Monetary Policy Rules for an Open Economy

Batini, Nicoletta; Harrison, Richard; Millard, Stephen P. Working Paper Monetary Policy Rules for an Open Economy Working Paper, No. 2001/4 Provided in Cooperation with: Norges Bank, Oslo Suggested Citation: Batini, Nicoletta; Harrison, Richard; Millard, Stephen P. (2001) : Monetary Policy Rules for an Open Economy, Working Paper, No. 2001/4, ISBN 82-7553-180-2, Norges Bank, Oslo, https://hdl.handle.net/11250/2498713 This Version is available at: https://hdl.handle.net/10419/209793 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. 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Millard Presented at the workshop ‘’The conduct of monetary policy in open economies’’ on 26–27 October 2000 ISSN 0801-2504 ISBN 82-7553-180-2 Working papers from Norges Bank can be ordered via the Internet: www.norges-bank.no/english/publications or from Norges Bank, Subscription service, P.O.Box. 1179 Sentrum, 0107 Oslo, Norway. Tel. +47 22 31 63 83, Fax. +47 22 41 31 05 Norges Bank's Working papers present research projects and reports (not usually in their final form), and are intended inter alia to enable the author to benefit from the comments of colleagues and other interested parties. Views and conclusions expressed in Working Papers are the responsibility of the authors alone. Monetary Policy Rules for an Open Economy Nicoletta Batini* Richard Harrison** and Stephen P. Millard< First draft: December 1999 This draft: November 2000 Abstract The most popular simple rules for the interest rate, due to Taylor (1993a) and Henderson and McKibbin (1993), are both meant to inform monetary policy in economies that are closed. On the other hand, their main open economy alternative, i.e. Ball’s (1999) rule based on a Monetary Conditions Index (MCI), may perform poorly in the face of specific types of exchange rate shocks and thus cannot offer guidance for the day-to-day conduct of monetary policy. In this paper we specify and evaluate a comprehensive set of simple monetary policy rules that are suitable for small open economies in general, and for the UK in particular. We do so by examining the performance of a battery of simple rules, including the familiar Taylor and Henderson and McKibbin rules and MCI-based rules à la Ball. This entails comparing the asymptotic properties of a two-sector open-economy dynamic stochastic general equilibrium model calibrated on UK data under different rules. We find that an inflation forecast based rule (‘IFB’), i.e. a rule that reacts to deviations of expected inflation from target is a good simple rule in this respect, when the horizon is adequately chosen. Adding a separate response to the level of the real exchange rate (contemporaneous and lagged) appears to reduce the difference in adjustment between output gaps in the two sectors of the economy, but this improvement is only marginal. Importantly, an IFB rule, with or without exchange rate adjustment, appears robust to different shocks, contrary to naïve or Ball’s MCIbased rules. * Research Adviser, MPC Unit, Bank of England, Threadneedle Street, London EC2R 8AH, United Kingdom. Tel: +44 20 76014354. Fax: +44 20 76013550 E-mail: [email protected] (corresponding author) ** Analyst, Monetary Assessment and Strategy Division, Bank of England, Threadneedle Street, London EC2R 8AH, United Kingdom. Tel: +44 20 76015662. Fax: +44 20 76014177. E-mail: [email protected] < Manager, Monetary Instrument and Markets Division, Bank of England, Threadneedle Street, London EC2R 8AH, United Kingdom. Tel: +44 20 76014115. Fax: +44 20 76015953. E-mail: [email protected] We would like to thank Nicola Anderson, Larry Ball, Spencer Dale, Shamik Dhar, Rebecca Driver, Chris Erceg, Neil Ericsson, Jeff Fuhrer, Andy Levin, Bennett McCallum, Katherine Neiss, Edward Nelson, Athanasios Orphanides, Glenn Rudebusch, Chris Salmon, Anthony Yates and seminar participants at the Bank of England for useful comments on earlier versions of this paper. Remaining errors, and the views expressed herein are those of the authors and not of the Bank of England nor of the Bank of England’s 2 Monetary Policy Committee. The work has still not been finalised and so results contained herein should only be quoted with the permission of the authors. 1. Introduction The literature on simple rules for monetary policy is vast.1 It contains theoretical research comparing rules that respond to alternative intermediate and final targets, backwardand forward-looking rules, and finally, rules which include or exclude interest rate smoothing terms. It also contains work on historical estimates of monetary policy rules for various countries. However, the literature does not contain a thorough analysis of simple rules for open economies, i.e. for economies where the exchange rate channel of monetary policy plays an important role in the transmission mechanism. The most popular simple rules for the interest rate  due to Taylor (1993a) and Henderson and McKibbin (1993)  for example, were both designed for the United States and, thus, on the assumption that the economy is closed. And the main open economy alternatives, (for example, the rule by Ball (1999) based on a Monetary Conditions Index (MCI)), may perform poorly in the face of specific types of exchange rate shocks and thus cannot offer guidance for the day-to-day conduct of monetary policy. So at present we only have a choice of ignoring the exchange rate channel of monetary transmission completely (Taylor, Henderson and McKibbin) or including it in an ad hoc way that may not always prove right (MCI-based rules). In this paper we specify and evaluate a family of simple monetary policy rules that may stabilize inflation and output in small open economies at a lower social cost than existing rules. These rules parsimoniously modify alternative closedor open-economy rules to analyse different ways of explicitly accounting for the exchange rate channel of monetary transmission. We compare the performance of these rules to that of a battery of alternative rules when the model economy is buffeted by various shocks. The alternatives we consider include the Taylor and Henderson and McKibbin closed-economy rules, naïve MCI-based rules as well as Ball’s MCI-based rule, and inflation forecast-based rules. Some of the rules in the family we consider appear to be robust across a set of different shocks, including shocks to the domestic economy emanating from the rest of the world. This is in contrast to closedeconomy rival simple rules, which ignore the exchange rate channel of monetary transmission, and naïve or Ball’s MCI-based rules, the performance of which can be highly shock-specific. To test the rules, we stylise the economy  that we calibrate to UK data  as a two-sector open-economy dynamic stochastic general equilibrium model. The export/non-traded sector split is important because it allows us to discern different impacts of the same shock on output and inflation in the two sectors. Identification of sectoral inflation and output dynamics is a key element on which to base the design of efficient policy rules. More generally, it also makes it possible for the monetary authority to consider the costs of price stabilization on each sector of the economy. Because it is theoretically derived on the assumption that consumers maximise utility and firms maximise profits, the model has a rich structural specification. This enables us to contemplate shocks that could not be analysed in less structural or reduced form small macro-models. 1 See Bryant et al (1993) and Taylor (ed.) (1999). 3 In particular, with our model, we can examine the implications of shocks to aggregate demand such as a shock to households’ preferences, or a shock to the rest of the world’s income. On the supply side, we can consider shocks overseas inflation. We can analyse the impact of a relative productivity shock on the two sectors and investigate how this affects the real exchange rate by altering the price of the non-tradables relative to export goods. We can also look at the effects of a change in the price of imported intermediate goods. We can examine the effects of shocks to the foreign exchange risk premium. Finally, we can look at the implications of a monetary policy shock, both home and abroad. The ability to examine all these different shocks is important when comparing alternative policy rules for an open economy, because the efficient policy response to changes in the exchange rate will typically depend on what shock has hit the economy  with different shocks sometimes requiring opposite responses. For this purpose our small economy general equilibrium model is sufficient. A two-country model would enable us to look at these same shocks, but we believe the small-economy assumption is more realistic for the UK. In short, this model is well suited to our analysis for three reasons. First it is a structural, theoretically based model. The structural nature of the model is important because it implies that our policy analysis (i.e. comparison of different rules/regimes) is less subject to the Lucas critique than a more reduced-form model. Second, it offers a more disaggregated picture of the economy than many existing models. This allows us to identify the different dynamics of output and inflation after a shock  a valuable input to the efficient design of rules. Third, because it is structural and built from micro-principles, it allows us to consider shocks (such as preference or relative productivity shocks) which are key for the design of a rule meant to be a ‘horse for all courses’ in an open economy setting. The rest of the paper is organised as follows. In section 2 we lay out the model that we employ throughout and describe its steady state properties. The solution and calibration of the model are discussed in section 3. In section 4 we study some properties of the model. In section 5 we specify a family of open-economy simple rules and present results comparing the stabilisation properties of these rules against those of a battery of alternative simple rules, in the face of various disturbances. Finally, section 6 concludes. The Technical Appendix contains further details about the model’s non-linear and log-linear specifications. 