- 1
- Baxter, M. and R. G. King. 1995. “Measuring Business Cycles: Approximate Band-Pass Filters for Economic Time Series.” Working Paper No. 5022.National Bureau of

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- Prefix
- As a result, various methods have been proposed to uncover the permanent and transitory components of output. One of these methods consists of using mechanical filters such as the Hodrick-Prescott (HP) filter or the band-pass filter (BK) proposed by
- Exact
- Baxter and King (1995).
- Suffix
- However, mechanical filters have been criticized. For example, Harvey and Jaeger (1993) and Cogley and Nason (1995) show that spurious cyclicality can be induced by the HP filter when it is used with integrated or nearly integrated data.

- In-text reference with the coordinate start=7683
- Prefix
- Guay and St-Amant (1996) reach the more general conclusion that the HP and BK filters perform poorly in identifying the cyclical component of time series that have a spectrum or pseudo-spectrum with Granger’s typical shape, i.e., the shape characteristic of most macroeconomic time series.
- Exact
- Baxter and King (1995) and
- Suffix
- others note that two-sided filters such as the HP and BK filters become ill-defined at the beginning and the end of samples. For this reason, they recommend discarding three years of quarterly data at both ends of the sample when using the HP filter.

- In-text reference with the coordinate start=33230
- Prefix
- ones discussed in this paper is that, unlike mechanical filters, they reflect at least some of the uncertainty. 3.2 Spectra analysis Chart 6 shows the estimated spectra of the CO, MBN (trivariate case) and LRRO (trivariate case) output gaps plus those resulting from the application of two mechanical filters: the Hodrick-Prescott filter (HP) and the band-pass filter (BK) proposed by
- Exact
- Baxter and King (1995).
- Suffix
- Loosely speaking, the spectrum of a series is that series expressed as the integral of random periodic components that are mutually orthogonal. The total area below the spectrum corresponds to the variance of the series.

- In-text reference with the coordinate start=36375
- Prefix
- It is difficult to compare the spectra of MBN output gaps with the others. 4 Conclusions In this paper, we compared different techniques that are used to measure potential output. We started with a brief explanation of why we think that mechanical filters such as the Hodrick-Prescott filter and the band-pass filter proposed by
- Exact
- Baxter and King (1995)
- Suffix
- perform poorly in accomplishing this task. We then compared the LRRO approach based on long-run restrictions with two alternative multivariate approaches: the one proposed by Cochrane (1994) and the multivariate Beveridge-Nelson methodology.

- In-text reference with the coordinate start=7098
- 6
- Blanchard, O. J. and D. Quah. 1989. “The Dynamic Effect of Aggregate Demand and Supply Disturbances.”American Economic Review 79: 655-73.

Total in-text references: 4- In-text reference with the coordinate start=10512
- Prefix
- In this paper, we compare some of the techniques briefly introduced above with the structural vector autoregression methodology based on long-run restrictions imposed on output (LRRO) proposed by
- Exact
- Blanchard and Quah (1989), Shapiro and Watson (1988), and King et al. (1991)
- Suffix
- in theory (Section 2) and in applications (Section 3). In Section 2, we note that one characteristic of the LRRO method is that it does not impose restrictions on the short-run dynamics of the permanent component of output.

- In-text reference with the coordinate start=14546
- Prefix
- From (2) and (3) we have: (4) CL() ΓL()Γ0 –1 = C1()ΩC1()′ C1()ΩC1()′ Γ1()Γ1()′= Γ0 C1() C1() This relation suggests that we can identify matrix with an appropriate number of restrictions on the long-run covariance matrix of the structural form.
- Exact
- Blanchard and Quah (1989) and Shapiro and Watson (1988)
- Suffix
- use long-run restrictions to identify shocks with having full rank. King et al. (1991) work in a context where the rank of is less than n1 and they use cointegration restrictions.

