Robust-Satisficing Monetary Policy Under Parameter Uncertainty

Akram, Q. Farooq; Ben-Haim, Yakov; Eitrheim, Øyvind Working Paper Robust-Satisficing Monetary Policy Under Parameter Uncertainty Working Paper, No. 2007/14 Provided in Cooperation with: Norges Bank, Oslo Suggested Citation: Akram, Q. Farooq; Ben-Haim, Yakov; Eitrheim, Øyvind (2007) : Robust- Satisficing Monetary Policy Under Parameter Uncertainty, Working Paper, No. 2007/14, ISBN 978-82-7553-416-1, Norges Bank, Oslo, https://hdl.handle.net/11250/2498259 This Version is available at: https://hdl.handle.net/10419/209890 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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Farooq Akram, Yakov Ben-Haim and Øyvind Eitrheim ISSN 0801-2504 (printed) 1502-8143 (online) ISBN 978-82-7553-415-4 (printed), 978-82-7553-416-1 (online) Working papers from Norges Bank can be ordered by e-mail: [email protected] or from Norges Bank, Subscription service, P.O.Box. 1179 Sentrum N-0107Oslo, Norway. Tel. +47 22 31 63 83, Fax. +47 22 41 31 05 Working papers from 1999 onwards are available as pdf-files on the bank’s web site: www.norges-bank.no, under “Publications”. Norges Bank’s working papers present research projects and reports (not usually in their final form) and are intended inter alia to enable the author to benefit from the comments of colleagues and other interested parties. Views and conclusions expressed in working papers are the responsibility of the authors alone. Working papers fra Norges Bank kan bestilles over e-post: [email protected] eller ved henvendelse til: Norges Bank, Abonnementsservice Postboks 1179 Sentrum 0107 Oslo Telefon 22 31 63 83, Telefaks 22 41 31 05 Fra 1999 og senere er publikasjonene tilgjengelige som pdf-filer på www.norges-bank.no, under “Publikasjoner”. Working papers inneholder forskningsarbeider og utredninger som vanligvis ikke har fått sin endelige form. Hensikten er blant annet at forfatteren kan motta kommentarer fra kolleger og andre interesserte. Synspunkter og konklusjoner i arbeidene står for forfatternes regning. Robust-satisficing monetary policy under parameter uncertainty∗ Q. Farooq Akram† , Yakov Ben-Haim‡and Øyvind Eitrheim§ December 22, 2007 Abstract We employ the robust-satisficing approach to derive robust monetary policy when parameters of a macro model are uncertain. There is a trade-off between robustness of policies and their performance. Hence, under uncertainty, the policy maker is assumed to be content with policy performance at some satisfactory level rather than a level thought to be optimal based on available information. Our empirical analysis illustrates key properties of robustsatisficing policies and compares them with min-max policies implied by the robust-control approach. Intuitively, our empirical results suggest that higher robustness can be achieved by overstating challenges to the economy and understating the abilities to meet them. How much to overstate the challenges or understate the abilities depends on the robustness sought. Robustness is achieved by lowering one’s aspirations regarding the performance of policies and is therefore costly. Moreover, costs of robustness increase with the level of robustness, making robustness to apparently extreme parameter values particularly costly. We also find that robust-satisficing policies are generally less aggressive than min-max policies. Keywords: Robust monetary policy, Knightian uncertainty, parameter uncertainty, info-gap decision theory. JEL Codes: D81, E52, E58. ∗The views expressed in this paper are those of the authors and do not necessarily represent those of Norges Bank (the Central Bank of Norway). We have received useful comments from participants at the International Conference on Computing in Economics and Finance 2007, the European Economic Associaton meeting 2007 and seminar participants at Norges Bank. †Corresponding author: faroo[email protected]. Research Department, Norges Bank, Box 1179 Sentrum, N-0107 Oslo, Norway. Phone: +4722316692; Fax: +4722424062. ‡[email protected]. Yitzhak Moda’i Chair in Technology and Economics, Technion – Israel Institute of Technology, Haifa 32000 Israel. §o[email protected]. Research Department, Norges Bank. 1 1 Introduction Studies of monetary policy decisions under uncertainty are mainly based on the Bayesian and the robust control approaches; see e.g. Hansen and Sargent (2001), Giannoni (2002), Onatski and Williams (2003), Levin and Williams (2003), Tetlow and von zur Muehlen (2001), Coenen (2007) and Leitemo and S¨oderstr¨om (2004). The Bayesian approach requires that one assigns a probability distribution on the uncertain aspect of a decision problem, e.g. model parameters. This enables one to choose an expected-loss-minimizing policy. The robust control theory suggests designing policies to perform relatively well in worst-case scenarios, i.e. when the underlying premises turn out to be false in the most unfortunate way. Thereby, this approach enables one to limit the potential loss if the underlying premises turn out to be false.1 However, both of the approaches require some assumption(s) about the probability distribution of the uncertain entity. Within the Bayesian approach, such an assumption is made explicit while it is made implicit under the robust control approach, by limiting the outcome space of the uncertain entity. This may not be innocuous since the outcome space assumed will generally affect the policy decision. One can also argue that by invoking some probability distribution, explicitly or implicitly, one would not be deriving policy response to Knightian uncertainty in the strict sense, since it implies lack of probabilistic information; see Knight (1921). It has also been argued that robust