Polynomial Adjustment Costs in FRB/US Flint Brayton, Morris Davis and Peter Tulip 1 May 2000

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1 Polynoial Adjusen Coss in FRB/US Flin Brayon, Morris Davis and Peer Tulip 1 May 2000 The adjusen dynaics of os ajor nonfinancial variables in FRB/US are based on a fraework of polynoial adjusen coss, or PAC. This is a odel, developed by Tinsle1993, in which decisions are driven by expecaions bu consrained by adjusen coss. This noe describes wha PAC is, how i is derived and how i is esiaed and ipleened wihin FRB/US. Descripion The PAC odel can be presened in differen ways, depending on he purpose. A for ha is siple o inerpre is he decision rule : y a 0 &1 & y &1 &1 % j a k y % E &k &1 j k1 j0 d j %j (1 where y is he dependen variable, (we use he ers inerchangeably, is he firs difference operaor and E&1 represens expecaions based on inforaion available a -1. Thus he equaion decoposes he deerinans of y ino hree eleens: he lagged gap beween he level of y and is equilibriu value, lagged values of y, and expeced fuure values of. The d j represens is desired, arge or equilibriu value coefficiens on leads of are ransforaions of he (k = 0,..., -1 coefficiens on lags of y, as discussed below. In FRB/US noaion, he level of he arge, y*, is represened by a variable in which he firs leer is Q. The expeced su of fuure values of is represened by a variable in which he firs leer is Z. This approach resebles FRB/US s precursor, MPS, and siilar odels in ha i a k Coens and quesions welcoe. To Peer Tulip, Sop 61A, Federal Reserve Board, Washingon DC Eail: pulip@frb.gov 1

2 cobines shor-er dynaics wih long-er error correcion echaniss. The close fi o he daa associaed wih his approach is ainained. The ain difference fro siple error-correcion odels is he inclusion of ers reflecing expeced growh in he arge. In radiional error-correcion odels, lagged explanaory variables ofen represen a ixure of expecaions and previous shocks o which agens are gradually adjusing. This abiguiy akes i difficul o address iporan quesions, such as he effec of changes in expecaions. In conras, he PAC fraework provides a eans of separaely idenifying he effecs of expecaions and adjusen coss. Derivaion Several observaionally-equivalen cos funcions can be used o derive he PAC specificaion. One such cos funcion, C j i0 i (y %i & %i 2 % j b k k1 (1&L k 2 y %i (2 penalizes boh deviaions of a variable y fro is desired value b k, k1,...,, (1982 and Nickell (1985. The exra ers peried by PAC iply inclusion of lagged changes in he dependen variable: he j &1 k1 a k y &kexpression in (1. Their inclusion provides a beer fi o he daa, paricularly in regard o shor-er dynaics. Many and changes in ie derivaives of he variable y. In his and following equaions, fuure-daed variables should be inerpreed as expeced values even when he expecaions operaor has been suppressed. is a discoun facor on fuure penalies, assued o equal 0.98, and are cos paraeers. Many oher researchers have worked wih cos equaions siilar o (2, bu resricing o equal 1. For exaples and discussion see Kennan (1979, Roeberg equaions in FRB/US have values of beween 2 and. The Appendix shows how he decision rule (1 is derived fro he cos funcion (2. In brief, iniizaion of coss yields he firs-order condiion, 2

3 (y & % j k1 b k [(1&L (1& F ] k y 0 (3 where L and F are he lag and lead operaors, respecively; wrien ore copacly as A( F A(L y c, (LF &1. Equaion (3 can be ( where A is a polynoial in he lag and lead operaors of order ; ha is, A(L 1 % 1 L %... % L and A( F 1 % 1 F %... % F ; and c is a consan. Afer soe algebra, his yields (1. The coefficiens in he decision rule (1 are ransforaions of he paraeers in A (which in urn are ransforaions of and he b paraeers in he cos equaion (2. Specifically, a 0 d 0 A(1 1 % j j1 j ; for k = 1, 2,..., -1 a k & j jk%1 j and, for j = 1, 2,... d j 1 & A(1A( j j&1 i0 G i where he arix G is also a funcion of he discoun facor and he adjusen coefficiens. Is srucure is provided in equaion (A. 38 of he Appendix. Pre and pos uliplicaion by he selecion vecor selecs he op lef eleen of. G i 3

