 1 ;! 2 ) 6= H (2;c) m;n (! 1 ;! 2 ), where H (1;c) (! 1 ;! 2 ) denotes the value of the Integrated Cospectrum in the rst sub-period t = 1; ::;, and H (2;c) (! 1 ;! 2 ) denotes its value in the second sub-period t = + 1; ::; T: Note that the value of these parameters depends on c, as indicated on the superscript. I de ne the Spectral Wald Statistic (Sm;n, W henceforth) for the element (m; n) as: S W m;n (! 1 ;! 2 ; c; T ) = h T H (1;c) m;n (! 1 ;! 2 ) ^ (1;c) m;n (! 1 ;! 2 ) + c H (2;c) i 2 m;n (! 1 ;! 2 ) (21) ^ (2;c) m;n (! 1 ;! 2 ) 1 c where ^ (1;c) m;n (! 1 ;! 2 ) is the estimate ^ m;n (! 1 ;! 2 ) computed in the rst sub-sample t = 1; ::;, and ^ (2;c) m;n (! 1 ;! 2 ) is the estimate computed in the second sub-sample t = + 1; ::; T. The time domain counterpart of the Wald statistics was rst introduced by Wald (1943), and it is largely used for statistical testing. As Engle (1983) shows, under the null hypothesis S W m;n (! 1 ;! 2 ; c; T ) has an approximate 2 distribution with 1 degree of freedom. However, in this paper I test the presence of a structural change at an unknown date, i.e. H 0 : H (1;c) m;n (! 1 ;! 2 ) = H (2;c) m;n (! 1 ;! 2 ) with c unknown vs H 1 : H (1;c) m;n (! 1 ;! 2 ) 6= H (2;c) m;n (! 1 ;! 2 ) : First, I will show that, when c is unknown, the Spectral Wald Statistics in (21) has the same asymptotic distribution as the sup-wald statistic (the time domain analouge of the S W m;n (! 1 ;! 2 ; c; T )) derived in Andrews (1993). Theorem 13 Assume that c 2 (0; 1) is unknown. For any interval of frequency [! 1 ;! 2 ], with 24

25 0! 1! 2 ; de ne the Partial-Sample estimator: ^H c (! 1 ;! 2 ) = 1 " T 1 ^(0) + # X ^( sin j) + ^(j) (!2 j) sin (! 1 j) j j=1 where ^(j) are the sample autocovariances estimated using the observation in the sample t = 1; ::; ; with = [T c] and with [] being the integer part operator. Under some regularity conditions as in Assumption 1 in Andrews (1993), and under the assumption of Near Epoch Dependence (NED), the Spectral Wald Statistic in (21) has the following asymptotic distribution: S W m;n (! 1 ;! 2 ; c; T ) =) Q (c) and sup c2 d! sup c2 Q (c) where Q (c) = (B (c) cb (1))2 c (1 c) where B (c) is a Brownian motion on [0; 1] restricted to : (For the proof see Appendix B). The Spectral Wald Statistics has therefore the same asymptotic distribution as its time domain counterpart. Intuitively, isolating the fraction of the variance attributable at a given interval of freqeuncies does not change the nature of the Wald-statics. Therefore, the de nition of the Spectral Wald Statistics is identical to its Partial-Sample GMM time-domain analogue. Given the analogy between the time-domain and the frequency-domain Wald statistics, I can follow the literature on structural break test to de ne optimal structural break tests in the frequency domain, in the sense that they maximize a weighted average power. For this purpose, I introduce the following three types of Spectral Covariance Instability tests: De nition 14 Let y t be a process as in (2) ; and 2 (0; 1) ; for any interval of frequency [! 1 ;! 2 ], with 0! 1! 2 ; we de ne the following Spectral Covariance Instability tests for each element (m; n) of the variance-covariance with m = 1; ::N, n = 1; ::N: 25

