The Effect of Past and Future Economic Fundamentals on Spending and Pricing Behavior in the FRB/US Macroeconomic Model

The Effect of Past and Future Economic Fundamentals on Spending and Pricing Behavior in the FRB/US Macroeconomic Model Peter von zur Muehlen (cid:3) February 16, 2001 Abstract Thispaperderivesandpresents meanleadsandlagsaswellaspatternsofrelative importanceweightsimpliedbythePAC(polynomial-adjustment-cost)error-correction equations which form the core of the FRB/US model at the Federal Reserve Board. Relative importance weights measure the contributions of past and future expected changesinfundamentalsoncurrentdecisions. Theseandtheassociatedmeanlagsand leadscanbeconsideredsummarymeasuresofkeydynamicpropertiesofFRB/US.The spending equations are those for total consumption, durable consumption, business equipment, residential housing, and private inventories. The pricing equations are thoseforthepricelevelandwagegrowth. Inaddition FRB/UShasonePACequation fordividends andoneforlaborhours. JEL Classification: C3,C5,E1 Keywords: Macro modeling,dynamics,expectations,polynomialadjustmentcosts,errorcorrection, meanlags and leads. (cid:3) Board ofGovernorsofthe FederalReserveSystem. Iwouldlike to thankDavid Reifschneiderforhis helpfulcomments.

LEAD & LAG DISTRIBUTIONS IN FRB/US– FEBRUARY 2001 2 1 Introduction Two principles guide the specification of behavioral equations in the FRB/US macroeconomicmodelof theU.S. One is theassumptionofoptimizingbehaviorby rational participantsintheeconomywhoseektoachievehighestlevelsofwelfareandprofits. Theotheris thepresenceoffrictionsthat(except inthecaseofpureassetprices)preventinstantaneous achievementof goals set by optimizingagents.1 A corollary of these two principles is that intheirdecisions,firmsandhouseholdslookforwardandmakeplansforthefuture,giving expectationsa pivotalroleinmodelingtheUSeconomy. To put these principles into effect for modeling the dynamic behavior of the US economy, FRB/US introduces the distinctive feature of explicitly separating expectations of future events from any delayed responses to them. Expectations are, of course, by nature unobservable, and so they need to be modeled. FRB/US generates estimates of expectations held by individuals and firms using the forecasts of two alternative representations of the economy: (1) either a small VAR model, auxiliary to the large model and assumed to be known to all agents, or (2) FRB/US itself. In the latter case, households and firms are assumed to have a detailed understanding of the entire economy, as represented by the model. TotheextentthatFRB/USisatruerepresentationoftheeconomy,such“modelconsistent” expectations are considered “rational.” Underlying the alternative VAR approach to expectations formation is the assumption that knowledge is more limited or segmented and that agents economize on information by focusing on historical interactions among three macroeconomic variables: the rate of inflation of consumption goods; the federal funds rate, reflecting monetary policy; and the gap between real output and its trend level, reflecting thecyclicalstateoftheeconomy.2 As noted above, the need for forming expectations in decision making other than asset pricingarisesfromdelaysincarryingoutplansthataredelayedbyfrictionsandconstraints 1The FRB/US macroeconomic model is documented in Brayton and Tinsley ((eds.), 1996), Brayton, Levin,TryonandWilliams(1997b),andBrayton,Mauskopf,ReifschneiderandWilliams(1997a) 2In principle, the VAR approachto modeling expectationsneed not be “irrational.” If decision makers engage in learning and revise their parsimonous “rest-of-the-world” VAR model in accordance with new information, the interplay between decisions based on forecasts from that model and its evolution implies eventualconvergencetowardexpectationthatareconsistentwiththelargeandpresumablytruemodelofthe economy.

LEAD & LAG DISTRIBUTIONS IN FRB/US– FEBRUARY 2001 3 in dynamic adjustments, which may differ across sectors. The presence of frictions in adjustment imposes a necessity to look ahead: the longer it takes to reach a desired (or targeted) level of expenditure, the longer will be the necessary forecast horizon; and if adjustmentisveryslow,decisionsmadetodaywillbemoreheavilyinfluencedbyexpectationsfartheroutinthefuture, meaningthatgreaterweightwillbegiventoexpectedevents inthedistantfuture. Frictions arise from a variety of sources. One is incompleteness of information on which to base decisions. If acquiring information is costly, efforts to learn about the environment, using, for example, signal extraction, search, and experimentation methods, will lead to a protraction of initially formed plans. Another source, affecting scheduling and delivery,ispurelymechanicalanddependson theeconomy’scommunications,transportation, and production infrastructure. Institutional arrangements, such as price and wage contracting over multiple periods also serve as impediments to instantaneous action. For example,Calvo(1983)showedthatifexpirationdatesofwagecontractsarerandomlydistributed over the economy, the aggregate of wages in any period is a weighted average of unexpired contract wages. Finally, as first proposed by Eisner and Strotz (1963), capital, labor, and otherbusinessexpendituresare subject toconvexcosts ofadjustment,costs that showupas reductionsin profit. Instances of frictions are familiar to economists. In the case of capital investment, the time-to-build argument is familiar: it takes time to procure and install equipment and to train workers to use it because often, many steps and negotiations come between initial conceptualizations, drawing of plans, developments of complete specifications, and final delivery of a satisfactorily engineered machine. In many instances, it is more expensive to get accelerated delivery from suppliers. In the case of residential housing expenditures, the time between a change in income or the mortgage rate and expenditure on a residence is filled with planning, search, construction, loan applications, etc.—all things that delay ultimate settlement. Adjustments in hourly compensation may be slowed by a number of factors, including such contractual arrangements as alluded to above, delays in workers’ recognitionofachangeintherealwage,fearofeffectsonmoraleinthecaseofdownward adjustments, and many similar factors familiar to labor economists. Even prices are sluggish. Under the assumptionof markup pricing over unit costs, delays in the adjustment of