- In-text reference with the coordinate start=15414
- Prefix
- Potential output based on the LRRO method is then: (6) ∆yt p μyΓ1 p ()ηLt p =+ Thus, “potential output” corresponds to the permanent component of output. The part of output due to transitory shocks is defined as the “output gap.” It is important to note that we do not talk in terms of “demand” or “supply” shocks as in
- Exact
- Blanchard and Quah (1989),
- Suffix
- but simply in terms of permanent and transitory shocks. 2.2 Comparison with other multivariate methods In this section, we examine the features of two alternatives to the LRRO approach: the multivariate Beveridge-Nelson decomposition (MBN) and Cochrane’s output-consumption decomposition (CO).4 The MBN decomposition defines potential output as the level of out

- In-text reference with the coordinate start=26960
- Prefix
- 2, models in which the permanent component of output is a random walk imply that the economy is below (above) potential in the transition period following a permanent positive (negative) shock to output. To the extent that the transition primarily reflects factors associated with an adjustment in the supply side of the economy, assuming that potential 7.
- Exact
- Blanchard and Quah (1989) and Gali (1992),
- Suffix
- among others, report similar results. output follows a random walk can be misleading. It could, in particular, provide misleading signals about the extent of inflationary pressures in the economy. Chart 3 shows the output gaps calculated on the basis of the LRRO methodology in the bivariate and trivariate cases.

- In-text reference with the coordinate start=10512
- 9
- Boschen, J. and L. Mills. 1990. “Monetary Policy with a New View of Potential GNP.”Federal

Total in-text references: 1- In-text reference with the coordinate start=8204
- Prefix
- Van Norden (1995) stresses the fact that this is a very significant limitation for policymakers interested in estimating 1. For a discussion of how the estimation of potential output can affect the formulation of monetary policy, see
- Exact
- Boschen and Mills (1990)
- Suffix
- or Laxton and Tetlow (1992). the current level of the output gap. Another strategy for identifying the permanent and transitory components of output involves the use of univariate techniques such as the unobserved components approach suggested by Watson (1986) and the BeveridgeNelson (1981) method.

- In-text reference with the coordinate start=8204
- 11
- Burns, A. M. and W. C. Mitchell. 1946.Measuring Business Cycles.

Total in-text references: 3- In-text reference with the coordinate start=12016
- Prefix
- This is consistent with recent results reported by Staiger, Stock and Watson (1996) for the estimation of the NAIRU. Another interesting result is that, of the methods we consider, only the LRRO-based one generates an output gap with a peak at business cycle frequencies as defined by
- Exact
- Burns and Mitchell (1946),
- Suffix
- i.e., cycles lasting between 6 and 32 quarters. 2 Methodologies used for estimating the trend in output 2.1 The approach based on the LRRO methodology In this section, we briefly present the LRRO decomposition methodology involving long-run identifying restrictions (LRRO) and explain how it can be used to estimate potential output.2 Let be a n x 1 stationary vector includ

- In-text reference with the coordinate start=33972
- Prefix
- We use a parametric estimator of the spectra for which ARMA processes were fitted.9 The spectra resulting from the LRRO, HP, and BK output gaps have their peak at business-cycle frequencies as defined by
- Exact
- Burns and Mitchell (1946),
- Suffix
- i.e., frequencies corresponding to cycles lasting between 6 and 32 quarters. Indeed, the peaks in the spectrum of the HP, BK and LRRO gaps correspond to cycles lasting around 20 quarters.

- In-text reference with the coordinate start=37502
- Prefix
- with permanent shocks and that the output gap series estimated on the basis of LRRO approach, Cochrane’s approach, and the multivariate Beveridge-Nelson methodology are different in the time and frequency domains. We note, in particular, that only the output gap associated with the LRRO approach has a peak at business-cycle frequencies as defined by
- Exact
- Burns and Mitchell (1946),
- Suffix
- i.e., cycles lasting between 6 and 32 quarters. However, the estimates are imprecise for specific points in time and it appears difficult to distinguish between these methodologies in that context.

- In-text reference with the coordinate start=12016
- 12
- Cochrane, J. H. 1994. “Permanent and Transitory Components of GNP and Stock Prices.”

Total in-text references: 3- In-text reference with the coordinate start=16260
- Prefix
- With reference to equation (2), where output is the first element of, we write the following decomposition: (7) Zt ∆ytμyC11()εtC1∗L()εt++= Potential output is defined by the first two terms on the right-hand side of (7): (8) ∆yt p =μyC11()εt+ 4. See Cogley (1995) for another comparison of the MBN and CO methodologies. Potential output is thus simply a random walk with drift.
- Exact
- Cochrane (1994)
- Suffix
- uses a two-variable VAR including GNP and consumption to identify the permanent and transitory components of GNP. The bivariate representation is augmented with lags of the ratio consumption to GNP.

