Modelling investment when relative prices are trending: Theory and evidence for the United Kingdom

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1 Modelling invesmen when relaive prices are rending: Theory and evidence for he Unied Kingdom Hasan Bakhshi Nicholas Oulon and Jamie Thompson 16 h January 2003 Corresponding auhor: Jamie.Thompson@bankofengland.co.uk, Bank of England, Threadneedle Sree, London UK EC2R 8AH The views are hose of he auhors and no necessarily hose of he Bank of England. We would like o hank Ian Bond, Colin Ellis, Jens Larsen, Rain Newon-Smih, Simon Price, Sally Srinivasan and an anonymous referee for heir helpful commens and suggesions. Thanks also o John Frain of he Cenral Bank of Ireland for kindly providing us wih he RATS code o run our inerpolaion procedures.

2 Conens Absrac 3 Summary 4 1. Inroducion 6 2. The heory 7 3. Sylised facs Economeric analysis Forecasing Conclusions Appendix 1: Consrucing he UK daase Appendix 2: Backing ou non-compuer invesmen from oal invesmen Appendix 3: Economeric resuls Appendix 4: Sensiiviy analysis 33 References 2

3 Absrac In recen work, Tevlin and Whelan (2002) argue ha aggregae economeric models fail o capure he US invesmen boom in plan and machinery in he second half of he 1990s, whereas a disaggregae approach does much beer. In paricular, hey show ha aggregae models do no capure he increase in replacemen invesmen associaed wih composiional shifs in he capial sock owards high depreciaion rae asses, such as compuers. And aggregae models invariably find lile or no role for he real user cos, so do no pick up he srong effecs of relaive price falls on invesmen in compuers. In his paper, a daase for he Unied Kingdom is consruced in order o invesigae he abiliy of differen equaions o accoun for he UK boom in plan and machinery invesmen in he second half of he 1990s. We repor similar findings o Tevlin and Whelan (2002). We exend Tevlin and Whelan s analysis in wo main ways. Firs, he failure of he aggregae equaions is explained more formally in erms of misspecificaion when relaive prices are rending. Second, he economeric analysis is conduced in a formal coinegraion framework. As in he Unied Saes, i is shown ha asse-level equaions can explain he invesmen boom in plan and machinery in he second half of he 1990s in he Unied Kingdom, whereas he aggregae equaion compleely fails. Keywords: invesmen, compuers, relaive prices JEL classificaion: C51, E22 3

4 Summary Recen research by Sacey Tevlin and Karl Whelan in he Unied Saes has shown ha aggregae economic models fail o explain he invesmen boom in real plan and machinery in he second half of he 1990s. In conras, a disaggregae modelling approach does much beer. This appears o reflec wo facors. Firs, aggregae models do no capure he increase in replacemen invesmen associaed wih composiional shifs in he capial sock owards shorer-lived asses, such as compuers. Second, aggregae models invariably find lile or no role for he real user cos of capial, so hey undersae he posiive effecs of falls in he relaive price of compuers on invesmen in compuers. The Unied Kingdom also experienced a boom in real plan and machinery invesmen in he second half of he 1990s. Bu underaking similar research is bese wih difficulies in he Unied Kingdom, no leas he relaive pauciy of disaggregae invesmen daa in he published Naional Accouns. In his paper, we carefully consruc a daase for he Unied Kingdom ha is consisen wih he Naional Accouns. We hen use his o invesigae he abiliy of differen invesmen equaions o accoun for he UK invesmen boom in plan and machinery. We repor similar resuls o Sacey Tevlin and Karl Whelan for he Unied Saes. In paricular, he radiional aggregae modelling approach compleely fails o explain he invesmen boom in plan and machinery in he second half of he 1990s. Our analysis consiss of wo main elemens: a heoreical secion seing ou he relaionship beween aggregae and disaggregae approaches o modelling invesmen; and an empirical analysis seing ou our economeric resuls. In our heoreical analysis, we firs derive he relaionship beween firms desired capial socks and he real user cos of capial ha is prediced by sandard economic heory. We show how ha relaionship breaks down in he presence of rend falls in he relaive price of invesmen goods. Such rends have been a paricularly imporan feaure of invesmen in recen years. In conras, we show ha well-specified relaionships exis a he disaggregae level. Our empirical exercise involves using ime-series coinegraion mehods o model invesmen a disaggregae and aggregae levels. We compare he abiliy of he wo approaches o explain he boom in plan and machinery invesmen. Recognising ha coinegraion echniques can have low power, paricularly in small samples, we furher evaluae he comparaive performance of he wo approaches by conducing ou of sample forecasing exercises. In all cases, our empirical resuls suppor he heoreically superior disaggregae modelling approach. Firs, composiional shifs in he capial sock owards shorer-lived compuer asses appear o have been imporan in he Unied Kingdom oo in he second half of he 1990s. Tha explains some, hough no all, of he inabiliy of he aggregae model o explain he invesmen boom. The second facor behind he srong invesmen growh has been falls in he relaive price of compuers. Echoing he finding of Sacey Tevlin and Karl Whelan for he Unied Saes, we find ha firms invesmen in compuers appears o be highly sensiive o falls in he real user cos for compuers. And ineresingly, our models sugges ha he increase in he size of firms 4

5 compuer capial socks in he second half of he 1990s are fully accouned for by he sharp falls in he real user cos for compuers. Given he grea uncerainies surrounding measures of he real user cos of capial and he price of invesmen goods in paricular, we invesigae he sensiiviy of our resuls o alernaive measures of he real user cos of capial. We find ha our resuls are reassuringly robus. In all hey provide srong suppor o aemps o model and forecas invesmen a he disaggregae level. 5

