Optimal Monetary Policy in an Operational Medium-Sized DSGE Model

Board of Governors of the Federal Reserve System International Finance Discussion Papers Number 1023 July 2011 Optimal Monetary Policy in an Operational Medium-Sized DSGE Model Malin Adolfson Stefan Laséen Jesper Lindé Lars E.O. Svensson NOTE: International Finance Discussion Papers are preliminary materials circulated to stimulate discussion and critical comment. References to International Finance Discussion Papers (other than an acknowledgment that the writer has had access to unpublished material) should be cleared with the author or authors. Recent IFDPs are available on the Web at www.federalreserve.gov/pubs/ifdp/. This paper can be downloaded without charge from the Social Science Research Network electronic library at www.ssrn.com.

Optimal Monetary Policy in an Operational Medium-Sized DSGE Model (cid:3) Malin Adolfsona, Stefan LasØena, Jesper LindØb, and Lars E.O. Svenssonc aSveriges Riksbank bFederal Reserve Board cSveriges Riksbank, Stockholm University, CEPR, and NBER July 2011 Abstract We show how to construct optimal policy projections in Ramses, the Riksbank(cid:146)s openeconomy medium-sized DSGE model for forecasting and policy analysis. Bayesian estimation of the parameters of the model indicates that they are relatively invariant to alternative policy assumptions and supports our view that the model parameters may be regarded as una⁄ected by the monetary policy speci(cid:133)cation. We discuss how monetary policy, and in particular the choice of output gap measure, a⁄ects the transmission of shocks. Finally, we use the model to assesstherecentGreatRecessionintheworldeconomyandhowitsimpactontheeconomicdevelopment in Sweden depends on the conduct of monetary policy. This provides an illustration on how Rames incoporates large international spillover e⁄ects. JEL Classi(cid:133)cation: E52, E58 Keywords: Optimal monetary policy, instrument rules, optimal policy projections, openeconomy DSGE models (cid:3)We are grateful for helpful comments from G(cid:252)nter Coenen, Lee Ohanian, Frank Smets, and participants in theSecondOsloWorkshoponMonetaryPolicy,theCentralBankWorkshoponMacroeconomicModelling,Oslo,the conferenceonNewPerspectivesonMonetaryPolicyDesign,Barcelona,theLindahlLectures,Uppsala,theconference onQuantitativeApproachestoMonetaryPolicyinOpenEconomies,Atlanta,andseminarsattheRiksbankandthe Institute for International Economic Studies. All remaining errors are ours. The views, analysis, and conclusions in thispaperaresolelytheresponsibilityoftheauthorsandshouldnotbeinterpretedasre(cid:135)ectingtheviewsofthoseof other members of the Riksbank(cid:146)s sta⁄or executive board, or the Board of Governors of the Federal Reserve System or of any other person associated with the Federal Reserve System.

Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2 The model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.1 Domestic goods (cid:133)rms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.2 Importing and exporting (cid:133)rms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.3 Households . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.4 Structural shocks, government, foreign economy . . . . . . . . . . . . . . . . . . . . . 8 2.5 Monetary policy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.6 Model solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 3 Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 3.1 Data, prior distributions, and calibrated parameters . . . . . . . . . . . . . . . . . . 12 3.2 Estimation results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 4 Monetary policy and the transmission of shocks . . . . . . . . . . . . . . . . . . . . . . . . 17 5 Optimal policy projections and the Great Recession . . . . . . . . . . . . . . . . . . . . . 20 5.1 Information and data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 5.2 The projection model and optimal projections . . . . . . . . . . . . . . . . . . . . . . 21 5.3 An application to the Great Recession . . . . . . . . . . . . . . . . . . . . . . . . . . 23 6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 A Ramses in some detail . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 B Optimal projections in some detail . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 B.1 Information and data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 B.2 Solving for the optimal projection. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 B.3 Determination of the initial Lagrange multipliers . . . . . . . . . . . . . . . . . . . . 36 B.4 Projections with an arbitrary instrument rule . . . . . . . . . . . . . . . . . . . . . . 37 C Flexprice equilibrium and alternative concepts of potential output . . . . . . . . . . . . . 38 C.1 Unconditional potential output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 C.2 Conditional potential output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 C.3 Projections of potential output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 C.4 Output gaps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 0

1. Introduction We study optimal monetary policy in Ramses, the main model used at Sveriges Riksbank for forecasting and policy analysis. Ramses is an estimated small open-economy dynamic stochastic general equilibrium (DSGE) model, developed by Adolfson, LasØen, LindØ, and Villani (ALLV) [4] and [3]. By optimal monetary policy we mean policy that minimizes an intertemporal loss function. The intertemporal loss function is a discounted sum of expected future period losses. We choose a quadratic period loss function that corresponds to (cid:135)exible in(cid:135)ation targeting and is the weighted sum of three terms: the squared in(cid:135)ation gap between 4-quarter CPI in(cid:135)ation and the in(cid:135)ation target, the squared output gap between output and potential output, and the squared quarterly change in the Riksbank(cid:146)s instrument rate, the repo rate. We interpret such a loss function as consistent with (cid:135)exible in(cid:135)ation targeting and the Riksbank(cid:146)s mandate, which implies that monetary policy is directed towards stabilizing both in(cid:135)ation around an in(cid:135)ation target of 2 percent and resource utilization around a normal level ([37]).1 AfundamentalassumptioninouranalysisisthatRamsesisastructuralmodelwhoseparameters are invariant to the changes in monetary policy we consider. We estimate the model parameters with Bayesian techniques under di⁄erent assumptions about the conduct of monetary policy. First, as in ALLV [3], we estimate the model under the assumption that the Riksbank has followed a simple instrument rule during the in(cid:135)ation-targeting period which started in 1993:1.2 Second, we estimate the model under the assumption that the Riksbank has minimized an intertemporal loss function during the in(cid:135)ation-targeting period. The estimates of the instrument-rule and lossfunction parameters provide benchmarks for the subsequent policy analysis. A (cid:133)nding in the empirical analysis is that whether past policy of the Riksbank until 2007:3 (the end of the sample used)isbetterexplainedasfollowingasimpleinstrumentruleorminimizingalossfunctiondepends on whether the simple instrument rule and the optimal policy rule has a white-noise policy shock (control error) or not. Without a shock in both rules, we (cid:133)nd that optimal policy (cid:133)ts the data 1 Insomesimplemodels,quadraticapproximationsofthewelfareofarepresentativehouseholdresultsinasimilar periodlossfunction(Woodford[39]). Suchapproximationsofhouseholdwelfareareverymodel-dependentandre(cid:135)ect theparticulardistortionsassumedinanygivenmodel. Householdwelfareisinanycasehardlyanoperationalcentralbankobjective,althoughitmaybeofinterestandrelevanttoexaminehow householdwelfareinparticularmodelsis a⁄ected by central-bank policy. Such an undertaking is beyond the scope of the present paper, though. 2 Thein(cid:135)ation-targetingperiodisassumedtostart1993:1(theestimationsampleinthispaperends2007:3). Prior to thisperiod (1986:1(cid:150)1992:4),theconductofpolicy isassumed to be given by a simple instrumentrule. Theswitch to the in(cid:135)ation-targeting regime in 1993:1 is assumed to be completely unanticipated but expected to be permanent once it has occurred. 1

equallywellasthesimpleinstrumentrule. Withashockinbothrules, thereisaclearimprovement in the empirical (cid:133)t of both rules. Furthermore, the simple instrument rule has a slight empirical advantagerelativetooptimalpolicy. Hence,fortheparticularsampleperiodinquestion,withpolicy shocks the simple instrument rule o⁄ers a slightly better characterization of monetary policy. Our results are hence slightly di⁄erent than those reported by Wolden Bache, Brubakk, and Maih [38] on Norwegian data, which stems from the fact that they do not allow for a control error in their analysis. One contribution of our paper is to provide a detailed analysis of how to do optimal policy projections in an estimated linear-quadratic model with forward-looking variables, extending on previousanalysisbySvensson[29]andSvenssonandTetlow[35]. Akeyissuefora(cid:135)exiblein(cid:135)ationtargetingcentralbankiswhichmeasureofresourceutilizationtostabilize. Weusetheoutputgapas measure of resource utilization and study alternative de(cid:133)nitions of potential output and the output gap, in order to make an assessment to what extent the formulation of the output gap in the loss functiona⁄ectstheconductofmonetarypolicyandpropagationofshocks. Moreprecisely,wereport resultsfromthreealternativeconceptsofoutputgaps(y y(cid:22)), deviationsofactual(log)output(y ) t t t (cid:0) from potential (log) output (y(cid:22)), in the loss function. One concept of output gap is the trend output t gap where potential output is the trend output level, which is growing stochastically due to the unit-root stochastic technology shock in the model. A second concept is the unconditional output gap, where potential output is unconditional potential output, which is de(cid:133)ned as the hypothetical outputlevelthatwouldexistiftheeconomywouldhavehad(cid:135)exiblepricesandwagesforalongtime and would have been subject to a subset of the same shocks as the actual economy. Unconditional potential output therefore presumes di⁄erent levels of the predetermined variables, including the capital stock, from those in the actual economy. A third concept is the conditional output gap, where potential output is conditional potential output, which is de(cid:133)ned as the hypothetical output level that would arise if prices and wages suddenly become (cid:135)exible in the current period and are expected to remain (cid:135)exible in the future. Conditional potential output therefore depends on the existing current predetermined variables, including the current capital stock. To illustrate how the policy maker(cid:146)s choice of output measure can in(cid:135)uence the transmission of shocks we study impulse response functions to a persistent but stationary technology shock. According to the estimated model, shocks to total factor productivity is a dominant driver of business cycles in Sweden. In addition to its economic importance, this shock is also particularly interesting to study since it a⁄ects the various output gaps di⁄erently. Conditional and unconditional poten- 2

tial output increases when a positive stationary technology shock hits the economy, whereas trend output by de(cid:133)nition is independent of such shocks and only depends on permanent (unit-root) technology shocks. With that analysis in hand, we report and discuss optimal policy projections for Sweden using data up to and including 2008:2. We use the model to interpret the economic development in Sweden just at the time when the world economy fell into a deep recession due to the collapse in the (cid:133)nancial markets. We choose this period to highlight the sensitivity of the Swedish economy to foreign shocks. We show projections for the estimated instrument rule and for optimal policy with di⁄erent output gaps in the loss function, which create di⁄erent trade-o⁄s for the central bank in stabilizing in(cid:135)ation and output. The paper is organized as follows: Section 2 presents Ramses in more general modeling terms. Section3discussesthedataandpriorsusedintheestimationandpresentsestimationresultsforthe various speci(cid:133)cations of the model and policy. Section 4 presents and discusses impulse response functions to a technology shock with di⁄erent output gaps in the loss function. Section 5 discusses how to construct optimal policy projections, and analyzes alternative projections for a policymaker during the recent Great Recession. Finally, section 6 presents a summary and some conclusions. Appendices A-B contain a detailed speci(cid:133)cation of Ramses and some other technical details. 2. The model Ramses is a small open-economy DSGE model developed in a series of papers by ALLV [4] and [3]. The model economy consists of households, domestic goods (cid:133)rms, importing consumption and importinginvestment(cid:133)rms,exporting(cid:133)rms,agovernment,acentralbank,andanexogenousforeign economy. Within each manufacturing sector there is a continuum of (cid:133)rms that each produces a di⁄erentiatedgoodandsetspricesaccordingtoanindexationvariantoftheCalvomodel. Domestic as well as global production grows with technology that contains a stochastic unit-root, see Altig et al. [8]. In what follows we provide the optimization problems of the di⁄erent (cid:133)rms and the households, and describe the behavior of the central bank. The log-linear approximation of the model is presented in appendix A. 3

2.1. Domestic goods (cid:133)rms The domestic goods (cid:133)rms produce their goods using capital and labor inputs, and sell them to a retailer which transforms the intermediate products into a homogenous (cid:133)nal good that in turn is sold to the households. The (cid:133)nal domestic good is a composite of a continuum of di⁄erentiated intermediate goods, each supplied by a di⁄erent (cid:133)rm. Output, Y , of the (cid:133)nal domestic good is produced with the t constant elasticity of substitution (CES) function (cid:21)d 1 t 1 Y t = (Y it )(cid:21)d t di ; 1 (cid:21)d t < ; (2.1) 2 3 (cid:20) 1 Z 0 4 5 where Y , 0 i 1, is the input of intermediate good i and (cid:21)d is a stochastic process that it t (cid:20) (cid:20) determines the time-varying (cid:135)exible-price markup in the domestic goods market. The production of the intermediate good i by intermediate-good (cid:133)rm i is given by Y = z1 (cid:11)(cid:15) K(cid:11)H1 (cid:11) z (cid:30); (2.2) it t(cid:0) t it it(cid:0) (cid:0) t where z is a unit-root technology shock common to the domestic and foreign economies, (cid:15) is a dot t mestic covariance stationary technology shock, K the capital stock and H denotes homogeneous it it labor hired by the ith (cid:133)rm. A (cid:133)xed cost z (cid:30) is included in the production function. We set this t parameter so that pro(cid:133)ts are zero in steady state, following Christiano et al. [11]. We allow for working capital by assuming that a fraction (cid:23) of the intermediate (cid:133)rms(cid:146)wage bill has to be (cid:133)nanced in advance through loans from a (cid:133)nancial intermediary. Cost minimization yields the following nominal marginal cost for intermediate (cid:133)rm i: 1 1 1 1 MCd = (Rk)(cid:11)[W (1+(cid:23)(R 1))]1 (cid:11) ; (2.3) it (1 (cid:11))1 (cid:0) (cid:11)(cid:11)(cid:11) t t t (cid:0) 1 (cid:0) (cid:0) z t 1 (cid:0) (cid:11)(cid:15) t (cid:0) where Rk is the gross nominal rental rate per unit of capital, R the gross nominal (economy t t 1 (cid:0) wide) interest rate, and W the nominal wage rate per unit of aggregate, homogeneous, labor H . t it Each of the domestic goods (cid:133)rms is subject to price stickiness through an indexation variant of the Calvo [10] model. Each intermediate (cid:133)rm faces in any period a probability 1 (cid:24) that it can d (cid:0) d;new reoptimize its price. The reoptimized price is denoted P . For the (cid:133)rms that are not allowed t to reoptimize their price, we adopt an indexation scheme with partial indexation to the current in(cid:135)ation target, (cid:25)(cid:22)c , since there is a perceived (time-varying) CPI in(cid:135)ation target in the model , t+1 4

and partial indexation to last period(cid:146)s in(cid:135)ation rate in order to allow for a lagged pricing term in the Phillips curve P t d +1 = (cid:25)d t (cid:20)d (cid:25)(cid:22)c t+1 1 (cid:0) (cid:20)dP t d; (2.4) (cid:16) (cid:17) (cid:0) (cid:1) where Pd is the price level, (cid:25)d = Pd =Pd is gross in(cid:135)ation in the domestic sector, and (cid:20) is an t t t+1 t d indexation parameter. The di⁄erent (cid:133)rms maximize pro(cid:133)ts taking into account that there might not be a chance to optimally change the price in the future. Firm i therefore faces the following optimization problem when setting its price max E t 1 ((cid:12)(cid:24) d )s(cid:29) t+s [( (cid:25)d t (cid:25)d t+1 :::(cid:25)d t+s 1 (cid:20)d (cid:25)(cid:22)c t+1 (cid:25)(cid:22)c t+2 :::(cid:25)(cid:22)c t+s 1 (cid:0) (cid:20)dP t d;new )Y i;t+s Pd;new s=0 (cid:0) (2.5) t P (cid:0) MCd (Y (cid:1)+z(cid:0) (cid:30)j)]; (cid:1) i;t+s i;t+s t+s (cid:0) where the (cid:133)rm is using the stochastic household discount factor ((cid:12)(cid:24) )s(cid:29) to make pro(cid:133)ts cond t+s ditional upon utility: (cid:12) is the discount factor, and (cid:29) the marginal utility of the households(cid:146) t+s nominal income in period t+s, which is exogenous to the intermediate (cid:133)rms. 2.2. Importing and exporting (cid:133)rms The importing consumption and importing investment (cid:133)rms buy a homogenous good at price P t(cid:3) in the world market, and convert it into a di⁄erentiated good through a brand naming technology. The exporting (cid:133)rms buy the (homogenous) domestic (cid:133)nal good at price Pd and turn this into t a di⁄erentiated export good through the same type of brand naming. The nominal marginal cost of the importing and exporting (cid:133)rms are thus S P and Pd=S , respectively, where S is the t t(cid:3) t t t nominal exchange rate (domestic currency per unit of foreign currency). The di⁄erentiated import and export goods are subsequently aggregated by an import consumption, import investment and export packer, respectively, so that the (cid:133)nal import consumption, import investment, and export good is each a CES composite according to the following: (cid:21)mc (cid:21)mi (cid:21)x 1 t 1 t 1 t 1 1 1 C t m = (C i m t )(cid:21)m t c di ; I t m = (I i m t )(cid:21)m t i di ; X t = (X it )(cid:21)x t di ; (2.6) 2 3 2 3 2 3 Z Z Z 0 0 0 4 5 4 5 4 5 where 1 (cid:21)j < for j = mc;mi;x is the time-varying (cid:135)exible-price markup in the import t (cid:20) 1 f g consumption (mc), import investment (mi) and export (x) sector. By assumption the continuum of consumption and investment importers invoice in the domestic currency and exporters in the foreigncurrency. Toallowforshort-runincompleteexchangeratepass-throughtoimportaswellas export prices we introduce nominal rigidities in the local currency price. This is modeled through 5

