Total Factor Productivity: An Unobserved Components Approach

Toal Facor Produciviy: An Unobserved Componens Approach Raul J. Crespo Discussion Paper No. 05/579 December 2005 Deparmen of Economics Universiy of Brisol 8 Woodland Road Brisol BS8 1TN

TOTAL FACTOR PRODUCTIVITY: AN UNOBSERVED COMPONENTS APPROACH Raul J. Crespo School of Economics, Finance & Managemen Deparmen of Economics Universiy of Brisol December, 2005

TOTAL FACTOR PRODUCTIVITY: AN UNOBSERVED COMPONENTS APPROACH Absrac This work examines he presence of unobserved componens in he ime series of Toal Facor Produciviy, which is an idea cenral o modern Macroeconomics. The main approaches in boh he sudy of economic growh and he sudy of business cycles rely on cerain properies of he differen componens of he ime series of Toal Facor Produciviy. In he sudy of economic growh, he Neoclassical growh model explains growh in erms of echnical progress as measured by he secular componen of Toal Facor Produciviy. While in he sudy of business cycles, he Real Business Cycle approach explains shor-run flucuaions in he economy as deermined by emporary movemens in he producion funcion, which are refleced by he cyclical componen of he ime series of he same variable. The economeric mehodology employed in he esimaion of hese differen componens is he srucural ime series approach developed by Harvey (1989), Harvey and Shephard (1993), and ohers. An applicaion o he ime series of Toal Facor Produciviy for he 1948-2002 U.S. privae non-farm business secor is presened. The paern described by echnical progress in his economy is characerised by imporan growh for he period immediaely afer War World II, which reaches is peak a he beginning of he 1960s o decline unil he earlies 1980s where i shows a modes rebound. On he oher hand, he cyclical componen of he series seems o be beer described by wo cycles wih periodiciy of six and welve years, respecively. Keywords: Produciviy, Business Cycles, Srucural Time Series Models, Unobserved Componens. JEL classificaion: E23, E32, C22 1

1. Inroducion The seminal work of Solow s (1957), which derives a mehodology o measure echnological progress, has been of major imporance in Macroeconomics. Firs, in he growh lieraure i has become he basis for an exensive heoreical body on growh accouning ha ries o quanify he sources of economic growh. Second, he main approach in he sudy of business cycles, he Real Business Cycle approach, assumes echnological innovaions (measured by Solow s procedure) as he main driving force of shor-run flucuaions in he economy, and employs i in he simulaions of quaniaive models. And hird, as i is believed ha echnological progress is an imporan source of economic growh many researchers have aemped o explain i as he endogenous oucome of economic decisions, which has served as he basis of a new body of lieraure on endogenous economic growh. Alhough he main approach in boh he sudy of economic growh and business cycles relies on he ime series behaviour of he same variable, echnological progress, heir ineres is focused on differen componens of he series. Hence, in he sudy of economic growh he aenion is cenred on he paern described by he non-saionary par of he series (which can keep seady, speed up or slow down), while in he sudy of business cycles, he ineres is on he saionary par of his series. This disincion is commonly ignored in he empirical esimaion of echnical progress, which someimes could have imporan effecs on our conclusions abou he paern displayed by he secular componen of he variable over ime. In his work he presence and characerisaion of unobserved componens in he ime series of Toal Facor Produciviy is examined. The srucure given o he paper is he following: in Secion 2 a brief descripion of he mehodology derived by Solow (1957) is presened, and some changes o he specificaion of he producion funcion are inroduced in order o give an explici accoun of he differen componens of he series in accordance wih he main approaches in he sudy of economic growh and he business cycle. In Secion 3 he economeric mehodology employed o ge he esimaes of he 2

differen componens of he ime series of echnological progress is described. Secion 4 shows he empirical resuls obained in he analysis of Toal Facor Produciviy in he U.S. economy under his mehodology. Finally, Secion 5 presens he conclusions of he paper. 2. Theoreical Background In he Growh Accouning lieraure, observed economic growh is pariioned ino componens associaed wih facor accumulaion and a residual ha reflecs echnical progress and oher elemens. This breakdown of he rae of growh of aggregae oupu ino differen componens has is foundaion in he pioneering work of Solow (1957). In his work, Solow derives a measure of echnical progress, and shows how o employ i o correc he esimaion of he producion funcion. He sars wih he Neoclassical producion funcion 1 Y () = FK ( (), L (), A ()) (2.1) where Y () is he flow of oupu produced a ime, K () is he physical capial sock accumulaed a ime, and L () is he labour inpu a ime. The producion funcion also depends on A (), he level of echnology, and he noaion makes explici ha i varies wih ime. Taking oal (logarihmic) differenial of equaion (2.1) and dividing hrough by Y yields, Y& Y FK. K K& FL. L L& = g +. +. Y K Y L (2.2) where given by F K and FL are he facor (social) marginal producs, and g (echnical progress) is 1 By Neoclassical producion funcion, we mean ha he funcion is concave, wice coninuously differeniable, saisfies he Inada (1964) condiions and ha boh facors are essenial in producion. 3

