Inflation Expectations and Monetary Policy Design: Evidence from the Laboratory

Finance and Economics Discussion Series Divisions of Research & Statistics and Monetary Affairs Federal Reserve Board, Washington, D.C. Inflation Expectations and Monetary Policy Design: Evidence from the Laboratory ˇ Damjan Pfajfar and Blaˇz Zakelj 2015-045 Please cite this paper as: Damjan Pfajfar and Blaˇz Zˇakelj (2015). “Inflation Expectations and Monetary Policy Design: Evidence from the Laboratory,” Finance and Economics Discussion Series 2015-045. Washington: Board of Governors of the Federal Reserve System, http://dx.doi.org/10.17016/FEDS.2015.045. NOTE: Staff working papers in the Finance and Economics Discussion Series (FEDS) are preliminary materials circulated to stimulate discussion and critical comment. The analysis and conclusions set forth are those of the authors and do not indicate concurrence by other members of the research staff or the Board of Governors. References in publications to the Finance and Economics Discussion Series (other than acknowledgement) should be cleared with the author(s) to protect the tentative character of these papers.

In(cid:135)ation Expectations and Monetary Policy Design: Evidence from the Laboratory (cid:3) (cid:181) Damjan Pfajfar Bla(cid:181)z Zakelj y z Federal Reserve Board Universitat Pompeu Fabra June 11, 2015 Abstract Using laboratory experiments within a New Keynesian framework, we explore theinteractionbetweentheformationofin(cid:135)ationexpectationsandmonetarypolicy design. The central question in this paper is how to design monetary policy when expectations formation is not perfectly rational. Instrumental rules that use actual rather than forecasted in(cid:135)ation produce lower in(cid:135)ation variability and reduce expectational cycles. A forward-looking Taylor rule where a reaction coe¢ cient equals 4 produces lower in(cid:135)ation variability than rules with reaction coe¢ cients of 1.5 and 1.35. In(cid:135)ation variability produced with the latter two rules is not signi(cid:133)cantly di⁄erent. Moreover, the forecasting rules chosen by subjects appear to vary systematically with the policy regime, with destabilizing mechanisms chosen more often when in(cid:135)ation control is weaker. JEL: C91, C92, E37, E52 Key words: Laboratory Experiments, In(cid:135)ation Expectations, New Keynesian Model, Monetary Policy Design. Theviewsexpressedinthispaperarethoseoftheauthorsanddonotnecessarilyre(cid:135)ectthoseofthe (cid:3) Federal Reserve Board. Address: Board of Governors of the Federal Reserve System, 20th and Constituy tion Ave, NW Washington, DC 20551, U.S.A. E-mail: damjan.pfajfar@frb.gov. Web: https://sites.google.com/site/dpfajfar/. A¢ liatedResearcher,LeeX,UniversitatPompeuFabra,RamonTriasFargas25-27,08005Barcelona, z Spain. E-mail: blaz.zakelj@gmail.com. 1

1 Introduction Withthedevelopmentofexplicitmicrofoundedmodels, expectationshavebecomepivotal in modern macroeconomic theory. Friedman(cid:146)s proposals (1948 and 1960) for economic stability postulate that the relationship between economic policies and expectations is crucialforpromotingeconomicstability. Friedmanarguesinfavorofsimplerulesbecause they are easier to learn and because they facilitate the coordination of agents(cid:146)beliefs. Several leading macroeconomists and policymakers, including Bernanke (2007), stress the importance of improving our understanding of the relationship between economic policies(cid:151)especially monetary policy(cid:151)agents(cid:146)expectations, and equilibrium outcomes. While the theoretical literature has expanded rapidly in the last two decades, less attention has been paid to empirical assessment of the relationship between expectations and monetary policy. Laboratory experiments provide an opportunity to explore these relationships, as one can control for the underlying model, shocks, and forecasters(cid:146)information sets. Thispaperanalyzesthee⁄ectivenessofalternativemonetarypolicyrulesinstabilizing the variabilityof in(cid:135)ationinasettingwhere in(cid:135)ationexpectation-formationprocesses are potentially non-rational. We study this question by employing several simple monetary policy rules in di⁄erent treatments and examining the relationship between the design of monetary policy and in(cid:135)ation forecasts. Based on prior reasoning we would expect that, under rational expectations (RE), a policy rule that reacts to contemporaneous data would result in lower in(cid:135)ation variability than under a forward-looking rule. We would also expect that the higher the reaction coe¢ cient attached to deviations of the in(cid:135)ation expectations from the target level, the lower should be the variability in in(cid:135)ation. Using simple nonparametric analysis of treatment di⁄erences, we (cid:133)nd that the variability of in(cid:135)ationis signi(cid:133)cantlya⁄ectedbythe aggressiveness of monetarypolicy. Indeed, we (cid:133)nd thatthehigherthereactioncoe¢ cientattachedtodeviationsofthein(cid:135)ationexpectations from the target level, the lower the variability in in(cid:135)ation. Our results con(cid:133)rm our prior that responding to contemporaneous in(cid:135)ation perform better than rules responding to in(cid:135)ation expectations. As pointed out by Marimon and Sunder (1995), the actual dynamics of an economy are the product of a complex interaction between the underlying stability properties of the model and agents(cid:146)behavior. Both in(cid:135)ation expectations and monetary policy in(cid:135)uence the variability. To con(cid:133)rm the e⁄ects of the monetary policy mentioned above, we have to (cid:133)rst determine how individuals form in(cid:135)ation expectations and then control for expectations formation. We (cid:133)nd that subjects form expectations using di⁄erent forecasting rules. The most often used by our subjects are trend extrapolation and a general model that, in some treatments, is of the form of Rational Expectations Equilibrium (REE) and includes all relevant information to forecast in(cid:135)ation in the next period. A 2

