A Critical Note on the Forecast Error Variance Decomposition

Transcription

1 A Critical Note on the Forecast Error Variance Decomposition Atilim Seymen This Version: March, 28 Preliminary: Do not cite without author s permisson. Abstract The paper questions the reasonability of using forecast error variance decompositions for estimating the role of different structural shocks in business cycle fluctuations. It is shown that the forecast error variance decomposition is related to a dubious definition of the business cycle. A historical variance decomposition approach is proposed to overcome the problems related to the forecast error variance decomposition. The new approach is implemented on two seminal structural VAR models with long-run restrictions by Gali (999) and King et al. (99) in order to document the amount of distortion caused by the forecast error variance decomposition. JEL classification: C32, E32 Keywords: Business Cycles, Structural Vector Autoregression Models, Forecast Error Variance Decomposition, Historical Variance Decomposition I am greately indebted to Bernd Lucke, Ulrich Fritsche and Olaf Posch for their comments and suggestions on earlier drafts of the paper. The work also benefited from participation in the following: a seminar at the University of Hamburg, 2nd Workshop on Macroeconomics and Business Cycles (Makroökonomik und Konjunktur) at ifo Institute for Economic Research in Dresden, the EC2 Conference 27 on Advances in Time Series Econometrics in Faro. Finally, I received a great deal of support while writing large parts of the paper during my tenure as a research assistant at the Institute for Growth and Business Cycles of the University of Hamburg. Any errors are my own. Center for European Economic Research (ZEW), P.O. Box 3443, D-6834 Mannheim, Germany. seymen@zew.de

2 . Introduction Forecast error variance decomposition (FEVD) is an econometric tool used by many researchers in the vector autoregression (VAR) context for assessing the driving forces of business cycles. This paper shows that the connection between the FEVD and the cycle 2 is not well established. Instead of the FEVD, we advocate using the historical variance decomposition (HVD), which, as we show, is compatible with different business cycle definitions. The HVD is implemented on two seminal structural VAR (SVAR) models à la Gali (999) and King et al. (99; henceforth KPSW) based on two different business cycle definitions. The results point to a large amount of distortion in the conclusions when the FEVD is employed for the analysis. There are different ways the business cycle is defined (measured) in the literature. One of the main arguments of this paper is that the FEVD is related to a dubious one of these definitions, which is that business cycles occur around a constant or a linear trend. This definition is particularly problematic if the estimated VAR process comprises nonstationary variables, since the FEVD would imply then a nonstationary cyclical component. There is a consensus among macroeconomists that the cycle is stationary, while many macroeconomic time series are nonstationary. 3 It is not one of the aims of this paper to discuss the most appropriate business cycle definition to be used by macroeconomists, but to emphasize merely that the FEVD is, together with other problems related to it, not consistent with widely used definitions of the business cycle, and to suggest a solution framework for this problem. 4 In order to present our point of view, we employ the filter proposed by Hodrick and Prescott (997; henceforth HP-filter) and the Beveridge-Nelson-Decomposition (BND) proposed by Beveridge and Nelson (98), which are two widely implemented approaches for measuring the business cycle. 5 In contrast to the FEVD, both of these approaches, like many others, imply that i) cyclical fluctuations occur around a nonlinear long-run trend, and ii) shocks occuring in the so-called business cycle horizon do not only contribute to the 2 We henceforth use the terms business cycle and cycle interchangeably. 3 See Baxter and King (995) on widely accepted time-series properties of the business cycle. 4 It must be clear that the business cycle definition chosen by an econometrician may have important implications on his results with respect to the role of different structural shocks in the cyclical fluctuations. 5 See Baxter and King (995) for some other very popular approches. 2