- In-text reference with the coordinate start=19093
- Prefix
- Indeed, the validity of the permanent-income hypothesis would imply that the last two terms of the consumption equation are equal to zero and that. It is not clear to what the CO decomposition corresponds if consumption is not a random walk.5
- Exact
- Cochrane (1994)
- Suffix
- notes that the measure of potential output obtained on the basis of the CO method would be equivalent to the one obtained from the LRRO approach if the transitory effect of permanent shocks to GNP and consumption were exactly the same, i.e., if and.

- In-text reference with the coordinate start=36598
- Prefix
- We started with a brief explanation of why we think that mechanical filters such as the Hodrick-Prescott filter and the band-pass filter proposed by Baxter and King (1995) perform poorly in accomplishing this task. We then compared the LRRO approach based on long-run restrictions with two alternative multivariate approaches: the one proposed by
- Exact
- Cochrane (1994) and
- Suffix
- the multivariate Beveridge-Nelson methodology. We argued that one advantage of the approach based on long-run restrictions is that it allows for estimated transitional dynamics following permanent shocks.

- In-text reference with the coordinate start=16260
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- Cogley, T. 1995. “Evaluating Non-Structural Measures of the Business Cycle.” Draft Paper.

Total in-text references: 1- In-text reference with the coordinate start=16144
- Prefix
- With reference to equation (2), where output is the first element of, we write the following decomposition: (7) Zt ∆ytμyC11()εtC1∗L()εt++= Potential output is defined by the first two terms on the right-hand side of (7): (8) ∆yt p =μyC11()εt+ 4. See
- Exact
- Cogley (1995)
- Suffix
- for another comparison of the MBN and CO methodologies. Potential output is thus simply a random walk with drift. Cochrane (1994) uses a two-variable VAR including GNP and consumption to identify the permanent and transitory components of GNP.

- In-text reference with the coordinate start=16144
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- Cogley, T. and J. Nason. 1995. “Effects of the Hodrick-Prescott Filter on Trend and Difference

Total in-text references: 2- In-text reference with the coordinate start=7193
- Prefix
- One of these methods consists of using mechanical filters such as the Hodrick-Prescott (HP) filter or the band-pass filter (BK) proposed by Baxter and King (1995). However, mechanical filters have been criticized. For example,
- Exact
- Harvey and Jaeger (1993) and Cogley and Nason (1995)
- Suffix
- show that spurious cyclicality can be induced by the HP filter when it is used with integrated or nearly integrated data. Guay and St-Amant (1996) reach the more general conclusion that the HP and BK filters perform poorly in identifying the cyclical component of time series that have a spectrum or pseudo-spectrum with Granger’s typical shape, i.e., the shape characteristic o

- In-text reference with the coordinate start=35330
- Prefix
- The latter result is not surprising, since it is well known that the MBN methodology gives a transitory component whose importance increases with the number of series used to identify it. CHART 6: Spectra of the output gaps 0.08 0.07 HP filter 0.06 0.05 BK filter CO 0.04 LRRO (3 variables) 0.03 MBN (3 variables) 0.02 0.01 0 00.050.10.150.20.250.30.35 fraction of pi
- Exact
- Cogley and Nason (1995) and Harvey and Jaeger (1993)
- Suffix
- show that the HP filter amplifies business-cycle frequencies when compared with the firstdifference of integrated or highly persistent time series. Guay and St-Amant (1996) extend this result to the BK filter but also note that, when compared with macroeconomic series in level terms, the HP and BK filters minimize the importance of business-cycle frequencies.

- In-text reference with the coordinate start=7193
- 19
- DeSerres, A. and A. Guay. 1995. “The Selection of the Truncation Lag in Structural VARs (or

Total in-text references: 1- In-text reference with the coordinate start=23881
- Prefix
- The autoregressive reduced-form VAR of the model is first estimated: q ZtΠiZti–et+ =∑ i1= withqthe number of lags and a vector of estimated residuals with . It is crucial that the estimated VARs include a sufficient number of et Eetet()Σ= lags. Indeed, Monte Carlo simulations by
- Exact
- DeSerres and Guay (1995)
- Suffix
- show that using a lag structure that is too parsimonious can significantly bias the estimation of the structural components. These authors also find that information-based criteria, such as the Akaike and Schwarz criteria, tend to select an insufficient number of lags, while Wald or likelihood-ratio (LR) tests, using a general-tospecific approach, perform much bett

- In-text reference with the coordinate start=23881
- 22
- Engle, R. F. and C. W. J. Granger. 1987. “Cointegration and Error Correction: Representation, Estimation and Testing.”Econometrica 55 (March): 251-76.

