Estimating the Output Gap: A Kalman filter approach 1

Transcription

1 Esimaing he Oupu Gap: A Kalman filer approach Draf version, please do no quoe. 3 June 4 Özer Karagedikli and L. Chrisopher Planier The oupu gap plays a crucial role in he hinking of many inflaion argeing cenral banks. A he same ime, real ime esimaes of he oupu gap undergo subsanial revisions as more daa become available. In his paper, we employ a sae space framework o augmen he simple Hodrick-Presco filer wih addiional srucural equaions o i) reduce he size of revisions and ii) reduce he uncerainy around he mean real ime oupu gap esimaes. In paricular we use he relaionship beween oupu and he capaciy uilisaion and oupu and unemploymen o infer he unobserved poenial oupu. Our echnique is demonsraed on daa for New Zealand, bu is applicable elsewhere as well. JEL: C3, E3 Key Words: Oupu Gap, Kalman filer, New Zealand Özer Karagedikli is wih he Economics Deparmen, Reserve Bank of New Zealand and Chris Planier is wih he Inernaional Affairs of he US Treasury. The views expressed here are he views of he auhors and do no necessarily represen he views of heir employers. We would like o hank Olivier Basdevan, Nils Björksen, Tim Hampon, Angela Huang, Saish Ranchhod and he seminar paricipans a he Reserve Bank of New Zealand and he Universiy of Oago. Corresponding auhor Özer Karagedikli. karagediklio@rbnz.gov.nz

2 Inroducion The oupu gap is defined as deviaions of acual oupu from poenial oupu, where poenial oupu is he level of oupu ha is consisen wih he producive capaciy of he economy. Ye his poenial oupu concep is no observed. Like many oher cenral banks, he Reserve Bank of New Zealand also relies on his unobserved measure in is policy making. Black, Cassino, Drew, Hansen, Hun, Rose and Sco (997) show ha in he Forecasing and he Policy Sysem (FPS), he Reserve Bank of New Zealand s main macroeconomic model, domesic inflaion is deermined wihin a Phillips curve framework using he oupu gap. In his paper, we esimaed an augmened Hodrick-Presco filer, proposed by Laxon and Telow (99) bu wih he Kalman filer. Because his approach encompasses he Reserve Bank of New Zealand s mulivariae filer, in his way, we are able o es he sensiiviies of he esimaes o differen assumpions. We demonsrae ha, here is a considerable uncerainy around he oupu gap esimaes. In addiion, we demonsrae ha, differen models, based on differen assumpions, can be presened in a hick modelling sense. The remainder of he paper is srucured as follows: Secion discusses he relaed lieraure. Secion 3 discusses he HPMV filer in general ( a la Laxon and Telow, LT hereafer) and in sae space. Secion 4 discusses he advanages of he Kalman filer approach over he ieraive LT approach. Secion 5 presens he resuls of our approach under differen assumpions and rend specificaions. Secion 6 briefly alks abou he revision properies of he oupu gap esimaes from our models. Secion 7 concludes. Lieraure review I has been widely discussed ha he cenral banks are no good predicors of he oupu gap in real ime. Wha seems a perfecly good predicor of inflaion ex-pos, does no work in real ime. Two very recen papers highlighed his: One by he Reserve Bank of Ausralia found ha he real ime oupu gaps, based on he Reserve Bank of Ausralia (Sone e al 3) filers, are no good forecaser of inflaion. Second is a more subsanial piece on he US economy. In his paper Orphanides and van Norden (3) look a he real ime inflaion predicing power of abou differen oupu gap esimaes, from Harvey (985) o Kuner (994). The conclusion is almos idenical ha he relaionship is very good ex-pos, bu no in real ime. On he oher hand, Graff (4) saes ha he Reserve Bank of New Zealand s oupu gap measure does predic he non-radable inflaion well, despie he oher problems associaed wih he mehodology. Because he oupu gap concep is no observable, economiss ry o esimae i. As is he case in many oher areas of economics, here are more han one ways of esimaing he oupu gap, ye probably no one knows which esimae is he correc one, especially in real ime. Hence, a differen echnique or model ha comes wih a differen oupu gap profile should no be disregarded; i should be regarded as one esimae. This approach of considering many specificaions, can be regarded as a hick modelling exercise proposed by Granger and Jeon (4).