policies can be quite costly in terms of forsaken performance in the normal course of events if the policy is geared towards limiting potential losses under extreme events; cf. Tetlow and von zur Muehlen (2001) and Cogley and Sargent (2005). In particular, the min-max policy implied by the robust control approach will be sub-optimal under all cases but the worst case. A similar objection can also be raised against the Bayesian approach where the “worst case scenario” can have undue influence on the policy decision; cf. Cogley and Sargent (2005). Relatively high potential costs of robust policies may therefore discourage one from adopting such policies. In this paper, we employ the robust-satisficing approach to derive monetary policy response under parameter uncertainty. This approach does not require any assumption about the probability distribution of the uncertain entity. And second, potential costs of robustness play a key role in defining robust policies; see e.g. Ben-Haim (2006). The robust-satisficing approach is quite general and can be easily employed to derive decisions under various kinds of uncertainties individually or jointly.2 1Accordingly, a fictitious malevolent agent who represents a policy maker’s worst fears concerning misspecification is introduced into the optimization problem and motivates her to minimize the loss function under the worst-case scenario. The level of uncertainty facing the decision maker can be regulated by adjusting the resources available to the malevolent agent. 2The robust-satisficing approach has been previously applied to a wide variety of decision problems with Knightian uncertainty, including financial risk assessment (Ben-Haim 2005); environmental regulation (Stranlund and Ben-Haim 2007); search behavior in animal foraging (Carmel and Ben-Haim 2005); policy decisions in marine reserve design (Halpern et al 2006); natural resource conservation decisions (Moilanen and Wintle 2006); inspection decisions by port authorities to detect terrorist weapons (Moffitt et al 2005) and to detect invasive species (Moffitt 2 The robust-satisficing approach bases decision making on two main premises. The first premise is that the decision maker faces uncertainty of the Knightian kind. Hence, it does not require one to specify either a probability distribution or bounds on the outcome space of the uncertain entity. The second premise is that the decision maker aims for performance at some satisfactory level rather than at a level which is deemed to be optimal based for instance on an estimated model; cf. Simon (1959) and (1979) and the references therein. The policy maker may still use such an optimal level as a reference, but is assumed willing to accept deviations from it to control her potential loss in case of faulty assumptions. The robust-satisficing policy maximizes robustness at a given level of acceptable performance. Robustness is measured as the extent of deviation from a decision’s underlying premises at which the performance will not deteriorate beyond some acceptable level. The robust-satisficing approach offers a trade-off between the robustness and the level of acceptable performance. Robustness of a policy can be raised by lowering one’s aspirations regarding its performance and accepting a higher level of loss; see e.g. Ben-Haim (2006). Common with the min-max approach, the robust-satisficing approach also allows one to cap one’s potential losses when the outcome space is given. Ben-Haim et al (2007) show that when the level of uncertainty is given, there would exist a robust-satisficing policy that is observationally equivalent to the min-max policy. Nonetheless, there are important differences between the two approaches, as shown in Ben-Haim et al (2007) and in this paper. The robust-satisficing approach is attractive when a decision maker would be content with performing relatively close to the optimal level derived under a specific set of scenarios, and with relatively poor performance under an alternative set of scenarios. For example, when a policy maker’s credibility depends on performing satisfactorily under ordinary events, but not under (apparently) extreme events, despite heavy losses. This could be the case for instance when private agents agree on the “ordinary-extreme classification” of events and are conscious of the potential costs of highly robust policies. We employ the robust-satisficing approach to derive monetary policy response when there is uncertainty about key parameters of an aggregate model of the US economy, estimated by Rudebusch and Svensson (1999).3These parameters represent degrees of persistence in the demand and supply shocks, the slope of the Phillip’s curve and the response of the output gap to interest rates. Uncertainty regarding persistence in shocks can be interpreted broadly as it can proxy uncertainty regarding omission of relevant variables as well as functional form misspecification, beside representing uncertainty regarding genuine shock persistence. Monetary policy is characterized by a simple Taylor-type interest rate rule, where the decision parameters are the response coefficients associated with inflation and output gaps as well as degree of interest rate smoothing; see Taylor et al 2007); technological fault diagnosis (Pierce et al 2006) and testing (Vinot et al 2005); and project management (Regev et al 2006). 3Results based on an alternative model with hybrid New Keynesian Phillips curve and IS curve are available upon request to the authors. 