Inflation Target Learning, Monetary Policy, and U.S. Inflation Dynamics

Transcription

1 Inflation Target Learning, Monetary Policy, and U.S. Inflation Dynamics Joachim Goeschel November 2007 Abstract This paper studies two different monetary policy regimes in an economy in which private agents are learning, as they cannot directly observe the central bank s inflation target and the persistence of inflation. Under the first regime, the policymaker uses a simple linear rule of the type commonly employed in the related literature. This policy rule disregards the non-linearities introduced by adaptive learning of the boundedly rational private sector. Under the second regime, the central bank has superior information and conducts policy optimally, taking into account private-sector learning. Based on a calibration exercise, I show that not only under the simple policy, but also under the optimal policy, adaptive learning increases the volatility and persistence of inflation as compared to the rational expectations benchmark. Using post W.W. II U.S. inflation and output data, I then estimate a simplified version of the model tractable enough to be taken to the data via minimum-distance estimation. The central finding is that the optimal policy fits the data better than the more commonly used linear policy rule. Contact information: Boston College, Department of Economics, 140 Commonwealth Ave, Chestnut Hill, MA ; goeschel@bc.edu. I am deeply indebted to Fabio Ghironi, Matteo Iacoviello, and Peter Ireland for their guidance and suggestions. I would also like to thank Marina Pavan and seminar participants at Goethe Universität for their helpful comments. All remaining errors are mine. This paper previously circulated under the title Learning about the inflation target.

2 1 Introduction This paper analyzes the conduct of optimal monetary policy when the private sector learns about an uncertain economic structure. The investigation centers on the question whether such a framework can capture features of post World War II U.S. inflation and output dynamics as well as models studied in the related literature which consider linear policies rules that are not fully optimal. For this purpose, I add parameter uncertainty to a simple Phillips curve model. The fundamental underlying assumption is that the private sector cannot directly observe the central bank s inflation target and the persistence of inflation. Following most of the recent learning literature, 1 private agents beliefs about the uncertain parameters evolve according to adaptive learning. I consider two different assumptions regarding the conduct of policy. Under the first assumption, the policymaker employs a simple rule corresponding to the linear rule that would be optimal if private agents had rational expectations (RE). Under the second, alternative assumption, the monetary policymaker understands the mechanism according to which private agents form inflation expectations and thus bases its policy on the actual law of motion of inflation under learning. To study the differences between these policies, I compare their dynamic implications using stochastic simulations. This calibration-based exercise suggests that, in terms of inflation variability, differences with the RE benchmark are of a similar magnitude for both policies. That is, even though the simple rule understabilizes inflation, which results in a higher welfare loss, inflation variability remains significantly higher than under RE under both policy regimes. The paper then takes a simplified version of the model in which the public learns only about the persistence of inflation to the post World War II U.S. data. The estimation uses a minimum-distance approach. 2 The central result is that the model featuring optimal policymaking fits the data better than the model in which the 1 For an overview, see Evans and Honkapohja (2001). 2 A simplified model as well a limited-information estimation approach is chosen to make the estimation particularly of the optimal monetary policy version feasible computationally. 2

3 central bank uses a simple rule (which would result from the more conventional linear-quadratic framework). Furthermore, while learning substantially improves the fit of the model when policy is conducted optimally, it does not improve the fit as compared to RE when the policymaker follows a simple rule. When policy is described by a simple, linear rule, backward-lookingness ( intrinsic persistence ) in the Phillips curve does just as well a job as adaptive learning. Taking the results from the simulations and the estimation together, the results show that learning affects the volatility of inflation in absense of any suboptimal behavior by the monetary authority, which has been identified in previous studies as a factor contributing to the high inflation of the 1960s and 1970s (see, for example, Boivin and Giannoni, 2002; Clarida, Gali, and Gertler, 2000, among others). Private-sector learning about the inflation target is motivated by a different strand of literature which stresses that the Fed s inflation target shifted over time (e.g., Cogley and Sargent, 2005a; Ireland, 2007, and the references therein). Although here, the true inflation target is fixed, the inflation target perceived by households may vary over time. The feedback from beliefs onto inflation and monetary policy provides an additional channel affecting the level and variance of inflation. Furthermore, learning about the persistence of inflation is motivated by private agents not knowing exactly the policy conducted by the central bank. 3 The way adaptive learning is introduced here is standard in the related literature (see, for example, Cogley and Sargent, 2005b; Milani, 2005a, 2006; Orphanides and Williams, 2003; Primiceri, 2006). While the aforementioned studies suggest that adaptive learning can be a powerful propagation mechanism, it is unclear whether or not it remains to be so under optimal policymaking. 4 In terms of the theoretical model, the paper is related to Orphanides and Williams (2003), who were among the first to investigate the performance of simple mon- 3 Uncertainty at least by the aggregate private sector about the structural parameters characterizing preferences and technology would provide an alternative rationale for private-sector learning. 4 Of the aforementioned references, Milani (2005a, 2006) and Orphanides and Williams (2003) concentrate on private-sector learning, whereas Cogley and Sargent (2005b) and Primiceri (2006) study central-bank learning. 3