6 1.Inroducion Tradiional aemps o model invesmen using ime-series mehods have for he mos par been unsuccessful (1).. A bes, only a limied role is found for he real user cos of capial. And he overall fi of invesmen equaions is invariably poor. Tha is paricularly rue of he 1990s, when hese models failed o predic he rapid real invesmen growh in he Unied Kingdom and in he Unied Saes. A hos of explanaions have been pu forward in he lieraure for he failure of ime-series invesmen equaions. These include (a) mismeasuremen of srucural variables such as he real user cos or marginal q (2) ; (b) aggregaion biases (3) ; and (c) esimaion biases (4). In his paper we sugges ha a leas par of he difficuly arises because aggregae invesmen equaions do no hold when differences across asses are imporan. In paricular, he long run of invesmen equaions is ypically designed o saisfy he properies of a one-secor model. Bu ha model canno accoun for he persisen shifs in he relaive prices of differen invesmen goods ha we observe in he daa (5). We argue in secion 2 ha he aggregae relaionship beween capial, oupu and he real user cos of capial breaks down in he presence of such rends in relaive prices. Having esablished misspecificaion of he aggregae equaion, we hen illusrae is empirical imporance for he Unied Kingdom. In secion 3 we summarise he main properies of our daase for plan and machinery invesmen, including and excluding compuers. In secion 4 we follow he work of Tevlin and Whelan (2002) in he Unied Saes by esimaing separae imeseries economeric equaions for he compuer and non-compuer componens of UK plan and machinery invesmen. We exend Tevlin and Whelan s work by conducing our economeric (1) See, for example, Oliner, Rudebusch and Sichel (1995). (2) See, for example, Chirinko (1993). (3) See, for example, Caballero (1999), in which he auhor illusraes ha, in general, informaion abou he crosssecional disribuion of capial imbalances is needed o explain aggregae invesmen. This has moivaed aemps o model invesmen a he micro level. See, for example, Bond e al (2002). (4) Caballero (1994) suggesed ha radiional esimaion approaches downwardly bias esimaes of he elasiciy of he cos of capial wih respec o he capial-oupu raio. The poor performance of radiional ime-series invesmen equaions has also been aribued by some o facors such as financing consrains and he irreversibiliy of invesmen under condiions of uncerainy. (5) As discussed in OECD (2001a), he relaive price of capial goods in mos counries has rended down, a leas since he early 1980s. See also Bakhshi and Larsen (2001), for a discussion of UK developmens, and Whelan (2001), for a US perspecive. 6

7 analysis wihin an explici coinegraing framework. Given he inheren measuremen difficulies, we ake paricular care o invesigae he sensiiviy of our resuls o alernaive measures of he real user cos. Finally, in secion 5, we compare ou-of-sample forecass from he disaggregae equaions wih forecass generaed from he radiional, aggregae model of plan and machinery invesmen. Like Tevlin and Whelan for he US, we find ha our asse-level invesmen equaions can ogeher generae much more accurae forecass of plan and machinery invesmen compared wih he (misspecified) aggregae approach. 2. The heory Invesmen in he Unied Kingdom in recen decades has been characerised by persisen shifs in he relaive price of invesmen goods. Tha has manifesed iself in wo main ways. Firs, here has been a decline in he price of invesmen goods, such as plan and machinery, relaive o prices in he economy as a whole (see char 1) (6). Second, here have been shifs in he price of high-ech invesmen goods relaive o oher invesmen goods. Char 2 demonsraes ha compuer prices have fallen dramaically relaive o he price of non-compuer plan and machinery. Char 1 Char 2 Raio of plan and machinery invesmen good deflaor o GDP deflaor (in logs) Raio Raio of compuer prices o noncompuer plan and machinery prices (in logs) Raio In his secion we show how easy i is for an aggregae invesmen equaion o go wrong when he relaive prices of capial goods are changing. Le he producion funcion for he whole economy (or for a secor) ake he consan elasiciy of subsiuion (CES) form: (6) This has also been a feaure of invesmen for many oher indusrialised counries. See OECD (2001a). For a full discussion of he UK rends see Bakhshi and Thompson (2002). 7

8 Y 1/ a ( K ) a ( K ) a ( e h ), a, a, a 0, 1 d d e e l d e l (1) Here we assume ha oupu is produced by wo ypes of capial, compuer capial ( exciing, labelled subscrip e) and non-compuer capial ( dull, labelled subscrip d), and by labour hours (h). (7) The K i are he service flows from he wo ypes of capial. The rae of labour-augmening echnical progress is denoed by. Denoe he elasiciy of subsiuion by 1/(1 ) 0, noing ha in he Cobb-Douglas case 1. Equaing he marginal producs of capial o he real user coss (denoed by afer rearrangemen r, r d e ), we obain Ki / Y ( ai / ri ), i d, e (2) Taking logs and hen differeniaing wih respec o ime, k i y ( d ln ri / d), i d, e (3) where lower case leers denoe logs and a do denoes a ime derivaive. The real user cos is given by he Hall-Jorgenson formula 8 : ri [ i i ] Pi, i d, e (4) where P i is he asse price, measured relaive o he price of oupu, 8 is he real rae of reurn (ie he nominal rae minus he growh rae of he price of oupu), i is he depreciaion rae, and d ln P d is he growh rae of he relaive price of asse i. i i / (7) The inerpreaion of he producion funcion as being for he whole economy can be rigorously jusified in wo ways. Eiher we can assume ha he wo ypes of capial good are impored, in which case domesic oupu is of some hird good. Or we can assume ha he home economy produces one of he capial goods while he oher is impored. If he home economy produces more han one good, hen an aggregae producion funcion can only be jusified if some resricions are pu on echnology. This is wha he model of Greenwood e al (1997) does. 8 See Jorgenson (1963) and Hall and Jorgenson (1967). The user cos expression is derived under profi maximisaion using he capial accumulaion ideniy (equaion A1 in appendix 1) and he assumpion of no adjusmen coss. I is also assumed ha he rae of reurn is equaed across asses, as is required for equilibrium.