3 3 I is quie a sandard pracice in economic modelling o disinguish he long-run properies of a sysem from is flucuaions around he long-run rend. There are various approaches o do his, bu wha is ineresing is ha mos of hese can be convenienly represened wihin a sae-space model. Apar from he mechanical filers such as he HP, ARMA and linear filers, unobserved componens models provide an alernaive framework for esimaing he oupu gap. Unobserved componens models decompose he observed oupu ino is rend and cycle componens. Wason (986), Harvey (985), Clark (989) and Harvey and Jeager (993) are examples of his approach. Kuner (994) adds a Phillips curve o he univariae model of Wason and Gerlach and Smes (997) models modifies Harvey (985) and Clark (987) approached by adding a Phillips curve equaion. Very recenly, Benes, J and P N Diaye (4), esimaed a so-called mulivariae filer for he Chech Republic. A he Reserve Bank of New Zealand, a lo of research has been underaken wih regard o he oupu gap. Sco (997) esimaed a common cycles model a la Clark (989). Claus (3) esimaes a srucural vecor auoregression o esimae he oupu gap. The main model he RBNZ uses for he oupu gap esimaion is a modified HP filer, a la Laxon and Telow (99), so-called he mulivariae (MV) filer. Conway and Hun (997) deail he approach and Hampon () modifies i slighly. This ype of modelling of he oupu gap is ofen referred as he, Hodrick-Presco Mulivariae, HPMV, filer. In his paper, we will pursue his HPMV filer approach. HPMV approach in sae space has been used in esimaing unobserved variables, such as non-acceleraing inflaion rae of unemploymen, NAIRU (OECD ) oupu gap or he neural real ineres rae previously (Basdevan e al 4). There are wo advanages of his approach: One, i is semi-srucural, i.e. i includes a mechanical filer, as well as some economic srucure. Second, because he HPMV is he way he Reserve Bank of New Zealand esimaes he oupu gap, puing he HPMV in sae-space and esimaing i wih differen assumpions, may ell us he advanages and disadvanages of cerain assumpions, and guide us owards a beer way of esimaing he oupu gap. This also enables us o es he sensiiviy of he Bank s MV filer o differen assumpions. As Boone () saes wo approaches are closely linked, and specifically. Esimaing he HPMV filer by using he Kalman filer has a leas wo advanages. Firs, while he radiional HPMV filer canno produce margins of uncerainy for he esimaed variables, he Kalman filer can. Secondly, i allows wider evaluaion of he impac of he differen assumpions underlying each echnique on he esimaes of he unobserved variables. 3 The HPMV framework and he Kalman filer LT argue ha he simple Hodrick Presco filer can be augmened wih residuals coming from srucural relaionships. Hence mimimising his new se of problem would be beer han a mechanical Hodrick-Presco (HP hereafer) filer. Because he HPMV is a an augmened HP filer, or is saring poin is an HP filer, le us sar wih he convenional Hodrick-Presco filer and pu i in sae-space The HP filer gives an esimae of he unobserved variable as he soluion o he following minimisaion problem: Min { y } ( y y ) + ( y ) T = σ () σ

4 4 Where y is he observed variable, y is he unobserved variable being filered, σ is he variance of he cyclical componen y y and σ is he variance of he growh rae of he rend componen. This problem is of course invarian o a homoheic ransformaion; σ herefore wha maers is he raio λ =. Hodrick and Presco sugges some σ parameerisaion of λ depending on he frequency of daa. Following Harvey (985), he HP filer can be wrien in a sae space form as follows. The measuremen equaion defines he observed variable as he sum of is rend and flucuaions around he rend: y = y + e () σ wih e ~ N(, ). The sae equaions define he growh rae of he rend ha is accumulaed o compue he rend iself: y = g + y +ν, (3) = + (4) g g ν, wih v, (no shock o he level of poenial oupu) and ν, ~ N (, σ λ ) v, is he change in he growh rae of he filered series or rend. In oher words, he change in he rend follows a random walk. Then o ge he HP filer esimae one has o use he whole se of informaion o derive y (as done in he minimisaion problem ()), ie o ake he smoohed esimae provided by he Kalman filer. = The Hodrick Presco Muli-Variae (HPMV hereafer) filer is an alernaive way of esimaing unobserved variables, developed a he Bank of Canada (see Laxon and Telow (99)) 3. This mehod sems from use of he sandard HP filer (see Hodrick and Presco (997)), augmened by relevan economic informaion. I has been used by he Cenral Banks of Canada and New Zealand o esimae poenial oupu (see Buler (996) and Conway and Hun (997)) and by OECD (999) o esimae he NAIRU. Boone () exends on Harvey (985) and pus he HPMV filer in sae space. This is done in wo seps. Firsly, he minimisaion problem is wrien as a sae space model. Secondly, resricions are imposed on he variances of he equaions of he sae space model, o reproduce he balance beween he elemens of he minimisaion programme. The HPMV filer seeks o esimae he unobserved variable as he soluion o he following minimisaion problem: = λ + λ (5) T ( y ) + ( ) { } y y y Min ζ Wih λ and λ given. This is a basic HP filer, augmened wih he residuals ζ aken from an esimaed economic relaionship: z = β y + dx + ζ (6) 3 for annual daa, 4 for semi-annual daa and 6 for quarerly daa. See S Aman and van Norden (997) for a criique of he ieraive Laxon and Telow approach.

5 5 where z is anoher explanaory variable can be explained by he unobserved variable y and X is a marix of oher exogenous variables. The residuals ζ are normal wih a covariance marix H. As for he simple HP filer, he smoohing consans λ and λ reflec he weighs aached o differen elemens of he minimisaion problem. The esimaed unobserved variable is no only a simple moving average going hrough he observed series, bu is also modelled o give a beer fi o he economic relaionship. The HPMV filer can also be reproduced by a Kalman filer, following a similar mehodology. In essence he problem is simply o add he addiional srucural equaion in he se of measuremen equaions: y z = y β + X d e + ζ (7) wih ( e ζ ) ~ N( H ),. The covariance marix H is defined as: σ H = (8) σ λ The wo sae-equaions are he same as for he sandard HP filer (see equaions (3) and (4)). The novely is he erm λ, which gives a balance beween he HP filer and he economic informaion embodied in he addiional equaion. A high value for λ corresponds o a beer fi of he economic relaionship, and an unobserved variable ha can depar significanly from he observed variable. There are wo advanages o be gained from reproducing an HPMV filer wih he Kalman filer. Firsly, he mehod is done of simulaneous naure (while he mehod proposed by LT is a muli-sep procedure 4 ), and also allows esimaion of he hyper-parameers. Thus, alhough i is rarely done in pracice, i is possible o freely esimae he parameers λ and λ insead of giving hese essenially arbirary values. The ype of HPMV filer we have esimaes has he following form: 5 Min { y } ( y y ) + ( y ) T = λ + λζ, + λ3ζ (9), Wih λ o λ 3 given, he residuals ζ i, corresponds o wo relaions: an Okun's law and a link beween capaciy uilisaion survey daa and he poenial oupu: ( y y ) ζ u = u + β +, () ( y y ) ζ cu = cu + γ +, () 4 5 See Hampon, Twaddle () and so on for his ieraive procedure. We should indicae ha his is a firs cu, a good firs cu we believe, which can be improved furher wih more condiioning informaion.

