Parameter Uncertainty and Non-Linear Monetary Policy Rules

Transcription

1 Parameter Uncertainty and Non-Linear Monetary Policy Rules Peter Tillmann 1 University of Bonn February 26, 2008 Abstract: Empirical evidence suggests that the instrument rule describing the interest rate setting behavior of the Federal Reserve is non-linear. This paper shows that optimal monetary policy under parameter uncertainty can motivate this pattern. If the central bank is uncertain about the slope of the Phillips curve and follows a min-max strategy to formulate policy, the interest rate reacts more strongly to inflation when inflation is further away from target. The reason is that the worst case the central bank takes into account is endogenous and depends on the inflation rate and the output gap. When inflation increases, the worst-case perception of the Phillips curve slope becomes larger, thus requiring a stronger interest rate adjustment. Empirical evidence supports this form of non-linearity for post-1982 U.S. data. Keywords: parameter uncertainty, robust control, non-linear Taylor rule, optimal monetary policy, Federal Reserve policy JEL classification: E43, E52 1 University of Bonn, Institute for International Economics, Lennéstr. 37, Bonn, Germany, tillmann@iiw.uni-bonn.de

2 1 Introduction The interest rate setting behavior of central banks is routinely described by estimated interest rate rules. In the baseline specification going back to Taylor (1993), for example, the policy instrument, i.e. the short-term interest rate, is linearly related to contemporaneous inflation and the output gap. These estimated rules perform remarkably well in replicating post-1982 Federal Reserve policy. 2 Moreover, these rules are essential to central bank communication and model building alike. Recent empirical evidence points to important non-linearities in interest rate setting that are neglected in the standard specification of estimated Taylor rules. For example, Dolado et al. (2004, 2005) include an interaction term between inflation and the output gap in an otherwise standard Taylor rule. They are able to show that policy behaves non-linearly after Their most successful model is one in which the inflation rate and the output gap interact in the policy rule. Kim et al. (2005) use a flexible nonparametric method to document non-linearity in the Fed s policy rule prior to 1979, but fail to show non-linearity thereafter. In addition, central banks frequently announce a target range around their inflation target, i.e. small deviations of inflation from target are tolerated while large deviations are fought vigorously. From a theoretical point of view, non-linearity in the policy rule can be motivated in three different ways. First, the underlying aggregate supply schedule might be nonlinear leading to a non-linear adjustment of the policy rate, see Nobay and Peel (2000) and Dolado et al. (2005). Second, the preferences of the policymaker might not be quadratic in output and inflation. 3 Surico (2007), among others, models asymmetric preferences in a standard New-Keynesian model. The resulting non-linear interest rate rule performs well in the pre-volcker period but shows less signs of asymmetry in the post-volcker era. Third, the policymaker might face uncertainty. Meyer et al. (2001) and Swanson (2006) show that non-linearities might stem from uncertainty about the natural rate of unemployment, formalized by a non-gaussian prior distribution and a non-linear updating rule. As a result of the signal extraction problem, the central bank is more cautious about adjusting interest rates in response to small output gaps than in a standard Taylor rule but more aggressive when they reach a certain threshold. 4 2 See, among others, Clarida et al. (1998, 2000), Judd and Rudebusch (1998), and Jondeau et al. (2004). 3 See also Ruge-Murcia (2003), Nobay and Peel (2003), and Cukierman and Muscatelli (2008) for the implications of non-standard, i.e. asymmetric, preferences in monetary policy models. A closely related literature proposes an opportunistic approach to monetary policy, see Orphanides and Wilcox (2002). According to this view, the Fed tolerates moderate levels of inflation above the target and waits for favorable circumstances to reduce inflation. 4 A series of speeches by Federal Reserve Governor Meyer provides narrative evidence for this kind 2