4 Esiaion The firs sep in he esiaion of a PAC equaion is consrucion of a odel deerining he arge,. This ypically consiss of a saionary coponen, y 0( and a rending coponen, y 1(. Tha is, y 0( % y 1( (5 The coefficien on he saionary coponen is esiaed wihin he PAC equaion, as discussed below. In conras, he rending coponen is specified before he PAC equaion is esiaed as a funcion of one or ore variables, wih coefficiens ha are consrained in accord wih heory or are esiaed fro coinegraing equaions. In he second sep, forecasing odels for he coponens of Specifically, suppose ha is an eleen of he inforaion vecor z, which also includes oher variables useful for forecasing. A vecor auoregression (VAR can hen be used o predic fuure levels of z as a linear cobinaion of pas levels. Tha is, are esiaed. z %1 Hz (6 Forecass ino he indefinie fuure are obained by repeaed applicaion of his equaion: z %i H i z (7 z 0 he arge y 0( and a vecor z 1 used for forecasing he rending coponen of he arge y 1( Decopose z ino a vecor used for forecasing he saionary coponen of. Subsiuing he VAR forecass for expeced changes in he arge in equaion (1 and convering he infinie leads ino a finie for gives an equaion ha is a funcion of observable variables: y a 0 y 1( &1 & y &1 % j &1 k1 a k y &k % h 0 z 0 &1 % h 1 z 1 &1 (8

5 he coefficien vecors h 0 and h 1 are consruced fro he VAR coefficiens H, he discoun facor and he adjusen coefficiens, as oulined in equaions (A. 7 and (A. 82 of he Appendix. h 0 and h 1 differ because h 0 z 0 reflecs conribuions fro boh he lagged level and expeced values of he saionary coponen of he arge whereas h 1 z 1 &1 reflecs conribuions fro expeced oveens -- he lagged level of he rending &1 z 0 and z 1 affec y syerically, hrough heir effec on. FRB/US equaions ofen values for a k ( k = 0 o -1, iniial esiaes of h 0 and h 1 can be consruced ( ĥ 0 and. Using hese, we can esiae he following linear regression ĥ 1 represens he conribuion of saionary eleens of he arge. The coponen being separaely idenified. PAC iposes nuerous resricions on equaion (8. These reflec assupions ha lead coefficiens are reparaeerizaions of he lag coefficiens (in urn reflecing he syery wih which he pas and he discouned fuure ener ( and ha he eleens of ipose addiional resricions reflecing assupions abou hoogeneiy and coinegraion. only A siple ehod of esiaing he resriced equaion is hrough an ieraive OLS procedure. Given values of he VAR coefficiens H, he discoun facor, and saring y & ĥ 1 z 1 &1 a 0 y 1( &1 & y &1 % j &1 k1 a k y &k % (ĥ 0 z 0 &1 (9 This provides an esiae of and revised esiaes of he a k coefficiens. ĥ 0 and ĥ 1 can hen be recalculaed and anoher ieraion perfored. Typically, paraeer esiaes converge in a few ieraions. The resricions iposed by PAC can be joinly esed by coparing he residuals fro (9 wih hose fro an unresriced regression. The unresriced regression resebles equaion (8 wihou resricions on he h vecors. The order of adjusen coss is deerined epirically by esing o see how any lags of he dependen variable are significan, and hen including all lags up hrough he las significan one. 5

6 In pracice, no all he series odeled in FRB/US exacly fi wihin he PAC fraework, involving soe odificaions o he specificaion above. For exaple, he presence of agens who do no opiize in he forward-looking anner assued by PAC (perhaps because of liquidiy consrains or bounded raionaliy is refleced in he inclusion of curren incoe in he consupion equaion and cash-flow in he invesen equaion. Teporary responses o variables ouside he arge, (for exaple payroll axes and he iniu wage in he wage equaion and differen speeds of adjusen o differen eleens of he arge (for exaple, ipors and energy coss in he price equaion is refleced in heir separae inclusion as regressors. Model-consisen expecaions Given a se of VAR-based expecaions, he odel can be esiaed as described above and hen used for forecasing and policy siulaions. The siulaions will norally yield differen values for he arge variables han iplied by he VARs. In oher words, he inforaion individuals use in foring heir expecaions is assued o be liied o he inforaion vecor z and does no encopass predicions of he odel. For soe applicaions, such as peranen changes in policy rules, VAR expecaions have he undesirable feaure ha persisen expecaional errors are ade. To avoid his, policy siulaions in FRB/US ofen assue odel-consisen expecaions. To siulae he odel wih odel-consisen expecaions, equaion (9 is iniially esiaed using VAR-based expecaions, as discussed above. This generaes an iniial se of coefficiens, VAR-based expecaions of and forecass of he oher endogenous variables of he odel. As explained below, new expecaions of can hen be copued. Subsiuing hese for he previous se of expecaions, while reaining he, and coefficiens fro he iniial esiaion (which reain consan hrough successive ieraions, new forecass and new expecaions can hen be copued. Ieraions coninue unil expecaions coincide wih forecass. Specifically, le Z represen he conribuion fro expeced oveens in he a k 6