3 business cycle, but to the long-run trend of the variables as well. 6 The HVD is, on the other hand, compatible with these properties of the cycle. Moreover, it is based on the idea that the cycles of macroeconomic variables can be decomposed with respect to (w.r.t.) structural shocks like with the FEVD. Yet the weights of different macroeconomic shocks on the variance of the cyclical components are computed with the HVD subject to a chosen business cycle definition. The incompatibility of the FEVD with convenient business cycle definitions is the first objection in this study to the implementation of it in the business cycle context. In order to investigate the connection between forecast errors and business cycles, we confront the historical forecast errors of output for different forecast horizons based on the models by Gali (999) and KPSW with the cyclical component of it computed with both the HP-filter and the BND. It is obtained that the historical forecast errors at the so-called business cycle horizon have hardly much to do with the business cycles. Another difficulty with the FEVD, even if it is assumed that structural shocks lead only to business cycle fluctuations, is that the business cycle is defined to be a macroeconomic phenomenon that occurs in a time span of 6 to 32 quarters. This definition makes it in many cases impossible to estimate, using the FEVD, which macroeconomic shocks are the main driving force of the busines cycle fluctuations over the entire business cycle horizon. Finally, we find that the historical forecast errors become nonstationary according to the ADF test when the forecast horizon exceeds quarters in both Gali and KPSW models. This finding implies that a FEVD analysis based on these models with long-run restrictions leads to spurious conclusions for a forecast horizon longer than ten quarters. We use the original data sets of the corresponding papers in our study. One striking result is that the HVD based on cycles computed using the BND implies, contrary to Gali s reported results, that technology shocks are the driving force of output cyclical fluctuations. Another interesting result is that the historical decomposition with both the HP-filter and the BND attributes only a small role to technology shocks in output cycles in the six-variable KPSW model, although the FEVD attributes them an important role especially for a forecast horizon longer than twelve quarters. 6 However, the FEVD is based on the implicit assumption that forecast errors occur within the business cycle horizon due to shocks that lead only to business cycle fluctuations. 3

4 The next section presents the FEVD technique and the arguments against it. Section 3 illustrates the HVD approach. Section 4 gives examples of the implementation of the HVD. Concluding remarks are given in Section The Forecast Error Variance Decomposition Consider the moving average (MA) representation of a stationary VAR(p) process with p being the order of the VAR, X t = CD t + Θ i w t i, () i= where X t is a (K ) vector of endogenous variables, Θ i is the i th (K K) MAcoefficient matrix, w t is a (K ) vector of orthogonal white noise innovations all with a unit variance, C is an (K M) coefficient matrix corresponding to the deterministic terms represented by the (M ) matrix D t. 7 Onecanwritetheh-step forecast error for the process as h X t+h X t (h) = Θ i w t+h i, (2) i= with X t (h) being the optimal h-step forecast at period t for X t+h. It is straightforward to compute the total forecast error variance of a variable in X t for the h-step forecast horizon and the corresponding shares of individual innovations to this variance, see Lütkepohl (25). What is traditionally done in the literature is to set h such that the computation is made for the business cycle horizon. 8 FEVD and Definition of the Business Cycle Users of the FEVD based on equation (2) make the implicit assumption that shocks occuring within the business cycle horizon contribute only to the business cycle and lead to forecast errors; therefore, forecast errors and business cycles are directly connected to each other. 9 This assumption is, however, not 7 M is here the number of deterministic variables. M = if there is only a constant, M =2ifthereisa constant and a linear trend, etc. 8 This means setting 6 h 32 if you work with quarterly data following the business cycle definition used by many macroeconomists. 9 A related consequence of this implicit assumption is the implication of the FEVD that business cycles occur approximately around a deterministic path. This deterministic path is typically assumed to be a constant or a linear time trend depending on the model specification. Moreover, VAR models often contain dummy variables. 4