2. A two-sector open-economy optimising model The model we use is a calibrated stochastic dynamic general equilibrium model of the UK economy with a sectoral split between exported and non-traded goods. Its specification draws on the literature on open-economy optimising models by Svensson and van Wijnbergen (1989), Correia, Neves and Rebelo (1994), Obstfeld and Rogoff (1996), and more recent work by McCallum and Nelson (1999). In this sense, the model is close in spirit to a number of open-economy models developed at or after the time of writing by Monacelli (1999), Gali and Monacelli (1999), Ghironi (2000), Smets and Wouters (2000), Benigno and Benigno (2000) and Devereux and Engle (2000). However, it extends upon all of these, individually (and other closed-economy optimising models), by introducing several novel features that are described in detail below. 4 The model describes an economy that is ‘small’ with respect to the rest of the world. In practice, this means that the supply of domestically produced traded goods does not affect the price of these goods internationally. It also means that the price of imported foreign goods, foreign interest rates and foreign income are exogenous in this model, rather than being endogenously determined in the international capital and goods markets, as would happen in a multiple-country, global-economy model. This assumption considerably simplifies our analysis; and because we are not interested here in studying either the transmission of economic shocks across countries or issues of policy interdependence, it comes at a relatively small price. As we are interested in evaluating alternative monetary policy rules, we specify monetary policy within the model as a rule for the nominal interest rate (the policy instrument). We look at alternative rules in order to see whether responding to some ‘open-economy’ variables such as the exchange rate or the balance of trade can improve the stabilisation properties of rules designed for a closed economy context. 2.1 Household preferences and government policy The economy is populated by a continuum of households of unit mass. Each household is infinitely lived and has identical preferences defined over consumption of a basket of (final) imported and non-traded goods, leisure and real money balances at every date. Households differ in one respect: they supply differentiated labour services to firms. Preferences are additively log-separable and imply that household j∈(0,1) maximises: ∑ ∞ = − −               Ω − +−+− 0 1 10 )( 1 ))(1ln())()(ln()exp( tt t ttctt t P j jhjcjcE ε ε χ δξνβ(1) where 0 < β < 1; δ, χ and ε are restricted to be positive and E0 denotes the expectation based on the information set available at time zero. In equation (1), t c(j) is total time t real consumption of household j, t ν is a white noise shock to preferences  essentially a demand shock, described in more detail in sections 3 and 4  and t h(j) is labour supplied to market activities, expressed as a fraction of the total time available. So the term ))(1(jht − captures the utility of time spent outside work. The last term tt Pj/)(Ω represents the flow of transaction-facilitating services yielded by real money balances during time t (more on this later). Hence here, as in the standard Sidrauski-Brock model, money enters the model by featuring directly in the utility function. In addition, since c ξ∈ [0,1), preferences over consumption exhibit habit formation, with the functional form used in (1) similar to that of Carrol et al. (1995) and Fuhrer (2000). This implies that preferences are not time-separable in consumption, so that households’ utility depends not only on the level of consumption in each period, but also on their level in the previous period. Total consumption is obtained by aggregating the consumption of imported and non-traded goods , Mt c and tN c, via the geometric combination 1 ,, tMtNt ccc γγ − =, where γ ∈ (0,1). Here , Mt c and tN c, represent imported and non-traded goods purchased by the consumer from 5 retailers at prices tM P, and tN P,, respectively. It is easily shown that the consumption-based price deflator is given by γγ γγ γγ − − − =1 1,, )1( tNtM t PP P.2 Households have access to a state contingent bond market. Bond b(s) in this market is priced in units of consumption, has price r(s) in period t, and pays one unit of consumption in state s in period t+1. In practice, this means that households within the domestic economy can insure themselves perfectly against idiosyncratic shocks. In equilibrium, consumption and real money balances are equal across households. So households differ only because labour supply varies across the population. In addition to this bond market, each household can also access a domestic and a foreign nominal government bond market at interest rates i and f i, respectively. For the time being, we assume that both kinds of bond are riskless, but we investigate alternative assumptions later (see sub-section 2.4). Money is introduced into the economy by the government. Under Ricardian equivalence, we can assume without loss of generality a zero net supply of domestic bonds. Then the public sector budget constraint requires that all the revenue associated with money creation must be returned to the private sector in the form of net lumpsum transfers in each period: tttt TMM τ−=− −1(2) where t M is end-of-period t nominal money balances, t T is a nominal lump-sum transfer received from the home government at the start of period t and τt is a lump sum tax levied on consumers. For simplicity we assume the tax is constant at its steady state level. The household’s dynamic budget constraint in each period is given by equations (3) and (4) below. Equation (3) describes the evolution of nominal wealth. Equation (4) defines the nominal balances available to consumers to spend at time t. This reflects the assumption that consumers participate in the financial markets before spending money on goods and services. As suggested by Carlstrom and Fuerst (1999), entering money balances as defined in (4) in the utility function, gives a better measure of period utility; one in which we account exclusively for the services of balances that are actually available to households when spending decisions are taken. 2 Formally, Pt defines the minimum cost of financing a unit of consumption, ct. See Obstfeld and Rogoff (1996, pp) for a simple example. 6 )()()(),( )( )1()()1()(),()( )( )( )()( 1 1, 1,111 , jcPTDjhjWdsjsbP e jB ijBijMdsjsbsrP je jB jBjM tttttttt t tf tftttttt t tf tt −+++ +++++=+++ ∫ ∫ − − −−−− (3) t tf t t tf tfttttt e jB jB e jB ijBiTjMj)( )( )( )1()()1()()( ,1, 1,111 −−+++++=Ω− −−−− (4) where 1−t M is nominal money balances at time t −1, )( 1jBt− and )( 1,jBtf − are time t −1 holdings of domestic and foreign bonds, respectively and t D are lump sum dividends from shares held in (domestic) firms. Household j’s holdings of (state contingent) bond bt(s) are bt(s,j). With t e we denote the nominal exchange rate, expressing domestic currency in terms of units of foreign currency.3 Finally, )( jWt is the nominal wage rate received by household j. Because each household supplies differentiated labour services, it has some market power over the wage rate. So we assume that household j chooses )( jct, )( 1jBt−, )( 1,jBtf −, )( j t Ω, )( jMt and b(s,j) to maximise (1) subject to (3) and (4). The choice of wage W(j) is discussed in section 2.3.2. 2.2 Technology and market structure This sub-section describes the supply side of the economy by sector. We assume that in our economy there are two kinds of producing firms: non-traded goods producers and export producers. By definition, non-traded goods are only consumed domestically, while we assume that exports produced at home are consumed only abroad. To produce, the exports and non-traded goods producers buy intermediate non-labour inputs for production (labour is purchased domestically from the households) from a group of ‘imported intermediate input retailers’. Since consumers also purchase their final imports and non-traded goods via ‘retailers’, the economy has a total of three groups of retailing firms: imported intermediates retailers, non-traded good retailers and final imports retailers. Finally, both final imports retailers and imported intermediates retailers originally purchase their ‘input’ from a group of ‘importers’, who in turn, acquire goods from the world markets. There are two types of importers, one for each import. We refer to the first group as ‘final goods importers’ and to the second group as ‘intermediate inputs importers’. Chart 1 depicts the goods market structure of the model. 3 So that an increase in et represents an appreciation of the domestic currency. 