3 (1999). In addition to illustrating key properties of robust-satisficing policies, we also point out differences and possible observational equivalence with min-max policies. This paper draws on but goes beyond Ben-Haim et al (2007) who considered the case of uncertainty in the persistence of supply shock to contrast robust-satisficing policies with the min-max policy. We find that higher robustness can be achieved by basing policy on relatively high degrees of persistence in the shocks and relatively weak effects of the output gap on inflation and of interest rates on the output gap. How much to raise the degree of persistence and lower the effects of the output gap and the interest rate depends on the level of robustness sought. Robustness is achieved by lowering one’s aspirations regarding the policy performance and is therefore costly. Such costs are found to increase with the level of robustness, making robustness to a wide set of parameter values as well as apparently extreme events particularly costly. This also implies relatively high costs of adopting min-max policies. We also show that a policy decision based on the robust satisficing approach offers a higher degree of robustness than min-max policies if one aims to perform relatively well under a subset of all possible parameter values conjectured rather than limiting the loss under the worst case values of the parameters. The policy implications of the two approaches may differ substantially in such cases. However, both approaches can suggest the same policy if the acceptable level of loss is equal or higher than the maximum level under the min-max policy. The robust-satisficing policies are found to be generally less aggressive than min-max policies. They may therefore be easier to reconcile with observed interest rate setting than min-max policies; see e.g. Giannoni (2002), Leitemo and S¨oderstr¨om (2004), Tetlow and von zur Muehlen (2001) and the references therein. The paper is organized as follows. The next section briefly presents the robust-satisficing approach. Section 3 presents the empirical model and characterizes monetary policy. Section 4 employs the robust-satisficing approach to deal with uncertainty in parameters individually and jointly. Section 5 presents the main conclusions followed by an appendix. 2 Robust-Satisficing Decisions This section presents the basic concepts related to the robust-satisficing approach, definitions of different decision strategies and their properties. 2.1 Uncertainty and Robustness We denote a policy maker’s decisions by the parameter vector Ω, which for instance may consist of parameters of a simple Taylor-type interest rate rule. The policy maker’s decisions are based on models and data. However, these models and data, including the probabilistic elements and 4 parameters, may be incomplete or erroneous in various unknown ways. There may be e.g. relevant variables missing from the models, the appropriate model specification could be unknown, estimates of key parameters could be unavailable because of lack of data or one may lack confidence in them because of measurement errors in the data and so on. We denote uncertain elements by θwhich can be e.g. specific parameters, functions, missing variables and/or probability distributions. e θsymbolizes some specific value of θwhich can be an estimate or one’s choice. We represent the uncertainty associated with θby the family of sets U(`, e θ). Each of these sets contains possible realizations of θin the “vicinity” `of e θ.U(`, e θ) is referred to as an information gap (info-gap) model of uncertainty. Info-gap models entail no probabilistic information and thus are one possible quantification of Knightian uncertainty; see Ben-Haim (2006) for details. An info-gap model obeys two axioms: Contraction: U(0,e θ) = {e θ}(1) Nesting: ` < `0=⇒ U(`, e θ)⊆ U(`0,e θ) (2) The contraction axiom asserts that e θis the only possibility when there is no uncertainty (`=0). Here, we consider `as an unbounded unidimensional indicator of parameter uncertainty. The nesting axiom asserts that the range of possible realizations increases as the level of uncertainty increases, ceteris paribus. That is, the set U(`, e θ) becomes more inclusive as `gets larger, implying that the range of possible realizations of θin the vicinity e θincreases with `. It can therefore be referred to as the level of uncertainty and is related to the level of robustness as explained later. The loss resulting from decision Ω when the uncertain elements take the values θis L(Ω, θ). The loss may be a statistical expectation or a deterministic value.4The satisficing policy maker desires low loss, and would prefer loss no greater than some satisfactory level Ls: L(Ω, θ)≤Ls(3) We treat Lsas a parameter which can be chosen small or large, so the satisficing requirement in eq.(3) includes minimizing the loss as a special case. The policy maker is satisficing if she does not aim to minimize the loss but would be content with a loss no larger than Ls, recognizing that the loss may exceed Lsfor some θ∈ U(`, e θ). 4Our discussion can be readily extended to multiple loss functions. 5 2.2 Decision strategies: Robust-Satisficing, Conditional Estimation, and Min-Maxing We consider three types of decision strategies for choosing a decision or policy Ω from a set Rof feasible policies. The robustness of decision Ω, with the satisficing requirement Lsof eq.(3), is the greatest level of uncertainty `up to which all realizations θwould result in a loss no greater than Ls: b `(Ω, Ls) = max (`: max θ∈U(`, e θ) L(Ω, θ)!