Total in-text references: 1- In-text reference with the coordinate start=13846
- Prefix
- If the rank of is less than n1, there exists at least one linear combination of the elements in that is I(0). In other words, there exists at least one cointegration relationship between these variables — see
- Exact
- Engle and Granger (1987).
- Suffix
- The LRRO approach assumes that has the following structural representation: (3) Zt Ztδt() ΓL()ηt+= where is a n-vector of structural shocks,, and (a ηtEηt()0=Eηtηt′()In= Γ0Γ0′Ω=εtΓ0ηt= simple normalization).

- In-text reference with the coordinate start=13846
- 23
- Evans, G. and L. Reichlin. 1994. “Information, Forecasts, and Measurement of the Business Cycle.”Journal of Monetary Economics 33: 233-54.

Total in-text references: 2- In-text reference with the coordinate start=9384
- Prefix
- One example is a decomposition method proposed by Cochrane (CO) which is based on the permanent-income theory and uses consumption to define the permanent component of output. Multivariate extensions of the Beveridge-Nelson decomposition method (MBN) have also been applied to identify the trend component of output
- Exact
- (Evans and Reichlin 1994).
- Suffix
- However, a major restriction in the univariate context, which is maintained in the multivariate extensions, is that the permanent component of output behaves like a random walk. This assumption is difficult to reconcile with the widely held view that the permanent component of output is, at least in part, driven by technological innovations.

- In-text reference with the coordinate start=13515
- Prefix
- Equation (1) can be decomposed into a long-run component and a transitory component: (2) ZtZtEεt()0=Eεtεt′()Ω= ΩCL() Ztδt()C1()εtC∗L()εt++= C1() Σi0= ∞ =CiC∗L()CL()C1()–= C11() X1tC11() X1t where and. This decomposition corresponds to the multivariate Beveridge-Nelson decomposition — see
- Exact
- Evans and Reichlin (1994) and King et al. (1991).
- Suffix
- We define as the long-run multiplier of the vector. If the rank of is less than n1, there exists at least one linear combination of the elements in that is I(0). In other words, there exists at least one cointegration relationship between these variables — see Engle and Granger (1987).

- In-text reference with the coordinate start=9384
- 24
- Gali, J. 1992. “How Well Does the IS-LM Model Fit Postwar US Data?”Quarterly Journal of Economics 107: 709-38.

Total in-text references: 1- In-text reference with the coordinate start=26960
- Prefix
- 2, models in which the permanent component of output is a random walk imply that the economy is below (above) potential in the transition period following a permanent positive (negative) shock to output. To the extent that the transition primarily reflects factors associated with an adjustment in the supply side of the economy, assuming that potential 7.
- Exact
- Blanchard and Quah (1989) and Gali (1992),
- Suffix
- among others, report similar results. output follows a random walk can be misleading. It could, in particular, provide misleading signals about the extent of inflationary pressures in the economy. Chart 3 shows the output gaps calculated on the basis of the LRRO methodology in the bivariate and trivariate cases.

- In-text reference with the coordinate start=26960
- 25
- Guay, A. and P. St-Amant. 1996.Do Mechanical Filters Provide a Good Approximation of

Total in-text references: 2- In-text reference with the coordinate start=7378
- Prefix
- For example, Harvey and Jaeger (1993) and Cogley and Nason (1995) show that spurious cyclicality can be induced by the HP filter when it is used with integrated or nearly integrated data.
- Exact
- Guay and St-Amant (1996)
- Suffix
- reach the more general conclusion that the HP and BK filters perform poorly in identifying the cyclical component of time series that have a spectrum or pseudo-spectrum with Granger’s typical shape, i.e., the shape characteristic of most macroeconomic time series.

- In-text reference with the coordinate start=35547
- Prefix
- CHART 6: Spectra of the output gaps 0.08 0.07 HP filter 0.06 0.05 BK filter CO 0.04 LRRO (3 variables) 0.03 MBN (3 variables) 0.02 0.01 0 00.050.10.150.20.250.30.35 fraction of pi Cogley and Nason (1995) and Harvey and Jaeger (1993) show that the HP filter amplifies business-cycle frequencies when compared with the firstdifference of integrated or highly persistent time series.
- Exact
- Guay and St-Amant (1996)
- Suffix
- extend this result to the BK filter but also note that, when compared with macroeconomic series in level terms, the HP and BK filters minimize the importance of business-cycle frequencies.