13 domestic economy can run a trade deficit in every period financed via the interest payments that it receives on the foreign assets held. In addition, since the economy is small, the foreign interest rate is exogenous in the model. So the supply of foreign government bonds is perfectly elastic at the exogenous world nominal interest rate. This means that steady state foreign bond holdings are indeterminate in our model. As a result, temporary nominal shocks can shift the real steady state of the model through the effects on nominal wealth (see Obstfeld and Rogoff (1996)). This means that the steady state around which log-linear approximations are taken is moving over time. This is a common feature of small open economy monetary models and can be avoided in a number of ways. One approach is to make assumptions about the form of the utility function (see, for example, Correia et al, 1995) or the way in which consumption is aggregated. This is difficult to implement in our model if we wish to retain a rich structural specification. Another approach is to impose a global equilibrium condition on asset holdings (and restrict the trade balance to be zero in all periods). But this seems too restrictive. So instead, we substitute foreign bond holdings out of the model and concentrate on the movements of the other variables, as in McCallum and Nelson (1999). 2.5 The Transmission Mechanism In an open economy, the exchange rate is an important channel of monetary transmission. This channel has a number of effects. First, and most obviously, the demand for exports is directly affected by exchange rate movements. Exporters also feel the effect of exchange rate changes through the price of imported intermediate goods. Importers of intermediate goods face an increase in their nominal unit costs as the nominal exchange rate depreciates. This is passed onto producers (including producers of non-traded goods) gradually, reflecting the fact that importers are required to set prices one period in advance and only a fraction of them are able to change price in any particular quarter. Exchange rate changes also affect the consumer price index through the direct impact on the prices of imported consumption goods. Again this occurs with a lag because of the assumptions reflecting importers’ pricing decisions. And the exchange rate affects consumer prices as non-traded goods producers pass on changes in production costs gradually (reflecting the Calvo pricing assumption). It is clear from this discussion that the exchange rate affects different sectors unevenly. In summary, there are two channels of monetary transmission in this model. There is a standard interest rate channel, that influences the consumption-saving decision and hence the output gap and inflation. In addition, there is an exchange rate channel that directly affects export sector prices; and indirectly affects exports and non-traded goods’ prices through changes in the cost of the intermediate imported inputs. 14 3. Model Solution and Calibration 3.1 Solving the Model To solve the model we first derive the relevant first order conditions discussed in section 2. We then solve for the non-stochastic flexible price steady state and take the log-linear approximation of each non-linear first-order condition around this steady state. This procedure is presented in the Technical Appendix. As shown in the Technical Appendix, the model can be cast in first order form: tttt Exzz 1CBA+= +(22) ttt ?+= +xx 1P(23) where A and B are 31× 31 matrices, while C is a 31 × 8 matrix. Ρ is an 8 × 8 matrix containing the first order cross-correlation coefficients of the exogenous variables, whose white noise i.i.d. innovations are expressed by the vector t ?. Let t f and t k denote the endogenous and pre-determined parts of the vector t z respectively. Then the rational expectations solution to (11)-(12), expressing the vector of endogenous variables t f as functions of predetermined ( t k) and exogenous ( t x) variables, can be written as: ttt xkf21 Ξ+Ξ=(24)       +       =       + + tt t t t ? 0 x k ? x k 1 1(25) In this paper we computed this solution using Klein’s (1997) algorithm. 3.2 Calibration We calibrate the model to match key features of UK macroeconomic data. For this purpose, we set the discount factor, β, to imply a steady-state annual real interest rate of 3.5%. This is equal to the average ten-year real forward rate derived from the index-linked gilt market in the United Kingdom since these were first issued in March 1983. The steady state inflation rate was set at 2.5% per year: the current UK inflation target. We assume that steady state foreign inflation was equal to steady state domestic inflation; that is, 2.5% per year. An implication is that the nominal exchange rate is stationary. We normalise the steady state prices of traded goods and intermediate goods (in foreign currency) to unity. To set the parameter in the utility function reflecting preferences for imports vis-à-vis nontraded goods, γ, we use data on consumption spending on traded versus non-traded goods. To do so, we equate consumption of non-traded goods with output of non-traded goods and set consumption of imports equal to output of traded goods less exports of traded goods. We 15 set γ equal to 0.103, so that the implied constant share of consumption spending on traded versus non-traded goods matched the average value seen in the available data.8 We set the habit formation parameter such that the persistence of the output response to shocks in the model is similar to that in the UK data. The value chosen is 7.0= c ξ. The weight on leisure vis-à-vis consumption in the utility function, δ, is set to ensure that steady-state hours were equal to 0.3 in the absence of ‘distortions’.9 The required value is 1.815. Though essentially a normalisation, this choice corresponds to an 18 hour day available to be split between work and leisure time and workers, on average, working fifty 40-hour weeks in a year. We set 165.0= W θ as this is consistent with steady state hours of 0.273 when habit formation and monopolistic supply of labour are accounted for. This level of hours represents a deviation from ‘distortion-free’ steady hours equal to 9% - the average level of UK unemployment using the LFS measure. We set 75.0= W φ as this implies that wage contracts are expected to last for one year. We set the weight on money in the utility function to χ = 0.005. This implies that the ratio of real money balances to GDP is around 30% in steady state. Though this is somewhat higher than the ratio of M0 to nominal GDP, it is not clear that ‘money’ in our model is best proxied by M0 in the data. The ratio of M4 to quarterly nominal GDP is larger – the average for 1963 Q1-2000 Q1 is around 1.4. So our calibration fixes the ratio of steady state real money balances to GDP at an intermediate level. We set ε=1 which implies a unit elasticity of money demand. This is consistent with findings for the UK (see QMA 1999). To calibrate parameters on the production side of the model requires sectoral data. A description of the assumptions needed to do this is given in the Appendix. We first calibrate the mark-ups that firms in each sector apply to unit marginal costs, using the results of Small (1997). Weighting these mark-ups with the respective shares in value added output,10 we obtain a value for the non-traded sector gross mark-up of 1.17. Gross mark-ups for the traded and intermediates goods sectors are found to be 1.183 and 1.270. These calibrations imply values for φ N, φ T and φ I of 0.17, 0.183 and 0.270, respectively. Computing elasticities of non-traded and traded goods output with respect to employment gives estimates of N αand X α, of 0.763 and 0.636, respectively. To calibrate the probabilities that firms in a particular sector receive signals allowing them to change price, we use data on the average number of price changes each year for different industries. Hall, Walsh and Yates (1997) find that the median manufacturing firm changes price twice a year, the median construction firm 3 or 4 times a year, the median retail firm 3 or 4 times a year and the median ‘Other Services’ firm once a year. On this basis, we assume an average duration of prices of six months for firms in the import goods and intermediate goods sectors and an average duration of four months for firms in the non-traded goods sector. This implies values for φM, φI and φN of 0.33, 0.33 and 0.43, respectively. 8 The only reliable data we could obtain on output in current prices by industry is annual and covers only the period 1989 to 1998. 9 This involved setting the habit formation parameter (ξ) to zero and assuming that the elasticity of substitution between labour types tended to infinity (θW=0). 