LEAD & LAG DISTRIBUTIONS IN FRB/US– FEBRUARY 2001 4 wages can affect the pattern of price adjustments over time. However, firms can and do excerciseconsiderablediscretionovermarkups,necessitatedforexamplebyinstantaneous fluctuationsintheprices ofintermediatecommoditiesused as inputs. Because of the variety of sources of delays, the paradigm for friction in FRB/US is a generalization of the standard theory of convex adjustment costs; this generalization was developedby Tinsley(1993). As is well known,standard econometricmodelsthat impose quadratic costs of adjustment have not done well in representing the dynamic behavior of macroeconomicvariables. Whereasquadraticadjustmentcostsarebydefinitionconfinedto first-orderdifferencingeffects, thegeneralizedmodel,namedPACfor“polynomialadjustment costs,” permits richer dynamics within a theoretically-based framework that remains parsimonious and is amenable to testable restrictions. The PAC model comprises a generalized class of specifications for frictions in economic activity, including not only costs of k-th-orderdifferencing butalso costs associatedwith changes in movingaverages, such as mightbeassociatedwithseasonalortermcontractsandtime-to-buildcapitalexpenditures. The implied Euler conditions lead to decision rules that can be represented as rational error-correction processes.3 Thesedifferfromconventionalerror-correction modelsintwo important ways. First, higher-order autoregressive lags of the decision variable will now appear in the error-correction model, where the degree of the autoregression is one less than the polynomial degree specified in the adjustment cost criterion. The PAC formulation therefore offers a plausible theoretical justification for high-order autoregressivity often foundin empiricalerror-correction modelsofimportantmacroeconomicvariablesin the US economy. Second, the term capturing forecasts of future changes in theplanned or targeted activity of the modeled variable now becomes subject to more complex discounting, involving the multiple roots of the higher-order polynomial adjustment cost function. Thismeans thattherelativeimportanceofexpected changes inthefutureequilibriumpath ofavariableto itscurrent behaviorhingesontheexactnatureoffrictionsthatcharacterize thebehaviorofthatvariable. This last comment suggests a useful way of viewing the implicationsof any particular 3A version of the theory of rational error correction, developed by Tinsley (1993), was published by KozickiandTinsley(1999).

LEAD & LAG DISTRIBUTIONS IN FRB/US– FEBRUARY 2001 5 dynamicspecification,onethatrevealsjusthowaseriesrespondstoitsownpastandfuture fundamental—its equilibrium path. The optimal decison rule requires that at any time, an action that is subject to friction will be influenced by partially fulfilled plans for adjustment formulated in earlier periods and to revisions to plans anticipated in future periods, based on current information. As a consequence, decisions and actions in the current period can be expressed as weighted averages of the equilibrium values inherited from past periods and the target path anticipated in the future, where the weights measure the relativecontributionsofpastandfutureexpectedfundamentalstocurrentlevelsofthedecision variable. It should come as no surprise that since different economic variables are subject to unique forces and frictions, each series will have its characteristic signature pattern of weights influencing its movements over time. These patterns are derived and discussed in theremainderofthispaper. 2 Adjustment Dynamics in FRB/US AsdescribedinBrayton andTinsley((eds.), 1996),firmsandhouseholdscalculatedesired equilibriumpathsofexpendituresorprices,havingformedexpectationsgeneratedbyeither of the two methods mentioned above. At time t , the equilibrium level of a variable y t is denoted y (cid:3) t . FRB/US adopts a singleparadigm to represent dynamic adjustments of major decision variables as follows: Participants seek to balance the expected cost of deviating fromtheequilibriumpath,definedastheinfinitediscountedsumofthesquareddifferences between y t+ j and y (cid:3) t+ j ,where j rangesfromzero(currenttime)toinfinity,againstthecosts ofmakingchanges: E (cid:0)t 1 1 j = X 0 (cid:12) j h ( y t+ j (cid:0) y (cid:3) t+ j ) 2 + b 1 ( (cid:1) t+ j y t+ j ) 2 + b 2 ( (cid:1) 2 y t+ j ) 2 + (cid:1) (cid:1) (cid:1) + b m ( (cid:1) m y t+ j ) 2 i ; (1) where E (cid:0)t 1 f : g is a forecast of future costs based on information available at the end of the preceding period, t (cid:0) 1 , m is the number of b coefficients in the cost function, and (cid:12) is a discount factor between zero and one. The first squared term is the cost incurred if the variable, y , deviates from its equilibrium level, y (cid:3) , in period t + j . The remaining

