Equivalence of several methods for decomposing time series into permananent and transitory components

Transcription

1 Equivalence of several methods for decomposing time series into permananent and transitory components Don Harding Department of Economics and Finance LaTrobe University, Bundoora Victoria 3086 and Centre for Applied Macroeconomic Analysis (CAMA) September 24, 2008

2 Abstract I show that several popular methods including the Hodrick Prescott lter, band pass lter, Butterworth lter, unobserved components model and the Beveridge Nelson decomposition all yield equivalent decompositions of time series into permanent and transitory components provided that the underlying model of the economy is held constant. Thus when practitioners obtain di erent permanent-transitory decompositions using these methods it is because they are varying the assumed model of the economy. JEL codes: C12; C13; C32; E32 Keywords: Permanent; transitory; unobserved components; Beveridge Nelson decomposition; Kalman lter; Butterworth Filter; Low Pass Filter; Band Pass Filter.

3 1 Introduction Macroeconomists separate time series into permanent and transitory components using one of the following methods, unobserved components (UC) models; Beveridge-Nelson (BN) decomposition; or Hodrick-Prescott (HP), Butterworth, Low Pass (LP) and Band-Pass (BP) lters. These methods all associate the permanent component with low frequency uctuations so it is of interest to ask how closely related are these methods. Previous work by Harvey and Jaeger (1993), Gomez (1999, 2001) and Harvey and Trimbur (2003) show that the HP, LP, BP and Butterworth lters all have UC representations so it su ces to focus attention on the relationship between the UC and BN permanent and transitory decompositions. Morely, Nelson and Zivot (2003) and Oh, Zivot and Creal (2007) show the equivalence of the BN and UC de nitions of permanent and transitory components in certain models. In this paper I extend these results in two ways. First, I extend the BN de nition of permanent component to I(2) processes. Second, I show that for a broad class of I(1) and I(2) processes the BN and UC permanenttransitory decompositions are equivalent in the sense that for a given reduced form model of the economy the BN permanent component is identical to the permanent component in the UC model. In addition to these extensions this paper contributes to the literature by virtue of the simplicity of the proofs and ease of access to practitioners. The paper also contributes to the literature by exploring two implications of the result just cited. The rst of these is that where practitioners obtain di erent permanent-transitory components using UC, BN, HP, BP or Butterworth methods, those di erence must be attributable to over identifying assumptions being imposed when particular methods are applied. Many of the methods for extracting permanent components are "black boxes" and this makes it di cult for practitioners to understand the identifying and over identifying restrictions that are imposed, implicitly, by the particular method that they have chosen use. To aid in understanding of this issue I set out the over identifying restrictions that are imposed in certain popular UC models. In future versions of the paper I propose to develop and implement tests of these restrictions. 2

4 The second implication that will be more fully explored in the future is that for a wide range of UC models it is possible to use the BN decomposition to write the model using a single source of error (SSOE) representation which simpli es the estimation and testing of these models without any loss of generality. The paper is structured as follows. In section 2 I generalize the de nitions of permanent and transitory components given by Beveridge and Nelson (1981) to I(2) processes. In section 3 I set out a general UC framework. That framework is used in section 4 to show that the Beveridge-Nelson and unobserved components approaches are equivalent for a wide range of processes. Over identifying restrictions imposed by popular UC models are explained in section 5. An application to US GDP is discussed in section 6. Conclusions are presented in section 7. 2 Generalization of the Beveridge Nelson de- nition of permanent and transitory components Macroeconomists spend much of their time studying how shocks move through the economic system. A useful question to ask about a variable in such a system is this, given the dynamics of the system, what value would that variable be expected to take after all of the shocks known at time t have moved through the economic system. This concept is formalized in the following de nition. De nition 1: Generalized Beveridge Nelson. Given a series y t, that is integrated of order two or less, the permanent component P t ; expected growth rate of the permanent component d t and transitory component T t are jointly de ned as follows P t = y t T t (1) T t = E t d t = E t 1 P t (2) 1 X j=1 (y t+j d t+j ) (3) Since P t is de ned as y t plus the sum of all the future changes in y t that are attributable to shocks known at time t: It is natural to name this concept as the permanent component of y as it is an estimate of the value that y will take at the in nite horizon conditional on what shocks we know 3