7 Z Z E &1 j arge. equals he er d j in he decision rule (1 and he er j0 h 0 z 0 &1 %h 1 z 1 &1 in he esiaion equaions (8 and (9 under VAR-expecaions. To represen as an expression conaining a finie nuber of leads, begin by assuing Z erinal condiions for Z%T... Z%T% %j where T is large. (These could be arbirary, hough i is ore efficien o assue a balanced growh pah consisen wih he exogenous variables of he odel. Coupled wih a projeced pah of fro he presen o he erinal sae, inerediae values of Z can hen be calculaed recursively fro he erinal sae back o period. The forula for calculaing his, as explained a equaion (A. 90 of he Appendix, is: Z j i1 i i Z %i % A(1 & j &1 k1 j &1 jk j%1 j%1 %k (10 An iniial esiae of Z will hen iply new forecass of he odel s endogenous variables. Fro hese, a new pah of can be forecas, fro which a new esiae of Z %k can be obained. This process ieraes unil he expeced growh of he arge coincides wih he forecas. Having obained one se of odel-consisen projecions, he exogenous variables of he odel, such as policy paraeers, can hen be changed. Repeiion of he above procedure leads o a new se of odel-consisen projecions. These can be copared wih he iniial se o assess he consequences of he policy change. 7

8 Algebraic Appendix Tinsle1993 concisely presens he algebra of PAC. This appendix provides a ore deailed presenaion. Derivaion of he decision rule Consider he cos funcion (2: C j i0 i y & 2 %i %i % j b k k1 (1&L k y %i 2 (A. 11 Differeniaing wih respec o y yields a firs order condiion: 0 2(y & % j i0 2 i M (1&Ly b %i M (1&L 2 2 y 1 %b %i M (1&L 2 y My 2 %...%b %i My. My (A. 12 Take he derivaives of he second par of equaion (A. 12 piece by piece. For k=1, j i0 2 i M (1&Ly b %i 1 2b My 1 (1&Ly & (1&L y %1 2b 1 (1&L (y & y %1 (A. 13 2b 1 (1&L (1& F y and for k=2 j i0 i M (1&L 2 2 y b %i 2 2b My 2 [(1&L 2 y &2 (1&L 2 y % 2 %1 (1&L 2 y ] %2 2b 2 [(1&L 2 (y &2 y %1 % 2 y %2 ] 2b 2 [(1&L 2 (1& F 2 y ]. (A. 1 8

9 Coninuing like his, we can derive he general expression for he derivaive of he er, which is k h j i0 i M (1&L k 2 y b %i k 2b My k (1&L k (1& F k y. (A. 15 Subsiuing (A. 15 ino (A. 12 gives us he firs order condiion, equaion (3: (y & % j k1 b k (1&L(1& F k y 0 (A. 16 which can be rewrien as 1 % j k1 b k [(1&L(1& F] k y. (A. 17 The expression in square brackes is a self-reciprocal polynoial, ha is, he coefficien on ( F k is he sae as he coefficien on L k. To see his, firs consider he expression b k [(1&L(1& F] k for k = 1 b 1 (1&L(1& F b 1 (1&L& F% b 1 (&L%(1% & F. (A. 18 The coefficiens on L and F are idenical (hey boh equal &b 1. Nex, consider k = 2 b 2 [(1&L(1& F] 2 b 2 [(1&L(1& F ][(1&L(1& F ] b 2 [ &L%(1% & F ][&L%(1% & F ] (A. 19 b 2 [ L 2 & 2(1% L % (1% % 2 & 2(1% F % ( F 2 ] The coefficiens on L 2 and ( F 2 are idenical (hey boh equal b 2, as are he coefficiens on L and F (hey equal &2b 2 (1%. In a siilar fashion, i is possible o show for all 9