26 The Spectral Average Wald Test (SAW): Z SAW m;n = Sm;n W (! 1 ;! 2 ; c; T ) dc: c2 The Spectral Exponential Wald Test (SEW): Z 1 SEW m;n = log exp c2 2 SW m;n (! 1 ;! 2 ; c; T ) dc: The Spectral Nyblom Test (SN): Z SN m;n = Sm;n W (! 1 ;! 2 ; c; T ) c(1 c2 c)dc: An opportune choice of is = [0:15; 0:85]. Since these statistics are based on the Spectral Wald Statistics, which has the same asymptotic distribution as its time domain counterpart, these three optimal tests are also equivalent to their time-domain analogues. As a consequence, the critical values for the three test presented above are equivalent to their counterpart in the time domain, and they are provided by Andrews, Lee, Ploberger (1996) for the SAW, and SEW test, and by Sowell (1996) for the SN test. 4 Monte Carlo Simulations In this section, I analyze the small sample proprieties of the Spectral Covariance Instability tests by using Monte Carlo simulations. First, I compare the empirical rejection frequencies and the power of the three Spectral Covariance Instability tests, and then I examine the proprieties of an alternative approach to test a break in the variances and covariances at particular frequencies using a Generalized Method of Moments (GMM) based test. 26

27 4.1 Empirical Rejection Frequencies and Power To study the small sample properties of the Spectral Covariance Instability tests, I conduct the following experiment. First, I consider a theoretical model shown to be able to generate macroeconomic variables with similar cyclical properties as their data counterpart. For this purpose I consider as a data-generating process a factor-hoarding real business cycle model introduced by Burnside and Eichenbaum (1995). This choice is motivated by the fact that this model has a well-functioning propagation mechanism that generates a relevant amount of low-frequency uctuations. A brief description of the model and its calibration is reported in Appendix C. Then I use this model to simulate two series, output and investment. Given any interval of frequencies, the parameters of interest are the variance of the two univariate processes, H 1;1 and H 2;2 ; respectively, for output and investment, and their covariance, H 1;2, attributable at those frequencies. Consistently with the de nitions used in the empirical part of this paper, I consider two intervals of frequencies: the one that de nes the Business Cycle (6-32 quarters), and the one that de nes the Medium-Cycle (6-80 quarters). The rst goal is to study the small sample properties of the Spectral Covariance Instability tests. Therefore, I compute the empirical rejection frequencies for the three tests when the nominal signi cance is 10 percent, 5 percent, and 1 percent. The reported rejection frequencies are based on 4000 Monte Carlo repetitions. Table 3 reports the empirical rejection frequencies for the SAW, SEW, and SN tests when the sample size is T = 250. The coverage of the three tests is adequate, although the SEW tests appears to perform worse than the other two tests. As Table 4 and Table 5 show, this result holds also when the sample size increase to T = 500 and T = 1000: as expected, the performance of the tests improves with increasing the sample size. In conclusion, the small sample properties of the Spectral Covariance Instability tests are rather satisfactory, even when the sample size is close to the length of quarterly macroeconomic series, which is about 250 observations. This result is particularly strong considering that the Medium Cycle contains medium-frequency cycles, which are less precisely estimated. Nevertheless, these tables show that it is reasonable to use asymptotic results for the Spectral Covariance Instability tests for inference purposes. 27