9 Now consider he problem of aggregaing equaion (3) across asse ypes. Theory suggess ha i is he flow of capial services, no he sock of capial, which belongs in he producion funcion, in jus he same way as hours worked are preferable o numbers employed as a measure of each ype of labour inpu. The correc way o aggregae he capial services of differen asse ypes is o use weighs based on he Hall-Jorgenson renal price formula. This amouns o calculaing he proporion of aggregae profi ha is generaed by flows from each asse ype. 9 Suppose we have an index (fixed-weigh or chain-linked) of he growh rae of aggregae capial services k wk d ( 1 w) k, 0 w 1 e (5) Plugging he capial services index ino equaions (3) we obain k y w( d ln r / d) (1 w)( d ln r / d) d e (6) This shows ha, if we wan o derive a relaionship for aggregae capial from he asse level relaionships (equaions (3)), we mus use he same paern of weighs o aggregae he growh of real user coss as we use o aggregae capial services. In pracice, his is no easy o do. The UK Naional Accouns do no immediaely provide us wih eiher aggregae capial services or an aggregae real user cos. Researchers have ofen employed aggregae measures of capial ha use weighs appropriae for a wealh raher han a services concep of capial. And hey have used measures of he aggregae real user cos ha differ from he heoreically correc measure in he las equaion. If asse prices are changing a very differen raes, i is unlikely ha he ypical aggregae measure of he real user cos of capial will be a good approximaion for he heoreically correc measure. We can see his even in he simplified case of a seady sae. In a seady sae, he growh of relaive prices, he depreciaion raes and he real rae of reurn are all consan. Hence d ln r / d d ln P d and plugging his ino he las equaion we ge i i / k y w( d ln P / d) (1 w)( d ln P / d) d e (7) 9 See OECD (2001b) and Oulon (2001b) for he heory of capial services measuremen. The alernaive aggregae measure is he aggregae capial sock, he wealh sock measure of aggregae capial. 9

10 The expression in square brackes looks like a price index for aggregae invesmen, bu i is no idenical o his laer concep since he weighs are no he same. The weigh w is he share of non-compuer capial in aggregae capial services. In a price index for invesmen, he corresponding share is he share of non-compuer invesmen in aggregae invesmen; hese shares are no he same. So if we use an aggregae index of capial services on he lef hand side and an aggregae price index for invesmen on he righ hand side, here will be an inconsisency and any economeric esimaes based on hese measures will be biased. The only case when he bias disappears is when he wo asse prices are growing a he same rae relaive o he price of oupu, ie relaive o each oher asse prices are consan, in which case he weighs do no maer. Oherwise, he ypical aggregae measure of he real user cos will be misleading. The derivaion of equaion (7) also assumes ha he elasiciy of subsiuion is he same for all inpus. If his assumpion is relaxed hen no measure of he aggregae real user cos derived from observable daa exiss. This provides a furher moivaion for considering disaggregaed invesmen equaions. In mos of our resuls, we mainain he CES assumpion. Bu we do also consider more general formulaions below. In pracice, researchers usually fi equaions for gross invesmen, no ne invesmen as in equaion (6), and his leads o furher difficulies. Expressing equaion (3) in erms of he ne invesmen rae,. i k and adding he depreciaion rae gives he gross invesmen rae: I / K y ( d ln r / d), i d e (8) i i i i, where I i is gross invesmen and i is he depreciaion rae for he ih ype of capial. Again, his can be urned ino an aggregae equaion by applying he same weighs o boh sides. Bu his is no he same as aggregaing gross invesmen and capial services separaely and hen dividing he one by he oher o obain I/K, since differen weighs would be normally used for numeraor and denominaor. So if a researcher akes aggregae invesmen and some measure of aggregae capial from he naional accouns, he raio of he wo aggregaes will no be appropriae for esimaing equaion (8). Noice oo ha he aggregae I/K raio is likely o drif up over ime if he paern of invesmen is shifing owards asses wih shor lives, since depreciaion will become increasingly imporan. 10

11 Hence i is a misake o rea he aggregae depreciaion rae as a consan when an aggregae version of equaion (8) is esimaed, a poin o which we reurn below. (10) These issues sugges here are imporan misspecificaion problems ha arise when esimaing aggregae invesmen equaions when relaive prices are rending. Tha poins o a disaggregae approach o modelling invesmen. 3 Sylised facs This secion considers he key feaures of our daase (deails of he consrucion of his daase are given in Appendices 1 and 2; summary saisics of he series are provided in able A3.8 of Appendix 3). Capial sock The capial sock as a share of GDP appears highly rended for aggregae plan and machinery and is compuer componen, bu less obviously so for non-compuer plan and machinery (char 3). Cos of capial Char 4 shows our esimaes of he real user cos of capial (as defined in equaion 4) for aggregae plan and machinery and, a he asse level, for compuers and non-compuers (11). In each case, here is evidence of a downward rend, alhough his is raher less pronounced in he non-compuer case. (10) See Whelan (2001). An example is he seady-sae growh pah of he wo-secor growh model where he relaive price of invesmen goods is rending down. In his case, under fixed weighs, he aggregae deprecaion rae is rending along he seady-sae pah and he aggregae invesmen equaion is misspecified. (11) Noe ha we proxy he real rae of reurn erm in equaion (4) wih a measure of he real cos of finance. The volailiy of our cos of capial measures in he earlier par of he sample reflecs he volailiy of his cos of finance measure, he raio of Privae Non-Financial Corporaions profis o he curren financial valuaion of he corporae secor. We use an alernaive, weighed average cos of capial measure in our sensiiviy analysis. The corresponding cos of capial measure is also smooher, bu alernaive cos of finance componens have no maerial impac on our economeric resuls. (See appendices 1 and 4 for furher deails.) 11

