The Golden Interest Rule: Robust Simple Interest Rate Rules for the Norwegian Economy

Hoen, Maria Brunborg Research Report The Golden Interest Rule: Robust Simple Interest Rate Rules for the Norwegian Economy Staff Memo, No. 16/2012 Provided in Cooperation with: Norges Bank, Oslo Suggested Citation: Hoen, Maria Brunborg (2012) : The Golden Interest Rule: Robust Simple Interest Rate Rules for the Norwegian Economy, Staff Memo, No. 16/2012, ISBN 978-82-7553-675-2, Norges Bank, Oslo, https://hdl.handle.net/11250/2507186 This Version is available at: https://hdl.handle.net/10419/210254 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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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. http://creativecommons.org/licenses/by-nc-nd/4.0/deed.no No. 16 | 2012 The golden interest rule Maria Brunborg Hoen, Monetary Policy Staff Memo Staff Memos present reports and documentation written by staff members and affiliates of Norges Bank, the central bank of Norway. Views and conclusions expressed in Staff Memos should not be taken to represent the views of Norges Bank. © 2012 Norges Bank The text may be quoted or referred to, provided that due acknowledgement is given to source. Staff Memo inneholder utredninger og dokumentasjon skrevet av Norges Banks ansatte og andre forfattere tilknyttet Norges Bank. Synspunkter og konklusjoner i arbeidene er ikke nødvendigvis representative for Norges Banks. © 2012 Norges Bank Det kan siteres fra eller henvises til dette arbeid, gitt at forfatter og Norges Bank oppgis som kilde. ISSN 1504-2596 (online only) ISBN 978-82-7553-675-2 (online only) The Golden Interest Rule Robust simple interest rate rules for the Norwegian economy Maria Brunborg Hoen Master thesis for the degree Master of Economic Theory and Econometrics Department of Economics UNIVERSITY OF OSLO Submitted May 14, 2012 This master thesis is part of a research project together with Pehlin Ilbas, Øistein Røisland, Tommy Sveen and Mathis Mæhlum on robust monetary policy in the research department of Norges Bank’s Monetary Policy Department. The research project has the objective to further explore the method of robustifying against model uncertainty by modifying a standard loss function with a small weight on a robust simple interest rate rule (Ilbas et al. 2012), and develop such robust rules tailored to the Norwegian economy. The latter is the goal of this master thesis, whereas Mathis Mæhlum extends the work on the modi…ed loss function to Norwegian models. I have worked for two years as an assistant on the research project, …rst one year contributing to Ilbas et al. 2012, and then one year on the analyses presented in this master thesis. I am most grateful to Norges Bank for letting me work on the project and for the …nancial support. I deeply thank my supervisor Tommy Sveen for invaluable guidance, help and instructive contributions throughout my whole period in Norges Bank. I also thank Øistein Røisland for his many and highly useful comments, and my fellow student and colleague Mathis Mæhlum for fruitful discussions, inspiration and pleasant company. A big thank goes to the other sta¤ in Norges Bank, who has helped me in some way or another, in particular Leif Brubakk, Martin Seneca, Junior Maih, Bjørn Naug, Ørjan Robstad, Kenneth Paulsen, and Klaus Blomli and Karsten Molværsmyr for rescuing my knocked-out computer, and those I might have forgotten to mention. At last I would like to thank my family and friends for support, and my father Helge Brunborg and Hannes Peinl for proofreading. I Contents 1 Summary and Introduction 1 2 Theory 4 2.1 In‡ation targeting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.2 A role for monetary policy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.3 Simple interest rate rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.4 Optimal monetary policy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.5 Robustifying monetary policy . . . . . . . . . . . . . . . . . . . . . . . . . . 13 3 Models 14 3.1 NEMO....................................... 14 3.1.1 Policy NEMO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 3.2 CreditNEMO................................... 18 3.3 LGM........................................ 19 3.3.1 Description of the model . . . . . . . . . . . . . . . . . . . . . . . . . 19 3.3.2 Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 3.4 NAM........................................ 22 4 Transmission mechanisms of monetary policy 25 5 Method 28 5.1 Optimal simple rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 5.2 Bayesian rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 5.3 Di¤erent loss functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 6 Results 34 6.1 Optimal simple rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 6.2 Bayesian rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 7 Robustness 40 7.1 Implied In‡ation variability Premium . . . . . . . . . . . . . . . . . . . . . . 41 7.1.1 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 7.2 Fault tolerance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 7.2.1 Bayesian rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 8 "The Golden Interest Rule" 50 III 9 Concluding remarks 55 References 58 Appendix 62 A Models 62 A.1 NEMO....................................... 62 A.1.1 List of variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 A.1.2 Model ................................... 63 A.2 LGM........................................ 66 A.2.1 List of variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 A.2.2 Model ................................... 67 A.2.3 Estimation results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 A.3 NAM........................................ 70 A.3.1 List of variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 A.3.2 Model ................................... 71 B Results 72 B.1 Rules with alternative loss functions . . . . . . . . . . . . . . . . . . . . . . . 72 B.2 Alternative rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 B.3 IIPresults..................................... 74 B.4 Fault tolerance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 IV There are also other ways of modelling price sluggishness, and the method proposed by Rotemberg (1982) is used in NEMO. It di¤ers from the Calvo pricing scheme in that all …rms set prices optimally every period based on complete information. Revising prices is costly, and price setters must be forward-looking and base current prices on expectations as well as on last period’s prices. The adjustment costs are modelled as quadratic deviations between the prices in two successive periods. At the aggregate level and up to a …rst order approximation, Rotemberg and Calvo pricing schemes are however equal (Lombardo and Vestin 2007). Due to sticky prices there are two sources of ine¢ ciencies in the canonical New Keynesian economy, namely ‡uctuations in the average output gap of …rms, and price dispersion between …rms. The output gap is the di¤erence between the actual production level and the e¢ cient level. The latter is the production level that would have been realized if prices were fully ‡exible, such that the marginal rate of substitution (MRS) is equal to the marginal rate of transformation (MRT). When there is monopolistic competition in a market, the markup on the prices set by …rms creates a wedge between MRS and MRT. This ine¢ ciency can however be completely o¤set by a proper subsidy. But even with such an optimal subsidy in place, production will in general di¤er from the e¢ cient level because prices and thus real wages do not fully adjust to changes in the marginal product of labour as shocks hit the economy. This leads to a suboptimal allocation of labour inputs and aggregate production level, which give monetary policy a motive for stabilizing the economy through responding to movements in the output gap. If, for example, a temporary positive demand shock occurs, production increases, and in consequence marginal costs go up and therefore also prices. But since prices cannot adjust instantaneously to the new marginal costs, monetary policy should increase the interest rate as a response to the positive output gap. A higher nominal interest rate increases real rates, thereby reducing demand and moving the economy faster back to steady state. This lowers the variability in the output gap and in‡ation. Price dispersion between …rms due to staggered price setting results in in‡ation because re-optimizing …rms choose a price that di¤ers from the average price in the last period. When …rms with equal production technology face di¤erent output prices, they produce at di¤erent output levels. This leads to an ine¢ cient goods combination, given that the consumers have a "taste for variety"6. The dispersion of production levels can be alleviated if monetary policy seeks to stabilize price in‡ation, because …rms would then have no incentives to change prices and hence produce at the same level. 6Preferences for variation in consumption is ensured through a constant elasticity of substitution function for the aggregation of the consumption good. 7 In the most basic New Keynesian model with no real rigidities, complete in‡ation stabilization is optimal because no ‡uctuations in prices automatically stabilize output as well. This is sometimes referred to as "the divine coincidence", namely that the trade-o¤ between in‡ation and output gap stabilization is absent when monetary policy stabilizes the marginal cost of …rms at a level consistent with their desired markup, given the prices in place. No …rms change their prices if they expect the policy to last forever, so the output gap remains closed and in‡ation stays at the target level. However, the presence of imperfect competition in labour markets and sticky nominal wages renders pure price in‡ation stabilization as suboptimal. Di¤erences in wages across workers lead to an ine¢ cient allocation of labour, making production e¢ ciency improvements possible. When wages respond slowly to macroeconomic distortions, the central bank should seek to stabilize a balance between price in‡ation, wage in‡ation and output gap movements, since they all lead to ine¢ cient resource utilization. Despite that, optimal policy can well be approximated by stabilizing a weighted average of wage and price in‡ation (in addition to the output gap), where the weights depend on relative price and wage stickiness (Galí 2008). This leaves (averaged) in‡ation and output gap as the two arguments entering the central bank’s loss function. In the staggered price-setting framework above, the slow response of prices and wages to economic disturbances is the central explanation for business cycle ‡uctuations. The resulting price dynamics creates a stabilizing role for monetary policy of the real economy beyond nominal variables. Movements in the policy rate lead to altered in‡ation expectations, but since agents know that prices and wages do not adjust immediately in proportion to the interest rate change, the real interest rate will be a¤ected, and thus consumption and investments, which in turn a¤ects employment and production. Albeit, in the long run all variables return to their steady state values or natural levels because wages and prices fully adjust, and monetary policy becomes neutral. Consequently, it is the presence of sticky prices and wages that makes e¢ cient responses to shocks and welfare enhancing macroeconomic policies possible: Even though nominal rigidities theoretically gives monetary policy a stabilizing role, it is, however, not obvious that it does so in the real world. But as Gali (2008) summarizes, there is ample evidence of both nominal rigidities and monetary non-neutrality from empirical studies and estimations. Signi…cant e¤ects of monetary policy on real variables have been found, which supports the theory that monetary policy can alleviate economic ine¢ ciencies that arises with nominal rigidities, in contrast to the ideas of the RBC advocates.7 7The e¤ects of monetary policy on real variables are of course di¢ cult to identify: changes in the policy rate are usually entirely or partly results of changes in the variables, and the causality thus goes in the opposite direction of the one to be estimated. 