Precautionary Learning, Endogenously Asymmetric. Loss Functions and Inflationary Biases

Transcription

1 Precautionary Learning, Endogenously Asymmetric Loss Functions and Inflationary Biases Chetan Dave New York University James Feigenbaum Utah State University September 2, 212 Abstract In a canonical model of monetary policy, if the central bank estimates the parameters of the Phillips curve while disregarding the role of the public s expectations then the bank s loss function is endogenously asymmetric with respect to these estimates. Depending on the true value of the Phillips-curve slope, overestimates may be more or less costly than underestimates, introducing a precautionary motive when estimating these parameters. Thus, e ciencydemands the use of a variance-adjusted least-squares estimator for learning about the inflation-unemployment trade-o. Under such precautionary learning the central bank sets low inflation targets, and the economy settles near a Ramsey equilibrium. We would like to thank John Du y, Tom Sargent and seminar participants at the Southern Economics Association Conference, Society for Nonlinear Dynamics and Econometrics Conference, the Federal Reserve Banks of Dallas and Richmond, and the Computational Economics and Finance Conference for insightful comments. We also thank Wouter den Haan and two referees for outstanding advice. The usual disclaimer applies. Address: New York University, Department of Economics, 19 W. 4th Street, 6FL, New York, NY, 112, USA. cdave@nyu.edu. Address: Utah State University, Jon M. Huntsman School of Business, 3565 Old Main Hill, Logan, UT., j.feigen@aggi .usu.edu.

2 1 Introduction In conventional macroeconomic policy models policymakers minimize a symmetric loss function over relevant variables subject to constraints imposed by the economic environment. In practice, a policymaker often determines optimal policy while simultaneously learning the parameters of her model. In this paper we show that, when learning the parameters of her model, if the policymaker uses the same loss function to evaluate policy results and econometric e ciency, the loss function need not be symmetric with respect to parameter estimates. This asymmetry induces a precautionary motive that favors a variance-adjusted learning algorithm, use of which can have important implications for policy dynamics. We focus on a canonical monetary policy context exemplified by the framework of Kydland and Prescott (1977) and Barro and Gordon (1983) (hereafter, KPBG), which assumes the central bank minimizes a quadratic loss function of inflation and unemployment subject to a Phillips curve. 1 Sargent (1999) and Cho et al. (22) have studied how learning impacts optimal decision-making in this framework when ordinary least-squares regressions are used to learn the parameters of the Phillips curve. As Leland (1968) and Sandmo (197) first established in the context of precautionary saving, an asymmetric loss function will cause optimal decisions to shift under uncertainty. If there is a single stochastic variable in the model, the direction of this shift depends on the sign of the third derivative of the loss function, and the size of the shift is proportional to the variance of the stochastic variable. In an econometric setting, this translates to employing an estimator with a bias proportional to the variance of an unbiased least-squares estimator (Varian (1975) and Zellner (1986)). While the third derivatives of the central bank s loss with respect to inflation and unemploy- 1 While we focus attention on a monetary policy context, models of least squares learning have yielded unique results in a fiscal policy context as well. For example, Sargent and Wallace (1973), employing perfect foresight, find that interest rates decrease initially and then return to the steady state in response to an announced, credible, one time increase in government spending given a balanced budget constraint. When Evans et al. (29) instead assume that agents adaptively learn about the path of future interest rates, they find that announced future permanent increases in government spending can lead to low interest rates before policy implementation and then higher than steady state interest rates after implementation before convergence to the steady state. Bullard (26) and Evans and Honkapohja (23) provide comprehensive surveys on learning and monetary policy design and we focus attention on that context in this paper. 1