4 etary policy rules when private agents are perpetually learning, since they have imperfect knowledge of the economy s structure. My work is perhaps most closely related to Gaspar, Smets, and Vestin (2006), who also study optimal monetary policy under adaptive learning. This paper differs from Gaspar, Smets, and Vestin (2006) in three aspects. First, I employ a Phillips curve and policy objective that permits persistent responses by the policymaker to inflation shocks even under rational expectations. Second, I allow for the possibility of a non-zero inflation target, which has implications for the economy s law of motion as perceived by private agents that differ from Gaspar, Smets, and Vestin (specifically, agents estimate an equation that includes a constant term). Moreover, here, agents perceived law of motion is constructed such that beliefs will be asymptotically unbiased. Third, and most importantly, unlike Gaspar, Smets, and Vestin, this paper contains an empirical investigation. This allows me to assess whether learning remains a quantitatively relevant propagation mechanism under optimal policymaking. The paper is organized as follows. Section 2 describes the model under RE (no learning) and characterizes its solution under discretion. Section 3 replaces RE with adaptive learning. I first discuss how the simple, RE policy fares in the learning environment and then characterize optimal monetary policy when the central bank has superior information and knows how the boundedly rational private sector forms expectations according to adaptive learning. Section 4 analyzes and compares the dynamic implications of both policies by stochastic simulations of the calibrated model. Section 5 explains the necessary simplification that makes estimation feasible and reports the empirical findings. Concluding remarks are in Section 6. A technical appendix discusses the numerical dynamic programming algorithm used to solve the policymaker s problem under learning. 4

5 2 The Model Under Rational Expectations 2.1 The Setup This section presents a simple economic structure that is designed to generate a non-trivial tradeoff between inflation and output variability and allows for a nonzero inflation target. The Phillips curve is taken from the backward and forwardlooking components model considered, among others, by Clark, Goodhart, and Huang (1999). In that model, inflation is determined by a Lucas-type supply function which is augmented to include some endogenous inflation persistence. This setup is similar to a version of the New Keynesian Phillips curve that is modified to include some endogenous inflation persistence proposed, among others, by Woodford (2003) and Clarida, Gali, and Gertler (1999). 5 I follow the standard practice of introducing learning at the level of the log-linearized model. The Phillips curve is given by: π t = (1 γ)ẽ t π t+1 + γπ t 1 + κx t + u t, (1) where π t is the period t inflation rate, x t the output gap, u t is a normal i.i.d. costpush shock with mean zero and variance σ 2, γ [0,1) measures the intrinsic persistence of inflation, and κ > 0 determines the sensitivity of inflation to excess demand. Ẽ t denotes the private-sector expectation conditional on time t information. In this section, private-sector expectations are rational, thus Ẽ t π t+1 = E t π t+1, the mathematical expectation of π t+1 conditional on the information set at t. Under rational expectations, there does not exist a long-run tradeoff between inflation and output variability in this model, as the coefficients on expected future inflation and lagged inflation sum to unity. To keep the model as simple as possible, I assume that the central bank can control the output gap directly. The policymaker s period loss takes the form L t = (π t π ) 2 + λxt 2, (2) 5 Woodford (2003, p ) fully develops such a New Keynesian variant from individual agents decision problems. 5

6 where π is the central bank s inflation target and λ > 0 is the relative weight on output gap stabilization. Considering the same model, Walsh (2003) suggests a period loss function of this form, albeit with π = 0. The central bank chooses a sequence for the output gap to minimize the expected, discounted infinite sum of L t, i.e., the central bank s objective is to minimize E 0 t=0β t L t, where β (0, 1) is the discount factor. As discussed in the Introduction, the focus is on optimal monetary policy under discretion. Thus, the policymaker reoptimizes every period taking the process by which the private sector forms (rational) expectations as given. The central bank s problem, however, does not reduce to a sequence of static problems because current policy actions affect future inflation (through the endogenous state variable π t 1 ), and policy takes this intertemporal link into account. 2.2 Some Analytical Intuition The central bank s problem from the previous subsection can be stated in recursive form as V (π t 1,u t ) = min x t [L t + βe t V (π t,u t+1 )] (3) subject to (1), where V denotes the value function. Since the policy problem is linear-quadratic, the value function will be quadratic. 6 Furthermore, due to the linear-quadratic nature of the problem, the optimal policy will be linear and we can guess the solution π t = a 0 + a 1 u t + a 2 π t 1, (4) where a 0, a 1, and a 2 are coefficients to be determined. Anticipating the learning problem in the next section, the goal here is to show that the constant term a 0 in 6 This problem is similar to Walsh (2003, p ). I follow the strategy taken there to find the policy rule. The slight complication here is that π appears in the objective. 6

7 general will be different from zero, unless π = 0. The reason is that below, I will follow a standard assumption in the learning literature which formulates the private sector s learning problem based on the functional form of the rational expectations equilibrium under discretion. Thus, showing that in equation (4), a 0 is different from zero unless π = 0 motivates why agents include a constant term in their perceived model (if they face uncertainty about the true value of π ). Using the proposed solution for inflation (4) and a quadratic value function (similarly with coefficients to be determined), we can take first-order and envelope conditions and equate coefficients to show that and where ψ 1 (1 γ)a 2. a 0 = κ2 π + γ[βγ{1 (1 γ)(1+a 2 )}+(1 γ)]a 0 κ 2 + βλγ(γ ψa 2 )+λψ 2, a 1 = a 2 = ψλ κ 2 + βλγ(γ ψa 2 )+λψ 2, ψλγ κ 2 + βλγ(γ ψa 2 )+λψ 2, The expression for a 2 is a cubic, which yields three solutions (exactly one of which is stable). It is, however, not handily written down analytically. Nevertheless, for later reference, note that the coefficient a 0 = 0 if π = 0 but, generally, a 0 0 if π 0. In other words, with a non-zero inflation target, there is a constant in the rational expectations solution for inflation. 2.3 Numerical Results Even though the policy problem under discretion lacks a convenient analytical solution, numerical methods are available to find the optimal policy and the rational expectations equilibrium. A suitable numerical algorithm is described in Söderlind (1999, Section 1.4). As in the previous subsection, this approach uses the linearquadratic nature of the problem: the value function will be quadratic and the relation between the forward-looking variable (i.e., π t ) and the state variables (i.e., a 7