10 k 1,...,, ha b k [(1&L(1& F] k is a self-reciprocal polynoial. Because j k1 b k [(1&L(1& F] k is a suaion over self-reciprocal polynoials, and a suaion over syeric ers produces syeric ers, i follows ha j k1 b k [(1&L(1& F] k is a self-reciprocal polynoial as well. The addiion of a consan 1 % j 1 does no affec he coefficiens on lags or leads, so [(1&L(1& F]k is also k1 b k self-reciprocal, as claied. I can be wrien explicily in self-reciprocal for as: 1 % j k1 b k [(1&L(1& F] k C (L, F, (A. 20 where C(L, F / c L %... % c 1 L % c 0 % c 1 ( F%... %c ( F. (A. 21 Tinsle1993 shows ha a self-reciprocal polynoial wih lags and discouned leads can be facored as C(L, F Â(LÂ( F (A. 22 where Â(z ˆ 0 % ˆ 1z % ˆ 2z 2 %... % ˆ z (A. 23 and for i = 0, 1,..., c i j ji ˆ %i&j ˆ &j &j. (A. 2 We can verify ha his is rue for a polynoial of degree wo by expanding Â(LÂ( F when Â(z ˆ 0 % ˆ 1z % ˆ 2z 2. In his case, 10

11 Â(LÂ( F ˆ 0%ˆ 1L%ˆ 2L 2 ˆ 0%ˆ 1( F%ˆ 2( F 2 ˆ 0ˆ2L 2 %(ˆ0ˆ1%ˆ 1ˆ2 L%(ˆ2 0 %ˆ 2 1 %ˆ %(ˆ0ˆ1%ˆ 1ˆ2 ( F%ˆ 0ˆ2( F 2 (A. 25 Coparing his wih (A. 21 indicaes c 2 ˆ 0ˆ 2 c 1 ˆ 0ˆ 1 % ˆ 1ˆ 2 c 0 ˆ 2 0 % ˆ 2 1 % ˆ (A. 26 consisen wih (A. 2. Given he resuls in (A. 22 o (A. 2, we can facor equaion (A. 17 as Â(L Â( F y. (A. 27 The relaionship beween he cos paraeers b 1,..., b and he polynoial ˆ 1,..., ˆ 1% j 2 coefficiens can be seen by expanding [(1&L(1& F]k for = 2. Using equaions (A. 18 and (A. 19, gives k1 b k 2 1% j b k [(1&L(1& F] k k1 1 % b 1 [ &L % (1% & F ] % b 2 L 2 &2(1% L%(1% % 2 &2(1% F%( F 2 (A. 28 b 2 L 2 & b 1 &2(1% L % 1%b 1 (1% %b 2 (1% % 2 & b 1 &2(1% F % b 2 ( F 2 A coparison of equaions (A. 21, (A. 2 and (A. 28 gives us c 2 ˆ 0ˆ 2 b 2 c 1 ˆ 0ˆ 1 % ˆ 1ˆ 2 &b 1 & 2b 2 (1% c 0 ˆ 2 0 % ˆ 2 1 % ˆ % b 1 (1% % b 2 (1% % 2 (A

12 Given, i is sraighforward o obain he cos paraeers b 1,..., b fro eiher ˆ 1,..., ˆ or c 1,..., c. However, expressions for ˆ 1,..., ˆ are non-linear and coplicaed. A firs sep in siplifying (A. 27 involves aking he polynoial onic, so he consan er is noralized o 1. Dividing boh sides of (A. 27 by gives a new polynoial A, wih coefficiens : ˆ 2 0 A(L A( F y ˆ 2 0 (A. 30 where A(z 1 % 1 z %... % z k ˆ k ˆ 0 (A. 31 Le c represen he arbirary consan, 1/ˆ 2. This can be fixed by assuing ha he agen 0 expecs o reach he arge pah in a seady sae; so ha, when L = F = 1, hen y = y*. Then, c 1ˆ 2 0 A(1 A( (A. 32 which gives us equaion (, A(L A( F y c (A. 33 This is an -order difference equaion, which can be siplified by expressing i as a firs 12

13 order difference equaion in arix for. As shorhand, le A(Ly x. Then he lef hand side of (A. 33 can be wrien as A( F x x % 1 x %1 % 2 2 x %2 %... % x % (A. 3 Pu fuure ers in arix for, wih naes of corresponding vecors wrien in bold underneah x %1 A( F x x % x %2! (A. 35 x % x % b g %1 Equaing his wih he righ hand side of (A. 33 gives us A( F x x % b g %1 c x c & b g %1 (A. 36 Because is he firs eleen of he vecor, his equaion can be re-expressed as a x firs-order difference equaion. Consruc an -vecor of zero s, wih 1 in he firs row. Tha is, [ ]. Then g g x c & b g %1 (A. 37 To solve his, we firs find an expression for firs equaion, wih oher eleens of g g. The following syse has (A. 37 as is being explained by ideniies. Naes of 13