6 6 where y is he oupu and y he poenial oupu, u is he unemploymen rae and cu he capaciy uilisaion rae ha are aken as differences from heir equilibrium value. 6 We do no use he Phillips Curve due o he problem of circulariy, suggesed in Graff (4). 4 Problems wih he Laxon and Telow mehod and he advanages of he Kalman filer approach In his secion, we will discuss he shorcomings of he LT approach and how Kalman filer is superior o he ieraive LT approach and can address some of he shorcomings of he LT approach. The equaions and are added o ge more informaion from he srucural economic relaionships, which hopefully would help us a he end of he sample. However, hese equaions have addiional problems oo. I is obvious ha he residuals ha come ou of hese equaions would be sensiive i) o he parameers, and ii) o he equilibrium values in hese srucural equaions, i.e. equilibrium capaciy uilisaion rae and unemploymen rae. Furhermore, how hose residuals are weighed in he minimisaion, namely λ and λ3 would also maer. The classic argumen in favour of hese augmening equaions is ha hese wo variables are no revised. This is no compleely rue. Alhough heir level values are no revised by he saisician, he reason why cu 7 is no revised is because we assume we know wha he rend or he equilibrium value is. I has been shown ha he ex-pos errors of a gap concep such as oupu gap and unemploymen gap do no largely come from he revisions o he daa; hey come from geing he rend wrong in real ime. Hence, saying ha we know he rend in cu or u, does no overcome our problems. The RBNZ s MV filer-deermined rend unemploymen rae was by means of an HP filer up unil (see Hampon ). Alhough he unemploymen rae is no revised by he saisician, he gap measured by means of an HP filer sill has very large revisions ex-pos. Figure below shows he revisions o he HP filered unemploymen gap in New Zealand. 8 Figure abou here Hampon () aemps o overcome his problem by augmening he HP filer wih daa from skill shorage surveys. The skill shorage daa is a weighed average of skilled and unskilled labour shorage daa by he Quarerly Survey of Business Opinion. 9 u = u + κ skill skill ) + ( φ () However, his augmenaion does no solve he problem. Again, his equaion requires an equilibrium assumpion. Equilibrium skill shorage, assumed by he curren MV filer is ploed agains an HP filered rend and he acual skill shorage series in figure below: See Appendix B for he sae-space model of our approach. Capaciy uilisaion being a survey measure may furher complicae he problems. The same problem of course exiss for he capaciy uilisaion equilibrium value. The Real Business Cycle school would probably argue ha he equilibrium capaciy uilisaion would be ime varying. This is a survey daa colleced by he New Zealand Insiue of Economic Research, NZIER.

7 7 Figure abou here As we can see, he way we filer can make a difference o he skill shorage gap. In fac, one can argue ha he equilibrium value is a series wih regime changes: Early 99s, where i was easy o find labour, mid o lae 99s, where i was relaively harder o ge labour. And finally he las few years, given very low unemploymen rae, i may be he case ha he pool of applicans for any given job ad is now smaller, hence he survey will have a lower negaive mean for a while. Of course he main quesion is wheher employers have o live wih his igher labour marke environmen and hence adjus or hey will reflec his on heir prices. The laer would mean a change in he equilibrium skill shorage series. The shorcoming of he LT approach is ha, i ries o solve he end-poin problem in one rend, by inroducing anoher variable wih an unobserved rend. In addiion, he rend unemploymen rae is exogenous in he minimisaion problem of he HPMV filer. The Kalman filer approach can overcome his problem in he following way: If we allow he unemploymen rae o be an unobserved variable wihin he whole sysem, where unemploymen is an unobserved, bu he rend unemploymen rae is included in he Okun s law relaionship. This is how we will approach o he modelling. Anoher assumpion in he RBNZ MV filer is ha he coefficiens are imposed, no esimaed. The sae space approach also allows us o examine he implicaions of imposing he coefficiens in he srucural equaions. We sar wih he capaciy uilisaion equaion. Le us esimae a few capaciy uilisaion equaions: cu = cu + ( y y ) γ + ε (3) Firs we esimae his wih he ordinary leas squares. The parameer γ is esimaed. However, we should be aware ha he OLS esimaor of he parameer γ would suffer from he simulaneiy bias, since cu and ( y y ) are deermined simulaneously in he economy. In addiion, here is a severe serial correlaion problem. The firs column of he Table repors he resuls of he OLS regressions of he equaion. The coefficien esimae is.69. The second column shows he wo sage leas square esimae by using he firs lags of he capaciy uilisaion and he oupu gap. Figure 3 shows he residuals ha would go ino he minimisaion problem of equaion 9, under differen esimaes. I is clear ha hey are differen a imes and hey may make difference o he final oupu gap esimaes. Table abou here Figure 3 abou here Alernaively, one can allow he capaciy uilisaion equaion o have a ime varying equilibrium value and esimae i wih he Kalman filer. We allowed for his wih he following specificaion: = + γ ( + ε (4) cu cu y y ), = 5) cu cu cu + g