LEAD & LAG DISTRIBUTIONS IN FRB/US– FEBRUARY 2001 6 terms measure the expected costs of frictions that arise with changes in y . The letter, (cid:1) , denotes change, so that (cid:1) y t+ j = y t+ j (cid:0) y t+ j (cid:0) 1 is the change in the level of y from period t + j (cid:0) 1 to t + j , and b 1 is the unit cost of making this change. In the next term, (cid:1) 2 y t+ j = (cid:1) ( y t+ j (cid:0) y t+ j (cid:0) 1 ) = ( y t+ j (cid:0) y t+ j (cid:0) 1 ) (cid:0) ( y t+ j (cid:0) 1 (cid:0) y t+ j (cid:0) 2 ) is the change in the change. Inthisinstance, b 2 representstheunitcostofaccelerating y . Inthefollowingterm, b 3 istheunitcostofchangingtherateofacceleration. The outcomeof this minimizationis an adjustmentprocess that acts to gradually eliminate the remaining gap between a price or a spending level and its desired level in a way thatdependsonfutureexpectedchangesinthetargetlevelaswellasonpastchangesinthe decisionvariable. Formally,thedecisionrulethatminimizes(1)is thefollowing,4 (cid:1) y t = (cid:0) a 0 ( y (cid:0)t 1 (cid:0) y (cid:3) (cid:0)t 1 ) + m j X (cid:0) = 1 1 a j (cid:1) y (cid:0)t j + E (cid:0)t 1 f 1 i= X 0 f i (cid:1) y (cid:3) t+ i g : (2) Thisisageneralizederrorcorrectionequationwhichisinfluencedbypastchangesin y and byexpectedchangesinthefutureequilibriumpath, y (cid:3) . Theoptimaladjustmentof y toward itsequilibriumisdeterminedby: (1)lastperiod’sdeviationof y from itsequilibriumlevel, y (cid:0)t 1 (cid:0) y (cid:3) (cid:0)t 1 ,(2)pastchangesinthelevelof y ,and(3)aweightedforecastoffuturechanges in equilibrium levels, y (cid:3) . The forecast weights, f i , are functions of the discount factor, (cid:12) , and the unit cost parameters, b 1 ; b 2 ; (cid:1) (cid:1) (cid:1) . It is important to note that lagged changes in y beyond t (cid:0) 1 would notappear in(2) wereit notfor thepresenceof adjustmentcost terms of order higher than quadratic, thoseinvolving b 2 ; b 3 and so on. Indeed, the autoregressive orderin(2)isexactly m (cid:0) 1 . IntheestimatedPACequationsfordurableequipmentspending, the price index for adjusted final sales (the principal deflator of the model), and the growth rate of employee compensation, lags exceeding 1 cannot be rejected, thus presentingaplausiblecaseforhigher-orderadjustmentcosts inat leastthesethreeinstances.5 The optimal level of activity, y t , can be represented as a two-sided moving average of 4SeeTinsley(1993)fordetails. 5The estimated higher-order terms in (cid:1) y can plausibly mimic more generalized moving average processes,suchaswouldbeimpliedbytime-to-build,learning-by-doing,orstaggeredcontracting.

LEAD & LAG DISTRIBUTIONS IN FRB/US– FEBRUARY 2001 7 pastand futureequilibriumvalues, y (cid:3) , y t = E (cid:0)t 1 f j 1 (cid:0) = X 1 w j y (cid:3) t+ j g ; (3) wheretheweights, w j ,sumtoone.6 Thisequationsaysthatthecurrentlevelof y is,inpart, thelegacyofearlierfundamentals, y (cid:3) ,and,inpart,theanticipationofitsfutureequilibrium path. One advantage of this equation is its parsimonious appearance. A second benefit is that the bell-shaped pattern of the weights plotted in Figures 1 and 2 providea simpleand intuitive view of the dynamic behavior that is characteristic to each activity. One would expectvariablessubjecttosignificantadjustmentcoststobefarmoreinfluencedbydistant eventsthan variablesthatmovemoreorlessunimpeded. 3 The Distribution of Lead and Lag Weights The FRB/US model contains nine PAC equations involving polynomial adjustment costs: spending on nondurablegoods and services; spending on consumerdurablegoods; investmentinresidentialhousing,producers’durableequipment,andtradeinventories;aggregate laborhours;thepricelevel;andthegrowthratesofemployeecompensationanddividends. Bycontrast,yieldsonequitiesandbonds,andtheexchangeratearemodeledasassetvaluationequations. Thismeansthatvaluations,whileforward-looking,arenotconstrainedby adjustmentfrictions: theyarepurely forward looking.7 Figures 1 and 2, drawnat identical scales to facilitatecomparisons,displaytherelative importance weights for all the PAC equations in FRB/US and of the yield on a ten-year Treasury bond. The area under each curve sums to 2, since each point on a plot measures theweight, w i ,relativetoitsbackwardorforwardsum. So,forexample,pointstotheright ofcenter,depictingforwardweights,arethe w i dividedbythesumofallweightstotheright of the center. Similarly, relative-importance lag weights are computed based on their sum goingbackward. Each pointtotherightofcentertherefore showsthecontributionof y (cid:3) to 6ThedistributionofweightsisderivedinAppendixA.3. 7FulldetailsaboutthespecificationoftheseequationsaregiveninBraytonandTinsley((eds.),1996).