14 corresponding arices are wrien in bold underneah. x c & 1 & & &1 &1 & x %1 x %1! x %&2 0! 0 % ! "! x %2! x %&1 (A. 38 x %& x % g f % G g %1 G is an x arix coprising he vecor &b in he firs row, an (-1x1 vecor of zeros in he res of he las colun and an ideniy arix, I&1 corner. Using (A. 38 o subsiue for successive leads of an infinie forward su: of rank -1 in he lower lef g, (A. 37 can be rewrien as x g f % Gg %1 f % G( f % Gg %1 %2 f % Gf % G 2 %1 (f % Gg %2 %3 f % Gf % G 2 %1 f % G 3 %2 (f % Gg %3 % j i0 G i f %i (A. 39 where G 0 1. (A. 39 provides a useful shorhand o which we reurn, when we discuss esiaion below. Bu o derive he decision rule, polynoial noaion is ore convenien. Firs, o pu he expression back in ers of y and y*, we can subsiue back for x A(Ly and f %i c : %i 1

15 A(L y j i0 G i c %i (A. 0 This replaces he forward operaor in (A. 33 wih an infinie forward su. To pu (A. 0 in error correcion for involves expressing boh sides as a level er and a weighed su of differences. The lef hand side can be reexpressed as A(Ly y % A(1 y &1 & A ( (L y &1 (A. 1 where A ( (L ( 1 % ( 2 L %... % ( &1 L &2 ( and for k = 1, 2,..., -1 k j jk%1 j To verify his, expand A(L wih = 3: A(Ly y % 1 y &1 % 2 y &2 % 3 y &3 y % 1 y &1 % 2 y &2 % ( 3 y &2 & 3 y &2 % 3 y &3 y % 1 y &1 % ( 2 % 3 y &2 & 3 y &2 y % ( 1 % 2 % 3 y &1 & ( 2 % 3 y &1 & 3 y &2 (A. 2 y & y &1 % (1% 1 % 2 % 3 y &1 & ( 2 % 3 y &1 & 3 y &2 y & y &1 % (1% 1 % 2 % 3 y &1 & ( 1 y &1 & ( 2 y &2 y % A(1y &1 & A ( (L y &1 ( ( wih and as claied. 1 2 % Subsiuing (A. 1 ino (A. 0 gives y % A(1y &1 & A ( (L y &1 j i0 G i c %i (A. 3 15

16 The nex sep is he decoposiion of he righ hand side ino levels and changes. I firs show his in scalar ers, which sees sipler and provides a represenaion ha can be inerpreed as a decision rule. I hen presen i in arix algebra, which is ore suied o esiaion. As pre and pos uliplicaion by he selecion vecor picks ou he op lef hand eleen of G i, he coefficiens on fuure values of are scalars. So, he righ hand side of (A. 3 can be represened as a polynoial in he forward operaor: j i0 G i c %i d 0 % d 1 F 1 % d 2 F 2 %... % d F D(F (A. D is an infinie order polynoial, wih ypical eleen d i c G i (A. 5 If = 2, as is he case wih any FRB/US equaions, hen G & 1 & (A. 6 and D F c [1 & 1 F % 2 1 & 2 % 3 1 % F 2 3 F 3 % 1 & % 2 2 F %... ] (A. 7 Subsiue (A. ino (A. 3. Then add and subrac A(1 &1 y a funcion of las period s disequilibriu ( &1 &y &1 : and rearrange o ake 16

17 y & A(1y &1 % A ( (L y &1 % D(F % A(1 &1 & A(1 &1 A(1( &1 & y &1 % A ( (L y &1 % D(F & A(1 &1 (A. 8 Group and difference he coefficiens on y* by defining D(F D(F & A(1 &1 (A. 9 where D(F d 0 % d 1 F % d 2 F 2 %.... (A. 50 wih d 0 A(1 1 % 1 % 2 %... % (A. 51 and, for j = 1, 2,... d j A(1 & j j&1 i0 d i (A. 52 To verify his, expand D(F & A(1 &1 as 17