≤Ls)(4) b `(Ω, Ls) is a robustness function indicating the robustness of a specific policy Ω at some acceptable loss level Ls. The robust-satisficing decision maximizes the robustness (4) while satisficing the loss at the value Ls: Ωs(Ls) = arg max Ω∈Rb `(Ω, Ls) (5) Maximization of b `(Ω, Ls) conditional on some Lsamounts to maximizing U(`, e θ), by the nesting axiom. Conditional optimization is the decision, Ω e θ, which minimizes the loss based on a specific value of the uncertain entities, e θ: Ω e θ= arg min Ω∈RL(Ω,e θ) (6) A special case of conditional optimization is optimization conditional on a value of θimplying the highest level of loss which defines a min-max policy. A min-max policy may be defined as follows. The min-max decision minimizes the maximum loss based on a conjecture of the greatest level of uncertainty, `m: Ωm(`m) = arg min Ω∈Rmax θ∈U(`m, e θ) L(Ω, θ) (7) The min-max policy Ωm(`m) would not lead to a loss higher than some specific level for any value of θfrom the parameter space defined by `m,U(`m,e θ). 2.3 Basic properties of the decision strategies Here, we note several basic properties of the three decision strategies: robust-satisficing Ωs(Ls), conditional optimization Ω e θ, and min-maxing Ωm(`m). These properties, presented as propositions 1–3, characterize the relationship between robustness b `(Ω, Ls) and acceptable loss Ls. Proposition 1 Performance trades-off against robustness, both at any fixed decision, Ω, and at the robust-satisficing decision Ωs(Ls), if L(Ω, θ)is uniformly continuous in θ. 6 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1 2 5 10 15 20 30 40 50 60 Figure 1: Robustness of all policies considered at different levels of acceptable loss when ρπis uncertain. Level of robustness b `(Ωρπ, dLs) is represented on the vertical axis, while ρπ-values are denoted on the horizontal axis. Robustness is measured by the length of an interval containing ρπ-values ∈[0, 0.99]. The degrees of persistence on which the optimal policies are conditioned, are presented on the horizontal axis. Different levels of acceptable loss dLsin per cent are indicated by line-style of the robustness curves. among the set of policies considered. The robust-satisficing policies at different loss levels are defined by the peaks of the different robustness curves. We note that the peaks correspond to optimal policies conditional on ρπ-values in the range 0.3–0.8. 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0 10 20 30 40 50 60 70 80 90 100 Figure 2: (a) Left frame: Robustness b `(Ωρπ, dLs) offered by robust-satisficing policies (and some other policies) is indicated on the vertical axis. Robustness is measured by the lengths of intervals containing ρπvalues ∈[0, 0.99]. The policies are identified by the degrees of persistence (ρπ) indicated on the horizontal axis. The set of robust-satisficing policies are conditional on the following set of ρπ-values: 0.30, 0.31, 0.32,...,0.80. (b) Right frame: Robustness (vertical axis) offered by the different policies considered in the left frame (a) at different levels of acceptable loss, dLs, which is represented on the horizontal axis in per cent. 13 Figure 2.a shows robustness offered by the robust-satisficing policies. The circled line represents maximal robustness offered by the corresponding policy. The bold face numbers in Table 1are points on this curve. It is seen that optimal policies conditional on values below 0.3 and above 0.8 are not robust-satisficing policies. The policy based on ρπ= 0.8 offers maximum robustness (indicated by 1) at the lowest level of loss compared with policies based on ρπoutside the range 0.3–0.8. Figures 2.a–b suggest that the policy conditional on ρπ= 0.8 would not lead to a loss higher than 50% under any value of ρπ∈[0,0.99]; see also Table 1. The other policies including those based on ρπ-values strictly larger than 0.8 require willingness to accept higher loss than 50% for maximum robustness and will therefore not be adopted. Figure 2.b also suggests that by accepting less than 10% per cent deviation from whatever would be the optimal loss level, one can perform satisfactorily under about 3/4 of the possible values of ρπ. However, a higher level of robustness requires willingness to accept a substantially higher loss. To perform satisfactorily under any value of ρπ∈[0,0.99] one would have to accept a loss of at least 50%. Table 2: Sets of ρπ-values for which selected policies are robust at different levels of dLs dLsU(Ωρπ, dLs) 50 0-.93 0-.94 0-.94 0-.94 0-.95 0-.95 0-.96 0-.97 0-.99 .47-.99 .89-.99 40 0-.91 0-.92 0-.92 0-.93 0-.93 0-.94 0-.95 0-.96 .1-.97 .55-.99 .91-.99 35 0-.9 0-.9 0-.91 0-.92 0-.92 0-.93 0-.94 0-.95 .17-.97 .6-.99 .92-.99 30 0-.88 0-.89 0-.9 0-.9 0-.91 0-.92 0-.93 0-.94 .24-.96 .64-.99 .93-.99 25 0-.86 0-.87 0-.88 0-.88 0-.89 0-.9 0-.92 0-.93 .32-.95 .67-.98 .94-.99 20 0-.83 0-.84 0-.85 0-.86 0-.87 0-.89 0-.90 .04-.92 .39-.94 .71-.98 .95-.99 15 0-.8 0-.81 0-.82 0-.83 0-.85 0-.86 0-.88 .16-.9 .47-.93 .75-.97 .96-.99 10 0-.73 0-.75 0-.77 0-.79 0-.80 0-.82 .01-.85 .29-.88 .56-.91 .79-.96 .97-.99 5 0-.61 0-.64 0-.67 0-.7 0-.73 0-.76 .23-.8 .44-.84 .65-.89 .83-.94 .98-.99 1 0-.35 0-.4 0-.46 0-.52 .15-.58 .31-.64 .46-.7 .61-.77 .75-.84 .88-.92 .99-.99 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.99 Ω0Ω0.1Ω0.2Ω0.3Ω0.4Ω0.5Ω0.6Ω0.7Ω0.8Ω0.9Ω0.99 Note: The policies, which are represented by Ωs containing the response coefficients in the Taylor rule, are optimal conditional on the subscripted values of persistence in the supply shock (ρπ). Bold faced sets of ρπ-values correspond to the robust-satisficing policies at different levels of acceptable loss (in per cent). Table 2presents sets of ρπ-values, U(Ωρπ, dLs), for which selected policies would be robust at different levels of (relative) loss. Specifically, at level of robustness/uncertainty b `=b `(Ωρπ, dLs), the uncertainty set U(b `, dLs) is an interval of ρπ-values of length b `, where U(b `, dLs) is alternatively represented as U(Ωρπ, dLs). These sets correspond to the robustness measures in Table 1and display similar properties. Table 2illustrates the contraction as well as the nesting axioms. It shows that sets corresponding to optimal policies for dLs= 0% only contain the single conditioning ρπ. Accordingly, highest aspirations have zero robustness: b `(Ωρπ, dLs) = 0 while U(Ωρπ, dLs) ={ρπ}when dLs= 0. The table also shows that robustness of policies considered increases as well as those of robust-satisficing policies with the (relative) loss levels, dLs. The uncertainty sets are non-decreasing as one moves 14 up each column of Table 2and across columns associated with the robust-satisficing policies (while raising dLs). Figure 3displays how the uncertainty sets vary with different robust-satisficing policies as well as with loss levels. 