12 Char 3 Char 4 Capial-oupu raios (consan prices) Raio plan and machinery (rhs) compuers (lhs) Raios 2.0 non-compuers (rhs) Cos of capial Compuers (lhs) Plan and machinery (rhs) Non-compuers (rhs) Expressed in logs, hese esimaes are he sum of wo componens: he relaive price of capial; and a non-relaive price componen. The firs componen is simply he price of capial relaive o he price of oupu. The second componen conflaes he cos of finance, depreciaion, capial gain, and ax facor erms described above. Chars 5 and 6 show ha for our asses, a boh he aggregae and disaggregae level, he price and non-price elemens exhibi very differen persisence rends. The price series are rended, as he price of capial goods has fallen relaive o general oupu prices. This is paricularly rue for compuers where qualiy-adjused prices have fallen dramaically, less so for non-compuer plan and machinery. Bu here is lile obvious rend o he non-price componens. 12 (12) Alhough (our alernaive measures of) he cos of finance declined in he laer half of he 1990s, he non-price cos of capial acually rose. This reflecs declines in our capial gain erm over his period, which ended o raise he cos of capial. Noe ha he sharp movemens beween 1992 and 1994 in he non-price cos of capial series for noncompuer plan and machinery reflec sharp movemens in his asse s deflaor, mos noably in 1992 Q4 when i fell by some 9%. This causes a sharp fall in our capial gain erm and a sharp rise in he non-price cos of capial a his ime. Since we measure our capial gain erm as a railing eigh-quarer moving average, here is also a corresponding fall in he non-price cos of capial eigh quarers laer, as can be seen in char 6. As discussed in appendix 4, alernaive capial gain erms have no maerial impac on he resuls we repor in secion 4. 12

13 Char 5 Char 6 Relaive price of capial Logs 2.5 Logs Plan and machinery (rhs) Compuers (lhs) Non-compuers (rhs) Non-price cos of capial Logs Compuers Plan and machinery Non-compuers Depreciaion Char 7 shows ha he esimaed aggregae depreciaion rae for plan and machinery has risen quie sharply in recen years (13). And ha is consisen wih he US experience oo (14). Char 7 Implied real depreciaion raes for plan and machinery in UK and US (1990=100) 118 US UK Economeric analysis In secion 2, we argued ha he aggregae relaionship beween sandard measures of he capialoupu raio and he real user cos end o break down when relaive invesmen prices are rending. And we illusraed ha such rends have been an imporan feaure of UK invesmen in recen decades. Bu equaion 2, reproduced below as equaion 9 for ease of reference, provides an esimable long-run relaionship a he disaggregae level. (13) This assumes a geomeric depreciaion rae see Appendix 1 for deails. (14) US daa o 1997 kindly provided by Sacey Tevlin. These series are backed ou using fixed-weigh Naional Accouns daa and so are no subjec o he bias idenified by Tevlin and Whelan for chain-weighed daa in an earlier version of heir paper. For char 7, we rebased boh UK and US real depreciaion raes o 1990=

14 Ki / Y ( ai / ri ), i d, e (9) We firs consider he saionariy properies of our variables. Table 1 shows he resuls from formal uni roo ess. Table 1: Augmened Dickey-Fuller and Phillips-Perron uni roo ess Consan, no ime rend Consan and ime rend ADF(1) ADF(4) PP(1) PP(4) ADF(1) ADF(4) PP(1) PP(4) Aggregae k/y plan and k/y -3.6** ** -7.3** -3.6* ** -7.3** machinery r * -3.8* r -9.4** -4.2** -13.7** -14.7** -9.3** -4.2** -13.6** -14.6** Compuers k/y k/y -3.4* ** -4.0** * -4.0* r r -8.3** -4.2** -12.6** -13.2** -8.3** -4.3** -12.6** -13.4** Non-compuers k/y k/y -3.4* ** -7.2** -3.5* ** -7.2** r r -8.4** -3.6** -12.7** -13.0** -8.4** -3.7** -12.7** -13.1** Noes: 1. Aserisks signify null hypohesis of uni roo rejeced a 5% (* )or 1% (**) level. 2. ADF() and PP() indicae Augmened Dickey-Fuller and Phillips-Perron ess, where number in brackes is number of lagged firs difference erms in es regression. Alhough he resuls are somewha mixed, depending on he model-dgp combinaion, aken ogeher he ess broadly sugges ha he capial-oupu raio and he real user cos are nonsaionary a boh he aggregae and disaggregae level. We proceed on he assumpion ha he variables are all I (1). Our esimaion sraegy is wo-fold. Firs, we examine he relaionship beween he capial-oupu raio and he real user cos in a coinegraion framework, given he non-saionariy of hese variables. Evidence of coinegraion means ha eiher invesmen or capial dynamics can be described by an error correcion mechanism (ECM), wih acceleraor and dynamic user cos erms describing dynamics around a long-run equilibrium relaionship beween he capial-oupu raio and he real user cos. Given our ineres in gross invesmen, we focus on invesmen raher han capial dynamics Noe ha on he assumpion of non-zero depreciaion, i follows from he capial accumulaion ideniy ha invesmen and he capial sock are inegraed of he same order and coinegrae. 14