10 Using weights from the 1985 ONS Blue Book. 16 The export demand function requires us to set the income and price elasticities. We set the income elasticity to unity and the price elasticity (η) to 0.2. The latter assumption approximates the one-quarter response of the UK export equation in the Bank of England’s Medium Term Macroeconomic Model (see Bank of England (1999, pp50-51)). To derive series for ‘total factor productivity’ in each sector, we use quarterly data on gross value added by industry at constant 1995 prices from 1983 onwards (ETAS Table 1.9) and ‘workforce jobs’ by industry for the same period.11 We calculate our productivity series as: tZZtZtZhyA,,, lnlnln α−= (26) where Z indexes the sector, y is value added and h is workforce jobs. An implicit assumption is that movements in intermediate inputs are ‘small’ relative to movements in output and employment. This is required to equate this measure of A with ‘total factor productivity’. After HP-filtering the two productivity series obtained from (15) we estimate the stochastic processes for the productivity terms using a vector autoregressive (VAR) system:         +         =         − − tN tT N t T t A N t T t A A R A A , , 1 1 ˆ ˆ ˆ ˆ ε ε(27) The disturbances tT, ε and tN, ε are normally distributed with variance-covariance matrix VD. Given that the model has zero productivity growth in steady state, Z A ˆ refers to ‘logdeviations of productivity in sector Z from a Hodrick-Prescott trend’. Our estimation results imply:         ×=         − =− 044.743.1 43.119.3 10 and 784.0066.0 227.0705.0 5 DA RV(28) To calibrate the forcing processes associated with overseas shocks we estimate another VAR. We derive processes for the shocks to the one-quarter change in the world price of traded goods and the world price of imported materials, as well as to foreign interest rates, the exchange rate risk premium and world demand. We construct a series for the foreign interest rate as a weighted average of three-month Euromarket rates for each of the other G6 countries, using the same weights used to construct the UK Effective Exchange Rate Index. For intermediate goods imports we follow Britton, Larsen and Small (1999) and construct an index based on the imported components of the Producer Price Index. For the world price of traded goods we use the G7 (excluding the United Kingdom) weighted average of exports of goods and services deflators where the weights match those in the UK Effective Exchange Rate index. For world output, we use the G7 (excluding the United Kingdom) average GDP weighted by the countries’ share in total UK exports of goods and services in 1996. We estimate the following VAR: 11We adjusted the workforce jobs series prior to 1995Q3 to take account of a level shift of about 350,000 in total workforce jobs when the series was rebased. To do this, we added to the figure for each industry a share of the 350,000 workers equal to the industry’s share in the published total. We combined the output data using the 1995 weights to get real value added for each of our two sectors (where, again, the traded goods sector consisted of ‘manufacturing’ and ‘transport and communications’). 17                   +                 ∆−∆ − − =                 ∆−∆ − − − − −− − ty P P ti tF t IttI ftf F tF t IttI ftf F t tI f y PP PPPP ii R y PP PPPP ii , , 1, **1 ***1 *1, 1, , ** **** , , * , ˆ loglog )/log()/log( ˆ loglog )/log()/log( ε ε ε ε (29) where variables without time subscripts refer to their averages in the data and tF y, ˆ is the logdeviation of world demand from its Hodrick-Prescott trend. The disturbances ti, ε, tP I, ε, tP, * ε and tyF, ε are normally distributed with variance-covariance matrix VF. The VAR is specified in this way because the rest of the world is modeled in a reduced form way that does not place restrictions on the long run behaviour of variables. In particular if we included inflation of foreign intermediates prices as a separate variable then there would be no reason to expect the long-run responses of foreign intermediates prices and the general foreign price level to be equal. If this restriction did not hold, then temporary shocks could shift the steady state relationships between (exogenous) world variables. This would destabilise the relationships between the endogenous variables in our model. Rather than place restrictions on a VAR including foreign inflation rates, we estimate the system in (29). Using data over the period 1977 Q3 − 1999 Q2 we obtained the following results:               − −−− − − = 962.0079.0003.0357.0 019.0711.0019.0359.0 07.1290.0902.0392.2 140.0083.0006.0448.0 F R               − ×= − 79.7 49.06.27 3.229.31760 54.008.347.482.3 10 6 F V. 18 We derived a measure of the sterling exchange rate risk premium derived from the Consensus Survey12 and estimated the following process: 009.0 ,261.0,1=+= −ζζ σεζζ ttt (30) Finally, in line with McCallum and Nelson (op. cit.) we assumed that the preference shock t ν is white noise, and, for simplicity, we set its standard deviation equal to 0.011 as they do for the US. 4. Properties of the model To analyse the dynamic properties of the model, we have derived impulse response functions for the key endogenous variables when the model is hit by shocks. Throughout, we closed the model with a policy rule for the nominal interest rate t i. The rule used here was estimated using UK data over the period 1981Q2-1998Q2. We estimated a reduced-form model in which there were also equations determining (log) aggregate output t y ˆ, (the log of) the annualised log-change in the RPIX index inflation measured in terms of deviations from target ( t P ˆ 4∆) and changes in the (log of the) nominal trade-weighted effective exchange rate ( t eln∆). The model which is similar to that in Batini and Nelson (2000a), also contains two dummies ( t DERM and 92 t D) to capture the years of the UK membership of the ERM and the shift in policy regime which occurred in 1992 Q4. To compute the impulse responses we need to identify the shocks. When the nominal interest rate is estimated as part of a VAR, a standard way of doing so is to orthogonalise the shocks using a Cholesky decomposition with a causal ordering that places the nominal interest rate last. Typically, however, estimating the equation for the nominal interest rate using a conventional VAR gives a reaction function where the nominal rate responds to lags of itself and lags of other variables in the VAR. This is unsatisfactory if we want to compare the estimated rule with Taylor-type rules that react to contemporaneous variables. To overcome this problem, Rotemberg and Woodford (1997) obtain a similar dynamic specification of the estimated policy rule by leading the other variables in the vector autoregression model (inflation and output in their case): in effect they estimate a VAR with a vector of endogenous variables equal to [ t i4 , 1 ˆ 4+ ∆t P, 1 ˆ+t y]. Even if it gives an estimated equation for the interest rate that responds to contemporaneous realisations of output and inflation, as we want, their approach may be unreliable. It, in fact, implies very restricted dynamic specifications for the other two variables in the model, where the leads of inflation and output depend only on lags of the interest rate and not also on the level of the interest rate at time t. 12 The measure is equal to the percentage point difference between the expected 24-month depreciation of the sterling ERI (derived from the responses of survey participants) and the two-year nominal interest rate differential. 19 For this reason, following the methodology in Ericsson, Hendry and Mizon (1998), we reparameterised the system Qt = [ t i,t P ˆ 4∆, t y ˆ, t eln∆] as the conditional and marginal models t i = f ( t P ˆ 4∆,t y ˆ,t eln∆,1−t Q) and ( t P ˆ 4∆,t eln∆,t y ˆ) = ( 1−t Q, χ), where χ is the vector of estimated parameters. In effect, this orthogonalises the shocks, so that the nominal interest rate is not affected by time-t changes in the other variables. However, contrary to a VAR estimation approach, this method allows us to derive an estimated equation for the nominal interest rate in which t i depends on contemporaneous values of inflation, output and changes in the exchange rate, rather than on lags of those variables. The model’s estimates are available on request. For convenience, we reproduce here the estimate of the nominal interest rate equation, which we interpret as being the monetary policy reaction function over that period: tittttttt DDERMeyPici,6543211 924ln ˆ ˆ 444 εκκκκκκ +++∆++∆++= −(31) where t i4 is the annualised interbank lending rate, and ti, ε are the equation’s estimated residuals. The estimated coefficients (standard errors in parenthesis) are: c = 0.0423, κ1 = 0.605, κ2 = 0.406, κ3 = 0.184, κ4 = − 0.065 , (0.008) (0.074) (----) (0.039) (0.027) κ5 = − 0.014, κ6 = − 0.015, (0.003) (0.004) with SE = 0.00821. To ensure that the log-run nominal interest rate response to inflation is larger than 1, we restrict )1/( 12 κκ −=1.01. For this reason, no standard error is reported for that coefficient. The LR test of over-identifying restrictions cannot reject the null implied by this restriction [χ2(1)=0.5032, p -value = 0.4781]. Since the endogenous variables in the model feature as deviations from their respective longrun values  or enter as first-differences  they are comparable to variables in the loglinearised first-order approximation version of the model. 20 Figure 1: Impulse responses following a 100 basis point monetary policy shock -0.4 -0.3 -0.2 -0.1 0 0.1 1 3 5 7 9 11 13 15 17 19 21 Panel 1: Output Response Quarter -0.4 -0.2 0 0.2 0.4 0.6 1 3 5 7 9 11 13 15 17 19 21 Panel 2: Inflation Response Quarter -0.5 0 0.5 1 1.5 1 3 5 7 9 11 13 15 17 19 21 Panel 3: Nominal Interest Rate Response Quarter Figure 1 shows output, (four-quarter) inflation and the nominal interest rate impulse response functions to a unit start shock to the monetary policy rule (31) over 20 periods (calendar quarters). The solid line depicts the analytical model’s responses and the dashed line gives the estimated model’s responses. Both the estimated and our model’s responses broadly agree with conventional wisdom: following a temporary rise in the interest rate, output declines, but ultimately reverts to base; and inflation also falls. Our estimated inflation equation exhibits no price puzzle (i.e. the finding in many empirically estimated models a rise in the nominal interest rate is associated with a rise  rather than a fall  in the rate of inflation in the periods immediately after the rise). However, we expect there to be rather wide error bands around the estimated model’s impulse responses (not shown here) indicating that these effects cannot be estimated with 21 great precision, particularly those on inflation. So, the comparison of the two sets of responses should not be taken too literally. Panel 1 indicates that, in our analytical model, output falls on impact by around 0.25%, following an unanticipated 