the same type of Calvo setup as above. The price setting problems of the importing and exporting (cid:133)rms are completely analogous to that of the domestic (cid:133)rms in equation (2.5).3 In total there are thusfourspeci(cid:133)cPhillipscurverelationsdeterminingin(cid:135)ationinthedomestic,importconsumption, import investment and export sectors. 2.3. Households There is a continuum of households which attain utility from consumption, leisure and real cash balances. The preferences of household j are given by E j 1 (cid:12)t (cid:16)cln(C bC ) (cid:16)hA (h jt )1+(cid:27)L +A z Q tP jt t d 1 (cid:0) (cid:27)q ; (2.7) 0 2 t jt (cid:0) j;t (cid:0) 1 (cid:0) t L 1+(cid:27) L q(cid:16)1 (cid:27)(cid:17) q 3 t=0 (cid:0) X 6 7 4 5 whereC , h andQ =Pd denotethejth household(cid:146)slevelsofaggregateconsumption, laborsupply jt jt jt t and real cash holdings, respectively. Consumption is subject to habit formation through bC , j;t 1 (cid:0) such that the household(cid:146)s marginal utility of consumption is increasing in the quantity of goods consumed last period. (cid:16)c and (cid:16)h are persistent preference shocks to consumption and labor supply, t t respectively. Households consume a basket of domestically produced goods (Cd) and imported t products (Cm) which are supplied by the domestic and importing consumption (cid:133)rms, respectively. t Aggregate consumption is assumed to be given by the following CES function: (cid:17) =((cid:17) 1) C t = (1 ! c )1=(cid:17) c(C t d)((cid:17) c(cid:0) 1)=(cid:17) c +! 1 c =(cid:17) c(C t m)((cid:17) c(cid:0) 1)=(cid:17) c c c(cid:0) ; (cid:0) h i where ! is the share of imports in consumption, and (cid:17) is the elasticity of substitution across c c consumption goods. Thehouseholdscaninvestintheirstockofcapital, saveindomesticbondsand/orforeignbonds and hold cash. The households invest in a basket of domestic and imported investment goods to form the capital stock, and decide how much capital to rent to the domestic (cid:133)rms given costs of adjusting the investment rate. The households can increase their capital stock by investing in additional physical capital (I ), taking one period to come in action. The capital accumulation t equation is given by K = (1 (cid:14))K +(cid:7) [1 S~(I =I )]I ; (2.8) t+1 t t t t 1 t (cid:0) (cid:0) (cid:0) 3 Total export demand satis(cid:133)es C t x + I t x = P P t t x (cid:3) (cid:0) (cid:17)f Y t(cid:3), where C t x and I t x is demand for consumption and investmentgoods,respectively;P t xtheexportpriceh;P t(cid:3)itheforeignpricelevel;Y t(cid:3)foreignoutputand(cid:17) f theelasticity of substitution across foreign goods: 6

where S~(I =I ) determines the investment adjustment costs through the estimated parameter t t 1 (cid:0) S~ , and (cid:7) is a stationary investment-speci(cid:133)c technology shock. Total investment is assumed to be 00 t given by a CES aggregate of domestic and imported investment goods (Id and Im, respectively) t t according to (cid:17) =((cid:17) 1) I t = (1 (cid:0) ! i )1=(cid:17) i I t d ((cid:17) i(cid:0) 1)=(cid:17) i +! 1 i =(cid:17) i(I t m)((cid:17) i(cid:0) 1)=(cid:17) i i i(cid:0) ; (2.9) (cid:20) (cid:21) (cid:16) (cid:17) where ! is the share of imports in investment, and (cid:17) is the elasticity of substitution across i i investment goods. Each household is a monopoly supplier of a di⁄erentiated labor service which implies that they can set their own wage, see Erceg, Henderson and Levin [16]. After having set their wage, households supply the (cid:133)rms(cid:146)demand for labor, (cid:21)w h = W jt 1 (cid:0) (cid:21)w H ; jt t W t (cid:20) (cid:21) at the going wage rate. Each household sells its labor to a (cid:133)rm which transforms household labor into a homogenous good that is demanded by each of the domestic goods producing (cid:133)rms. Wage stickiness is introduced through the Calvo [10] setup, where household j reoptimizes its nominal wage rate Wnew according to the following4 jt max E ((cid:12)(cid:24) )s[ (cid:16)h A (hj;t+s)1+(cid:27)L + Wnew t 1s=0 w (cid:0) t+s L 1+(cid:27)L jt (2.10) (cid:29) t+s( (1 1 (cid:0) + (cid:28) (cid:28) y t w +s ) ) (cid:25)c t :::(cid:25)c t+s 1 (cid:20)P w (cid:25)(cid:22)c t+1 :::(cid:25)(cid:22)c t+s (1 (cid:0) (cid:20)w) (cid:22) z;t+1 :::(cid:22) z;t+s W j n t ew h j;t+s ]; t+s (cid:0) (cid:16) (cid:17) (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1) y where(cid:24) istheprobabilitythatahouseholdisnotallowedtoreoptimizeitswage,(cid:28) alaborincome w t tax, (cid:28)w a pay-roll tax (paid for simplicity by the households), and (cid:22) = z =z is the growth rate t zt t t 1 (cid:0) of the unit-root technology shock. The choice between domestic and foreign bond holdings balances into an arbitrage condition pinningdownexpectedexchangeratechanges(thatis, anuncoveredinterestrateparitycondition). To ensure a well-de(cid:133)ned steady-state in the model, we assume that there is premium on the foreign bondholdingswhichdependsontheaggregatenetforeignassetpositionofthedomestichouseholds, see, for instance, Schmitt-GrohØ and Uribe [24]. Compared to a standard setting the risk premium is allowed to be negatively correlated with the expected change in the exchange rate (that is, the expected depreciation), following the evidence discussed in for example Duarte and Stockman [15]. 4 For the households that are not allowed to reoptimize, the indexation scheme is W = j;t+1 ((cid:25)c t )(cid:20)w((cid:25)(cid:22)c t+1 )(1 (cid:0) (cid:20)w)(cid:22) z;t+1 W jt , where (cid:20) w is an indexation parameter. 7

For a detailed discussion and evaluation of this modi(cid:133)cation see ALLV [3]. The risk premium is given by: E S S (cid:8)(a ;S ;(cid:30)~ ) = exp (cid:30)~ (a a(cid:22)) (cid:30)~ t t+1 t 1 +(cid:30)~ ; (2.11) t t t (cid:0) a t (cid:0) (cid:0) s S S (cid:0) t t t 1 (cid:18) (cid:18) (cid:0) (cid:19) (cid:19) where a (S B )=(P z ) is the net foreign asset position, and (cid:30)~ is a shock to the risk premium. t t t(cid:3) t t t (cid:17) To clear the (cid:133)nal goods market, the foreign bond market, and the loan market for working capital, the following three constraints must hold in equilibrium: Cd+Id+G +Cx+Ix z1 (cid:11)(cid:15) K(cid:11)H1 (cid:11) z (cid:30); (2.12) t t t t t t(cid:0) t t t(cid:0) t (cid:20) (cid:0) S B = S Px(Cx+Ix) S P (Cm+Im)+R (cid:8)(a ;(cid:30) )S B ; (2.13) t t(cid:3)+1 t t t t t t(cid:3) t t t(cid:3) 1 t 1 t 1 t t(cid:3) (cid:0) (cid:0) (cid:0) (cid:0) (cid:23)W H = (cid:22) M Q ; e (2.14) t t t t t (cid:0) where G is government expenditures, Cx and Ix are the foreign demand for export goods which t t t follow CES aggregates with elasticity (cid:17) , and (cid:22) = M =M is the monetary injection by the f t t+1 t central bank. When de(cid:133)ning the demand for export goods, we introduce a stationary asymmetric (or foreign) technology shock z~ = z =z , where z is the permanent technology level abroad, to t(cid:3) t(cid:3) t t(cid:3) allow for temporary di⁄erences in permanent technological progress domestically and abroad. 2.4. Structural shocks, government, foreign economy The structural shock processes in the model are given by the univariate representation ^& = (cid:26) ^& +" ; " iid N 0;(cid:27)2 (2.15) t & t 1 &t &t & (cid:0) (cid:24) (cid:0) (cid:1) where & = (cid:22) , (cid:15) ;(cid:21) j ; (cid:16)c; (cid:16)h; (cid:7) ; (cid:30)~ ; " ;(cid:25)(cid:22)c; z~ , j = d;mc;mi;x ; and a hat denotes the t zt t t t t t t Rt t t(cid:3) f g f g deviation of a log-linearized variable from a steady-state level (v^ dv =v for any variable v , where t t t (cid:17) j v is the steady-state level). (cid:21) and " are assumed to be white noise (that is, (cid:26) = 0; (cid:26) = 0). t Rt (cid:21)j "R The government spends resources on consuming part of the domestic good, and collects taxes from the households. The resulting (cid:133)scal surplus/de(cid:133)cit plus the seigniorage are assumed to be transferred back to the households in a lump sum fashion. Consequently, there is no government debt. The (cid:133)scal policy variables (cid:150)taxes on labor income ((cid:28)^ y ), consumption ((cid:28)^c), and the payt t roll ((cid:28)^w), together with (HP-detrended) government expenditures (g^) (cid:150)are assumed to follow an t t identi(cid:133)ed VAR model with two lags, (cid:2) (cid:28) = (cid:2) (cid:28) +(cid:2) (cid:28) +S " ; (2.16) 0 t 1 t 1 2 t 2 (cid:28) (cid:28)t (cid:0) (cid:0) 8

where (cid:28) t ((cid:28)^ y t ;(cid:28)^c t ;(cid:28)^w t ;g^ t ) 0 , " (cid:28)t s N (0;I (cid:28) ), S (cid:28) is a diagonal matrix with standard deviations and (cid:17) (cid:2) (cid:0)0 1S (cid:28) " (cid:28)t s N (0;(cid:6) (cid:28) ). Since Sweden is a small open economy we assume that the foreign economy is exogenous. Foreign in(cid:135)ation, (cid:25) , output (HP-detrended), y^ ; and interest rate, R , are exogenously given by (cid:3)t t(cid:3) t(cid:3) an identi(cid:133)ed VAR model with four lags, (cid:8) X = (cid:8) X +(cid:8) X +(cid:8) X +(cid:8) X +S " ; (2.17) 0 t(cid:3) 1 t(cid:3) 1 2 t(cid:3) 2 3 t(cid:3) 3 4 t(cid:3) 4 x (cid:3) x (cid:3) t (cid:0) (cid:0) (cid:0) (cid:0) where X t(cid:3) (cid:17) ((cid:25) (cid:3)t ;y^ t(cid:3) ;R t(cid:3) ) 0 , " x (cid:3) t s N (0;I x (cid:3) ); S X (cid:3) is a diagonal matrix with standard deviations and (cid:8) (cid:0)0 1S x (cid:3) " x (cid:3) t s N (0;(cid:6) x (cid:3) ). Given our assumption of equal substitution elasticities in foreign consumption and investment, these three variables su¢ ce to describe the foreign economy in our model setup. 2.5. Monetary policy Monetarypolicyismodeledintwodi⁄erentways. First,weassumethatthecentralbankminimizes an intertemporal loss function under commitment. Let the intertemporal loss function in period t be 1 E (cid:14)(cid:28)L ; (2.18) t t+(cid:28) (cid:28)=0 X where 0 < (cid:14) < 1 is a discount factor, L is the period loss that is given by t L = (pc pc (cid:25)(cid:22)c)2+(cid:21) (y y(cid:22) )2+(cid:21) (i i )2; (2.19) t t t 4 y t t (cid:1)i t t 1 (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) where the central bank(cid:146)s target variables are; model-consistent year-over-year CPI in(cid:135)ation rate, pc pc , where pc denotes the log of CPI and (cid:25)(cid:22)c is the 2 percent in(cid:135)ation target; a measure of t (cid:0) t (cid:0) 4 t the output gap, y y(cid:22); the (cid:133)rst di⁄erence of the instrument rate, i i , where i denotes the t t t t 1 t (cid:0) (cid:0) (cid:0) Riksbank(cid:146)s instrument rate, the repo rate, and (cid:21) and (cid:21) are nonnegative weights on output-gap y (cid:1)i stabilization and instrument-rate smoothing, respectively.5 Wereportresultsfromthreealternativeconceptsofoutputgaps(y y(cid:22))asmeasuresofresource t t (cid:0) utilization in the loss function. One concept of output gap is the trend output gap where potential output (y(cid:22)) is the trend output level, which is growing stochastically due to the unit-root stochastic t technology shock in the model. A second concept is the unconditional output gap, where potential output is unconditional potential output, which is de(cid:133)ned as the hypothetical output level that 5 We use the 4-quarter CPI di⁄erence as a target variable rather than quarterly in(cid:135)ation since the Riksbank and other in(cid:135)ation-targeting central banks normally specify their in(cid:135)ation target as a 12-month rate. 9

would exist if the economy would have had (cid:135)exible prices and wages for a long time and would have been subject to the same shocks as the actual economy except mark-up shocks and shocks to taxes which are held constant at their steady-state levels. Unconditional potential output therefore presumes di⁄erent levels of the predetermined variables, including the capital stock, from those in the actual economy. A third concept is the conditional output gap, where potential output is conditional potential output, which is de(cid:133)ned as the hypothetical output level that would arise if prices and wages suddenly become (cid:135)exible in the current period and are expected to remain (cid:135)exible inthefuture. Conditionalpotentialoutputthereforedependsontheexistingcurrentpredetermined variables,includingthecurrentcapitalstock. Inpreciseformthethreedi⁄erentconceptsofpotential output are y(cid:22)trend = z ; t t y(cid:22)cond = FfX ; t y t (cid:1) y(cid:22)uncond = FfX f ; t y t (cid:1) f where z is the unit-root technology shock, the row vector F expresses output as a function of t y (cid:1) the predetermined state variables in the (cid:135)ex-price economy, X is the vector of predetermined state t f variables in Ramses, and X is the state vector in the economy with (cid:135)exible prices and wages. t Second,weassumethatmonetarypolicyobeysaninstrumentrule,followingSmetsandWouters [26]. Insteadofoptimizinganintertemporallossfunction,thecentralbankisthenassumedtoadjust the short term interest rate in response to deviations of CPI in(cid:135)ation from the perceived in(cid:135)ation target, the trend output gap (measured as actual minus trend output), the real exchange rate (x^ S^ +P^ P^c) and the interest rate set in the previous period. The log-linearized instrument t t t(cid:3) t (cid:17) (cid:0) rule follows: i = (cid:26) i +(1 (cid:26) ) (cid:25)(cid:22) c +r (cid:25)^c (cid:25)(cid:22) c +r y^ +r x~ (2.20) t R t 1 R t (cid:25) t 1 t y t 1 x t 1 (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) h i +r (cid:25)^c (cid:25)^c +r (y^(cid:0) y^ )+(cid:1)" ; (cid:1)(cid:25) t (cid:0) t (cid:0) 1 b (cid:1)y t (cid:0) t (cid:0) 1 b Rt b (cid:0) (cid:1) where i R^ (the notation for the short nominal interest rate in Ramses), and " is an uncort t Rt (cid:17) related monetary policy shock. Since (cid:25)^c and y^ are forward-looking variables, this is an implicit t t instrument rule (see appendix B.4).6 6 As reported in ALLV [3], the output gap resulting from trend output seems to more closely correspond to the measure of resource utilization that the Riksbank has been responding to historically rather than the unconditional output gap. Del Negro, Schorfheide, Smets, and Wouters [14] report similar results for the US. 10