15 Second, we exploi any covariance beween he shocks driving compuer and non-compuer invesmen by esimaing he asse-level equaions using Seemingly Unrelaed Regression (SUR) echniques. Tha also allows us o es explicily he CES cross-equaion resricion ha he elasiciy of subsiuion beween capial inpus is equal in he wo equaions. We ake he significance of he ECM coefficien in our SUR equaions a he asse level o be evidence for a coinegraing relaionship, even hough he criical values on he -saisic will no in general have precisely he same disribuion as in Banerjee e al (1986) s analysis of single equaion coinegraion ess based on OLS (16) (17). Given he well-known low power of coinegraion ess in small samples, we also implemen he sysem-based Johansen coinegraion es (18) and he dynamic OLS mehod, as robusness checks on our resuls (19). Equaion (9) summarises he sysem we esimae using SUR echniques. Noe here is a crossequaion resricion on he elasiciy of subsiuion, being he same in he wo equaions, which we es in our empirical work. To faciliae comparison wih Tevlin and Whelan (2002) for he Unied Saes, we also repor resuls for heir specificaion (we call hese invesmen rae equaions (equaion 11) as disinc from our dynamic invesmen growh equaions (equaion (10)). These should in principle be inferior o our dynamic equaions as hey make no use of he informaion conained in he long-run relaionship beween he variables. i i 8 8 i ij gdp ik rcc i ( ki / y) rcc ], i = d, e (10) j 1 j k 1 i k [ 1 i i / ki 1 i ij gdp j ik rcci k, i = d, e (11) j 1 k 1 i (16) Single equaion ess are valid in his conex, as he invesmen equaions we esimae are bivariae and so here can a mos be one coinegraing vecor. And he regressors are always found o be weakly exogenous wih respec o he parameers of ineres. (17) For all our single equaions, we adoped a general-o-specific esimaion approach in order o obain a parsimonious dynamic specificaion. Specifically, we esed down from a general model conaining eigh lags of he dynamic erms, o a parsimonious dynamic specificaion conaining only erms significan a he 10% level. (18) In order o deermine he appropriae lag srucure for our VECMs in he Johansen es, we firs esimaed unresriced VARs over our full sample period (1978Q1-1999Q4). Lag lengh was hen deermined by lag lengh selecion crieria, subjec o he VAR passing serial correlaion ess. (19) Specifically, we implemened he dynamic OLS mehod suggesed by Saikkonen (1991) and Sock and Wason (1993) of adding dynamic leads and lags of regressors o a levels regression, in order o check he robusness of esimaed elasiciies. We also applied he Banerjee e al (1998) ECM version of his es, in which leads of he regressors are added o an ECM specificaion, as a furher es of coinegraion. We repor resuls from regressions including wo leads, bu resuls were no sensiive o number of leads chosen. 15

16 Of course, he esimaion resuls repored below are only as good as he daase underlying i. Given he absence of official esimaes in he UK for many of he series in our economeric analysis, we need o make a number of assumpions (discussed in Appendix 1). Tha leaves our resuls poenially open o he criicism ha hey reflec mismeasuremen of he variables of ineres. We go some way o address his in a horough sensiiviy analysis. In paricular, we invesigae he robusness of our resuls o a range of alernaive measures of he real user cos, o compuer price mismeasuremen and o differen inerpolaion echniques used o derive a quarerly invesmen daase. Appendix 4 shows how hese changes reassuringly have no maerial impac on our resuls. Aggregae plan and machinery In our equaion for plan and machinery, we find only weak evidence of coinegraion using boh single-equaion and sysem echniques (for full resuls see ables A1 and A2 in Appendix 3). In he single equaion case, he ECM coefficien has a -saisic of 2.2 (similar resuls were obained using he Banerjee e al (1998) dynamic OLS procedure). This -saisic is respecable compared o he usual suden- criical values, bu is low compared wih he criical values abulaed by Banerjee e al (1986). Using he Johansen approach, neiher he maximumeigenvalue es nor he race es indicae coinegraion a he 20% significance level (see able A7 in appendix 3) (20). Table 2 Esimaed ECM coefficien and elasiciy of subsiuion 1 Single-ECM equaion, Leas Squares 2 Single-ECM equaion, dynamic OLS 2 Long-run equaion, dynamic OLS Long-run equaion, Johansen Plan & machinery ECM (-2.16) (-2.18) - - elasiciy 0.32 (5.13) 0.32 (5.46) 0.28 (21.54) 0.78 (8.4) Compuers ECM (-3.78) (-3.69) - - elasiciy 1.33 (24.26) 1.33 (23.59) 1.35 (71.53) 1.33 (28.43) Non-compuer plan & machinery ECM (-2.04) (-2.15) - - elasiciy (-0.23) (-0.14) (-1.77) (-1.18) Noe: 1. T-saisics in parenheses; 2. For compuers and non-compuer plan & machinery, he firs wo columns refer o esimaes in individual asse-level equaions in SUR sysem. (20) We esimaed our sysem over five lags, given ha here was evidence of serial correlaion a four lags, he number suggesed by lag order selecion crieria. The equaion diagnosics were saisfacory. 16

17 Char 8 repors fied values agains acual ouurns of invesmen growh from he model. The esimaed response of invesmen o he real user cos of capial is small: is elasiciy wih respec o he capial-oupu raio is only -0.3 (21). (This is in line wih he esimaes from he dynamic OLS mehod, repored in able 2 above). The equaion performs reasonably for much of he sample, bu fails compleely o capure he srengh of invesmen in he second half of he 1990s, mirroring Tevlin and Whelan s (2002) finding for he Unied Saes. We find no role for he real user cos in he invesmen rae equaion, so in his case i collapses o a simple acceleraor model: see char 9. Char 8 Char 9 Invesmen growh equaion Annual growh, per cen 30 ouurns fied values Invesmen rae equaion Invesmen rae 0.06 ouurns fied values Tevlin and Whelan (2002) sugges wo main reasons why heir aggregae equaion fails o capure he srengh of plan and machinery invesmen in he Unied Saes in he second half of he 1990s. Firs, he aggregae equaion ignores increases in replacemen invesmen associaed wih composiional shifs in he capial sock owards compuers, which have a higher deprecaion rae han oher plan and machinery. Second, in finding no significan role for he real user cos of capial, he aggregae equaion misses he srong effec ha relaive price declines appear o have had on invesmen in compuers. We find ha hese wo facors are behind our resuls oo. Firs, following Tevlin and Whelan (2002) we esimae an alernaive version of he invesmen equaion where he dependen variable is ne invesmen (ha is gross invesmen ne of replacemen invesmen). Char 10 compares he residuals from he gross invesmen equaion wih he residuals from he ne invesmen equaion. Clearly he underpredicion in he ne equaion is much less marked, hough (unlike in Tevlin and Whelan (2002)) i is sill apparen. (21) If we drop he levels erms because of insufficien evidence of coinegraion, he real user cos of capial plays even less of a role: only one erm is significan a he 10% level in he equaion s dynamics, and ha erm is incorrecly signed. 17