100 basis point rise in the nominal interest rate  the same order of magnitude of that of the estimated model. The policy shock response in the data is slightly more sluggish than that in the model and in the data, the trough in output following the shock occurs later than in our model. The speedier response of output in our model reflects the volatility of the net trade component of aggregate output in our model. The consumption component of aggregate output is sluggish and ‘hump shaped’, which reflects the high value of the habit formation parameter (ξ). This result accords with the findings of Fuhrer (2000). Panel 2 compares the RPIX inflation responses of the theoretical and estimated models. In our model inflation responds earlier and more intensely than the estimated model. There, inflation touches its nadir around ten quarters after the shock, and returns smoothly back on track over a period of about two to three years. The difference between the two responses probably reflects the fact that our model, even accounting for the built-in persistence, is still a forward-looking, ‘jumpier’ model, whereas the estimated model is entirely backward looking. Panel 3 depicts how the (nominal) interest rate responds. While it rises by a full 1% in the estimated model, the nominal interest rate by slightly less in our model. There are two reasons why this happens. First, in our model, inflation expected at time t + 1 falls on impact one period after the shock; by contrast, in the estimated model, inflation is almost unchanged in the first quarters after the shock. This implies that, in practice, the real interest rate response is harsher on impact in our model than in the estimated model. Second, inflation and output (the feedback variables in the estimated policy rule) are forward-looking in our model; thus the interest rate response will be more muted than in the estimated model, inasmuch as those variables will themselves have already adjusted pre-emptively to the shock. A second way of evaluating the correspondence between UK data and our model is to compare the dynamic cross-correlations of key variables from the data with those from the model.13 Figure 2 shows this comparison for (log deviations of) aggregate output (y), value added sectoral outputs ( Nv y, and Xv y,), annual CPI inflation ( 4 π), the nominal interest rate (i) and the real exchange rate (q). In each of the thirty-six panels, the solid line illustrates the theoretical cross-correlation function and the dashed line the cross-correlation function from the data. Figure 2 indicates that our model seems to account for the auto-correlations of the data to a reasonable extent (see charts on the diagonal). In particular, our model can in part replicate the degree of persistence of inflation seen in the data, although this is mainly driven by persistence in the exogenous shocks. The model is perhaps less successful at capturing cross-correlations: for example, the dynamic relationship between the real exchange rate and some of the other variables in the panel. 13 For the model, the cross-correlations were computed using a variant of the Hansen-Sargent doubling algorithm discussed in section 5. 22 Figure 2: Cross-correlations of selected endogenous variables kt y−ktNv y−,, ktXv y−,, kt−,4 πkt i−kt q− t y -1 -0.5 0 0.5 1 0 5 10 -1 -0.5 0 0.5 1 0 5 10 -1 -0.5 0 0.5 1 0 5 10 -1 -0.5 0 0.5 1 0 5 10 -1 -0.5 0 0.5 1 0 5 10 -1 -0.5 0 0.5 1 0 5 10 tNv y,, -1 -0.5 0 0.5 1 0 5 10 -1 -0.5 0 0.5 1 0 5 10 -1 -0.5 0 0.5 1 0 5 10 -1 -0.5 0 0.5 1 0 5 10 -1 -0.5 0 0.5 1 0 5 10 -1 -0.5 0 0.5 1 0 5 10 tXv y,, -1 -0.5 0 0.5 1 0 5 10 -1 -0.5 0 0.5 1 0 5 10 -1 -0.5 0 0.5 1 0 5 10 -1 -0.5 0 0.5 1 0 5 10 -1 -0.5 0 0.5 1 0 5 10 -1 -0.5 0 0.5 1 0 5 10 t.4 π -1 -0.5 0 0.5 1 0 5 10 -1 -0.5 0 0.5 1 0 5 10 -1 -0.5 0 0.5 1 0 5 10 -1 -0.5 0 0.5 1 0 5 10 -1 -0.5 0 0.5 1 0 5 10 -1 -0.5 0 0.5 1 0 5 10 t i -1 -0.5 0 0.5 1 0 5 10 -1 -0.5 0 0.5 1 0 5 10 -1 -0.5 0 0.5 1 0 5 10 -1 -0.5 0 0.5 1 1.5 0 5 10 -1 -0.5 0 0.5 1 0 5 10 -1 -0.5 0 0.5 1 0 5 10 t q -1 -0.5 0 0.5 1 0 5 10 -1 -0.5 0 0.5 1 0 5 10 -1 -0.5 0 0.5 1 0 5 10 -1 -0.5 0 0.5 1 0 5 10 -1 -0.5 0 0.5 1 0 5 10 -1 -0.5 0 0.5 1 0 5 10 5. Results: a comparison of alternative simple rules In this section we present results from the model when it is closed with alternative monetary policy rules. In what follows we assume that deviations of the nominal interest rate from base are a linear function of deviations of endogenous variables (current, lagged or expected) from base. So we consider rules of the form, tt Rgi=, tt gz⊆.(32) where g is the set of feedback variables in the rule and R is a row vector of coefficients.14 A simple rule therefore consists of two components, the vector of feedback variables, g, and the vector of coefficients, R. We define generic classes of rules by the g vector, that is, by the set of variables on which they feed back. To carry out the comparison, for each rule we consider two kinds of coefficients vectors, R. First, we look at the rules in their original specification. In this case, the vector of coefficients, R, is that suggested for those rules. For example, the first group of rules includes a Taylor rule 14 Note that by using lag and lead identities within the model, the set of variables that could be included in the rule is large. For example, for the inflation forecast based rule considered below, we include conditional expectations of inflation up to five quarters ahead. 29 5.2 Results Table 1 below contains values of the two loss functions ( 1 L and 2 L) and asymptotic second order moments of inflation, output, the nominal interest rate, sectoral outputs and the real exchange rate. These are reported for the estimated rule, the Taylor and Henderson and McKibbin rules, the IFB rule, the naïve MCI-based rule and Ball’s rule with the original weights under our model specification. Table 2, in turn, reports analogous statistics for these rules (excluding the estimated rule) and for the OE1, OE2, OE3 and OE4 rules when coefficients are optimally derived. This table also reports corresponding optimised rules’ coefficients. Finally, Table 3 offers a test for relative robustness, by showing the same statistics for each rule when the model is hit by individual shocks rather than by a combination of shocks. 5.2.1 Results under an ‘all shocks’ scenario Table 1 suggests that, when coefficients are not optimised and all the shocks in the model are operative, the best performing rule according to loss function 1 L is, surprisingly, the naïve MCI-based rule, which ensures the lowest volatility of the real exchange rate and of exports. This in turn gives better inflation control than Taylor and Henderson and McKibbin rules. Among these two, the latter comes second being more successful than the Taylor rule at stabilising both output and inflation  a consequence of its stronger feedback coefficients. The estimated rule ranks fourth. Thanks to its term for interest rate smoothing, this rule responds gradually to inflationary pressures, and thus minimises interest rate and output volatility compared to Taylor/Henderson-McKibbin rules. On the other hand, this makes the estimated rule less successful at stabilising inflation than those rules. When coefficients are not optimally derived for this model, Ball’s rule gives a high loss (abstracting from the results under the IFB rule). This is mainly a consequence of its huge coefficient on deviations of output from potential: something that turns out not to be optimal for this model (sees table 3a and 3b below). This, in turn, leads to large interest rate gyrations and a high variability in the first difference of the nominal interest rate  a variable that 1 L penalises. As expected, the IFB rule, with a horizon and feedback coefficients originally suggested for a model with significantly more inflation inertia than ours, triggers considerable volatility in the interest rate, the result of which is destabilised output and, thus, a very large loss. Finally, with non-optimal coefficients, the naïve MCI-based rule and the estimated rule give a lower probability of hitting a zero bound with the nominal interest rate, followed by the IFB rule.24 All these rules imply a low variability of the level of the nominal interest rate. Table 1: Comparison of simple monetary policy rules (non-optimised coefficients) Rules: Taylor H-McK IFB MCI Ball Est’d Coeffs: t π1.5 1.5 -12.51 0.407 t y0.125 0.5 - - 1.93 0.046 24 The probability of hitting the lower bound on interest rates was derived as follows. We first note that, in the log-linearised model with Gaussian shocks, the nominal interest rate is normally distributed. It has a mean equal to the steady state value and the variance is given by the relevant element of V (see equation (37)). This estimate is likely to understate the true probability of hitting the lower bound because, once interest rates have fallen to zero and the economy is in a liquidity trap, it becomes less likely than implied by the asymptotic distribution that interest rates will be moved away from zero. 30 1−t i- - 0.5 - - 0.597 5 tt E π + - - 5--- t q----0.33 -0.43 - 1−t q- - - - 0.33 - t e∆------0.016 Welfare Loss 1 L6.399 5.290 117.3 5.006 7.049 5.736 Loss 2 L8.951 9.088 213.6 1.761 10.236 3.332 Avars π0.339 0.265 4.573 0.221 0.277 0.339 y0.301 0.297 27.436 0.592 0.168 0.198 i ∆ 0.169 0.187 4.177 0.219 0.611 0.029 Nv y,0.436 0.529 42.585 1.139 0.386 0.301 Xv y,1.077 1.058 14.125 0.719 0.920 0.987 c0.374 0.486 34.022 1.295 0.409 0.327 i0.721 0.754 22.981 0.308 0.847 0.208 q5.157 4.776 338.863 1.423 3.171 4.545 Prob(i < 0), % 3.9 4.2 37.7 0.3 5.2 0.1 Table 2 below lists analogous statistics and optimal coefficients for these rules and for the alternative open-economy rules (OE1 to OE4) when coefficients are optimally selected. The optimisation indicates that Taylor and Henderson-McKibbin rules for the UK economy, as modelled here, require stronger weights on inflation relative to output than those suggested for the US. This suggests that a mechanical application of the Taylor and/or Henderson and McKibbin rules in the UK context with coefficients designed for the US is not ideal. Moreover, our model seems to favour a stronger weight on inflation relative to output, even when the policymakers’ preferences are symmetric between inflation and output stabilisation. Similarly, for our model economy, the optimal coefficient for the MCI-based rule is smaller than one third – the value commonly used in the MCI literature – suggesting that a greater weight than that used in practice should be placed on interest rates vis-à-vis the exchange rate when altering monetary conditions. 