8 A low and stable in‡ation rate is also desirable for other reasons than the theoretical e¢ ciency arguments in the New Keynesian models. A small positive in‡ation rate renders real wage adjustments possible without adjusting nominal wages. It appears self-evident that frequent nominal wage increases accompanied by equal increases in the consumer prices are preferred by workers to a constant nominal and real wage (due to the "illusion of money" (Fisher 1928)). A positive in‡ation target also helps to prevent the nominal interest rate from hitting the "zero lower bound"8, in which communication about future policy intentions becomes especially important. By credibly committing to higher in‡ation rates in the future and thereby in‡uencing expectations about future consumer prices, the central bank stimulates the economy today and thus mitigates the e¤ects of the lower bound. When equilibrium in‡ation is positive, the nominal interest rate supporting this equilibrium will also be positive, because it is the real interest that a¤ects economic agents’ decisions, and depending on the size of this equilibrium real rate, there will be more room for monetary stimuli in economic downturns. Yet, high in‡ation rates could create macroeconomic instabilities by causing large and varying in‡ation expectations, rent-seeking and irrational behavior, uncertainty about the conduct of monetary policy, and undermine the role of money, etc. Some in‡ation is preferable, but there is no consensus about the optimal level. The goal of Norges Bank is to keep the growth in the consumer price index in the medium run at 2.5 percent, but other central banks have di¤erent in‡ation targets, most of them below the Norwegian. The European Central Bank, Sveriges Riksbank and Bank of England all have goals of 2 percent. A 2 percent in‡ation target is commonly viewed to be su¢ cient to avoid signi…cant macroeconomic instabilities of the type explained above (Taylor and Williams 2010). 2.3 Simple interest rate rules "Why does the Bank make things so complicated? Why doesn’t it just follow the Taylor rule?" [Interruption by a distinguished macro economist at an American university, when Lars E. O. Svensson was presenting Bank of Sweden’s approach to in‡ation targeting.] (Svensson 2000: p. 1) Setting the policy rate is a demanding engagement. There are often several con‡icting considerations in the evaluation procedure, and it is important to have a consistent and comprehensive way of doing it. There are two main approaches to the conduct of monetary policy: Optimal policy or simple instrument rules. Optimal policy refers to when the central 8A negative nominal interest rate is obviously not possible for more than very short periods of time, as it could lead to a collapse of the bank sector if all agents withdrew their deposits. 9 bank sets the policy rate in order to minimize a speci…c objective, like Equation 2 below. Simple instrument rules are, as the name suggests, easy rules of thumb to be followed by the monetary authority. The simple rules consist of a few selected variables that the nominal interest rate should respond to ‡uctuations in, and can hence be implemented without further knowledge about the economy’s functioning or the realization of shocks. The rules are thus robust to all kinds of uncertainties, in particular model uncertainty, which is also the main advantage of such rules. In addition, they make monetary policy transparent and predictable due to their simple structure. The original and famous "Taylor rule" (Taylor 1993) is a classical example of a simple interest rate rule. The rule dictates that the nominal interest rate moves 1.5 percentage point when in‡ation deviates one percentage point from the in‡ation target and 0.5 percentage point when output deviates one percentage from its trend or potential level: rT t= 1:5t+ 0:5yt, (1) where tis year-on-year in‡ation and ytis the output gap, and the targets for in‡ation and output are set to zero for simplicity. As highlighted by Taylor, the crucial feature of the interest rate feedback rule is that it dictates monetary authorities to "lean against the wind" in the sense that if the output gap is positive or in‡ation rises above target, the nominal interest rate is increased in order to dampen economic overheating. The principle that the nominal interest rate should react more than one-to-one to in‡ationary movements in order to avoid indeterminacy of the price level, has become known as "The Taylor principle" (Woodford 2001). This principle should always be followed when instrument rules are used to set the interest rate. For example, for an increase in demand without any change in economic fundamentals (e.g. a sunspot shock), production and marginal costs increase, creating in‡ationary pressure. If the nominal interest rate did not respond enough to the rise in in‡ation, the real interest rate would actually go down and the e¤ects on the economy would be the opposite of the intended, justifying the initial increase in consumption. Therefore, monetary policy must be designed so that this type of self-ful…lling prophecies cannot happen. Taylor (1993) and others (e.g. Clarida et al. 2000) argues that the Fed did not follow the Taylor principle before Volcker was appointed as Chairman for the Federal Reserve in 1979, and that this is the reason for the great in‡ation and macroeconomic instability in the USA in the 1960s and 1970s. Responding to in‡ation and output solely though, without taking all other available information into consideration, may seem too simple and sub-optimal. This is indeed the case, but the two variables are such good indicators of the state of the economy that reacting to 10 movements in them yields su¢ cient macroeconomic stabilization. Taking the robustness and transparency qualities of simple policy rules into account, they seem like a good alternative to optimal monetary policy. It has been shown that simple rules do describe actual monetary policy in most countries quite well and …t data in a satisfying manner (Kuester and Wieland 2010), which was actually the main goal of rule (1). Taylor’s famous paper was meant as a contribution to the discussion of optimal monetary policy with discretion versus using a time-invariant policy rule. The proposed rule is able to explain the Federal funds rate movements remarkably well from the 1970s to 1992. Thus, the policy conducted by the Fed could be approximated by such a simple rule. It was, however, not intended as a rule for the actual conduct of monetary policy, and it is unlikely that policy makers literally follow the simple rules. But they anyhow work well as a benchmark for assessing monetary policy. 2.4 Optimal monetary policy In‡ation targeting could well be described as optimizing behaviour of the central bank, in the sense that it takes all relevant information into account and responds with the policy rate to shocks in order to minimize the expected value of some objective. The central bank commits to the prescribed policy for all future periods, and acts as if it made the commitment a long time ago by treating all periods equal. This is referred to as conducting optimal monetary policy in a timeless perspective.9 The objective function for most in‡ation targeting central banks consists of the unconditional variance of in‡ation, output gap and change in the nominal interest rate: Lt=Et[(t)2+y(yty)2+r(rtrt1)2], (2) where is the in‡ation target, yis output target or trend, and rtis the nominal interest rate. Etis the mathematical unconditional expectation operator, and yand rrepresent the central bank’s relative preferences for output gap stabilization and interest rate smoothing, respectively. The arguments entering the loss function and the relative weights assigned each of them is frequently debated in the literature, but most seem to agree that some concerns for output and interest rate smoothing should be included in addition to in‡ation. As argued above, 9I refer to the loss generated by optimal policy as "Ramsey loss" in order to distinguish it from "optimal simple rule loss", even though the policies are not the same. The di¤erence is that Ramsey policy exploits the gain in the …rst period after a shock by promising to overshoot the in‡ation target in future periods and thereby reduce current in‡ation due to the forward-looking behaviour of price setters. The losses are however the same when they are calculated from theoretical variances and the discount factor of the central bank is unity, such that all periods are treated equal, which is what I use. 11 the preferences of the monetary authority should be to maintain low in‡ation variability and a steady output growth in order to secure a stable economic environment, and it appears natural to set the policy rate in a manner that minimizes deviations of in‡ation and GDP away from target or trend. Including the output gap has been found to be necessary in order to attain robust monetary policy (Levin and Williams 2003), which is the goal of this thesis. Consequently, excluding output from the loss function appears to be counterproductive.10 By including the variability of the change in the nominal interest rate in the loss function, the central bank ensures that the interest rate is moved gradually and with caution, which is preferable when there is general uncertainty about the structure of the economy and the e¤ects of monetary policy (Levin et al. 1999).It appears intuitively correct that economic agents prefer predictability in the direction of movements of the nominal interest rate, rather than frequent small ‡uctuations around a constant level, which could be the case if the level of the interest rate entered as an argument11. Such frequent adjustments in alternate directions may cause policy makers to appear uninformed and whimsical, and undermine the role of monetary policy as an anchor for nominal variables. As explained in Holmsen et al. (2008), using the …rst di¤erence of the policy rate delivers reasonable paths for the nominal interest rate: "paths that do not look unacceptable to the policy maker at …rst glance". In addition, Svensson (2000) …nds that the case of strict CPI targeting does not converge unless a small weight on interest rate smoothing is added to the loss function. Further, interest rate smoothing12 leads to higher macroeconomic stabilization by providing a better control over long term rates which are the main driving forces of in‡ation expectations, actual in‡ation and production. When movements in the policy rate are expected to be long-lived, as in the case of gradualism in interest rate setting, the e¤ect on long term rates is signi…cantly larger. A smooth interest rate path makes real variables move in an even fashion, preventing undesirable jumps in consumption and labour. Besides, substantial and sudden changes in the interest rate amplify …nancial market volatilities, which may lead to instabilities in the real economy. Taylor and Wieland (2009) and many others include the change in the nominal interest rate in the monetary policy objective. Because 10 Taylor and Wieland (2009) and Kuester and Wieland (2008) …nd, however, that the insurance against model uncertainty can be done at a relatively low cost even if the objective function does not contain any output stabilization concern. 11 It is, of course, possible to include both the level of and change in the nominal interest rate. This is in fact done in Norges Bank’s most recent monetary policy report (Norges Bank 2012), in order to mitigate the risk of a buildup of …nancial imbalances, where the deviations from a "normal" interest rate level is one of the arguments. 12 I shall use the term "interest rate smoothing" to refer to a policy reaction function where there is a high degree of persistence in the nominal interest rate, i.e. that the coe¢ cient on the lagged interest rate is large, and not as a policy that leads to a smooth interest rate path of other reasons. 12 low interest rate volatility is found in data, preferences for interest rate smoothing by policy makers appears to be justi…ed empirically (Levin et al. 1999). For all these reasons it is wise to conduct monetary policy with caution and "stodginess", it being due to preferences for interest smoothing in a loss function or used in a monetary policy reaction function. 