3 ment vanish in the conventional KPBG framework, 2 we show that, after expressing inflation and unemployment in terms of the Phillips curve parameters, the loss function is no longer symmetric. We then explore the inflation dynamics that result if the central bank employs such precautionary learning, using a variance adjusted estimator to learn the Phillips curve parameters. It is well known that the KPBG model with complete information about the Phillips curve delivers one of two outcomes depending on the modeler s choice of equilibrium concept: a time-inconsistent low inflation outcome that corresponds to a Ramsey equilibrium, and a time-consistent high inflation outcome that corresponds to a Nash equilibrium. The Ramsey equilibrium yields a smaller loss function, but realizing this lower loss is feasible only if the central bank can tie its hands and refrain from exploiting the Phillips curve to lower unemployment. Under adaptive learning 3 of a misspecified Phillips curve in which the e ect of the public s expectations (which are di cult to model) are disregarded, the long-run outcome depends on how the policy maker treats new data. For a decreasing-gain learning algorithm in which each new datum is given less weight than new data were at previous iterations, inflation will converge to its Nash equilibrium value. For a constant-gain learning equilibrium in which each new datum is weighted the same with each iteration, 4 Cho et al. (22) found under least-squares learning that inflation will spend most of its time near its Nash equilibrium value, but occasionally escapes to the Ramsey outcome. Escapes occur because a sequence of unusual shocks can lead the policy maker to lower her estimate of the Phillips-curve slope. This reduces the apparent trade-o between un- 2 The central bank s loss function could also be made explicitly asymmetric. Cukierman (22) argues that central banks target positive inflation, in spite of theory suggesting a Friedman rule with optimal deflation, because their loss function is asymmetric, putting more weight on evading deflation than on evading inflation. Recent concern over the zero lower bound in interest rates indicates another example of an asymmetry in central banks loss functions. For the dynamic consequences of an explicitly asymmetric loss function in a monetary policy context see Ruge-Murcia (23). 3 See Evans and Honkapohja (1999, 21), who provide a seminal treatment of this alternative to rational expectations. 4 Constant gain implies that more recent data is weighted more heavily than past data. If, for example, the central bank only included a fixed window of data when updating its estimates, the learning algorithm would be constant gain. 2

4 employment and inflation, so the policy maker lowers her inflation target, which produces data confirming that the trade-o has indeed diminished and is even smaller than originally thought. The policy maker further reduces the inflation target and the process continues until inflation is driven to zero and the economy replicates the Ramsey outcome. Once the targeted level of inflation is at zero, the negative correlation between inflation and unemployment becomes apparent again. The central bank then ratchets up inflation, and the economy heads back toward the high-inflation Nash outcome. However, assuming the misspecified model is correct, the recursive ordinary least-squares (ROLS) estimator employed by the central bank in Cho et al. (22) is ine cient because the econometric loss function is endogenously asymmetric. This stems from the nonlinear nature of the central bank s reaction function. Because the central bank s optimization problem is set up as a linear-quadratic problem, the solution is linear in the public s expectation of inflation (which gets set to zero in the misspecified model where the public s expectations are disregarded). However, the solution is nonlinear with respect to the parameters of the model, specifically the slope of the Phillips curve. Consequently, when the reaction function is substituted back into the objective function to obtain the econometric loss as a function of the parameter estimates, the third derivative of this econometric loss function is in general not zero. Note that this endogenous asymmetry in the econometric loss function is not specific to the KPBG model but will apply to any policy model with learning about the model parameters in which the policy maker s problem is linear-quadratic. 5 Precautionary learning should be more e cient than ordinary least-squares learning in a wide variety of policy models. For the monetary-policy example, the third derivative depends on the true value of the Phillips curve slope. If the central bank is behaving optimally, it will not employ an ROLS estimator and instead will follow Varian (1975) and Zellner (1986) by using a varianceadjusted least-squares (VALS) estimator. This augments the ROLS estimator by a term 5 Of course, if the policy maker s problem is more complicated than linear-quadratic, there will usually be an exogenous asymmetry to begin with. 3

5 proportional to the variance of the ROLS estimator. We term the constant of proportionality the precautionary parameter, thoughitisnotanexogenous parameterofthemodel. Rather, it is a choice variable of the econometrician. If the econometrician is behaving e ciently, the precautionary parameter will be set to minimize her loss function just as she would set the gain of the algorithm (or an experimental scientist would set the parameters of his experiment to produce the most accurate results). The optimal choice for the precautionary parameter depends on the third derivative of the loss function. This in turn depends on the true value of the Phillips-curve slope, which is unknown a priori. Just as the econometrician has to learn the parameters of the Phillips curve by observation, she will have to learn how to calibrate the optimal precautionary parameter through experimentation. Returning to a world in which the actual law of motion di ers from the central bank s perceived law of motion, we simulate the economy and plot the expected loss as a function of the precautionary parameters, allowing for asymmetries in both Phillips curve parameters. Consistent with the preceding argument, the loss function is not minimized when the precautionary parameters are set to zero. In fact, ROLS, which is nested in the VALS family of estimators, is near a ridge line of the loss function rather than being close to a minimum. An upward bias of slope estimates reduces long-run inflation since the central bank believes asmallincreaseininflationwillproduceabigdecreaseinunemployment. Italsoreduces the frequency of escapes since zero slope estimates will be more infrequent. A downward bias of slope estimates increases the long run level of inflation and the frequency of escapes. Within the neighborhood of ROLS, biasing the slope estimates downward increases the central bank s loss function. However, we find that the loss function exhibits a non-monotonic dependence on the precautionary parameters. For a su ciently large downward bias, the escape frequency is so high that the economy hardly returns to its high long-run level, and average inflation is very low. With a large enough bias in either direction, the policymaker achieves average outcomes close to the Ramsey outcome. This can occur with a smaller slope bias if, in addition, the econometrician biases down the intercept estimate as well, leading 4