2.6. Model solution After log-linearization, Ramses can be written in the following state-space form, X X C t+1 = A t +Bi + " : (2.21) Hx x t 0 t+1 t+1t t (cid:20) j (cid:21) (cid:20) (cid:21) (cid:20) (cid:21) Here, X is an n -vector of predetermined variables in period t (where the period is a quarter); t X x is an n -vector of forward-looking variables; i is an n -vector of instruments (the forwardt x t i looking variables and the instruments are the nonpredetermined variables);7 " is an n -vector of t " i.i.d. shocks with mean zero and covariance matrix I ; A, B, and C, and H are matrices of the n" appropriate dimension; and, for any variable y , y denotes E y , the rational expectation of t t+(cid:28) t t t+(cid:28) j y conditional on information available in period t. The variables are measured as di⁄erences t+(cid:28) from steady-state values; thus their unconditional means are zero. The elements of the matrices A, B, C, and H are estimated with Bayesian methods and are considered (cid:133)xed and known for the policy simulations. Then the conditions for certainty equivalence are satis(cid:133)ed. Thus, we abstract from any consideration of model uncertainty in the formulation of optimal policy.8 The upper block of (2.21) provides n equations that determine the n -vector X in period X X t+1 t+1 for given X , x , i and " ; t t t t+1 X = A X +A x +B i +C" ; (2.22) t+1 11 t 12 t 1 t t+1 where A and B are partitioned conformably with X and x as t t A A B A 11 12 ; B = 1 : (2.23) (cid:17) A 21 A 22 B 2 (cid:20) (cid:21) (cid:20) (cid:21) The lower block provides n equations that determine the n -vector x in period t for given x , x x t t+1t j X , and i ; t t x = A 1(Hx A X B i ): (2.24) t (cid:0)22 t+1 j t (cid:0) 21 t (cid:0) 2 t We hence assume that the n n submatrix A is nonsingular.9 x x 22 (cid:2) We assume that the central bank(cid:146)s n -vector of target variables, measured as the di⁄erence Y from an n -vector Y of target levels; Y (pc pc (cid:25)(cid:22)c;y y(cid:22) ;i i ), can be written as a Y (cid:3) t (cid:17) t (cid:0) t (cid:0) 4 (cid:0) t (cid:0) t t (cid:0) t (cid:0) 1 0 7 A variable is predetermined if its one-period-ahead prediction error is an exogenous stochastic process (Klein [21]). For(2.21),theone-period-aheadpredictionerrorofthepredeterminedvariablesisthestochasticvectorC" . t+1 8 Onatski and Williams [23] provide a thorough discussion of model uncertainty. Svensson and Williams [33] and [34]showhowtocomputeoptimalpoliciesforMarkovJump-Linear-Quadraticsystems,whichprovideaquite(cid:135)exible way to model most kinds of relevant model uncertainty for monetary policy. Levin, Onatski, Williams and Williams [20]studyoptimalpolicywhenthecentralbankfacesuncertaintyaboutthetruestructureoftheeconomy(i.e.,they look at the entire posterior distribution of the model parameters). 9 Withoutlossofgenerality,weassumethattheshocks" onlyenterintheupperblockof(2.21),sinceanyshocks t inthelowerblockof(2.21)canberede(cid:133)nedasadditionalpredeterminedvariablesandintroducedintheupperblock. 11

linear function of the predetermined, forward-looking, and instrument variables, X X t t Y = D x [D D D ] x ; (2.25) t t X x i t 2 3 (cid:17) 2 3 i i t t 4 5 4 5 whereDisann (n +n +n )matrixandpartitionedconformablywithX ,x ,andi . Assuming Y X x i t t t (cid:2) optimization of (2.18) under commitment in a timeless perspective, the resulting intertemporal equilibrium can then be described by the following di⁄erence equations, x X t = F t ; (2.26) i (cid:4) t t 1 (cid:20) (cid:21) (cid:20) (cid:0) (cid:21) X X C t+1 = M t + " ; (2.27) (cid:4) (cid:4) 0 t+1 t t 1 (cid:20) (cid:21) (cid:20) (cid:0) (cid:21) (cid:20) (cid:21) for t 0, where X and (cid:4) are given. This system of di⁄erence equations can be solved with 0 1 (cid:21) (cid:0) several alternative algorithms, for instance those developed by Klein [21] and Sims [25].10 The choice and calculation of the initial (cid:4) is further discussed in footnote 14 and appendix B.3. 1 (cid:0) When policy instead is described by the simple instrument rule in (2.20), there exists F and M matrices such that the intertemporal equilibrium di⁄erence equations (2.26) and (2.27) are still valid but with (cid:4) = 0 for t 0. t (cid:21) 3. Estimation 3.1. Data, prior distributions, and calibrated parameters WeusequarterlySwedishdatafortheperiod1980:1-2007:3andestimatethemodelusingaBayesian approach by placing a prior distribution on the structural parameters.11;12 As in ALLV [3], we include the following n = 15 variables among the observable variables: Z GDP de(cid:135)ator in(cid:135)ation ((cid:25)d), real wage (W =Pd), consumption (C ), investment (I ), real exchange t t t t t rate (x~ ), short interest rate (R ), hours worked (H ), GDP (Y ), exports (X~ ), imports (M~ ), CPI t t t t t t in(cid:135)ation ((cid:25) cpi ), investment-de(cid:135)ator in(cid:135)ation ((cid:25)def;i ), foreign (trade-weighted) output (Y ), foreign t t t(cid:3) in(cid:135)ation ((cid:25) ), and foreign interest rate (R ). We use (cid:133)rst di⁄erences of the quantities and the real (cid:3)t t(cid:3) 10 See Svensson [29] and [30] for details of the derivation and the application of the Klein algorithm. 11 All data are from Statistics Sweden, except the repo rate which is from the Riksbank. The nominal wage is de(cid:135)ated by the GDP de(cid:135)ator. Foreign in(cid:135)ation, output, and interest rate are weighted together across Sweden(cid:146)s 20 largest trading partners in 1991 using weights from the IMF. 12 In the data, the ratios of import and export to output are increasing from about 0:25 to 0:40 and from 0:21 to 0:50, respectively, during the sample period. In the model, import and export are assumed to grow at the same rate as output. We have removed the excess trend in import and export in the data to make the export and import shares stationary. For all other variables we use the actual series (seasonally adjusted with the X12-method, except the variables in the GDP identity which were seasonally adjusted by Statistics Sweden). 12

wage, since the unit-root technology shock induces a common stochastic trend in these variables, and derive the state-space representation for the following vector of observed variables, ((cid:25)d;(cid:1)ln(W =P );(cid:1)lnC ;(cid:1)lnI ;x~ ;R ;H^ ;::: Z t t t t t t t t (3.1) t (cid:17) (cid:1)lnY ;(cid:1)lnX~ ;(cid:1)lnM~ ;(cid:25)cpi;(cid:25)def;i ;(cid:1)lnY ;(cid:25) ;R ): t t t t t t(cid:3) (cid:3)t t(cid:3) 0 b The growth rates are computed as quarterly log-di⁄erences, while the in(cid:135)ation and interest-rate series are measured as annualized quarterly rates. It should be noted that the stationary variables x~ and H^ are measured as deviations around the mean and the HP-(cid:133)ltered trend, that is, x~ t t t (cid:17) (x~ x~)=x~ and H^ H HHP =HHP, respectively.13 Finally, all real variables are measured in b t (cid:0) t (cid:17) t (cid:0) t t b per-capita units. (cid:0) (cid:1) We estimate 13 structural shocks, of which 8 follow AR(1) processes and 5 are assumed to be i.i.d. (as described in section 2.4). In addition to these, there are 8 shocks provided by the exogenous (pre-estimated) (cid:133)scal and foreign VARs, whose parameters are kept (cid:133)xed throughout the estimation of the model (uninformative priors are used for these stochastic processes). The shocks enter in such a way that there is no stochastic singularity in the likelihood function. To compute the likelihood function, the reduced-form solution of the model (2.26-2.27) is transformedintoastate-spacerepresentationthatmapstheunobservedstatevariablesintotheobserved data.14 The posterior mode and Hessian matrix evaluated at the mode is computed by standard numerical optimization routines (see Smets and Wouters [26] and the references there for details). The parameters we choose to estimate pertain mostly to the nominal and real frictions in the model and the exogenous shock processes.15 Table 3.1 shows the assumptions for the prior distribution of the estimated parameters. For the model with a simple instrument rule, we choose identical priors for the parameters in the instrument rule before and after the adoption of an 13 The reason why we use a smooth HP-(cid:133)ltered trend for hours per capita, as opposed to a constant mean, is that there is a large and very persistent reduction in hours worked per capita during the recession in the beginning ofthe 1990s. Neglectingtakingthisreduction intoaccountimpliesthattheforecastingperformanceforhourspercapitain themodeldeterioratessigni(cid:133)cantly,asdocumented in theforecastingexercisesin ALLV [3]. Ratherthan imposinga discreteshiftinhoursinaspeci(cid:133)ctimeperiod,wethereforedecidedtoremoveasmoothHPtrendfromthevariable. This choice is not particularly important for the parameter estimates, but has some impact on the 2-sided (cid:133)ltered estimates of the unobserved states of the economy. 14 We use the Kalman (cid:133)lter to calculate the likelihood function of the observed variables. The period 1980:1(cid:150) 1985:4 is used to form a prior on the unobserved state variables in 1985:4, and the period 1986:1-2007:3 is used for inference. During estimation the Lagrange multipliers, (cid:4) ; are updated through the Kalman (cid:133)lter just as the other t statevariables. Whentheinstrumentruleisactive(cid:4) equalszero,andin1993:1,whenpolicy(unexpectedly)switches t to minimizing the loss function, we assume commmitment from scratch so that the initial Lagrange multipliers are zero. 15 We choose to calibrate those parameters that we think are weakly identi(cid:133)ed by the variables that we include in thevectorofobserveddata. Theseparametersaremostlyrelatedtothesteady-statevaluesoftheobservedvariables (that is, the great ratios: C=Y, I=Y, and G=Y). The parameters that we calibrate are set as follows: the money growth (cid:22)=1:010445; the discount factor (cid:12) =0:999999; the steady state growth rate of productivity (cid:22) =1:005455; z the depreciation rate ~(cid:14) =0:025; the capital share in production (cid:11)=0:25; the share of imports in consumption and investment ! =0:35 and ! =0:50, respectively; the share of wage bill (cid:133)nanced by loans (cid:23) =1; the labour supply c i elasticity (cid:27) =1; the wage markup (cid:21) =1:30; in(cid:135)ation target persistence (cid:26) =0:975; the steady-state tax rates on L w (cid:25) labour income and consumption (cid:28)y = 0:30 and (cid:28)c = 0:24, respectively; government expenditures-output ratio 0:30; and the subsitution elasticity between consumption goods (cid:17) =5. c 13

in(cid:135)ation target in 1993:1. For the model with optimal policy during the in(cid:135)ation-targeting regime, we use very uninformative priors for the loss-function parameters ((cid:21) and (cid:21) ), as indicated by the y (cid:1)i high standard deviations. As mentioned in the introduction, the switch from the simple instrument rule to the in(cid:135)ation-targeting regime in 1993:1 is modelled as unanticipated and expected to last forever once it has occurred. Relative to other estimated small open-economy DSGE models (for instance, Justiniano and Preston [17]), the international spillover e⁄ects are relatively large due to the inclusion of a worldwide stochastic technology shock. This means that the open-economy aspects are of particular importance in our setting. 3.2. Estimation results Intable3.1,wereportthepriorandestimatedposteriordistributions. Threeposteriordistributions are reported. The (cid:133)rst, labeled (cid:147)Simple inst rule(cid:148), is under the assumption that the Riksbank has followed a simple instrument rule during the in(cid:135)ation-targeting period (see equation (2.20)). The second, labeled (cid:147)Loss function(cid:148), is under the assumption that the Riksbank has minimized a quadratic loss function under commitment during the in(cid:135)ation-targeting period, with the output gap in the loss function being the trend output gap (see equation (2.19)). In this case, the optimal policy rule does not include a policy shock. The third, labeled (cid:147)Loss fn params(cid:148), only estimates the two parameters in that loss function.16 16 The estimations are based on allowing the in(cid:135)ation target to be time-varying. The parameter estimates are, however, robust to keeping the in(cid:135)ation target (cid:133)xed at 2% during the the in(cid:135)ation-targeting period. 14