2.5 Robustifying monetary policy In order to avoid disastrous outcomes and reduce the risk of total failure of monetary policy, but without switching from optimal policy to a simple interest rate rule, it is possible to robustify optimal policy by extending the loss function with a fourth argument, namely deviations from the interest rate implied by a simple instrument rule (Ilbas et al. 2012): Lt= (1 )((t)2+y(yty)2+r(rtrt1)2) + (rtrSR t)2, (3) where rSR tis the nominal interest rate that would have been set if the central bank followed the simple rule SR. This rule would be a type of rule like Equation 1, which does not perform outstandingly in any single model, but does on average quite well and thus serves as an insurance against model uncertainty. Optimal policy and simple instrument rules are combined in a framework that robusti…es monetary policy. As argued in Ilbas et al. (2012), this modi…ed loss function is realistic, ‡exible and sensible. Central banks can easily adopt to it and there are no limits to which or how many simple rules that can be included. The last term in (3) could easily be substituted or modi…ed when new knowledge about robust simple instrument rules is obtained or as experiences about which rules best serve the robustness purposes develop. The authors further argue that simple rules are not realistic since central banks rarely follow them in a strict manner. Conducting optimal monetary policy gives no room for simple rules which are proved to be a lot more robust. Thus, a combination of the two monetary policy approaches appears to be a good solution. Robustness is a big topic within monetary policy research, and there are numerous papers discussing di¤erent ways of insuring against mis-speci…cations, and model and parameter uncertainties. However, the method developed by Ilbas et al. (2012) is as far as we know the only approach combining optimal policy with a simple interest rate rule. The large literature on parameter uncertainty and model mis-spe…ciation supplies ways of robustifying when there are uncertainties about details of a speci…c model, but not if the true economy is completely di¤erent. As Levin and Williams (2003, p. 958) notes: "...the results suggest that simply designing a rule to be robust in the neigh- 13 borhood of a given reference model does not ensure that the rule will perform robustly in competing reference models." Therefore, by extending the loss function with a small weight on deviations from a simple interest rate rule, the central bank insures against uncertainties of several forms, without sacri…cing much. In this thesis I search for the best simple rule(s) to modify the loss function of Norges Bank with. The rule(s) I end up with should in particular do well in the models that Norges Bank seeks to insure against disastrous outcomes in, i.e. NAM, as optimal NEMO policy yields instability in this model (Mathis Mæhlum 2012). 3 Models Following is a description of the models used in this thesis. Since it is part of a larger project on which my fellow student Mathis Mæhlum also works, we have cooperated on this chapter. He has written the parts describing the LGM model and NAM, and I have written about the NEMO models. He does however not consider "Policy NEMO", so this model is only included in my analysis. I hence use four main models, but extend to …ve whenever appropriate. Credit NEMO, NAM and the LGM model are those I refer to as "reference models" because they are the models I compare NEMO with. The models di¤er in many important aspects, although four are New Keynesian and all are DSGE models, albeit with highly di¤erent speci…cations. Lower case letters represent the log-deviation from a variable’s steady state value, except from growth rates and interest rates. 3.1 NEMO The Norwegian Economy Model, NEMO (Brubakk et al. 2006), is a New Keynesian DSGE model used by Norges Bank for policy evaluation and forecasting . It is a model of a small open economy consisting of two countries, home and foreign, interpreted as Norway and its trading partners, and two sectors, one producing intermediate goods and one producing a single …nal good. The model economy is a representation of the Norwegian mainland economy, with the petroleum sector entering as an exogenous process for oil investments. The foreign economy is modelled symmetrically to the home economy, but enter as exogenous variables, such that Norway has no in‡uence on its trading partners. All variables in NEMO are detrended with a common stochastic growth trend. We use a …rst-order Taylor approximation of the model. 14 The economy consists of a continuum of in…nitely lived households that are divided into two types, savers and spenders, who both supply labour services to the intermediate goods sector. The share slc of spenders are rule of thumb consumers and spend their total labour income every period, whereas the share (1 slc)of savers have access to a credit market and choose consumption and saving plans that maximize expected utility over the lifetime subject to a budget constraint, which leads to the following Euler equation: csa t=f191Etcsa t+1 +f192csa t1f193Etfrtt+1g  f194Z t+f195zU t, (4a) where csa tis consumption done by savers, Et(rtt+1)the real interest rate, Z ta shock to the growth trend and zU ta preference shock raising the marginal utility of consumption relative to leisure. Savers are forward-looking and wish to smooth consumption over time, and due to habit persistence, current consumption also depends on last period’s consumption. A temporary rise in growth reduces the value of (detrended) consumption and households thereby postpone consumption. Forward-looking households invest in domestic and foreign bonds, receive all dividends from …rms, pay lump sum taxes and set nominal wages taking …rms’labour demand into account. They have some degree of monopoly power in the labour market such that the resulting wages are above the competitive wages, whereas spenders receive the average wage rate of the savers and simply supply the amount of labour demanded from them at this wage. There are quadratic costs of adjusting wages that makes wage growth, w t, respond sluggishly to shocks, which thus depends on past and future wage growth, deviations of the actual wage from the optimal wage (equal to the marginal rate of substitution between consumption and leisure), (wtmrst), and the degree of bargaining power represented by the substitution elasticity between labour inputs, !t: W t= 1 + EtW t+1 +1 1 + W t1f231(wtmrst)f232!t. (4b) Production of the …nal good, A, is done using a combination of imported and domestically produced intermediates, respectively M and Q, with the shares given by the degree of "home bias", i.e. the relative preferences for input factors produced in the home economy. The …nal good is used for consumption, C, capital investments in the intermediate sector, I, government spending, G, and oil investments, IOIL. The only source of imports in the economy are the imported intermediate goods, T*, and exports consist purely of domestically produced intermediate goods, M*. In the intermediate goods sector, monopolistically competitive …rms produce di¤erentiated goods tt, utilizing capital services, kt=ut+kt1Z t, and labour in a Constant 15 Figure 1: The production structure of NEMO. The …gure is taken from Brubakk et al. (2006). Elasticity of Substitution (CES) production function: tt=f61(lt+zL t) + f62kt, (4c) where zL tis a labour augmenting productivity shock that temporary increases the level of production, and drives the exogenous total factor productivity. The amount of capital services depends on the capital stock and the utilization rate, whereas the stock itself is determined by investments done one period earlier and capital depreciation. There are convex adjustment costs of changing both the level of the investment to capital ratio, (invtkt1), and the rate of change in this ratio, which together with variable capital utilization and habit persistence are the real rigidities in NEMO. The investment to capital ratio is thus a slowly moving variable that reacts positively to increases in the expected real return to capital, EtrK t+1, and negatively to the expected real interest rate since it reduces the discounted value of the return. A somewhat simpli…ed version of the investment Euler equation: invtkt1=f111(invt1kt2)+f112Etfinvt+1 ktgf113Et(rtt+1)f114rK t+1+shockinv t. (4d) Intermediate …rms set prices as a markup above the competitive price, and prices respond sluggishly to shocks due to convex adjustment costs à la Rotemberg (1982). In‡ation on intermediate goods, Q t, increases with real marginal costs and decreases with a cost push shock represented by the substitution elasticity between the domestically produced intermediate goods, H t, by the following Phillips curve: 16 3.4 NAM The Norwegian aggregated model (NAM) is a quarterly macroeconometric model developed speci…cally for the Norwegian economy by Bårdsen and Nymoen (2001), Bårdsen, Jansen, and Nymoen (2003), and Bårdsen (2005). The version used in this thesis is the one documented in Bårdsen and Nymoen (2009). As opposed to the other models we consider, it does not assume that the economy is a system in general equilibrium and there are no forward-looking rational agents modelled. Instead, di¤erent parts of the economy are modelled separately, relying partly on theory and partly on data to identify the relevant variables in each Equation. The model is formulated in error correction form. First, starting from a general vector autoregression, a cointegrating relationship between variables in levels is identi…ed as a longrun steady state. Then the short-run dynamic structure is estimated, using the long-run relationships as error correction terms. When the system is out of equilibrium, i.e. when the long-run relationship between endogenous variables does not hold, this cointegrating term will make sure that the relevant variables move back towards their long-run values. The model can be written on the form: yt=+ j X i=1 iyti+ k X i=1 iyti+ut;(6) where ytis a vector of (logged) endogenous variables, is a vector of constants, iand i are parameter matrices 8i, and utis a vector of error terms. Here the second term on the right hand side is the error correction term, which in each equation describes a cointegrating relationship between the left hand side variable and a linear combination of other variables. The short run dynamics is described by lags of di¤erenced variables. The model consists of equations for the wage, prices, productivity, output, unemployment, household credit, money market interest rates, and the nominal exchange rate. Wages are modelled in a Nash bargaining framework meant to capture the high degree of coordination in Norwegian wage setting. In the long run nominal wages will move one-for-one with the general price level and productivity, and it will also depend to some extent on the unemployment rate. Domestic prices are set by …rms engaged in monopolistic competition. Thus the general price level will in the long run depend on wages relative to productivity, as well as imported prices. Long run equilibrium unemployment is determined by the growth of the real wage, as well as the real interest rate and output. The long run behavior of the nominal exchange rate is derived assuming that expected depreciation depends on deviations of the exchange rate from its long-run value, and that there is a constant long run risk premium in the foreign exchange market. Movements in relative real interest rates do not lead to one-for-one changes in the real exchange rate. 23 Total production is in the long run determined to a large extent by government demand, which in the original system is exogenous and will be assumed constant in our model (see below). In addition, depreciations of the real exchange rate and decreases in the real interest rate both a¤ect output positively in the long run. In the short run, output growth is sig- ni…cantly a¤ected by its own lag, changes to government expenditures and changes in real credit. The latter e¤ect may be due to frictions in the credit market. The growth of real credit is in turn determined in the long run by the growth of output and - to a smaller extent - by interest rate di¤erentials. Since output a¤ects credit and vice versa, there is a simple …nancial accelerator mechanism at work. Labour productivity depends in the long run both on real wages, the unemployment rate and a linear trend. In the short run it is a¤ected by the change in real wages. Most of NAM is estimated equation-by-equation using OLS, but the wage and price block is estimated as a system with full information maximum likelihood. Identi…cation of the system is achieved by means of theoretical and ad hoc overidentifying restrictions on the short run dynamics. Seasonal dummies are added for better …t. The original model’s longrun growth is driven by neutral technological progress, approximated by the linear trend in labour productivity. Simulations show that the model induces a constant (apart from seasonal variations) growth rate of output, nominal wages and prices, and constant unemployment rate and nominal exchange rate