31 Table 2: Comparison of simple monetary policy rules (optimised coefficients) Rules Taylor/ H-McK IFB MCI Ball OE1 OE2 OE3 OE4 Coeffs: t π5.94 -15.117 -4.111 5.193 5.869 t y0.23 - - - 0.036 -0.053 -0.228 1−t i-0.913 - - 0.947 - - - 1+tt Eπ-0.808 - - 1.093 - - - t q- - - 0.220 - 0.110 - 0.015 - 0.071 - 0.064 - 1−t q- - - - 0.047 - 0.024 - 0.054 - 0.039 - tNv y,, - - - - - - - 0.035 - tXv y,, - - - - - - 0.148 - t BT - - - - - - - - 0.029 t q∆- - - - - 6.476 - - Welfare Loss 1 L2.68 2.08 4.82 2.59 1.987 2.548 2.580 2.654 Loss 2 L2.32 -3.04 2.31 2.26 -2.30 1.807 2.123 2.211 Avars π0.029 0.029 0.234 0.031 0.029 0.033 0.030 0.029 y 1.608 1.540 0.449 1.489 1.425 1.452 1.522 1.614 i ∆ 0.151 0.018 0.156 0.152 0.023 0.144 0.147 0.145 Nv y,2.958 2.832 0.859 2.742 2.646 2.708 2.799 2.972 Xv y,1.178 1.137 0.768 1.120 1.078 1.105 1.108 1.172 c 2.679 2.668 1.001 2.525 2.525 2.517 2.583 2.692 i 0.613 0.170 0.287 0.582 0.198 0.544 0.581 0.607 q 6.431 5.332 1.923 5.503 4.592 5.257 5.512 6.327 Prob(i < 0), % 2.8 0.0 0.3 2.5 0.0 2.1 2.5 2.7 Looking at the actual stabilisation properties of each rule, the first thing to notice is that Taylor/HendersonMcKibbin, IFB and Ball’s rules perform much better than their counterparts with non-optimised coefficients. This is not true of the naïve MCI-based rule, since its performance is almost unaffected by the optimisation of its (unique) coefficient, making it the worse rule in the set. Indeed, when coefficients are optimised, rules in the Taylor/Henderson-McKibbin class now outperform the naïve MCI-based rule by ensuring lower inflation volatility. Second, with optimally chosen horizon and feedback coefficients (both significantly reduced from the non-optimal case), the IFB rule performs extraordinarily well. When coefficients are optimal, the IFB appears to be extremely successful at minimising inflation volatility for a level of output volatility that is now comparable, if not lower, than that of other rules in the table. This is a consequence of the fact that, under this rule, the interest rate now moves optimally and solely to correct low-frequency changes in inflation. (The asymptotic standard deviation of changes in the nominal interest rate for this rule is, in fact, around a tenth of that under all 32 other rules if we exclude the OE1 rule, a modification of the IFB.) This result is not altogether surprising. IFB rules have now been found to perform well in a number of studies: essentially for the reasons noted in Batini and Haldane (1999). Third, Ball’s rule provides a lower than average variability of output when compared to Taylor/Henderson-McKibbin and to the IFB rule. Relative to them, it also reduces the disparity between output sector volatilities in the two sectors, other things being equal. However, it produces more interest rate, exchange rate and ultimately inflation variability than the IFB rule because it reacts to current, rather than expected inflation and hence necessitates greater aggressiveness. This rule is, unsurprisingly, more efficient than the simplistic MCIbased rule which does not react either to inflation nor output, and which thus is unsuited to cope with inflationary shocks that did not originate in a shock to the exchange rate. Additional interesting results emerge when we look at our ‘family’ of open economy rules. In general, it seems as if parsimonious modifications of either Ball’s or the IFB rule do not gain much in terms of inflation or output control. The reduction in the value of 1 L is indeed negligible and most certainly ascribable to the fact that rules in the OE family typically react to more state variables. By construction, this gives them a performance ‘bonus’ relative to nonOE rules. By the same logic, the opposite is true of rule OE4  a restricted version of Ball’s rule  which hence does worse than Ball’s rule itself. However, on the whole, rules OE2 (Ball plus response to balance of trade), OE3 (Ball with separate response to sectoral outputs) and OE1 (IFB with additional exchange rate terms) do seem capable of reducing further the disparity between output sector volatilities in the two sectors, others things being equal. When we look at each rule individually, the following emerges. In rules OE2 and OE3 which feed back on the balance trade and on sectoral output gaps respectively, introducing extra terms has the effect of lowering slightly the response to exchange rate terms as the optimised Ball’s rule would imply in the absence of those terms. And in OE2, adding a feedback term on the trade balance inverts the sign on the output gap term (which was, somewhat counterintuitively, negative in Ball’s optimised rule). This is possibly a consequence of the fact that when policymakers respond separately to the net trade component of the aggregate output gap, policy no longer needs to give a procyclical response to the output gap. The implications of these changes in existing coefficients combined with the effect of new coefficients are that: (i) OE2 gives a marginally better control than Ball’s rule of output and interest rates, but a slightly worse control of inflation; (ii) OE3 gives a marginally better control of inflation and interest rates than Ball’s rule, but worsens slightly the control of output; and (iii) only OE2 ensures more symmetry than Ball’s rule in the adjustment of sectoral outputs after a shock. Adding separate exchange rate terms to the IFB rule (‘OE1’) also moderately improves its stabilisation properties. The rule now delivers a lower volatility of output than without the exchange rate terms. Exchange rate volatility also falls. OE1 is in fact the best rule in the table. Compared to non-OE rules, OE1 gives considerably lower output, exchange rate and interest rate variability than Taylor/Henderson-McKibbin rules and also than Ball’s rule. If we abstract from the naïve-MCI that smooths the costs of adjustment across sectors but does 33 badly in term of inflation variability, OE1 produces the minimum disparity between the volatility of the output gap in the two sectors. Note that, thanks to their ability of minimising the volatility of the policy instrument, IFB and OE1 rules also give the lowest probability of hitting a zero bound with the nominal interest rate. These results on the relative performance of the rules are confirmed by our second measure of loss, the utility-based loss function 2 L. According to this metric, households would be better off if policymakers followed an OE1 rule or an IFB rule with coefficients optimised over the objective function 1 L, rather than other rules that we consider. The worse possible rule according to 2 L is instead the Taylor/Henderson-McKibbin. In summary: • IFB rules appear to be efficient open economy rules because they capture all channels of transmission. They outperform closed economy rules like Taylor and HendersonMcKibbin in terms of both output and interest rate control. But they also prove superior to Ball’s rule, which reacts to current, rather than expected inflation; this makes policy myopic rather than pre-emptive, and hence requires more aggressive changes in the interest rate, which in turn affect the exchange rate and thereby inflation. For these reasons, IFB rules also help stabilising the economy in a more ‘symmetric way’, demanding less adjustment from the the internationally exposed sector than that required by closed-economy rules which ignore differences in adjustment across sectors; • Modifications to these rules to include explicit feedbacks on the level of the exchange rate (contemporaneous and lagged) improve the performance of these rules only marginally. But since they help reduce further the disparity in adjustment between traded and exports sectors, they may be desirable if the authorities have a specific preference for symmetry in adjustment, other things equal. • Relative to other rules in the battery, IFB and exchange-rate-adjusted IFB (OE1) minimise the probability of hitting a zero bound with nominal interest rates, and thereby increase the chances of policy remaining operational under particularly severe deflationary shocks; • Ball’s rule and variants of it (notably OE2 which allows a response to disequilibria in the trade balance) are second-best options. Because they also account for the exchange rate channel of transmission, as expected they are significantly more efficient than Taylor/Henderson-McKibbin rules in stabilising inflation and output in our open economy model of the UK. In this respect they are indeed by far preferable to naïve MCI-based rule, which gives more stable output outcomes at the price of massive inflation variance (the loss associated with Ball’s rule is a sixth of the loss associated with the naïve MCIbased rule). 