18 D(F & A(1 &1 d 0 % d 1 F% d 2 F 2 % d 3 F 3 %... & A(1 &1 % A(1 & A(1 A(1 & A(1 % d 0 % d 1 F% d 2 F 2 % d 3 F 3 %... A(1 & A(1& d 0 % d 1 %1 % d 2 F 2 % d 3 F 3 %... (A. 53 A(1 % A(1& d 0 %1 & A(1& d 0 & d 1 %1 % d 2 F 2 % d 3 F 3 %... A(1 % A(1& d 0 %1 % A(1& d 0 & d 1 %2 & A(1& d 0 & d 1 & d 2 %2 % d 3 F 3 %... Fro coninuing he expansion of equaion (A. 53 in his anner he coefficiens on lags of can be seen o ach (A. 52. Subsiuing (A. 9 ino (A. 8 hen yields y A(1( &1 &y &1 & A ( (L y &1 % D(F. (A. 5 which can be rewrien as he decision rule (1 y a 0 ( &1 &y &1 % j &1 k1 a k y &k % E &1 j j0 d j %j (A. 55 Where, as shown above, a 0 d 0 A(1 1% 1 %...% 1% j j1 j &1 jk1 a k y &A ( &k (L y &1 a k & ( k & j for k = 1, 2,..., where and, for j = 1, 2,... d j A(1 & j j&1 i0 d i A(1 & A(1A( j j&1 i0 G i jk%1 j 18

19 A arix represenaion (A. 5 provides an equaion ha is siple o inerpre, bu canno be esiaed. One reason for his is ha expeced values of he arge y* are no observable. Anoher is ha (A. 5 conains an infinie nuber of coefficiens (albei reflecing +n underlying paraeers. We can address boh hese issues afer expressing he forward leads in arix algebra. Reurn o (A. 39, which is an infinie su of leads of he vecor f. Each lead of f can be decoposed ino he lagged level and a su of changes. Tha is, i f %i j f %j % f &1. To see his, add and subrac lagged levels fro he righ hand side: j0 f f %1 f & f &1 % f &1 f % f &1 f %1 % f (A. 56 f %1 % f % f &1 And so on. The weighed suaion of leads of f can hen be expressed as: j G i f %i i0 G 0 f % G 1 f %1 %... % G f % G 0 f % f &1 % G 1 f %1 % f % f &1 %... % G j j G j f &1 % j G j f % j G j f %1 %... % j j0 j0 j1 jk j0 f % f %j &1 G j f %k %... [I&G] &1 f &1 % [I&G]&1 f % [I&G] &1 G 1 f %1 %...% [I&G]&1 G k f %k %... (A. 57 [I&G] &1 f &1 % j k0 [I&G] &1 G k f %k 19

20 where he second las line uses he forula for he su of an infinie series j jk G j [I&G] &1 G k wih G 0 I. Subsiuing (A. 57 ino (A. 39 gives: x [I&G]&1 f &1 % j k0 [I&G] &1 G k f %k (A. 58 Fro he definiion of f c we have: f c & c &1 c (A. 59 Subsiuing hese ino (A. 58 and subsiuing (A. 1 for of levels and differences of y*: x gives an expression in ers y %A(1 y &A ( &1 (L y c &1 [I&G]&1 &1 % c j k0 [I&G] &1 G k %k (A. 60 In a seady sae, y = y* and y 0. This gives A(1 c [I&G]&1 (A. 61 Subsiuing his ino (A. 60 gives: y %A(1 y &A ( &1 (L y A(1 &1 &1 % c j k0 [I&G] &1 G k %k (A. 62 A(1 &1 % Z where Z represens he conribuion fro he expeced growh of he arge. We reurn o his forulaion in considering odel-consisen expecaions, below. Rearranging gives: 20

21 y A(1 &1 & y &1 % A ( (L y &1 % Z (A. 63 This differs fro (A. 5 in ha a arix expression, Z, is subsiued for he forward polynoial D, which was defined recursively. VAR Expecaions Suppose ha he arge, y* can be idenified ouside he PAC equaion. In he erinology used earlier, his iplies ha here is no saionary coponen of he arge, so y 1(. Also suppose ha expecaions of y* are generaed as if by a vecorauo-regression (VAR. Le z be an n-vecor of variables coprising he deerinans of expecaions of he arge y*. Define as he firs eleen of, so: z n z (A. 6 where n is a selecion vecor coprised of 1 as he firs eleen hen n-1 zeros. Then, define H as he n x n arix of coefficiens generaing one-period ahead forecass of z. Tha is: z %1 Hz (A. 65 If z conains ore han one lag of a variable he corresponding row of H will conain zeros excep for a 1 on he relevan lag. H is ofen also resriced o be consisen wih assupions abou hoogeneiy and orders of inegraion. Fuure levels of z are obained by repeaed subsiuion: z %2 Hz %1 H(Hz H 2 z (A