5 compatible with the relationship between VAR models and widely accepted business cycle definitions. The bivariate model of Gali (999) is a good example to illuminate our point of view. The VAR comprises the labor productivity (x t ) and the total hours worked (n t ). There are two types of identified macroeconomic shocks called technology and nontechnology. Since this VAR includes nonstationary variables, it is estimated in first differences, and equation () should be written as X t = µ + Θ i w t i i= where X t =(x t,n t,y t ), µ is the constant vector, and is the difference operator, such that X t = X t X t. Rewriting equation (3) leads to X t = X + µt +Θ w t +Θ w t + +Θ t w (4) with X being the vector containing the initial values of the model variables and Θ i = i j= Θ j. Note that the h-step forecast error following from the representation in equation (3) is different than the representation in equation (2) due to the nonstationarity of the processes in X t and reads h i= Θ i w t+h i. It is clear from equation (4) that the user of the FEVD in such a framework 2 implicitly takes the business cycle to be a phenomenon that occurs approximately around a linear trend, since the difference between Θ i and Θ j is very small if i and j are big enough. 3 Macroeconomic time series are, however, usually assumed to have neither deterministic nor linear long-run trends. For example, the trend component of X t is a random walk with drift according to the BND, which reads T t = X + µt +Θ() t w i (5) i= Hence, output (y t ) enters the estimation indirectly, as it is just a linear combination of the labor productivity and the total hours worked; i.e., y t = x t + n t. Notice that Gali estimates a bivariate model with X t =(x t,n t ). It is trivial to add y t to X t after the structural estimation is carried out, because y t is only a linear combination of x t and n t. The Θ i matrices are accordingly of order (3 2), of which third rows are the sum of the first two. 2 That is, in a framework with nonstationary variables, where the nonstationarity is due to at least one unit root. 3 This follows from the stability of the system estimated in equation (3). (3) 5

6 with T t being the trend component and Θ () = Θ +Θ +... the matrix of long-run multipliers for the model in equation (3). 4 The HP-filter implies a non-linear and non-deterministic trend, too. Recall that equation (4) provides an exact representation of the variables in X t, according to which X t has three components: a constant (X ), a linear time trend (µt) and a stochastic component ( Θ w t +Θ w t + +Θ t w ). Since the HP-filter is a linear filter, it is applied to all of these three components for computing the cyclical component of X t. It removes the constant and the linear time trend entirely from X t and the stochastic component partly. Therefore, structural innovations contribute clearly not only to the cycles of the variables in X t, but to their long-run trends as well according to the HP-filter. Figure shows the trend and cyclical components of the output data used by Gali (999) computed with a linear trend, the HP-filter and the BND. 5 Estimated trend and cyclical components of a variable are obviously very sensitive to how they are measured, as the reported features of the cyclical components in Table shows. Those differ quite a bit w.r.t. their volatility, amplitude, persistence and comovement properties. 3.5 Panel A: Trend component.2 Panel B: Cyclical component Linear Trend HP filter BND Figure : Cyclical and trend components of output according to linear-trend, HP-filter and Beveridge-Nelson-decomposition in Gali model 4 Beveridge and Nelson (98) consider a univariate model, but the implementation in the multivariate case is straightforward. 5 Note that the cycles around the linear trend have a non-zero mean, while the HP-cycles and the BND cycles do have a zero mean. We have normalized the linear trend and the cycles around it accordingly by subtracting the mean from the cyclical component and adding it to the trend component. 6

7 Table : Characteristics of cyclical components according to Gali model Relative volatility Amplitude Persistence Linear-Trend-Cycles HP-Cycles BN-Cycles Linear-Trend-Cycles HP-Cycles HP-Cycles.42 (.4) BN-Cycles.7.29 (.5) (.9) Notes: Relative volatility is the standard deviation of one series divided by the standard deviation of the cycles around the linear trend. Amplitude stands for the difference between the maximum and minimum values of the cycles shown in Figure. The persistence measure is the estimated coefficient of the first lag of the corresponding cycle in an AR() model. Standard error of a correlation coefficient is given in parenthesis. To summarize, both the HP-filter and the BND imply a nondeterministic long-run trend and neither is based on the assumption that shocks occuring in any forecast horizon contribute only to business cycles, in contrast to the implications of the FEVD. Since the FEVD is inconsistent with widely accepted business cycle definitions, it should not be used for analyzing the driving forces of business cycles. The Ambiguity about the Connection between Forecast Errors and Business Cycles Another issue when using the FEVD in the business cycle context is the ambiguity about why setting h to a value within the business cycle horizon should render a business cycle analysis. The estimated cross-correlation coefficients between the historical i-horizon forecast errors, computed based on both Gali (999) and KPSW models, and the cyclical component of output are displayed in Figure 2. A closer relationship between the historical forecast errors at business cycle frequencies and the business cycle fluctuations of output cannot be established; that is, the correlations are not particularly higher at the so-called business cycle frequencies (i.e., at a forecast horizon of six to thirty-two quarters) than at lower frequencies. 7