8 8 = + (6) cu cu g g ε, where, he firs equaion is he signal and he las wo are he sae equaions. The choice of he signal o noise raio of course does maer here. We chose a very siff raio of 3,. The esimaed γ is.56. Figure 4 below compares a few differen equilibrium capaciy uilisaion raes: The RBNZ s MV filer s assumpion, our simple sae space model of equaions 4-6 and wo HP filered rends wih differen smoohing parameers. I is clear ha he rend can be so differen. Hence he minimisaion problem would be a differen one, under a differen rend assumpion or esimaion. Figure 4 abou here Anoher major assumpion in he original LT approach is ha here is an explici assumpion on he rend oupu growh, called he siffener. The siffener explicily assumes a rend growh rae for he end of he sample, hence disallows he filer o be very flexible in is view of he rend. This can be a dangerous approach o employ, as a imes, he rend growh may be changing oo fas before we realise i. This is anoher advanage of he Kalman filer approach is ha here is no need for an explici siffener in esimaion procedure. Since mos of he end poin problem is more severe when he rend growh rae is changing fas, his assumpion may increase he size of real ime errors. I is also imporan how he filer weighs hese addiional informaion. If we hink in erms of signal o noise raio, hese weighs can make significan changes o he final esimaes of he oupu gap. The Bank s MV filer fixes hese weighs a some arbirary numbers. The HPMV approach wih he Kalman filer allows us o esimae hese weighs, given ha we already impose one of he hyperparameers. Original LT work suggess ha i is difficul o esimae confidence inervals for heir approach. However, despie is difficulies hey come up wih a confidence inerval which is 4 per cen on eiher side of he mean esimae. This gives a range of 8 per cen. Because he sae space approach enables us o esimae he poenial oupu wih sandard errors, and because we know ha he hisorical revisions o he level of GDP is very minor (which can be ignored), we can creae confidence inervals for our oupu gap esimaes. Below, we show he confidence inervals for our preferred model. 5 Esimaing HPMV filers wih he Kalman filer Having discussed all he hings ha may maer for he final oupu gap esimaes, we now presen ou HPMV filer in sae space. We sar wih he following HPMV filer, which is he simples form: Signal equaions y = + ε (7) y ( y y ) ζ u = u + β +, (8)

9 ( y y ) ζ cu = cu + γ +, Sae equaions 9 (9) = y + g y () g = + ε () g where equaions 7-9 are signal equaions, and equaions and are sae equaions. We sar wih a baseline model, where our baseline model ries o replicae he Reserve Bank of New Zealand s inernal MV filer, as much as possible. We will hen relax he assumpions of he MV filer as we discussed in he previous chaper. The baseline model, Model assumes an equilibrium capaciy uilisaion (ime invarian) and equilibrium unemploymen (ime varying). These are deermined ouside he model we esimae here. Equilibrium capaciy uilisaion is 88 per cen unil 99 and hen 99 per cen afer 99, which is exacly he same as wha he Reserve Bank s MV filer uses. This model also assumes ha he equilibrium unemploymen is an ex-pos HP filered unemploymen rae and he equilibrium capaciy uilisaion is known wih precision. In oher words, here is no uncerainy wih hese wo equilibrium values. The advanage of his base model is ha because i is very close o he curren filering approach used by he Reserve Bank, we can hen relax some of he assumpions, or change hings here and here o see if hey make any difference and how much difference hey make. Table below shows he esimaion resuls from he basic HPMV filer. The rend unemploymen rae is deermined by an HP filer. Unemploymen is filered ouside he sysem and hen enered ino he sysem here. There are hree variaions of his base model. Model imposes he coefficiens on he Okun s law and he capaciy uilisaion. The coefficien β is he so-called Okun s coefficien, which maps he changes in oupu gap o he changes in unemploymen from is equilibrium. The coefficien suggess ha for every per cen deviaion of oupu from is poenial, he unemploymen deviaes from is naural rae by.33 per cen, which is a relaionship of o 3. This coefficien is very similar o he original esimaes of Okun (96) and slighly differen ha o relaionship found by Sco (997). Coefficien γ is he relaionship beween he capaciy uilisaion and he oupu gap. The calibraed coefficien of means ha a per cen deviaion of oupu from is rend leads he capaciy uilisaion o deviae per cen from is rend. Table abou here The model also imposes he rends in cu and u equaions. The signal o noise raios are also imposed a 6, and respecively. These are all consisen wih he Reserve Bank s official filer. This is also he case in he original Laxon and Telow (99) approach, where he equilibrium unemploymen rae was assumed o be known wih no uncerainy around i.