34 5.2.2 Robustness analysis to individual shocks In order to provide more intuition about why certain rules perform better than others, we have re-assessed the performance of the rules assuming that the economy was hit by one type of shock at a time. In particular, we are looking to see which rules seem to produce ‘sensible’ responses to each of the different shocks and analyse whether or not the rules that perform well do so because they are robust to many different shocks. In each case, the coefficients in the rules are again those optimally derived for the ‘all-shocks’ case (shown in table 2), so this is a test of robustness of the exact rule specification.25 Results from this experiment are summarised in Tables 3a and 3b below. The tables suggest that the OE1 (i.e. the modified IFB) rule and the IFB rule itself are still the ‘best’ rules under most shocks. The OE1 (and to some extent also the original IFB) rule seems to perform particularly well in the face of shocks from overseas. However, both the OE1 and the IFB rules are outperformed by the OE4 (restricted Ball) rule and by their ‘closed-economy’ counterparts under productivity shocks to the exports sector. (OE1 and IFB are also inferior to OE4 and Taylor/Henderson-McKibbin under a shock to non-traded goods sector productivity but this is less severe than in the case of a shock to productivity to the export sector). The reason why this happens is that a shock to productivity in the export sector will affect both export prices and output. Since export prices do not enter the calculation of CPI inflation, rules like OE1 or IFB that respond only to consumer price inflation will not perform well because they fail to respond to the first round effects of this shock. This is in line with the general intuition that a simple rule can be a good guide for policy in the face of some  but not all  shocks. Crucially, however, our rule seems more robust to different shocks than a naïve and Ball’s MCI-based rules. This is particularly evident for overseas shocks (e.g. foreign interest rate shocks and shocks to the risk premium), but also to shocks to the price of intermediate inputs and to shocks to world output and inflation. In the case of Ball’s rule, this is also true for shocks to productivity to both the export sector and the non-traded goods sector. A comparison of the losses associated with each shock in turn reveals that the most costly shock by far is that to intermediates prices. This is because this shock not only has a higher variance than other shocks but it is also highly cross-correlated with other overseas shocks. In fact, intermediate prices are a large proportion of unit costs in both sectors. For instance, since non-traded producers set prices as a mark up over unit costs, changes in these prices feed directly through non traded price inflation. On the other hand, shocks to the export sector seem to be relatively unimportant given the size of this sector and the openness of the economy. This is because shocks to this sector are largely absorbed by the price of exports which is not a component of CPI inflation. Given that a shock to intermediate prices is the most costly of our shocks, we would ideally wish to use a monetary policy rule that generated the appropriate response to this shock. In particular, we know that optimal policy would want to absorb the first-round effects of this shock but would want to make sure that there was no long-run effect on inflation. A standard 25 To perform this test, we have re-derived losses and asymptotic second moments of the variables of interest by setting the variances of the remaining shocks to zero. 35 rule that feeds back off current inflation may lead to policy that was too tight and cause a fall in output. By contrast, the IFB and OE1 rules act to stabilise future inflation rather than current: exactly the policy response that seems appropriate for this sort of shock. The results of this section suggest that OE1 and IFB rules manage to dominate all other rules in an ‘all-shocks’ scenario because they are efficient at stabilising the economy in the face of overseas shocks (among which are, notably, shocks to intermediate prices). 36 Table 3a: Comparison of simple monetary policy rules (individual shocks) Taylor/ H-McK IFB MCI Ball OE1 OE2 OE3 OE4 Non-traded productivity shock Loss 1 L0.0233 0.0240 0.0267 0.0277 0.0238 0.0268 0.0277 0.0233 Avars π0.0000 0.0000 0.0005 0.0000 0.0000 0.0000 0.0001 0.0000 y0.0226 0.0237 0.0188 0.0243 0.0233 0.0254 0.0235 0.0227 i ∆ 0.0000 0.0000 0.0001 0.0007 0.0000 0.0002 0.0008 0.0000 Nv y,0.0046 0.0054 0.0007 0.0052 0.0048 0.0088 0.0042 0.0047 Xv y,0.0117 0.0119 0.0100 0.0121 0.0118 0.0126 0.0118 0.0117 c0.0017 0.0021 0.0000 0.0018 0.0017 0.0045 0.0012 0.0017 i0.0001 0.0001 0.0004 0.0005 0.0001 0.0004 0.0006 0.0001 q0.0036 0.0050 0.0001 0.0069 0.0045 0.0069 0.0060 0.0036 Export productivity shock Loss 1 L0.0492 0.0507 0.0487 0.0518 0.0505 0.0579 0.0562 0.0485 Avars π0.0000 0.0000 0.0000 0.0000 0.0000 0.0003 0.0000 0.0000 y0.0445 0.0507 0.0483 0.0517 0.0505 0.0359 0.0547 0.0445 i ∆ 0.0010 0.0000 0.0000 0.0000 0.0000 0.0044 0.0002 0.0008 Nv y,0.0001 0.0004 0.0000 0.0005 0.0003 0.0013 0.0009 0.0001 Xv y,0.0550 0.0507 0.0510 0.0507 0.0508 0.0615 0.0487 0.0549 c0.0001 0.0001 0.0000 0.0002 0.0001 0.0015 0.0005 0.0001 i0.0005 0.0000 0.0000 0.0000 0.0000 0.0022 0.0001 0.0004 q0.0023 0.0002 0.0000 0.0003 0.0002 0.0146 0.0012 0.0022 FX risk premium shock Loss 1 L0.1013 0.0187 0.2486 0.1287 0.0146 0.1150 0.1204 0.0891 Avars π0.0006 0.0009 0.0007 0.0004 0.0005 0.0004 0.0004 0.0007 y0.0013 0.0019 0.0010 0.0007 0.0033 0.0008 0.0008 0.0015 i ∆ 0.0226 0.0006 0.0592 0.0305 0.0009 0.0269 0.0282 0.0190 Nv y,0.0006 0.0001 0.0001 0.0004 0.0045 0.0003 0.0004 0.0008 Xv y,0.0372 0.0483 0.0309 0.0300 0.0389 0.0316 0.0310 0.0363 c0.0004 0.0002 0.0003 0.0004 0.0052 0.0004 0.0004 0.0004 i0.0219 0.0008 0.0440 0.0377 0.0023 0.0324 0.0347 0.0232 q0.9686 1.2620 0.8071 0.7787 1.0156 0.8226 0.8067 0.9445 Preference shock Loss 1 L0.1310 0.1253 0.1336 0.1275 0.1258 0.1347 0.1277 0.1281 Avars π0.0001 0.0000 0.0001 0.0000 0.0000 0.0001 0.0000 0.0001 y0.1121 0.1252 0.1324 0.1272 0.1257 0.1330 0.1274 0.1121 i ∆ 0.0044 0.0000 0.0000 0.0000 0.0000 0.0002 0.0000 0.0037 Nv y,0.1718 0.1893 0.2000 0.1923 0.1900 0.2004 0.1926 0.1718 Xv y,0.0004 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0004 c0.1867 0.2041 0.2148 0.2072 0.2048 0.2152 0.2074 0.1867 i0.0022 0.0000 0.0001 0.0000 0.0000 0.0001 0.0000 0.0019 q0.0117 0.0007 0.0000 0.0004 0.0006 0.0002 0.0004 0.0112 37 38 Table 3b: Comparison of simple monetary policy rules (individual shocks) Taylor/ H-McK IFB MCI Ball OE1 OE2 OE3 OE4 Intermediate relative (world) price shock Loss 1 L1.3742 1.1519 2.5288 1.3274 1.1515 1.3131 1.3100 1.3777 Avars π0.0227 0.0253 0.1337 0.0248 0.0226 0.0254 0.0223 0.0223 y0.7648 0.6859 0.1690 0.6804 0.7089 0.6519 0.7122 0.7682 i ∆ 0.0616 0.0152 0.0552 0.0623 0.0205 0.0637 0.0601 0.0631 Nv y,1.5420 1.4270 0.4322 1.3934 1.4660 1.3441 1.4511 1.5496 Xv y,0.1502 0.0693 0.0340 0.1310 0.0756 0.1210 0.1340 0.1484 c1.4231 1.4029 0.5089 1.3070 1.4372 1.2740 1.3594 1.4305 i0.4815 0.1388 0.1548 0.4373 0.1637 0.4083 0.4337 0.4715 q2.3738 0.5994 0.2366 1.9364 0.7385 1.7180 2.0106 2.3236 World inflation shock Loss 1 L0.4298 0.2179 0.2712 0.4605 0.1912 0.4914 0.4887 0.4340 Avars π0.0031 0.0032 0.0093 0.0028 0.0026 0.0035 0.0030 0.0032 y0.1894 0.1590 0.0550 0.2042 0.1402 0.2285 0.2183 0.1912 i ∆ 0.0477 0.0020 0.0170 0.0531 0.0024 0.0517 0.0558 0.0481 Nv y,0.3721 0.3120 0.1230 0.3988 0.2797 0.4410 0.4238 0.3753 Xv y,0.0119 0.0232 0.0169 0.0109 0.0198 0.0121 0.0108 0.0116 c0.3690 0.3198 0.1419 0.3962 0.2914 0.4357 0.4195 0.3719 i0.0906 0.0126 0.0212 0.0957 0.0113 0.0913 0.1010 0.0926 q0.3366 0.6156 0.2875 0.3219 0.4713 0.3941 0.3332 0.3340 World interest rate shock Loss 1 L0.1251 0.1187 0.6265 0.1398 0.1119 0.1530 0.1484 0.1242 Avars π0.0014 0.0012 0.0319 0.0025 0.0012 0.0036 0.0031 0.0015 y0.0827 0.0974 0.0048 0.0674 0.0862 0.0548 0.0610 0.0816 i ∆ 0.0049 0.0007 0.0278 0.0081 0.0016 0.0101 0.0093 0.0047 Nv y,0.1418 0.1686 0.0087 0.1159 0.1488 0.0952 0.1051 0.1402 Xv y,0.1711 0.1699 0.1123 0.1663 0.1702 0.1604 0.1643 0.1704 c0.1066 0.1310 0.0111 0.0859 0.1134 0.0701 0.0772 0.1054 i0.0263 0.0053 0.0573 0.0383 0.0126 0.0424 0.0425 0.0261 q0.3897 0.4337 0.2006 0.3429 0.4063 0.2959 0.3255 0.3841 World output shock Loss 1 L0.2636 0.1730 0.3944 0.1879 0.0962 0.1419 0.1528 0.2516 Avars π0.0022 0.0051 0.0225 0.0012 0.0038 0.0010 0.0009 0.0022 y0.0849 0.0771 0.0320 0.0563 0.0274 0.0479 0.0477 0.0859 i ∆ 0.0357 0.0034 0.0005 0.0280 0.0022 0.0196 0.0227 0.0325 Nv y,0.1494 0.1180 0.0877 0.1055 0.0518 0.0943 0.0954 0.1513 Xv y,0.6645 0.7375 0.5664 0.6544 0.6804 0.6437 0.6392 0.6633 c0.1033 0.0766 0.1437 0.0779 0.0468 0.0743 0.0753 0.1048 i0.0815 0.0155 0.0011 0.0578 0.0099 0.0513 0.0555 0.0818 q1.3761 1.8257 0.5741 1.2551 1.3493 1.1713 1.1489 1.3707 ‘money’ (A2). We let the Lagrange multipliers on these two constraints be denoted λ1 and λ2, respectively. Suppressing the j index throughout, we differentiate to get: ÷ ÷ ø ö ç ç è æ − ÷ ÷ ø ö ç ç è æ =− ÷ ÷ ø ö ç ç è æ −+ + −− −tt t t tM tN tMt tM tN tt t cc E c c P c c cc ξ ν βγξλ ξ νγ γγ 1 1 1 , , ,,1 1 , , 1 )exp()exp( (A5) ÷ ÷ ø ö ç ç è æ − ÷ ÷ ø ö ç ç è æ −=− ÷ ÷ ø ö ç ç è æ − − + + −tt t t tM tN tNt tM tN tt t cc E c c P c c cc ξ ν ξγβλ ξ νγ γγ 1 1 , , ,,1 , , 1 )exp( )1( )exp()1( (A6) ( ) 1,21,1,2,1 )1( ++ ++=+ tttttt Ei λλβλλ (A7) ÷ ÷ ø ö ç ç è æ+ += + + ++ 1 1,21,1 , ,2,1 )1( t tt ttf t tt e Ei e λλ β λλ (A8) ε χ λ − ÷ ÷ ø ö ç ç è æΩ = t t t tPP ,2 (A9) ( ) 1,21,1,1 ++ += tttt E λλβλ (A10) The choice of the nominal wage discussed in section 2.3.2. The first order condition