FA. A A& g. Y A (2.3) Solow assumed echnological change o be Hicks-Neural, so ha i could be facored ou of he producion funcion in he following way 2, Y () = AFK () ( (), L ()) (2.4) In his paricular case echnological change would be given by A& g = (2.5) A Equaion (2.2) suggess ha he rae of growh of real oupu can be decomposed ino he growh raes of capial and labour, weighed by heir oupu elasiciies, and he rae of growh of echnical progress. Consequenly, he rae of echnical progress can be obained from his equaion as a residual, Y& K& L& g = K. L (2.6) Y K L where K is he oupu elasiciy wih respec o capial and L is he oupu elasiciy wih respec o labour. In pracice, as hese elasiciies are no observable, o compue echnical change researchers usually assume ha each inpu is paid heir (social) marginal producs, so ha F K = r (he renal price of capial) and F L = w (he wage rae). This subsiuion allows he rae of change of echnical progress o be expressed in erms of observable income shares as 2 By assuming Hicks-Neural echnological change, as saed by Solow (1957, p. 312), shifs in he producion funcion leave marginal raes of subsiuion unouched bu simply increase or decrease he oupu aainable from given inpus. 4

Y& K& L& gˆ = sk. sl. (2.7) Y K L where sk and sl are he respecive shares of each facor paymen in oal oupu, and ĝ is ofen described as an esimae of Toal Facor Produciviy (TFP) or he Solow residual. Solow made i explici ha in applied work he residual would pick up any facor shifing he producion funcion. However, he labelled i echnical progress under he presumpion ha echnological change would be he main influence being capured by i. He found some ground for his asserion in his esimaes of he facor A () for he US economy, which showed a srong upward rend during he period 1909-1949. 3 The producion funcion specified by Solow (1957) o measure echnological progress is he same specificaion given o he producion funcion in he Solow-Swan model or Neoclassical model of economic growh. In his model he facor A () is inroduced in he producion funcion in order o enable he modelled economy o reproduce he observed paern of some macroeconomic variables ha regiser growh in per capia erms over he years. Therefore, he specificaion of he producion funcion is inended o pick up hose driving forces ha bring abou economic growh under he Neoclassical model of economic growh. I is imporan o noice, however, ha such a specificaion for he producion process does no provide an explici accoun of any oher forces ha drive shor-run flucuaions in he economy as hose ones claimed by he Real Business Cycle approach. From his perspecive, a more appropriae specificaion for he producion process seems o be one ha explicily disinguishes hose forces ha drive economic growh from hose associaed wih business cycles. 3 A negaive rend in A() would imply he unreasonable case of echnical regress, somehing ha would have discouraged Solow from wriing his paper (see, Solow 1957, p.316). 5

In modern Macroeconomics he producion funcion is specified in such erms ha i is allowed o pick up forces ha drive boh economic growh and business cycles, and i is described as follows Y ( ) = λ( ) F( K( ), L( ), A( )) (2.8) Here he producion process is similar o ha one specified in equaion (2.1) excep ha here is an explici accoun of emporary changes in he producion funcion hrough a random variable λ (), while secular improvemens in echnology are measured by A (). Hence, he producion funcion esablishes a clear disincion beween forces ha drive economic growh from hose ha drive shor-run flucuaions. 4 In he economic growh lieraure he specificaion given o he producion process ignores he erm λ (), while in he business cycle lieraure growh is omied or i is simply sared wih a ransformed economy. 5 Therefore, λ () and A () sand for processes whose driving forces are compleely differen, and consequenly hey require differen specificaions. In he business cycle lieraure λ () is commonly described as a saionary process, which displays considerable serial correlaion, wih firs-differences nearly serially uncorrelaed, while in he economic growh lieraure A () is usually specified as a non-saionary process ha can be expressed eiher as a rend-saionary process or a difference-saionary process. Even hough economiss have considered i appropriae o separae hese differen processes according o he subjec of sudy (i.e. economic growh or shor-run flucuaions), i seems clearly inappropriae o ignore hem in an empirical esimaion of echnological progress. For ha reason, if equaion (2.8) is employed and he same reasoning is carried ou as before, we arrive a an expression for TFP for he paricular case of Hicks-Neural echnological change given by 4 This specificaion is found in papers such as King, Plosser and Rebelo (1988) and King and Rebelo (1999). 5 In he analysis of business cycles, models wih seady sae growh are ransformed ino saionary economies. This ransformaion is inroduced o he Neoclassical growh model by scaling all he rending variables by he growh componen A (). 6

& λ gˆ = + λ A& Y& = A Y s K K&. K L& sl. L (2.9) Equaion (2.9) esablishes an explici disincion beween flucuaions of he producion funcion ha occur in he shor-run from hose of a more permanen naure such as echnological progress. This discrepancy beween TFP and changes in echnology, which is commonly ignored in he growh accouning lieraure, is he one ha will be addressed in his paper by employing he srucural ime series approach. 3. Economeric Mehodology The economeric mehodology employed in his paper is he srucural ime series approach developed by Harvey (1989) and Harvey and Shephard (1993), which builds on early work such as Nervole, Greher and Carlvalho (1979). The essence of his approach is o se up a model, which regards he observaion as being made up of a rend (or permanen) componen and an irregular (or emporary) componen. Consequenly, srucural ime series models are nohing more han regression models in which he explanaory variables are funcions of ime and he parameers are ime varying. The esimaion is conduced by seing he model in sae space form, wih he sae of he sysem represening he various unobserved componens. In he case of linear models, he Kalman filer is employed, which provides he means of updaing he sae as new observaions become available. 6 The simples srucural ime series model, usually referred o as he local level model, is given by a rend componen and an irregular erm, which is a whie noise process. The model can be wrien in he following way, y = µ + ε = 1, 2,... T (3.1) 6 A horough discussion of he mehodological and echnical ideas underlying his approach is found in Harvey, A. (1989). 7