in steady state (Bårdsen and Nymoen 2009, p. 879-883). In order to make numerical simulations of the model tractable by making also nominal variables stationary, we remove all trends and constant terms so that all variables are zero in steady state. The original model can be viewed as a log-linearization. Under this interpretation, the variables in our modi…ed model will be interpreted as deviations of the actual (logged) variables from either a deterministic balanced growth path (for some variables, such as the output gap and productivity) or constant steady state values (for other variables, including the in‡ation rate and the unemployment rate). This corresponds roughly to the log-linearization used to make NEMO and Leitemo stationary, and we will thus interpret the relevant variables in the same way across models. NAM contains several exogenous variables, including the oil price and foreign variables such as consumer prices and interest rates. This poses a problem for our simulations. Instead of assuming dynamic processes for all these variables, we set the domestic exogenous variables equal to zero (their steady state values) in all periods. This is clearly unrealistic, and it means that the total variation in the endogenous variables will be smaller than what is observed in the data. However, we do not want to change the original model dynamics in any important ways by adding new equations, and thus this approach is the most convenient for our purposes. As for the foreign variables, we tried to model these in the same way as 24 in NEMO, but the AR(1) process for foreign in‡ation created stability-problems in NAM, leading to in…nite variance of several important variables, including the domestic in‡ation rate. For this reason, we let foreign in‡ation be constant, but model the foreign interest rate as in the other models. Because Dynare has problems in dealing with models in which some variables have in…nite variance - which is the case for the nominal prices in NAM - we use a stationarized version when calculating optimal policy rules. In this version, growth rates of price variables and the cointegrating relationships are de…ned as new variables. 4 Transmission mechanisms of monetary policy Generally in a small open economy like the Norwegian, changes in the policy rate works on in‡ation through three main channels: the demand channel, the exchange rate and the expectation channel.16 When the nominal interest rate increases, the real interest rate also increases due to sticky prices and wages. A higher real interest rate reduces investments done by …rms through lower pro…ts and availability of credit; it decreases households’demand through a higher relative price of current consumption to future and lower income of households with negative net savings. This reduces production and prices set by …rms due to lower marginal costs, slowing down price growth by the demand channel. Increased Norwegian interest rates raises the attractiveness of investments in NOK and drives up the price, leading to an appreciation. A lower exchange rate, meaning a higher value of NOK, increases imports and reduces exports, which lowers production and in‡ationary pressure in domestic …rms, adding to the demand channel. Total in‡ation is dampened by reduced prices (in domestic currency) on imported goods, both directly through the e¤ect of lower imported in‡ation on consumer goods and indirectly through prices set by domestic producers utilizing imported input factors. These two e¤ects constitute the (direct and indirect) exchange rate channel to in‡ation. The expectation channel works through the e¤ects of all forward-looking variables on current in‡ation by changing agents’expectations about future economic developments. When …rms expect smaller price increases because of higher real interest rates, they set lower prices today due to adjustment costs. Firms decrease investments when future return to capital is expected to be lower, thereby reducing wage pressure, and hence production costs and in‡ation. Households reduce current demand when they expect future consumption to be lower, due to consumption smoothing, contributing to the fall in in‡ation. Expectations are also very important in the determination of the exchange rate, since this 16 See http://www.norges-bank.no/en/price-stability/in‡ation/e¤ect-of-interest-rate-changes/. 25 is a forward-looking variable. Expectations give monetary policy a way of in‡uencing in‡ation that works with a shorter lag than the demand channel, since forward-looking variables respond immediately to policy changes. 510 15 20 25 30 -15 -10 -5 0 5x 10 -3 inflation 510 15 20 25 30 -0.06 -0.04 -0.02 0 0.02 output 510 15 20 25 30 -0.5 0 0.5 1interest NEMO 510 15 20 25 30 -15 -10 -5 0 5x 10 -3 inflation 510 15 20 25 30 -0.06 -0.04 -0.02 0 0.02 output 510 15 20 25 30 -0.5 0 0.5 1interest Credit NEMO 510 15 20 25 30 -0.1 -0.05 0 0.05 0.1 inflation 510 15 20 25 30 -1 -0.5 0 0.5 output 510 15 20 25 30 -0.5 0 0.5 1interest NAM 510 15 20 25 30 -0.03 -0.02 -0.01 0 0.01 inflation 510 15 20 25 30 -0.1 -0.05 0 0.05 0.1 output 510 15 20 25 30 -0.5 0 0.5 1interest LGM Figure 2: Impulse responses (in percent) of year-on-year in‡ation, the output gap and the quarterly annualized nominal interest rate to a one percentage point temporary increase in the latter when monetary policy follows the original Taylor rule. Periods along the horizontal axis are quarters. In order to visualize the e¤ects of monetary policy on the key variables, Figure 2 displays the impulse response functions (IRF) to a one period one percent increase in the nominal 26 interest rate in period t=0 of each of the four main models when the original Taylor rule is applied: rt= 1:5t+ 0:5yt. In all models in‡ation decreases less than output, and is more persistent, although the magnitude and length of the disturbance varies substantially across the models. Output in NEMO falls immediately nearly 0.06 percent below trend and rises gradually for two years until it reaches the steady state growth path. In‡ation reaches the minimum after one year and then returns to target after three and a half years after the nominal interest rate shock. Due to nominal rigidities, a higher nominal interest rate leads to a higher real interest rate, reducing investments, consumption and the exchange rate. The lower demand and investments reduces production, leading to a downward pressure on prices and nominal wages through lower marginal costs. Real wages fall as well, further contributing to the demand channel of monetary policy, but it is dampened by reduced utilization of capital causing marginal costs to rise. The exchange rate channel works through the e¤ect of a real appreciation following the higher interest rate on lower prices of imported intermediaries and higher export prices. Both imports and exports are reduced — the former because of reduced demand and investments despite the lower import prices causing a substitution towards foreign inputs. Lower demand will in turn lead to lower export prices, and exports pick up again after the initial decrease. The expectation channel is present through the e¤ects on expected future marginal costs. CN has relatively similar responses to a monetary policy shock as NEMO, although the variables move in smoother way and the e¤ects are slower and last longer. The persistence of the disturbance is due to the …nancial accelerator. The additional credit channel present in this model is due to the fact that a higher real interest rate leads to lower house prices through lower demand and investments, reducing the collateral value of the credit constrained households and thereby dampening their demand. Lower housing prices also reduces consumption of the patient households, further dampening GDP growth. In NAM the variables have more high frequent variation than in the other models, but the duration of the disturbance to the variables is roughly the same as in NEMO, albeit a more hump-shaped and larger e¤ect. An increase in the nominal interest rate leads to an immediate nominal and real appreciation of the NOK which a¤ects domestic prices and wages through decreased import prices. This is the direct exchange rate channel to in‡ation, but there is also an indirect e¤ect in the demand channel through lower competitiveness of exporting …rms following an appreciation, and therefore lower GDP and higher unemployment. Due to lower in‡ation, the increase in the nominal interest rate is translated into an even bigger increase in the real interest rate17 which a¤ects production with a lag and unemployment 17 The real interest rate in NAM is de…ned as the di¤erence between the nominal interest rate and current 27 with two lags, contributing to the demand channel. A third channel is the credit channel working through the e¤ects of the interest rate on output due to reduced availability of real credit. In NAM there is no expectation channel since the model is purely backward-looking. The Leitemo-Gali-Monacelli model appears in Figure 2 to be the most persistent model, but this is just an artifact of the policy rule used, which is not hard-hitting enough to stabilize the economy as good as in the other three models. Due to the rule, output falls substantially more, and moves slowly in dampened cycles back to the long run equilibrium. The monetary policy shock immediately reduces demand and production, which lowers domestic in‡ation through the demand channel with a lag. The expectation channel works through expected future in‡ation that lowers current in‡ation, and the complicated lag structure causes GDP and consumer prices to move in a hump-shaped fashion. A higher nominal interest rate decreases the exchange rate, creating a negative law-of-one-price gap, ceteris paribus. But the reduced terms of trade reduces the gap because importing …rms lowers prices on imported goods, further reducing the output gap and in‡ation. 5 Method The program used for the technical part of the analysis is MatLab with the software Dynare. Dynare solves the models using a …rst order approximation around (a zero) steady state and calculates the theoretical variances of the endogenous variables. In order to …nd the …rst-best policy reaction function of a speci…ed form, the variances of the variables entering the objective function is minimized. The size of the shocks in the models are given by their standard deviation, and the resulting value of the objective function, or loss, is the sum of the unconditional variances of the included variables. The estimated size of the standard deviations of identical shocks could be di¤erent across models because it depends on the other shocks included in the estimation. If a model has few shocks, each of them will play a signi…cant role in explaining the variability in the data, but as the number of shocks increases, the relative importance of each shock decreases. The standard deviations determines the size of the loss, as more variability in the exogenous variables in the simulations naturally leads to larger ‡uctuations in the endogenous variables. This can indeed be seen in the results, where NEMO yields losses of a magnitude far above the other models, and NAM quite a lot below. CN is the model with the greatest number of shocks (17), but has the second largest loss conducting optimal monetary policy. NEMO has 15 shocks, NAM 11, and the LGM model only 7 estimated shocks. It may seem that the shocks included in NAM have less explanatory power than the shocks in the other models, in‡ation: t=itt: 28 and therefore estimated to be smaller, but as explained in Chapter 3.4 about NAM, the small loss is partly due to the fact that we have removed some of the original exogenous variables, in particular government spending, which is an important driver of short run ‡uctuations in GDP. We have, of course, also removed the interest rate shock in all models, and since the models are estimated with di¤erent numbers of shocks, the variability explained by the excluded shock di¤ers. Hence, the removal of it could possibly explain some of the di¤erence in the size of losses. Below is a presentation of the simple interest rate rules in this thesis, both …rst-best rules and Bayesian rules, and an explanation of how they are derived. Bayesian rules are relevant for my analyses because they are found to be more robust than simple rules optimized within a single model. The chapter ends with a discussion of the di¤erent loss function speci…cations. 