3 This paper adds to the analysis of the third source of non-linearity, i.e. to monetary policy under uncertainty. In contrast to Meyer et al. (2001), we assume that the central bank is uncertain about a key parameter governing the transmission process of monetary policy, which is the slope of the Phillips curve in an otherwise standard New- Keynesian model. 5 In this paper, the linearity of the Phillips curve and the quadratic nature of the loss function are retained. The key contribution is to show that nonlinearity results from optimal monetary policy if the central bank follows a min-max strategy to take account of parameter uncertainty. Policymakers aim at setting interest rates optimally given a particular reference model but, at the same time, admit that they cannot be completely certain about the true model specification. As a result, central banks want to formulate robust policies that are to some extent immune with respect to model disturbances. They set interest rates so as to minimize the maximum harm to the economy. Such a policy concept is known as a robust control approach to policymaking and was pioneered by Hansen and Sargent (2007). 6 Given this policy approach, the resulting optimal interest rate rule includes not only the inflation rate and the output gap, but also an interaction term between output and the squared inflation rate. If the central bank is uncertain about the slope of the Phillips curve and follows a worst-case strategy to formulate policy, the interest rates react more strongly to inflation when inflation is further away from target. The reason is that the worst case the central bank takes into account is endogenous and depends on the size of the inflation rate. When inflation is high, the loss from a misspecified parameter is particularly high. Hence, the central banks becomes more vigorous in fighting inflation. A robustness-concerned central bank tolerates small deviations of inflation from target, but strongly counteracts larger movements of inflation. We provide empirical evidence that supports this form of non-linearity for post-1982 U.S. data. This paper is organized as follows. Section two presents the model and solves for optimal min-max policy under uncertainty. Section three studies the properties of the resulting non-linear instrument rule and section four provides empirical support for the form of non-linearity analyzed here. Section five draws some conclusions. of non-linearity, see Meyer (2000). 5 See Giannoni (2002), Rudebusch (2001), and Söderström (2002) for an analysis of monetary policy rules under parameter uncertainty. 6 The special attention policymakers pay to the worst-case outcome is supported by narrative evidence, see Greenspan (2004) or recently Mishkin (2008). 3

4 2 Optimal policy rules under uncertainty The section outlines the role of parameter uncertainty and robust monetary policy in an otherwise standard New-Keynesian model. 2.1 The model We employ the standard New Keynesian model as a laboratory, see e.g. Woodford (2003) for a complete derivation. The forward-looking Phillips curve (1) and the IS curve (2) represent log-linearised equilibrium conditions of a simple sticky-price general equilibrium model π t = βe t π t+1 + κx t + ξ t (1) x t = E t x t+1 σ 1 (i t E t π t+1 r n t ) (2) where π t is the inflation rate, x t the output gap, i t the risk-free nominal interest rate controlled by the central bank, and E t is the expectations operator. All variables are expressed in percentage deviations from their respective steady state values. The discount factor is denoted by β<1, σ is the coefficient of relative risk aversion, and κ, theslopecoefficient of the Phillips curve, depends negatively on the degree of price stickiness. Shocks to the Wicksellian natural real rate of interest are i.i.d. and are denoted by rt n N (0, 1). The precise nature of the shock process plays no particular role for the subsequent analysis. The central bank is uncertain about the slope coefficient κ. 7 In particular, the policymaker knows that his reference value κ might be subject to model distortions z to be explained below κ = κ + z (3) The central banker also faces an i.i.d. control error ξ t with mean zero. Thus, policy is unable to use observations on inflation and the output gap to back out k. Monetary policy is assumed to set the policy instrument, i.e. the short term interest rate, in order to minimize the welfare loss due to sticky-prices which is described in terms of inflation volatility, output gap volatility, and interest rate variance weighted by the parameters λ x,λ i > 0 1 min i t 2 E 0 X h i β t (π t π ) 2 + λ x x 2 2 t + λ i i t (4) t=0 7 As in Leitemo and Söderström (2008), the model is simple enough to facilitate an analytical solution of the policy problem. However, in contrast to their contribution, the central bank is uncertain about a particular parameter of the model with the z distortion directly affecting the parameter value instead of affecting the disturbance terms. 4