5 today. Similarly, it is natural to de ne as transitory that part of y t which is not permanent. If y t is I(1) with a deterministic trend then we have the standard Beveridge Nelson decomposition. But the extended de nition can handle the I(2) case where P t = t 1 + t And t = t 1 + t In this case d t = E t 1 ( t 1 + t ) = t 1 : The de nition also encompasses the case where y t is stationary about a deterministic trend. In this case g (t) is a deterministic function of time, d t = g (t) g (t 1), P t = g (t) and T t = (y t g (t)) : Thus a desirable feature of the generalized Beveridge-Nelson de nition is that it encompasses I(0), I(1) and I(2) cases in one setup. 3 A general unobserved components framework Consider a general UC representation of the DGP for y t in which y t is comprised of three unobservable components viz t ; z t and " t via the measurement equation (4). y t = t + z t + " " t 0 " < 1 (4) The component t evolves as follows Where t follows a random walk. t = t 1 + t 1 + t 0 < 1 (5) t = t 1 + t 0 < 1 (6) And s t = " t ; t ; t 0 is a 3 1 vector of mean zero shocks with unit variance that are mutually uncorrelated and are serially uncorrelated. Technically, Es t = 0; Es 2 t = I 3 and Es t s = 08t 6= : (7) The component z t is assumed to be an ergodic covariance stationary process that converges in mean to zero. The latter condition implies that lim E tz t+j = 0 (8) j!1 No restriction is placed on the covariation between z t and s t. 4

6 3.1 In what sense is this UC representation general? The system (4) to (8) is a reduced form model and thus the BN permanent and transitory components are reduced form concepts. A natural question is whether the dynamics of the model (4) to (8) are restricted in any way other than that which is necessary to ensure that the model is just identi ed. To address this question consider the process y t = t + z t + " " t 0 " < 1 (9) t = t 1 + t 1 + w t + t 0 < 1 (10) Where t evolves according to (6), zt and wt are ergodic covariance stationary processes. Many authors suggest that (9) and (10) represents a more general UC model than (4) and (5) because in the former t contains a cyclical component whereas t does not. For example, Quah (1992, p 110) writes By contrast, Shapiro and Watson (1998) and Blanchard and Quah (1989) have considered models where permanent components have richer dynamics than those in a random walk. But since it is covariance stationary w t can be represented as w t = (1 L) u t where u t has strictly positive but nite spectral density at the zero frequency. Thus we can replace w t in (10) and rewrite it as t u t = t 1 u t 1 + t 1 + t (11) De ning t = t u t and z t = z t + u t we can rewrite (9) and (10) as y t = t + z t + " " t (12) t = t 1 + t 1 + t (13) con rming that the representation (9) and (10) and the representation (4) and (5) are equivalent. It is worth observing here that this is the sense in which the UC representation (4) to (8) is general. In the representation (9) and (10) t u t is the BN permanent component and zt + u t is the BN transitory component. While in the representation (4) and (5) t is the BN permanent component and z t is the transitory component. Since t = t u t and z t = zt + u t the two representations yield identical BN decompositions. 5

7 4 Equivalence of Beveridge Nelson and Unobserved Components decompositions Consider a variable y t generated by a DGP the reduced form of which can be represented as the UC model (4) to (8) Then, proposition 1 states that the Beveridge Nelson and UC de nitions of permanent and transitory components are equivalent for that variable. Proposition 1 Equivalence of Beveridge Nelson and Unobserved Components decompositions of y t. The unobserved component model (4) to (6) satisfying restrictions (7) and (8) has the property that it produces an permanent-transitory decomposition identical to that of the Beveridge-Nelson decomposition (1), (2) and (3). Speci cally, t P t and T t z t " " t : Proof. Using the BN de nition of P t P t = y t + E t 1 X j=1 (y t+j d t+j ) Substituting in for y t from (4) and using the facts that d t = t 1 yields So, (y t+j d t+j ) = tt+j + t+j 1 + z t+j + " " t+j t+j 1 = tt+j + z t+j + " " t+j P t = t + z t + " " t + E t 1 X j=1 t+j + z t+j + " " t+j = t + z t + " " t + lim j!1 E t z t+j z t " " t But lim j!1 E t z t+j = 0 by the ergodic stationarity of z t : The remaining terms cancel leaving P t = t The result that T t z t " " t is obtained immediately by subtracting P t from y t. Proposition 1 shows when the same DGP is used the BN and UC permanent and transitory components are equivalent. The implication is that where practitioners obtain permanent and transitory components with UC models that di er from those obtained via a BN decomposition then the di erence must arise because in one of the cases the practitioner implicitly imposed over identifying restrictions on the model. 6