5.1 Optimal simple rules To …nd the …rst-best simple interest rate rules in a model, the following loss function18 Lt=Et[(t)2+y(yty)2+r(rtrt1)2] is minimized in order to …nd the coe¢ cients r,;,yand y1in the following equations: rt=t+yyt;(Rule 2) rt=rrt1+ (1 r)(t+yyt), or (Rule 3) rt=rrt1+ (1 r)(t+yyt+y1yt1). (Rule 4) Here tis the current year-on-year in‡ation rate and ytis the output gap. The resulting value of the loss function is what I refer to as "osr loss" (optimal simple rule loss), denoted by Losr, and the rules as the two-, three- and four-parameter interest rate rules, respectively. Rule 2 is of the same form as the classical Taylor rule, capturing the important aspect of central banks’tendency to "lean against the wind", but not allowing for responses to further information. Rule 3 incorporates gradualism in the adjustments of the policy rate, as this is found in data and therefore appears to better represent actual monetary policy than Rule 2, as in Levin et al. (1999). Following Taylor and Wieland (2009), I include Rule 4, where more lagged information is used for monetary policy decisions. I also consider two variations 18 The loss function is the same as for optimal monetary policy, but is minimized subject to the simple policy rule. In optimal policy the unconditional variances are minimized by reacting in a highly complex manner with the interest rate to economic disturbances, whereas it is restricted to respond only to a limited set of variables in the above optimization. 29 of Rule 3: One where the lagged interest rate has been replaced by the lagged output gap, and one where it has been replaced by lagged in‡ation. The values and yrepresent the long-run response of the nominal interest rate to deviations in in‡ation and output from trend or target, i.e. when the smoothing of the interest rate is completed, and are thus the "true" responses. They are the values I report in the tables, if nothing else is stated. The short run coe¢ cients are attained by multiplying the bracket in front of the long run rule with the coe¢ cients in this rule, and indicate how much the central bank reacts in the same period as deviations is observed. The rationale for not including forward-looking variables, only contemporaneous and lagged variables in the interest rate rules, is that the former are highly model dependent and should not be used in robustness analyses. Basing the policy decision in a model on a future variable naturally implies the use of a forecast within the model, and transferring a rule with a model-speci…c forecast to another model in order to analyze the rule’s performance would not be fully valid since the variable that the rule responds to would be di¤erent across the models. Also, a forward-looking rule may be less robust due to the fact that it uses more model-speci…c information (Brubakk and Natvik 2010). Contemporaneous values of in‡ation and output gap are, of course, di¢ cult to observe (in particular the latter which may not be revealed until many years later when the trend has been properly estimated) and must be estimated. If the models are used to …nd the values, the same argument as for future variables would hold true. But in Norges Bank they use SAM (System of Aggregated Models)19 for nowcasts, so real time observations are done outside the model, and can hence be utilized in an instrument rule. 5.2 Bayesian rules The optimal Bayesian interest rate rules is found by combining the three competing models20 Credit NEMO, the Leitemo-Gali-Monacelli model and NAM into one model script, and minimizing a weighted sum of the losses generated in them when using the rule found to be optimal for the "model average": 19 SAM has since 2008 been used to produce contemporaneous and short run CPI and GDP forecasts, and has signi…cantly improved forecasting in Norges Bank (Jore 2012). It consists of 167 di¤erent models, including NEMO and a slightly di¤erent version of NAM as the only two structural models. But their importance in the nowcasts are minor due to the large number of models, so the use of current variables is still valid. 20 NEMO is not included because it would lead to a "false" picture of the performance of the Bayesian rules in this model, as the rule would do arti…cially well by construction. The evalutation of a rule based on only the three altrenative models is more clear-cut for robustness purposes. 30 LBayesian;abs t=NAM LNAM t+LGM LLGM t+CN LCN t:(Bayesian "absolute loss" function) Here NAM ,LGM and CN are the probabilities assigned by the monetary authority to each of the models, and LNAM t,LLGM tand LCN tthe resulting loss from applying the optimized Bayesian rule in the seperate models. The form of the Bayesian loss function is clearly a subject for discussion. First of all, it is not obvious that the sum of the seperate loss functions is the proper form, it could just as well be a quadratic or some other function. Apart from that, the relative weights depend on both the subjective beliefs about the probability of the di¤erent models and on how the losses of these models are scaled. The losses of each model could either enter with their absolute size (in level) or be normalized in some way in order to have a common reference for the performance of the policy rule in the optimization, since they are of such di¤erent magnitude, as already discussed. Because NAM yields signi…cantly smaller losses than the two other models for all policies, using absolute losses could be viewed as "unfair" to this model, which is seen from the optimized rules in Chapter 6.2. It is natural to question whether the losses are the "true" ones when we have removed some variability in the models and changed a few of the exogenous processes. They may not have the same interpretation and are thus inappropriate as arguments in the objective function, LBayesian;abs t. But on the other hand, the losses are supposed to be expressions of the actual harm caused by macroeconomic ‡uctuations, and it could thus be argued that they correctly represent the preferences of the central bank. Another approach is to weight the Bayesian objective with the loss in the seperate models generated by the models’…rst-best simple rules of the same form as the Bayesian rule to be found, LModel;osr;to re‡ect how well the Bayesian rules do relative to the models’own rules. NAM now receives relatively more importance in the optimization, and the Bayesian rule using this osr-loss-weighted objective is more similar to the rules from NAM than with the absolute size of the losses.21 LBayesian;osr t=NAM LNAM t LNAM;osr +LGM LLGM t LLGM;osr +CN LCN t LCN;osr :(Bayesian osr loss) I have looked at four di¤erent combinations of relative weighting of the models: First 21 A third alternative would be to scale the Bayesian objective with the individual Ramsey losses, re‡ecting concerns about the damage of using a Bayesian rule instead of conducting optimal monetary policy in each model. As will be seen, the di¤erence between Ramsey losses and osr losses are roughly of the same magnitude across the models. Using osr losses appears to be more intuitive since I am looking at the use of simple interest rate rules, and I therefore present results with the osr losses only. 31 NAM =LGM =CN = 1=3and then 1/6 for two of the models and 2/3 for the last, for each of the three possible combinations. Optimized coe¢ cients are reported in the next chapter. This type of rule is commonly referred to as a "Bayesian rule" in the literature, although the term is somewhat misleading as it rather is a "model averaging" rule than "Bayesian" in the original meaning. The weights assigned to each model are …xed "priors" and are not updated using Bayes’rule when policy makers acquire additional knowledge about the probability distribution of the models. Assuming that beliefs are never updated is highly debatable. It is natural to believe that new information about the structure of the economy will a¤ect the priors, and the rules should in consequence be changed.22 Nevertheless, the Bayesian rules can be interpreted as an upper bound of the insurance possibilities, when the central bank commits to the prescribed policy and never updates the relevance of the competing models. 5.3 Di¤erent loss functions The variables entering the loss function are the unconditional variances of yearly in‡ation, output deviations relative to trend and the change in the nominal interest rate. As the models are quarterly, the prescribed interest rate rules are also in quarterly terms and had to be converted into yearly terms, either year-on-year or annualized. Year-on-year in‡ation is less noisy and appears to be a better variable for policy making than quarterly annualized in‡ation which includes a lot of short run ‡uctuations that are impossible and undesirable to mitigate23. The optimal coe¢ cients in the simple interest rate rules increase when quarterly in‡ation is used, since the central bank then needs to react more aggressively. The goal of monetary policy is after all to ensure macroeconomic stabilization, so responding to large shocks that moves the economy substantially away from the trend, and not to short run ‡uctuations, seems natural. I therefore choose year-on-year in‡ation in the objective functions and simple interest rate rules, in line with the existing literature (e.g. Taylor and Wieland 2009, Taylor 1993, Levin et al. 1999). I consider three variations of the loss function in Equation 2, which I refer to as L1,L2 and L3. The …rst is the benchmark loss function that I present throughout the thesis. The 22 Thanks to Lars Svensson for pointing this out to me. 23 Looking at the autocorrelation coe¢ cients con…rms this. All models have a high degree of persistence in yearly in‡ation when optimal monetary policy is conducted, around 0.8-0.9, but the correlation between two succeeding quarters is substantially lower. In NAM the autocorrelation coe¢ cient decreases from 0.85 for yearly to neglible 0.0014 for quarterly, and in the LGM model from 0.8 to 0.3. The decreases are large in Credit NEMO and NEMO as well, 0.11 and 0.29, respectively. Thus, quarterly in‡ation ‡uctuates more randomly and is harder to stabilize than year-on-year in‡ation in the set of models. 32 not worth considering if the central bank believes in the set of models used in this thesis, but rather to the historic output gap if NAM is attached a high probability. For optimal simple instrument rules within each seperate model, it appears that threeparameter rules are quite good. But as will be seen in Chapter 7.1, they do poorly in other models and are hence little robust. Instead Bayesian rules should be considered for robustness purposes, because they accommodate the con‡ict between the preferred policies in the models. 6.2 Bayesian rules Below are interest rate rules prescribed by averaging over the outcomes of the three reference models, NAM, the LGM model and CN. Table 5 display the two, three and four-parameter optimized Bayesian rules with equal relative weights attached to the models, both for absolute and relative losses in the objective function. Again, the di¤erences in loss between four- and three-parameter rules are relatively small (albeit somewhat larger than the di¤erence between two and three for the "osr loss"-rules due to NAM), and the smaller rule appears to be su¢ cient for robustness purposes. Coe¢ cients B2abs B3abs B4abs B2osr B3osr B4osr r0.66 0.53 0.63 0.40 2.16 3.53 3.24 1.96 2.87 2.55 y0.65 1.98 1.24 0.59 1.68 0.84 y10.53 0.54 LBayesian 5.571 4.697 4.626 1.299 1.245 1.172 Table 5: Optimal Bayesian interest rate rules with equal relative weighting of the three models. The rules are less aggressive when osr losses are used to normalize the Bayesian loss function, since NAM then receives more importance because of its small losses that the Bayesian objective is scaled with. As already explained, the …rst-best rules from NAM are quite mild and little inertial, whereas the LGM model prefers aggressive rules. The con‡ict in the Bayesian optimization is mostly between these two models, and the optimal coe¢ cients seem to be "averages" of the two individual rules. NAM is relatively more important in the optimization over the output gap coe¢ cient, because larger values of this may lead to instability, as will be seen from the performance of the LGM-rules in NAM in Chapter 7.1. Similar patterns with di¤erent relative weights on the three models can be seen in Table 6. When NAM enters in the objective function with four times the probability of the two other models, the coe¢ cients in the Bayesian rule decreases substantially, and again, particularly 39 the coe¢ cient on the lagged interest rate. The rules tailored more to CN and the LGM model are similar to each other, although the CN-rules in the …rst three columns have somewhat larger coe¢ cients, except for the output gap coe¢ cient. Coe¢ cients B2oC B3oC B4oC B2oL B3oL B4oL B2oN B3oN B4oN r0.65 0.50 0.73 0.59 0.52 0.24 2.28 3.75 3.32 2.01 3.49 2.99 1.65 1.98 1.89 y0.59 1.78 1.04 0.74 2.82 1.57 0.48 1.04 0.52 y10.43 0.65 0.45 LBayesian 1,185 1,192 1,168 1,326 1,176 1,049 1,244 1,210 1,157 Table 6: Di¤erent relative weights on the alternative models, using relative losses in the Bayesian loss function. The Bayesian loss generated by B3oC is actually larger than B2oC, which is a consequence of the con‡ict between the preferred policy preferences in NAM and CN when the nominal interest rate responds to three variables. NAM is the model driving most of the results. Without this model, i.e. averaging over the two other reference models only, the rules become more aggressive (see Appendix B.2, Table 16). Not surprisingly, they perform signi…cantly better in all models but NAM, in which the loss is tripled relative to the …rst-best four-parameter rule. Switching to the three-parameter rule generated by LGM and CN, the improvement becomes substantial, however — the loss in NAM is "only" doubled. This result may appear surprising since the outcome in NAM is found to be better with Rule 4 than Rule 3. The superiority of the larger rule hinges crucially on a positive reaction to the lagged output gap. CN, on the other hand, strongly prefers the opposite, such that the optimized rule for the average of CN and LGM adopts this characteristic, and hence the performance in NAM is poor. 7 Robustness ”Simple monetary policy rules are designed to take account of only the most basic principle of monetary policy of leaning against the wind of in‡ation and output movements. Because they are not …ne tuned to speci…c assumptions, they are more robust to mistaken assumptions.”