22 ore generally, z %i H i z (A. 67 And, o obain differences: z %i H i z & H i&1 z H& I H i&1 z (A. 68 This can be lagged an exra period, o pu i in ers of predeerined variables: z %i H& I H i&1 Hz &1 H& I H i z &1 (A. 69 H i H& I z &1 An esiaion equaion Z Le represen he conribuion fro he expeced growh of he arge, as in (A. 62. Tha is, Z c j k0 [I &G] &1 G k %k (A. 70 As y* is an eleen of he inforaion vecor z, (see equaion (A. 6, his can be reexpressed in ers of fuure levels of z : Z c j k0 [I &G] &1 G k n z %k (A. 71 and, using (A. 69, in ers of lagged levels of z : 22

23 Z c j h z &1 k0 [I &G] &1 G k n H k H& I n z &1 (A. 72 h is a 1 x n row vecor. Is ranspose, an n x 1 colun vecor, is h c j k0 [I &G] &1 G k n H k H& I n c H& I n j c H & I n j [I &G] &1 G k k0 k0 (H k n n H k (G k [I &G] &1 (A. 73 This can be siplified by use of colun sacks. One propery of colun sacks is ha he colun sack of a colun vecor equals he vecor iself. So vec(hh. Anoher is ha he colun sack of a produc of hree arices is vec(abc (C Avec(B. Repeaed applicaion of hese properies gives: 23

24 h [vec(h] vec c H & I n j (H k n k0 (G k [I &G] &1 c c [I &G]&1 H & I vec n j (H k n k0 [I &G]&1 H & I n j G k (H k vec k0 (G k n (A. 7 c c [I &G]&1 H & I n j G H k n k0 [I &G]&1 H & I n I n & G H &1 n This represens he infinie sus of he H and G arices as a finie expression ha can be calculaed. Subsiuing (A. 71 and (A. 61 ino (A. 60 and rearranging gives: y A(1 &1 & y &1 % A ( (L y &1 % h z &1 (A. 75 where h is consruced as in (A. 7. As discussed in he ex, his equaion can be esiaed boh in resriced for (wih he resricions iplied by he consrucion of he G and H arices and as a siple linear regression. Alernaive specificaions of he arge The previous discussion has assued he arge, y* is a known scalar. More usually, he arge will be a linear cobinaion of several variables, wih weighs ha need o be esiaed. For rending variables, i is possible o esiae hese weighs fro a saic regression - he firs sep in he Engle-Granger 2-sep procedure. These esiaes are inefficien, bu super-consisen. This approach is inuiive, siple o progra and faciliaes he iposiion of cross-equaion resricions. For saionary variables, 2-sage esiaion is no longer super-consisen. 2

25 In FRB/US, arges coonly coprise a rending coponen, denoed y 1(, which has a coefficien of 1 and conains eleens wih weighs ha are consrained or esiaed ouside he PAC equaion and a saionary coponen, denoed y 0( wih a coefficien vecor ha is esiaed wihin he PAC equaion. Tha is, y 0( % y 1(. Siilarly, he VAR forecass above can be decoposed as h z h &1 0 z 0 &1 % h 1 z 1 &1. We separaely idenify he lagged level of he rending coponen as a variable beginning wih a Q, expeced values of he rending coponen as a variable beginning wih a Z and cobine lagged and expeced values of he saionary coponen as one variable beginning wih a Z. The algebra and copuer code for hese ransforaions is se ou below. Alernaive copuer code for he h vecor FRB/US codes expecaions of he arge in hree differen ways. Equaion (A. 7 applies when he VAR is esiaed wih he level of y* as he dependen variable, as in (A. 6. Then he vecor h is coded as he vecor PV_COF in he file ge_pv as follows: PV_COF = SUMA1*SUMAB*W2*W1_DL*KRON(IG,IH where W2 = INVERSE{ EYENM-KRON[MAT_G,TRANSPOSE(MAT_H ] } and W1_DL = KRON{ TRANSPOSE(IG*INVERSE(EYEM-MAT_G, TRANSPOSE(MAT_H- EYEN } SUMA1 = A(1 and SUMAB = A( are scalars, whose produc is c IG and IH correspond o he selecion vecors EYEN, EYEM and EYENM are ideniy arices of order n, and nx respecively MAT_G and MAT_H correspond o he arixes G and H. Noaional correspondences are close bu inexac. For exaple, he order of rows in MAT_G and IG is reversed in G and sign o hose in G. and n and eleens of MAT_G have he opposie 25