18 Tha suggess ha composiional shifs in he capial sock owards higher depreciaion rae compuers can explain some, hough no all, of he puzzle. Char 10 Residuals from invesmen rae equaion Invesmen rae gross equaion ne equaion Second, an analysis of he fied values from he aggregae invesmen growh equaion in char 8 suggess ha he conribuion o invesmen growh in he second half of he 1990s from relaive price falls, hrough he dynamic user cos erms, is minimal. In paricular, according o ha equaion, he fall in he price of invesmen relaive o final oupu explains around only one-fifh of he 71% rise in invesmen over ha period. Tha is a leas suggesive ha an imporan source of invesmen growh, he rapid decline in compuer prices in char 2, is missing in he aggregae equaion. Compuers Tables A3 and A5 in Appendix 3 summarise he resuls from SUR esimaion of invesmen equaions for compuers and non-compuer plan and machinery separaely. For compuers, he ECM coefficien has a -saisic of 3.5 (and 3.7 in he Banerjee e al (1998) dynamic OLS case). Tha -saisic is high relaive o he usual suden -criical values, bu is modes compared wih he criical values abulaed by Banerjee e al (1986) in heir analysis of single equaions esimaed using OLS. In any case, he appropriae criical values would be sensiive o he paricular DGP-model combinaion in hand. And, as discussed in Maddala and Kim (1998), he Banerjee e al criical values do no ake ino accoun he fac ha a resricion ha he ECM coefficien equals zero also implies ha he coefficien on he real user cos of capial equals zero. For compuers, he Johansen coinegraion ess also provide some evidence for coinegraion 18

19 beween he capial-oupu raio and he real user cos (22). Trace and maximum eigenvalue ess poin owards coinegraion a he 20% level (see able A7 in appendix 3). Considered ogeher, we conclude from hese resuls ha here is some evidence of coinegraion in he case of compuers (23). As in Tevlin and Whelan (2002), he esimaed response of invesmen o he real user cos of capial for compuers is very high, wih an elasiciy of 1.3 (24). (This is robus o he alernaive dynamic OLS and Johansen esimaion approaches see able 2.) The equaion now broadly capures he paern of invesmen over he sample, hough sill sruggles o mach he precise dynamics and he 1998 boom in paricular. Consisen wih hese resuls, here is also a significan role for he real user cos of capial in he invesmen rae equaion (10) for compuers. The coefficiens on he dynamic real user cos of capial erms his ime sum o a much higher 3.3. Bu given he evidence above for a long-run equilibrium relaionship beween he real user cos and he sock of compuers, we would cerainly pu less weigh on an equaion ha makes no use of his informaion. Non-compuer plan and machinery In conras wih he resuls for compuers, here is lile evidence of coinegraion in he noncompuer case. In he SUR model, he ECM and cos of capial erms are insignifican in he coinegraing vecor; he cos of capial erm is also incorrecly signed. These resuls are echoed in he Johansen and dynamic OLS ess (see ables 5 and 6 in Appendix 3). While in he hree-lag sysem, he race es suppors coinegraion a he 20% level, he real user cos of capial is again incorrecly signed. Given he lack of evidence for coinegraion, we esimae a varian of he invesmen growh equaion (9) ha excludes he error correcion erm. In his case oo, here is lile role for real user cos erms (25). (22) We esimae our sysem over hree lags, given ha here was evidence of serial correlaion a wo lags, he number suggesed by lag order selecion crieria. Allowing for an oulier, he equaion diagnosics are saisfacory. (23) Noe ha he single equaion ECM approach is a robus mehod of esimaing a coinegraing relaionship only if he variables oher han he dependen variable are weakly exogenous wih respec o he parameers of ineres. To es ha, we esimae a VAR in k/y and r wih he esimaed long-run relaionship included as an exogenous variable. The esimaed long-run relaionship is only significan in he k/y equaion, indicaing ha r is indeed weakly exogenous. (24) Ineresingly, he esimaed long run suggess ha he acual capial-oupu raio was close o is equilibrium level in he lae-1990s. In oher words, our equaion suggess ha he rise in he compuer capial-oupu raio is fully accouned for by he sharp falls in he compuer cos of capial. (25) As in he aggregae case, we sill sruggle o find a role for he cos of capial when we exclude he ECM erm: only one erm is significan a he 10% level in he equaion s dynamics, and ha erm is posiively signed. 19

20 The negaive evidence of coinegraion for non-compuers appears robus o model specificaion: here is no role for he real user cos of capial in Tevlin and Whelan s invesmen rae specificaion. Though ineresingly, even for non-compuer plan and machinery, he underesimaion of he invesmen rae in he lae-1990s is much less pronounced han in he aggregae case. The CES assumpion: a diagnosic es Our earlier assumpion ha firms producion echnologies can be described by a CES producion funcion has he effec of imposing a cross-equaion resricion on our equaions for compuers and plan and machinery excluding compuers. In paricular, he elasiciy of subsiuion for boh ypes of capial should be equal. A LR es of his resricion on our equaions esimaed using SUR is no rejeced a convenional significance levels. Bu i is apparen ha ha resul reflecs he poorly deermined elasiciy of non-compuers wih respec o he real user cos. In absolue erms he esimaed elasiciy of subsiuion for compuers, a 1.3, is much greaer han ha esimaed for non-compuers (which is no significanly differen o zero). Tha resul again echoes Tevlin and Whelan s finding for he Unied Saes. Given he poorly deermined elasiciy ouside compuers, we offer wo poenial explanaions for boh he UK and US resuls. The firs, discussed by Tevlin and Whelan (2002), is consisen wih he CES producion funcion. They argue ha he esimaed elasiciy of subsiuion will be greaer for capial inpus where shocks o he real user cos are more persisen, compared wih shocks ha are emporary. Inuiively, profi-maximising firms will respond by a greaer amoun in heir invesmen decisions o shocks ha are perceived o be more persisen. Tevlin and Whelan (2002) furher argue ha shocks o he real user cos of capial for compuers are likely o reflec he long-run endency for echnological progress in he compuer secor o exceed ha in oher plan and machinery. In ha case reducions in he real user cos are likely o be driven by rend declines in he relaive price of compuers. We have seen ha he evidence for he UK is clearly also consisen wih ha explanaion: see Chars 5-6. An alernaive explanaion is ha he CES producion funcion iself is an inadequae descripion of he subsiuabiliy beween compuers and oher forms of plan and machinery. As a diagnosic on his hypohesis we experimened including he real user cos measure for compuers in a single-equaion for non-compuer plan and machinery. Ineresingly, ha variable is 20