LEAD & LAG DISTRIBUTIONS IN FRB/US– FEBRUARY 2001 8 thecurrentdecisionasafractionofthecontributionsofallfutureexpected y (cid:3) . 8 Weightsfor pastquarters,showntotheleftofthepeakscenteredatzero,indicatetherelativeimportance ofpastequilibriumlevelstocurrentdecisions. Because,astimepasses,olderplansbecome completed, the relative weights for past planning periods approach zero. Conversely, the relativeweightsofmoredistantfutureweights,showntotherightofzero,graduallyvanish becauseofdiscountingandbecausemoredistantfutureneedscanbesatisfiedbycorrective actionsin futurequarters. In Figures 1 and 2, we can broadly distinguish two types of weight distributions. One shape, typified by expenditures on consumption, equipment investment, the price level, wage growth, and, most notably, dividends, tends to be relatively flat, indicating a strong influence of planning considerations in the evolution of these variables. In the case of durable equipment expenditures, for example, the shape of the curve conforms to well knownpriorsaboutthesluggishnessofcapitalgoodsinvestment: itisslowedconsiderably by frictions and adjustment costs, leading to a pattern that gives much weight to both past changes and to expected future planned changes. Although firms, in principle, plan over an infinite horizon, the effective length of the planning period is limited by the extent of frictions associated with a firm’s actions. The similarity of the patterns for consumption and equipment expenditures at frequencies in excess of one year is likely a reflection of firms’settingtheircapital spendingplansin anticipationoffuturesales. The second group of shapes, typified by inventories, residential housing expenditures, and during the first year, durable consumption, exhibits relatively large weights at neartermfrequencies. Inventoryaccumulationtendstobeafastreactercomparedtowagesand prices, with very little adjustment needed after the first three to four quarters. Although firms have, at least theoretically, the ability to adjust wages, hours, prices, and inventories in response to demand shocks, it appears to be typically inventories that bear the initial brunt, withhours followingas thenext option.9 Likewise,housingexpendituresreact very quickly,indeed,fullywithintwoyears. Thisisconsistentwiththewellknowntendencyof 8Under the asssumption that agents in various sectors plan their actions rationally, the required rate of return, includinga risk premium, receivedby investorsand householdsis approximatedby the realrate of returntoequity. 9Theemergenceofjust-in-timeinventorymanagementmaychangethisorderingsothatmanipulationof inventoriesmaybecomelessofanoption.

LEAD & LAG DISTRIBUTIONS IN FRB/US– FEBRUARY 2001 9 housingexpenditurestobeamongthefirsttoreacttochangesinmonetarypolicy. Therelativelyfastmovementofhousingexpendituresisconsistentwithaslowlychanginghousing stock characterized by a low depreciation rate. One might note that a certain disparity between stock and flow adjustment also describes other forms of capital. A comparison betweentheweightpatternsofpricesandwagesshowsfurtherthattheformerhaveamuch shorter effective time horizon than the latter. This apparent lack of dynamic coordination suggests that mark-ups over unit labor costs are flexible, allowing prices to adjust more quickly than wages, which tend to be set contractually over several periods. Inventories and thepricelevelas wellas wagegrowthexperiencesomedegreeofovershootingbehavior within one or two years: in the first two cases, the weight density includes negative values for the influence of equilibrium stocks and prices, respectively, beyond two years fromthecurrentdate,whilewagesreactnegativelytoequilibriumwagesmorethan4years away. In Figure2 thecharacteristicweightpattern ofapurely foward-lookingassetprice, the ten-year treasury bond yield, has been added for illustrative purposes. The ten-year yield is determined by a term structure equation which forecasts the federal funds rate over the maturity of the bond. Note that the relative importance weights of expected future funds rates declinein astraightlineovertheten-year planningperiod. Theassociatedmean lead isabout fouryears, roughlyhalfitsmean duration. 4 Implied Mean Leads and Lags A summary measure of the effective average length of the forward planning period for a variable is the mean lead implied by its relative-importance weights. The mean lag of a series is useful as a measure of its characteristic average speed of response to unexpected shocks, i.e., events, for which planning is, by definition, impossible. Mean leads measure the average responsivenessto future expected events and are a reflection of the typical decision horizon for a variable. The mean lags and leads for the six series discussed here