is: 0)( )(1 )1( )()1( )( 0,1 = ú û ù ê ë é − +−Λ + + ∞ =+ + + åjh jhP jW Est sst Wst st t s s Wt δ θ π βφ . (A11) Equation (A12) features the real marginal utility of consumption, 1 Λ, which is related to the marginal utility of nominal consumption in a simple manner: 11 λ P=Λ . This is discussed in more detail below. Non-Traded Sector As described in section 2.3.1 producer k∈(0,1) in the non-traded sector choose prices to solve the following problem. )( )()1( )(max , 0 , ,1 kyV P kP EjtN jjt jt tN j jt j Nt + ∞ =+ + + å÷ ÷ ø ö ç ç è æ− + Λ π βφ subject to jtN jtN jtN j jtN y P kP ky N N + − + + + + ÷ ÷ ø ö ç ç è æ+ =, , , , )1( )()1( )( θ θ π . The first order condition is: 0)()1( )()1( )( , 0 , ,1 = ÷ ÷ ø ö ç ç è æ++ +− Λ+ ∞ =+ + + åkyV P kP EjtN jjtN jt tN j N jt j Nt θ πθ βφ . (A12) The real unit cost, V, in units of final consumption is given by: 1)()( subject to )()(min )1( ,,,, , ,= ï þ ï ý ü ï î ï í ì+= − ++++ + + + + + +NN kIkhAkI P P kh P W VstNstNstNstN st stI stN st st st αα The first order conditions to this problem imply that: )( )( 1, , ,kh kI P W tN tN N N tI t α α − =, for all k∈(0,1) at all dates t. Because non-traded producers are price takers in the factor market, the equilibrium ratio of intermediates to labour is constant across firm in this sector: tN tN N N tI t h I P W , , ,1 α α − =. (A13) The constancy of the intermediate:labour ratio implies that the aggregate output in the non-traded producers is given by: N Ndkkyy tNtN θ θ + +ú û ù ê ë é =ò 1 1 0 )1/(1 ,, )( N N N N NNN N N dkkI I h AdkkIkhA dkkyy tN tN tN tNtNtNtN tNtN θ θ α θ θαα θ θ + + + +− + + ú û ù ê ë é ÷ ÷ ø ö ç ç è æ = ú û ù ê ë é = ú û ù ê ë é = òò ò 1 1 0 )1/(1 , , , , 1 1 0 )1/(1)1( ,,, 1 1 0 )1/(1 ,, )(])()([ )( So, NN tNtNtNtN IhAy αα − =1,,,, .(A14) The minimised unit cost for all firms in the non-traded sector is found to be: ttN tIt NN tPA PW VNN NN , 1 , 1 )( )1( 1 αα αα αα − − − =. (A15) Export sector As described in section 2.2.2 exports are produced using a Cobb-Douglas technology: XX tXtXtXtX IhAy αα − =1,,,, (A16) Efficient production implies that factor demands are given by: X tX tX tXX tX t h I A P W α α − ÷ ÷ ø ö ç ç è æ = 1 , , , , (A17) X tX tX tXX tX tI I h A P P α α ÷ ÷ ø ö ç ç è æ −= , , , , ,)1( (A18) Export demand is: btf t tXt ty P Pe X, * , η − ÷ ÷ ø ö ç ç è æ =. (A19) Intermediate goods sector Producers in both the non-traded and export sectors purchase imported intermediates from retailers who solve a pricing problem described in section 2.3.1. The first order condition is: 0)()1( )()1( )( , 0 * ,, ,11 = ÷ ÷ ø ö ç ç è æ++ +− Λ+ ∞ =++ + + +− åky Pe P P kP EstI sstst stI I st tI s I st s It θ πθ βφ (A20) Final imports sector The first order condition for the pricing problem of retailers of final imported goods is given by: 0)()1( )()1( )( , 0 * , ,11 = ÷ ÷ ø ö ç ç è æ++ +− Λ+ ∞ =++ + + +− åky Pe P P kP EstM sstst st M st tM s M st s Mt θ πθ βφ . (A21) Government The government operates monetary policy by setting nominal interest rates according to a rule (described below) and prints as much money as is demanded at this level of nominal interest rates. Any seignorage revenue is distributed as a lump-sum transfer to consumers. For simplicity, we assume a zero supply of domestic bonds. Hence: tttt TMM τ −=− −1. (A22) Market Clearing We have the following market clearing conditions in factor markets, goods markets and asset markets: tNtXt hhh ,, += (A23) tNtN yc ,, =(A24) tXt yX , =(A25) 0),( = òò djdsjsbt(A26) Net foreign assets The evolution of net foreign assets can be found by evaluating the household’s budget constraint (A2) at market equilibrium and then aggregating across households. As discussed in section 2.4, the net foreign asset position (under our assumptions this is equal to the domestic holdings of foreign bonds) is non-stationary. To deal with this problem we do not include this equation in our system. Instead we use the equation to substitute foreign bond holdings out of the definition of ‘money’ (A3). Annex B: Flexible-price steady state We use the following notation. Variables without time subscripts are the steady state values. Lower case letters represent nominal variables expressed relative to the CPI (we also define the real value of foreign bond holdings as bf=Bf/eP). We express nominal variables relative to the general price level in order to solve for steady state variables that are not trended (in steady state all nominal variables will follow the same trend path). In addition, the Lagrange multipliers 1 λ and 2 λ are homogenous of degree -1 so we scale them by the CPI, to give stationary multipliers 11 λ P=Λ and 22 λ P=Λ . Throughout we use the real exchange rate definition, * t tt tP Pe q=. To construct a steady state, we first assume that all domestic nominal variables are growing at an annual rate of 2.5%. This means that, in steady state, the government is meeting an inflation target of 2.5%. For simplicity, we also assume that the steady state growth of foreign nominal variables is 2.5%. The implied steady state value of nominal interest rates at home and abroad will be given by: 1 1− + == β π f ii . In what follows, we use equations (A23) and (A27) before evaluating the steady state. We assume that steady state taxes are set to exactly offset steady state dividends. Finally, we choose a flexible price equilibrium so that, although price setters retain some monopoly power, they simply set prices as a mark up over unit costs. Then, the first order conditions imply the following equations defining steady state values of the variables: () () γ ξ γβξ λ − ÷ ÷ ø ö ç ç è æ −1 − = 1 1 1 M N Mc c c p(A27) () () γ ξ γβξ λ ÷ ÷ ø ö ç ç è æ −1 −− = N M Nc c c p)1(1 1(A28) 2 λχω ε = −(A29) π λλβ λ +1 )+( =21 1(A30) () )1( 1 1h wW − + = δθ λ (A31) () NXIMMXf IIpcpXpb +−−= − β β 1(A32) f bm β β ω −1 += (A33) γγ − =1 NM ccc (A34) γγγγ γγ −1− )−= 1( 1MN pp (A35) vp NN )1( θ += (A36) N N N N Ih I p w α α − =1(A37) NN NNNN IhAy αα − =1(A38) NN NN NN I pw v αα αα αα − − − =1 1 )1( (A39) XX XXXX IhAy αα − =1(A40) X X X XX Xh I A p w α α − ÷ ÷ ø ö ç ç è æ = 1 (A41) X X X XX X I I h A p p α α ÷ ÷ ø ö ç ç è æ −= )1( (A42) b f X y p q X η ÷ ÷ ø ö ç ç è æ =(A43) q p II I p* )1( θ += (A44) q MM p1 )1( θ += (A45) NN yc =(A46) XN hhh += (A47) X yX =(A48) Annex C: A log-linear representation of the model To solve the model we log-linearise the first order conditions of the model around the non-stochastic steady state defined by equations (A27) to (A48). As described in the main text we use (A2) evaluated at market equilibrium to substitute foreign bond holdings out of the model. As in Annex 2, we also substitute out for taxes, transfers and dividends. Log-linearising the consumers’ first order conditions (equations (A5) to (A10)) gives us: ()() ()()()() 1 2 ,,,111 ˆ 11 ˆ 11 1 1 ˆ 1 1 ˆ ˆ 1 ˆ 11 − ++ −− − ÷ ÷ ø ö ç ç è æ −− + −− + − −+Λ= ÷ ÷ ø ö ç ç è æ − − −− tt tMttMtttt cc cpcE ξβξ ξ ξβξ βξ ν βξ ν βξ βξ ξβξ βξ (A49) ()() ()()()() 1 2 ,,,111 ˆ 11 ˆ 11 1 1 ˆ 1 1 ˆ ˆ 1 ˆ 11 − ++ −− − ÷ ÷ ø ö ç ç è æ −− + −− + − −+Λ= ÷ ÷ ø ö ç ç è æ − − −− tt tNttNtttt cc cpcE ξβξ ξ ξβξ βξ ν βξ ν βξ βξ ξβξ βξ (A50) t tttttt iiE ,2 21 2 ,1 21 1 1,2 21 2 1,1 21 1 1 ˆ ˆ )( ˆˆ ˆ Λ Λ+Λ Λ − Λ Λ+Λ Λ −−= ÷ ÷ ø ö ç ç è æΛ Λ+Λ Λ −Λ Λ+Λ Λ −+++ π (A51) tt ttftfttttt qiiqE ζ π +Λ Λ+Λ Λ − Λ Λ+Λ Λ −+−= ÷ ÷ ø ö ç ç è æΛ Λ+Λ Λ −Λ Λ+Λ Λ −+ ++++ ,2 21 2 ,1 21 1 ,1,2 21 2 1,1 21 1 1 *1 ˆ ˆ ˆ )( ˆˆ ˆˆ (A52) 0 ˆ ˆ,2 =+Λ tt ωε (A53) ttttt E,11,2 21 2 1,1 21 1 1ˆˆˆ ˆΛ= ÷ ÷ ø ö ç ç è æΛ Λ+Λ Λ −Λ Λ+Λ Λ −+++ π (A54) where for any variable x, ÷ ø ö ç è æ =x x xt ln ˆ where x is its steady state value and ζ is an exogenous ‘foreign exchange risk premium’ shock. The definition of Ω (equation (A3)) becomes: 0 ˆˆ ˆ ˆˆ =+−++− ttttt c c w wh h wh m m ωωωω ω . (A55) The definitions of consumption and the price indices are: tNtMt ccc ,, ˆ 1( ˆˆ )−+= γγ , (A56) tNtM pp ,, ˆ 1( ˆ 0)−+= γγ . (A57) Wage setting is given by the following two equations (the derivation follows Erceg et al (1999, p25)). t h h W WW t h h W WW t h h W WW ttt w h h h WEW W W W W W W ˆ ]1[ )1)(1( ˆ ]1[ )1)(1( ˆ ]1)[1( )1)(1( ˆˆ )1( )1( ,1 )1( )1( )1( )1( 1 − + − + − + + + −− − Λ + −− − +− −− +∆=∆ θ θ θ θ θ θ φ βφφ φ βφ φ φ βφ φ β (A58) tttt Www π ˆ ˆ ˆˆ 1−∆+= −. (A59) Pricing decisions by non-traded goods producers are described by: tN N NN t N NN tNttN pvPEP ,1,, ˆ )1)(1( ˆ )1)(1( ˆˆ φ βφφ φ βφφ β −− − −− +∆=∆ +, (A60) ttNtNtN Ppp π ˆ ˆ ˆˆ ,1,, −∆+= −, (A61) tNtINtNt Apwv ,, ˆ ˆ )1( ˆˆ −−+= αα . (A62) Efficient production by non-traded producers implies that: tNtNtIt hIpw ,,, ˆ ˆ ˆˆ −=− , (A63) tNNtNNtNtN IhAy ,,,, ˆ )1( ˆ ˆ ˆ αα −++= . (A64) The first order conditions for export producers become: 0 ˆ )1( ˆ )1( ˆ ˆˆ ,,,, =−+−−−− tXXtXXtXtXt hIApw αα , (A65) 0 ˆ ˆ ˆ ˆˆ ,,,,, =++−− tXXtXXtXtXtI hIApp αα . (A66) Export production is given by: 0 ˆ )1( ˆ ˆ ˆ,,,, =−−−− tXXtXXtXtX IhAy αα . (A67) Export demand can be written as: 0 ˆˆˆ ˆ,, =−++ tFtXtt ybpqX ηη . (A68) Pricing of intermediates is described by: tIt I II tt I II tIt I II tIttI pEqE pEPEP ,11 *,11,1, ˆ )1)(1( ˆ )1)(1( ˆ )1)(1( ˆˆ −− −+− −− − −− − −− +∆=∆ φ βφφ φ βφφ φ βφ φ β (A69) ttItItI Ppp π ˆ ˆ ˆˆ ,1,, −∆+= −. (A70) Pricing of final imports is described by: tt I II tMttM qEPEP ˆ )1)(1( ˆˆ 11,1, −+− −− −∆=∆ φ βφφ β , (A71) ttMtMtM Ppp π ˆ ˆ ˆˆ ,1,, −∆+= −. (A72) The relevant market-clearing conditions can be written as: 0 ˆˆˆ ,, =−+ ttN N tX Xhh h h h h h, (A73) 0 ˆˆ ,, =− tNtN yc , (A74) 0 ˆ ˆ,=− tXt yX . (A75) Together with some obvious lag identities and log-linearised definitions (for example the GDP identity) the model can be cast in the form of equations (22) and (23) in the main text. The calibration of the forcing processes is described in section 3.2 of the main text.