5 where π is the constant inflation target. In the absence of misspecifications z, minimizing (4) subject to the model in (1) and (2) would give a set of first-order conditions, from which the optimal policy response to shocks could be computed. The task is to reformulate the central bank s optimization problem such that the resulting policy rule performs well even if the model deviates from the reference model. We transform the minimization problem into a min-max problem. The central bank wants to minimize the maximum welfare loss due to model misspecifications by specifying an appropriate policy. To illustrate the problem, we introduce a fictitious second rational agent, the evil agent, whose only goal is to maximize the central bank s loss. The evil agent chooses a model from the available set of alternative models and the central bank chooses its policy optimally. Hence, the equilibrium is the outcome of a two-person game. Note that the evil agent is a convenient metaphor for the planner s cautionary behavior. Let z denote the evil agent s control variable, i.e. the parameter misspecification. The only constraint imposed upon the fictitious evil agent is his budget constraint requiring X E t β τ z 2 ω (5) τ=0 Hence, the parameter ω measures the amount of misspecification the evil agent has available. The standard rational expectations solution for optimal monetary policy corresponds to ω =0, such that the evil agent s budget is empty. 2.2 The policy problem Throughout the paper we assume that policy is unable to commit to the optimal inertial plan. Instead, policy is conducted under discretionary optimization. The policymaker solves X h i min max E 0 β t (π t π ) 2 + λ x x 2 2 i t z t + λ i i t (6) t=0 subject to (1), (2), and (3). The Lagrangian of the policy problem can be written as follows min i t max z L = (π t π ) 2 + λ x x 2 t + λ i i t 2 θ (z) 2 (7) μ π t (π t βe t π t+1 ( κ + z) x t ξ t ) μ x t xt E t x t+1 + σ 1 (i t E t π t+1 rt n ) where μ π t and μ x t denote the Lagrange multipliers associated to the inflation adjustment equation and the consumption Euler equation, respectively. The Lagrange parameter θ is inversely related to ω. Hence, the rational expectations case corresponds to θ. 8 8 In this case, the evil agent maximizes the welfare loss by choosing z =0. 5

6 Alowerθ means that the central bank designs a policy which is appropriate for a wider set of possible misspecifications. Therefore, a lower θ is equivalent to a higher degree of robustness. The central bank plays a Nash game against the evil agent, who wants to maximize the welfare loss. Optimization under discretion results in the following set of first-order conditions λ x x t +( κ + z) μ π t μ x t = 0 π t π μ π t = 0 λ i i t μ x t σ 1 = 0 θz + μ π t x t = 0 Together with the second condition, the fourth condition states that z =(π t π ) x t θ 1. The larger is the central bank s concern for robustness, i.e. the lower θ, thelargerthe model distortion. Likewise, the evil agent s choice of z positive depends on both the output gap and inflation. Hence, the worst case policy outcome against the central bank wishes to shield the economy is endogenous. Intuitively, model uncertainty matters most if inflation and output exhibit large deviations from their steady state values. 9 The first order conditions can be combined to eliminate the Lagrange multipliers λ x x t + κ (π t π ) σλ i i t =0 with κ = κ +(π t π ) x t θ 1 (8) When the inflation rate is above target and κ is known, the central bank has to raise the interest rate to contract the economy. When the central bank fears κ to be misspecified, a higher inflation rate also affects the slope coefficient κ. Sonotonly does the central bank face an increase in inflation, but also witnesses an increase in κ, i.e. its instrument becomes less effective in dampening aggregate demand. As a result, the size of the interest rate adjustment depends non-linearly on the inflation rate. 3 The non-linear instrument rule Equation (8), which links all three endogenous variables, can be solved for i t to obtain an expression that resembles a conventional Taylor rule augmented by a non-linear term i t = κ (π t π )+ λ x x t + 1 h x t (π t π ) 2i (9) σλ i σλ i θσλ i 9 These first order conditions link the three endogenous variable irrespective of whether the misspecification of the underlying model actual occurs, i.e. whether the reference model turns out to be in fact the undistorted. 6