26 In FRB/US i is coon o ipose he resricion ha he rending coponen of he arge is inegraed of order 1. In such cases, he VAR is esiaed wih dependen variable insead of as he. This specificaion faciliaes he iposiion of oher resricions (for exaple, inflaion neuraliy and could ake inference easier (hough he VAR s are no used for his purpose. Differencing he arge involves replacing (A. 6 wih: n z (A. 76 Using (A. 65 and (A. 67, which are unchanged, his iplies: %k n z %k n H k%1 z &1 (A. 77 Subsiuing his ino (A. 70 iplies a odified for of (A. 72 Z 1 c j k0 [I &G] &1 G k n H k%1 z &1 (A. 78 h 1 z &1 which differs fro (A. 72 hrough he disappearance of gives a new version of (A. 7: I n. The sae seps as above h 1 c [I &G]&1 H I n & G H &1 n (A. 79 and W1_DL in he copuer code is replaced wih: 26

27 W1_DD = KRON(TRANSPOSE(IG*INVERSE(EYEM-MAT_G, TRANSPOSE(MAT_H Anoher variaion is o cobine in one er he presen value of expeced values of he arge (in FRB/US noaion, a Z variable, wih is lagged level (a Q variable. This is a coon approach for coding of saionary conribuions o he arge. I is also coon o drop he resricion ha he elasiciy of y wih respec o his coponen of he arge is one. Supposing he coefficien on he saionary coponen of he arge is eans rewriing (A. 39 as: x j i0 G i f %i (A. 81 Subsiuing he definiions of x and f ino his gives: A(Ly j G i i0 cy0( %i (A. 82 Leing y 0( be he firs eleen of a vecor z 0 and using his in a VAR as above iplies y 0( %i n z 0 %i n H i%1 z 0 &1 subsiuing his in, hen using he sae seps as above gives 27

28 A(Ly c j G i i0 n H i%1 z 0 &1 c j i0 G i n H i%1 z 0 &1 c j H H i i0 n G i z 0 &1 c vec H j i0 c H H i n G i vec j H i i0 n G i z 0 &1 z 0 &1 (A. 8 c H j vec H i i0 n G i z 0 &1 c H j i0 G i H i vec n z 0 &1 c h 0 z 0 &1 H I n & G H &1 n z 0 &1 and W1_DL in he copuer code is replaced wih: W1_LL = KRON( TRANSPOSE(IG, TRANSPOSE(MAT_H Model consisen expecaions To find an expeced pah for he arge, y* ha is consisen wih odel forecass, firs subsiue (A. 1 ino (A. 62: A(Ly A(1 &1 % Z (A

29 Muliply hough by A( F A( FA(Ly A( FA(1 &1 % A( FZ (A. 87 The lef hand side can be pu in ers of y* by using (A. 32 and (A. 33 A(1A( A( FA(1 &1 % A( FZ (A. 88 Then rearranging and expanding he polynoials gives A( FZ A(1 A( & A( F &1 Z % j i1 i i Z %i A(1 1 % j i1 i i & &1 % j i1 i i &1%i (A. 89 Z j i1 i i Z %i % A(1 % j i1 i i & &1%i The las er is a weighed su of higher order differences. This can be rewrien as a reweighed su of firs differences. To see his add and subrac inerediae levels of %k : %k & %k & %k&1 % %k&1 %... & k j i0 %i (A. 90 I follows ha 29

30 j i1 i i & &1%i % 2 2 & %1 % 3 3 & %2 % & %3. %... % & %&1 & 2 2 %1 & 3 3 %1 % %2 &... & &1 j k0 %k (A. 91 &1 & j k1 &1 j jk j%1 j%1 %k Subsiuing his ino (A. 87 gives Z j i1 i i Z %i % A(1 &1 & j k1 &1 j jk j%1 j%1 %k (A. 92 As discussed in he ex, given for k = 1,... T and for j = T,... T+, hen his can be used o solve for. Z %k Z %j 30

31 References J. Kennan, (1979 The Esiaion of Parial Adjusen Models wih Raional Expecaions, Econoerica, Volue 7, Issue 6, pp S. Nickell (198 Error Correcion, Parial Adjusen and All ha: An Exposiory Noe Oxford Bullein of Econoics and Saisics, 7, J. J. Roeberg (1982 Price Adjusen and Aggregae Oupu The Review of Econoic Sudies Volue 9, Issue (Oc 1982, P.A. Tinsle1993 Fiing boh daa and heories: Polynoial adjusen coss and error-correcion decision rules. Federal Reserve Board of Governors, Finance and Econoics Discussion Series,