LEAD & LAG DISTRIBUTIONS IN FRB/US– FEBRUARY 2001 10 are shown in Table 1.10 As indicated in the table, inventories, prices, and hours, for which frictions are among the least, compared with those of other series considered here, exhibit very similar mean leads and lags of less than a year, apparently reflecting firms’ abilities toadjustthesedecisionsvariablesquicklyandinterchangeablyaspredictedbytheory. The shortest lags are those for residential housing expenditures, whose mean lead and lag is 1/2 year. This may be explainedby production function features in construction: typically low capital intensity in the residential construction industry allows relatively quicker responsestochangesininterestratesandinexpectedpricesandincome. Likewise,financing at theresidentiallevelis fairlyqick and efficient. Durableequipmentand consumptionexpenditures, characterized by typically longer planning horizons, have mean leads of 1-1/2 years, as does wage growth. Interestingly, it is dividend payouts that exhibit the greatest persistence. Although earnings tend to be volatile, firms have incentives to smooth dividendpayoutstoavoidgivingtheirstockholdersfalsesignalsoffundamentaleventsthatare likely not as variable. Investors who own equity for their income stream may be discouraged from owning a stock if its dividends are unreliable; for the same reason, firms may be disinclined to raise dividends unless they are sure that earnings will remain high in the future. This suggests that dividend payouts can be interpreted as signals of the underlying fundamentals that drive cyclical earnings. Another explanation for highly persistent dividendpatterns maybefirms’desireto maintainmaximumflexibilitywithretainedearnings tofinance investment. Forprices,themeanlagisfourquarters,thesameasitsmeanlead. Themeanresponse lag fordurableequipmentexpendituresis roughlyoneand ahalfyears. 10An exact method of calculating mean leads and lags is presented in the appendix. For the truncated horizon, T (cid:20) 1 , anapproximationofthemeanleadofa seriesistheproductofthesequentialnumberof eachquarterintheforwardplanningperiodandthecorrespondingrelative-importanceweight, w(cid:22) = T i= X 0 i P w T i= i 0 w i ; (4) where P w T i= i 0 w i is therelative-importanceweightforthe i-thquarterin theplanningperiod. Meanlagsare calculatedsimilarly.

LEAD & LAG DISTRIBUTIONS IN FRB/US– FEBRUARY 2001 11 Table1: Equation Coefficients andImpliedMean Leads and Lags t (cid:0) 1 t (cid:0) 2 t (cid:0) 3 Equation (cid:0) a 0 a 1 a 2 a 3 Mean Lag Mean Lead DurableEquipment .095 .092 .232 6.16 5.56 Inventories .110 .544 3.14 3.07 Consumption .119 .081 6.70 5.81 DurableConsumption .197 (cid:0) .147 4.80 4.30 HousingExpenditures .155 .478 2.35 2.32 Price Deflator .082 .339 .258 3.95 3.95 Wage Growth .058 .192 .237 .184 5.70 5.70 Hours .124 .402 3.83 3.60 Dividends .043 .399 13.0 10.4 5 Implied Polynomial Cost Parameters AsshowninAppendixA.2,itispossibletoretrievetheimpliedpolynomialadjustmentcost parametersfromtheestimatederror-correctionequations. TheseareshowninTable2. The presence of second and third lags in only durable equipment expenditures, wage growth, andthepricedeflatorsuggeststhathigher-orderfrictionsarenotcommonintheU.S.economy. Observe that a high order in a variable’s polynomial adjustment cost function does notnecessesarily implyahighmean lead orlag ofitsseries,sincethesearedeterminedby theimpliedeigenvaluesofthepolynomial,showninTable3. 6 Two Caveats For each decision variable, theestimated lead and lag patterns in Figures 1and 2 are based on a single-equation, unitary error-correction model, estimated independently of other decisionvariablesthatmightinfluenceitsbehavior.11 11SomepreliminaryempiricalfindingsbyvonzurMuehlen(1993)suggestthatforageneralequilibrium vector rational error-correction model of multiple factor demands by a firm that maintains both final and intermediate inventories, the movements in factor inputs and inventories are influenced by disequilibrium gapsofallotherdecisionvariablesunderthecontrolofthefirminvaryingdegrees.