6 the policy-maker to lower her inflation target. 6 The paper is structured as follows. In Section 2 we review the Cho et al. (22) framework that we adopt; show that the loss function is asymmetric when expressed as a function of the estimated Phillips curve parameters, and discuss the VALS estimator. In Section 3 we show how the expected loss function varies with the precautionary parameters and explain how the behavior of unemployment and inflation are impacted by such precautionary learning. We conclude in Section 4. 2 The Model We adopt the model of Cho et al. (22) in which a central bank chooses its target inflation rate x t to minimize a quadratic loss function defined over realized inflation ( t )and unemployment (u t ) L = E[ 2 t + u 2 t ], (1) where > is the weight placed by the central bank on u t. 7 The relationship between u t and t is governed by a Phillips curve (that is, the actual law of motion (ALM)) for the economy u t = u NR! ( t bx t )+ 1 W 1t,u NR >,! >, 1 >, (2) in which t is determined by t = x t + 2 W 2t, 2 >. (3) 6 There is no principal-agent tension between the policy-setting division of the central bank and the econometrics division since they are both minimizing the same loss function. Willful ignorance is not entirely without theoretical precedence (see Carrillo and Mariotti (2)). Relatedly, within a managerial compensation context, Sinclair-Desgagné and Spaeter (211) show that optimal incentive pay is concave in performance if a principal exhibits precaution over net final wealth. Here we explore the role of precaution in learning. 7 For simplicity, the central bank is by assumption myopic in that it only cares about the e ect of its current policy on impending inflation and unemployment. Its policy will a ect future inflation and unemployment by influencing the stream of future data that will be used to estimate the Phillips curve in later periods, but the central bank does not consider these e ects when it sets policy. 5

7 The mutually independent shocks (W 1t,W 2t ) are i.i.d. and normally distributed with zero means and unit variances. In the Phillips curve (2), bx t is the private sector s expectation of inflation at t and the parameters u NR and! are respectively the actual natural rate of unemployment and slope. Since the public s expectations are rational, bx t will ultimately be set equal to x t in equilibrium. The outcomes that arise when the central bank knows the parameters u NR and! are well known: if x t is fixed after the public forms bx t, the highinflation Nash outcome is a targeted inflation rate of x t =! u NR ;ifx t is fixed before the public forms bx t, the low-inflation Ramsey outcome would have the central bank set x t =. Since the Nash outcome, unlike the Ramsey outcome, is time-consistent, the central bank su ers from an inflationary bias absent a technology that allows it to commit to the Ramsey rule. Under adaptive learning, the central bank believes instead in a misspecified version of the Phillips curve: u t = u NR,t! t t + t, (4) where t is a mean-zero noise variable uncorrelated with the other time-t dated variables. The intercept and slope terms in this perceived law of motion (PLM) are indexed by t since every period the central bank re-estimates the Phillips curve. One motivation for why the central bank might estimate (4) instead of (2) is that it avoids the di cult problem of modeling how the public forms its expectations. Given (3), (4) and dropping time subscripts, the central bank believes the loss function is L(x u NR,!) =x 2 + (u NR!x) (1+! 2 ) 2 2 (5) when expressed as a function of its policy variable, the inflation target x. Given the current estimates bu NR,t and b! t,thecentralbankchoosesx t to minimize L(x t bu NR,t, b! t ),whichgives 6

8 the optimal target inflation rate x(b! t, bu NR,t )= b! tbu NRt 1+ b! 2. (6) t Note that (6) is nonlinear in b! t. It is this nonlinearity that drives our ensuing results. If the central bank believed it was constrained by (2) instead of (4), we would still obtain a reaction function for the central bank that is nonlinear in b! t.thediscussionwouldbemore complicated, however, since we would need to address how the central bank models bx t Precaution in Statistical Decision Making Following Berger (1985), suppose that a hypothetical statistical decision maker (SDM) is estimating a parameter 2 R so as to minimize the loss function f(, b ), where b is his estimate of. ArationalSDMoperatinginafrequentistparadigmwillspecifyanestimator b (y), whichisafunctionofthedatay, withtheobjectiveofminimizingtheexpectedloss function E h f(, b i (y)), (7) where the expectation is taken over y. Toensurethatfindingthetruthisoptimal,weassume that f is positive definite so f(, b ) for all (, b ) and f(, b )=if and only if = b. Further assuming f is b (, ) = (8) 2 b 2 (, ) (9) for all. 8 Branch and McGough (25) discuss misspecification and nonlinear dynamics within the context of restricted perceptions equilibria. 7