(Taylor and Wieland 2009) Next the robustness properties of some of the interest rate rules considered in this thesis is presented. I use two di¤erent tools for checking how well the rules perform in the set of models, implied in‡ation variability premium (IIP) and fault tolerance. Below I describe 40 both measures and how to derive them. In the Discussion section the results are related to the existing literature on robust monetary policy rules. I start out by analyzing IIP in order to eliminate those rules that generate particularly high in‡ation variability in the models. These rules are not robust and not worth considering further. I then analyze the fault tolerance properties of the remaining rules. It is desirable to have a rule that is both robust in the sense of generating little ‡uctuations in consumer prices across models, and robust to changes in the parameter values. If a rule is fault tolerant, the policy maker does not cause too much harm if he "misses" on the optimal value of a coe¢ cient. Even though I have found that four-parameter rules do not signi…cantly outperform smaller rules in the seperate models, they could still be more robust, and are therefore included in the analyses below. 7.1 Implied In‡ation variability Premium Implied In‡ation variability Premium (IIP) is used to evaluate the performance of a policy rule relative to another rule, proposed by Kuester and Wieland (2010). It is de…ned as the percentage point (pp) increase in the standard deviation of the in‡ation rate for a given increase in absolute loss, keeping output and interest rate variability constant, i.e. the difference in loss caused by the interchange of two rules, from A to B, translated into in‡ation variation: IIPB= 100 [qLB tyvar(yA t)rvar(rA t)s:d:(A t)]: For constant variance of ytand rt, the increase in loss generated by rule B relative to rule A will be equal to the relative increase in the variance of in‡ation, but since the variances of the former two variables in general di¤er between the rules, the variance dispersion is converted into variance of in‡ation in order to make the rules’achievements comparable. IIP is then the square root of this "altered in‡ation variance". For example, the optimized four-parameter rule in NEMO yields a loss of 22.40217, while the rule from NAM yields a loss in NEMO of 45.68388, which is an increase of 104 percentage. The standard deviation of the in‡ation must increase by 2.67 pp in order to yield the same increase in loss in NEMO switching from NEMO’s to NAM’s …rst-best rule. IIP is measured in absolute terms and is thus easily comparable across models, whereas the relative increase in loss in a model depends on the size of this loss. Due to the large di¤erence in losses in the models, the relative increase is a less "neutral" measure. IIP is directly interpretable in terms of economic consequences, namely how much more consumer prices will ‡uctuate if the central bank uses the wrong model. 41 Table 7 illustrates the dispersion of the preferred monetary policy in the …ve models. The rules from NAM generally creates the biggest increase in in‡ation variability across models, with an IIP in CN of 3.68 pp in the case of Rule 4 and 3.60 pp in LGM with Rule 2. This is a big price to pay in form of increased in‡ation ‡uctuations if the economy turns out to be best described by CN or the LGM model, but the central bank believes it is NAM. Even larger is the IIP of Rule 3 from CN applied to NAM, of a striking 7.14 pp. This outlier is due to the fact that NAM is little tolerant to inertial policy, and in particular when the possibility to respond to the lagged output gap is absent. The three-parameter rule from CN holds both of these properties and therefore yields a miserable outcome in NAM. The extreme parameter values excludes it from the set of candidate rules for insurance against model uncertainty, because no policy maker would apply such an aggressive rule. NAM is intolerant to policies from the other models because it is self-stabilizing, so reacting with large changes in the policy rate to movements in the key variables might do more harm than good. The IIP to be paid if NAM is the "true" economy, and the central bank acts as if it were another model is thus relatively high. On average it is 1.17 for Rule 4, 2.42 for Rule 3 and "in…nity" for Rule 2. The original Taylor rule, shown in the bottom row of Table 7, performs the best, because it is the rule with the smallest parameter values and therefore resembles most NAM’s …rst-best policy. The other models also appear to be relatively tolerant to Taylor’s policy compared to the optimized rules from the other models, particularly the LGM model. Table 7 reveals that Rule 2 is signi…cantly less robust than Rule 3, but more than Rule 4 in most of the cases, due to the smaller amount of "model-tailoring". The IIP increases on average a lot more between Rule 3 and Rule 2 than Rule 3 and Rule 4, and again, particularly in the LGM model. Rule 2 from the latter model actually creates instability in NAM because of the large output response, which emphasizes the con‡ict between the preferred policies of the two models. However, reducing the output coe¢ cient with 0.5 removes the instability it creates in NAM, because the relation between the in‡ation and output gap coe¢ cients then becomes more equal to NAM’s Rule 2, albeit twice the size in magnitude. A large in‡ation response can somehow "outweigh" the output reaction such that a rule’s performance in NAM improves, which is why the four- and two-parameter rule from CN are better than Rule 3. Again, the con‡ict between the models’preferences for reaction patterns of the nominal interest rate becomes clear. In particular, it seems to be NAM versus the other models, as expected, due to this model’s highly di¤erent features. On average across all the rules, Policy NEMO has the lowest IIP of the …ve models. This model is the most robust with respect to the design of monetary policy, closely followed 42 IIP [%L] Rule 4 NEMO NAM LGM CN PN NEMO 0 [8.0] 1.21 [193.2] 2.16 [355.5] 0.06 [14.7] 0.41 [28.8] NAM 2.67 [120.3] 0 [12.6] 1.34 [191.4] 3.68 [473.3] 0.80 [111.7] LGM 1.04 [43.5] 0.88 [128.1] 0 [13.9] 1.00 [83.6] 2.46 [42.0] CN 0.06 [9.7] 1.86 [352.5] 1.71 [259.8] 0 [11.5] 0.83 [42.8] Rule 3 NEMO 0 [8.0] 1.21 [207.6] 2.26 [380.2] 0.06 [14.9] 0.38 [28.4] NAM 2.07 [89.3] 0 [25.7] 1.92 [302.1] 2.50 [266.9] 0.78 [88.1] LGM 1.03 [43.3] 0.82 [133.0] 0 [13.9] 1.00 [83.3] 1.93 [42.0] CN 0.06 [9.8] 7.14 [3325.4]* 1.84 [287.3] 0 [11.9] 0.71 [39.6] Rule 2 NEMO 0 [12.1] 0.66 [112.8] 2.09 [384.7] 0.15 [31.4] 0.01 [17.2] NAM 1.80 [79.8] 0 [33.0] 3.60 [806.6] 2.24 [238.2] 0.92 [77.3] LGM 1.21 [54.1] Instability* 0 [46.9] 1.09 [102.2] 1.69 [6.2] CN 0.17 [74.3] 0.84 [142.8] 0.95 [165.6] 0 [23.2] 0.16 [21.5] Taylor rule 0.66 [33.4] 0.15 [47.5] 0.92 [160.6] 0.70 [68.6] 0.51 [32.9] Table 7: IIP: increase in the standard deviation of in‡ation changing between each model’s optimized four-parameter rules. Percentage increase in loss relative to Ramsey loss is displayed in parentheses. by NEMO. The least robust model is LGM, having the highest average IIP, disregarding the two "outlier" results (marked with an asterisk). LGM needs strong responses to output ‡uctuations, and since none of the other models entail similar properties, their rules lead to signi…cantly less stability. As expected, NEMO is tolerant to CN policy, and vice versa, because they have similar core structure and main transmission mechanisms for monetary policy. This holds true for PN as well, even though it is signi…cantly more backward-looking than the former two. The values in parentheses in Table 7 show the percentage increases in loss relative to conducting optimal policy, in order to highlight the di¤erence between the two measures. A high increase in loss in a model could either be due to the fact that simple rules generally perform poorly, or because the model is little tolerant to rules from other models. A high IIP can also be caused by two di¤erent factors: either due to general policy intolerance or because the new rule creates larger ‡uctuations in interest rate changes and/or the output gap relative to in‡ation ‡uctuations. When the increase in loss resulting from the interchange of two policy rules is translated into in‡ation variability, the IIP is higher if the relative variability in the other two variables caused by the new rule is larger than with the old rule. The ordering of the rules’performances across models actually changes in some instances, but the supremacy of PN as the most tolerant model is strenghtened. NEMO is however almost as tolerant as PN when the relative increase in loss is used as the robustness measure 43 instead of IIP. Rule 4 and rule 3 from the LGM model now performs better in PN than in NAM, and CN is no longer the model that is most tolerant to LGM’s Rule 2 — again it is PN. All rules perform best in this model, except Rule 4 and Rule 3 interchanged between NEMO and CN. The latter is a consequence of the high degree of interest rate smoothing in these rules, which is not desirable in PN due to the backward-lookingness of this model. The reason for the change in the ranking of the models may be that LGM-rules are so aggressive that they signi…cantly increase interest rate movements in order to stabilize in‡ation and GDP, but since Policy NEMO is so persistent, the dampened in‡ation ‡uctuations are small relative to the increased interest rate volatility. The trade-o¤ between outcomes in the di¤erent models displayed in Table 7 calls for more robust monetary policy. Using either of the …rst-best individual rules appears to be a bad idea if a positive probability is attached to all …ve models, as the rules could potentially lead to very high variances of key variables. Therefore, Bayesian rules should be considered. The trade-o¤ between the outcomes in the models has already been taken into account, such that the resulting in‡ation variability premium is lower compared to the individual …rst-best rules transferred between models. Since NEMO and Policy NEMO are quite robust to policy speci…cations, the IIPs generated by the Bayesian rules are relatively low in these models as well. The improvements in outcome across all models from Table 7 to Table 8 is indeed striking. IIP [%L] Rule NEMO NAM LGM CN PN B2abs 0.16 [17.0] 0.52 [92.7] 0.54 [106.8] 0.18 [33.2] 0.07 [18.9] B2osr 0.21 [18.5] 0.39 [74.4] 0.60 [115.8] 0.26 [37.6] 0.11 [20.2] B3abs 0.28 [16.6] 0.45 [75.7] 0.26 [39.0] 0.30 [29.3] 0.20 [23.0] B3osr 0.41 [20.9] 0.35 [62.3] 0.25 [38.6] 0.42 [37.4] 0.29 [25.7] B4abs 0.34 [18.7] 0.42 [57.6] 0.26 [39.3] 0.34 [31.7] 0.19 [21.6] B4osr 0.48 [23.2] 0.27 [39.5] 0.29 [42.4] 0.47 [40.6] 0.26 [23.8] Table 8: IIP: increase in