LEAD & LAG DISTRIBUTIONS IN FRB/US– FEBRUARY 2001 12 Table2: PolynomialCostParameters Impliedby Estimates Equation b 1 b 2 b 3 b 4 DurableEquipment 0.30 1.01 (cid:0) 0.23 Inventories 0.04 0.54 Consumption 0.72 0.08 DurableConsumption 1.14 (cid:0) 0.15 Housing 0.05 0.48 Price Deflator 0.00 1.29 (cid:0) 0.26 WageGrowth 88.7 (cid:0) 19.8 (cid:0) 1.30 0.18 Hours 0.19 0.40 Dividends 0.31 0.40 Theforegoingdiscussionalsoabstracts from thelikelypresence ofasymmetriesindynamic responses to shocks with respect to sign and phase of business cycle if frictions and/or adjustment costs for any of the variables treated here asymmetric with respect to business cycle phases. For example, it is easier to raise prices, even in the face a potential loss of customer base, than to replenish stocks when economy-wide capacity is strained. Similarly, downward nominal wage adjustment is less constrained when unemployment is high than it would be under normal conditions. The individual estimated error-correction equations on which the weight patterns are based, represent average responses over businesscycles and thusdonotreflect asymmetry,per se.12 12Tinsley and Krieger(1997)examineasymmetriesin price and outputadjustmentsin SIC two-digitindustriesand find thatasymmetric pricingappearsto be structural, with pricingdecisionsdependingon the positionoftheindustrypriceinrelationtoitstrend.

LEAD & LAG DISTRIBUTIONS IN FRB/US– FEBRUARY 2001 13 Table3: Eigenvalues ofDynamicEquations Equation Eigenvalues DurableEquipment .83 .62 (cid:0) .45 Inventories .74 .74 Consumption .87 .09 DurableConsumption .83 (cid:0) .18 Housing .69 .69 Price Deflator .83 .83 (cid:0) .37 WageGrowth .88 .88 (cid:0) .49 (cid:0) .49 Hours .72 .56 Dividends .92 .43 A Appendix This appendix describes a methodology for deriving the backward and forward weight distributionsimpliedbyerror-correction equationsofaPAC model. A.1 The Model TheEulerequationforthelossfunctionin (1)is,as shownin Tinsley(1993),13 y (cid:3) t = " 1 + k X m = 1 b k [ ( 1 (cid:0) L ) ( 1 (cid:0) (cid:12) F ) k ] # y t ; (5) whichcan beequivalentlyexpressedas, y (cid:3) t = (cid:17) (cid:13) C A ( ( L (cid:12) ; F (cid:12) ) F A ) ( y L t ) y t ; (6) 13 L and F aretheconventionallagandleadoperator,respectively.

LEAD & LAG DISTRIBUTIONS IN FRB/US– FEBRUARY 2001 14 where (cid:13) (cid:0) 1 = A ( 1 ) A ( (cid:12) ) is a constant and ensures that y t = y (cid:3) t in equilibrium. The lag polynomial, A ( z ) , is afunctionofthecost coefficientsin (1), and,expanded, isgivenby, A ( x ) = a 0 + a 1 x + a 2 x 2 + (cid:1) (cid:1) (cid:1) + a m x m : (7) The Euler condition implies the following PAC decision rule for for the I(1) decision variable, y t , (cid:1) y t = (cid:17) (cid:0) (cid:0) A A ( ( 1 1 ) ) ( ( y y (cid:0)t (cid:0)t 1 1 (cid:0) (cid:0) y y (cid:3) (cid:0)t (cid:3) (cid:0)t 1 1 ) ) + + A A (cid:3) (cid:3) ( ( L L ) ) (cid:1) (cid:1) y y (cid:0)t (cid:0)t 1 1 + + A S ( 1 t (cid:12) ( G ) A ; ( (cid:1) 1 ) y f (cid:3) 1 i= X ; ) 0 (cid:19) 0 m [ I (cid:0) G ] (cid:0) 1 G i (cid:19) m (cid:1) y (cid:3) t+ i g (8) where A ( L ) = A ( 1 ) L + [ 1 (cid:0) A (cid:3) ( L ) L ] ( 1 (cid:0) L ) , G isan m m (cid:2) m companionmatrixofthelead polynomial, A ( (cid:12) F ) ,14 and the m -dimensional vector (cid:19) m = [ 0 ; : : : ; 0 ; 1 ] 0 selects elements of thebottomrowof G . Givenaconsistentestimateofthecoefficients A ( 1 ) and A (cid:3) ( L ) in(8), thePACpolynomial A ( L ) is uniquelyidentified. A.2 Deriving Cost Parameters from the Lag Polynomial An important property is that A ( (cid:12) F ) A ( L ) is “self-reciprocal” with the property that the coefficient on each term L j is the same as the coefficient on ( (cid:12) F ) j . Expanding C ( L ; (cid:12) F ) yields, c m L m + (cid:1) (cid:1) (cid:1) + c 1 L + c 0 + c 1 ( (cid:12) F ) + (cid:1) (cid:1) (cid:1) + c m ( (cid:12) F ) m ; (9) where c i = (cid:13) k X m = j a m + j (cid:0) k a m (cid:0) k (cid:12) m (cid:0) k : (10) Thecostcoefficients, b k canberetrievedfromtheestimatedlagpolynomialcoefficients 14 G isthe“bottom-row”companionmatrixfeaturingtherequisitezerosandonesinthetopportion,and elements, (cid:0) a m (cid:12) m ; (cid:0) a m (cid:0) 1 (cid:12) m (cid:0) 1 ; :::; (cid:0) a (cid:12) inthebottomrow,wherethe a ’sarethecoefficientsin A ( L ) .