8 4.1 Interpreting the BN permanent and transitory components It is commonplace for practitioners to associate the BN transitory component with demand or to associate the variance in of the transitory component with a measure of the maximum contribution that demand uctuations make to the variation in y t : Both of these interpretations are false as is explained by Quah (1992, p110) who states that... permanent components turn out to have variances that can no longer be identi ed from just the second moments of the original sequence. To fully understand this issue it is useful to use the representation (4) to (8) and associate t with the permanent component from the supply side of the economy. Then, the BN transitory component is comprised of two parts viz z t and the measurement error " t : Here the variance of y t is Suppose to simplify the discussion = 0 so we are dealing with a I(1) process. The we can decompose the variance of y t as follows. V ar (y t ) = V ar ( t ) + V ar (z t ) + var (" t ) + 2Cov ( t ; z t ) +2Cov (" t ; z t ) + 2Cov (" t ; t ) But we saw in section 3.1 that z t is comprised of two parts zt that we can think of as the demand component and u t that we can view as the transitory part of the supply component. In many instances it is convenient to assume that demand and supply components are uncorrelated ie zt and u t ; and zt and t are uncorrelated so that V ar (z t ) = V ar (zt ) + V ar (u t ) and Cov ( t ; z t ) = 0: the measurement error " t is uncorrelated with (z t ; u t ; t ) With these assumptions we can express the variance of y t as, V ar (y t ) = 2 + V ar (z t ) + V ar (u t ) + 2 " + 2Cov ( t ; u t ) The term Cov ( t ; u t ) is the covariance between the innovation into the permanent part of the supply component and the change in the transitory part of the supply component. This covariance can have either a positive or negative sign and so we cannot use our estimates of 2 ; 2 " and V ar (y t ) 7

9 to place an upper bound on V ar (z t ) : Thus, even with the assumptions made earlier it is not possible to place an upper bound on the contribution made by demand to the variance of y t : Further, restrictions from economic theory are required to identify the respective contributions of demand and supply shocks. 1 5 Over identifying restrictions imposed in popular UC model Clarke (1987), Harvey (1985) and Harvey Trimbur and Van Dijk (2007) all use UC models to obtain permanent-transitory decompositions that di er substantially from that obtained when the BN decomposition is applied using an unrestricted ARIMA model as the reduced form. The implication of proposition 1 is that this nding must arise because the UC models are estimated imposing over identifying restrictions that are not supported by the data. In this section I investigate those over identifying restrictions. A maintained assumption here is that, using the notation of section 3, the shocks s t = 0 t t " t are mean zero iid and mutually uncorrelated. Speci cally 2 Es t = 0; Es t s 0 t = " 3 5 (14) The remaining identi cation assumptions relate to the nature of the process z t : I look at three cases in the literature. The rst studied in section 5.1 allows z t to follow a stationary ARMA (p; q) process. The second speci cation studied in section 5.2 is favoured by Andrew Harvey and his coauthors imposes some additional over identifying restrictions on z t : The third speci cation studied in section is of interest because it provides a UC motivation for BP lters. 5.1 ARMA (p; q) speci cation of z t Here we assume that z t follows the ARMA(p,q) process z t = (L) (L) $ t (15) 1 The UC model (4) to (6) is some time referred to as being a structural time series model but this name is very misleading as the shocks do not necessarily have an economic interpretation and thus do not have meaning derived from economic theory. 8