the standard deviation of in‡ation changing from the optimized rule of each model to a Bayesian rule of the same speci…cation. Relative increase to Ramsey loss are in parentheses. Switching from a four- to a three-parameter Bayesian rule improves the outcome in all models except for NAM, and marginally PN, con…rming the robustness superiority of smaller rules. In the LGM model these two rule speci…cations achieves roughly equally good outcomes, while in NEMO and CN the improvement is more pronounced. The three-parameter rule is more robust for the other loss functions considered as well. In NAM though, the IIP 44 increases from 0.27 to 0.35 pp between B4osr and B3osr, revealing that the larger rule is more tailored to this model. The increase in NAM is, however, not enough to balance the reduced IIPs in the other models. The relative increases in loss between four- and three-parameter rules in NAM are at a much larger level than in the other models. If the central bank seeks to avoid large potential losses, four-parameter rules should be considered for robustness checks. But since the losses in NAM generally are very small, the percentage increase will be higher even though the absolute increase is of the same magnitude as in the other models. Since IIP is measured in absolute terms, it is more neutral, and therefore serves the purpose of analyzing robustness properties better. Not surprisingly, the Bayesian rules optimized over relative losses perform better in NAM and worse in CN than those with the level of losses. The rules are about equally good in the LGM model, although absolute losses is best with the four- and two-parameter rules due to the con‡ict with NAM in the three-parameter rule. I therefore conclude that the Bayesian rules based on the relative outcome in the three models are more robust than those based on the absolute outcome. If the central bank changes its beliefs about the likelihood of the three alternative models, the Bayesian rules with di¤erent relative weights could be considered instead. The performance of the three-parameter rules for the benchmark loss function is displayed in Table 9. The IIP is naturally smallest in the more important model, particularly in the LGM model. In NEMO the IIP is substantially lower for the rule with the highest weight on CN, which also performs quite well in the other two models. The con‡ict appears to be smaller inbetween NAM and the LGM model, than between the NEMO versions and these two models. IIP [%L] Rule NEMO NAM LGM CN PN B3osr 0.41 [20.9] 0.35 [62.3] 0.25 [38.6] 0.42 [37.4] 0.29 [25.7] B3oC 0.17 [13.1] 0.48 [80.0] 0.38 [52.5] 0.19 [22.6] 0.14 [21.1] B3oL 0.65 [29.0] 0.52 [85.0] 0.07 [20.3] 0.63 [52.2] 0.48 [31.8] B3oN 0.64 [28.5] 0.15 [40.2] 0.47 [63.2] 0.63 [52.7] 0.47 [31.5] Table 9: IIP in the individual models generated by the Bayesian three-parameter rules with di¤erent relative weighting of the osr losses. The last letter in the name of the rule indicates which model is assigned the highest probability in the Bayesian objective. No capital letter means equal weights. Increases in loss relative to Ramsey loss is displayed in parantheses. On average of the …ve models and the eight variations of the Bayesian rules (four with osr losses and four with absolute losses), the three-parameter rule yields lower IIPs than both four- and two-parameter rules. Since it performs best across the models, it is the most robust 45 rule. First-best rules of the seperate models were shown to be little robust, and I therefore do not investigate them further. 7.1.1 Discussion The results above con…rm existing results from the literature on robust monetary policy in the face of model uncertainty. Levin et al. (1999) …nd that adding more variables than three does not signi…cantly improve the performance of Bayesian rules in their four models of the U.S. economy. Bayesian rules are designed to the "average" behaviour of the models, so adding more variables in order to …ne-tune the optimization to one model will not be fruitful because the models behave so di¤erently. Removing the lagged interest rate as well, deteriorates the rule’s performance. Interest rate smoothing is optimal in the simple interest rate rules, both because it yields smaller losses, and because it is normatively desirable to avoid large jumps in the interest rate. This is somewhat contrary to the results of Taylor and Wieland (2009), who analyze the robustness of …rst-best rules from three di¤erent models of the U.S. economy. The authors conclude that rules responding to only in‡ation and output are more robust than rules that in addition respond to either just the lagged interest rate or the lagged output as well. Yet, the optimal three-parameter rules are preferred within each model, and the gain from extending them with a fourth variable is minimal. A well-known result in the literature is that policy inertia is favorable in forward-looking models. The performance of the inertial rules from CN, NEMO and the LGM model is poor in the backward-looking model NAM, equivalent to what is found by many authors, e.g. Taylor and Williams (2010) and Kuester and Wieland (2010). Backward-looking models lacks the expectation channel of monetary policy and interest rate smoothing is thus not necessary. In addition, rules from backward-looking models may not be active enough to anchor expectations if agents indeed are forward-looking (Kuester and Wieland 2010): The threeparameter rules from NAM generate large IIPs in the other models, in particular in CN. Adalid et al. (2005), however, conclude that the optimized simple rules from backwardlooking models perform better in models with forward-looking features than the opposite, contrary to my results about NAM (except the extreme Rule 3 from CN). Akram and Nymoen (2009) …nd that the use of a suit of models for robustness purposes is not optimal when monetary policy is based on the average of the policy prescribed by three equally likely macroeconometric models for the Norwegian economy, of which one is more probable but the other two have more desirable properties. The generated loss in the most valid model is by far larger than with optimized rules utilizing sub-optimal policy horizons (which is the uncertainty measure used). Akram and Nymoen (2009) conclude that because 46 the performance of the "average rule" is miserable, empirical validity should not be traded o¤ against other properties, such as consistency with economic theory or data, transparency, resemblance to other models commonly used, parsimony, etc. Averaging over policies and not outcomes in the models will necessarily deteriorate performance, since the behaviour of each model is not taken into account. The Bayesian rules in this thesis are not the average of the three individual …rst-best rules, as such a rule could indeed potentially lead to disastrous outcomes. 7.2 Fault tolerance Fault tolerance is a tool used to analyze how robust a model is to changes in the monetary policy rule, for example due to the "trembling hand" of the policy maker, originating from Levin and Williams (2003). It is a widely used concept in engineering, and is de…ned as "the ability of a system or component to continue normal operation despite the presence of ... faults"(IEEE Standard Computer Dictionary 1990). By varying one coe¢ cient in a rule at a time, we …nd how robust a certain model is to the speci…cation of the policy rule. If the loss increases only slightly for large changes in a parameter, we say that the model is fault tolerant with respect to this parameter, and if the model is relatively insensitive to alterations of all parameters in the rule, the model is fault tolerant (Levin and Williams 2003). If, on the other hand, small deviations lead to a large increase, the model is fault intolerant. Fault tolerance is thus a way to display the degree of curvature in the loss function. The advantage of fault tolerance as a robustness measure is that it renders evaluation of the same interest rate rule’s performance in di¤erent models possible, and thus comparisons of the models’robustness properties. By breaking it down into each single parameter, one can easily "track" the causes of robustness. If one variable in the rule is more sensitive to variations in the associated parameter, the central bank should be extra careful setting its value. Fault tolerance can be used both for models and for policy rules, as noted by Kuester and Wieland (2010). An obvious drawback about the method described above, is that it measures the robustness of one speci…c parameter, ceteris paribus, and is hence not properly comparable across models. As a parameter in a rule that is applied in another model than the rule-generating model is changed, the optimal responses to the other variables in that rule will change as well, and hence the fault tolerance properties of the model for that rule. Instead it is possible to measure how well an interest rate rule performs across a set of competing models by looking at the fault tolerance of the policy rather than the models. If perturbations in the policy parameters have only marginal e¤ect on the performance across models, the rule is said to be fault tolerant. I therefore only consider the Bayesian rules’ 47 performance in the models, and not the robustness properties of the models for the individual …rst-best policies. 7.2.1 Bayesian rules Fault tolerance for the Bayesian rules are constructed in the following way: In each separate model I apply the instrument rule found by minimizing a weighted average of the outcomes in the three alternative models NAM, LGM and CN. I then …x the coe¢ cients at their optimal values while gradually altering one coe¢ cient at a time, plotting the increase in loss relative to the loss generated by the …rst-best individual rule of the same speci…cation as the Bayesian rule along the vertical axis and the parameter values along the horizontal axis.26 In Figure 3 the fault tolerance of the Bayesian three-parameter rule with equal weighting of the osr losses in each of the three reference models is showed. It is highly fault tolerant in the NEMO models, and somewhat less in the LGM model. NAM is the least tolerant model to this rule, at least with respect to the coe¢ cients on the output gap and lagged interest rate. The in‡ation coe¢ cient can however vary more in the LGM model than in NAM. Due to NAM, the output response is restricted to the close neighborhood of 1-2, and interest rate smoothing should be kept at the level of roughly 0.6, in order to insure against model uncertainty. The LGM model demands that in‡ation ‡uctuations must not result in too large movements in the nominal interest rate, but the Taylor principle must by all means be ful…lled. At best, a one percentage increase in the in‡ation rate should lead to a policy rate increase in the range of 2-4 percentage points. The other models are more fault tolerant with respect to the in‡ation responses, and for the three NEMO versions it could be increased signi…cantly without causing much harm. 26 Plotting relative to Ramsey losses gave similar …gures since all models have a roughly equal relation between the loss from optimal policy and the optimized simple rule. 48 9 Concluding remarks Uncertainty in various forms will always be an issue in monetary policy. The "true" e¤ects of monetary policy and how the mechanisms in the actual economy function can never be precisely identi…ed due to the impossibility of conducting macroeconomic experiments. The best solution is to conduct experiments in the closed environment of a model. But this just adds to the general uncertainty, since a model is a highly stylized and simpli…ed representation of the economy, and will always be mis-speci…ed in some way or another. Which model is best suited for "experimenting" with monetary policy will therefore remain unknown. But policy makers can use the knowledge obtained from research within the …eld to assess which of the available models are most propriate. Because all models have strengths and weaknesses, using several di¤erent models may be a sensible approach. Due to uncertainty, the members of a monetary policy committee may disagree about the speci…cation of the (main) model used by the central bank. The model depends at all times on the sta¤ doing the model revisions and on which research is given priority by the central bank. This could be accommodated by conducting robust monetary policy, either by using extra-model information, cross-checking optimal policy with a robust simple interest rate rule, or using a modi…ed loss function of the form proposed