- In-text reference with the coordinate start=7378
- 27
- Hamilton, J. D. 1994.Time Series Analysis. Princeton University Press.

Total in-text references: 1- In-text reference with the coordinate start=34807
- Prefix
- The spectrum of the gap resulting from the two-variable MBN application (not shown on the graph) has the same shape as the three-variable 9. For an introduction to spectral analysis see
- Exact
- Hamilton (1994).
- Suffix
- The order of these processes was determined on the basis of the Akaike criteria. case, although with a lower peak and a smaller total variance. The latter result is not surprising, since it is well known that the MBN methodology gives a transitory component whose importance increases with the number of series used to identify it.

- In-text reference with the coordinate start=34807
- 28
- Harvey, A. C. and A. Jaeger. 1993. “Detrending, Stylized Facts and the Business Cycle.” Journal of Applied Econometrics8: 231-47.

Total in-text references: 2- In-text reference with the coordinate start=7193
- Prefix
- One of these methods consists of using mechanical filters such as the Hodrick-Prescott (HP) filter or the band-pass filter (BK) proposed by Baxter and King (1995). However, mechanical filters have been criticized. For example,
- Exact
- Harvey and Jaeger (1993) and Cogley and Nason (1995)
- Suffix
- show that spurious cyclicality can be induced by the HP filter when it is used with integrated or nearly integrated data. Guay and St-Amant (1996) reach the more general conclusion that the HP and BK filters perform poorly in identifying the cyclical component of time series that have a spectrum or pseudo-spectrum with Granger’s typical shape, i.e., the shape characteristic o

- In-text reference with the coordinate start=35330
- Prefix
- The latter result is not surprising, since it is well known that the MBN methodology gives a transitory component whose importance increases with the number of series used to identify it. CHART 6: Spectra of the output gaps 0.08 0.07 HP filter 0.06 0.05 BK filter CO 0.04 LRRO (3 variables) 0.03 MBN (3 variables) 0.02 0.01 0 00.050.10.150.20.250.30.35 fraction of pi
- Exact
- Cogley and Nason (1995) and Harvey and Jaeger (1993)
- Suffix
- show that the HP filter amplifies business-cycle frequencies when compared with the firstdifference of integrated or highly persistent time series. Guay and St-Amant (1996) extend this result to the BK filter but also note that, when compared with macroeconomic series in level terms, the HP and BK filters minimize the importance of business-cycle frequencies.

- In-text reference with the coordinate start=7193
- 30
- King, R. G., C. I. Plosser and S. Rebelo. 1988. “Production, Growth, and Business Cycles: II

Total in-text references: 1- In-text reference with the coordinate start=20309
- Prefix
- Working in a univariate framework, Lippi and Reichlin must constrain the dynamic of the trend to follow a particular shape (S-shape dynamic) in order to identify the trend and cyclical components. Again, a decisive advantage of the LRRO approach is that it lets the data determine the shape of the diffusion process of permanent shocks.6 5. Stochastic growth models — such as in
- Exact
- King et al. (1988)
- Suffix
- or King et al. (1991) — imply that the ratio of the log of GNP to the log of consumption is stationary but that consumption is not a random walk because the real interest rate is not constant.

- In-text reference with the coordinate start=20309
- 32
- King, R. G., G. H. Plosser, J. H. Stock, and M. W. Watson. 1991. “Stochastic Trends and

Total in-text references: 6- In-text reference with the coordinate start=9768
- Prefix
- However, a major restriction in the univariate context, which is maintained in the multivariate extensions, is that the permanent component of output behaves like a random walk. This assumption is difficult to reconcile with the widely held view that the permanent component of output is, at least in part, driven by technological innovations. As underlined by
- Exact
- King et al. (1991),
- Suffix
- “productivity shocks set off transitional dynamics, as capital is accumulated and the economy moves towards a new steady-state.” Lippi and Reichlin (1994) go even further, arguing that modelling the trend in output as a random walk is inconsistent with standard views concerning the dynamics of productivity shocks.

- In-text reference with the coordinate start=10512
- Prefix
- In this paper, we compare some of the techniques briefly introduced above with the structural vector autoregression methodology based on long-run restrictions imposed on output (LRRO) proposed by
- Exact
- Blanchard and Quah (1989), Shapiro and Watson (1988), and King et al. (1991)
- Suffix
- in theory (Section 2) and in applications (Section 3). In Section 2, we note that one characteristic of the LRRO method is that it does not impose restrictions on the short-run dynamics of the permanent component of output.