9 Given, ataylorexpansionoff(, b ) for b near is f(, b )=f(, )+ 2 f b 2 (, )(b ) f b 3 (, )(b ) 3 + O ( b ) 4. (1) Thus the SDM wishes to choose a function b (y) that minimizes E 2 f b 2 (, )(b (y) ) f b 3 (, )(b (y) ) 3, where again the expectation is with respect to y. 2 b 2 (, ) h i E b (y) b 3 (, )E h( b i (y) ) 2 =. (11) If the third derivative of f at b = vanishes, (11) reduces to the condition that the SDM h i should choose a consistent estimator such that E b (y) =. Inthemoregeneralcasewhere the third derivative does not vanish, however, an optimal estimator will satisfy h i E b (y) = 3 f/@ b 3 (, ) 2 f/@ b 2 E h( b i (y) ) 2 3 f/@ b 3 (, ) (, ) 2 f/@ b 2 V (, ) h b (y) i (12) h i since, if (12) holds, E b (y) for i =1,..., m. 3 f/@ b 3 (, ) >, thelossfunctionis higher if b (y) = + " than if b (y) = ", where" >. Consequently, it is optimal for the SDM to bias down his estimate, erring on the side of caution, since it is less costly to underestimate than it is to overestimate. The more imprecise his estimate b (y) is, the larger the bias should be. Conversely, 3 f/@ b 3 (, ) <, heshouldbiashisestimatesin the opposite direction. Suppose that b u (y) is an unbiased estimator with variance 2 (y). Then(12)suggeststhe SDM replace the unbiased estimator with a variance-adjusted estimator b v i (y) = b u i (y)+ 1 2 a 2 (y). (13) 8

10 The parameter a in (13) is what we call the precautionary parameter. Optimally, given(12), the SDM should choose a f/@ b 3 (, 2 f/@ b 2. (14) (, ) For Varian s (1975) LINEX loss function f(, b )=exp a( b ) + a( b ) 1, (15) the right hand side of (14) is constant and is determined by the loss function. For a more general loss function as in our monetary-policy example, (14) may be di cult to compute, in which case the choice of a will have to be calibrated through experimentation. 2.2 Asymmetry in the KPBG Loss Function What are the properties of the central bank s econometric loss function, assuming it seeks to minimize its loss function (1) when estimating the parameters of the Phillips curve? Since the choice of what policy to set at time t given the latest estimates is not a dynamic problem, we suppress time indices in this section. We also treat the misspecified Phillips curve (4) as the actual law of motion for the economy since that is what the central bank believes to be the case as it tries to determine the most e cient estimator. Thus the econometric loss function is obtained by substituting the optimal inflation target (6) given the current parameter estimates (b!, bu NR ) into the perceived policy loss function (5), which depends on the true, albeit unknown parameters (!,u NR ): bl(b!, bu NR,!,u NR )=L (x(b!, bu NR )!,u NR ) = 1+! 2 2 b!bu NR 1+ b! 2 2u NR! 2 b!bu NR 1+ b! 2 + (u2 NR + 2 1)+(1+! 2 ) 2 2 (16) 9

11 The first partial derivatives of (16) with respect to the estimates b =2(1+!2 ) 2 b!bu 2 NR (1 + b! 2 ) 2 4(1 +!2 ) 3 b! 3 bu 2 NR (1 + b! 2 ) 3 2u NR! 2 bu NR 1+ b! 2 +4u NR! 3 b! 2 bu NR (1 + b! 2 ) 2 If b! =! and bu NR = u b NR =2 1+!2 (1 + b! 2 ) 2 2 b! 2 bu NR 2u NR 2 b!! 1+ b! 2. b (!,u NR)=2 2!u 2 NR 1+! 2 2 2!u 2 NR 1+! 2 4 3! 3 u 2 NR (1 +!) ! 3 u 2 NR (1 +! 2 ) 2 b NR (!,u NR )= 2 2! 2 u NR 1+! 2 2u NR 2! 2 1+! 2 =. Since the loss function is quadratic in bu 2 b 2 NR (!,u NR )= 2 2! 2 1+! 2 and all higher-order derivatives with respect to bu NR vanish, so it is optimal that the central bank should learn the truth about u NR,andaconsistentestimatorofu NR is optimal. Thus in the following we can set bu NR = u NR.Thissimplifiesthepartialderivativewithrespectto b!, (17),whichisthenexactlyproportionalto 2 u 2 NR : L b b! 2 u 2 =2(1+!2 ) (1 + b! 2 ) 4(1 + b! 3 2!2 ) (1 + b! 2 ) 2! 3 1+ b! 2 + 4!b!2 (1 + b! 2 ) 2 The second derivative of the loss function is then given by 1 2 u 2 2 b 2 (!) =2+4!2 +2 2! 4 8! 2 (1 +! 2 ) 3 = 2 4!2 +2 2! 4 (1 +! 2 ) 3 = 2(1!2 ) 2 (1 +! 2 ) 3, 1