Table 3.1: Prior and posterior distributions Parameter Priordistribution Posterior distribution Simple inst rule Loss function Loss fn params type mean std.d. mode std.d. mode std.d. mode std.d. /df (Hess.) (Hess.) (Hess.) Calvo wages (cid:24) beta 0.750 0.050 0.719 0.045 0.719 0.042 w Calvo domestic prices (cid:24) beta 0.750 0.050 0.712 0.039 0.737 0.043 d Calvo import cons. prices (cid:24) beta 0.750 0.050 0.868 0.018 0.859 0.016 mc Calvo import inv. prices (cid:24) beta 0.750 0.050 0.933 0.010 0.929 0.011 mi Calvo export prices (cid:24) beta 0.750 0.050 0.898 0.019 0.889 0.025 x Indexation wages (cid:20)w beta 0.500 0.150 0.445 0.124 0.422 0.115 Indexation prices (cid:20)d beta 0.500 0.150 0.180 0.051 0.173 0.050 Markup domestic (cid:21)f truncnormal 1.200 0.050 1.192 0.049 1.176 0.050 Markup imported cons. (cid:21)mc truncnormal 1.200 0.050 1.020 0.028 1.021 0.029 Markup.imported invest. (cid:21)mi truncnormal 1.200 0.050 1.137 0.051 1.154 0.049 Investment adj. cost S~ normal 7.694 1.500 7.951 1.295 7.684 1.261 00 Habit formation b beta 0.650 0.100 0.626 0.044 0.728 0.035 Subst. elasticity invest. (cid:17) invgamma 1.500 4.0 1.239 0.031 1.238 0.030 i Subst. elasticity foreign (cid:17) invgamma 1.500 4.0 1.577 0.204 1.794 0.318 f Risk premium (cid:30)~ invgamma 0.010 2.0 0.038 0.026 0.144 0.068 UIP modi(cid:133)cation (cid:30)~ beta 0.500 0.15 0.493 0.067 0.488 0.029 s Unit root tech. shock (cid:26) beta 0.750 0.100 0.790 0.065 0.765 0.072 (cid:22)z Stationary tech. shock (cid:26) beta 0.750 0.100 0.966 0.006 0.968 0.005 " Invest. spec. tech shock (cid:26) beta 0.750 0.100 0.750 0.077 0.719 0.067 (cid:7) Asymmetric tech. shock (cid:26) (cid:30)~ beta 0.750 0.100 0.852 0.059 0.885 0.041 Consumption pref. shock (cid:26) beta 0.750 0.100 0.919 0.034 0.881 0.038 (cid:16)c Laboursupply shock (cid:26) beta 0.750 0.100 0.382 0.082 0.282 0.064 (cid:16)h Risk premium shock (cid:26) beta 0.750 0.100 0.722 0.052 0.736 0.058 z~(cid:3) Unit root tech. shock (cid:27)(cid:22)z invgamma 0.200 2.0 0.127 0.025 0.201 0.039 Stationary tech. shock (cid:27)" invgamma 0.700 2.0 0.457 0.051 0.516 0.054 Invest. spec. tech. shock (cid:27)(cid:7) invgamma 0.200 2.0 0.441 0.069 0.470 0.065 Asymmetric tech. shock (cid:15)z~(cid:3) invgamma 0.400 2.0 0.199 0.030 0.203 0.031 Consumption pref. shock (cid:27)(cid:16)c invgamma 0.200 2.0 0.177 0.035 0.192 0.031 Laboursupply shock (cid:27)(cid:16)h invgamma 1.000 2.0 0.470 0.051 0.511 0.053 Risk premium shock (cid:27) (cid:30)~ invgamma 0.050 2.0 0.454 0.157 0.519 0.067 Domestic markup shock (cid:27)(cid:21)d invgamma 1.000 2.0 0.656 0.064 0.667 0.068 Imp. cons. markup shock (cid:27)(cid:21)mc invgamma 1.000 2.0 0.838 0.081 0.841 0.084 Imp. invest. markup shock (cid:27)(cid:21)mi invgamma 1.000 2.0 1.604 0.159 1.661 0.169 Export markup shock (cid:27)(cid:21)x invgamma 1.000 2.0 0.753 0.115 0.695 0.122 Interest rate smoothing (cid:26) beta 0.800 0.050 0.912 0.019 0.900 0.023 R1 In(cid:135)ation response r(cid:25)1 truncnormal 1.700 0.100 1.676 0.100 1.687 0.100 Di⁄. in(cid:135)response r(cid:1)(cid:25)1 normal 0.300 0.100 0.210 0.052 0.208 0.053 Realexch. rate response rx1 normal 0.000 0.050 0.042 0.032 0.053 0.036 (cid:0) (cid:0) Output response ry1 normal 0.125 0.050 0.100 0.042 0.082 0.043 Di⁄. output response r(cid:1)y1 normal 0.063 0.050 0.125 0.043 0.133 0.042 Monetary policy shock (cid:27)R1 invgamma 0.150 2.0 0.465 0.108 0.647 0.198 In(cid:135)ation target shock (cid:27)(cid:25)(cid:22)c1 invgamma 0.050 2.0 0.372 0.061 0.360 0.059 Interest rate smoothing 2 (cid:26) beta 0.800 0.050 0.882 0.019 R2 In(cid:135)ation response 2 r(cid:25)2 truncnormal 1.700 0.100 1.697 0.097 Di⁄. in(cid:135)response 2 r(cid:1)(cid:25)2 normal 0.300 0.100 0.132 0.024 Realexch. rate response 2 rx2 normal 0.000 0.050 0.058 0.029 (cid:0) Output response 2 ry2 normal 0.125 0.050 0.081 0.040 Di⁄. output response 2 r(cid:1)y2 normal 0.063 0.050 0.100 0.012 Monetary policy shock 2 (cid:27)R2 invgamma 0.150 2.0 0.135 0.029 In(cid:135)ation target shock 2 (cid:27)(cid:25)(cid:22)c2 invgamma 0.050 2.0 0.081 0.037 Output stabilization (cid:21)y truncnormal 0.5 100.0 1.091 0.526 1.102 0.224 Interest rate smoothing (cid:21)(cid:1)i truncnormal 0.2 100.0 0.476 0.191 0.369 0.061 Log marg likelihood laplace 2631.56 2654.45 (cid:0) (cid:0) 15

There are two important facts to note in the (cid:133)rst two posterior distributions. First, it is clear that the version of the model where policy is characterized with the simple instrument rule with an exogenous policy shock is a better characterization of how the Riksbank has conducted monetary policy during the in(cid:135)ation-targeting period. The di⁄erence between log marginal likelihoods is almost 23 in favor of the model with the instrument rule compared to the model with optimal policy. In terms of Bayesian posterior odds, this is overwhelming evidence against the loss-function characterization of the Riksbank(cid:146)s past policy behavior. However, this result crucially depends on the assumption that the simple instrument rule includes a policy shock (control error) and that the optimal policy rule does not. If we follow Wolden Bache, Brubakk, and Maih [38] and treat the two cases symmetrically by not including a policy shock in the simple rule during the in(cid:135)ation targeting period, that is, we impose (cid:27) = 0, then the log marginal likelihood falls to 2654:5, R2 (cid:0) which is almost the same as for the optimal policy, implying that optimal policy (cid:133)ts the data equally well in this case. Alternatively, we can treat the two cases symmetrically by including a shock in the optimal policy rule. This improves the (cid:133)t of optimal policy, but the log marginal likelihood ( 2636:8) is then still lower than the log marginal likelihood for the simple instrument (cid:0) rule with a policy shock in table 3.1. Hence, for the particular sample period in question, with policy shocks in both rules the simple instrument rule o⁄ers a slightly better characterization of monetary policy. Compared to optimal policy the Bayesian posterior odds speak clearly in favor of the simple instrument rule, but in practice we have seen that log marginal likelihood di⁄erences (Laplace approximation) of these magnitudes do not necessarily generate very large discrepancies in terms of the forecasting accuracy, see ALLV [3] and Adolfson, LindØ and Villani [7]. Thus, the improvement in (cid:133)t relative to optimal policy is moderate. Second,perhapsmorerelevantforourpurpose,thenon-policyparametersofthemodelarefairly invariant to the speci(cid:133)cation of how monetary policy has been conducted during the in(cid:135)ationtargeting period. This is also true for the non-policy parameters of the model with the simple instrument rule without the policy shock.17 One way to assess the quantitative importance of the parameter di⁄erences between the instrument rule and the loss function (columns (cid:147)Simple inst rule(cid:148)and (cid:147)Loss function(cid:148)) is to calibrate all parameters except the coe¢ cients in the loss function to the estimates obtained in the simple rule speci(cid:133)cation of the model, and then reestimate the coe¢ cients in the loss function conditional on 17 It should be noted that data are informative about the parameters as the posteriors are often di⁄erent, and moreconcentrated,thanthepriordistributions. Moreover,AdolfsonandLindØ[6]useMonteCarlomethodstostudy identi(cid:133)cation in a very similar model, and (cid:133)nd that while a few parameters are weakly identi(cid:133)ed in small samples, all parameters are unbiased and consistent. 16

these parameters. If the loss function parameters are similar, we can conjecture that the di⁄erences in deep parameters are quantitatively unimportant. The result of this experiment is reported in the last column in table 3.1 labeled (cid:147)Loss fn params(cid:148), and, as conjectured, the resulting loss function parameters are very similar to the ones obtained when estimating all parameters jointly. We interpret this result as support for our assumption that the non-policy parameters are una⁄ected by the alternative assumptions about the conduct of monetary policy we consider below. Inthesubsequentanalysisweusetheposterior-modeestimatesofthenon-policyparametersfor the model with the simple instrument rule. In most cases, the model with the simple instrument rule is also used to generate the (partly) unobserved state variables. For consistency reasons, the associated estimated loss function parameters (cid:21) = 1:102 and (cid:21) = 0:369 are therefore used in the y (cid:1)i analysis below. 4. Monetary policy and the transmission of shocks According to the estimated model, shocks to total factor productivity play a dominant role in explaining business cycle variations in Sweden, as these shocks can explain the negative correlation between GDP growth and CPI in(cid:135)ation in our sample. In Adolfson, LasØen, LindØ and Svensson [2], we show that stationary technology shocks are the single most important driver of output (cid:135)uctuations around trend. To interpret and understand the development of the Swedish economy, it is therefore of key importance to analyze and understand the e⁄ects of this type of technology shock. To illustrate how optimal policy projections are a⁄ected by which output measure the central bank tries to stabilize we therefore start by looking at impulse response functions to a stationary technology shock. Although this type of technology shock (" ) is estimated to be quite persistent t ((cid:26) = 0:966) it does not a⁄ect trend output in the model (which by de(cid:133)nition is only in(cid:135)uenced " by the permanent technology shock, (cid:22) ). In contrast, the output level under (cid:135)exible prices and z wages, (cid:135)exprice potential output, increases with the shock. Abstracting from the policy response to the shock this means that the trend- and (cid:135)exprice-output gaps, by de(cid:133)nition, will behave quite di⁄erently following this shock. Figure 4.1 shows impulse response functions to a positive (one-standard deviation) stationary technology shock under the instrument rule and under optimal policy with di⁄erent output gaps. The plots show deviations from trend, where by trend we mean the steady state. Output thus equals the trend output gap. The real interest rate is de(cid:133)ned as the instrument rate less 1-quarter- 17

Figure 4.1: Impulse responses to a (one-standard deviation) stationary technology shock under optimal policy for di⁄erent output gaps and under the instrument rule 4 qtr CPI inflation Output gap Potential output Output 0.2 0.8 0.8 0.8 0.6 0 0.6 0.6 r y 0.4 / % 0.2 0.4 0.4 ,% 0.2 0.4 0.2 0.2 0 0.6 0.2 0 0 0 10 20 0 10 20 0 10 20 0 10 20 Instrument rate Real interest rate Real exchange rate 0.2 0.8 0.8 0.6 0.6 0.1 r y 0.4 0.4 / 0 % ,% 0.2 0.2 0.1 0 0 0.2 0.2 0.2 0 10 20 0 10 20 0 10 20 Conditional Unconditional Trend Instrument rule ahead CPI in(cid:135)ation expectations. All variables are measured in percent or percent per year. The impulse occurs in quarter 0. Before quarter 0, the economy is in the steady state with X = 0 and t (cid:4) = 0 for t 0 and x = 0 and i = 0 for t 1. t 1 t t (cid:0) (cid:20) (cid:20) (cid:0) The dashed curves show the impulse responses when policy follows the instrument rule (which responds to the trend output gap). For the central bank, the stationary technology shock creates a trade-o⁄between balancing the induced decline in in(cid:135)ation and the improvement in the (trend) outputgap. Sincetheshockisverypersistent((cid:26) = 0:966)thistrade-o⁄willlastformanyquarters, " and it takes time before in(cid:135)ation can be brought back to target. The instrument rule keeps the nominal interest rate below the steady state for the 20 periods plotted and longer. The real interest increasesandremainspositiveforaboutayear, whereasthereisarealdepreciationofthecurrency. The result is a relatively large positive output gap and a negative in(cid:135)ation gap between in(cid:135)ation and the target. These gaps remain for a long time. The instrument rule is obviously not successful 18

in closing these gaps with the time-period plotted. The dashed-dotted curves show the responses under optimal policy when the trend output gap enters the loss function of the central bank. The weights in the loss function, as for all responses under optimal policy in this (cid:133)gure, are (cid:21) = 1:1, and (cid:21) = 0:37, the estimated weights in table 3.1 y (cid:1)i when the non-policy parameters are kept at their posterior mode obtained under the instrument rule. We see that this optimal policy stabilizes in(cid:135)ation and the output gap more e⁄ectively over time than the instrument rule, although optimal policy (with the trend output gap in the loss function) initially allows for a larger fall in CPI in(cid:135)ation. This requires initially tighter monetary policy than the instrument-rule, as demonstrated by initially higher nominal and real interest rates and a real appreciation of the currency. The solid curves show the impulse responses under optimal policy when the output gap in the loss function is the conditional output gap. Hence, potential output plotted in the (cid:133)gure is conditional potential output, and the output gap plotted is the conditional output gap. We see that this optimal policy successfully stabilizes 4-quarter CPI in(cid:135)ation around the in(cid:135)ation target. It also successfully stabilizes the output gap, and we see that the conditional output gap is much smaller than the trend output gap and initially even negative. Due to sticky prices and wages, the stationary technology shock a⁄ects potential output quicker than actual output and the (cid:135)exprice (conditional and unconditional) output gaps are therefore initially negative, whereas the trend output gap is positive. This implies that the interest rate responses will di⁄er depending on which output gap the central bank tries to stabilize. With the (cid:135)exprice output gap in its loss function, the central bank does not face an unfavorable trade-o⁄ between stabilizing in(cid:135)ation and output after a technology shock. Optimal policy takes into account that conditional potential output is high because productivity is temporarily high and therefore allows for more expansionary policy (as shown by lower nominal and real interest rates) and higher actual output than when the trend output gap is the target variable or when the instrument rule is followed. This in e⁄ect implies that in(cid:135)ation can be stabilized much quicker than for policy with the trend output gap, even though the weights in the loss function are the same in the two cases. Thattheinstrumentruleisspeci(cid:133)edintermsofthetrendoutputgap(ratherthan, forinstance, the conditional output gap) is also one of the main reasons why the instrument rule does not bring in(cid:135)ation quickly back to target in this particular situation. Had the rule instead been speci(cid:133)ed with a stronger in(cid:135)ation response than the estimated one or using a response to the conditional or unconditional output gap, the ine¢ cient trade-o⁄between in(cid:135)ation and output stabilization would 19

be less pronounced. The dotted curves in (cid:133)gure 4.1 show the impulse responses when the output gap in the loss function is the unconditional output gap. Comparing with the impulse responses with the conditional output gap in the loss function, we see that potential output levels di⁄er from period 1 and onwards. This occurs because the conditional and unconditional potential output levels are computed from di⁄erent predetermined variables (those in the actual sticky-price economy, and those in the hypothetical economy with (cid:135)exible prices and wages in the past and present, respectively). Thus, unconditionalpotentialoutputisindependentofpolicy, whereasconditionalpotentialoutput depends on policy through the endogenous predetermined variables. When the shock hits the economy in quarter 0, the two output-gap de(cid:133)nitions will be equal (since the economy by assumption starts out in steady state in quarter 1, which is the same for both the actual economy and the (cid:0) hypothetical (cid:135)exprice economy), but in quarter 1 they will diverge. The predetermined variables in quarter 1 in the sticky-price economy will di⁄er from those in the (cid:135)exprice economy because the forward-looking variables and the instrument rate in quarter 0 will di⁄er between the sticky-price andthe(cid:135)expriceeconomies. Evenifnonewinnovationshaveoccurredbetweenquarter0andquarter t; the levels of the predetermined variables used for computing the two potential output levels willthusdi⁄er. Sinceactualoutputandconditionalpotentialoutputsharethesamepredetermined variables in each period, we would expect the conditional output gap will normally be smaller than the unconditional output gap. 5. Optimal policy projections and the Great Recession Having estimated the model and obtained an understanding of the role of di⁄erent monetary policy assumptions can play for the propagation of shocks, we now turn to a discussion about how to compute optimal policy projections with the model. We also consider an application to the Great Recession in the world economy that was initiated during 2008. 5.1. Information and data To calculate the optimal policy projection for a policy maker in period(quarter) t we assume the following. The information set, , in the beginning of period t, just after the instrument setting t I for quarter t has been announced, is speci(cid:133)ed as i ;Z ;Z ;::: ; t t t 1 t 2 I (cid:17) f (cid:0) (cid:0) g 20