7 The interest rate responds not only to the level of inflation and the output gap, but also to the product of the squared inflation deviation and the output gap. Note that the non-linear term disappears once we approach the rational expectations benchmark, i.e. θ. Suppose that the central bank observes an increase in inflation. Equation (10) shows that the interest rate response depends on the level of inflation and the output gap i t (π t π ) = κ + 2 [x t (π t π )] (10) σλ i θσλ i The interest rate response grows in the inflation rate. The higher the level of inflation, the stronger (for a positive output gap) the central bank adjusts interest rates to fightanincreaseininflation. 10 Furthermore, when the output gap is positive, the interest rate adjustment is stronger for positive inflation rates than for corresponding (in absolute terms) negative inflation rates. Hence, uncertainty not only introduces non-linearity, but also asymmetry into the optimal policy stance. Likewise, the interest rate response to the output gap depends on the squared level of inflation i t = λ x + 1 (π t π ) 2 (11) x t σλ i θσλ i If inflation is high, the interest rate is raised stronger to contract the economy than in a situation with moderate inflation. The precise interest rate step in this case depends on the parameterization. To visualize the degree of non-linearity in the Taylor rule, we choose standard parameter values to calculate the coefficients. In order to derive the interest rate rule, a positive interest weight in the central bank s loss function is essential. We choose to set λ x =0.25, which is a frequently used benchmark parameterization, and set the penalty on interest rate changes to λ i =0.10. The rational expectations case corresponds to θ RE =. We choose θ robust =10to illustrate the effect of uncertainty. We assume an inflation target of zero, i.e. π =0. The other parameters are set to κ =0.10, β =0.99, andσ =1.80. The resulting interest rate response to inflation and output gap movements is depicted in figure (1). The non-linear response to inflation is clearly evident. A robustness-concerned central bank tolerates small fluctuations of inflation around the target, but forcefully counteracts larger deviations from target. Hence, the model also rationalizes that central banks frequently announce a target zone, typically π ± 1%, around their inflation target. Inflation is fought mildly inside the zone, but strongly once it leaves the target range As in Giannoni (2002), the interest rate response to inflation within the Tayor rule increases as the central bank s degree of uncertainty becomes larger. 11 See Orphanides and Wieland (2000) for another model of inflation zone targeting. 7

8 The period loss function of the form L =(π t π ) 2 + λ x x 2 t + λ i i 2 t can be derived as an approximation to the households utility function in the presence of transactions frictions that motivate a demand for money. Woodford (2003, p ) shows that the optimal weights λ x and λ i depend on the underlying model structure. In particular, they depend on κ perceived by the central bank λ x = Ω 1 κ and λ i = Ω 2 λ x (12) where Ω 1, Ω 2 > 0 depend on the model parameters, including the interest rate semielasticity of money demand. This expression clearly shows the cross-equation restriction implied by the underlying theory. Any variation in κ should be reflected in variations of the weights λ x and λ i. 12 As a consequence, the misspecification z affects the weights the central bank attaches to conflicting objectives. If inflation increases, κ = κ+(π t π ) x t θ 1 also increases for a positive output gap leading to larger weights λ x and λ i. This dampens the degree of non-linearity in (9). 4 Empirical Evidence Is the non-linear instrument rule derived above empirically supported? To answer this question, we rewrite (9) in a form that corresponds to the large literature on estimated Taylor-type interest rate rules i t = φ i ī + φπ π t + φ x x t + φ π 2 xx t π 2 t ª +(1 φi )i t 1 (13) where ī is a constant and φ i, φ π, φ x,andφ π 2 x are reduced form coefficients to be estimated. 13 In accordance to the large literature on estimated Taylor rules, we add the lagged interest rate to account for the high degree of inertia in the policy instrument. Again, the inflation target is set to zero. Let G t 1 denote a vector of instruments that are dated t and earlier and, thus, are orthogonal to the inflation forecast error. Imposing rational expectations defines the following orthogonality conditions E t {f (Θ) G t 1 } =0,wheref (Θ) is a function of the parameter vector Θ. In this context, the reduced form specification of the GMM estimation is given by E t it φ i ī + φπ π t + φ x x t + φ π 2 xx t π 2 t ª (1 φi )i t 1 Gt 1 ª =0 (14) We estimate this equation using U.S. data for the period 1982:3-2006:4. The inflation rate is the annualized rate of change of the personal consumption expenditure deflator 12 See Walsh (2005) for a detailed analysis of the consequences of endogenous weights for optimal monetary policy. 13 Clarida et al. (1998, 2000), Judd and Rudebusch (1998), and Jondeau et al. (2004) estimate similar, though linear, specifications. 8