LEAD & LAG DISTRIBUTIONS IN FRB/US– FEBRUARY 2001 15 embedded in A ( L ) by expanding 1 + P m k = 1 b k [ ( 1 (cid:0) L ) ( 1 (cid:0) (cid:12) F ) ] k and equating like terms in L inthisand (9). A.3 Mean Leads and Lags Letting T ( L ; (cid:12) F ) = C ( L ; (cid:12) F ) (cid:0) 1 , themean lagand mean lead aredefined by, M e a n L a g = T T L ( 1 ( 1 ; ; 0 0 ) ) = (cid:0) A ( 0 A ) A ( 1 L ) ( 1 ) = (cid:0) A A L ( ( 1 1 ) ) (11) M e a n L e a d = T T F ( 0 ( 0 ; (cid:12) ; (cid:12) ) ) = (cid:0) A F ( A (cid:12) ( ) (cid:12) A ) ( 0 ) = (cid:0) A A F ( ( (cid:12) (cid:12) ) ) ; (12) where T i is the derivativeof T ( L ; (cid:12) F ) with respect to i = L o r F . For theprice equation, themeanlead and mean lagare thesame: 3.9.15 A.4 The Lead and Lag Weights Equation(8)maybesolvedas follows. Write, A ( L ) y t = A ( (cid:12) F ) (cid:0) 1 A ( (cid:12) ) A ( 1 ) y (cid:3) t (cid:17) y(cid:22) e t : (13) Define the vector z t = [ z ( t m ) ; z ( t m (cid:0) 1 ) ; (cid:1) (cid:1) (cid:1) ; : : : ; y t ] 0 , where z ( t m ) = F m y t is the m -th lead of y t . Using the previously defined companion matrix, G , the right hand side of (13) can accordinglyberewrittenas thedifferenceequation z t = G z t+ 1 + Y (cid:3) t : (14) Equation(14)canbesolvedbyforwardrecursion,where Y (cid:3) t = [ 0 ; 0 ; (cid:1) (cid:1) (cid:1) ; y (cid:3) t ] 0 . Repeated substitutionof y t ontoitselfyields, z t = A ( (cid:12) ) A ( 1 ) 1 i= X 0 G i Y (cid:3) t+ i ; (15) 15For (cid:12) = :9 8 , A ( L ) = 1 (cid:0) 1 :2 6 L + :0 8 L 2 + :2 6 L 3 , A ( (cid:12) F ) = (cid:0) 1 :2 3 L + :0 8 L 2 + :2 4 L 3 .

LEAD & LAG DISTRIBUTIONS IN FRB/US– FEBRUARY 2001 16 y(cid:22) e t = A ( (cid:12) ) A ( 1 ) (cid:19) 0 m 1 i= X 0 G i (cid:19) m Y (cid:3) t+ i : (16) Combiningequation(13)with(16), wehave, A ( L ) y t = y(cid:22) e t ; (17) which can be solved by backward recursion. Again, define an m (cid:2) m companion matrix H ofthelag polynomial, A ( L ) 16. Using H and thevector, x t = [ x ( t m ) ; x ( t m (cid:0) 1 ) ; (cid:1) (cid:1) (cid:1) ; : : : ; y t ] 0 , where x ( t m ) = L m y t ,(17)is castintofirst order, x t = H x (cid:0)t 1 + z t ; (18) whichhas thesolution, x t = 1 j = X 0 H j z (cid:0)t j (19) y t = (cid:19) 0 m 1 j = X 0 H j (cid:19) m z (cid:0)t j ; (20) = A ( (cid:12) ) A ( 1 ) (cid:19) 0 m 1 j = X 0 H j (cid:19) m (cid:19) 0 m 1 i= X 0 G i (cid:19) m Y (cid:3) t+ i ; (21) = A ( (cid:12) ) A ( 1 ) (cid:19) 0 m 1 j = X 0 1 i= X 0 H j (cid:19) m (cid:19) 0 m G i (cid:19) m Y (cid:3) t+ i : (22) From this we may cull the weights associated with y (cid:3) for all dates, 1 ; (cid:1) (cid:1) (cid:1) ; t (cid:0) 1 ; t ; t + 1 ; (cid:1) (cid:1) (cid:1) ; t + k (cid:0) 1 ; t + k g f t (cid:0) k ; t (cid:0) k + ,where k isarbitrarilylarge. Theaboveequation consistsofthefollowingsum, y t = + (cid:1) (cid:19) (cid:19) 0 m 0 m H H k k (cid:20) (cid:0) [ 1 G (cid:20) 0 G [ (cid:19) m 0 Y (cid:19) Y (cid:3) (cid:0)t (cid:3) (cid:0)t k k + + G G 1 (cid:19) m 1 (cid:19) m Y Y (cid:3) (cid:0)t (cid:3) (cid:0)t k + k + 1 1 + + (cid:1) (cid:1) (cid:1) + (cid:1) (cid:1) (cid:1) + G G k (cid:19) m k (cid:19) m Y Y (cid:3) ] t (cid:3) t+ 1 ] 16 H isthe“bottom-row”companionmatrixsimilarto G ,featuringtherequisitezerosandonesinthetop portion,andelements, (cid:0) (cid:11) m ; (cid:0) (cid:11) m (cid:0) 1 ; :::; (cid:0) (cid:11) 1 inthebottomrow.