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0 10 20 30 40 50 60 70 80 90 100 Figure 3: (a) Left frame: Robustness intervals offered by robust-satisficing policies (and some others) in terms of sets of ρπ-values. The sets are defined by the minimum and the maximum value in their ranges; the values within the extreme values are also parts of the sets. The robust-satisficing policies are identified by values of ρπ(= 0, 0.30, 0.31,...,0.80) indicated on the horizontal axis. (b) Robustness (vertical axis) in terms of sets of of ρπ-values indicated by the minimum and maximum values offered by the different policies considered in (a) at different levels of acceptable loss, dLs, which is represented on the horizontal axis in per cent. Notably, the costs of robustness increase with the level of robustness and asymmetrically around the persistence values conditioned upon. Table 2as well as Figure 3show that the parameter sets, U(Ωρπ, dLs)s, expand at a decreasing rate when we raise the relative loss levels, implying increasing costs of robustness. Hence, inclusion of relative extreme values of ρπin the parameter sets demands willingness to accept relatively high costs. It also appears that when the relative loss level increases, the parameter sets do not expand symmetrically around the parameter values conditioned on. Hence, the costs of expanding the parameter sets to include particularly high or low parameter values can be quite high. We also note that the increase in robustness, i.e. expansions of the sets U(Ωρπ, dLs), is highly policy dependent. For example, we see in Table 2that an increase in the loss from 1% to 5% expands the parameter set associated with policy Ω0.1from [0, 0.35] to [0, 0.61] while the parameter set associated with Ω0.9changes from [0.88, 0.92] to [0.83, 0.94]. Regarding policies that are not robust-satisficing, we note that policies conditional on degrees of persistence below 0.3 seem to be more robust, i.e. the associated sets U(Ωρπ, dLs) are larger, than policies conditional on degrees of persistence higher than 0.8.9 9The robust-satisficing policies indicated in the Table 2are relative to policies evaluated in this table. Some of 15 The min-max policy Ωmis defined as the optimal policy conditioned on ρπ= 0.8, Ω.8, if we assume that the level of uncertainty `mcoincides with the range 0–0.99. Then, this policy would offer robustness against any ρπ∈[0,0.99] at the lowest loss level, which is 50%. The other policies do not offer robustness against the complete set of ρπ-values at this level of relative loss. Specifically, optimal policies conditioned on ρπ∈[0,0.8) imply higher loss than 50% if ρπturns out to be e.g. 0.99, while the policies conditioned on ρπ∈(0.8,0.99] imply relatively higher loss than 50% if ρπturns out to be particularly low. Notably, robust-satisficing polices defined by ρπ∈[0.3, 0.8) would imply higher loss than the min-max policy for ρπ-values slightly below and including 0.99. Except for these values, the robust-satisficing policies will imply lower relative loss than the min-max policy. The min-max policy and the robust satisficing policy coincide, i.e. Ωm= Ω0.8, if dL(Ω e ρπ, ρπ)≤ dLs= 50%. Moreover, at a given level of uncertainty, the robust-satisficing policy will coincide with the min-max policy even for dL(Ω e ρπ, ρπ)> dLs= 50%. This is because any policy different from the min-max policy will imply higher loss than necessary for complete robustness and hence not be selected. Thus, a min-max policy and a robust satisficing would be observationally equivalent if the robust satisficer may not incur more than the maximum loss level under the min-max policy. The robust-satisficing policy may, however, deviate from the min-max policy and offer higher robustness if relatively lower levels of loss are required, as shown above. Theoretically, by assuming away uncertainty a min-max policy can be equated to any robustsatisficing policy. Moreover, by raising the acceptable level of loss to dLm, any robust-satisficing policy can be equated to the min-max policy implying dLm; see proposition 3. Such an exercise may be unreasonable and hence seem artificial in practice, though. Finally, robust-satisficing policies are found to be less aggressive than the min-max policy, in general. Figure 4shows the response coefficients of inflation, ωπ, in the coefficient vectors Ω0–Ω0.99 defining the Taylor-type rule. The figure shows that ωπincreases with ρπ. The range of ωπ-values corresponding to the robust satisficing policies, which are defined by ρπ∈[0.3,0.8], is about 3– 4.5. Hence, except for the robust-satisficing policy that coincides with the min-max policy, the robust-satisficing policies will be generally less aggressive than the min-max policy, for which ωπ = 4.5. In particular, robust-satisficing policy for relatively low levels of loss which would be based on relatively low persistence values, will imply relatively weak response to the inflation gap. The response coefficient of the output gap (ωy), associated with the response coefficients of the inflation gap (ωπ), displays similar properties. This varies in the range 1.45–2.30 for the robust-satisficing policies (not shown). the policies are robust-satisficing at several levels of relative losses. This is an artefact of not evaluating policies based on a finer grid of ρπ. When we use a finer grid, robust-satisficing policies would vary continuously over the range 0.30–0.80 with the different relative loss levels, as shown in Figures 1–3. 