21 saisically significan and posiive - an increase in he real user cos for compuers leads o an increase in demand for non-compuer capial. Including his variable also has dramaic implicaions for he equilibrium properies of he equaion, in ha here is now evidence for coinegraion: he -saisic on he ECM erm jumps o 4.2, and he non-compuer cos of capial erm is correcly signed and significan. Experimening wih producion echnologies ha are more flexible han he CES case is clearly an area worhy of furher research. 5 Forecasing Given he raher mixed evidence for coinegraion, even a he disaggregae level, a criical es of our analysis is wheher he disaggregae equaions can beer capure he srengh of invesmen in he second half of he 1990s when compared wih he aggregae equaion. We invesigae his by re-esimaing he equaions over he period 1978Q1 o 1994Q4, and generaing ou-of-sample forecass for he second half of he 1990s, condiional on acual ouurns for each of he explanaory variables. Chars 11 and 12 shows how he disaggregae model can now explain he invesmen boom of he laer half of he 1990s. Boh aggregae and disaggregae models sruggle o capure he precise dynamics of plan and machinery invesmen. Bu unlike he forecass from he aggregae equaions, he aggregae invesmen profiles implied by he disaggregae equaions are broadly in line wih he srong invesmen ouurns during his period. Char 11 Char 12 Ou-of-sample forecass from invesmen growh equaions Annual growh, per cen 30 ouurns disaggregae model -5 aggregae model Ou-of-sample forecass from invesmen rae equaions disaggregae model Invesmen rae 0.06 ouurns aggregae model

22 6 Conclusions In his paper we have argued ha radiional mehods of modelling aggregae invesmen are incapable of explaining he invesmen boom in he Unied Kingdom in he second half of he 1990s. The aggregae long-run relaionship beween he capial-oupu raio and he real user cos of capial breaks down in he presence of rending relaive invesmen good prices - and such rends have been an imporan feaure of invesmen in recen decades. In conras, well specified, esimable, long run relaionships exis a he disaggregae level. Empirical evidence appears o suppor his heoreically superior, disaggregae modelling approach, as Tevlin and Whelan (2002) find for he Unied Saes. Firs, composiional shifs in he plan and machinery capial sock owards compuers have led o increases in replacemen invesmen ha are no capured in aggregae invesmen equaions. Second, we find evidence of coinegraion beween he capial-oupu raio and he real user cos of capial in he compuers case, and he real user cos eners significanly and quaniaively more imporanly for compuers han for oher asses. The sharp falls in relaive compuer prices ha lie behind he persisen falls in he compuer real user cos of capial appear o have played an imporan role in he real invesmen boom of he laer half of he 1990s. Bu his role is hidden in he aggregae modelling approach, in which he real user cos of capial plays only a small role. Ou-of-sample forecass sugges ha our asse-level invesmen equaions can ogeher generae far more accurae forecass of plan and machinery invesmen compared wih he (misspecified) aggregae model. These findings suppor aemps o model, and forecas, invesmen a he asse level. 22

23 Appendix 1: Consrucing he UK daase Our daase consiss of quarerly series over 1978 Q Q4 for real invesmen, real capial sock and he real user cos of capial all calculaed separaely for aggregae plan and machinery, and a he asse level, for compuers and non-compuer plan and machinery. Our preferred oupu measure would be he value added of jus hose secors using plan and machinery. Bu ha is no available. We use insead GDP a consan (1995) prices (ONS alias code: ABMM). A brief discussion of how we consruced his daase is provided below; ONS alias codes are provided in brackes where relevan. Invesmen Our real plan and machinery invesmen series is published by he ONS in he Quarerly Naional Accouns as oher machinery and equipmen invesmen (DLWO). Deriving quarerly disaggregaed series is problemaic, as he ONS does no publish compuer invesmen daa on a quarerly basis. We follow he approach of Oulon (2001a), who consruced an annual, nominal compuer invesmen series from Inpu-Oupu Supply and Use ables for 1989 onwards; prior o 1989, he series is consruced from he various IO ables, wih missing years inerpolaed. We updae he esimaes in Oulon (2001a) using he supply and use ables consisen wih he 2001 Blue Book. We hen inerpolae hese annual daa o ge a quarerly series, using he saisical procedure described in Chow and Lin (1971). Tha involves using an indicaor variable o inform he quarerly profile of our known annual series. Given he high correlaion beween nominal compuer invesmen and plan and machinery invesmen a he annual frequency - growh raes have a correlaion coefficien of over 0.6 for our sample period - we use published, quarerly, nominal plan and machinery invesmen series as our indicaor (TLPW). Furher deails are provided in appendix 2. We hen deflae he (inerpolaed) quarerly, nominal compuer invesmen series by he corresponding quarerly, ONS producer price index (PQEK) o obain our real compuer invesmen series. The ONS does no publish consan price invesmen daa for he level of plan and machinery invesmen excluding compuers. And deriving his componen is non-rivial, given he nonaddiiviy of published ONS fixed-weigh series prior o The approach we ake is o 23