12 which is strictly positive as long as! 6= 1/2.Thethirdderivativeis 3 L b 12! 2 u 2 3 (!) = (1 +! 2 ) (3 4!2 )(1! 2 ), (18) which only vanishes for the knife-edge cases where! 2 =1or! 3 =3. This means there generally will be an asymmetry in the econometric loss function for the Phillips curve slope, and optimal estimation will require a biased estimator for!. Thesignoftheasymmetryis ambiguous and depends on the true Phillips curve slope, which a priori is unknown. 2.3 The VALS Estimator Under the constant-gain ROLS learning algorithm, at each period t the estimates t = (bu NRt, b! t ) are updated using the least-squares formula t =(Z tz t ) 1 Z tu t, (19) where U t =(u,..., u t 1 ) and Z t =[1 (,..., t 1 ) ]. Let R t = gz tz t be the matrix of second moments. Defining z t =(1, t ),theestimatesandr t evolve according to t = t 1 + gr 1 t z t 1 (u t 1 z t 1 t 1 ), (2) R t = R t 1 + g z t 1 z t 1 R t 1, (21) in which, unbeknownst to the central bank, the u t 1 in the evolution of t above is determined by the ALM and g denotes the gain, which can be interpreted as the inverse of the learning horizon T. 9 Aconstant-gainlearningalgorithmisappropriateifthepolicymakerisconcerned about structural change to the Phillips curve and believes data from say, 1 years ago, is not informative about what is happening today. 9 If the learning algorithm was of the decreasing gain variety the gain would decrease with t and the economy would converge to the Nash equilibrium. 11

13 The constant gain ROLS estimator departs from a decreasing-gain algorithm in that escape dynamics can arise. If by chance the Phillips curve in the last T periods appears flat, the inflation rate can temporarily drop to the low value of the Ramsey equilibrium. Specifically, under escape dynamics a sequence of unusual shocks causes a policy maker to lower the estimate of the trade-o between unemployment and inflation. As the trade-o diminishes, the policy maker responds by lowering the inflation target. This response produces data confirming that the trade-o has indeed diminished, and, in fact, suggests that the trade-o is even smaller than originally thought. The policy maker then further reduces the inflation target and the process continues until inflation is driven to zero and the economy replicates the Ramsey outcome. Once the targeted level of inflation is at zero, the negative correlation between inflation and unemployment eventually becomes apparent again. Witnessing this return through their constant-gain ROLS learning process, the government begins ratcheting up inflation, and the economy heads back toward the high inflation Nash outcome. It is this constant-gain ROLS learning case that we generalize. The first step in our analysis is the assumption of precaution in the interpretation of data when the specification of an estimator is viewed from the perspective of statistical decision theory (Berger (1985)) to obtain a constant-gain variance-adjusted least-squares (VALS) estimator of (u NRt,! t ). This estimator, denoted by a t,isdefinedby a t = t ta, (22) where t is the variance-covariance matrix of the least squares estimator t and (a unr,a! ) are the precautionary parameters that determine whether the estimator biases up or down the two Phillips curve parameters u NR and!. If either a unr or a! is negative (positive) then the bias for that variable is against overestimation (underestimation). The variance covariance matrix is given by t = T 1 R 1 t 2 t, (23) 12

14 where 2 t = 1 T 1 is the most recent estimate of the variance of t.wedefine Xt 1 (u i zi t ) 2 (24) i= S t = 1 T Xt 1 u i zi, (25) i= which satisfies the di erence equation S t = S t T (u t 1x t 1 S t 1 ). (26) Then we can rewrite 2 t as 2 t = 1 T 1 Xt 1 (ui zi t 1 ) 2 +2(u i t 1z i )zi( t 1 t )+( t 1 t )z i zi( t 1 t ), i= which simplifies to 2 t = 2 t T 1 [(u t 1 zt 1 t 1 ) 2 1+T 1 zt 1R t 1 z t+1 +2 t 1R t S t R 1 t z t 1 (u t 1 z t 1 t 1 )]. (27) 3 Simulation Results 3.1 Loss Function Surfaces Given an environment in which agents adaptively learn, the usual procedure is to stochastically approximate the system of equations that characterize model dynamics, including the Ricatti equations associated with ROLS regressions. Such an approximation for mean dynamics delivers a system of di erential equations that can be examined for stability of model equilibria. Cho et al. (22) defined escape dynamics as well for the case of ROLS 13