10 Model, esimaes he coefficiens alhough sill imposed he equilibrium values in capaciy uilisaion and unemploymen. Hence he equilibrium unemploymen is he same in hese models. One major change in his model is he change in he coefficienγ. This implies ha he relaionship beween capaciy uilisaion gap and he oupu gap may no be o. The Wald es of γ = is rejeced a 5 per cen significance level. The coefficien urns ou.77. In oher words, a per cen oupu gap is associaed wih a capaciy uilisaion deviaion of.7 per cen from is rend. The esimaed Okun s law coefficien is also differen, -.7. This indicaed ha a per cen oupu gap implies unemploymen o be differ by -.7 per cen from is equilibrium. In Model 3, in addiion o esimaing he coefficiens, we also esimae he equilibrium capaciy uilisaion as a consan. This model makes furher changes in coefficiens. Esimaed γ is now closed o,.97. Furhermore, he Okun s law coefficien is now much lower, indicaing a weaker relaion o he unemploymen deviaions from is rend from he oupu gap. The equilibrium capaciy uilisaion urns ou o be.884 per cen, which is very close o he Model, where cu was calibraed a.89. As we can see he poenial oupu growh can be somehow sensiive he way we use he same informaion, which are he same capaciy uilisaion and unemploymen rae equaions. The final saes of he growh rae of he poenial oupu are.879,.864 and.878 per quarer in Models, and 3 respecively. Figure 5 below shows he Bank s oupu gap esimaes versus hese hree models we presened above. I is clear ha hey have he idenical paern and hey are very similar. The difference comes from he way he coefficiens and he rend in he srucural equaions are reaed. Figure 5 abou here One crucial hing missing from he Laxon and Telow approach, as we discussed, is he uncerainy around he oupu gap esimaes. We can esimae he sandard errors wih he Kalman filer approach. There are more han one kind of uncerainy wih regard o he final oupu gap esimaes: One is he uncerainy around he esimaes of he poenial oupu. Second is he uncerainy around he parameers used/esimaed. Third is he uncerainy abou he addiional rend valued in he sysem. Wihou going ino deail, we would like o show he imporance of his: Figure 6 shows he oupu gap esimaed from he model, where he coefficiens and he equilibrium values are fixed and hence assumed o exhibi no uncerainy. Hence he only uncerainy would come from he poenial oupu gap. Figure 6 abou here As we have shown in he firs hree models, he way we rea he srucural informaion can make a difference for he final oupu gap esimaes. Moreover, in he firs hree models, we have no esed he relaive sensiiviy of he oupu gap esimaed o he rend unemploymen assumpions. Trend unemploymen has been declining in New Zealand since 99. Therefore, esimaing he rend for his is much harder han esimaing a consan rend of he capaciy uilisaion. As far as he inflaion predicing power of he models are concerned, i is obvious from Figure 5 ha, hee are as good/bad as he Reserve Bank s official filer is.

11 We will now ry o esimae models, where unemploymen rend is also reaed as unobserved variable. One nice and flexible feaure of he HPMV approach is ha, since i fixes one of he signal o noise raios, he oher variances in he srucural equaions can be freely esimaed. This enables us o see if he assumpions made above are similar o a mode daa driven approach. Table 3 abou here As we saed before, he weighs placed o he srucural residuals in he minimisaion are he hyperparameers in our sae-space model. Models, and 3 assumed he weighs imposed by he RBNZ s MV filer where λ = λ 3 = We run he same models by allowing he Kalman filer o choose he opimal weighs. Because, we have signal o noise raio, λ, which is fixed a 6, he variances of u cu e and e can be esimaed. Table 3 and Figure 7 shows he resuls from his. Figure 7 abou here As we have shown above, he way he unobserved equilibrium values are reaed can make some imporan difference o he final esimaes. As a resul of his conclusion, we now esimae he model in way ha all unobserved rends and he coefficiens are esimaed simulaneously. All rend/equilibrium values are reaed as unobserved variables. Hence, we wan o invesigae his furher. Wha we would like o do is o allow he unemploymen rend and he capaciy uilisaion equilibrium o be unobserved variables, as well as he poenial oupu. In fac, his is he approach aken in a recen paper by Benes and N Diaye (4). They esimae anoher class of mulivariae filer, where boh NAIRU and he poenial oupu are reaed as unobserved componens. We call hese Models 4 and 5 respecively. In hese models, we rea he equilibrium rae as unobserved variable in he following way and esimae he equilibrium capaciy uilisaion as a consan. The only difference beween models 4 and 5 is ha, in model 4, he coefficiens are esimaed and in model 5, hey are imposed. = () u u u + g g u = g + ϕ (3) u The reason for reaing he capaciy uilisaion as a consan, comes from he naure of he capaciy uilisaion survey. is sensible, as we believe he equilibrium capaciy uilisaion does no move around much. The rend unemploymen on he oher hand is a random walk wih drif. We fix he hpyerparameer of he unemploymen rae a 6 as he oupu gap one. In oher words, ζ, ~ N(, σ u ) and ϕ ~ N(, σ u λ ) ϕ. Table 4 and 5 shows he esimaes from hese wo kind of models. The difference beween models 4 and 5 and models 4a and 4b is he imposiion and esimaion of he