28 The second goal of this section is to compare the power of the three tests. For this purpose I impose that the model generates a break in the variance and covariance matrix of the vector composed by output, investment, and consumption at the middle of the sample. I assume that the break causes a given percentage decline in the variance and covariance between the rst half of the sample and the second half of the sample. I consider three di erent magnitudes of the break, i.e. a 25 percent, 50 percent, and 75 percent decline in the variance and covariance. In Table 6 I display the power of the tests at the 0.05 signi cance level, using a sample size of length T = 500: Note that when the break in the variance covariance matrix is small (25 percent ) the SEW test slightly dominates the SAW test, which dominates the SN tests. However, since the break has a modest magnitude, the powers are overall low. However, as expected, when the magnitude of the break increases, the power of the test signi cantly improves: in particular, the SAW and SN tests are more powerful than the SN test when the decline of the covariance is 50 or 75 percent. Note that, as discussed in Section 2 and showed in Table 2, in the post mid-80s the decline of the variance of output at Business Cycle frequencies was about 75 percent. Therefore, when considering breaks of similar magnitude the three tests have power greater than 95 percent. In conclusion, the Monte Carlo simulations suggest that the SCI tests have good small sample size performances, especially the SAW and SN test. Moreover, the three tests have a large power when considering breaks in the variance (or covariance) of similar magnitude as experienced in the data. 4.2 SCI Test Versus GMM: A Comparison In this section I discuss an alternative approach for testing a break in the variance and covariance of a series at particular frequencies using a GMM approach. I show that the small sample proprieties of this approach are worse than for the SCI tests. The GMM approach requires the following steps: rst, the series of interest should be ltered at a particular interval of frequencies using a bandpass lter. Then, their variance, or covariance, and their standard errors are computed using a GMM estimator: note that in order to calculate the optimal weighing matrix with the Newey and West (1994) procedure, a bandwidth and a smoothing 28

29 window must be selected. Finally, the time domain equivalent of the SCI test, namely the Average LM test (ALM), the Exponential LM test, or the Nyblom test (NYB), can be directly applied to test whether these parameters have experienced a structural break at an unknown date 13. The small sample proprieties, namely the empirical rejection frequencies, of these tests are presented in Table 7, Table 8, and Table 9. respectively for T = 250; T = 500, and T = I use a Bartlett window and its corresponding optimal bandwidth. In particular Newey and West (1994) shows that asymptotically the optimal bandwidth for this window is given by: b = " 4 # 2 T As the Tables show, the GMM approach performs considerably worse than the SCI test in the small sample, since its empirical rejection frequencies are far from their nominal values for the three sample sizes considered, both at Business Cycle and Medium-Cycle frequencies, and for any of the sample sizes considered. This result should not be surprising. den Haan and Levin (1996), and Kiefer, Vogelsang, and Bunzel (2000) have discussed the unsatisfactory small sample proprieties of GMM estimators, related in particular to the choice of bandwidth. In fact, whereas the Bartlett windows have been shown to have satisfactory properties, the choice of the bandwidth is a problematic issue. In fact, only asymptotic results related to the optimal rate of convergence of a bandwidth have been proposed in the literature, whereas there are no similar guidelines for the small sample problem. Second, the choice of the bandwidth implicitly implies a trade-o between the bias of the estimator and its variance. Therefore, the choice of the bandwidth in a small sample is not a trivial concern in practice, and with my calculation I show that although the choice of the bandwidth has been conducted considering an asymptotic optimal rule, the imprecision of the test statistics is evident. Similar results are obtained if the choice of the bandwidth is guided by the Andrew s (1991) procedure. On the other hand, the Spectral Covariance Instability tests do not su er from the same problem. In fact, as stated in Priestley (1982), the Integrated Cospectrum does not require any 13 See Nyblom (1989), Andrew (1993), Andrew et al. (1996). 29