where y is he observed value, µ is a rend and ε is a whie noise disurbance erm, ha is, a sequence of serially uncorrelaed random variables wih consan mean, in his case zero, and consan variance, σ. The rend componen, µ, may ake a variey of 2 ε forms, he simples being a level ha flucuaes up and down according o a random walk µ = µ 1 + η =... 1, 0, 1,... (3.2) where η is a whie noise disurbance wih variance 2 σ η, which is uncorrelaed wih he sochasic erm ε. No saring value needs o be specified for µ since i is assumed o have sared a some poin in he remoe pas. An alernaive specificaion for he rend componen is he following = µ 1 + β 1 η (3.3) µ + β = β 1 + ς =... 1, 0, 1, where η and ς are muually uncorrelaed whie noise disurbances wih zero means and 2 2 variances σ η and σ ς, respecively. Togeher, (3.1) and (3.3) form wha is ofen referred o as he local linear rend model. The effec of η is o allow he level of he rend o shif up and down, while ς allows he slope o change. The longer he variances he greaer are he sochasic movemens in he rend. We should noice ha he rend specificaion given in (3.3) ness differen processes such as, he random walk wih drif rend ( 2 2 2 σ = 0 ) and he deerminisic linear rend ( σ σ = 0 ). ς η = ς A cycle can be inroduced o (3.1) in order o formulae a model more in line wih economiss radiional view ha he movemens of an annually recorded ime series for a 8

macroeconomic variable are deermined by a rend componen, a cyclical componen and a noise componen. Formally, y = µ + ψ + ε = 1,..., T (3.4) where ψ is he cyclical componen ha is a funcion of ime, and he oher componens have been specified above. Modelling he cyclical process akes he form ψ cosλ = ρ * ψ sin λ sin λ ψ * cosλ ψ 1 1 ω + * ω (3.5) where σ 2 ω* ω and, respecively, and * ω are uncorrelaed whie noise disurbance erms wih variance 2 σ ω and * ψ appears by consrucion in order o form ψ. The disurbance erms make he cycle sochasic raher han deerminisic. The parameer 0 λ π is he frequency of he cycle, which is measured in radians. The period of a cycle corresponding o a frequency of λ is 2π / λ years. The coefficien 0 ρ 1 is a damping facor on he ampliude of he cycle. If 0 < ρ < 1 he process is a damped sine or cosine, wave. While if ρ = 1 he process is again a sine or cosine wave, bu no damping movemen is presen. A single equaion for ψ can be obained by wriing he model as (1 ρ cosλl) ω + ( ρ sin λl) ω * ψ = (3.6) 2 2 1 2ρ cos λl + ρ L where L is he lag operaor. Equaion (3.6) shows ha he process described by ψ is an ARMA(2,1), which becomes an AR(2) whenever σ 2 = 0. A final poin o noe is ha he sochasic cycle collapses o an AR(1) process when λ = 0 or π. ω In he model described by equaion (3.4) he cycle is inroduced by adding i o a rend componen and an irregular componen. Such a model is usually referred o as he 9

rend plus cycle model. An alernaive way of inroducing a cycle is by incorporaing i ino he rend. This specificaion is usually known as he cyclical rend model. In his case, rend and cycle are no separable, and he model can be formally wrien as y = µ + ε = 1, 2,... T (3.6) β = β 1 + ς =... 1, 0, 1, = 1 + 1 + 1 µ µ ψ β + η The rend plus cycle model (3.4) and he cyclical rend model (3.6) are he mos imporan formulaions of srucural ime series models ha exhibi cyclical process. 4. Empirical Resuls In his secion he empirical resuls of he paper will be presened. The ime series o be analysed is he widely cied measure of Toal Facor Produciviy for he U.S. economy produced by he Bureau of Labour Saisics (BLS). 7 Figure 1 shows he annually recorded TFP series in logarihmic erms for he period 1948-2002. 7 Series Id: MPU750023 (K) 10

Figure 1 Toal Facor Produciviy: US Privae Non-farm Business Secor 1948-2002 4.7 4.6 4.5 4.4 Log 4.3 4.2 4.1 4 3.9 3.8 1948 1952 1956 1960 1964 1968 1972 1976 Year 1980 1984 1988 1992 1996 2000 The series compued by he BLS uses for real oupu he naional accouning daa from he Bureau of Economic Analysis (BEA). The privae non-farm business secor includes all of gross domesic produc excep he oupu of general governmen, governmen enerprises, non-profi insiuions, he renal value of owner-occupied real esae, he oupu of paid employees of privae households, and farms from he privae business secor, bu includes agriculural services. The oupu index, which is supplied by BEA, is compued as chained superlaive index (Fisher Ideal Index) of componens of real oupu, and hen adjused by he BLS. Labour inpu is obained by Tornqvisaggregaion of he hours a work by all persons, classified by educaion, work experience, and gender wih weigh deermined by heir shares of labour compensaion. Finally, he capial inpu measures he services derived from he sock of physical asses and sofware. The asses included are fixed business equipmen, srucures, invenories and land. The BLS produces an aggregae inpu measure obained by Tornqvis aggregaion of he capial sock of each asse ype using esimaed renal prices. 8 8 More deailed informaion on mehods, limiaions, and daa sources is provided BLS Bullein 2178 (Sepember 1983), Trends in Mulifacor Produciviy, 1948-81. 11