9 (PCE). We also report results based on the PCE inflation rate net of changes in food andenergyprices,whichisthefed spreferredinflation measure. The output gap is the deviation of real GDP from the trend estimated by the Congressional Budget Office (CBO). The interest rate is the Federal Funds rate. The set of instruments includes six lags of π t and i t, and four lags of the output gap. The results for the post-1982 period and the Greenspan-Bernanke era, i.e. post 1987, are presented in table (1). Most importantly, the non-linear term x t π 2 t enters positively in both periods and for both inflation series. 14 When the non-linear term enters the rule, however, the output gap coefficient is no longer significant. In line with the theory outlined above, the Fed has adjusted interest rate more aggressively the further inflation was away from steady state. The size of this non-linear response is lower in the post 1987 sample than in the sample that includes the aggressive disinflation post Conclusions This paper showed that optimal monetary policy under parameter uncertainty can motivate a non-linear interest rate rule that is supported by U.S. data. While the linearity of the Phillips curve and the quadratic nature of the loss function are retained, the nonlinearity of the policy rule solely stems from the assumption of a min-max approach to parameter uncertainty. The crucial idea is that if the policymaker tries to avoid particularly bad outcomes, i.e. if she sets policy according to a min-max strategy, the maximum harm is endogenous and depends on the size of the output gap and the inflation rate. As a result, the policy response to inflation becomes stronger, the higher the inflation rate and the larger the output gap. The resulting non-linear Taylor rule is supported by U.S. data from the post-1982 period. In contrast to the bulk of the literature, these results do not stem from non-linearity in the Phillips curve or non-quadratic central preferences. Certainly, the nature of parameter uncertainty analyzed here is overly simplistic. Not only is the central bank uncertain about a key parameter, but also gains no information about this parameter even if the central bank repeatedly plays against the evil agent. However, the basic principle appears to be relevant to interpret actual policy decisions. 14 Note that existing evidence, e.g. Dolado et al. (2004, 2005), include the product of the levels of inflation and the output gap, i.e. π tx t, as a regressor. 9

10 References [1] Clarida, R. J. Galí, and M. Gertler (1998): "Monetary policy rules in practice: Some international evidence", European Economic Review 42, [2] Clarida, R. J. Galí, and M. Gertler (2000): "Monetary policy rules and macroeconomic stability: evidence and some theory", Quarterly Journal of Economics 115, [3] Cukierman, A. and A. Muscatelli (2008): "Nonlinear Taylor rules and asymmetric preferences in central banking: Evidence from the United Kingdom and the United States", unpublished, Tel-AvivUniversity. [4] Dolado, J., R. M.-D. Pedrero, and F. J. Ruge-Murcia (2004): "Non-linear monetary policy rules: Some new evidence for the U.S.", Studies in Nonlinear Dynamics & Econometrics 8, Issue 3, Article 2. [5] Dolado, J., R. Maria-Dolores, and M. Naveira (2005): "Are monetary policy reactions functions asymmetric? The role of non-linearity in the Phillips curve", European Economic Review 49, [6] Giannoni, M. P. (2002): "Does model uncertainty justify caution? Robust optimal monetary policy in a forward-looking model", Macroeconomic Dynamics 6, [7] Greenspan, A. (2004): "Risk and uncertainty in monetary policy", American Economic Review 94, [8] Hansen, L. P. and T. J. Sargent (2007): "Robustness", book manuscript, available at [9] Jondeau, E., H. Le Bihan, and C. Gallès (2004): "Assessing Generalized Methodof-Moments estimates of the Federal Reserve reaction function", Journal of Business and Economic Statistics 22, [10] Judd, J. P. and G. D. Rudebusch (1998): "Taylor s rule and the Fed ", Federal Reserve Bank of San Francisco Economic Review 1998, No.3. [11] Kim, D. H., D. R. Osborne, and M. Sensier (2005): "Non-linearity in the Fed s monetary policy rule", Journal of Applied Econometrics 20, [12] Leitemo, K. and U. Söderström (2008): "Robust monetary policy in the New- Keynesian framework", Macroeconomic Dynamics 12,