15 learning under constant gain. Since this analytical procedure is not easily generalized to account for the dynamics of the variance, we instead opt to provide evidence via simulation on whether Ramsey or Nash inflation is obtained under precautionary learning. The simulation results prompt the intuition described above: precautionary learning, instantiated by the use of a VALS estimator on the part of a central bank, can lead to a higher frequency of escape and a lowering of the long run average rate of inflation. The key to this intuition is the nature of the loss function (L)surface,discussedbelow. We note that we are plotting the loss function as a function of the precautionary parameters that inflation and unemployment implicitly depend upon; the expected loss function is computed as the average loss over a simulation of 1 periods. Since the econometrician will not know a priori the Phillips curve slope that he is estimating and he may also realize that his (misspecified) model is at best an approximation to reality, we suppose that he will experiment with di erent values for the precautionary parameters, trying each pair for some interval of time and then settling on the pair that produces the smallest loss. We generate each of these figures (and all reported simulations below) having fixed at 1,! at 2 and u NR at 5 so that the Nash level of inflation from x t =! u NR is 1 while the Ramsey equilibrium inflation rate is zero. The decreasing gain learning algorithm will converge to an equilibrium in which the Phillips curve is perceived to have the actual slope of 2 and an intercept of 25, whichaddstou NR the e ect of the public s expectations about inflation. The standard deviations of the shocks, 1 and 2,aresetto1.5. Westart each simulation looking at data generated by a predecessor policy maker who targeted the inflation rate at its Nash value. In our baseline calibration, we set the horizon for the constant gain algorithm to T =2 periods, implying a gain of g = 1 T =.5. The loss surface is shown in Figure 1. We also use this calibration for the time series plots of targeted inflation that we report below. 14

16 Loss Loss Least Squares Loss a unr a ω Figure 1. Loss Function (L(x t ),.5gain). AhighgaintranslatesintoashortertimespanofdataemployedinlearningthePhillips curve parameters. Thus the probability of observing a history in which the Phillips curve is flat is high, raising the probability of escape. Given our interest in examining dynamics with constant gain VALS vs ROLS learning, a high gain of.5 is a natural choice. For robustness, we also plot the loss surface (L) foralowvalueof thegaininfigure2andata higher value in Figure 3 (corresponding to 4 and 1 periods respectively) Least Squares Loss a unr a ω Figure 2. Loss Function (L(x t ),.25gain). 15

17 Loss Least Squares Loss a unr a ω Figure 3. Loss Function (L(x t ),.1gain). In each of Figures 1, 2 and 3, we see that least squares learning is close to a ridge line of the loss function. Allowing for a skew in the estimation of either of the two Phillips curve parameters (i.e., non-zero values for (a unr,a! ) ), suggests that better policy results from overestimating the slope and underestimating the intercept relative to least squares. However, the improvement that comes from moving away from least squares in this direction is a purely local phenomenon. Whereas slight underestimates of the slope lead to a worsening of policy, large underestimates improve policy, and likewise for overestimates of the intercept. The policymaker will be nearly indi erent between receiving very high or very low estimates of the parameters as long as she does not receive estimates in the vicinity of least squares. 3.2 Time-Series Simulations Why do biased estimates of the Phillips curve parameters improve policy? Do they cause the policymaker to target rates of inflation lower than those consistent with a Nash outcome? We investigate this question by looking at the time series generated by our model under various values for the precautionary parameters, holding the shock processes constant. For 16

18 our baseline parameters, the period loss function for the Ramsey equilibrium is 38. In contrast, the loss function for the Nash equilibrium is the much higher value of 138. First, we consider what happens for the familiar case of constant gain ROLS learning, in which (a!,a unr )=(, ). The time series for targeted inflation and the estimated Phillips curve parameters are shown in Figure x t ω t u NR t 2 15 u t π t L t Period Period Figure 4. Time Paths under Constant Gain ROLS Learning. Consistent with Cho et al. (22), the inflation target, shown in the top panel, remains around the Nash equilibrium value of 1 most of the time but there are escapes where the target falls close to the Ramsey level of zero. The plots of! t and u NRt in the second and third panels reflect the intuition of Cho et al. (22), as follows. The average estimate of! is and of u NR is , bothclosetothevaluesthatwouldbeobtainedbyregressing 1 Those panels of Figure 4 for which the horizontal axis runs from 1 to 1, the plotted time series are the actual values for the 1 long simulation. Those panels for which the horizontal axis runs from 1 to 5, the plotted time series are 2-period averages. We plot 2-period average values for realized inflation and unemployment in order to smooth out the noise from the two sets of shocks. 17