- In-text reference with the coordinate start=13515
- Prefix
- Equation (1) can be decomposed into a long-run component and a transitory component: (2) ZtZtEεt()0=Eεtεt′()Ω= ΩCL() Ztδt()C1()εtC∗L()εt++= C1() Σi0= ∞ =CiC∗L()CL()C1()–= C11() X1tC11() X1t where and. This decomposition corresponds to the multivariate Beveridge-Nelson decomposition — see
- Exact
- Evans and Reichlin (1994) and King et al. (1991).
- Suffix
- We define as the long-run multiplier of the vector. If the rank of is less than n1, there exists at least one linear combination of the elements in that is I(0). In other words, there exists at least one cointegration relationship between these variables — see Engle and Granger (1987).

- In-text reference with the coordinate start=14676
- Prefix
- (2) and (3) we have: (4) CL() ΓL()Γ0 –1 = C1()ΩC1()′ C1()ΩC1()′ Γ1()Γ1()′= Γ0 C1() C1() This relation suggests that we can identify matrix with an appropriate number of restrictions on the long-run covariance matrix of the structural form. Blanchard and Quah (1989) and Shapiro and Watson (1988) use long-run restrictions to identify shocks with having full rank.
- Exact
- King et al. (1991)
- Suffix
- work in a context where the rank of is less than n1 and they use cointegration restrictions. Let us assume that the log of output is the first variable in the vector . It is then equal to: (5) Z1t ∆ytμyΓ1 p ()ηLt p Γ1 c ()ηLt c =++ ηt p ηt c where is the vector of permanent shocks affecting output and is the vector containing shocks having only a transitory effect on output.

- In-text reference with the coordinate start=20331
- Prefix
- Again, a decisive advantage of the LRRO approach is that it lets the data determine the shape of the diffusion process of permanent shocks.6 5. Stochastic growth models — such as in King et al. (1988) or
- Exact
- King et al. (1991)
- Suffix
- — imply that the ratio of the log of GNP to the log of consumption is stationary but that consumption is not a random walk because the real interest rate is not constant. In these models, the transitory component of permanent shocks to consumption is not equal to zero.

- In-text reference with the coordinate start=22991
- Prefix
- The use of this variable, which we suppose to be stationary, makes the comparison between the different methodologies easier. We also consider a case where we add a nominal variable to the information set as recommended by
- Exact
- King et al (1991).
- Suffix
- In a vector form, the structural shocks and the variables used in the VARs can be expressed in the following way: ∆y εPεT yc ηt εP εT =Zt ∆y yc– = and or and ηt εP εT1 εT2 =Zt ∆y yc– ∆i = We use quarterly data on U.

- In-text reference with the coordinate start=9768
- 34
- Kuttner, K. N. 1994. “Estimating Potential Output as a Latent Variable.”Journal of Business and Economic Statistics 12: 361-68.

Total in-text references: 1- In-text reference with the coordinate start=20710
- Prefix
- al. (1988) or King et al. (1991) — imply that the ratio of the log of GNP to the log of consumption is stationary but that consumption is not a random walk because the real interest rate is not constant. In these models, the transitory component of permanent shocks to consumption is not equal to zero. The LRRO decomposition is compatible with the prediction of these models. 6.
- Exact
- Kuttner (1994)
- Suffix
- proposes a method based on the univariate unobserved stochastic trend decomposition of Watson (1986) augmented with a Phillips-curve equation. As with the Beveridge-Nelson decomposition, Kuttner’s approach constrains potential output to follow a random-walk process. μΓy p ()η1t p + Γ p∗ ()L Γy p () Γ1c p =()1 Γy p∗ () ΓLc p∗ =()LΓy p () Γ1c p =()1 One implication of def

- In-text reference with the coordinate start=20710
- 35
- Laxton, D. and R. Tetlow. 1992.A Simple Multivariate Filter for the Measurement of Potential Output. Technical Report No. 59. Ottawa: Bank of Canada.

Total in-text references: 1- In-text reference with the coordinate start=8232
- Prefix
- Van Norden (1995) stresses the fact that this is a very significant limitation for policymakers interested in estimating 1. For a discussion of how the estimation of potential output can affect the formulation of monetary policy, see Boschen and Mills (1990) or
- Exact
- Laxton and Tetlow (1992).
- Suffix
- the current level of the output gap. Another strategy for identifying the permanent and transitory components of output involves the use of univariate techniques such as the unobserved components approach suggested by Watson (1986) and the BeveridgeNelson (1981) method.