8 Panel A: Gali model Panel B: KPSW model HP filter BND Figure 2: Cross-correlation between i-horizon historical forecast errors and cyclical component of output according to Gali and KPSW models The Share of Structural Shocks in Fluctuations over the Entire Business Cycle Even if the hitherho mentioned issues do not present a problem when using the FEVD in the business cycle context, the technique does not give an answer to the question of which structural shocks drive the business cycle fluctuations over the entire business cycle horizon. The result obtained by KPSW, who identify balanced-growth, inflation and real-interest-rate shocks, provides a good example. It is reported in their paper that the fraction of the forecast error variance of output attributable to the real-interest-rate shock is 74 percent in 4 quarters, 55 percent in 8 quarters and 25 percent in 24 quarters. Moreover, the contribution of the technology shock to output fluctuations is forecasted to be 5 percent in 4 quarters, 22 percent in 8 quarters and 62 percent in 24 quarters. Given that a typical definition of the business cycle horizon is from 6 to 32 quarters, it is not possible to have an idea about the weight of real-interest-rate and technology shocks in the fluctuations of output over the entire business cycle horizon. Is, for instance, the share of real-interest-rate shocks 55 percent, 22 percent, or something in between? Similarly, the question of whether technology shocks have a bigger weight than real-interest-rate shocks over the whole business cycle horizon can also not be answered. 8

9 Nonstationarity and FEVD Using FEVD in any context becomes even more problematic when the SVAR model comprises nonstationary variables as in the model represented by equations (3) and (4). In this case, the stochastic component of the variables, and therefore the forecast errors, become nonstationary when the forecast horizon approaches infinity and the variance decomposition leads to only spurious conclusions after a certain forecast horizon. We check what the critical forecast horizon is, i.e. after which forecast horizon the historical output forecast errors based on Gali and KPSW models exhibit nonstationarity. Although the two models have very different structures, the historical forecast error series of output become nonstationary after a forecast horizon of ten according to the Augmented- Dickey-Fuller test in both models. In suh a case, it does not make much sense to carry out a FEVD analysis of output in these models, with the given data set, for a forecast horizon of h> as the reported error variance shares would then become spurious. Historical FEVD The FEVD is typically computed based on the crucial assumption that the structural shocks w t in equation (2) are uncorrelated contemporaneously and across time among each other. Moreover, they are assumed not to exhibit an autocorrelation. When these assumptions hold, the forecast error variances of the variables in X t are the diagonal elements of the matrix E ( (X t+h X t (h)) (X t+h X t (h)) ) h h = Θ i Σ w Θ i = Θ i Θ i, (6) i= where Σ w is the variance-covariance matrix of the structural shocks, which is a K-order identity matrix by construction. Moreover, due to these assumptions, the contribution of the k th structural shock to the forecast error variance of the j th variable for a given forecast horizon is computed as i= h ( ) e j Θ 2 i e k i= (7) where e k is the k th column of the identity matrix of order (K K). It is straightforward to compute the share of a structural shock in the fluctuations of a variable in the VAR, yet this procedure has the important drawback that only the contamporaneous orthonormality of the structural shocks is imposed on the model but the orthogonality of them across time. In 9