19 u t on t according to (4) in the Nash equilibrium. 11 However, a string of data does occur in which the correlation of u t and t falls, resulting in smaller estimates of both! and u NR. The policymaker thinks there is a smaller trade-o between unemployment and inflation, so according to the targeted rate under the PLM x PLM t =! tu NR,t, (28) 1+! 2 t she reduces the inflation target. As a consequence of the lower target, the slope and intercept estimates remain depressed, so it takes time for the system to return to Nash equilibrium values. The period loss function falls during the escape but most of the time is around the Nash equilibrium value of 138. Now let us consider what happens as we vary the precautionary parameters. To facilitate comparison, we use the same sequence of shocks as those that generated the time series reported in Figure 4. Under rational expectations the actual law of motion (2) eliminates any dependence of unemployment on the central bank s policy decisions, so the unemployment series is purely a function of the shocks. Consequently, all of the following experiments have the same unemployment series as in Figure 4. Only the series for the target inflation rate and the realized inflation rate will depend on the central bank s policy decisions, which themselves depend on the choice of precautionary parameters. Since the realized inflation rate di ers from the target inflation only by the addition of noise, we only plot the target inflation rate series below. Further, since the contribution of unemployment to the loss is the same for each experiment, the loss function associated with each experiment will move with the variability of inflation around its target value of zero. Intuitively, one might expect the policymaker to achieve better results if the slope is underestimated since that will increase the probability of finding there is no trade-o between unemployment and inflation, which should produce a Ramsey-like outcome. The time series 11 The misspecification of the model will not bias estimates of the slope but will produce consistent estimates of u NR + x rather than u NR. 18

20 for targeted inflation that result from setting a! < are therefore shown in Figure However, whether policy will improve if a! is increased above or below zero depends on the parameters of the model. Di erentiating (28) by!, we PLM t = u NR,t(1! 2 t ) (1 +! 2 t ) 2, (29) which is of ambiguous sign. For our parameters,! 2 =4> 1, where! is the actual slope of the Phillips curve. Thus a small decrease in the slope estimate will actually increase the target inflation rate. a ω =, a unr = x x x t t t a = - 1, 5 a = ω unr a = - 2, 5 a = ω unr a = - 15, a = ω unr x t Period (in Thousands) Figure 5. Targeted Inflation (a! <, a unr =). 12 Corresponding values of the slope and intercept (as in Figure 4) are available from the authors. 19

21 x t x t x t x t a ω =, a unr = a = 1, 5a = ω unr a = 2, 5a = ω unr a = 15, a = ω unr Period (in Thousands) Figure 6. Targeted Inflation (a! >, a unr =). Consistent with our expectation, escapes occur more frequently as a! drops below zero in Figure 5. However, because the inflation rate is higher when the economy is not escaping, the loss function increases as we decrease a! for small a!. When a! = 1, theeconomy escapes away from the Nash outcome about half the time. The period loss function can fall almost to zero during these many escapes, but these gains are dominated by the higher values of the period loss function that arise during normal periods when inflation is targeted around 15 instead of 1. Overall the loss with a! = 1 averages to 152, which is higher than the loss of 138 if the economy stays at the Nash equilibrium. On the other hand, when! = 2, escapesbecomesofrequentthattheeconomyneverhasachancetoconverge to a steady state. These frequent escapes yield a loss of 8 that is much less than the loss under the Nash equilibrium. In pushing a! so low, we have passed over the ridge line of the loss surface in Fig. 1. Decreasing a! even further can push the loss down almost to 2

22 its Ramsey-equilibrium value. 13 In the bottom panel of Fig. 6, we see that for a! = 1, target inflation is quite steady at a value slightly less than. Summary statistics for target inflation, actual inflation, unemployment, and the period loss function are summarized in Table 1. In contrast, when the policy maker overestimates the Phillips curve slope as in Fig. 6, she thinks the trade-o between unemployment and inflation is so large that she does not need much inflation to achieve the desired low unemployment. Escapes never happen in these simulations because she never convinces herself that the trade-o has gone away. Instead, target inflation remains fairly constant at a value that decreases with a!.consequentlythe loss also decreases with a! for a! (at least over the interval a! 2 [ 1, 1] that we investigated). When a! =2,targetinflationaveragesto5.4, andtheperiodlossfunction averages to 68. Asa! increases all the way to 1 the average target inflation falls to 1.85,and the period loss function averages to 42, whichisnearlythevalue oftheramseyequilibrium. Indeed, the economy behaves quite similarly for both a! = ±1. Table 1. Simulation Statistics (a! Q, a unr =) x t! t u NR,t Loss a! Mean Std. Dev. Mean Std. Dev. Mean Std. Dev. Mean Std. Dev The econometrician can also apply a precautionary bias to estimation of the intercept of the PLM, i.e. the natural rate of unemployment. What happens then? What of VALS 13 This will be the minimum average value of the loss function over long periods in a model where the public has rational expectations. 21