The U.S. TFP series has been widely analysed and a growing body of research has emerged around i. Among he mos salien and well-known feaures of he series are he paerns of produciviy slowdowns afer 1973, which has been associaed by some researchers wih he oil price shocks of he 1970s, and rebounds afer 1995. Addiionally, i has been recognised ha TFP ends o move pro-cyclically; in periods of economic expansion, TFP is unusually large, while during recessions, i is low or even negaive. In he economic lieraure here are very few cases of an explici reamen of he presence of differen componens in he TFP series. An excepion o his is found in King and Rebelo (1999), where he produciviy series is specified in erms of wo componens; a rend which is assumed o be linear and deerminisic, and a cyclical componen which follows an firs-order auoregressive process, AR(1). Employing quarerly daa of TFP for he U.S. economy during he period 1947 (firs quarer) o 1996 (fourh quarer) hey fi a linear rend o he series, and hen use he residuals o esimae an AR(1) model he resuling poin esimae of he persisence parameer is 0.979. I is his decomposiion of he TFP series ha is addressed in his work, bu by employing a formal economeric mehodology in he specificaion process in order o ge esimaes of he differen componens of he series and o deermine heir main characerisics. In order o narrow down he number of suiable srucural ime series models for he U.S. TFP series some saisics have been compued, which provide addiional informaion in relaion o he main characerisics of he differen componens of he variable. In relaion o he rend of he variable, uni roo ess can provide a valuable insigh ino he presence of eiher a deerminisic or sochasic secular componen in he series. To deermine wheher or no he U.S. TFP series is characerised by having a uni roo in heir auoregressive represenaions, a modified Augmened Dickey-Fuller es (hereafer ADF-GLS τ ) developed by Ellio, Rohenberg and Sock (1996), which has difference-saionary [or I(1)] as he null hypohesis will be employed. An imporan 12

propery of his es is ha i has more power han he original ADF ess, and is approximaely uniformly mos power invarian. Similarly, a second es ha is a version of he Kwiakowski, Phillips, Schmid and Shin (1992) ess developed by Leybourne and McCabe (1994), which has rend-saionary [or I(0)] as he null hypohesis [hereafer KPSS(LM)] will be conduced. The KPSS(LM) resuls will be used o corroborae he informaion obained by applying he ADF-GLS τ es, and vice versa. Consequenly, if he ADF-GLS τ es rejecs he uni roo hypohesis and he KPSS(LM) es fail o rejec he saionary null hypohesis hen, hese resuls will be considered as srong evidence in favour of a rendsaionary process. By conras, if he ADF-GLS τ es fails o rejec he null hypohesis bu he KPSS(LM) rejecs i, we will consider his as srong evidence supporing he view of he presence of a difference-saionary process. If boh ess fail o rejec heir respecive null hypohesis hen, i will be considered ha he daa does no conain sufficien informaion o discriminae beween hese wo kinds of processes. 9 Null specific criical values for he ADF-GLS τ ess using a preferred differencesaionary specificaion following he approach specified by Cheung and Chinn (1997) have been generaed. 10 Similarly, for he KPSS(LM) ess null specific criical values using a preferred rend-saionary specificaion following he procedure suggesed in Leybourne and McCabe (1996) have been compued. 11 In Table 1 he ADF-GLS τ saisic and he KPSS(LM) saisic ogeher wih heir associaed 10%, 5% and 1% criical values for he U.S. TFP series are presened. 9 In cases where boh ess rejec heir respecive null hypohesis, as argued by Cheung and Chinn (1997), i migh be an indicaion ha he daa generaing mechanism is more complex han ha capured by sandard linear ime series models. 10 Cheung and Chinn (1997) generae null specific criical values using a seleced difference-saionary specificaion, which is chosen from models wih lag parameers p and q ranging from 0 o 5 using he BIC saisic. 11 Leybourned and McCabe (1996) generae null specific criical values by fiing an ARIMA (p,1,1) model 1/2 wih p se iniially a 5, and hen reducing i o 4 if he saisic z( p = 5) = T $ ϕθ $ <1.645, and so on. 5 Once he value of p has been deermined a preferred rend-saionary descripion is obained by reesimaing an ARIMA (p,0,0) model wih a ime rend. 13