8 constant, u t, and π t 1 ) will be linear. The policy rule takes the form x t = c 0 + c 1 u t + c 2 π t 1. (5) After writing down the Bellman equation and taking first-order and envelope conditions, we can derive a Riccati equation which we can iterate until convergence. To compute the numerical solution, I calibrate the parameters as shown in Table 1. The calibration of β, κ, and σ is reasonably standard. I calibrate γ to be 0.3, which is somewhat smaller than estimates closer to one-half found under RE (e.g., Fuhrer and Moore, 1995) in light of Milani (2005b), who shows that the need for the backward-looking term in the Phillips curve is reduced under adaptive learning. I set λ = 0.05, which is in line with what normative analysis suggests (see, for example, Rotemberg and Woodford, 1997). However, I discuss the sensitivity of the results to this choice of λ below, as part of the related literature considers substantially higher weights on output stabilization (e.g., Orphanides and Williams, 2003). Finally, I set π = 0.009, which approximately corresponds to the average (quarterly) inflation rate in the post W.W. II U.S. data. The optimal policies for π t and x t are shown in Figure 1 and a simulation of the responses to a cost-push shock are shown in Figure 2. Not surprisingly, in Figure 1 we see that, under the optimal policy, inflation is increasing in the cost-push shock u t, whereas the output gap is decreasing in u t. Thus, the policymaker counteracts cost-push shocks by pushing aggregate demand in the opposite direction. Since the calibration puts a relatively small weight on output stabilization λ, policy x t responds vigorously to cost-push shocks. Moreover, policy is set tighter (i.e., x t is lower) when π t 1 is high. Starting from an initial situation where π 0 = π and x 0 = 0, the simulation in Figure 2 depicts the dynamic responses to a one standard deviation cost-push shock in period 0. Recall that u t is an i.i.d. shock. Nevertheless, in this model with intrinsic inflation persistence, both π t and x t display a persistent response to a transitory shock. After roughly 5 periods, the effects of the shock have largely levelled off, inflation is very close to π, and output is almost back to potential. Note that, at least for moderate degrees of backward-lookingness γ, this model provides less 8

9 persistence than observed in the U.S. data, which motivates the introduction of the learning as a persistence-generating mechanism below. 7 3 The Model Under Adaptive Expectations In this section, I move away from the assumption that the private sector has rational expectations and consider adaptive learning as an alternative mechanism of expectations formation. Following a common assumption in the learning literature (see, for example, Evans and Honkapohja, 2001), private agents estimate a model of the same form as the rational expectations solution. Forecasts of inflation are then based on the latest parameter estimates. As outlined in the introduction, I entertain two different assumptions with respect to central bank policy. Under the first assumption, the central bank incorrectly believes that the private sector has rational expectations and therefore uses policy (5), where the coefficients are determined as described in Section 2.3. Under the second, alternative assumption, the central bank has full information about the model (i.e., the central bank is aware that expectations are formed according to adaptive learning). It thus chooses policy optimally, knowing the true model Learning Under rational expectations, the private sector has perfect knowledge of the model s parameters, and the solution for inflation is of the form (4). If, on the other hand, the private sector has imperfect knowledge of the model s parameters, it needs to use the data to learn about the economic structure. In order to obtain estimates about the law of motion of inflation, agents recursively estimate the equation which is of the same form as (4). π t = b 0 + b 1 π t 1 + ε t, (6) 7 See, for example, Benati (2007) for a recent study estimating inflation persistence in the U.S. 8 The definition of the optimal policy follows Gaspar, Smets, and Vestin (2006); it is also similar in spirit to Orphanides and Williams (2003). 9