12 hyperparameers. In hese wo ables, we do only focus on he equilibrium values, raher han he coefficiens. I is clear ha he imposiion or esimaion of parameers make a big difference for he key equilibrium values. Models 4 and 4a, imposes he coefficiens β and γ In hese models, he curren equilibrium unemploymen raes are esimaed as 4.9 and 4.79 per cen respecively. In models 5 and 5a on he oher hand, rend unemploymen is much lower, hence he rend growh rae is much higher, above.9 per cen per quarer. The esimaed equilibrium capaciy uilisaion is jus under 89 per cen, almos idenical o he RBNZ s MV filer assumpion. Table 4 abou here Table 5 abou here To highligh his difference, we plo he oupu gap esimaed from models 4 and 5. Because he model 5 believes he fall in unemploymen is due o rend, i gives almos no gap a he end of he sample. Model 4 on he oher hand gives an oupu gap of around.5 per cen. We have o cauion he reader wih one hing abou he model 5. In capaciy uilisaion equaion, because we ry o esimae he consan rend, as well as he coefficien and he variance (hence nohing is imposed), we ge an exremely small coefficien. Hence ha is also he reason, why oupu gap urns ou o be zero in his model, despie a capaciy uilisaion, which is well above is esimaed equilibrium value a he momen. Figure 8 shows he resuls from models 4 and 5 Figure 8 abou here I is clear ha as he way we rea he equilibrium capaciy uilisaion does no maer, as long as i is reaed as a consan. The equilibrium capaciy uilisaion is around.885 in mos models. However, he rend unemploymen rae depends on how we rea i and i also affecs he final oupu gap esimaes. In models, and 3 we used a rend unemploymen rae ha was esimaed ouside he sysem. In Models 4 and 5, we esimaed he rend unemploymen rae as an unobserved variable. Figure 9 below compares he four differen rend unemploymen raes we used/esimaed in his paper. An HP filered rend, he MV filered rend which uses he shill shorage series o augmen an HP filer and wo oher equilibrium unemploymen rae series, we esimaed from models 4 and 5. Their evoluions are raher differen. 3 6 Revisions Figure 9 abou here In his secion, we will look a he revision properies of he models we presened above. We believe his issue deserves more aenion. This is as imporan as he inflaion predicing abiliy of he oupu gap. Orphanides and van Norden (3) have shown ha no oupu gap model can escape from subsanial revisions in ex-pos. Hence, using he 3 If i is allowed o be a ime varying equilibrium, hen we ge a differen sory. We do no follow ha avenue for he reasons we saed above. See Guy and Szeo (4) for a Phillips curve based NAIRU esimaes for New Zealand. They are compleely differen han he ones we esimaed in his paper.

13 3 oupu gap in real ime, can lead o wrong policy conclusions. (See Orphanides and Nelson and Nikolov 3). Previously, Twaddle () argued ha he revisions o he real ime oupu gap esimaes of he RBNZ s MV filer are no as bad as an HP filer, bu sill very high. Furhermore, Graff (4) saes ha he esimaes for any given quarer do no converge o any final value, even afer abou quarers. Those errors are around per cen. However, as we argued above, he MV filer assumed no uncerainy in parameers, equilibrium values in he condiioning informaion. Therefore, once we allow uncerainy around hese, he revisions can be subsanially larger. Figure abou here Figure above, shows he ex-pos real ime difference, in oher words, he revisions o he oupu gap for he five differen models, we esimaed in his paper. The revisions are much smaller in models, where he poenial oupu is he only unobserved variable. These resuls are similar o he Twaddle (). However, as we rea he oher rends as unobserved variables as well, he revisions ge much larger. 7 Fuure Direcions and conclusions The oupu gap concep plays a crucial role in he hinking and behaviour of many inflaion argeing cenral banks. Ye, his concep is difficul o measure, due o he unobserved naure of he poenial oupu. In addiion, he real ime esimaes of he oupu gap are subjec o massive revisions. We have shown ha he resuls of he LT approach can be dependen on he assumpions abou he srucural relaions. In paricular, he parameers and he equilibrium values assumed in he srucural equaions can make a significan difference o he oupu gap esimaes. We also showed, hough did no discuss in grea deail, ha he revision properies ge worse, as he unobserved equilibrium values are no assumed o be known. We argued ha esimaing he HPMV filer wih he Kalman filer has many advanages over he ieraive procedure proposed by LT. Based on differen assumpions on rends and coefficiens, he policy makers can see an envelope of oupu gap in hick modelling sense. References Basdevan, O (3), On he Applicaions of Kalman Filer in Macroeconomic Time Series, Reserve Bank of New Zealand Discussion papers Series, 3/ Basdevan, O, N Björksen and Ö Karagedikli (4), Esimaing a ime varying neuralreal ineres rae for New Zealand, Reserve Bank of New Zealand Discussion papers Series, 4/ Benes, J and P N Diaye (4), A Mulivariae Filer for Measuring Poenial Oupu and he NAIRU: Applicaion o he Czech Republic, IMF Working Paper WP/4/45.