Table 1 ADF-GLS τ and KPSS(LM) Tess: U.S. TFP (1948-2002) Saisic Acual Criical Values 10% 5% 1% ADF-GLS τ -1.2697-2.8583-3.1873-3.8360 KPSS(LM) 1.1635 0.8648 1.0005 1.1569 In he firs row of Table 1 he resuls obained from applying he ADF-GLS τ es is shown. I is possible o see ha he acual saisic is well below he rejecion area of he null hypohesis of a uni roo. Addiionally, in he second row of he able he resuls of he KPSS(LM) ess is presened. According o his resul here is a clear rejecion of he null hypohesis of a rend-saionary process as i is rejeced a a 1% significan level. Based on he resuls obained in boh he ADF-GLS τ ess and he KPSS(LM) ess we find srong evidence o disregard he possibiliy of having a deerminisic linear rend in he imes series of he TFP series for he U.S. economy. I is known ha uni roo ess are sensiive o he presence of srucural breaks in a series. Perron (1989) demonsraed ha when here are srucural changes in a series he sandard ess for uni roo hypohesis agains he rend-saionary alernaives are biased owards he non-rejecion of a uni roo. Considering his possibiliy srucural change ess following he mehodology suggesed by Perron (1997) have been conduced. Perron s echnique consiss of examining he likelihood of hree differen kinds of changes in he srucure of a series: one ha permis an exogenous change in he level of he series (Model A), one ha allows an exogenous change in he slope (Model B), and finally one ha considers changes in boh level and slope (Model C). 12 Table 2 shows he resuls obained by conducing srucural break ess on he ime series of he U.S. TFP. 12 Perron s (1997) mehodology involves esimaing he regressions for he hree models for all possible break poins, and selecing ha poin where he -saisic of he null hypohesis of a uni roo is he highes in absolue value. 14

Table 2 Srucural Break Tess: U.S. TFP (1948-2002) Model Time Break Saisic Criical Values 10% 5% 1% Model A 1962-4.232-4.92-5.23-5.92 Model B 1970-3.485-4.44-4.74-5.41 Model C 1962-4.011-5.29-5.59-6.32 The able above shows hose years in which he -saisics of he null hypohesis of a uni roo were found o be he highes in absolue value. For boh models, he one ha allows a change in level and he one ha allows a change in level and slope, he suggesed ime break was a he early 1960s, while for he model wih an exogenous change in slope he ime break was a he beginning of he 1970s. The criical values were obained from Perron s ables (1997) wih a sample size seleced according o he one ha is closes o he size of he series under sudy. As can be seen from he able he ess fail o rejec he null hypohesis of a uni roo a 10% significan level for all he specificaions. Consequenly, hese resuls seem o corroborae he absence of a deerminisic linear rend in he ime series of TFP in he U.S. economy. In order o evaluae he possibiliy of he presence of a cyclical componen in he U.S. TFP series some descripive saisics such as he correlogram and he power specrum can provide useful informaion. Figure 2 presens he esimaes of hese saisics for he series in firs-differences (i.e. he U.S. TFP rae of growh). 15

Figure 2 U.S. Toal Facor Produciviy (Firs-Differences): Correlogram and Power Specrum The correlogram shows small individual auocorrelaions no providing srong evidence of he presence of cyclical movemen in he series, alhough here seems o be some evidence of cyclical movemen buried wih noise. However, a much clearer message emerges from he examinaion of he power specrum, which shows wha appears o be a cycle wih a period beween 6 o 7 years, and he possibiliy of addiional cyclical movemens. 13 Based on he informaion gahered by conducing uni roo ess and he descripive saisics employed o evaluae he presence of cyclical movemens in he series, some likely specificaion for he rend and he cyclical componens of a srucural ime series model for he daa have been esimaed. 14 Table 3 shows some basic diagnosic and goodness-of-fi saisics for hese differen srucural ime series models. 13 On his graph he period is obained as 2 divided by he frequency. 14 Srucural ime series models were esimaed using he economeric sofware Samp 5.0. 16

Table 3 U.S. Toal Facor Produciviy Srucural Time Series Models Diagnosics and Goodness-of-Fi Saisics Model Log-Lik. P.E.V. H(h) Q(p,q) RSQ AIC BIC Random Walk wih Drif 211.03 3.07E-4 0.323 9.874 0.063 4.26E-4 5.91E-4 Smooh Trend 213.09 2.84E-4 0.332 7.848 0.132 3.94E-4 5.48E-4 Local Linear Trend 213.11 2.78E-4 0.321 8.475 0.151 4.00E-4 5.76E-4 Q(p,q) is Box-Ljung saisics based on firs p residual auocorrelaions and 6 degrees of freedom. H(h) is a heeroskedasiciy es wih 17,17 degrees of freedom. An aserisk indicaes a significan value a 5% level. All hese models assume he presence of a rend, wo cycles and an irregular componen. The able shows diagnosic and goodness-of-fi saisics for hree srucural ime series models wih differen specificaions for he rend or secular componen of he series. The firs saisical specificaion assumes ha he rend componen follows a random walk wih drif, which is specified by employing equaion (3.3) and a deerminisic slope (i.e. σ 2 = 0 ). The second saisical specificaion for he long-run ς componen is a varian of he local linear rend model, which inroduces a somewha smooher rend by employing equaion (3.3) wih a deerminisic level (i.e. σ 2 = 0 ) and a sochasic slope. Finally, he las specificaion for he long-run componen is he local linear rend model, which sipulaes he level and he slope o be sochasic (i.e. equaion 3.3). η Diagnosic checking ess are conduced by compuing he Box-Ljung Q(p,q) saisic for serial correlaion, which is based on he firs p residual auocorrelaions and esed agains a χ 2 disribuion wih q (i.e. p + 1 minus he number of esimaed parameers) degree of freedom. A simple diagnosic es for heeroskedasiciy H(h), which is he raio of he squares of he las h residuals o he squares of he firs h residuals, where h is se o he closer ineger of T/3. This saisic is compared wih he appropriae significan poin of an F disribuion wih (h,h) degrees of freedom. The Predicion Error Variance (P.E.V.), he coefficien of deerminaion ( R ) and he informaion crieria (Akaike Informaion Crierion, AIC, and Bayesian Informaion Crierion, BIC) provide he goodness-of-fi saisics. The Predicion Error Variance is he 2 D 17