24 esimae he series ha he ONS would have arrived a, had hey decided o exclude compuers from plan and machinery invesmen. See appendix 2 for furher deails. Capial sock We sar by deriving annual, raher han quarerly, consan-price capial sock daa. Specifically, we use he sandard, consan-price capial accumulaion ideniy: K ( 1 ) K i 1 (A1) where he variables are: K I Consan-price capial sock Real depreciaion rae (assumed geomeric) Consan-price invesmen The real invesmen daa are he annual equivalen of he daa described above, daing back o 1948 for aggregae plan and machinery and o 1976 for compuers. For depreciaion raes, we follow Oulon (2001a) and use US depreciaion raes from Fraumeni (1997). For non-compuer plan and machinery, we assume a (consan) annual depreciaion rae of 13%. For compuers, we assume a (consan) depreciaion rae of 31.5%. The (ime-varying) real aggregae implied depreciaion rae for plan and machinery is a weighed average of hese raes, he weighs being heir consan-price shares in he aggregae capial sock in he previous period. For aggregae plan and machinery, he iniial sock in 1947 is provided by he ONS. This value is also used for non-compuer plan and machinery. For compuers, he iniial value in 1975 is from Oulon (2001a). The iniial socks for our quarerly series are hen aken from hese endyear annual observaions, and quarerly capial socks calculaed using he above capial accumulaion approach (26). Real user cos of capial We consruc Hall-Jorgensen real user cos of capial measures for aggregae plan and machinery and, a he asse level, for compuers and non-compuers. These measures ake he form: (26) For aggregae plan and machinery, he quarerly real invesmen series begins in 1965Q1, so he annual capial sock series provides he 1964 Q4 observaion. For compuers, he annual series provides he 1975 Q4 observaion. 24

25 P K RCC ( r E( PK / PY )) T (A2) P Y where he variables are: r Real cos of finance Depreciaion rae P k P y T The price of capial goods The price of oupu Tax facor E( P K / PY ) Expeced change in he relaive price of capial goods. For aggregae plan and machinery, he price of capial goods is he ONS deflaor implici in he published consan and curren price invesmen series (TLPW/DLWO). For compuers, we again use he relevan ONS PPI series. The price of oupu is measured by he GDP deflaor (ABML/ABMM). Given ha he expeced change in he relaive price of invesmen goods is unobservable, we use a proxy: an eigh-quarer average of he acual relaive price. The real cos of finance measure is he raio of Privae Non-Financial Corporaions profis (specifically, gross operaing surplus less ax and depreciaion) o he curren financial valuaion of he corporae secor (27). A he asse level, depreciaion raes are as discussed above. For aggregae plan and machinery, we follow he sandard approach and assume a consan depreciaion rae. A ax facor series, capuring he impac of axes and allowances on he cos of capial, is supplied by HMT. (27) This approach is aken in Flemming e al (1976). As discussed by Whiaker (1998), he real cos of finance is he rae a which a company s fuure real earnings are discouned by he capial marke in valuing he securiies upon which hose earnings will accrue. Assuming ha (unobservable) real earnings in fuure years are equal o curren earnings, a measure of he real cos of finance is herefore he raio of curren real earnings o he marke value of a firm s liabiliies. 25

26 Appendix 2: Backing ou non-compuer invesmen from oal invesmen This annex considers how o consruc he series ha he ONS would have arrived a, had hey decided o exclude compuer invesmen from oal invesmen in Oher machinery and equipmen (OME). Hence we aim o produce a series which is fully consisen wih he naional accouns. We have ONS daa on oal invesmen in Oher machinery and equipmen (OME) and also on a componen of OME, compuer invesmen, in boh consan and curren prices, for he period We wan o derive invesmen in OME excluding compuers (OMEXC). There is no problem in doing his in curren prices by simple subracion, bu how o do i in consan prices is no so sraighforward. The chain-linked soluion For he period 1994 o he presen, he ONS uses 1995 prices. So for his period we can indeed calculae OMEXC by subracing compuer invesmen in 1995 prices from oal OME in 1995 prices. Bu prior o 1994 he ONS used differen weighs: successively 1990, 1985, 1980 and 1975 prices as we go back in ime. In oher words he ONS does no use fixed base indices bu insead a ype of chain index in which he weighs are periodically updaed (abou every 5 years in pracice) (28). For each of he periods over which he weighs are consan, he index of OME invesmen is in effec consruced by he ONS as follows: QOME w QCOMP ( 1 w) QOMEXC (A3) where QOME is he index of oal invesmen, se equal o 1 in he base year, QCOMP is a similar index for compuer invesmen, QOMEXC is he index for oher plan and machinery, and w is he weigh for compuers. This weigh is he nominal share of compuer invesmen in he oal in he base year (successively 1975, 1980, 1985, 1990 and 1995). We can find he OME index for (say) 1984 relaive o 1985 by dividing OME invesmen in 1995 prices for 1984 by OME invesmen in 1995 prices for This works because rebasing o 1995 prices does no change growh raes for earlier periods. We can calculae he COMP index similarly. Therefore for each period covered by a singe base, we can solve his equaion for QOMEXC: (28) For his reason saying ha such indices are in 1995 prices or in consan prices is poenially misleading. I migh be beer o say ha hese series are in chained 1995 pounds (copying he BEA usage of chained 1996 dollars ). 26

27 QOMEXC [ QOME w QCOMP]/(1 w) (A4) We can hen link all hese fixed base index numbers ogeher, so ha we have a ype of chain index which covers he whole period. This chain index can be referenced o any year we choose, wihou changing is growh rae. Suppose we choose 1995 as he reference year when he index akes he value 1. Then we can muliply he chain index in each year by he nominal value of OMEXC in 1995, hus obaining OMEXC in consan 1995 prices. To illusrae he process, consider he following imaginary daa for an OMEXC index calculaed using equaion (A4). Here he base periods are assumed o be periods 1 and 4 and he link period is 3. Illusraive calculaion of chain index from sequence of fixed base indices Fixed base index Chain index Period Base: Period 1 Base: Period 4 Reference: period 1 Reference: period When he reference period for he chain index is period 4, he value of he index in eg period 2 is calculaed as ( ) x Non-addiiviy In general, chain indices are non-addiive: he componens do no necessarily sum o he oal. In oher words, if we add OMEXC in 1995 prices o COMP in 1995 prices, he resul will no be equal o OME in 1995 prices, excep for he period 1994 o he presen when he ONS has used 1995 as he base. If he componen (compuers) which is growing more rapidly has a falling relaive price, as is he case here, hen he ONS s chain index of OME grows more rapidly han he sum of he componens before he base year, here This is because he sum of he componens in 1995 prices is jus a fixed base index of invesmen, he base being I hen 27