where Z is the n -vector of observable variables that satis(cid:133)es the measurement equation, t Z X t Z = D(cid:22) x +(cid:17) ; t t t 2 3 i t 4 5 and where D(cid:22) is a given matrix and (cid:17) is an n -vector of i.i.d. period-t measurement errors with t Z distribution N(0;(cid:6) ). (cid:17) For the monetary-policy decision at the beginning of quarter t, the matrices A, B, C, and H as well as the state vectors, denoted X , x , C" , and (cid:17) for (cid:28) 1, are formed by the t (cid:28) t t (cid:28) t t (cid:28) t t (cid:28) t (cid:0) j (cid:0) j (cid:0) j (cid:0) j (cid:21) posterior mode estimates in table 3.1: We also specify the estimate of X , denoted X , as t tt j X = A X +A x +B i ; (5.1) tt 11 t 1t 12 t 1t 1 t 1t j (cid:0) j (cid:0) j (cid:0) j where the estimated shocks C" = 0 since Z is not in the information set .18 tt t t j I With respect to projections in period t, we regard the matrices A, B, C, and H as certain and known. Then we can rely on certainty equivalence(cid:151)under which conditional means of the relevant variables are su¢ cient for determining the optimal policy(cid:151)and compute the optimal projections accordingly.19 5.2. The projection model and optimal projections Let yt y denote a projection in period t for any variable y , a mean forecast conditional (cid:17) f t+(cid:28);t g 1(cid:28)=0 t on information in period t. The projection model for the projections (Xt;xt;it;Yt) in period t is X X t+(cid:28)+1;t = A t+(cid:28);t +Bi ; (5.2) Hx x t+(cid:28);t t+(cid:28)+1;t t+(cid:28);t (cid:20) (cid:21) (cid:20) (cid:21) X t+(cid:28);t Y = D x (5.3) t+(cid:28);t t+(cid:28);t 2 3 i t+(cid:28);t 4 5 for (cid:28) 0, where (cid:21) X = X ; (5.4) t;t tt j 18 Thus, the estimated/expected shock C" and (cid:17) for (cid:28) 0 are zero, whereas the estimated shocks C" and (cid:17) for (cid:28) 1 are given by C" t+(cid:28) j t = X t+(cid:28) j t A X (cid:21) A x B i and (cid:17) = Z t 1(cid:0) t 9 (cid:0) (cid:28) (cid:28) I (cid:0) t j t s D(cid:22) h ( o X ul t0 d(cid:0) t (cid:28) (cid:0) bj t (cid:28) e ; j t x n 0t o(cid:0)t (cid:28) ejd t ; t i (cid:21) t h (cid:0) a (cid:28) t )0 t , h a e n s d et a u r p e n h o er r e m d a t l (cid:0) i l ⁄ y (cid:28) e j r t n s on co z m er t p (cid:0) o. a (cid:28) r j t ed (cid:0) to 1 w 1 ha t (cid:0) t (cid:28) w (cid:0) a 1 j s t u (cid:0) sed 12 in, t (cid:0) f (cid:28) o (cid:0) r 1 e j t xa (cid:0) mp 1 le, t (cid:0) A (cid:28) (cid:0) L 1 LV [3] a t n (cid:0) d (cid:28) j t [5], whichexamineforecastsusinganinstrumentrule. Thereuncertaintyaboutbothparameters,thecurrentstateofthe economy, the sequence of future shocks as well as the measurement errors were allowed for (see Adolfson, LindØ and Villani [7] for a description). However, this uncertainty is additive so certainty equivalence holds. Also our timing convention for the projections di⁄ers. In ALLV [3], [5], and [7] it is assumed that the projections are carried out at theend ofperiod t(using theestimated instrumentrule). Thatis,Z isobserved and considered to beknown atthe t time of the projection in ALLV [3], [5], and [7]. 21

where X is given by (5.1). Thus, we let (cid:147);t(cid:148)and (cid:147)t(cid:148)in subindices refer to projections and tt j j estimates (rational expectations) in the beginning of period t, respectively. The reason for this separate notation for the projections is that they are conceptually distinct from the equilibrium rational expectations and include possible hypothetical projections contemplated by the central bankduringitsdecisionprocess. Thefeasiblesetofprojections forgivenX isthesetofprojections tt j that satisfy (5.2)-(5.4). The policy problem in period t is to determine the optimal projection in period t, denoted (X(cid:20)t;x(cid:20)t;(cid:20)it;Y(cid:20)t). The optimal projection is the projection that minimizes the intertemporal loss function, 1 (cid:14)(cid:28)L ; (5.5) t+(cid:28);t (cid:28)=0 X where the period loss, L , is speci(cid:133)ed as t+(cid:28);t L = Y WY ; (5.6) t+(cid:28);t t+(cid:28);t0 t+(cid:28);t where W is symmetric positive semide(cid:133)nite matrix with diagonal (1, (cid:21) (cid:21) ). The minimization y; (cid:1)i 0 is subject to the projection being in the feasible set of projections for given X .20 tt j When the policy problem is formulated in terms of projections, we can allow 0 < (cid:14) 1, since (cid:20) the above in(cid:133)nite sum will normally converge also for (cid:14) = 1. The optimization is done under commitment in a timeless perspective (Woodford [39]). The optimization results in a set of (cid:133)rstorder conditions, which combined with the model equations, (2.21), yields a system of di⁄erence equations(seeS(cid:246)derlind[28]andSvensson[30])thatcanbesolvedwithseveralalternativenumerical algorithms (see section 2.6 for references).21 Under the assumption of optimization under commitment in a timeless perspective, one way to describe the optimal projection is by the following di⁄erence equations, x(cid:20) X(cid:20) t+(cid:28);t = F t+(cid:28);t ; (5.7) (cid:20)i (cid:4) t+(cid:28);t t+(cid:28) 1;t (cid:20) (cid:21) (cid:20) (cid:0) (cid:21) X(cid:20) X(cid:20) t+(cid:28)+1;t = M t+(cid:28);t ; (5.8) (cid:4) (cid:4) t+(cid:28);t t+(cid:28) 1;t (cid:20) (cid:21) (cid:20) (cid:0) (cid:21) for (cid:28) 0, where X(cid:20) = X . The matrices F and M depend on A, B, H, D, W, and (cid:14), but are t;t tt (cid:21) j independentofC. TheindependenceofC demonstratesthecertaintyequivalenceoftheprojections. The n -vector (cid:4) consists of the Lagrange multipliers of the lower block of (5.2), the block X t+(cid:28);t 20 PolicyprojectionswhenmonetarypolicyischaracterizedbyasimpleinstrumentrulearedescribedinappendixB. 21 As discussed in Woodford [39] and Svensson [30] and [31], optimization under commitment in a timeless perspective allows optimal policy that is consistent over time. Svensson [31] also discusses optimal projections under discretion. 22

determining the projection of the forward-looking variables. We assume that the optimization is under commitment in a timeless perspective, so that the initial Lagrange multiplier, (cid:4) , is nont 1;t (cid:0) zero. Commitment is thus considered having occurred some time in the past, and more precisely we assume policy has been optimal since the start of the in(cid:135)ation targeting in 1993:1.22 5.3. An application to the Great Recession Next, we use the model to interpret the future economic development in Sweden for a policy maker standing in 2008:3. This period is especially interesting to analyze since the world economy were just about to enter a deep recession in the following quarters. Figure 5.1 show projections from Ramses using data up to and including 2008:2, for 10 key variables: foreign output, foreign annualized 1-quarter CPI in(cid:135)ation, foreign interest rate, the real exchange rate, 4-quarter CPI in(cid:135)ation, the output gap, the instrument rate, the real interest rate (the instrument rate less 1-quarter-ahead CPI in(cid:135)ation expectations), output and potential output. The plots show deviations from steady state. The solid vertical line marks 2008:3, which is quarter 0inthe(cid:133)gure. Thecurvestotheleftofthelineshowsthehistoryofactualdatauptoandincluding 2008:2, which is quarter -1 in the (cid:133)gure. The state in 2008:2, and its history, is estimated using policy with the instrument rule. The solid curves to the right of the solid vertical line show the projections for a policy maker following the estimated instrument rule. We see that Ramses(cid:146)VAR-model for the three foreign variables did not forecast the large drop in the world economy that occurred in the next few quarters after 2008:3, following the turbulence in the (cid:133)nancial markets. In reality, world output diminishedbyalmost4percentinthenextthreequartersandworldin(cid:135)ationapproached0percent. GiventheVAR-model(cid:146)scounterfactualviewoftheforeigneconomy, RamsesprojectedSwedishCPI in(cid:135)ation to be close to 2 percent and output to be less than 0:5 percent below its trend level. Ex post, we know Swedish GDP decreased by almost 6 percent in the next year and consumer prices actually declined. We know from VAR evidence that international spillover e⁄ects to small open economies are generally large. LindØ [19] shows, using a block exogenity assumption in a VAR model on Swedish data, that foreign shocks account for around 50 percent of the (cid:135)uctuations around trend in Swedish output and domestic in(cid:135)ation. Can then the substantial drop in the world economy explain the economic development in Sweden during the (cid:133)nancial crisis? To analyze this we compute a forecast 22 See appendix B.3 for a discussion about alternative methods to calculate the initial Lagrange multipliers. 23

with Ramses that is conditional upon the decline in foreign output during 2008:4 - 2009:2. After those three quarters the forecast is endogenously determined by the model, which is marked by a dashed vertical line. It is assumed that the agents in the model fully anticipate the path of foreign output during 2008:4 - 2009:2. The expectations in the model are thus consistent with the conditional path of foreign output. We use the method in LasØen and Svensson [18] to calculate how large time-varying intercepts in the VAR equation for foreign output are needed to replicate the drop in foreign output. Foreign in(cid:135)ation and the foreign interest rate is then determined endogenously within the foreign VAR model. The dashed curves in (cid:133)gure 5.1 show this conditional projection when policy follows the estimated instrument rule. The cumulative drop of about 3:7 percent in world output implies that world in(cid:135)ation drops 1 percent below steady state and that the world interest rate almost hits the zero lower bound. The steady state level of the nominal interest rate in both Sweden and the rest of the world is assumed to be 4:25 percent, and, as stated above, the plots show deviations from steady state. The substantial fall in foreign output implies less demand for Swedish export goods, and domestic output drops below trend by about 1:5 percent. We see that the spillover to Swedish production is approximately 50 percent of the drop in the world economy. However, CPI in(cid:135)ation in Sweden decreases by more than the decline in the world economy. From (cid:133)gure 5.1 we also see that the instrument rule implies that the Swedish interest rate is lowered by about 2 percentage points below its steady state level. Since the foreign interest rate falls by more, this implies that the nominal (not depicted) and real (depicted) exchange rate appreciates, due to the UIP condition in the model. In reality we saw a large (cid:135)ight from the Swedish krona, however. To account for the large depreciation of the real exchange rate, the model needs risk premium shocks that can capture the deviations from interest rate parity. In reality, substantial declines in investmentandexportsalsooccurred. TocaptureallcomponentsintheGDPidentity(andthereby describe the entire fall in Swedish GDP), the model additionally requires shocks to investment technology and the export markup. The large fall in foreign output can thus not alone explain all aspects of the development of the Swedish economy during the (cid:133)nancial crisis. There are several reasons for why the model scenario under consideration cannot fully capture the fall in domestic GDP that actually occurred. The scenario above, with a decline in foreign output only, works through the trade and real exchange rate channels of the model. In reality, the recession was created in the (cid:133)nancial sector which caused the spread between inter-bank rates and the instrument rate to increase dramatically. As in most DSGE models, Ramses lacks such a 24

Figure 5.1: Projections in 2008:3 for the instrument rule World output World inflation World interest rate 0 0 2 r y 1 1 / 1 % 2 ,% 2 0 3 3 1 4 0 10 20 30 0 10 20 30 0 10 20 30 4 qtr CPI inflation Output gap Potential output Output 1 1 0.5 0.5 r 0.5 y / 0 0 0 % 0 ,% 1 0.5 0.5 1 0.5 1 2 1.5 1 1.5 0 10 20 30 0 10 20 30 0 10 20 30 0 10 20 30 Quarters Instrument rate Real interest rate Real exchange rate 2 0 1 0 r y / 0.5 0 2 % ,% 1 2 1 4 1.5 6 3 2 8 0 10 20 30 0 10 20 30 0 10 20 30 Quarters Quarters Quarters Endogenous projection Conditional projection (cid:133)nancial sector that includes a (cid:133)nancing premium which could have further dampened domestic GDP.23 If we would have been able to consider both shocks simultaneously, this would have caused the zero bound on nominal interest rates to bind for Sweden(cid:146)s most important trading partners and in all likelihood also for the Swedish economy. This channel, per se, would have ampli(cid:133)ed the fall in domestic output even further. Finally, we assess how the projections during the (cid:133)nancial crisis could have been a⁄ected by alternative monetary policies. Depending on the state of the economy (and thus the current size of the shocks that persistently die out over the projection horizon), the policy assumption of which output measure to stabilize can matter quite a bit for the projections, as shown in section 4. In (cid:133)gure 5.2 we therefore compare projections with the instrument rule and projections with policy for di⁄erent output gaps in the loss function. In all the optimal policy projections in this (cid:133)gure, we 23 Christiano, Trabandt and Walentin [12] estimate a similar model to ours extended with (cid:133)nancial frictions on Swedish data and (cid:133)nd that shocks stemming from the (cid:133)nancial sector indeed contributed signi(cid:133)cantly to the contraction in output during 2008-2009. A variant of their model has recently replaced the version of Ramses we consider in the policy process at the Riksbank. 25

Figure 5.2: Di⁄erence between unconditional and conditional projections in 2008:3 for optimal policy with di⁄erent output gaps ((cid:21) = 1:1;(cid:21) = 0:37) y (cid:1)i World output World inflation World interest rate 0 0 2 r y 1 1 / 1 % 2 ,% 2 0 3 3 1 4 0 10 20 30 0 10 20 30 0 10 20 30 4 qtr CPI inflation Output gap Potential output Output 1 1 1 2 r y 0 0 0 / % 1 1 ,% 1 0 2 1 2 1 3 2 0 10 20 30 0 10 20 30 0 10 20 30 0 10 20 30 Quarters Instrument rate Real interest rate Real exchange rate 4 5 2 r y 2 0 / % 0 5 0 ,% 10 2 2 15 4 0 10 20 30 0 10 20 30 0 10 20 30 Quarters Quarters Quarters Conditional gap Unconditional gap Trend gap Instrument rule assumecommitmentinatimelessperspectiveandcomputetheinitialvectorofLagrangemultipliers for the equations for the forward-looking variables ((cid:4) ) under the assumption that policy during 1 (cid:0) the in(cid:135)ation-targeting period has been optimal (see appendix B.3).24 As in (cid:133)gure 5.1 we condition all the projections on the decline in foreign output during the (cid:133)rst three quarters after 2008:3. The dashed curves in (cid:133)gure 5.2 reprints the conditional projections from (cid:133)gure 5.1 where policy follows the instrument rule. The solid curves show optimal policy projections with the conditional output gap in the loss function. The dotted curves show the optimal policy projections when the output gap in the loss function is the unconditional output gap (which is computed using equation (C.7) with (cid:135)exible prices from 1993:1 and onwards to form the state vector under (cid:135)exible prices). The dashed-dotted curves show the optimal policy projections with the trend output gap in the loss function. In (cid:133)gure 5.2, we see that 4-quarter CPI-in(cid:135)ation and the output gap falls quite a bit when 24 It should, however, be noted that the results do not change much to setting the Lagrange multipliers to zero. 26

policy follows the instrument rule. Optimal policy, with the trend output gap in the loss function, is shown with the dash-dotted curves. It is able to stabilize both in(cid:135)ation and the output gap much better. With the instrument rule(cid:146)s slow adjustment of the instrument rate, the real interest rate initially increases because in(cid:135)ation falls so quickly and is then followed by a compensating modest fall in the real interest rate. Optimal policy initially increases the real interest rate less and then lowers the real interest rate substantially. This implies substantially more expansionary policy and succeeds in preventing large falls in both CPI in(cid:135)ation and output. In spite of the non-negligible weight on instrument-rate smoothing, optimal policy allows the instrument rate to (cid:133)rst increase and then fall substantially, a much more e⁄ective policy than the instrument rule(cid:146)s smooth fall and rise of the instrument rate. It is important to understand that the initial increase in the nominal instrumentrateforthelossfunctionwiththetrendgapisdrivenbytheinitialstateoftheeconomy. Had the economy been in steady state when the negative foreign demand shocks hit the economy, the instrument rate for the loss function with the trend gap would have been lowered directly. Optimal policy with the unconditional output gap in the loss function and potential output given by unconditional (cid:135)exible-price output is shown with the dotted curves. The fall in world output and the other foreign shocks imply a large fall in potential output. Optimal policy results in a small positive output gap. This requires a large fall in output, which requires contractionary monetary policy in the form of a high real interest rate and a real appreciation of the currency. This results in an intermediate fall in CPI-in(cid:135)ation. Optimal policy with the conditional output gap in the loss function and potential output given by conditional (cid:135)exible-price output is shown with the solid curves. The shocks from the world economy results in a fall and then a rise in potential output. Optimal policy results in (cid:133)rst a positive and then a negative output gap, corresponding to positive output falling to the steadystate level and then rising above. CPI in(cid:135)ation falls modestly below target and then rises above. This requires expansionary monetary policy, with a signi(cid:133)cant fall in the real interest rate and initially much smaller real appreciation of the currency than the other policies. Clearly, it matters quite a bit for optimal policy what output gap enters the loss function, with di⁄erent consequences for the dynamics of output and in(cid:135)ation and the optimal policy response. Some policies result in considerably more output and CPI stabilization than others. Clearly it is important to determine what potential-output de(cid:133)nition is the most appropriate. 27