16 2 3 4 5 6 7 8 9 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Figure 4: Optimal values of the response coefficient associated with the inflation gap in the Taylor rule, ωπ, conditional on different degrees of persistence in the supply shock ρπ= 0, 0.01, 0.02,...,0.99 (horizontal axis). 4.2 Uncertain slope of the Phillips curve In the following we apply the robust-satisficing approach to the case when the coefficient corresponding to the output gap in the inflation equation (13), cy, is uncertain. The results regarding uncertainty in the response of the output gap to interest rates in the demand equation (14) were found to the comparable to the case with uncertain slope coefficient cyand are therefore not reported. We let the parameter space of cybe unbounded and investigate the robustness of different policies at different levels of relative loss. For illustration, we evaluate three polices Ω e cy= Ω.13, Ω.25 and Ω.30,where Ω e cyis the optimal policy conditional on a specific cyvalue.10 To calculate the robustness of a policy Ω e cywe find the range and the set of cyvalues for which dL(Ω e cy, cy) ≤dLsholds, where dLs= 0, 5, 10, 20%. The range of cy-values for which dL(Ω e cy, cy)≤dLsholds defines the degree of robustness of policy Ω e cy,`(Ωcy, dLs), while the corresponding set of cy-values defines the uncertainty set U(Ωcy, dLs). Table 3presents the results where bold-faced numbers correspond to robust-satisficing policies. Table 3illustrates the characteristics of robust-satisficing policies consistent with propositions 1–2, as above; see Figures 2.a and 3.a and Table 2. The results are also consistent with the other properties of robust-satisficing policies observed above. In particular, the costs of robustness increase with the level of robustness and to some extent asymmetrically around the parameter values conditioned upon. In greater detail, among the three policies evaluated, Ω0.30 has relatively higher robustness for acceptable loss up to 10%. The policy defined by Ω0.25, however, becomes slightly more robust than Ω0.30 when the acceptable loss is raised to 20%. The right panel of the table shows the cy 10Note that defining the set of feasible policies Ris not the dual of defining the parameter space of cy.A policy maker has more information about the set of feasible policies than the parameter space. Hence, it is not unreasonable to assume that the policy maker compares the robustness of feasible policies, without specifying the boundaries of the parameter space. 17 Table 3: Robustness and corresponding sets of cy-values at different levels of dLs dLsb `(Ωcy, dLs)U(Ωcy, dLs) 20 0.310 0.430 0.420 .005–.305 .050–.480 .080–.500 10 0.240 0.315 0.345 .015–.255 .100–.415 .135–.480 5 0.170 0.220 0.245 .050–.220 .145–.365 .180–.425 0 0 0 0 .13 .25 .30 ΩcyΩ.13 Ω.25 Ω.30 Ω.13 Ω.25 Ω.30 Note: The policies, which are represented by Ωs containing the response coefficients in the Taylor rule, are optimal conditional on the subscripted values of slope coefficients (cy). Bold faced cy-values correspond to the robust-satisficing policies at different levels of acceptable loss (in per cent). values for which the losses will not exceed the acceptable levels. When the acceptable loss is 0%, i.e. one aspires for the optimal levels, the robustness of all policies is zero as any deviation from the value conditioned on, will lead to a higher loss than aspired. However, by being willing to accept up to 5% deviation from optimal levels, one can raise the robustness of all policies to a quite large range of possible cyvalues. We also observe that robustness, i.e. expansion of U(Ωcy, dLs), does not increase symmetrically around the parameter values conditioned upon. Moreover, the policy conditioned on cy= 0.13,Ω0.13, which is the econometrically estimated value of 0.13, has relatively lower robustness than the other two policies. In general, a robustsatisficing policy would not be conditioned on the estimated value of a parameter. This is because when we assume Knightian parameter uncertainty, the estimated value of a parameter does not receive more weight than any other parameter value. One may also say that the choice of the parameter value for policy making is based on “strategic” rather than econometric considerations, in the robust-satisficing approach as well in the robust-control approach. To ease comparison with the min-max policy, let us now assume that the slope coefficient cy takes on a value in the range [0.005, 0.5], which is fairly broad suggesting a relatively high level of uncertainty `m. Figure 5presents robustness curves implied by optimal policies conditional on every value of the slope coefficient in the interval 0.005–0.5, differing from each other by just 0.005. The results support the characteristics of robust-satisficing policies noted above; see Figure 5and Table 4, which reports uncertainty sets for selected policies. At a given loss level, the robustness curves in Figure 5also suggest a strongly concave relationship between robustness and the different values of the slope coefficient on which the policies have been conditioned on. Specifically, it is seen that policies based on relatively low values of cyare more robust than those based on relatively higher values of cy. The robust-satisficing policies are defined by cyvalues in the range [0.25, 0.44]. The policy based on cy= 0.25 offers complete