variance of he one-sep-ahead predicion errors in he seady sae. These saisics have been employed o compue he informaion crieria, which are he appropriae saisics o compare models ha have differen numbers of parameers. 15 The coefficien of deerminaion, R, is he saisic recommended by Harvey (1989, chaper 5), which 2 D enables he fi of he esimaed model o be compared direcly o a random walk wih drif. For 0 R 2 1 he model is giving a beer fi han he random walk wih drif; for < D R 2 D = 0 he fi is he same; while for R 2 D < 0 he fi of he model is worse han he random walk wih drif. Table 1 also presens informaion relaed o he Log-Likelihood. The srucural ime series model ha regisers beer goodness-of-fi based on boh informaion crieria is he smooh rend plus cycle and irregular componens. The diagnosic ess of his model indicae ha he fi is fine. Figure 3 shows he differen componens of he srucural ime series model for he TFP series of he U.S. economy. 15 The informaion crieria have been compued using he procedure suggesed in Harvey (1989), pp.269-270. 18

Figure 3 U.S. TFP Unobserved Componens 1948-2002 US TFP and Trend:1948-2002 TFP Slope 4.8 4.6 4.4 4.2 4 3.8 3.6 1948 1954 1960 1966 1972 1978 1984 1990 1996 2002 TFP Trend 0.025 0.02 0.015 0.01 0.005 0 1948 1954 1960 1966 1972 1978 1984 1990 1996 2002 Cyclical Componen Irregular Componen 0.04 0.03 0.02 0.01 0-0.01 1948 1954 1960 1966 1972 1978 1984 1990 1996 2002-0.02-0.03-0.04 0.015 0.01 0.005 0 1948 1954 1960 1966 1972 1978 1984 1990 1996 2002-0.005-0.01-0.015 Cycle 1: Period 6.42 years Cycle 2: Period 11.74 years 0.025 0.02 0.015 0.01 0.005 0-0.0051948 1954 1960 1966 1972 1978 1984 1990 1996 2002-0.01-0.015-0.02-0.025-0.03 0.015 0.01 0.005 0 1948 1954 1960 1966 1972 1978 1984 1990 1996 2002-0.005-0.01-0.015 From he figure above i can be seen how he secular componen of echnological progress has evolved over he years. The esimaes of his componen sugges ha echnological progress slows down in he U.S. economy long before he oil price shocks of he 1970s. Technological progress seems o have reached a peak a he beginning of he 1960s when i sars o slow down unil early 1980s o rebound hen afer. The esimaed sandard error of he disurbances driving he slope ( σˆ ) is 0.0026. For he cyclical componen he model suggess he presence of wo cycles wih frequencies ς 19

λ 1 = 0.979 (6.42 years period) and λ 2 = 0. 535(11.74 years period). The esimae of ρ for he firs cycle is 0.810, while for he second cycle i is 0.998, which is very close o 1 indicaing he presence of a deerminisic cycle. The esimaed sandard errors of he disurbances driving hese wo cycles are 0.0068 and 0.0006, respecively. Finally, he irregular componen seems o be he mos volaile par of he model wih an esimaed sandard deviaion of 0.0075. An imporan issue o address a his sage is o compare he resuls obained in he sudy of he U.S. TFP series wih hose of he U.S. real oupu series. If business cycles are mainly driven by shor-run flucuaions in he producion funcion, as i is claimed by he Real Business Cycles approach, hen we should expec close similariies beween he cyclical movemens shown by he TFP series wih hose shown by he real oupu series. Similarly, if he secular componen of he TFP series drives economic growh, hen i should be found ha boh he TFP series and he real oupu (per labour) series share a single common rend. In order o compue he correlogram and he specrum of he real oupu series i is imporan o deermine he main characerisic of he rend o conduc he proper de-rending procedure. In Table 4 he ADF-GLS τ saisic and he KPSS(LM) saisic ogeher wih heir associaed 10%, 5% and 1% criical values for he U.S. real oupu series are presened. 16 Table 4 ADF-GLS τ and KPSS(LM) Tess: U.S. Real Oupu (1948-2002) Saisic Acual Criical Values 10% 5% 1% ADF-GLS τ -2.8717-2.8544-3.1386-3.7726 KPSS(LM) 0.0568 0.1674 0.3536 0.6278 In he able above i is possible o observe ha he ADF-GLS τ saisic is below he rejecion area as i is no possible o rejec he null hypohesis of he presence of a uni roo in he real oupu series a 5% significan level. Addiionally, he resuls obained by conducing he KPSS(LM) es does no allow he rejecion of he null hypohesis of a 16 The real oupu series is he same employed by BLS in he compuaion of he U.S. TFP series. 20