10 where (L) = L + + q L q and (L) = 1 1 L p L p are lag polynomials and (L) has all its roots inside the unit circle. The shocks $ t are Gaussian with mean zero variance 2 $ and cov(; $) = $ :The covariances with all other shocks are zero ARIMA representation and identi cation Given the assumptions above 2 y t can be written as 2 y t = t 1 + t t 1 + (L) ($ t 2$ t 1 + $ t 2 ) (L) + " t 2" t 1 + " t 2 (16) Multiplying both sides of (16) by (L) yields an AR(p) in 2 y t on the left hand side. The rst term of the right hand side is (L) t 1 a p order moving average, the second terms is (L) (1 L) t a p + 1 order moving average, the third term is (L) (1 L) 2 $ t a (2 + q) order moving average and the nal term is (L) (1 L) 2 " t a p + 2 order moving average. Thus letting q = max (q; p) + 2 and appealing to Grangers theorem, 2 y t has an ARIMA(p; 2; q) representation. The ARIMA representation yields p q parameters. The state space representation has 5 + p + q parameters. A necessary condition for identi cation is that the number of parameters in the ARIMA model equals the number of parameters in the UC model, That is, There are two cases to consider. p q = 5 + p + q (17) ) q q = 4 Case 1: q = 2 + q This is the case where there are more MA parameters than AR parameters (q > p) Making use of (17) we see that identi cation requires that 2 = 4 which is impossible so the parameters of the UC model are unidenti ed unless further restrictions are placed on the parameters. Case 2: q = p + 2 This is the case where there are at lease as many AR parameters as MA parameters. Again making use of (17) we see that the UC model is just identi ed if there are two more AR parameters in the cycle than there are MA parameters (p q = 2) : The UC model is over identi ed if (p q > 2) It is useful to explore the issue of identi cation in the context of the models made popular by Clarke (1989) and Harvey (1985). In this model 9

11 p = 2 and q 1: There are two ways to ensure that this model is just identi ed. The rst is to set q = 0 so that there are no MA terms in the cycle. This identi cation was pursued by Clarke (1987). The second is to set q > 0 but write the model so that the MA parameters are completely determined as functions of the AR parameters. This later identifying assumption is made in the local linear trend plus cycle models and the cyclical trend models that have been made popular by Harvey et al(1993). 5.2 Periodic cycle speci cation of z t In this model z t is generated by, zt cos sin zt = 1 + sin cos z t z t 1 t t (18) where, 0 1 to impose stationarity, is frequency in radians, t and t are mutually independent i:i:d N (0; 2 ) and i:i:d N (0; 2 ) respectively. To establish the nature of these restrictions write the moving average representation of (18) as, zt z t = 1 cos L sin L sin L 1 cos L 1 t t (19) Now applying a standard formula for matrix inversion 1 cos L sin L sin L 1 cos L 1 = 1 cos L sin L sin L 1 cos L (20) 1 2 cos L + 2 cos 2 L 2 Thus z t has a representation as, 1 2 cos L + 2 cos 2 L 2 z t = (1 cos L ) t + sin L t (21) The right hand side in (21) is the sum of a MA(1) process and Gaussian white noise. Thus by Granger s theorem the right hand side is an MA(1). The left hand side of (21) is an AR(2) with coe cients 1 = 2 cos and 10

12 2 = 2 : Thus z t is an ARMA(2; 1): Focusing on the AR(2) part we can see that the roots are cos p cos 2 1 and since 0 cos 2 < 1 they are complex so z t must have a periodic cycle in this speci cation. One can obtain an equivalent restriction on (15) that involves restricting the AR parameters to the region de ned by the intersection of the regions ; and 1 2 0:. The last two regions coming from the de nitions of 1 and 2 together with the restrictions on and : Thus, the identifying assumption used by Harvey imposes a periodic cycle by placing restrictions on the AR parameters. In addition it also imposes a restriction that MA coe cient is completely determined by the AR parameters. It is this identifying assumption that allows the model to be estimated. To obtain the restriction on the MA parameters let w t = (1 cos L ) t + sin L t then E (wt 2 ) = (1 + 2 ) 2, E (w t ; w t 1 ) = cos 2 : Thus, w t has a representation as w t = (1 + L) u t where u t ~N (0; 2 u) and in this representation E (wt 2 ) = u and E (w t ; w t 1 ) = 2 u. Thus equating the expectations we have u = E w 2 t = (22) 2 u = E (w t ; w t 1 ) = cos 2 (23) Dividing (23) by (22) yields Rearranging gives cos = = cos cos 2 (24) Solving the quadratic for yields = q (1 + 2 ) cos 2 2 cos And, the invertibility condition that jj < 1 selects (25) 11