robustness to any cy∈[0.005, 0.5] at a relative loss level of 40%, while the policy based on cy= 0.44 is the robust-satisficing policy at the 1% level of loss and also the least robust policy, among the set of robust-satisficing policies. The policy based on cy= 0.25 is also the min-max policy for the full range of parameter 18 .0 .1 .2 .3 .4 .0 .1 .2 .3 .4 .5 .6 1 2 5 10 15 20 30 40 50 60 Figure 5: Robustness of all policies considered at different levels of acceptable loss when the slope of the Phillips curve, cy, is uncertain. Level of robustness b `(Ωcy, dLs) is represented on the vertical axis, while cy-values are indicated on the horizontal axis. Robustness is measured by the length of an interval containing cy-values ∈[0.005, 0.50]. Policies that are optimal conditional on these slope coefficient values are represented by these values on the horizontal axis. Different levels of acceptable loss dL in per cent, are indicated by patterns of the robustness curves. Table 4: Sets of cy-values for which selected policies are robust at different levels of dLs dLsU(Ωcy, dLs) 40 .005–.190 .005–.370 .005–.5 .015–.5 .035–.5 .055–.5 .100–.5 30 .005–.170 .005–.345 .015–.5 .040–.5 .065–.5 .090–.5 .140–.5 20 .005–.140 .005–.305 .050–.48 .080–.5 .110–.5 .135–.5 .195–.5 15 .005–.125 .005–.285 .075–.450 .105–.5 .135–.5 .170–.5 .230–.5 10 .005–.105 .015–.255 .100–.415 .135–.480 .170–.5 .205–.5 .275–.5 5 .005–.075 .050–.220 .145–.365 .180–.425 .220–.490 .260–.5 .335–.5 2 .005–.050 .075–.185 .180–.320 .225–.380 .270–.435 .310–.490 .395–.5 1 .005–.035 .095–.170 .200–.300 .245–.355 .290–.410 .335–.465 .425–.5 0 0.005 0.13 0.25 0.3 0.35 0.4 0.5 Note: The policies, which are represented by Ω s containing the response coefficients in the Taylor rule, are optimal conditional on the subscripted values of slope coefficients (cy). Bold faced sets of cy-values correspond to the robust-satisficing policies at different levels of acceptable loss (in per cent). uncertainty conjectured, as any other policy implies relatively higher loss for some of the possible values of cywithin its assumed range. For example, the optimal policies conditional on cy= 0.30 and cy= 0.13 would imply a higher loss than 40% for parameter values in the ranges [0.005, 0.015) and (0.370, 0.5], respectively; see Table 3. It should be noted that when the level of uncertainty, here represented by the range [0.005, 0.5], is given, the robust-satisficing policies (as well as the min-max) policy may be affected. This is because a rise or reduction in the level of uncertainty, here widening or narrowing of the range, increases or reduces the set of parameter values for which the robustness of a policy is evaluated. This is, however, not the case when the parameter space is unbounded. 19 1.6 2.0 2.4 2.8 3.2 3.6 .0 .1 .2 .3 .4 1.0 1.2 1.4 1.6 1.8 2.0 2.2 .0 .1 .2 .3 .4 Figure 6: Left frame: Optimal values of the response coefficient associated with the inflation gap in the Taylor rule, ωπ, conditional upon different values of the slope coefficient cy= 0.005, 0.01, 0.015,...,0.50 (horizontal axis). Right frame: Optimal values of the response coefficient associated with the output gap in the Taylor rule, ωy, conditional on the cyvalues (horizontal axis). Subsets of these response coefficients define robust-satisficing policies. Uncertainty in the slope coefficient implies that the policy is more influenced by the inflation gap than the output gap in the Taylor rule. Figure 6presents (optimal) values of the response coefficients associated with the inflation and the output gaps. These values are optimal conditional on values of the slope coefficient denoted on the horizontal axes. We note that the response coefficient associated with the inflation gap increases with the quest for robustness while the response coefficient associated the output gap decreases. The response coefficients associated with cy∈[0.25, 0.44] increase from 2 to 2.5 for the inflation gap and decrease from 2 to 1.3 for the output gap. The reason for these choices of the response coefficients is that the policy is conditioned on a lower value of the slope coefficient cy, within the range 0.25–0.45, the higher robustness one seeks. Accordingly, policy becomes more effective if interest rates are less influenced by the output gap than the inflation gap. Thus, the weight on the output gap declines with the values of the slope coefficient conditioned on. The relatively higher weight on the inflation gap at the expense of relatively lower weight on the output gap in the interest rate rule is consistent with earlier studies based on the min-max approach; see e.g. Smets (2002) and the references therein. However, within the robust-satisficing approach alteration of the weights depends on the how much robustness one seeks, which depends on the acceptable level of loss. As noted above, results for the case when the interest rate effect on the output gap is uncertain are comparable to those for uncertainty in the slope of the Phillips curves. In the former case, 20 robustness was increased by basing policy on a relatively weak response of the output gap to interest rates. In both cases, there is uncertainty regarding the policy maker’s ability to control inflation. And in both cases, the degree of robustness increases if policy is based on understating the policy maker’s ability to control inflation. Moreover, the higher the robustness one seeks, the weaker control does one assume; up to some limit, though, as the extreme case would be to assume no control at all. 4.3 Uncertain persistence in the demand and the supply shocks This section employs the robust satisficing approach to deal with uncertainty in two parameters, %πand %y. In this case, θbecomes a vector and U(`, e θ) becomes