rend saionary process eiher. Therefore, we should conclude ha for he ime series of real oupu in he U.S. economy he daa does no conain sufficien informaion o discriminae beween a difference-saionary process and a rend-saionary process. As in he U.S. TFP series, srucural change ess have been conduced on he U.S. real oupu series. Table 5 shows he resuls obained from hese ess. Table 5 Srucural Break Tess: U.S. Real Oupu (1948-2002) Model Time Break Saisic Criical Values 10% 5% 1% Model A 1962-4.358-4.92-5.23-5.92 Model B 1971-3.554-4.44-4.74-5.41 Model C 1962-4.380-5.29-5.59-6.32 Ineresingly, he resuls shown by he able above indicae likely ime breaks similar o hose obained in he examinaion of he U.S. TFP series. However, as in he case of he U.S. TFP series, he ess fail o rejec he null hypohesis of a uni roo a 10% significan level for all possible specificaions. Based on he previous resuls i is necessary o esablish an assumpion in relaion o he kind of process described by he rend of he series in order o render saionariy in he series and compue boh he correlogram and he specrum. In Figure 4 esimaes of hese descripive saisics for he de-rended U.S. real oupu series under he assumpion of a rend-saionary process are shown. 21

Figure 4 U.S. Real Oupu (Linear De-Trending): Correlogram and Power Specrum The informaion provided by he correlogram shows clear cyclical movemens in he saionary componen of he series. The daa generaing mechanism seems o be ha of a second order auoregressive process, AR(2), wih complex roos. Neverheless, he message given by he power specrum suggess he presence of a cycle wih a very long period ( λ is close o cero), which is no in accordance wih he evidence of cyclical flucuaions observed in he economy. By conras, under he assumpion of a differencesaionary process for he U.S. oupu series he resuls are more in accordance wih he empirical evidence on business cycles. In Figure 5 he correlogram and he power specrum for he U.S. real oupu growh are shown. 22

Figure 5 U.S. Real Oupu (Firs-Differences): Correlogram and Power Specrum The figure above shows he correlogram and power specrum for he firs-differences of he U.S. real oupu (i.e. he growh rae of real oupu). Similarly o he case of he TFP series, he auocorrelaions are small providing weak evidence of cyclical movemen in he series. However, an examinaion of he specrum indicaes a clear cycle wih a period beween 5 o 6 years, and he possibiliy of an addiional cycle of longer periodiciy. I is ineresing o noice he close similariy beween he power specrum of he firsdifferences of TFP and he one obained for he real oupu series. Based on hese resuls, i seems reasonable o disregard he presence of a deerminisic linear rend in he U.S. real oupu series. Table 6 shows some basic diagnosic and goodness-of-fi saisics for suiable srucural ime series models for he U.S. real oupu series. Table 6 U.S. Real Oupu Srucural Time Series Models Diagnosics and Goodness-of-Fi Saisics Model Log-Lik. P.E.V. H(h) Q(p,q) RSQ AIC BIC Random Walk wih Drif 188.28 6.64E-4 0.117 6.356 0.256 9.21E-4 12.8E-4 Smooh Trend 186.51 6.91E-4 0.201 6.279 0.225 9.59E-4 13.3E-4 Local Linear Trend 188.28 6.64E-4 0.117 6.364 0.256 9.55E-4 13.8E-4 Q(p,q) is Box-Ljung saisics based on firs p residual auocorrelaions and 6 degrees of freedom. H(h) is a heeroskedasiciy es wih 17,17 degrees of freedom. An aserisk indicaes a significan value a 5% level. As in he case of he U.S. TFP series all hese models assume he presence of a rend, wo cycles and an irregular componen. The srucural ime series model wih he 23

bes goodness-of-fi based on boh informaion crieria is he random walk wih drif plus cycle and irregular componens. The diagnosic ess indicae no problem wih he fi of he model. Figure 6 displays he differen componens of he srucural ime series model for he real oupu series of he U.S. economy. Figure 6 U.S. Real Oupu Unobserved Componens (1948-2002) US Oupu and Trend: 1948-2002 Slope Log 4.9 4.4 3.9 3.4 2.9 2.4 1948 1954 1960 1966 1972 1978 1984 1990 1996 2002 Oupu Trend 0.036 0.0358 0.0356 0.0354 0.0352 0.035 0.0348 0.0346 0.0344 0.0342 0.034 1948 1954 1960 1966 1972 1978 1984 1990 1996 2002 Cyclical Componen Irregular Componen 0.06 0.04 0.02 0 1948 1954 1960 1966 1972 1978 1984 1990 1996 2002-0.02-0.04-0.06 0.015 0.01 0.005 0 1948 1954 1960 1966 1972 1978 1984 1990 1996 2002-0.005-0.01-0.015 Cycle 1: Period 5.75 years Cycle 2: Period 11.14 years 0.05 0.04 0.03 0.02 0.01 0-0.011948 1954 1960 1966 1972 1978 1984 1990 1996 2002-0.02-0.03-0.04 0.03 0.02 0.01 0 1948 1954 1960 1966 1972 1978 1984 1990 1996 2002-0.01-0.02-0.03 Figure 6 shows he significan differences ha exis beween he long-run componens of he TFP series and he real oupu series of he U.S. economy. For he 24