6. Conclusions In this paper, we have shown how to construct operational optimal policy projections in the Riksbank(cid:146)s model Ramses, a linear-quadratic open-economy DSGE model. By optimal policy projections we mean projections of the target variables and the instrument rate that minimize a loss function. We have illustrated the use and consequences of di⁄erent output-gap concepts in the loss function and clari(cid:133)ed the di⁄erence between output gaps relative to conditional potential output, trend output, and unconditional potential output, where conditional refers to the dependence on existing endogenous predetermined variables, such as the capital stock. When productivity is temporarily high, conditional potential output exceeds trend output. Then optimal policy projections in this case di⁄er substantially depending on whether conditional or trend output gaps enter the loss function. We have also illustrated how the model interpreted the economic development in Sweden with the Great Recession ahead in 2008:3, using di⁄erent policy assumptions. We studied the direct impact of the fall in the world economy, but not the (cid:133)nancial spillover that must have accounted for a large share of the Swedish output decline. An important challenge ahead for the profession is to develop DSGE models with elaborate (cid:133)nancial frictions. Withthetoolswehavedemonstrated, webelieveoptimalpolicyprojectionsinRamsesandsimilar DSGE models can now be applied in real-time policy processes and provide policymakers with useful advice for their decisions(cid:151)together with the usual input of detailed analysis and estimation oftheinitialstateoftheeconomy, policysimulationswithhistoricalpolicyreactionfunctions, other forecasting models, judgment, concerns about model uncertainty, and so forth. 28

Appendix A. Ramses in some detail Thisappendixpresentstheloglinearapproximationofthemodel. Foramoredetaileddescriptionof the complete model and the derivation of the loglinear approximation, see ALLV [4]. A hat denotes the deviation of a loglinearized variable from a steady-state level (v^ dv =v for any variable v , t t t (cid:17) where v is the steady-state level), and delta denotes the log-di⁄erence of a variable ((cid:1)v v v t t t 1 (cid:17) (cid:0) (cid:0) for any variable v ). Since all real variables grow with the non-stationary technology shock z , t t we have to divide all quantities with the trend level of technology to make them stationary. We denote the resulting stationary variables by lower-case letters, that is, c = C =z ; ~{ = I =z (we t t t t t t denote investment by ~{ to avoid confusion with the monetary-policy instrument i ), k = K =z t t t+1 t t (capital-services), k(cid:22) = K(cid:22) =z (physical capital stock), w = W =(P z ) (real wage). t+1 t+1 t t t t t j In(cid:135)ation,(cid:25)^ ;inthefoursectorsof(cid:133)rmsj d;mc;mi;x : domestic(production)(d),consumert 2 f g goods import (mc), investment-goods import (mi), and export (x), is determined from the Phillips curve (cid:24) (cid:12) (cid:24) (cid:20) j c j j c j j j c (cid:24) ((cid:25)^ (cid:25)(cid:22) ) = ((cid:25)^ (cid:26) (cid:25)(cid:22) )+ ((cid:25)^ (cid:25)(cid:22) ) (A.1) j t (cid:0) t 1+(cid:20) (cid:12) t+1t(cid:0) (cid:25)(cid:22)c t 1+(cid:20) (cid:12) t 1(cid:0) t j j j (cid:0) (cid:24) (cid:20) (cid:12)(1 (cid:26) ) (1 (cid:24) )(1 (cid:12)(cid:24) ) b j j (cid:0) (cid:25)(cid:22)c (cid:25)(cid:22) c +b (cid:0) j (cid:0) j (mbc j +(cid:21)^j ); (cid:0) 1+(cid:20) (cid:12) t 1+(cid:20) (cid:12) t t j j j b c The (cid:133)rms(cid:146)marginal costs, mc , are de(cid:133)ned as t mcdc (cid:11)((cid:22)^ +H^ k^ )+w(cid:22) +R^f ^" ; t zt t t t t t (cid:17) (cid:0) (cid:0) mcmc P^ +S^ P^mc mcx (cid:13)^x (cid:13)^mcd; ct (cid:17) t(cid:3) t (cid:0) t (cid:17) (cid:0)b t (cid:0) t(cid:3) (cid:0) t mcmi P^ +S^ P^mi mcx (cid:13)^x (cid:13)^mid; ct (cid:17) t(cid:3) t (cid:0) t (cid:17) (cid:0)ct (cid:0) t(cid:3) (cid:0) t mcx P^d S^ P^x mcx +(cid:25)^ (cid:25)^x (cid:1)S^; c t (cid:17) t (cid:0) t (cid:0) t (cid:17) tc (cid:0) 1 t (cid:0) t (cid:0) t respectively; where R^f dcenotes the e⁄ective nominalcinterest rate paid by the (cid:133)rms, t (cid:23)R (cid:23)(R 1) R^f = R^ + (cid:0) (cid:23)^ ; t v(R 1)+1 t (cid:0) 1 v(R 1)+1 t (cid:0) (cid:0) where(cid:23)^ isashocktothefractionof(cid:133)rmsthathasto(cid:133)nancetheirwagebillinadvance(throughout t 29

we set (cid:23) = 1). Furthermore, (cid:13)^x , (cid:13)^mcd, (cid:13)^mid, and (cid:13)^ f are relative prices de(cid:133)ned as t(cid:3) t t t (cid:13)^x P^x P^ (cid:13)^x +(cid:25)^x (cid:25)^ ;; t(cid:3) t t(cid:3) t(cid:3)1 t (cid:3)t (cid:17) (cid:0) (cid:17) (cid:0) (cid:0) (cid:13)^mcd P^mc P^d (cid:13)^mcd+(cid:25)^mc (cid:25)^d; t t t t 1 t t (cid:17) (cid:0) (cid:17) (cid:0) (cid:0) (cid:13)^mid P^mi P^d (cid:13)^mid +(cid:25)^mi (cid:25)^d; t t t t 1 t t (cid:17) (cid:0) (cid:17) (cid:0) (cid:0) (cid:13)^ f P^d S^ P^ mcx+(cid:13)^x : t t t t(cid:3) t t(cid:3) (cid:17) (cid:0) (cid:0) (cid:17) CPI in(cid:135)ation, (cid:25)^c, satis(cid:133)es c t (cid:25)^c t = (1 ! c )((cid:13)cd) (cid:0) (1 (cid:0) (cid:17) c )(cid:25)^d t +! c ((cid:13)cmc) (cid:0) (1 (cid:0) (cid:17) c )(cid:25)^m t c; (cid:0) The real-wage equation can be written 0 = b (cid:24) w(cid:22) +[(cid:27) (cid:21) b 1+(cid:12)(cid:24)2 ]w(cid:22) +b (cid:12)(cid:24) w(cid:22) w w t 1 L w w w t w w t+1t (cid:0) (cid:0) j b (cid:24) (cid:25)^d (cid:25)(cid:22) c +b (cid:12)(cid:0)(cid:24) (cid:25)^d (cid:1) (cid:26) (cid:25)(cid:22) c (cid:0) w bw t (cid:0) t w w t+1 (cid:0)b (cid:25)(cid:22)c t b +b (cid:24) (cid:20) (cid:16) (cid:25)^c (cid:17) (cid:25)(cid:22) c b (cid:12) (cid:16) (cid:24) (cid:20) (cid:25)^c (cid:26) (cid:17) (cid:25)(cid:22) c w w w t (cid:0) b1 (cid:0) t (cid:0) w w w t (cid:0)b (cid:25)(cid:22)c t +(1 (cid:21) )(cid:0) ^ (1 (cid:1)(cid:21) )(cid:27) H^ (cid:0) (cid:1) (cid:0) w zt (cid:0) b(cid:0) w L t b (1 (cid:21) ) (cid:28)y (cid:28)^ y (1 (cid:21) ) (cid:28)w (cid:28)^w (1 (cid:21) )(cid:16)^h ; (cid:0) (cid:0) w 1 (cid:28)y t (cid:0) (cid:0) w 1+(cid:28)w t (cid:0) (cid:0) w t (cid:0) where (cid:21) (cid:27) (1 (cid:21) ) w L w b (cid:0) (cid:0) ; w (cid:17) (1 (cid:12)(cid:24) )(1 (cid:24) ) w w (cid:0) (cid:0) ^ denotes the marginal utility of income. zt The Euler equation for consumption expenditures is, 0 = b(cid:12)(cid:22) c^ + (cid:22)2+b2(cid:12) c^ b(cid:22) c^ +b(cid:22) (cid:22)^ (cid:12)(cid:22)^ z t+1t z t z t 1 z zt z;t+1t (cid:0) j (cid:0) (cid:0) (cid:0) j + ((cid:22) b(cid:12))((cid:22) (cid:0) b) ^ + (cid:1) (cid:28)c ((cid:22) b(cid:12))((cid:22) (cid:16) b)(cid:28)^c (cid:17) z (cid:0) z (cid:0) zt 1+(cid:28)c z (cid:0) z (cid:0) t + ((cid:22) b(cid:12))((cid:22) b)(cid:13)^cd ((cid:22) b) (cid:22) (cid:16)^c b(cid:12)(cid:16)^c ; z z t z z t t+1t (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) j (cid:16) (cid:17) where (cid:13)^cd P^c P^d (cid:13)^cd +(cid:25)^c (cid:25)^d t t t t 1 t t (cid:17) (cid:0) (cid:17) (cid:0) (cid:0) denotes the relative price between consumption and domestically produced goods. The household(cid:146)s (cid:133)rst-order conditions with respect to investment ~{ , the physical capital stock t k(cid:22) , and the utilization rate t+1 u^ k^ k(cid:22) ; t t t (cid:17) (cid:0) b 30

are, in their loglinearized forms, given by P^ +(cid:7)^ (cid:13)^id (cid:22)2S~ ~{ ~{ (cid:12) ~{ ~{ +(cid:22)^ (cid:12)(cid:22)^ = 0; (A.2) kt t t z 00 t t 1 t+1t t zt z;t+1t (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) j (cid:0) (cid:0) j h(cid:16) (cid:17) (cid:16) (cid:17) i (cid:12)(1 (cid:14)) b b (cid:22) (cid:12)b(1 (cid:14))b (cid:28)k (cid:22) (cid:12)(1 (cid:14)) ^ +(cid:22)^ ^ (cid:0) P^ +P^ z (cid:0) (cid:0) r^k + z (cid:0) (cid:0) (cid:28)^k = 0; zt z;t+1 j t (cid:0) z;t+1 j t (cid:0) (cid:22) z k;t+1 j t kt (cid:0) (cid:22) z t+1 j t 1 (cid:28)k (cid:22) z t+1 j t (cid:0) 1 1 (cid:28)k u^ = r^k (cid:28)^k; (A.3) t (cid:27) t (cid:0) (cid:27) (1 (cid:28)k) t a a (cid:0) where P is the price of capital goods in terms of domestic goods, (cid:14) is the depreciation rate, (cid:27) kt a is a parameter determining the capital utilization rate (here calibrated so that there is no variable capital utilization (cid:27) = 106) and r^k denotes the log-linearized expression for the real rental rate of a t capital and satis(cid:133)es r^k = (cid:22)^ +w(cid:22) +R^f +H^ k^ : t zt t t t t (cid:0) The loglinearized law of motion for capitabl is given by 1 1 1 1 k(cid:22) = (1 (cid:14)) k(cid:22) (1 (cid:14)) (cid:22)^ + 1 (1 (cid:14)) (cid:7)^ + 1 (1 (cid:14)) ~{ : (A.4) t+1 (cid:0) (cid:22) t (cid:0) (cid:0) (cid:22) zt (cid:0) (cid:0) (cid:22) t (cid:0) (cid:0) (cid:22) t z z (cid:20) z(cid:21) (cid:20) z(cid:21) b b b The output gap satis(cid:133)es, y^ = (cid:21) ^(cid:15) +(cid:21) (cid:11)k^ (cid:21) (cid:11)(cid:22)^ +(cid:21) (1 (cid:11))H^ ; (A.5) t d t d t d zt d t (cid:0) (cid:0) where (cid:21) is the steady-state markup. d By combining the (cid:133)rst-order conditions for the holdings of domestic and foreign bonds, we obtain the modi(cid:133)ed uncovered interest parity (UIP) condition, 1 (cid:30)~ E (cid:1)S^ (cid:30)~ (cid:1)S^ R^ R^ (cid:30)~ a^ +(cid:30)~ = 0: (A.6) s t t+1 s t t t(cid:3) a t t (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) (cid:16) (cid:17) (cid:16) (cid:17) b The real exchange rate, x~ , satis(cid:133)es t x~ S^ +P^ P^c x~ +(cid:25)^ (cid:25)^c. t t t(cid:3) t t 1 (cid:3)t t (cid:17) (cid:0) (cid:17) (cid:0) (cid:0) b b Theloglinearizedversionofthe(cid:133)rst-orderconditionsformoneybalancesm andcashholdings t+1 q are, respectively, t (cid:28)k (cid:22) ^ +(cid:22) ^ (cid:22)(cid:22)^ + (cid:22) (cid:12)(cid:28)k R^ (cid:22)(cid:25)^d + ((cid:12) (cid:22))(cid:28)^k = 0; (cid:0) zt z;t+1 j t (cid:0) z;t+1 j t (cid:0) t (cid:0) t+1 j t 1 (cid:28)k (cid:0) t+1 j t (cid:16) (cid:17) (cid:0) q^ = 1 (cid:16)^q + (cid:28)k (cid:28)^k ^ R R^ ; (A.7) t (cid:27) q t 1 (cid:28)k t (cid:0) zt (cid:0) R 1 t (cid:0) 1 (cid:20) (cid:0) (cid:0) (cid:21) where (cid:22) is steady-state money growth and (cid:16)^q is a cash preference shock, to be speci(cid:133)ed below. t 31