10 order to check the dimension of the measurement problem related to this issue, we confront the within- sample FEVD with the conventional one. The within-sample FEVD is based on the historical h-horizon forecast errors that can be computed with h ˆX t k (h) = ˆΘ k,iŵk,t i, t= h +,h+2,..., T (8) i= where T is the number of observations in the sample excluding the initial observations necessary for the estimation, and ˆX k t (h), ˆΘ k,i and ŵ k,t are the historical h-horizon historical forecast errors of the variables in X t due to the k th structural shock, the k th column of the i th estimated structural MA coefficient matrix and the estimated realization of the k th structural shock at period t. The total historical h-horizon historical forecast error is accordingly given by ˆX t (h) = h i= ˆΘ i ŵt i. The estimate of the share of the k th structural shock in the within-sample forecast error variance of the j th variable follows from dividing the variance of the corresponding time series generated by (8) by the variance of the total forecast error component of the relevant variable. Figure 3 shows the shares of structural shocks in output fluctuations according to the FEVD and the historical FEVD (HFEVD) computed based on Gali and KPSW models. The divergence between the FEVD and the HFEVD for a forecast horizon longer than periods in the Gali model is striking and has two basic reasons. First, there is a lot of uncertainty related to the estimation of the ˆΘ i matrices 6 and this problem is further exacerbated in the estimation of the coefficients that relate to output, since output is not directly included in the Gali model. Second, the estimated technology shocks deviate very much from their mean value around the th quarter after the beginning of the sample. Since the ˆΘ i matrices reflect accumulated impulse responses, the effect of this large deviation is accumulated in the HFEVD estimations with a forecast horizon longer than periods. The second issue applies to the KPSW model as well, for which a big discrepancy between the FEVD and the HFEVD estimations w.r.t. the balanced-growth and the real-interest-rate shocks occurs 6 See Faust and Leeper (997) and Christiano et al. (25) on this issue. (9)

11 especially for a forecast horizon longer than 8 quarters. Given that the sample of KPSW starts at the first quarter of 954 excluding the initial observations, the discrepancy between the estimations is obviously due to the effects of the first oil price shock. 7 FEVD HFEVD Technology Share Gali Model 5 Nontechnology Share Gali Model 5 Balanced Growth Share KPSW Model 5 Inflation Share KPSW Model 5 Real Interest Rate Share KPSW Model 5 Figure 3: FEVD vs. historical FEVD of output at different forecast horizons according to Gali and KPSW models The exercise reflected in Figure 3 is to be evaluated with some caution, since the variance decomposition analysis is carried out with nonstationarity processes for a forecast horizon longer than quarters as noted above. Comparing the FEVD and the HFEVD of output in first differences, which is based on equation (3), provides an analysis based on stationary processes. Moreover, it sheds light on the accumulation-of-the-forecast-errors problem mentioned in the previous paragraph. Figure 4 makes clear that the discrepancy between the FEVD and the HFEVD estimates in Figure 3 is in 4 of the 5 cases largely due to the error accumulation problem. Yet the FEVD share of the real-interest-rate shock deviates to a large extent from its HFEVD counterpart even for the output growth rate. Therefore, it can be concluded that using the FEVD even with stationary processes is a questionable practise. 7 This observation may be a sign for that there was a structural break in the economy, which has not been accounted for by the empirical model.

12 FEVD HFEVD Technology Share Gali Model 5 Nontechnology Share Gali Model 5 Balanced Growth Share KPSW Model 5 Inflation Share KPSW Model 5 Real Interest Rate Share KPSW Model 5 Figure 4: FEVD vs. historical FEVD of first-differenced output at different forecast horizons according to Gali and KPSW models 3. Historical Variance Decomposition The representation of the process in (4) implies that the variables in X t can be written as a linear combination of the structural shocks that are identified from the beginning of the sample until period t plus the initial value and a deterministic trend. The stochastic component represented by equation (6) can be decomposed with respect to the structural shocks, namely t Θ i w t i = i= t i= N Θ i,j w t i,j () j= where Θ i,j is the j th column of Θ i and w i,j is the j th value of w i. N is the number of structural shocks in the system and K N. 8 Let X k be the T vector containing the T observations in the sample of the k th variable in the VAR. The historical decomposition of the stochastic part of the k th variable with respect to the structural shocks can be written 8 In many examples, K = N. One example for where this is not the case is the bivariate model of Gali (999) with three variables and two structural shocks. 2