10 In our departure from rational expectations, beyond assumptions regarding the econometric model agents estimate, we must make a further, informational assumption as to which information is used to forecast π t+1. Here, I assume that π t is used during forecasting but not while updating the parameters. This assumption implies that agents can forecast using real time data but update their forecasting model with a one-period lag. The assumption that agents update the parameters at t using only data up to t 1 simplifies the characterization of the actual law of motion (ALM) discussed below, but has negligible quantitative consequences, as long as changes in agents parameter estimates are small (i.e., the gain is not very large). 9 Formally, agents forecast according to Ẽ t π t+1 = b 0,t 1 + b 1,t 1 π t, (7) where b i,t 1 denotes the estimate of b i using observations of inflation up to t 1, and Ẽ t corresponds to the expectation under adaptive learning. It is instructive to substitute (7) into the Phillips curve (1) to obtain in the terminology of Evans and Honkapohja (2001) the ALM under adaptive learning: π t = 1 1 (1 γ)b 1,t 1 [(1 γ)b 0,t 1 + γπ t 1 + κx t + u t ]. (8) Comparing the ALM (8) with the estimation equation (6) illustrates where bounded rationality comes into play: The coefficients in the ALM are time-varying (because agents parameter estimates and thus their inflation forecasts systematically change over time), whereas the coefficients in the estimation equation are constant. Thus, agents estimate an econometrically misspecified model. The coefficient estimates b 0,t 1 and b 1,t 1 are updated via recursive least squares. I use a constant-gain algorithm mainly because it implies a lower-dimensional state space. Although a decreasing-gain algorithm, which gives equal weight to all observations, might be theoretically more appealing (since the model without learning does not imply any structural change), other studies have shown in similar contexts 9 This timing assumption follows Gaspar, Smets, and Vestin (2006). The implications of different timing assumptions are discussed in Evans and Honkapohja (2001, ch. 3), albeit the focus there is on decreasing-gain learning. 10

11 that this choice has only a minor impact on the results (see, for example, Primiceri, 2006). 10 The parameters in the least-squares regression (6) can be estimated recursively via the updating equations b t = b t 1 + φr 1 t z t 1 [ πt b t 1 z t 1], (9) R t = R t 1 + φ [ z t 1 z t 1 R t 1], (10) [ ] [ ] [ ] b0,t R0,t R 01,t 1 where b t =, R t =, z t =, and φ > 0 is the gain. b 1,t R 01,t R 1,t π t In the numerical analysis of the model, I calibrate φ to be 0.025, which is roughly equivalent to running rolling regressions with a window length of 160 quarters. This choice is conservative when one compares the length of the sample employed by agents at each point in time to the amount of data an econometrician would use in estimating a macroeconomic model. Note that under any reasonable initialization of the learning statistics, and both for the constant-gain and decreasing-gain algorithms, R 0,t = 1 for all t since the first element of z t is unity (and R t is a weighted sum of squares). 3.2 The Rational Expectations Monetary Policy Rule Before turning to optimal monetary policy, it is instructive to lay out some properties of the model with learning if monetary policy employs the same policy as under RE. This policy takes the form (5), where the coefficients are determined numerically as discussed in Section 2.3 above. Following Evans and Honkapohja (2001), the model under RE policy can be analyzed by studying a stochastic recursive system that represents the evolution of the learning statistics. To that end, substitute equation (5) into the ALM (8) to 10 A constant-gain algorithm corresponds to an updating of beliefs following an adaptive expectations formula. Decreasing-gain learning, under appropriate initialization of beliefs, corresponds to recursive ordinary least-squares. Under decreasing-gain learning, the gain is 1/t, so the additional variable t enters the updating equations directly. 11

12 obtain the actual law of motion under the RE policy: where π t = T(b t 1 ) z t 1 + T(b t 1 ) = (1+κc 1 ) 1 (1 γ)b 1,t 1 u t, (11) κc 0 +(1 γ)b 0,t 1 1 (1 γ)b 1,t 1 κc 2 +γ 1 (1 γ)b 1,t 1 Before using equation (11), a slight modification of the updating equations (9) and (10) is necessary to make the system recursive: Iterate (10) one period forward and carry out the change of variable S t 1 = R t. Then substitute (11) in the rewritten updating equations. This yields the stochastic recursive algorithm [ b t = b t 1 + φst 1 1 z t 1 z t 1 (T(b t 1) b t 1 )+. (1+κc 1 ) 1 (1 γ)b 1,t 1 u t ], (12) S t = S t 1 + φ [ z t z t S t 1]. (13) The stochastic approximation literature provides the basis for the analysis of the system (12) - (13). Under appropriate regularity conditions (see Evans and Honkapohja, 2001, ch. 7), the convergence properties of this system can be (approximately) determined by studying an associated ordinary differential equation. In the following, I present only the solutions for b but not those for S. The above system has two equilibria which are the fixed points of T(b t 1 ). The solutions are available in closed form but are complicated, so I report only numerical solutions (under the calibration in table 1): 11 [ ] [ ] b S =, and b US = The conditions determining stability depend only on the estimate of b 1 and are given by and 1 γ 1 (1 γ)b 1 < 1 (κc 2 + γ)(1 γ) (1 (1 γ)b 1 ) 2 < Analytical solutions are available upon request. 12

13 Thus, b S is stable, whereas b US is not. Since the private sector learns using a constant-gain algorithm, the interpretation of this finding is not that convergence will occur to b S almost surely, but rather that beliefs will converge to a distribution which is centered around b S. With constant-gain learning, information is not accumulated. That is, the private sector forgets the realizations of inflation that lie very far in the past. Of course, the rate of oblivion is determined by the gain parameter φ. Note that the rational expectations solution for inflation (4) under the chosen calibration is π t = u t π t 1. Thus, even though the private sector cannot observe u t (and therefore does not estimate a 1 ), its estimates b 0 and b 1 are unbiased asymptotically and the meansquared deviations of b 0 and b 1 from a 0 and a 2, respectively, are small if φ is small. Note that the shock is i.i.d. and therefore, knowledge of a 1 is unnecessary to make a forecast of inflation in t + 1 based on information available in t. As the sample size available to agents approaches infinity and φ 0, agents would make the same inflation forecasts as under RE Optimal Monetary Policy As an alternative assumption, consider the case in which the monetary policy maker is aware of how agents form expectations under learning. The optimal policy problem under learning has four more state variables than the problem without learning discussed above. These additional states are the learning statistics b 0,t 1, b 1,t 1, R 01,t, and R 1,t. As in Section 2 above, the policymaker s period objective is (2). Since the policymaker has perfect knowledge, his minimization is now subject to the Phillips curve under learning and the updating equations. The policymaker s dynamic pro- 12 For that reason, RE may be interpreted as a restricted model (Sargent, 1999). 13