13 = q (1 + 2 ) cos 2 2 cos (26) Now letting 1 = 2 cos and 2 = terms of the AR parameters viz, 2 we can rewrite completely in = 1 2 q (1 2 ) (27) Which demonstrates the earlier claim that the periodicity restriction completely determine 1 in terms of the AR parameters: 5.3 n th order cycles, Butterworth and low pass lters The low pass Butterworth lter is B lp m (L) = As m! 1 B lp m (L)! q 1 j1 Lj 2m ; q = 2 sin 2 2m ; 0 < Harvey and Trimbur (2003) and Harvey Trimbur and Van Dijk (2007) consider an extension of the a periodic cycle model (18). Speci cally, they de ne z 1t and z1t as follows. z1t cos sin z1t = 1 t + (28) sin cos 0 z 1t z 1t 1 where, 0 1 to impose stationarity, is frequency in radians, t and t are mutually independent i:i:d N (0; 2 ) and i:i:d N (0; 2 ) respectively. The components z it and zit are de ned recursively as, zit z it cos sin = sin cos zit 1 z it 1 + zi 1;t 1 0 i = 2; 3; ::: (29) Here there are two di erences between (28) and (29) and the periodic model discussed earlier. First, in (28) the innovation t is set to zero. Second in (29) z i 1;t 1 is set to zero. 12

14 Following Harvey and Trimbur (2003) we can write where c (L) = z nt = [c (L)] n t 1 cos L 1 2 cos L + 2 L 2 Harvey and Trimbur (2003) de ne a m th order stochastic trend as follows. De nition 2 An unobserved component 1 is an m;t t= 1 mth order stochastic trend, for positive integers m, if 1;t = 1;t 1 + t i;t = i;t 1 + i 1;t ; i = 2; : : : ; m where the disturbance term t is serially uncorrelated with mean zero and constant variance 2 ; denoted hereafter by writing t~w N 0; 2 Harvey and Trimbur (2003) write the measurement equation associated with the UC model that has a m th order stochastic trend and a n th order cycle and measurement error " t as y t = m;t + z nt + " t ; " t ~W N 0; 2 " The low pass Wiener-Kolmogorov lter associated with this model is q GBm;n lp j1 Lj L; ; ; q ; q = 2m q + q j1 Lj 2m + 1 ; q = jc(l)j 2n 2 2 " ; and q = 2 2 " (30) Harvey and Timbur state that as m and n go to in nity GBm;n lp L; ; ; q ; q tends to an ideal low pass lter. The important point to make here is that the equivalence between certain UC models and low pass or band pass ltering arises because potentially testable assumptions are being imposed on the UC models. Testing those assumptions directly may prove to be di cult. Proposition 1 provides us with a simpler way to proceed. That is, estimate a parsimonious reduced form model that is not rejected by the data. Use that model to obtain the Beveridge-Nelson permanent-transitory decomposition. If that BN decomposition di ers substantially from the permanent transitory decomposition from the UC model or the band pass lter then we can conclude that the UC model implied by the band pass lter imposes over identifying restrictions that are not supported by the data. 13