13 as X k = X k w + Xk w 2 + Xk w N, () where X k w is the T vector, of which elements are the k th entries in t j i= Θ i,jw t i,j. Equation () stands for a historical decomposition of fluctuations around the linear trend. Note that the variance of X i is given by the statistical identity, var ( X i) = cov ( X i,x i w ) + cov ( X i,x i w 2 ) + + cov ( X i,x i w N ), (2) where var (a) stands for the variance of variable a and cov (a, b) stands for the covariance between variables a and b. AslongasacomponentX k w j does not exhibit a much smaller variance than X i, the covariance terms in equation (2) are all positive. Hence, cov ( X i,x i w j ) /var (X i ) denotes an estimation of the share of the j th structural shock in the fluctuations of variable i around a linear trend. This type of a variance decomposition has the advantage of being compatible with many different business cycle measures. For example, an HP-filter based variance decomposition can be computed based on X i,hp = X i,hp w + X i,hp w + X i,hp, (3) 2 w N where X i,hp is the HP cyclical component of X i and X i,hp is the HP cyclical component of w j X i w. This is possible since the HP-cycle of a variable which is the sum of multiple components j is equal to the sum of the HP-cycles of the components. It is then straightforward to write var ( X hp,i) ( ) ( ) ( ) = cov X hp,i,x hp,i w + cov X hp,i,x hp,i w + + cov X hp,i,x hp,i, (4) 2 w N ( ) and cov X hp,i,x hp,i /var ( X hp,i) is estimate of the share of the j th structural shock in w j cyclical fluctuations of the ith variable according to the HP-filter. The BND is also compatible with a HVD. The historical cyclical component is written as t i= Ψ i+w t i with Ψ j = i=j Θ i+ and w i =fori>according to the BND. t i= Ψ i+w t i is decomposed then like in () and (3), and the HVD follows in the same way. The HVD provides an econometrician two advantages over the FEVD. First, it is compatible with many business cycle measures. Second, it solves all the problems related to the FEVD. 3

14 4. Empirical Applications In this section, the HVD approach is implemented to the bivariate model of Gali (999) and the six-variable model of KPSW, where the cycles are computed based on three different measures: the linear trend, the HP-filter, and BND. Note that the cycles around the linear trend are nonstationary according to both Gali and KPSW models. Although a variance decomposition analysis based on nonstationary processes does not make sense, the results of this section gives an idea about the amount of distortion that the FEVD causes, since the FEVD and the cycles around a linear trend are closely related as argued in Section 2. Gali (999) investigates in his paper whether technology or nontechnology shocks drive the business cycles. He first establishes the high correlation between the cyclical components of output and hours worked according to the HP-filter, and then shows that the nontechnology shocks lead to a strong positive comovement between the two variables but the technology shocks. The HVD with the HP-filter implies the same conclusion and attributes a share of.89 to nontechnology shocks, as reported in Table 2. But technology shocks are the driving force of output fluctuations with a share of.74 according to the HVD with the BND. Note that the conclusion Gali (999) arrives at in his study has a lot to do with the business cycle definition used by him. When the analysis of Gali is based on the BND, the cyclical components of output and hours are still very highly correlated with a coefficient of.87. Yet this strong positive comovement does not follow only from a strong comovement of nontechnology components in this case, but the technology components are highly correlated as well. The former comovement corresponds to a correlation coefficient of.96 and the latter to a coefficient of.95. The reason behind the larger share of technology shocks in the BNcycles is that the technology BN-component of output has a variance, which is 2.8 times larger than the variance of its nontechnology component. KPSW estimate only the structural shocks with long-run effects and their identification scheme is different than Gali s in the sense that their six-variable model embodies cointegrating relationships. Table 2 reports that the balanced-growth shocks have a share of merely ten percent in the output cycle and the real-interest-rate shocks are the driving force of the cyclical fluctuations, when the HVD is computed with the HP-cycles. On the other hand, 4