LEAD & LAG DISTRIBUTIONS IN FRB/US– FEBRUARY 2001 17 (cid:1) + (cid:19) 0 m H 0 (cid:20) [ G 0 (cid:19) m Y (cid:3) t + G 1 (cid:19) m Y (cid:3) t+ 1 + (cid:1) (cid:1) (cid:1) + G k (cid:19) m Y (cid:3) t+ k ] ; (23) where (cid:20) = A ( (cid:12) ) A ( 1 ) (cid:19) m (cid:19) 0 m . Foreach datethen,thedistributedlead and lagweightsare, w w w w (cid:0)t (cid:0)t w t+ t+ k s t s k = = (cid:1) (cid:1) = (cid:1) (cid:1) = = (cid:19) (cid:19) (cid:19) (cid:19) (cid:19) 0 m 0 m 0 m 0 m 0 m H (cid:0) k i= X k i= X (cid:0) k i= X H k (cid:20) G s H 0 H 0 s H 0 0 (cid:20) G 0 i+ i (cid:20) i (cid:20) k (cid:19) m s (cid:20) G G (cid:19) m G i (cid:19) i+ ; i m s (cid:19) (cid:19) m m where P 1 s = (cid:0) 1 w t+ s = 1 .

LEAD & LAG DISTRIBUTIONS IN FRB/US– FEBRUARY 2001 18 References Brayton, F., E. Mauskopf, D. Reifschneider, and J.C Williams, “The Role of Expectations in the FRB/US Macroeconomic Model,” Federal Reserve Bulletin, 1997, 83, 227–245. Brayton, Flintand Peter A. Tinsley, “A Guideto FRB/US: A MacroeconomicModel of theUnitedStates,”FinanceandEconomicsDiscussionSeries96-42,Federal Reserve Board, Washington,D.C. (eds.), 1996. , Andrew Levin, Ralph Tryon, and John C. Williams, “The Evolution of Macro Models at the Federal Reserve Board,” Carnegie-Rochester Conference Series on PublicPolicy,1997,43. Calvo, Guillermo A., “Staggered Prices in a Utility-MaximizingFramework,” Journal of MonetaryEconomics,1983,12,383–398. Eisner, R. and R. Strotz, Determinantsof BusinessInvestment, Impacts of MonetaryPolicy,CMC, EnglewoodCliffs,N.J.: Prentice-Hall, 1963. Kozicki, Sharon and Peter A. Tinsley, “Vector Rational Error Correction,” Journal of EconomicDynamicsand Control,1999,23, 1299–1327. Tinsley, Peter A., “Fitting Both Data and Theories: Polynomial Adjustment Costs and Error-Correction Decision Rules,” Finance and Economics Discussion Series 93-21, Federal Reserve Board, Washington,D.C. 1993. and Reva Krieger, “Asymmetric Adjustments of Price and Output,” Economic Inquiry,1997,35,631–632. von zur Muehlen, Peter, “DynamicFactor Demand with Final and Intermediate Inventories,”1993. UnpublishedManuscript,Federal ReserveBoard, Washington,D.C.

LEAD & LAG DISTRIBUTIONS IN FRB/US– FEBRUARY 2001 19 Figure1: Distributionsofweightsin spendingequations Quarters sthgieW 3.0 2.0 1.0 0.0 -30 -25 -20 -15 -10 -5 0 5 10 15 20 25 30 3333....0000 2222....0000 1111....0000 0000....0000 <-------- Inventories <--------- Residential Housing <--------- Durable Consumption <-------- Consumption <------------- Equipment

LEAD & LAG DISTRIBUTIONS IN FRB/US– FEBRUARY 2001 20 Figure2: Distributionsofweightsinpriceand dividendequations Quarters sthgieW 3.0 2.0 1.0 0.0 -30 -25 -20 -15 -10 -5 0 5 10 15 20 25 30 3333....0000 2222....0000 1111....0000 0000....0000 <------ Hours <-------------- Price level <-------- Wage growth <--------- Dividends <--------- Bond yield