The following aggregate resource constraint must hold in equilibrium (cid:17) c (cid:17) ~{ g y (1 (cid:0) ! c ) (cid:13)cd c y c^ t +(cid:17) c (cid:13)^c t d +(1 (cid:0) ! i ) (cid:13)id i y ~{ t +(cid:17) i (cid:13)^i t d + y g^ t + y (cid:3) y^ t(cid:3) (cid:0) (cid:17) f (cid:13)^x t(cid:3) +z~(cid:3) t (cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17) b k(cid:22) 1 b = (cid:21) [^" +(cid:11)(k^ (cid:22)^ )+(1 (cid:11))H^ ] 1 (cid:28)k rk (k^ k(cid:22) ); d t t (cid:0) zt (cid:0) t (cid:0) (cid:0) y(cid:22) t (cid:0) t z (cid:16) (cid:17) where(cid:17) ,(cid:17) ,and(cid:17) areelasticitiesofsubstitutionbetweendomesticandimpobrtedconsumergoods, c i f domesticandimportedinvestmentgoods,anddomesticandforeigngoods(inforeignconsumption), respectively;c,~{,y,y ,g,andk(cid:22)aresteady-statelevelsofconsumption,investment,domesticoutput, (cid:3) foreign output, government expenditure, and the capital stock, respectively (when scaled with z , t the technology level); g^ is government expenditure, and t (cid:13)^id P^i P^d (cid:13)^id +(cid:25)^i (cid:25)^d t t t t 1 t t (cid:17) (cid:0) (cid:17) (cid:0) (cid:0) is the relative price between investment and domestically produced goods. Wealsoneedtorelatemoneygrowth(cid:22) torealbalances(whererealbalancesm(cid:22) (M =Pd)=z t t+1 t+1 t t (cid:17) are scaled by the technology shock z and M denote nominal balances) and domestic in(cid:135)ation, t t+1 M m(cid:22) z Pd m(cid:22) (cid:22) (cid:25)d (cid:22) t+1 t+1 t t t+1 zt t: t (cid:17) M (cid:17) m(cid:22) z Pd (cid:17) m(cid:22) t t t 1 t 1 t (cid:0) (cid:0) Loglinearizing, we have m(cid:22) = m(cid:22) (cid:22)^ +(cid:22)^ (cid:25)^d: (A.8) t+1 t zt t t (cid:0) (cid:0) To clear the loan market, the dbemand fbor liquidity from the (cid:133)rms (which are (cid:133)nancing their wage bills) must equal the supplied deposits of the households plus the monetary injection by the central bank: (cid:22)m(cid:22) (cid:23)w(cid:22)H (cid:23)^ +w(cid:22) +H^ = (cid:22)^ +m(cid:22) (cid:25)^d (cid:22)^ qq^; (A.9) t t t (cid:25)d(cid:22) t t (cid:0) t (cid:0) zt (cid:0) t z (cid:16) (cid:17) (cid:16) (cid:17) where w(cid:22), m(cid:22) and q are steady-stbate levels of real wages,breal balances, and cash holdings (when scaled by z ), H is the steady-state level of hours worked, and (cid:25)d is the steady-state in(cid:135)ation of t domestic goods. The evolution of net foreign assets at the aggregate level satis(cid:133)es a^ t = y (cid:3) mcx t (cid:17) f y t(cid:3) (cid:13)^x t(cid:3) +y (cid:3) y^ t(cid:3) +y (cid:3) z~(cid:3) t +(cm+~{m)(cid:13)^ f t (cid:0) (cid:0) (1 (cid:17) ) cm (cid:17) (1 ! ) (cid:13)cd (cid:0) (cid:0) c (cid:13)^mcd+c^ (A.10) (cid:0) c(cid:0) c (cid:0) c b t t (cid:20) (cid:21) (cid:16) (cid:17) (1 (cid:17) ) R ~{m (cid:17) (1 ! ) (cid:13)id (cid:0) (cid:0) i (cid:13)^mid+~{ + a^ ; (cid:0) (cid:20) (cid:0) i (cid:0) i t t (cid:21) (cid:25)(cid:22) z t (cid:0) 1 (cid:16) (cid:17) b 32

where cm and im are steady-state levels of consumption and investment of imported goods (when scaled by z t ), R is the steady-state nominal interest-rate level, and z~(cid:3) t is a stationary shock that measures the degree of asymmetry in the technological level between the domestic economy and b the foreign economy. The exogenous shocks of the model are AR(1) given by the representation ^& = (cid:26) ^& +" ; t & t 1 &t (cid:0) where" isi.i.d. andN 0;(cid:27)2 ,for& (cid:22) ," ;(cid:21) j ;(cid:16)c;(cid:16)h;(cid:16)q;(cid:7) ;(cid:30)~ ;(cid:25)(cid:22)c;z~ andj d;mc;mi;x . &t & t zt t t t t t t t t(cid:3) 2 f g 2 f g Furthermore,asdiscusse(cid:0)dinA(cid:1)LLV[3],forestimationpurposesitisconvenienttorescalethemarkup shock (cid:21)^j j = d;mc;mi;x in the Phillips curves so as to include the coe¢ cient (1 (cid:0) (cid:24) j )(1 (cid:0) (cid:12)(cid:24) j ) in t f g (cid:24) j (1+(cid:20)j(cid:12)) the markup shock. Then the new coe¢ cient on them in the Phillips curve is unity. Similarly, we rescaletheinvestmentspeci(cid:133)ctechnologyshock(cid:7)^ ,thelaborsupplyshock(cid:16)h,andtheconsumption t t preference shock (cid:16)c, so that these shocks enter in an additive fashion as well.25 t The(cid:133)scalpolicyvariablesareassumedtobeexogenouslygivenbyanidenti(cid:133)edVARmodelwith two lags and an uninformative prior. Let (cid:28) ((cid:28)^k;(cid:28)^ y ;(cid:28)^c;(cid:28)^w;g^), where g^ denotes HP-detrended t t t t t t 0 t (cid:17) government expenditures. The (cid:133)scal policy VAR(2)-model is given by (cid:2) (cid:28) = (cid:2) (cid:28) +(cid:2) (cid:28) +S " ; (A.11) 0 t 1 t 1 2 t 2 (cid:28) (cid:28)t (cid:0) (cid:0) where " (cid:28)t s N (0;I (cid:28) ) S (cid:28) is a diagonal matrix with the standard deviations, and (cid:2) (cid:0)0 1S (cid:28) " (cid:28)t s N (0;(cid:6) ). (cid:28) The foreign economy is exogenously given by an identi(cid:133)ed VAR model with four lags using an uninformative prior, (cid:8) X = (cid:8) X +(cid:8) X +(cid:8) X +(cid:8) X +S " ; (A.12) 0 t(cid:3) 1 t(cid:3) 1 2 t(cid:3) 2 3 t(cid:3) 3 4 t(cid:3) 4 x (cid:3) x (cid:3) t (cid:0) (cid:0) (cid:0) (cid:0) where X ((cid:25) ;y^ ;R ), where (cid:25) and R are quarterly foreign in(cid:135)ation and interest rates, and t(cid:3) (cid:3)t t(cid:3) t(cid:3) 0 (cid:3)t t(cid:3) (cid:17) y^ t(cid:3) is foreign output; " x (cid:3) t s N (0;I x (cid:3) ); S X (cid:3) is a diagonal matrix with the standard deviations; and (cid:8) (cid:0)0 1S x (cid:3) " x (cid:3) t s N (0;(cid:6) x (cid:3) ).26 When estimating the VAR, we assume and do not reject that (cid:8) 0 in 25 Although this is not of any major importance for the baseline estimation of the model in ALLV [3], it is important when carrying out the sensitivity analysis there because the e⁄ective prior standard deviation of the shocks changes with the value of the nominal and real friction parameters. Smets and Wouters [27] adopt the same strategy. Therefore,toobtainthesizeofthefourtrulyfundamentalmarkupshocks,theestimatedstandarddeviations reported in table 3:1 should be divided by their respective scaling parameter (for instance, (1 (cid:0) (cid:24)d)(1 (cid:0) (cid:12)(cid:24)d) in the case (cid:24)d(1+(cid:20)d(cid:12)) of the domestic markup shock). 26 ThereasonwhyweincludeforeignoutputHP-detrendedandnotingrowthratesintheVAR isthattheforeign output gap enters the log-linearized model (for instance, in the aggregate resource constraint). This also enables identi(cid:133)cation of the asymmetric technology shock. 33

(A.12) has the following structure 1 0 0 (cid:8) 0 1 0 : 0 (cid:17) 2 3 (cid:13) (cid:13) 1 (cid:0) (cid:3)(cid:25)0 (cid:0) (cid:3)y0 4 5 The model can then be written on the form (2.21), where the speci(cid:133)cation of the predetermined variables X , the forward-looking variables x , the policy instrument i , and the shock vector " can t t t t be found in the technical appendix [1]. B. Optimal projections in some detail B.1. Information and data First we clarify the information assumptions underlying the conditional expectation E used in the t loss function (2.18). Let E [ ] E[ ] where denotes the information set in period t. In the t t t (cid:1) (cid:17) (cid:1)jI I standard and simple case when all variables are observed, we can specify " ;X ;i ;" ;X ;x ; i ;" ;X ;x ;i ;::: : t t t t t 1 t 1 t 1 t 1 t 2 t 2 t 2 t 1 I (cid:17) f (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) g This can be interpreted as information in the beginning of period t. We can understand this as the agents of the model (the central bank, the private sector, the (cid:133)scal authority, and the rest of the world) entering the beginning of period t with the knowledge of past realizations of shocks, variables, and instruments, C" ;X ;x ; i ;C" ;X ;x ;i ,...27 Then, at t 1 t 1 t 1 t 1 t 2 t 2 t 2 t 2 (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) the beginning of period t, the shock " is realized and observed, and X is determined by (2.22) t t and observed. We also assume that the agents know the model, including the matrices A, B, C, D, H, and W and the scalar (cid:14), so either C" or X is su¢ cient for inferring the other from (2.22), t t given that previous realizations are known. Then the central bank determines, announces, and implements its instrument setting, i , which is hence observed by the other agents. After this, the t expectations x are formed, and x is determined by (2.24).28 In equilibrium, both i and x t+1t t t t j will be a function of X and previous realizations of X , consistent with this speci(cid:133)cation of the t t information set. However, Ramses makes more elaborate and realistic information assumptions as shown in section5.1. ThevariablesX andx include(seriallycorrelated)shocksandsomeotherunobservable t t variables for which no data exists. Furthermore, the elements of X , x , and i are in many cases t t t quarterly averages, which have not been realized until the end of quarter t. 27 Only the linear combination C" of the shocks " matters, not the individal shocks " . t t t 28 More precisely, the expectations x and x are simultaneously determined. t+1t t j 34

B.2. Solving for the optimal projection As shown in Svensson [30], the (n +n +n ) (cid:133)rst-order conditions for minimizing (2.18) under X x i commitment subject to (2.21) can be written X (cid:24) t 1 (cid:24) A(cid:22) t+1t = W(cid:22) x + H(cid:22) t ; (B.1) 0 (cid:20) (cid:4) t j (cid:21) 2 i t t 3 (cid:14) 0 (cid:20) (cid:4) t (cid:0) 1 (cid:21) 4 5 where the elements of the n -vector (cid:24) are the Lagrange multipliers for the upper block of (2.21) X t+1 (the dating of (cid:24) emphasizes that this is a restriction that applies in period t+1), the elements t+1 of the n -vector (cid:4) are the Lagrange multipliers for the lower block of (2.21) (the dating of (cid:4) x t t emphasizes that this is restriction that applies in period t), and the matrices A(cid:22), W(cid:22) , and H(cid:22) are de(cid:133)ned by I 0 0 A(cid:22) [A B]; W(cid:22) D WD; H(cid:22) : (B.2) (cid:17) (cid:17) 0 (cid:17) 0 H 0 (cid:20) (cid:21) The(cid:133)rst-orderconditionscanbecombinedwiththen +n modelequations(2.21)togetasystem X x of 2(n +n )+n di⁄erence equations for t 0, X x i (cid:21) X X t+1 t C x x H(cid:22) 0 2 t+1 j t 3 A(cid:22) 0 2 t 3 0 0 A(cid:22) i t+1t = W(cid:22) 1H(cid:22) i t +2 0 3" t+1 : (B.3) (cid:20) 0 (cid:21)6 (cid:24) j 7 (cid:20) (cid:14) 0 (cid:21)6 (cid:24) 7 6 t+1t 7 6 t 7 6 0 7 6 (cid:4) j 7 6 (cid:4) 7 6 7 6 t 7 6 t (cid:0) 1 7 4 5 4 5 4 5 Here, X and (cid:4) are predetermined variables (n + n in total), and x , i , and (cid:24) are nont t 1 X x t t t (cid:0) predetermined variables (n +n +n in total). This system can be rewritten as x i X H~ y +H~ y A~ A~ y C~ 11 1;t+1 12 2;t+1t = 11 12 1t + " ; (B.4) H~ y +H~ y j A~ A~ y 0 t+1 (cid:20) 21 1;t+1t 22 2;t+1t (cid:21) (cid:20) 21 22 (cid:21)(cid:20) 2t (cid:21) (cid:20) (cid:21) j j wherey (X ;(cid:4) ) X~ isthevectorofpredeterminedvariablesandy (x ;i ;(cid:24) ) (x~ ;(cid:24) ) 1t (cid:17) t0 0t (cid:0) 1 0 (cid:17) t0 2t (cid:17) 0t 0t 0t 0 (cid:17) 0t 0t 0 is the vector of non-predetermined variables. The matrices are de(cid:133)ned as 0 0 H 0 0 I 0 0 0 0 H~ ; H~ ; H~ 0 A ; H~ 0 0 A ; 11 (cid:17) (cid:20) 0 A 022 (cid:21) 12 (cid:17) (cid:20) 0 0 A 012 (cid:21) 21 (cid:17) 2 0 B 021 3 22 (cid:17) 2 0 0 B 011 3 20 10 4 5 4 5 A 0 A 0 A B 0 21 A~ 11 ; A~ 12 1 ; A~ W(cid:22) 0 ; 11 (cid:17) (cid:20) W(cid:22) xX 1 (cid:14) H 0 (cid:21) 12 (cid:17) (cid:20) W(cid:22) xx W(cid:22) xi 0 (cid:21) 21 (cid:17) 2 W(cid:22) X iX X 0 3 4 5 A B 0 22 2 C A~ W(cid:22) W(cid:22) 1I ; C~ : 22 (cid:17) 2 W(cid:22) X ix x W(cid:22) X ii i (cid:14) 0 3 (cid:17) (cid:20) 0 (cid:21) 4 5 35

The solution to (B.4) can be written x~ F y t = F y X~ ; (B.5) 2t (cid:17) (cid:20) (cid:24) t (cid:21) 2 1t (cid:17) (cid:20) F (cid:24) (cid:21) t y X~ = My +H~ 1C~" MX~ +H~ 1C~" (B.6) 1;t+1 (cid:17) t+1 1t 1(cid:0)1 t+1 (cid:17) t 1(cid:0)1 t+1 for t 0, where y X~ (X ;(cid:4) ) is given. We note that our assumption that A is (cid:21) 10 (cid:17) 0 (cid:17) 00 0 (cid:0) 1 0 22 nonsingular implies that H~ is nonsingular. 11 B.3. Determination of the initial Lagrange multipliers As discussed in Svensson [29, Appendix A],29 the value of the initial Lagrange multiplier, (cid:4) , is t 1;t (cid:0) zero, if there is commitment from scratch in period t, that is, if any previous commitment is disregarded. This re(cid:135)ects a time-consistency problem when there is reoptimization and recommitment in later periods, as is inherently the case in practical monetary policy. Instead, we assume that the optimization is under commitment in a timeless perspective. Then the commitment is considered having occurred some time in the past, and the initial value of the Lagrange multiplier satis(cid:133)es (cid:4) = (cid:4) ; (B.7) t 1;t t 1;t 1 (cid:0) (cid:0) (cid:0) where (cid:4) denotes the Lagrange multiplier of the lower block of (5.2) for the determination of t 1;t 1 (cid:0) (cid:0) x in the decision problem in period t 1. The dependence of the optimal policy projection t 1;t 1 (cid:0) (cid:0) (cid:0) in period t on this Lagrange multiplier from the decision problem in the previous period makes the optimal policy projection depend on previous projections and illustrates the history dependence of optimal policy under commitment in a forward-looking model shown in Backus and Dri¢ ll [9] and Currie and Levine [13] and especially examined and emphasized in Woodford [39]. We now discuss how to compute the initial Lagrange multiplier, (cid:4) when no explicit optit 1;t (cid:0) mization was done in previous periods. In doing so, we assume that past policy has been optimal.30 Note that (5.8) and (B.5) imply that, in real time, the Lagrange multiplier (cid:4) satis(cid:133)es t T (cid:4) = M X +M (cid:4) = (M )(cid:28)M X ; t;t (cid:4)X tt (cid:4)(cid:4) t 1;t 1 (cid:4)(cid:4) (cid:4)X t (cid:28) t (cid:28) j (cid:0) (cid:0) (cid:0) j (cid:0) (cid:28)=0 X if the commitment occurred from scratch in period t T, so (cid:4) = 0. Here M is partitioned t T 1;t T (cid:0) (cid:0) (cid:0) (cid:0) conformably with X and (cid:4) so t t 1 (cid:0) M M M XX X(cid:4) : (cid:17) M (cid:4)X M (cid:4)(cid:4) (cid:20) (cid:21) 29 The appendix is availabale at www.larseosvensson.net. 30 In the working paperversion ofthe paper,we also consideranothermethod which assumesthatpastpolicy has been systematic but not necessarily optimal. This method seems less restrictive but is not used in this paper. 36