23 learning on the intercept of the PLM? Figure 7 shows what happens given overestimates of the intercept with a unr > and Figure 8 what happens given underestimates of the intercept with a unr <. 4 a ω =, a unr = 2 x t a =, a = 1 ω unr 4 2 x t a =, a = 1 ω unr 4 2 x t Period (in Thousands) Figure 7. Targeted Inflation (a! =, a unr > ). 22

24 a ω =, a unr = 1 x t a =, a = - 1 ω unr 1 x t a =, a = - 1 ω unr 1 x t Period (in Thousands) Figure 8. Targeted Inflation (a! =, a unr < ). Overestimation of the intercept leads indirectly to lower values of the slope estimate. As aconsequence,theprobabilityofescapeincreases. However,ifwedi erentiate (28) by u NR, we PLM NR,t =! t 1+! 2 t >, (3) so overestimation of the intercept also produces higher values for target inflation. The benefit of more frequent escapes will be overwhelmed by the cost of higher inflation during normal times for small a UNR,andtheperiodlossfunctionaveragesto21 for a UNR =1.In contrast, Figure 8 shows that escapes are eliminated when we underestimate the intercept, but target inflation averages to a lower value of 7.4 when a UNR = 1. In that case, the period loss function averages to 88. Thus experimentation around a unr =will cause the econometrician to underestimate the intercept. This pattern persists if we increase a UNR even further as evidenced by Table 2 below. 23

25 Table 2. Simulation Statistics (a! =, a unr Q ) x t! t u NR,t Loss a unr Mean Std. Dev. Mean Std. Dev. Mean Std. Dev. Mean Std. Dev min L Finally, Figure 9 shows the time series with the parameters (a UNR,a! )=( 1, 76) that produce the minimum loss function value of for the region of the parameter space over which we searched. The last row of Table 2 provides the associated statistics for the variables of interest. For this last specification of a UNR and a!, which produces the minimum loss, we are across the ridge line from ROLS learning, and the targeted rate of inflation averages to (a slight deflation). This suggests that a monetary policymaker who learns about the Phillips curve using an e cient constant gain VALS estimator can attain aramsey-likeequilibrium,incontrasttowhathappensifsherestrictsherselftoanrols estimator as in Cho et al. (22). 24

26 a ω =, a unr = 1 x t a ω = - 76, a unr = x t Period (in Thousands) Figure 9. Targeted Inflation (a unr = 1, a! = 76). 4 Conclusion In replacing expectations with regressions in canonical monetary policy models, escape from a high inflation steady state is possible if a constant gain recursive least squares estimator is employed by a policy maker. In this paper we note that least-squares estimation will generally not be e cient because the econometric loss function is asymmetric so overestimates and underestimates are not equally costly. We then explore the escape dynamics that result if the policy maker uses a more e cient variance-adjusted least squares estimator that biases her estimates in the more advantageous direction. Our results point towards situations in which such precaution on the part of the learner could result in the lowering of the long run rate of inflation in an economy characterized by the time inconsistency of discretionary policy. Specifically, given the analytically intractable nature of the model, we focus on simula- 25

27 tions that indicate that if there is overestimation of the slope relative to least squares, we see an increased frequency of the escapes analyzed by Cho et al. (22). This is because estimates of the slope of the perceived law of motion (a misspecified Phillips curve) fall often, along with wide variation in estimates of the intercept. This leads to overshooting dynamics in the targeted rate of inflation. However, this behavior is not monotonic. For high degrees of slope underestimation, negative slope estimates push down estimates of the natural rate to near zero leading to low values of the targeted rate of inflation. These complex learning dynamics are rich enough to warrant further analytical investigation since they suggest a possible tension between the pull of time consistency and the degree to which a policy maker is cautious about interpreting information, reflected in the estimator employed. We reemphasize that, while we are introducing new parameters, we are not modifying the economic model that delivers the time-inconsistency tension, so this does not cut against Occam s Razor. The more complex dynamics we present are not obtained by expanding the economic framework; we are focused on examining parameter values for the tools employed to analyze the model, rather than parameters of the model itself. Indeed, the space of possible time series estimators is always infinite dimensional, regardless of whether a policy-maker ties his hands by only considering the recursive least squares in this space. With the variance adjusted least squares (or precautionary ) learning we have described, the policy-maker is e ciently processing the data that comes out of the model given that the usual least squares case is nested within the broader framework we consider. Doing so suggests that accounting for the variance of estimates may well assist a policy-maker to escape the tragically high inflation usually expected in economies characterized by the time inconsistency of discretionary monetary policy. Indeed, precaution in learning about economic fundamentals, motivated by the potentially asymmetric nature of the policy maker s objective function and instantiated via a constant gain VALS estimator, should cause a policy maker to target low inflation. 26

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