14 gram can be written as V (π t 1,u t,b 0,t 1,b 1,t 1,R 01,t,R 1,t ) = min[l t + βe t V (π t,u t+1,b 0,t,b 1,t,R 01,t+1,R 1,t+1 )] x t [ Z ] = min L t + β V (π t,u t+1,b 0,t,b 1,t,R 01,t+1,R 1,t+1 ) f(u t+1 )du t+1 x t (14) subject to (8), (9), and (10). This problem is recursive after eliminating π t from (9) by using (8). Since the policymaker knows the model, the expectation in (14) is taken with respect to the distribution of u t+1, as it would be in a standard rational expectations model. I denote the (i.i.d. normal) distribution of u by f(u). The constraints in (14) are non-linear, therefore the dynamic program cannot be solved with techniques suitable for linear-quadratic control. However, approximations to the value and policy functions associated with the central bank s dynamic decision problem can be obtained via numerical dynamic programming methods. The numerical algorithm starts with an initial guess for policy and value functions and obtains successively better approximations in an iterative procedure. Solving this problem is technically challenging due to the large number of state variables. Several innovations, including parallelized computing, are therefore used to speed up the algorithm. Details are in the appendix. The optimal policy function resulting from this numerical dynamic programming exercise is a six-dimensional object. To visualize the policy, Figure 3 depicts the policy instrument x t as a function of one state variable at a time, holding the other states at their long-run means. For comparison, I also show the simple rule (which only responds to u t and π t 1 ) in the first row of the figure. Note that, while the policy action depends non-linearly on all other state variables, when varying one state variable at a time, the central bank tightens policy as the public s belief about the mean b 0,t 1 increases and as the persistence of inflation b 1,t 1 increases. Moreover, the optimal policy responds more strongly to lagged inflation than the simple rule. Furthermore, in Figure 3, I expect little response from the policymaker with 14

15 respect to R 01,t and R 1,t, as beliefs b t 1 are close to their means. I interpret the pronounced response of the policymaker to R 01,t when it is large in absolute value as a slight numerical inaccuracy. This, however, has only negligible consequences for the simulations, as it turns out that the probability of observing a value of R 01,t in the simulations of such magnitudes is extremely low. 13 The next section focuses on stochastic simulations, to further understand the implications of the model. 4 Simulation Results This section compares the performance of the RE policy with the optimal policy in the model featuring adaptive learning. I consider the dynamic responses of the economy to cost-push shocks by means of simulation. I compute several statistics to describe the distribution of the model s variables and assess the relative stabilization performance of both policies. Note that the RE policy is feasible for the policy maker that knows how the private sector forms expectations. In part, I therefore attempt to evaluate how much more successful the optimal policy is at stabilizing inflation and output. In other words, does optimal policymaking almost bring us back to RE or can the optimal policy model still account for higher volatility and persistence of inflation? The evolution of the economy depends non-linearly on the initial states b t 1 and R t. The simulation results are therefore based on the following setup. First, I simulate the economy for 200,000 periods. I discard the data on the first 50,000 periods, and use the remaining data to obtain empirical distributions of beliefs and other states. Then, initial conditions for the state vector are obtained by drawing from this empirical distribution using residual-based bootstrapping. I then perform 13 To interpret the responses to R 01,t and R 1,t more generally, note from (10) that the covariance matrix associated with agents beliefs b t 1 is approximately the inverse of R t, up to a factor of proportionality (i.e., the inverse of R t would precisely be the covariance matrix of b t up to a factor of proportionality under decreasing-gain learning). By taking the inverse of R t, it can be shown that the uncertainty agents face about the coefficient vector b t 1 decreases with R 1,t and increases as R 01,t gets larger in absolute value. 15

16 5000 simulations over 1000 periods each, where each simulation uses a different, randomly drawn series of the exogenous shock u t. The results below are averages over these 5000 time series. 4.1 Baseline Calibration Table 2 provides summary statistics for the simulated series (for ease of comparison, the statistics are also provided for the rational-expectations model discussed in Section 2). Monetary policy finds it optimal to accept higher output variability in order to attain lower inflation variability. Given the low weight on output stabilization (i.e., λ = 0.05), the differences in the intertemporal loss between the simple, linear rule and the optimal policy are large. 14 The optimal monetary policy takes into account that cost-push shocks affect not only the endogenous state variable π t 1 but also have persistent effects on the private-sector learning statistics. The optimal monetary policymaker, being aware of movements in private-sector inflation expectations through this channel, conducts additional stabilization policy to prevent drifting private-agents estimates drifting away too much from target. Nevertheless, as measured by the correlation of π t and π t 1, the learning dynamics increase the persistence of inflation above the value under RE (0.41) under both policies, albeit less so under the optimal policy. The reason is that agents forecasting model, and thus their forecasts of inflation, change slowly. This is indicated by the correlation of χ t and χ t 1, where χ t Ẽ t π t+1. The correlation of inflation expectations is significantly higher than it would be in the rational expectations model discussed in Section 2 above. Learning still acts as a propagation mechanism even under optimal monetary policy, which provides the motivation for the more rigorous empirical investigation in the next section. The finding that the policy trade-off is worsened under learning is also reflected in the intertemporal loss: While being lower under the optimal policy than under the simple rule, it is still roughly one and a half times as large as under RE. This substantially higher loss indicates that agents misperceptions about the inflation target and the intrin- 14 I discuss the sensitivity of the results to changes in λ below. 16