by Ilbas et al. (2012). In both latter cases, the rule referred to as "The Golden Interest Rule" could be a good candidate for Norges Bank as a replacement of the Taylor rule, which in any case is suboptimal. The main …ndings in my thesis is that rules optimized in a speci…c model do not perform well in other models, and are hence little robust. Bayesian rules where an average of the three reference models NAM, CN and the LGM model is optimized improves robustness signi…cantly, as expected. I …nd that rules with three variables are more robust than those with four, as they utilize less model-speci…c information. They are also more robust than two-parameter rules where interest rate smoothing is removed. Gradualism is preferable in interest rate setting due to uncertainty about the true structure of the economy, and monetary policy is conducted in a cautious manner with three-parameter rules. Among the Bayesian rules, those with four times higher probability attached to CN naturally performs best on average across all models, since CN has similar structure and transmission mechanisms as NEMO and Policy NEMO. Apart from these rules, the threeparameter Bayesian rule based on absolute losses do quite well, because the weight attached to NAM is tuned down. However, all of the Bayesian rules lead to half a percentage point or more increase in the in‡ation variability compared to the …rst-best rules in NAM. If the rationale for using a robust rule is to avoid large movements in in‡ation in the face of model 55 uncertainty, the Bayesian rules should not be used when the belief in NAM is strong. Rather, the "Golden Interest Rule" (GIR) serves the purpose of a robust simple rule of thumb for cross-checking optimal monetary policy in Norway. For all the rules I have considered in my work, GIR yields the lowest average implied in‡ation variability premium (IIP) across the three reference models. It is however outperformed on average of all …ve models by rules that reduces variability in the NEMO models, such as the CN-rules. The reduction in IIP in these models are, however, from already low levels, but the increase in NAM and the LGM model are at higher levels. The large potential increase in consumer price ‡uctuations could be avoided by robustifying monetary policy with GIR. The rule does well on average, in addition to signi…cantly reduce the IIP in NAM compared to the other Bayesian and "robust candidate" rules. The analysis done in this thesis is by all means not complete, and there are numerous extensions and further robustness checks that could have been done. It would be interesting to look at other variables in the simple rule, in particular key foreign variables, e.g. the exchange rate or foreign interest rates. The latter is included in one of the simple rules that Norges Bank uses for robustness checks, and would thus be interesting to investigate further. The exchange rate is relevant for a small open economy like the Norwegian, in particular because the exporting sector plays a key role in the wage determination. Norges Bank often takes the exchange rate channel of monetary policy explicitly into consideration in the interest rate decision. In this respect, allowing for a direct response of the nominal interest rate to movements in foreign variables in a simple rule used for robustness purposes would indeed be relevant. There is no consensus in the literature whether exchange rate rules improve the performance of monetary policy and are more robust than other simple interest rate rules. Many analyses (e.g. Galí 2008 and Dennis, Leitemo, and Søderstrøm 2006) …nd that responding to exchange rate movements in a simple rule gains little, because the variable is highly correlated with the interest rate itself and with variables already included in the rule, namely in‡ation and GDP. However, the exchange rate is a forward-looking variable, so including it may improve the outcome compared to a "contemporaneous rule" in models where it is advantageous to respond to future developments (Dennis 2000). Other possible extensions of my work would be the inclusion of the unemployment rate and the wage in‡ation. Wage in‡ation is not a variable in the LGM model, so the robustness analysis could not be executed for a rule including this variable. Targeting a low and stable unemployment rate is desirable as it may possibly represent the welfare of the households in a better way than GDP, but unemployment is not a variable in any of the NEMO versions or LGM neither. Reestimating the models to include these two variables is beyond the scope 56 of this thesis. Due to time and space constraints, I had to limit my analysis to rules utilizing the lagged interest rate, in‡ation, and the current and lagged output gap, which are standard rules in the literature. 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NBER Working Papers 11898, National Bureau of Economic Research, Inc. 61 Appendix A Models A.1 NEMO A.1.1 List of variables ytoutput gap cttotal consumption csa tconsumption savers csp tconsumption spenders at…nal good ttintermediate production qtdomestically produced intermediates used in production of the …nal good mtimported intermediates used in production of the …nal good ltlabour hours ktcapital stock ututilization rate of capital invtcapital investments invoil toil investments gtgovernment spending tCPI in‡ation Q tin‡ation on intermediates M tin‡ation on imported intermediates W tnominal wage in‡ation pQ treal price on domestic intermediate goods pM treal price on imported intermediate goods wtreal wage mctreal marginal costs mrstmarginal rate of substitution between consumption and leisure rtnominal interest rate rK treal return to capital btdomestic holdings of foreign bonds !telasticity of substitution between labour inputs H telasticity of substitution between domestically produced intermediates F telasticity of substitution between foreign intermediates used in domestic production tshare of domestic intermediates used in domestic production streal exchange rate 62 y tforeign output gap M tin‡ation on exported intermediates pM treal price on exported intermediates m texported intermediates F telasticity of substitution between domestic intermediates in foreign production mc tforeign real marginal costs r tforeign nominal interest rate  tshare of foreign intermediates used in foreign production Z tshock to the growth rate of technology zL ttemporary labour augmenting productivity shock zI tshock to investments zB trisk premium shock zU tshock to consumption preferences zM tshock to imports A.1.2 Model Final goods at=qt+ (1 )mt(7) qt=atpQ t(8) mt=atpM t(9) m t=y t[pM t+f51(pM tst)]  1 t(10) 63 Intermediate goods Production tt=f61(lt+zL t) + f62[ut+kt1Z t](11) kt=inv kinvt+f81(kt1Z t)(12) rK t=mct+1 [tt(ut+kt1Z t)] (13) mct=wt+1 (lttt) + 1 zL t(14) invt+Z tkt1=f111(invt1+Z t1kt2) + f112Etinvt+1 +Z t+1 kt(15) f113Et(rtt+1)f114rK t+1f115EtzI t+1 zI t Uut=rK t(16) Domestic prices Q t= (1 + )EtQ t+1 +1 (1 + )Q t1+f131(mctpQ t)f132H t(17) Q t=t+pQ tpQ t1(18) Export prices M t= 1 + EtM t+1 +1 1 + M t1+f151(mctpM tst)f152F t(19) M t= t+pM tpM t1(20) Import prices M t= 1 + EtM t+1 +1 1 + M t1+f171(mc tpM t+st)f172F t(21) M t=t+pM tpM t1(22) Households Forward-looking optimizers (savers) csa t=f191Etcsa t+1 +f192csa t1f193Etfrtt+1g  f194Z t+f195zU t(23) st=f201Etst+1 Etfrtt+1g+Etr t t+1+zB t(24) mrst=lt+f211(Zcsa tbccsa t1) + f212Z t(25) 64 A.3.2 Model In the simulations we keep oil prices, energy prices in CPI, government spending, taxes, productivity trend and all foreign variables but the nominal interest rate constant, and set constants equal to zero. vt=f11f(vt1+p t1pt1) + f12[(Rt1t1)(R t1 t1)]g(59) f13(RtR t)f142pot1+v t pit=f21[(pit1vt1pi t1)f22(pt1vt1p t1)] (60) +f23vt+f25pi t+pi t pt=f31[pt3f32(wt1zt1)f33pit1]f34zt+f35pt2(61) +f36pit+f37pet+f38yt1+p t wt=f41[(wt1pt2zt1) + f42ut4] + f43pt+f44pt1(62) f45(2ut1+ ut3) + f46T1t+w t zt=f51[zt3f52(wt1pt1)f53Trendtf54ut2](63) +f55(wtpt)f562zt1+z t ut=f61fut1f62(wt2pt2)f63[(RL;t2t2)1004yt2]g(64) +f64ut1f65ut4f66ut5+u t yt=f71[yt2f72gt1f73(vt1+p t1pt1) + f74(RL;t1t1)] (65) f75yt1+f76gt+f77(lt1pt1) + y t (ltpt) = f81[(lt3pt3)f82yt4+f83(RL;t4RB;t4)] (66) +f842yt2+f852(wtpt) + (lp) t RL;t =f91(RL;t1f92RB;t1f93Rt1) + f94Rt+R;L t(67) RB;t =f101(RB;t1f102Rt1f103R B;t1) + f104Rt+f105R t+R;B t(68) xtxtxt1(69) tptpt4 pt4 (70) 71 B Results B.1 Rules with alternative loss functions Loss function Coe¢ cients NEMO NAM LGM CN L22.07 1.11 2.13 2.94 y1.11 0.40 2.61 1.41 %L12,5 45,4 41,8 30,3 L32.35 1.15 2.72 3.87 y0.36 0.39 2.43 0.93 %L9,0 34,5 35,1 21,6 Table 12: Coe¢ cients in the optimized two parameter rule with the alternative loss functions. Loss function 2 has a weight on output stabilization of 1.5, and Loss function 3 has a weight on interest rate smoothing of 0.1, but are in other aspects equal to the benchmark. Loss function Coe¢ cients NEMO NAM LGM CN L2r1.05* 0.34 0.77 1.02* 0.64 1.14 4.24 1.04 y0.82 0.63 7.05 0.86 %L6,6 36,6 12,9 15,9 L3r0.66 0.33 0.79 0.95 3.55 1.20 5.99 23.94 y0.95 0.62 6.90 9.55 %L7,3 28,1 12,6 13,2 Table 13: Coe¢ cients in the optimized three-parameter rules with the alternative loss functions. Loss function 2 has a weight on output stabilization of 1.5, and Loss function 3 has a weight on interest rate smoothing of 0.1, but are in other aspects equal to the benchmark. Rules marked with an asterisk are displayed with short run (net) coe¢ cients due to super inertia. 72 Loss function Coe¢ cients NEMO NAM LGM CN L2r1.08* 0.01 0.78 1.05* 0.64 1.14 4.33 0.80 y1.41 0.37 7.57 1.31 y1-0.60 0.46 -0.40 -0.70 %L6.2 16.7 12.8 14.8 L3r0.67 0.02 0.81 0.97 3.57 1.21 6.34 46.54 y0.98 0.37 8.27 36.81 y1-0.02 0.45 -1.01 -19.27 %L7.3 12.8 12.5 12.4 Table 14: Coe¢ cients in the optimized four-parameter rules with the alternative loss functions. Loss function 2 has a weight on output stabilization of 1.5, and Loss function 3 has a weight on interest rate smoothing of 0.1, but are in other aspects equal to the benchmark. B.2 Alternative rules Robust candidate rules (CR) Coe¢ cients 1=GIR2345678 r0.6 0.6 0.6 0.6 0.6 0.7 0.7 0.7 3.0 2.8 2.8 2.9 3.0 3.0 3.1 2.5 y1.5 1.5 1.6 1.6 1.6 1.6 1.6 1.5 Table 15: A limited selection of the di¤erent candidate rules for the best simple robust rule. Number 1, GIR, is the one referred to as "The Golden Interst Rule" in the main text. Coe¢ cients Rule 2 Rule 3 Rule 4 r0.84 0.88 2.80 8.04 10.17 y1.28 5.71 9.46 y1-2.58 Lt1.126 1.135 1.026 Table 16: Bayesian rules optimized over Credit NEMO and the LGM model only, with equal weights and osr losses in the objective function. 73 Rule-generating model IIP [%Losr] Rule 3b NEMO NAM LGM CN NEMO 0[3.82] 0.79 2.09 0.17 NAM 2.96 0[-10.24] 1.40 3.96 LGM 1.34 0.88 0[19.88] 1.20 CN 0.23 0.79 0.85 0[8.99] Rule 3c NEMO 0[2.64] 0.79 2.09 0.17 NAM 0.05 0[4.01] 1.40 3.96 LGM 1.34 0.88 0[25.03] 1.20 CN 0.23 0.79 0.85 0[9.04] Table 17: IIP for the two alternative three-parameter rules, Rule 3b with in‡ation, current and lagged output as variables, and Rule 3c with current and lagged in‡ation and current output. In parentheses are the relative losses to the loss generated by the standard Rule 3. B.3 IIP results Average IIP of IIP in Rule 3 models 4 models 5 models NEMO NAM LGM CN GIR 0.339 0.322 0.294 0.270 0.360 0.354 0.302 CR 5 0.338 0.334 0.309 0.321 0.370 0.300 0.343 CR 7 0.375 0.343 0.321 0.248 0.428 0.409 0.286 B3osr 0.342 0.360 0.347 0.414 0.348 0.254 0.424 B3abs 0.337 0.232 0.300 0.282 0.452 0.258 0.302 B3oC 0.353 0.307 0.274 0.168 0.483 0.382 0.194 B3oL 0.406 0.467 0.470 0.650 0.519 0.070 0.628 Table 18: Average IIP of all models, the four main models and the three reference models for a selection of the best performing rules. 74 B.4 Fault tolerance -0.5 -0.4 -0.3 -0.2 -0.1 00.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 20 40 60 80 100 120 140 160 180 200 Coefficient on lagged interest rate % Increase in loss relative to osr loss NEMO NAM LGM Credit NEMO Policy NEMO -1 0 1 2 3 4 5 6 0 20 40 60 80 100 120 140 160 180 200 Coefficient on the output gap % increase in loss relative to osr loss NEMO NAM LGM Credit NEMO Policy NEMO -1 0 1 2 3 4 5 6 0 20 40 60 80 100 120 140 160 180 200 Coefficient on inflation % Increase in loss relative to osr loss NEMO NAM LGM Credit NEMO Policy NEMO -1 0 1 2 3 4 5 6 0 20 40 60 80 100 120 140 160 180 200 Coefficient on the lagged output gap % increase in loss relative to osr loss NEMO NAM LGM Credit NEMO Policy NEMO Figure 5: Fault tolerance of B4osr, the Bayesian four-parameter rule with ‡at prior and osr losses in the objective function (relative increase in loss to the …rst-best four-parameter rule in each model). 75