a two-dimensional parameter space, defined in Appendix A. This space will be defined by the optimal policy based on value e θ of θ, Ω e θ. We represent the robustness of a policy, b `(= `(Ω e θ, dLs)), by the fraction of all possible vectors θfor which the given policy does not imply a higher loss than some specific level dLs. 1 1 1 1 2 2 2 2 5 5 5 5 10 10 10 10 15 15 15 15 15 20 20 20 20 20 25 25 25 25 25 30 30 30 30 35 35 35 40 40 40 45 45 45 50 50 50 ρπ ρy 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 5 10 15 20 25 30 35 40 45 (a) e θ= (0.3,0.3) 1 1 1 1 1 22 2 2 2 5 5 5 5 10 10 10 10 10 15 15 15 15 15 20 20 20 20 20 25 25 25 30 30 30 35 35 35 40 40 40 45 45 45 50 ρπ ρy 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 5 10 15 20 25 30 35 40 45 (b) e θ= (0.4,0.4) 1 1 1 1 1 2 2 2 2 2 2 55 5 5 5 5 10 10 10 10 10 15 15 15 15 20 20 20 25 25 25 30 30 30 35 35 35 40 45 ρπ ρy 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 5 10 15 20 25 30 35 40 (c) e θ= (0.5,0.5) 1 1 1 2 2 2 2 2 2 5 5 5 5 5 5 10 10 10 10 10 15 15 15 20 ρπ ρy 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 2 4 6 8 10 12 14 16 18 (d) e θ= (0.7,0.7) Figure 7: Sets of ρπand ρy(vertical axis) for which selected policies are robust at different levels of dLs in per cent, noted on curves defining the boundaries of the sets. The policies are optimal conditional on the given values of ρπand ρyin parentheses below the figures. In the two-dimensional case considered here, b `is represented by the fraction of the area defined by all possible parameter values: %π×%π. In our simulations, we let each of the degrees of persistence take on the following values: 0, 0.1, 0.2,...0.9 and evaluate robustness of policies conditional 21 on 100 different values of the θvector. ρπ ρy ∆Ls = 1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.0 0.2 0.4 0.6 0.8 1.0 (a) e θ= (ρπ, ρy); dLs= 1 ρπ ρy ∆Ls = 2 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.0 0.2 0.4 0.6 0.8 1.0 (b) e θ= (ρπ, ρy); dLs= 2 ρπ ρy ∆Ls = 5 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.0 0.2 0.4 0.6 0.8 1.0 (c) e θ= (ρπ, ρy); dLs= 5 ρπ ρy ∆Ls = 15 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.0 0.2 0.4 0.6 0.8 1.0 (d) e θ= (ρπ, ρy); dLs= 15 Figure 8: Robustness of all policies considered at indicated levels of acceptable losses. The policies considered are optimal conditional on 100 different combinations of ρπand ρyvalues, where both ρπand ρytake on the following values: 0, 0.1, 0.2,...,0.9. The policies are identified by the ρπ- and ρy-values indicated on the axes. Robustness is measured as the share of the 100 different parameter combinations for which the loss remains below the acceptable losses. Figures 7.a–d show the parameter spaces for which the losses do not exceed acceptable loss levels when policies are conditional on given parameter vectors. The parameter spaces vary highly with the policies as well as with the relative loss levels; see also Figure 8. The figures display the main properties of the robust-satisficing approach consistent with the results for the case of uncertainty in single parameters. For example, U(b `, e θ) representing areas associated with a given robustness level increase with relative loss level dLsand at a decreasing rate. This suggests increasing costs of robustness. In particular, one would have to accept a relatively large loss to perform satisfactorily under every possible combination of %π- and %y-values within their assumed ranges. In particular, the cost of the robust-satisficing policy coinciding with the min-max policy would be high. Figures 8.a–d compare the robustness of different policies at selected levels of relative loss. Robustness is portrayed on the z-axis, while the x- and the y-axis indicate the parameter values on which policies Ω e θs are conditioned. The surface plots correspond to the strongly concave robustness curves in Figures 1and 4. 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(ed.), 1999, Monetary Policy Rules, A National Bureau of Economic Research Conference Report, University of Chicago Press, Chicago. Tetlow, R. and von zur Muehlen, P., 2001, Robust monetary policy with misspecified models: does model uncertainty always call for attenuated policy?, Journal of Economic Dynamics and Control, 25 (6-7): 911–949. Rudebusch, G.D. and Svensson, L.E.O., 1999, Policy rules for inflation targeting, in J. Taylor, ed., Policy Rules for Inflation Targeting, University of Chicago Press for NBER, pp.203–246. Vinot, P., Cogan, S. and Cipolla, V., 2005, A robust model-based test planning procedure, Journal of Sound and Vibration, 288: 571–585. 29 30 WORKING PAPERS (ANO) FROM NORGES BANK 2003-2007 Working Papers were previously issued as Arbeidsnotater from Norges Bank, see Norges Bank’s website http://www.norges-bank.no 2003/1 Solveig Erlandsen Age structure effects and consumption in Norway, 1968(3) – 1998(4) Research Department, 27 p 2003/2 Bjørn Bakke og Asbjørn Enge Risiko i det norske betalingssystemet Avdeling for finansiell infrastruktur og betalingssystemer, 15 s 2003/3 Egil Matsen and Ragnar Torvik Optimal Dutch Disease Research Department, 26 p 2003/4 Ida Wolden Bache Critical Realism and Econometrics Research Department, 18 p 2003/5 David B. 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Farooq Akram What horizon for targeting inflation? Research Department, 45 p 2007/14 Q. Farooq Akram, Yakov Ben-Haim and Øyvind Eitrheim Robust-satisficing monetary policy under parameter uncertainty Research Depatrment, 33 p Q. Farooq Akram, Yakov Ben-Haim and Øyvind Eitrheim: Robust-satisficing monetary policy under parameter uncertainty Working Paper 2007/14 KEY WORDS: Robust monetary policy Knightian uncertainty Parameter uncertainty Info-gap decision theory - 43902