15 6 Application The important implication for practitioners is that one does not need to use an UC model to implement a valid permanent transitory decomposition. All that one needs to do is nd an ARIMA model that is an adequate approximation to the DGP for y t and then apply (1), (2) and (3) to obtain the permanent-transitory decomposition. For example a reasonable approximation to the DGP of y t = ln(usgdp t ) is 2 y t = + y t 1 + " t So the permanent component is given by (31) X 1 P t = y t + E t j y t 1 j=1 And the transitory component is given by = y t + 1 y t (31) T t = 1 y t (32) The permanent and transitory components of US GDP are shown in Figure 1 where it is evident that almost all of the movement in that series is attributable to the permanent component. The transitory component of US GDP is shown in Figure 2. The facts that an AR (1) in growth rates ts US GDP well and that it implies the decomposition above is well known. But there are a variety of other approaches that yield very di erent permanent transitory decompositions and the question that has been asked is why should the decomposition shown in Figures 1 and 2 be chosen over the other decompositions? This paper gives the following answer to that question First, the AR (1) model is selected by standard signi cance testing so it is the most parsimonious model that is consistent with the data. Second we know from proposition 1 that all of the methods yield equivalent decompositions if the underlying model is held constant. Thus, the permanent transitory decompositions that di er from that obtained from (31) and (32) must be based on models that have a reduced form which is very di erent from an AR(1). This can only happen if those other methods involve the imposition of assumptions that are not supported by the data. 2 The estimated parameters are = 0: and = 0:3259: 14

16 Figure 1: Beveridge-Nelson permanent and transitory components of US GDP, 1947Q1 to 2008Q Ln(US GDP) ln(us GDP) Permanent component (BN) 7 Apr 47 Apr 52 Apr 57 Apr 62 Apr 67 Apr 72 Apr 77 Apr 82 Apr 87 Apr 92 Apr 97 Apr 02 Apr 07 Figure 2: Transitory component of US GDP, 1947Q1 to 2008Q BN Transitory component in ln(us GDP) Transitory component (BN) 0.02 Apr 47 Apr 52 Apr 57 Apr 62 Apr 67 Apr 72 Apr 77 Apr 82 Apr 87 Apr 92 Apr 97 Apr 02 Apr 07 15

17 7 Conclusion I have shown that permanent-transitory decompositions obtained from a wide range of methods are all equivalent if the underlying reduced form model is held constant. The implication from this is that practitioners can obtain valid permanent-transitory decompositions through the following steps. 1) Find a parsimonious reduced form model that is not rejected by the data. 2) Use that model to obtain the Beveridge-Nelson decomposition. 3) Justify the use of this as the permanent - transitory decomposition by arguing that any method which provides a markedly di erent permanent-transitory decomposition must involve restrictions on the reduced form model that are not supported by the data. 4) Observe that using a model that is more profligate with parameters will incur ine ciency but should not yield a markedly di erent permanent - transitory decomposition.. References Baxter, M. and R. King (1999), Measuring Business Cycles: Approximate Band-Pass Filters for Economic Time Series, Review of Economics and Statistics, 81, pp Beveridge, S. and C.R. Nelson (1981), "A new approach to the decomposition of economic time series into permanent and transitory components with particular attention to measurement of the business cycle", Journal of Monetary Economics, 7, Blanchard, O.J. and S. Fischer (1989), Lectures on Macroeconomics, M.I.T. Press Cambridge, MA, Canova, F. (1998), Detrending and business cycle facts, Journal of Monetary Economics, Volume 41, Issue 3, 20 May, Hamilton, J.D., (1989), A New Approach to the Economic Analysis of Non-Stationary Times Series and the Business Cycle, Econometrica, 57, pp Harvey, A.C., (1985), "Trends and cycles in macroeconomic time series", Journal of Business and Economic Statistics,3, Harvey, A.C., and A. Jaeger (1993). Detrending, Stylized Facts and the Business Cycle, Journal of Applied Econometrics, 8,. pp King, R.G. and S.T. Rebelo (1993), Low Frequency Filtering and Real Business Cycles, Journal of Economic Dynamics and Control, 17, Kaiser, R and A. Maravall (2001), "Measuring Business Cycles in Economic Time Series", Lecture Notes in Statistics 154, New York, Springer- Verlag 16

18 Morley, J.C. (2002), "A state-space approach to calculating the Beveridge- Nelson decomposition", Economics Letters 75, Morley, J.C., C.R. Nelson, and Zivot, E. (2003), "Why are the Beveridge_Nelson and Unobserved Components Decompositions of GDP so Different", The Review of Economics and Statistics, Vol LXXV, May, No 2, pp Pierce, D.A. (1979), Signal extraction error in nonstationary time series. Annals of Statistics, 7, pp Sargent, T.J. (1979), Macroeconomic Theory (Academic Press) 17