11 [13] Meyer, L. H. (2000): "Structural change and monetary policy", Remarks before the Joint Conference of the San Francisco Fed and the Stanford Institute of Economic Policy Research, March 3, [14] Meyer, L. H., E. T. Swanson and V. W. Wieland (2001): "NAIRU uncertainty and non-linear policy rules", American Economic Review 91, [15] Mishkin, F. S. (2008): "Monetary policy, flexibility, risk management, and financial disruptions", Speech at the Federal Reserve Bank of New York, January 11, [16] Nobay, A. R. and D. A. Peel (2000): "Optimal monetary policy with a nonlinear Phillips Curve", Economics Letters 67, [17] Nobay, A. R. and D. A. Peel (2003): "Optimal discretionary monetary policy in a model with asymmetric central bank preferences, TheEconomicJournal113, [18] Orphanides, A. and V. Wieland (2000): "Inflation zone targeting", European Economic Review 44, [19] Orphanides, A. and D. W. Wilcox (2002): "The opportunistic approach to disinflation", International Finance 5, [20] Rudebusch, G. D. (2001): "Is the Fed too timid? Monetary policy in an uncertain world", The Review of Economics and Statistics 83, [21] Ruge-Murcia, F. J. (2003): "Inflation targeting under asymmetric preferences", Journal of Money, Credit, and Banking 25, [22] Söderström, U. (2002): "Monetary policy with uncertain parameters", Scandinavian Journal of Economics 104, [23] Surico, P. (2007): "The Fed s monetary policy rule and U.S. inflation: The case of asymmetric preferences", Journal of Economic Dynamics and Control 31, [24] Swanson, E. T. (2006): "Optimal non-linear policy: signal extraction with a nonnormal prior", Journal of Economic Dynamics and Control 30, [25] Taylor, J. B. (1993): "Discretion vs policy rules in practice", Carnegie-Rochester Conference Series on Public Policy 39,

12 [26] Walsh, C. E. (2005): "Endogenous objectives and the evaluation of targeting rules for monetary policy", Journal of Monetary Economics 52, [27] Woodford, M. (2003): Interest and Prices, Princeton University Press: Princeton. 12

13 Table 1: Results from reduced form GMM estimates of the non-linear contemporaneous-variable Taylor rule Parameters Test ī φ i φ π φ x φ π 2 x J core PCE (1.253) PCE (1.380) 1982:3-2006: (0.031) (0.584) (1.163) (0.247) (0.947) (0.026) (0.395) (0.193) (1.081) (0.019) (0.493) (0.838) (0.150) (0.018) (0.405) (0.258) 1987:3-2006:4 core PCE (0.732) (0.032) (0.392) (0.594) (0.128) (0.591) (0.024) (0.262) (0.152) PCE (0.680) (0.021) (0.325) (0.325) (0.057) (0.943) (0.023) (0.386) (0.230) Notes: Newey-West corrected standard errors in parenthesis. The last column reports p-values from J-tests of the validity of overidentifying restrictions. 13

14 4 interest rate inflation output 2 10 interest rate inflation output 2 Figure 1: The interest rate as described by a linear (upper panel) policy rule and a non-linear (lower panel) policy rule 14