- In-text reference with the coordinate start=8232
- 36
- Lippi, M., and L. Reichlin. 1993. “The Dynamic Effects of Aggregate Demand and Supply Disturbances: Comment.” American Economic Review83: 644-52.

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- Prefix
- C0In=εt in given information on lagged values of,, and with positive definite. We suppose that the determinantal polynomial has all its roots on or outside the unit circle, which rules out the non-fundamental representations emphasized by
- Exact
- Lippi and Reichlin (1993).
- Suffix
- Equation (1) can be decomposed into a long-run component and a transitory component: (2) ZtZtEεt()0=Eεtεt′()Ω= ΩCL() Ztδt()C1()εtC∗L()εt++= C1() Σi0= ∞ =CiC∗L()CL()C1()–= C11() X1tC11() X1t where and.

- In-text reference with the coordinate start=13173
- 40
- Quah, D. 1992. “The Relative Importance of Permanent and Transitory Components: Identification and Some Theoretical Bounds.”Econometrica 60: 107-18.

Total in-text references: 1- In-text reference with the coordinate start=8554
- Prefix
- Another strategy for identifying the permanent and transitory components of output involves the use of univariate techniques such as the unobserved components approach suggested by Watson (1986) and the BeveridgeNelson (1981) method. However,
- Exact
- Quah (1992)
- Suffix
- has shown that “without additional ad hoc restrictions those [univariate] characterizations are completely uninformative for the relative importance of the underlying permanent and transitory components.

- In-text reference with the coordinate start=8554
- 45
- Shapiro, M. D. and M. W. Watson. 1988. “Sources of Business Cycle Fluctuations.” Working Paper 2589. National Bureau of Economic Research, Cambridge (MA).

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- Prefix
- In this paper, we compare some of the techniques briefly introduced above with the structural vector autoregression methodology based on long-run restrictions imposed on output (LRRO) proposed by
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- Blanchard and Quah (1989), Shapiro and Watson (1988), and King et al. (1991)
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- in theory (Section 2) and in applications (Section 3). In Section 2, we note that one characteristic of the LRRO method is that it does not impose restrictions on the short-run dynamics of the permanent component of output.

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- From (2) and (3) we have: (4) CL() ΓL()Γ0 –1 = C1()ΩC1()′ C1()ΩC1()′ Γ1()Γ1()′= Γ0 C1() C1() This relation suggests that we can identify matrix with an appropriate number of restrictions on the long-run covariance matrix of the structural form.
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- Blanchard and Quah (1989) and Shapiro and Watson (1988)
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- use long-run restrictions to identify shocks with having full rank. King et al. (1991) work in a context where the rank of is less than n1 and they use cointegration restrictions.

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- Staiger, D., J. Stock, and M. W. Watson. 1996. “How Precise are Estimates of the Natural Rate of Unemployment?” Presented at the Monetary Policy and Low Inflation Conference.

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- This suggests that permanent shocks have more complex dynamics than a random walk, which is the basic assumption of the CO and MBN approaches. However, it is also found that the estimation of the output gap on the basis of an estimated VAR is imprecise, which is consistent with results obtained by
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- Staiger, Stock and Watson (1996)
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- with a different methodology. The spectra of the transitory components (output gaps) resulting from the empirical applications of the CO, MBN and LRRO methodologies differ from one another. Indeed, only the LRRO transitory component has a peak at business-cycle frequencies, i.e., cycles lasting between 6 and 32 quarters.

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- S'il faut en croire ces résultats, la dynamique des chocs permanents serait plus complexe qu'une marche aléatoire, laquelle est au coeur des méthodes CO et MBN. Les auteurs constatent cependant que l'écart de production calculé à l'aide d'un vecteur autorégressif estimé manque de précision, ce qui est conforme aux résultats obtenus par
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- Staiger, Stock et Watson (1996)
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- au moyen d'une autre méthode. Les spectres des composantes transitoires (écarts de production) qui résultent de l'application empirique des méthodes CO, MBN et LRRO diffèrent les uns des autres.