30 choice of a bandwidth. In fact, as shown in equation (14) ; the Integrated Cospectrum is estimated as the integral of the sample periodogram and the integration procedure along the frequencies works directly as a smoothing function. However, since the integration does not require the speci cation of any bandwidth parameter, the Integrated Cospectrum does not su er from any trade-o between its bias and its variance of the estimation. In conclusion, although a GMM approach can be followed to test for a break for the variances and covariances at particular frequencies, this procedure requires a not trivial choice of the bandwidth and has worse small sample properties than the frequency domain approach presented in this paper. 5 Application of the SCI Tests The empirical evidence discussed in this paper indicates that the macroeconomic variables experienced a larger decline at higher frequencies (High-Frequency component and Higher-Business Cycle component) than at lower frequencies (Lower-Business Cycle component and Medium-Frequency component). A natural question to answer is whether these drops in volatility are statistically signi cant. Using the Spectral Covariance Instability tests introduced in this paper, I formally test whether the macroeconomic variables display a break on each of their element of variance and covariance at these four intervals of frequency: the Business Cycle, the Medium Cycle, the Higher-Business Cycle and the Lower-Business Cycle components. A similar question as been largely studied in the Great Moderation literature ( for example McConnell and Perez-Quiros (2000) and Stock and Watson (2002)). However, my analysis brings two additional contributions to this topic: rst, it considers interval of frequencies not yet analyzed in this context, and second, it uses a tool, the Spectral Covariance Instability test, which I showed to have better performance in small-samples. The results I obtain shade additional lights to the temporal evolution of the macroeconomic volatility. In fact, analyzing output, consumption, investment, and their disaggregated components, I nd that whereas the Business Cycle component showed a signi cant change in their variance, the tests fail to reject the moderation when adding the Medium-Frequencies into the analysis. Also, once we split the Business Cycle into its higher and lower frequency component, 30

31 the signi cant change in their variance is present only for the former. In more details, rst I consider three important U.S. real per-capita macroeconomic variables, namely, output (measured as the gross domestic product), consumption (measured as personal consumption expenditure), and investment (measured as gross private investment). I then apply the Spectral Covariance Instability tests presented in Section 3 to these series, to test, element by element, the null hypothesis that there was no break in their variance (or covariance) at an unknown period. In the top-panel of Table 10 I consider separately their Business Cycle and Medium Cycle components, whereas in the bottom panel I consider the Higher-Business Cycle and the Lower-Business Cycle components. First consider the tests on the variance of these series, i.e. the diagonal element of the 3x3 matrices. When the Business Cycle is considered, the three tests detect a break on the variance and the covariance of output, consumption, and investment at a 5 percent level of signi cance, consistent with the theory on the Great Moderation. However, when the Medium-Frequency component is taken into account these results change dramatically. In fact, all the tests fail to reject at 5 percent the hypothesis of a stable variance (and covariance) at these frequencies. As a rst result, I obtain that consumption, investment, and output did not experienced any signi cant change of their variance when the Medium-Frequencies are included in the analysis This result is consistent with the empirical evidences showed in the previous section. Now consider the results when the business cycle frequencies are split into the Higher- and Lower Business Cycle component: we obtain a similar result. The three tests detects the presence of a breaks at the higher business cycle frequencies for all the elements of the variance covariance matrix, with pvalues lower than In contrast, when we consider the lower part of the business cycle, all tests fail to reject the signi cant break. This is a new result that this paper bring to the literature: the moderation of macroeconomic variables at business cycle frequency largely studied in the literature is actually due only by the higher frequencies of business cycle, that is by uctuations with periodicity up to 4 years (6-16 quarters), whereas it is not present when studying the uctuations with periodicity between 4 and 8 years (16-32 quarters). Next, I consider the consumption series and its disaggregated components, namely Durable, Non-Durable, and Services. Table 11displays same ndings as described above: at Business Cycle 31