17 sic persistence of inflation continue to have quantitatively important consequences even under the sophisticated policy. 4.2 Sensitivity Analysis Increasing the value of λ clearly increases losses under learning relative to RE, regardless of the policy employed. This is intuitive as, given the tradeoff present in the model, a higher (relative) weight on output stabilization increases inflation variability, and fluctuations of inflation are amplified by private-sector learning dynamics. Furthermore, differences between the two policy regimes under learning are less pronounced if λ is large. This is due to the fact that while both policymakers find it more costly to stabilize inflation in that instance, only the optimal policymaker finds it more costly to stabilize private agents beliefs (the simple policymaker, of course, ignores beliefs). Besides of the sensitivity of the simulations to λ, further experiments suggest that the results also strongly depend on the calibration of φ. Specifically, under both policies, a larger φ is associated with a larger variance of inflation, even though somewhat less so under the optimal policy. Instead, under the optimal policy, the variance of the output gap tends to increase more strongly with φ, as the central bank is aware of the fact that movements in private agents beliefs can amplifiy the effects of shocks. The simple, linear rule, on the other hand, does not depend on φ; a central bank following it does not stabilize preemptively but only reacts to deviations of inflation from target the variance of output does increase with φ, too, but only because the variance of inflation increases with φ. 5 Estimation This section presents a simplified version of the model that can be estimated, and discusses the estimation strategy and results. To arrive at a model that can be solved sufficiently fast computationally and thus taken to the data, I make an additional assumption to further simplify the learning algorithm and hence the central bank s 17

18 problem under perfect knowledge originally presented in Section The Simplified Model Specifically, I assume that agents face uncertainty only about the perceived persistence of inflation b 1 and know the constant b 0 in (6). As discussed in Section 2.2 above, b 0 depends on the inflation target π, so implicitly uncertainty about the target is shut off. To this end, I set b 0 equal to its long-run mean for t large, a 0. Agents then can regress π t a 0 on π t 1. This assumption of learning exclusively about b 1 preserves the non-linearity of the model. If instead, b 1 is certain and b 0 uncertain, agents would merely have to update their beliefs about a mean, which would make the central banks problem linear-quadratic. 15 On the other hand, if the aggregate private sector does not know b 1, a slope parameter, it must keep track both of the estimate of the slope itself and of the associated second moment. The optimal policy will therefore react to both the point estimate and the uncertainty associated with it, thus keeping the model as closely aligned as computationally feasible with the full model of Section 3 above, where both coefficients (i.e., b t ) and the uncertainty about them (i.e., R t ) are contained in the central bank s state vector. Agents model thus simplifies to π t = a 0 + b 1 π t 1 + η t, (15) where only b 1 is uncertain, and inflation expectations are given by Ẽ t π t+1 = a 0 + b 1,t 1 π t. (16) The updating equations corresponding to agents perceived model (15) are b 1,t = b 1,t 1 + φr 1 1,t 1 π t 1(π t a 0 b 1,t 1 ), (17) 15 Recursively updating a mean under constant-gain learning sets the updated estimate of the mean by taking a weighted average of its estimate last period and the current observation, where the weights are 1 φ and φ, respectively. This equation would constitute agents only and linear updating equation. Given b 1 is known, the ALM becomes linear as well, and the central bank then faces a linear-quadratic control problem. 18

19 R 1,t = R 1,t 1 + φ ( πt 1 2 ) R 1,t 1. (18) Equations (17) and (18), together with the ALM (8), in which b 0,t 1 is replaced by a 0 for all t, along with the policy rule govern the dynamics of the simplified model. By making b 0 certain, the dimensionality of the problem is reduced substantially, as the optimal policy problem now features only four state variables: u t, π t 1, b 1,t 1, and R 1,t. Analogously to the full problem of Section 3 above, the alternative, simple rule policy which disregards learning dynamics continues to be given by equation (5). 5.2 Estimation Strategy The estimation procedure employed here uses simulated vector autoregressions (for an early, related approach to the estimation of non-linear time series models see Smith, 1993). I estimate the parameter vector θ = [σ,γ,λ,φ,κ], containing the main parameters of interest, whereas the remaining two models parameters, β and π, are fixed at the same values used in the calibration exercise of Table 1. I calibrate β = 0.99, as preliminary estimation attempts suggest that it is difficult to estimate a reasonable value for it; and calibrating β is common practice in the related literature (see, for example, Milani, 2005a). I set the target π = to match the average inflation rate in the U.S. data over the sample discussed below. The estimator of θ, denoted θ, minimizes the weighted, squared distance between the model-based responses, ξ, and the U.S. data-based responses, ˆξ, to an identified inflation shock, obtained from a structural vector autoregression (VAR). The mimimum-distance estimator θ minimizes C = (ξ(θ) ˆξ) ˆΩ 1 (ξ(θ) ˆξ), (19) where ˆΩ 1 is a diagonal weighting matrix containing the inverses of the (analytical) standard errors associated with the orthogonalized impulse responses from the data, and C is the value of the criterion function at θ. The data responses ˆξ are obtained from a bivariate VAR for inflation and output. I detrend inflation (as measured by the log difference in the GDP deflator) 19