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- We find that the answer is “yes” when the entire output gap series is considered, but “not really” when one is interested in estimating the output gap at a specific point in time. In the latter case, the estimation of potential output and the output gap is indeed imprecise. This is consistent with recent results reported by
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- Staiger, Stock and Watson (1996)
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- for the estimation of the NAIRU. Another interesting result is that, of the methods we consider, only the LRRO-based one generates an output gap with a peak at business cycle frequencies as defined by Burns and Mitchell (1946), i.e., cycles lasting between 6 and 32 quarters. 2 Methodologies used for estimating the trend in output 2.1 The approach based on the LRRO methodology In this section,

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- CHART 5: Uncertainty surrounding the estimation of the output gap MBN Confidence interval LRRO Confidence interval The main message of Chart 5, which would apply to other estimates of the output gap reviewed in this paper, is that there is a substantial amount of uncertainty surrounding the estimation of the output gap.
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- Staiger, Stock and Watson (1996),
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- using a different methodology, reach a similar conclusion concerning the estimation of the NAIRU. This uncertainty should probably be taken into account by policymakers who use the output gap to guide their decisions.

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- However, the estimates are imprecise for specific points in time and it appears difficult to distinguish between these methodologies in that context. This later result is consistent with the conclusions of
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- Staiger, Stock and Watson (1996).
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- Van Norden, S. 1995. “Why Is It So Hard to Measure the Current Output Gap?” Unpublished. Bank of Canada.

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- Baxter and King (1995) and others note that two-sided filters such as the HP and BK filters become ill-defined at the beginning and the end of samples. For this reason, they recommend discarding three years of quarterly data at both ends of the sample when using the HP filter. Van
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- Norden (1995)
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- stresses the fact that this is a very significant limitation for policymakers interested in estimating 1. For a discussion of how the estimation of potential output can affect the formulation of monetary policy, see Boschen and Mills (1990) or Laxton and Tetlow (1992). the current level of the output gap.

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- Watson, M. W. 1986. “Univariate Detrending Methods with Stochastic Trends.”Journal of

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- of how the estimation of potential output can affect the formulation of monetary policy, see Boschen and Mills (1990) or Laxton and Tetlow (1992). the current level of the output gap. Another strategy for identifying the permanent and transitory components of output involves the use of univariate techniques such as the unobserved components approach suggested by
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- Watson (1986) and
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- the BeveridgeNelson (1981) method. However, Quah (1992) has shown that “without additional ad hoc restrictions those [univariate] characterizations are completely uninformative for the relative importance of the underlying permanent and transitory components.

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- In these models, the transitory component of permanent shocks to consumption is not equal to zero. The LRRO decomposition is compatible with the prediction of these models. 6. Kuttner (1994) proposes a method based on the univariate unobserved stochastic trend decomposition of
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- Watson (1986)
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- augmented with a Phillips-curve equation. As with the Beveridge-Nelson decomposition, Kuttner’s approach constrains potential output to follow a random-walk process. μΓy p ()η1t p + Γ p∗ ()L Γy p () Γ1c p =()1 Γy p∗ () ΓLc p∗ =()LΓy p () Γ1c p =()1 One implication of defining potential output as a random walk with drift is that when the contemporary effect of a positive permanent sh

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- Watson, M. W. 1993. “Vector Autoregressions and Cointegration.”Handbook of Econometrics, vol. 4. Bank of Canada Working Papers 1997 97-1Reconsidering Cointegration in International Finance:

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- vector including a n1-vector of I(1) variables and a n2-vector of I(0) variables such that.3 By the Wold decomposition theorem, can be expressed as the following reduced form: (1) Zt Zt∆X1t′X2t′,()′= Zt Ztδt()CL()εt+= δt()CL() Σi0= ∞ CiL i = where is deterministic, is a matrix of polynomial lags, is the identity matrix, the vector is the one-step-ahead forecast errors 2. See
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- Watson (1993)
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- for a more detailed presentation of the LRRO approach. 3. I(d) denotes a variable that is integrated of order d. C0In=εt in given information on lagged values of,, and with positive definite. We suppose that the determinantal polynomial has all its roots on or outside the unit circle, which rules out the non-fundamental representations emphasized by Lippi and Reichlin (1993).

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- Nevertheless, the correlation between the output gaps identified on the basis of the CO and MBN methodologies is rather small, indicating that consumption may not be a random walk. This is consistent with results reported in
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- Watson (1993),
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- showing that the spectrum of the first difference of consumption has a peak at business-cycle frequencies. In Section 2, we noted that if consumption followed a random-walk process, the CO and MBN methodologies would give identical results. 8.

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