14 4 Billmeier, A (4), Measuring a Roller Coaser: Evidence on he Finnish Oupu Gap, IMF Working Paper WP/4/57. Black, R e al (997), The Forecasing and Policy Sysem; The core model, Reserve Bank of New Zealand, Research Paper, 43 Boone, L (), "Comparing semi-srucural mehods o esimae unobserved variables: he HPMV and Kalman filer approaches," OECD Working Paper #4. Buler, L (996), A Semi-Srucural Mehod o Esimae Poenial Oupu: Combining Economic Theory wih a Time-Series Filer, Bank of Canada Technical Repor No 77. Ciu, F and J Twaddle (3), The oupu gap and is role in moneary policy decisionmaking, Reserve Bank of New Zealand Bullein, 66 (), 5 4. Clark, P K (987), The Cyclical Componens of US Economic Aciviy, Quarerly Journal of Economics (4), Clark, P K (989), Trend reversion in real oupu and unemploymen Journal of Economerics 4, 5-3. Claus, I (3), Esimaing poenial oupu for New Zealand Applied Economics, 3, vol. 35, issue 7, pages 75-6 Claus, I, P Conway and A Sco (), The Oupu Gap: Measuremen, Comparisons and Assessmen, Reserve Bank of New Zealand Research Paper No 44, Wellingon, June. Conway, P and B Hun (997), "Esimaing poenial oupu: a semi-srucural approach," Reserve Bank of New Zealand Discussion Paper D97/9. De Brouwer, G (998), Esimaing Oupu Gaps, Reserve Bank of Ausralia Research Discussion Paper 989. Elekdag, S (), Kalman filer and is applicaions Cenral Bank of Republic of Turkey, Discussion papers series. Gerlach, S and F Smes (997), Oupu Gaps and Inflaion: Unobserved Componens Esimaed for he G-7 Counries, BIS mimeo Gerlach, S and F Smes (999), "Oupu gaps and moneary policy in he EMU area," European Economic Review, 43: 8-8. Graff, M (4), Esimaes of he oupu gap in real ime: how well have we been doing? Reserve Bank of New Zealand Discussion Papers Series, DP 4/4 Granger, C and Y Jeon (4), Thick Modelling, Economic Modelling,, pp

15 5 Gruen, D, Robinson, T and A Sone (), Oupu Gaps in Real Time: Are They reliable Enough o Use for Moneary Policy, Reserve Bank of Ausralia Discussion Paper, /6 Hamilon, J (995), Sae-space models in Engle, R F and D Mc Fadden (eds) Handbook of Economerics Volume 4, Hampon, T (), Revisiing MV filer, Reserve Bank of New Zealand, mimeo Harvey, A (989), Forecasing, Srucural Time Series Models and he Kalman Filer, Cambridge Universiy Press, Cambridge. Harvey, A C (985), "Trends and cycles in macroeconomic ime series" Journal of Business and Economic Saisics 3: 6-7. Harvey, A C (985), Trends and Cycles in Macroeconomic Time Series, Journal of Business and Economic Saisics, 3, 6-7 Harvey, A C and A Jaeger (993), Derending, Sylised Facs and he Business Cycle, Journal of Applied Economerics, 8, 3-47 Hodrick, R J and E Presco (997), Pos-war Business Cycles: An Empirical Invesigaion, Journal of Money, Banking and Credi, 9, -6 Kalman, R E (96), "A new approach o linear filering and predicion problems," Journal of Basic Engineering, Transacion of he SAME, 59: Kuner, K N (994), Esimaing Poenial Oupu as a Laen variable, Journal of Business and Economic Saisics, (3) Lansing, K (), Can he Phillip Curve Help Forecas Inflaion? Federal Reserve Bank of San Francisco Economic Leer No -9 Laxon, D and R Telow (99), A simple mulivariae filer for he measuremen of oupu gap, Bank of Canada echnical repor, 59 Nelson, E and K Nikolov (3), UK Inflaion in he 97s and 98s: he Role of Oupu Gap Mismeasuremen, Journal of Economics and Business, 55 (4), OECD (), The Concep, Policy Use and Measuremen of Srucural Unemploymen: Esimaing a Time Varying NAIRU Across OECD Counries, Economics Deparmen Working Paper 5 Okun, A (96), Poenial GDP: Is measuremen and significance, Proceedings of he Business and Economics Saisics Secion, American Saisical Associaion. Orphanides, A and S van Norden (3) The Reliabiliy of Inflaion Forecass Based on Oupu Gap Esimaes in Real Time, mimeo Orphanides, A () Moneary policy rules based on real-ime daa, American Economic Review 9,

16 6 Orphanides, A (3), Hisorical Moneary Policy Analysis and he Taylor Rule, Journal of Moneary Economics, 5 (5), 983. Orphanides, A, R D Porer, D Reifschneider, R Telow and F Finan (999), "Errors in he measuremen of he oupu gap and he design of moneary policy," Federal Reserve Board, Finance and Economics Discussion Series, Orphanides, A and S van Norden (), The Unreliabiliy of Oupu-gap Esimaes in Real Time, Review of Economics and Saisics 84, Razzak, W (997), The Hodrick-Presco echnique: A Smooher versus a filer. An Applicaion o New Zealand GDP, Economics Leers, 75, Razzak, W (), Moneary policy and forecasing inflaion wih and wihou he oupu gap, Reserve Bank of New Zealand Discussion Paper /3. Robinson, T, Sone, A and M van Zyl (3), The Real-ime Forecasing Performance of Phillips Curves, Reserve Bank of Ausralia Discussion Paper, 3/ Sco, A (996), A Mulivariae unobserved componens model of he cyclical aciviy Reserve Bank of Discussion Papers Series, /4 S Aman, P and S van Norden (998) Measuremen of he oupu gap: A discussion of recen research a he Bank of Canada, Bank of Canada Technical Repor No 79 Twaddle, J (), The MV Filer and real-ime daa, Reserve Bank of New Zealand, inernal mimeo Van Norden, S (), Filering for Curren Analysis, Bank of Canada Working Paper -8.