20 using the HP filter with parameter The output gap is the difference between log real GDP and log potential GDP (for potential GDP, I use the measure by the Congressional Budget Office as provided by the Federal Reserve Economic Data (FRED) web site). The sample is 1960:I 2006:IV, and the lag length in the VAR is set to three (based on the Akaike information criterion). I choose to detrend inflation to remove low-frequency components from the series. Preliminary estimation attempts using unfiltered inflation led to unplausibly persistent responses of inflation and output to cost-push shocks (e.g., output is only half-way back to baseline after four years). However, the main finding below, namely that the optimal policy version provides the best fit among the models considered, is robust to a specification in which inflation is not detrended. 16 Identification is achieved through a Cholesky decomposition with inflation ordered first, consistent with the model under both policies considered. Figure 4 plots the responses of inflation and output gap to a unit inflation shock along with 95% confidence intervals. 17 The results in Table 4 appear reasonably standard. Particularly, a trivariate structural VAR of inflation, output gap, and a short-term interest rate yields quantitatively similar responses to an inflation shock. I compute model-based responses as follows. Given θ, I simulate the model for T + N periods, where T is the length of the sample used to estimate the VAR from U.S. data. I discard the first N = 1000 periods of artificial model data to eliminate the dependence on initial conditions. Furthermore, I retain only the simulated series m t = (π t,x t ),t = N + 1,...,N + T (i.e., I do not use simulated series for the other three state variables in obtaining model-based responses). The idea is then to estimate a VAR for the artificial data, making the same identifying assumption as for the historical U.S. data, and compute the responses to a shock to inflation. However, even though the model has tight implications for π t and x t, it is driven by a single shock and thus stochastically singular. Therefore, to avoid a degenerate 16 Results are available upon request. 17 I compute confidence intervals and ˆΩ using bootstrapping. Analytical standard errors for the orthogonalized impulse-response functions computed following Hamilton (1994, ch. 11.7) yield slightly smaller confidence intervals. 20

21 conditional likelihood function, I follow Sargent (1989) in adding a small amount of measurement error to the system, the justification being that the actual data is at least minimally error-ridden as well. More specifically, I augment the simulated path of m t by a white noise residual ν t. I assume that Eν t ν s = { S if s = t 0 otherwise, Eν t = 0 for all t, Eν t u s = 0 for all t, s, and S = [ S1,1 0 0 S 2,2 ]. For simplicity, I calibrate the standard deviations of the measurement errors to S1,1 = S 2,2 = The error-ridden data is then given by m t = m t + ν t. Preliminary estimation attempts suggest that these small measurement errors make the data m t sufficiently well-conditioned. 18 For given θ, the model responses ξ(θ) are then obtained by approximating the central bank s policy rule using numerical dynamic programming, generating 5000 series of artificial, error-ridden data, m t, obtaining the orthogonalized impulse responses for each series, and averaging over the 5000 orthogonalized impulse responses. Finally, in the estimation, I restrict the parameter φ to be in the range [0, 0.05], as preliminary estimation attempts showed that, for the simple rule version, the estimation routine inevitably pushed the gain parameter to unreasonably large values. However, a very large gain poses a problem under the simple rule as it may render the model non-stationary. 5.3 Estimation Results I carry out the estimation separately for the two policy regimes. The results are reported in Table 3, which, as a point of reference, also provides estimates of the 18 In principle, the elements of S could be estimated as suggested by Ireland (2004). 21

22 RE model in the last column. 19 The learning models (results are provided in the first two columns of Table 3) introduce an additional free parameter, φ, relative to the RE benchmark. I find that with adaptive learning and the simple policy, I obtain almost the same fit (as measured by C) and parameter estimates as under RE. Consequently, the model responses to a cost-push shock are also very similar, as shown in Figure 5. The reason is that under the simple rule, the data prefers that agents are almost rational, as indicated by the gain which is estimated to be As argued in Section 4 above, the variance and persistence of inflation increases with φ under both policies. But, when policy is conducted according to the linear rule, the estimation procedure prefers intrinsic inflation persistence (as measured by a high γ) over boundedly rational agents (as measured by φ) in matching the data responses of inflation and output to an identified cost-push shock. The central finding from Table 3 is that the model featuring optimal policy fits the data better than the version of the model featuring the simple rule (and the RE model), as indicated by the value of the criterion function C. Under the optimal policy, the output response is much more persistent, matching the data better (see Figure 5). To interpret this result, recall that this paper studies discretionary monetary policy. Even though the optimal monetary policymaker is free to reoptimize every period and cannot commit, he is aware that his policy actions influence the data observed by boundedly rational agents. Understanding the mechanism of inflation expectations formation, the policymaker realizes that unless policy action is taken cost-push shocks have a prolonged effect on private-sector beliefs, which move back towards their means more slowly than inflation. 21 Since, for example, a very high b 1,t makes future shocks more costly, the optimal policy contracts output more strongly and more persistently. The response of output the period after the 19 The standard errors reported in parentheses are computed using the asymptotic delta function method applied to the first-order condition associated with the minimization problem (see Greene, 2003, equation (18-11)). 20 Recall that when φ 0, and the sample available to the boundedly rational agents gets large, agents would make the same forecasts as under RE, as discussed in Section 3.2 above. 21 This, of course, assumes that agents do not discount past observations too heavily, which only occurs if φ is unreasonably large relative to γ. 22