32 frequencies the test detects the presence of a signi cant break, except for the variance of Non- Durable consumption. However, the break is not present studying the Medium Cycle. A similar contrast is showed by comparing the Higher-Business Cycle frequencies, where the variance of Services are is the only parameter that does not show a break, whereas the elements of the variance-covariance at Lower-Business Cycle frequencies are stable, except few exception. Finally, I consider the investment series and its disaggregated component, i.e. the Residential and Non-Residential investment. As Table 12 shows, the variance of Residential investment does not display a signi cant break at Business Cycle frequencies, unlike Investment and Non- Residential investment (at 10 percent of signi cance). However, when we study the Higher-Business Cycle frequencies, the tests detect a signi cant breaks for all the element (expect for the variance of Non-Residential investment using the SN test), but the Lower-Business Cycle variance and covariance does not experience any break at 10 percent of signi cance. In conclusion, the application of the Spectral Covariance Instability Test to macroeconomic variables provides two important results: rst, the break in the variance (and covariance) of macroeconomic series is present at Business Cycle, but now when we consider Medium Cycles. Also, only the highest frequencies of business cycle display a signi cant change in variance. 6 Conclusion The contribution of this paper is twofold. First, from an empirical point of view I investigate the behavior di erent economic cycles, which depart from the conventional de nition. In particular, I consider the role of the Medium-Frequency component, which includes uctuations between 32 and 80 quarters, and I split the Business Cycle (6-32 quarters) in two sub-components: the Higher-Business Cycle component (6-16 quarters), and the Lower-Business Cycle component (16-32 quarters). I show that the Medium-Frequency component captures the largest fraction of the uctuation of many macroeconomic variables, in particular in the last two decades. Therefore, in order to fully characterize the properties of the economic cycle, a researcher should include the Medium-Frequencies in her analysis. In addition, the empirical investigation of the cyclical components of the U.S. macroeconomic variables provides other interesting stylized facts: whereas 32

33 the volatility of the Business Cycle component has declined from the early 1980s, the so called Great Moderation, phenomenon, the volatility of the Medium Cycle (which include the Business Cycle and the Medium-Frequency component) does not have a similar pattern. Moreover, whereas the Higher-Business Cycle variance declined sharply, the Lower-Business Cycle displays an inverse U-shaped dynamics. The second contribution of this paper provides a tool to analyze whether a set of macroeconomic series displayed a signi cant break in their variance (or covariance) at a particular interval of frequencies. In particular, I de ne the Spectral Covariance Instability tests, which are useful to test whether a set of variables experienced a break in the elements of the variance-covariance matrix at any given interval of frequencies. This battery of tests is based in the frequency domain and are built starting from the Integrated Cospectrum. After deriving the asymptotic properties of the Spectral Covariance Instability tests, I investigate their small sample properties, namely the empirical rejection frequency and the power, using Monte Carlo simulations. I show that the test has a good small sample behavior, in particular if compared with a time-domain GMM-based alternative. In fact, the procedure proposed in this paper does not require the choice of any bandwidth parameter, a problematic choice in the GMM approach. Finally, I apply the Spectral Covariance Instability tests to the some important U.S. macroeconomic variables, namely consumption, investment, and output, and their disaggregated components in the postwar periods. The test formally detects a break in the variance of the three variables at Business Cyle, consistently with the Great Moderation literature, but the three test are unable to detect a similar break when considering the Medium Cycle. Moreover, when we consider the sub-components of the Business Cycle, a statistical signi cant break is detected only on the Higher-Business Cycle component and not on the Lower-Business Cycle component. This formal results show that the dynamics of the macroeconomic volatilty are qualitatively di erent when considering more speci c measure of cycles, and, as macroeconomists, we should analyze in a greater detail the evolution of the economic cycles at di erent frequencies. 33

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37 Figure 1: GDP: Level and Trend Note: The solid line plots U.S. GDP de ned in real per-capita terms from NIPA. The dashed line plots the trend identi ed by isolating the Business Cycle component (6-32 quarters). The sample includes quarterly observation from 1947:1 to 2007:4. 37

38 Figure 2A: GDP: Cyclical Components Note: The cyclical components, which are the High-Frequencies (2-6 quarters, HF, solid line), Higher-Business Cycle component (6-16 quarters, HBC, dotted line), Lower-Business Cycle component (16-32 quarters, LBC, star-line), and Medium-Frequencies (32-80 quarters, MF, dashed line) are isolated using a band-pass lter. Figure 2B: GDP: Cyclical Components Note: The cyclical components, which are the Business Cycle component (6-32 quarters, BC, solid line), and the Medium-Cycle component (6-80 quarters, MC, dashed line) are isolated using a band-pass lter. 38