15 Table 2: Shares of structural shocks in fluctuations of output according to historical variance decomposition Technology Nontechnology Cycles around linear trend.79.2 Cycles around HP-trend.4.86 Beveridge-Nelson-cycles PanelA:Galimodel Balanced-growth Inflation Real-interest-rate Temporary Cycles around linear trend Cycles around HP-trend Beveridge-Nelson-cycles Panel B: KPSW model the fluctuations around the linear trend are basically driven by balanced-growth shocks with a share of sixty-five percent. Note that the typical application of the FEVD in KPSW attributes to both technology and real-interest-rate shocks some importance as the driving force of the business cycles. The HVD with the BND renders similar results to the HVD with the HP-filter. Finally, the HVD with the HP-filter and the BND both attribute only a limited share to shocks with temporary effects in the cyclical fluctuations of output in the KPSW model. A 29 percent share belongs to shocks with only short-run effects when the HP-filter is used and 28 percent when the BND is used. Thus, shocks with long-run effects have a larger weight than the shocks with short-run effects in the cyclical fluctuations. 5. Summary An important research topic of modern macroeconomics is the driving forces of cyclical fluctuations. The most important challenge is that the cycle and the shocks, which are the driving forces of the cycles, are not directly observable. When the driving forces of the business cycle is investigated, three core questions need to be answered: i) How should the structural shocks and their dynamic effects be identified? ii) How should the business cycle be defined? iii) How should the contribution of structural shocks to the cycle be computed? The focus of this paper has been the third question criticising the FEVD technique as a 5

16 tool of macroeconomic analysis. The core criticism has been that the FEVD is based on a primitive business cycle definition that the cycles around a linear trend represent the business cycle. It has been shown that the FEVD leads to important distortions when analysing the role of different shocks in the cycles and that it produces spurious results when applied in models with nonstationary variables.the HVD has been proposed to overcome the problems related to the FEVD. This approach has the advantage of being compatible with different business cycle definitions. The HVD, when applied with a typical cycle measure like the HP-filter or the BND, implies by definition that the cycle is a stationary process. This is important in particular when an SVAR comprising nonstationary variables underlies the analysis. Furthermore, structural shocks are allowed to contribute not only to the cycle but to the long-run component as well, as typical business cycle measures imply. The HVD has been implemented with both the HP-filter and the BND for illustrating the amount of distortion that the FEVD techniqe causes. Note that the models used in the implementations comprise different variables and use different identification approaches. The models have been taken as they are and have been estimated with their original data sets. The results with the HVD have been confronted then with the orinigal results of Gali (999) and KPSW. In other words, the first question above has also been out of scope of this paper. References Baxter, M. and R. G. King, 995, Measuring Business Cycles: Approximate Band-Pass Filters for Economic Time Series, NBER Working Paper, No. 522 Beveridge, S. and C. R. Nelson, 98, A New Approach to Decomposition of Economic Time Series into Permanent and Transitory Components with Particular Attention to Measurement of the Business Cycle, Journal of Monetary Economics, Vol. 7, 5-74 Christiano, L.J., Eichenbaum, M. and R. Vigfusson, 26, Assessing Structural VARs, NBER Working Paper, No

17 Faust, J. and E. Leeper, 997, When Do Long-Run Identifying Restrictions Give Reliable Results, Journal of Business and Economic Statistics, 5, Gali, J., 999, Technology, Employment and the Business Cycle: Do Technology Shocks Explain Aggregate Fluctuations?, The American Economic Review, Vol. 89, No., Hodrick, R.J. and E.C. Prescott, 997, Postwar U.S. Business Cycles: An Empirical Investigation, Journal of Money, Credit, and Banking, Vol. 29, No., -6 King, R., Plosser, C., Stock, J. and M. Watson, 99, Stochastic Trends and Economic Fluctuations, The American Economic Review, Vol. 8, No. 4, Lütkepohl, H., 25, New Introduction to Multiple Time Series Analysis, Springer Verlag, Berlin 7