23 shock hits is almost identical to the response in the period the shock hits, better matching the output response in the data than the close to geometric decay under the simple rule and under RE. This is consistent with the strong role the optimal policymaker assigns to stabilizing beliefs, as the propagation of a shock to inflation through beliefs onto inflation expectations is characterized by a one-period lag. A central coefficient estimated is the gain parameter φ. Using data on inflation expectations by the Survey of International Forecasters, Orphanides and Williams (2006) find values of the gain ranging from I estimate a gain in this range under the optimal policy, under which the data prefers modeling agents behaving like econometricians. Under the simple rule, instead, the data prefers modeling agents being (almost fully) rational. It is interesting to note that Milani (2005a) also finds a gain in this range under the assumption that the policymaker follows a simple, Taylor-type rule in which the nominal interest rate is set in response to deviations of the private sector s boundedly rational expectations of next period s inflation and output gap from their respective targets. Here, a model in which agents behave as econometricians fits best, and also, the policymaker takes private-sector expectations formation into account. Lastly, in both the two learning models and the RE benchmark, λ, the weight on the output gap in the central bank s objective function, and κ, the elasticity of inflation with respect to the output gap, are estimated imprecisely. However, this has only minor consequences on the estimated model responses. For example, if in the RE benchmark, I restrict the relative weight on output stabilization to be λ = 0.1, κ is estimated to be 0.049, with little change in the estimates ˆσ and ˆγ and only a moderate increase of C to If, on the other hand, I set κ = 0.05 and estimate the remaining parameters, the estimate of λ is 0.12, again with only a small increase of C to 38.1 and minor changes in the estimates of σ and γ. This suggests that the data contains little information that would allow for obtaining precise estimates of both parameters. 23

24 6 Conclusions I study monetary policy under private-sector learning and characterize differences between the optimal policy when the central bank understands private-sector expectations formation and a particular simple rule, namely the policy that would be optimal under RE (which therefore responds only to state variables relevant under RE). Replacing RE by adaptive learning increases inflation forecast errors by the private sector, and learning acts as a propagation mechanism. The paper argues that this propagation mechanism is quantitatively important when the policymaker has superior information and policy is conducted fully optimally. A calibration exercise demonstrates that, as compared to the simple rule, the optimal policy reduces the welfare loss by making inflation less persistent at the cost of making output respond more persistently and in stronger magnitude to costpush shocks. The results suggest that even under the optimal policy, the difference between the adaptive learning framework and the RE benchmark remains nontrivial. When estimating a simplified setup using minimum-distance estimation, I find that the optimal policy version fits the data better than the simple rule version, as well as the standard framework in which expectations are rational. While the simulation results demonstrate that learning can generate inflation persistence, the estimation results indicate that adaptive learning is preferred as a propagation mechanism over structural (or, intrinisc) persistence in the Phillips curve only when policy is conducted optimally. The estimation exercise suggests that learning is quantitatively important in complete absense of policy mistakes because the central bank under the optimal policy vigilantly responds to cost-push shocks in order to stabilize private-sector expectations the prolonged output response under the optimal policy matches the U.S. data better than the corresponding responses under both the simple rule and under RE. While the assumption that the central bank has perfect knowledge is simplifying, the idea that the central bank observes private-sector beliefs has intuitive appeal if one takes as a given that expectations are boundedly rational. As pointed out by Orphanides and Williams (2006), under RE there is no special merit for 24

25 the central bank in observing the public s expectations. But just as central banks in practice do seem to follow movements in inflation expectations, in the optimal policy version of the model the public s beliefs are taken into consideration (in addition to the natural state variables, the cost-push shock and lagged inflation), as they do matter for policy. Given the different empirical results under both policies, optimal policymaking when the public is boundedly rational deserves further attention in future research. Appendix The optimal policy for the central bank s problem in Section 3 is computed by iterating over the Bellman equation (14) employing a discrete state space method (see, for example, Judd, 1998). Associated with this Bellman equation is the contraction mapping { TW = min L(π,u,b 0,b 1,x) x Z + β W ( π ( ),u,b 0( ),b 1( ),R 01( ),R 1 ( ) ) f(u )du }, where next period s state variables except for u have been eliminated using the constraints (8), (9), and (10) (it is not shown explicitly that next period s states are functions of the current state and x). In the functional equation, T denotes the functional operator, L( ) is the period loss (where R 01 and R 1 do not enter directly), and W( ) is a continuous function defined on the state space. Starting from any W, repeated applications of the operator T lead to successively better approximations of the value function. The computational algorithm therefore involves guessing W for a discrete set of values in the state space and storing this guess in an array W 0. I then apply the operator T to W 0 to obtain a new array W 1. Next, an upper bound to the approximation error of the value function can be computed as a function of W 0 and W 1. If the approximation error is too big, I apply T to W 1 to obtain W 2, compute the approximation error based on W 1 and W 2, etc. 25