Recall that that M will depend on the weight matrix W in the period loss function and hence vary with the assumed or estimated loss function. Given this, one possible initial value for (cid:4) = (cid:4) is t 1;t t 1;t 1 (cid:0) (cid:0) (cid:0) T (cid:4) = (M )(cid:28)M X : (B.8) t 1;t 1 (cid:4)(cid:4) (cid:4)X t 1 (cid:28) t 1 (cid:28) (cid:0) (cid:0) (cid:0) (cid:0) j (cid:0) (cid:0) (cid:28)=0 X This treats estimated realized X ((cid:28) = 0;1;:::;T) as resulting from optimal policy under comt (cid:28) (cid:0) mitment. The lag T may be chosen such that (M )TM is su¢ ciently small. (cid:4)(cid:4) (cid:4)X B.4. Projections with an arbitrary instrument rule With a constant (that is, time-invariant) arbitrary instrument rule, the instrument rate satis(cid:133)es X i = [f f ] t t X x x t (cid:20) (cid:21) for (cid:28) 0, where the n (n + n ) matrix [f f ] is a given (linear) instrument rule and i X x X x (cid:21) (cid:2) partitioned conformably with X and x .31 If f 0, the instrument rule is an explicit instrument t t x (cid:17) rule; if f = 0, the instrument rule is an implicit instrument rule. In the latter case, the instrument x 6 rule is actually an equilibrium condition, in the sense that in a real-time analogue the instrument rate in period t and the forward-looking variables in period t would be simultaneously determined. An arbitrary more general (linear) policy rule (G;f) can be written as G x +G i = f X +f x +f i ; (B.9) x t+1t i t+1t X t x t i t j j where the n (n +n ) matrix G [G G ] is partitioned conformably with x and i and the i x i x i t t (cid:2) (cid:17) n (n +n +n ) matrix f [f f f ] is partitioned conformably with X , x , and i . This i X x i X x i t t t (cid:2) (cid:17) general policy rules includes explicit, implicit, and forecast-based instrument rules (in the latter the instrument rate depends on expectations of future forward-looking variables, x ) as well as t+1t j targetingrules(conditionsoncurrentorexpectedfuturetargetvariables). Whenthisgeneralpolicy rule is an instrument rule, we require the n n matrix f to be nonsingular, so (B.9) determines x i i (cid:2) i for given X , x , x , and i . t t t t+1t t+1t j j The general policy rule can be added to the model equations (2.21) to form the new system to be solved. With the notation x~ (x ;i ), the new system can be written t 0t 0t 0 (cid:17) X X C t+1 = A~ t + " ; H~x~ x~ 0 t+1 (cid:20) t+1 j t (cid:21) (cid:20) t (cid:21) (cid:20) (nx+ni) (cid:2) n" (cid:21) 31 For projections with arbitrary time-varying policy rules, see LasØen, and Svensson [18]. 37

X X Y = D t ; Z = D(cid:22) t ; t x~ t x~ t t (cid:20) (cid:21) (cid:20) (cid:21) where A A B H 0 11 12 1 H~ ; A~ A A B ; (cid:17) (cid:20) G x G i (cid:21) (cid:17) 2 f X 21 f 2 x 2 f i 2 3 4 5 where H~ is partitioned conformably with x and i and A~ is partitioned conformably with X , x , t t t t and i . The corresponding projection model can then be written t X X t+(cid:28)+1;t = A~ t+(cid:28);t ; H~x~ x~ t+(cid:28)+1;t t+(cid:28);t (cid:20) (cid:21) (cid:20) (cid:21) X X Y = D t+(cid:28);t ; Z = D(cid:22) t+(cid:28);t t+(cid:28);t x~ t+(cid:28);t x~ t+(cid:28);t t+(cid:28);t (cid:20) (cid:21) (cid:20) (cid:21) for (cid:28) 0, where X = X . t;t tt (cid:21) j Then, (under the usual assumption that the policy rule gives rise to the standard saddlepoint property for the system(cid:146)s eigenvalues) there exist matrices M and F such that the resulting equilibrium projection satis(cid:133)es X = MX ; t+(cid:28)+1;t t+(cid:28);t x F x~ t+(cid:28);t = FX x X t+(cid:28);t (cid:17) i t+(cid:28);t t+(cid:28);t (cid:17) F i t+(cid:28);t (cid:20) (cid:21) (cid:20) (cid:21) for (cid:28) 0, where the matrices M and F depend on A~ and H~, and thereby on A, B, H, G, and f.32 (cid:21) C. Flexprice equilibrium and alternative concepts of potential output Under the assumption of (cid:135)exible prices and wages and an additional equation that determines nominal variables (in(cid:135)ation, the price level, the exchange rate, or some other nominal variable), the (cid:135)exprice model can be written X C X t t+1 = Af x + 0 " ; (C.1) (cid:20) Hfx t+1 j t (cid:21) 2 i t t 3 2 0 n n x i (cid:2) n n " " 3 t+1 (cid:2) 4 5 4 5 32 Equivalently, the resulting equilibrium projection satisfy X(cid:20) X(cid:20) x(cid:20) t t + + (cid:28) (cid:28) + + 1 1 ; ; t t =B(cid:22) x(cid:20) t t + + (cid:28) (cid:28) ; ; t t " (cid:20)i # " (cid:20)i # t+(cid:28)+1;t t+(cid:28);t for (cid:28) 0, where (cid:21) X(cid:20) = X ; t;t tt j x (cid:20) (cid:20) i t t ; ; t t = B B (cid:22) (cid:22) x i " X (cid:20)i x(cid:20) (cid:20) t t (cid:0) (cid:0) 1 1 ; ; t t (cid:0) (cid:0) 1 1 # + (cid:8) (cid:8) x iX X (X t j t(cid:0) X(cid:20) t;t (cid:0) 1 ): h i h i t (cid:0) 1;;t (cid:0) 1 h i 38

with the same variables X , x , and i and the same i.i.d. shocks " as in the sticky-price model but t t t t withthenew(n +n ) n matrixHf and(n +n +n ) (n +n +n )matrixAf. Therearehence x i x X x i X x i (cid:2) (cid:2) n extraequationsaddedtothelowerblockoftheequations(theblockofequationsdeterminingthe i forward-looking variables), as many equations as the number of policy instruments. The discussion here is restricted to the case n = 1 (as is Ramses), so only one equation needs to be added, such i as (cid:25)^ cpi = 0 or (cid:25)^d = 0: (C.2) t t In the latter case, A A B H 11 12 1 Hf ; Af A A B ; (cid:17) (cid:20) 0 1 (cid:2) nx (cid:21) (cid:17) 2 0 1 2 n 1 X e 2 1 2 0 2 3 (cid:2) 4 5 where e here denotes a row n -vector with the (cid:133)rst element equal to unity and the other elements 1 x equal to zero (re(cid:135)ecting that domestic in(cid:135)ation, (cid:25)^d, is the (cid:133)rst forward-looking variable). t The matrix Af has been modi(cid:133)ed so there is no e⁄ect on the endogenous variables of the four time-varying markups (cid:21)^ , j d;mc;mi;x , since these time-varying distortions would introduce j 2 f g undesirable variation in the di⁄erence between the (loglinearized) e¢ cient and (cid:135)exprice output. This is achieved by setting the corresponding elements in A and A equal to zero. With the 11 21 same argument one may also want to eliminate the e⁄ect on the endogenous variables of timevarying tax rates. f f f f Let x~ t (x t0;i t0) 0 and X t denote the realizations of the nonpredetermined and predetermined (cid:17) variables, respectively, in a (cid:135)exprice equilibrium for t t , where t is some period in the past with 0 0 (cid:21) given predetermined variables X from which we compute the (cid:135)exprice equilibrium. The (cid:135)exprice t0 equilibrium is the solution to the system of di⁄erence equations (C.1) for t t . It can be written 0 (cid:21) f f x F x~ f t = FfX f x X f ; t (cid:17) i f t (cid:17) F f t " t # " i # X f = MfX f +Cf" ; t+1 t t+1 for t t , where Ff and Mf are matrices returned by the Klein [21] algorithm and X f = X . In (cid:21) 0 t0 t0 f f f particular, one of the elements of x (and x~ ) is y^ , output in the (cid:135)exprice equilibrium. We can t t t therefore write (cid:135)exprice output in period t t as 0 (cid:21) y^ f = FfX f ; t y t (cid:1) where the row vector F f is the row of the matrix Ff that corresponds to output. y (cid:1) 39

C.1. Unconditional potential output Considernowahypothetical(cid:135)expriceequilibriumthathaslastedforever,theunconditional(cid:135)exprice f equilibrium. Thehypotheticalrealizationofthepredeterminedvariablesinperiodt,denotedX , t; (cid:0)1 then satis(cid:133)es X f = MfX f +Cf" = 1 (Mf)sCf" ; t; t 1; t t s (cid:0)1 (cid:0) (cid:0)1 (cid:0) s=0 X and hence depends on the realizations of the shocks " , " , ... The corresponding (cid:135)exprice output t t 1 (cid:0) is y^ f FfX f = Ff 1 (Mf)sCf" : (C.3) t; y t; y t s (cid:0)1 (cid:17) (cid:1) (cid:0)1 (cid:1) (cid:0) s=0 X We refer to this output level as unconditional potential output. It hence corresponds to the output level in a hypothetical economy that has always had (cid:135)exible prices and wages but is subject to the same shocks as the actual economy. It hence has a di⁄erent capital stock and di⁄erent realizations of the endogenous both predetermined and nonpredetermined variables compared to the actual economy. It corresponds to the natural rate of output consistent with the de(cid:133)nition of the natural rate of interest in Neiss and Nelson [22]. C.2. Conditional potential output Consider also the hypothetical situation in which prices and wages in the actual economy unexpectedly become (cid:135)exible in the current period t and are then expected to remain (cid:135)exible forever. f The corresponding (cid:135)exprice output in this economy is denoted y^ and given by t;t y^ f FfX : (C.4) t;t y t (cid:17) (cid:1) Werefertothisoutputlevelasconditional potential output (conditionalonpricesandwagesbecoming (cid:135)exible in the same period and therefore conditional on the existing predetermined variables, including the capital stock). It corresponds to the de(cid:133)nition of the natural rate of output presented in Woodford [39, section 5.3.4]. We then realize that we can de(cid:133)ne conditional-h potential output, where h 0;1;::: refers to 2 f g prices and wages unexpectedly becoming (cid:135)exible and expected to remain (cid:135)exible forever in period f t h, that is, h periods before t. This concept of potential output is denoted y^ and is given (cid:0) t;t ht (cid:0) j by h 1 y^ f FfX f Ff(MfX f +Cf" ) Ff[(Mf)hX + (cid:0) (Mf)sCf" ]: (C.5) t;t (cid:0) h (cid:17) y (cid:1) t;t (cid:0) h (cid:17) y (cid:1) t (cid:0) 1;t (cid:0) h t (cid:17) y (cid:1) t (cid:0) h t (cid:0) s s=0 X 40

Conditional-h potential output hence depends on the state of the economy (the predetermined variables) in period t h and the shocks from t h+1 to t. Conditional-0 potential output is (cid:0) (cid:0) obviously the same as conditional potential output. Unconditional potential output is the limit of conditional-h potential output when h goes to in(cid:133)nity, f f y^ = lim y^ : t; t;t h (cid:0)1 h (cid:0) !1 C.3. Projections of potential output Consider now projections in period t of these alternative concepts of potential output. The projecf tion in period t of unconditional potential output, y^ (where the (cid:133)rst subindex, t+(cid:28), f t+(cid:28); (cid:0)1 ;t g 1(cid:28)=0 refers to the future period for which potential output is projected; the second subindex, , in- (cid:0)1 dicates that unconditional potential output is considered; and the third subindex, t, refers to the period in which the projection is made and for which information is available), is related to the f projection of the unconditional (cid:135)exprice predetermined variables, X , and is given by f t+(cid:28); (cid:0)1 ;t g 1(cid:28)=0 y^ f FfX f Ff(Mf)(cid:28)X f (C.6) t+(cid:28); (cid:0)1 ;t (cid:17) y (cid:1) t+(cid:28); (cid:0)1 ;t (cid:17) y (cid:1) t; (cid:0)1j t f for (cid:28) 0, where X denotes the estimated realization of the unconditional (cid:135)exprice predeter- (cid:21) t; t (cid:0)1j mined variables in period t conditional on information available in period t. f Here, X can be estimated from t; t (cid:0)1j t 1 t0 X f = (Mf)t t0X + (cid:0) (cid:0) (Mf)sCf" ; (C.7) t; t (cid:0) t0t t st (cid:0)1j j (cid:0) j s=0 X wheretheunconditional(cid:135)expriceequilibriumisapproximatedbya(cid:135)expriceequilibriumthatstarts in a particular period t < t, X denotes the estimate conditional on information available in 0 t0t j period t of the predetermined variables in period t , and " denotes the estimate conditional in 0 t st (cid:0) j information available in period t of the realization of the shock in period t s. (cid:0) The projection in period t of conditional potential output (that is, conditional-0 potential outf put), y^ , is given by f t+(cid:28);t+(cid:28);t g 1(cid:28)=0 y^ f FfX (C.8) t+(cid:28);t+(cid:28);t y t+(cid:28);t (cid:17) (cid:1) for (cid:28) 0. Note that the projection of conditional potential output in period t+(cid:28) then refers to the (cid:21) (cid:135)exprice output for a (cid:135)exprice equilibrium that starts in the future period t+(cid:28), not in the current period period t. In the latter case, it would instead be the projection of conditional-(cid:28) potential output in period t+(cid:28). Therefore, the projection of the predetermined variables in period t+(cid:28) of 41

the actual economy, X , enters in (C.8), not the projection of the predetermined variables in t+(cid:28);t f period t+(cid:28) of the (cid:135)exprice equilibrium starting in period t, X . t+(cid:28);t;t f The projection in period t of conditional-h potential output, y^ , is related to the f t+(cid:28);t+(cid:28) h;tg 1(cid:28)=0 (cid:0) f projection of the conditional-h (cid:135)exprice predetermined variables, X , and is given f t+(cid:28);t+(cid:28) h;tg 1(cid:28)=0 (cid:0) by y^ f FfX f (C.9) t+(cid:28);t+(cid:28) (cid:0) h;t (cid:17) y (cid:1) t+(cid:28);t+(cid:28) (cid:0) h;t for(cid:28) 0. Thatis, theprojectionoftheconditional-hpredeterminedvariablesinperiodt+(cid:28) enters (cid:21) in (C.9), the predetermined variables in the (cid:135)exprice equilibrium that starts h periods earlier, in period t+(cid:28) h. Furthermore, this projection is given by (cid:0) h 1 X f (Mf)hX + (cid:0) (Mf)sCf" ; t+(cid:28);t+(cid:28) (cid:0) h;t (cid:17) t+(cid:28) (cid:0) h;t t+(cid:28) (cid:0) s j t s=(cid:28) X where the summation term is zero when (cid:28) h. Thus, the projection in period t of the realization (cid:21) f of conditional-h potential output in period t + (cid:28), y^ , depends on the projection of the t+(cid:28);t+(cid:28) h;t (cid:0) predetermined variables in period t+(cid:28) h of the actual economy (with sticky prices and wages), (cid:0) X , and when (cid:28) < h also on the estimated shocks " for s = 0, 1, ..., h (cid:28) 1. (For (cid:28) > h, t+(cid:28) h;t t st (cid:0) (cid:0) j (cid:0) (cid:0) " = 0). t+(cid:28) ht (cid:0) j C.4. Output gaps We can then consider several concepts of output gaps: We have the trend output gap, the gap between actual output and trend output, y^ (recall that y^ is the deviation from trend). For each t t concept of potential output, we have a corresponding concept of output gap: The unconditional output gap, y^ y f y^ FfX f ; t t; t y t; (cid:0) (cid:0)1 (cid:17) (cid:0) (cid:1) (cid:0)1 the conditional output gap, y^ y f y^ FfX ; t t;t t y t (cid:0) (cid:17) (cid:0) (cid:1) and the conditional-h output gap, y^ y f y^ FfX f : t (cid:0) t;t (cid:0) h (cid:17) t (cid:0) y (cid:1) t;t (cid:0) h The projections of the di⁄erent output gaps are then de(cid:133)ned in analogy with the projections of the di⁄erent potential outputs. 42

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