17 7 Appendix A: The Kalman filer and he smoohed esimaes 4 Many dynamic models can be wrien and esimaed in sae-space form. The Kalman filer is he algorihm ha generaes he minimum mean square error forecass for a given model in sae space. If he errors are assumed o be Gaussian, he filer can hen compue he log-likelihood funcion of he model. This enables he parameers o be esimaed by using maximum likelihood mehods. For simpliciy le us consider a measuremen equaion ha has no fixed coefficiens: Y = Γ X + ε (A) where Y is a vecor of measured variables, Γ is he sae vecor of unobserved variables, N, H. The sae equaion is given as: X is a marix of parameers and ε ~ ( ) Γ = Γ + η (A) where η ~ N (,Q). 5 Le γ be he opimal esimaor of Γ based on he observaions up o and including Y, γ he esimaor based on he informaion available in, and γ T he esimaor based on he whole sample. We define he covariance marix P of he sae variable as follows: E ( Γ γ )( Γ γ ) = P (A3) The prediced esimae of he sae variable in period is defined as he opimal esimaor based on informaion up o he period, which is given by: 4 This secion draws heavily on Basdevan (3). 5 Q and H are referred o as he hyperparameers of he model, o disinguish hem from he oher parameers.

18 γ γ = 8 (A4) while he covariance marix of he esimaor is: ( Γ γ )( Γ ) = P Q P = E γ + (A5) The filered esimae of he sae variable in period is defined as he opimal esimaor based on informaion up o period and is derived from he updaing formulas of he Kalman filer: 6 γ γ + P X ( X P X + H ) ( Y X a ) = and (A6) P ( X P X H ) X P = P P X + (A7) The smoohed esimae of he sae variable in period is defined as he opimal esimaor based on he whole se of informaion, i.e. on informaion up o period T (he las poin of he sample). I is compued backwards from he las value of he earlier esimae γ T T =γ T, P T T =P T wih he following updaing relaions: ( γ ) γ P + γ T = γ + + T (A8) P ( + ) P P P = P + P + T + T (A9) where P PP. = + Depending on he problem sudied one can be ineresed in any one of hose hree esimaes. In our paricular case, looking a smoohed values is more appropriae, as he poin is no o use he Kalman filer o produce forecass bu o give he mos accurae informaion abou he pah followed by he ime-varying coefficiens. Therefore i is more informaive o use he full daase o derive each value of he sae variables. 6 This esimaor minimises he mean square errors when he expecaion is aken over all he variables in he informaion se raher han being condiional on a paricular se of values (see Harvey 989 for a deailed discussion). Thus he condiional mean esimaor, γ, is he minimum mean square esimaor of Γ. This esimaor is uncondiionally unbiased and he uncondiional covariance marix of he esimaor is he P marix given by he Kalman filer.

19 9

20 Table Esimaion resuls of differen capaciy uilisaion specificaions OLS SLS OLS- Consan NA NA -. (.) γ.69 (.9).858 (.) R DW All sandard errors are Newey-Wes heeroscedasiciy consisen sandard errors

21 Table Esimaion resuls for he Firs 3 Models Model y + ε = y u = u.33( y y ) + ζ, ( y y ) + cu =.89 + ζ, Model y + ε = y u = u + β ( y y ) + ζ, ( y y ) cu =.89 + γ + ζ, Model 3 y + ε = y u = u.33( y y ) + ζ, ( y y ) + cu = cu + ζ, β (.5) γ.77 (.83) -.37 (.68).97 (.9) Final saes y g (.3) (.5) (.8) u cu (.) Saisics Log Likelihood AIC SC H-Q

22 Table 3 Esimaion resuls wih differen hyper parameers Model a Model a Model 3a β (.4) γ.547 (.4) -.38 (.46).675 (.5) Final saes y g (.4) (.4) (.4) u cu (.3) Saisics Log Likelihood AIC SC H-Q

23 3 Table 4 Model Comparison Model 4 Model 5 y g.889 (.5) u.49 (.38) cu.8854 (.34).944 (.8).436 (.48).8854 (.9) Log Likelihood AIC SC H-Q Crier Table 5 Model Comparison Model 4a Model 5a y g.873 (.6) u.479 (.39) cu.8853 (.5).96 (.5).58 (.4).8854 (.63) Log Likelihood AIC SC H-Q Crier

24 4

25 Figure : Revisions o he HP filered unemploymen gap DIFFERENCE_UGAPNZ Figure Skill shorage survey series and is assumed equilibrium STAFF_HPTREND STAFFSHORT STAFF_MEAN

26 Figure CU_RESID_LS CU_RESID_OLS CU_RESID_OLS Figure 4 Time varying equilibrium of he capaciy uilisaion Capaciy uilisaion MV filer rend HP-6 HP-3

27 Figure 5 HPMV filered oupu gaps and he Bank s MV filer RBNZ M V filer M odel M odel M odel 3 Figure 6 Mean ou pu gap esimae and sandard errors sd dev mean gap esimae - sd dev -5

28 Figure RBNZ M V filer M odel M odel M odel 3 Figure M odel 4 M odel

29 9 Figure 9 Equilibrium Unemploymen raes MV filred rend unemp HP filered rend unemp rend unemp from Model 4 rend unemp from Model 5 Figure Revisions o 5 models presened revisions o Model revisions o Model revisions o Model 3 revisions o M odel 4 revisions o M odel 5