laer he rend is beer described as a random walk wih a drif of 0.036. The sandard error of he disurbances of he level ( σ η ) is 0.0149 making his componen he mos volaile par of he model. The cyclical componen, on he oher hand, shows srong similariies wih hose found for he U.S. TFP series. The model suggess he presence of wo cycles wih frequencies λ 1 = 1. 093 (5.75 years period) and λ 2 = 0. 564 (11.14 years period). The esimae of ρ for he firs cycle is 0.817, while for he second cycle i is 1, indicaing he presence of a deerminisic cycle. The esimaed sandard deviaion of he disurbances for he firs cycle is 0.0103. The correlaion beween he cyclical componen of he U.S. TFP and he cyclical componen of he real oupu series is 0.86. Finally, he irregular componen shows an esimaed sandard error of 0.0095. In order o evaluae he exisence of a single common rend beween he ime series of TFP and real oupu (per labour) for he U.S. economy, as suggesed by he Neoclassical growh model, coinegraion ess have been conduced. 17 The economeric invesigaion of his opic is based on he concep of coinegraion inroduced by Engle and Granger (1987). Is aim is o deermine he number and shape of saionary linear combinaions -named coinegraing relaions- of ime series which are hemselves nonsaionary. In order o conduc he coinegraion ess he mehodology developed by Johansen (1988, 1991, 1995) will be employed, which is based on maximum-likelihood esimaion wihin a Gaussian vecor auoregression. Table 5 shows he resuls of applying Johansen coinegraion ess for he series under sudy. λ race λ max Table 5 Johansen Coinegraion Tess Null Alernaive Saisic 95% C.V. 90% C.V r = 0 r 1 21.03 25.77 23.08 r 1 r 2 5.03 12.39 10.55 r = 0 r = 1 16.00 19.22 17.18 r 1 r = 2 5.03 12.39 10.55 The eigenvalues in descending order are 0.26058 and 0.090453. Superscrips * indicaes ha he es saisic is significan a 10%. 17 The BLS series Id number is MPU750021. 25

The unresriced vecor auoregressive (VAR) model for he variables in level was se wih wo lags as suggesed by he BIC. The diagnosic ess for his model did no show problems of auocorrelaion, heeroscedasiciy or normaliy in he residuals. The specificaion given o he deerminisic componens of he model was ha of unresriced inerceps and resriced rend in he coinegraion space. The resuls show ha boh saisics, he λ race and he λ max saisic, fall in he non-rejecion area of he nullhypohesis of no coinegraion. Consequenly, he resuls obained do no provide evidence of he presence of a single common rend for he series of TFP and real oupu of he U.S. economy as i is suggesed by economic heory. Alhough, i should be said ha boh saisics are relaively close o he 10% significan level suggesing he presence of one coinegraing relaion. 5. Conclusions In his work he presence of unobserved componens in he ime series of Toal Facor Produciviy is considered. This idea is cenral o modern Macroeconomics as he main approach in boh he sudy of economic growh and he business cycle relies on cerain feaures of he differen componens belonging o he ime series of his variable. The economeric mehodology employed in order o ge he esimaes of he differen componens of Toal Facor Produciviy is he srucural ime series approach developed by Harvey (1989) and Harvey and Shephard (1993) ha build on early works such as Nervole, Greher and Carlvalho (1979). In he examinaion of he 1948-2002 annually recorded U.S. Toal Facor Produciviy series compued by he Bureau of Labour Saisics he resuls indicae he presence of differen unobserved componens (i.e. rend, cycle and irregular componen) as economic heory suggess. The secular componen of he series seems o be beer represened as a smooh rend, ha is, a process given by a deerminisic level and a sochasic slope. The esimaes of his componen sugges ha echnical progress in he U.S. economy reached a peak a he beginning of he 1960s when i sared o decline unil he early 1980s, o rebound aferward. This resul conradics he idea ha 26

echnology in he U.S. economy slowed down in he 1970s as a resul of he oil price shocks during his decade. Similarly, evidence supporing he view of he presence of a deerminisic linear rend as i is someimes assumed in he business cycles lieraure was srongly rejeced. In relaion o he cyclical componen of he series, i seems o be bes represened by wo cycles wih a period of 6.42 years and 11.74 years, respecively. The resuls obained in he analysis of he Toal Facor Produciviy series were compared wih hose obained from a similar analysis of he U.S. real oupu series. Economic heory suggess ha boh he secular componen of Toal Facor Produciviy and real oupu (per labour) should be he same. In addiion, if shifs in he producion funcion are he main driving forces generaing shor-run flucuaions in he economy, hen he cyclical componens of Toal Facor Produciviy and real oupu should share some of heir main characerisics. The empirical resuls for he U.S. economy seem o sugges differen secular componens for he wo series, as here is no evidence of he exisence of coinegraion beween he series, alhough i should be menioned ha he acual saisics are relaively close o he 10% criical values. By conras, he resuls were more in accordance wih economic heory for he case relaed o shor-run flucuaions. The cyclical componen of he U.S. real oupu is beer represened by wo cyclical movemens wih periods 5.75 years and 11 years, respecively. Consequenly, i has been found ha he periodiciy of he cyclical componen of he wo series is very similar one o anoher. 27

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