signi(cid:133)cant share of the subjects also use adaptive expectations, adaptive learning, and sticky-information type models.1 Furthermore, we have to be aware that under the trend extrapolation rule and adaptive expectations(cid:151)rules that we characterize as potentially destabilizing(cid:151)policy prescriptions are altered. Under these rules, a higher reaction coe¢ cient attached to deviations of in(cid:135)ation expectations from the target level may result in a higher volatility of in(cid:135)ation. However, even when controlling for the expectationformation mechanism, we are still able to identify signi(cid:133)cant e⁄ects of monetary policy: (i) when monetary policy attaches a higher weight to the deviation of expected in(cid:135)ation from the in(cid:135)ation target, we observe lower in(cid:135)ation variability and (ii) instrumental rules that respond to contemporaneous in(cid:135)ation (as opposed to in(cid:135)ation expectations) reduce in(cid:135)ation variability. We also (cid:133)nd that the interaction between monetary policy and in(cid:135)ation expectations is important. In particular, we (cid:133)nd that the volatility of in(cid:135)ation is signi(cid:133)cantly higher when more subjects use trend extrapolation rules. At the same time, the design of monetary policy signi(cid:133)cantly a⁄ects the composition of forecasting rules used by subjects in the experiment(cid:151)especially the proportion of subjects who use trend extrapolation rules, which are identi(cid:133)ed as the ones most dangerous to the stability of the main macroeconomic variables. The proportion of subjects using trend extrapolation rules increases in an environment characterized by excessive in(cid:135)ation variability and expectational cycles; this rule then further ampli(cid:133)es the cycles. Our experiment relates to previous studies that investigate the expectation-formation process. Learning-to-forecast experiments have been conducted within a simple macroeconomic setup (e.g., Williams, 1987; Marimon et al., 1993; Evans et al., 2001; Arifovic and Sargent, 2003) and also within an asset pricing framework (see Hommes et al., 2005 and Anufriev and Hommes, 2012).2 Marimon and Sunder (1995), and Bernasconi and Kirchkamp (2000) (cid:133)nd that most subjects behave adaptively, although the latter provide evidence of a more complex form of adaptive expectations than argued by the former. Both papers also investigate the e⁄ects of di⁄erent monetary policies on in(cid:135)ation volatility. Marimon and Sunder (1995) compare di⁄erent monetary rules in an overlapping generations (OLG) framework to explore their in(cid:135)uence on the stability of in(cid:135)ation expectations. In particular, they focus on a comparison between Friedman(cid:146)s k-percent money rule and the de(cid:133)cit rule where the government (cid:133)xes the real de(cid:133)cit and (cid:133)nances it through seigniorage. They (cid:133)nd little evidence that Friedman(cid:146)s rule could help coordinate agent beliefs or help stabilize the economy. A similar analysis is performed in Bernasconi and Kirchkamp (2000). They argue that Friedman(cid:146)s money growth rule produces less in(cid:135)ation volatility but higher average in(cid:135)ation compared to a constant real de(cid:133)cit rule.3 1Adaptive learning assumes that the subjects are acting as econometricians when forecasting, i.e., reestimating their models each time new data become available. See Evans and Honkapohja (2001). 2See Du⁄y (2012) and Hommes (2011) for a survey of experimental macroeconomics. 3The e⁄ects of monetary policy design on expectations are also examined by Hazelett and Kernen 3

Adam (2007) conducts experiments in a sticky-price environment where in(cid:135)ation and output depend on expected in(cid:135)ation, and analyzes the resulting cyclical patterns of in(cid:135)ation around its steady state. These cycles exhibit signi(cid:133)cant persistence, and he argues that they closely resemble a Restricted Perception Equilibrium (RPE) where subjects make forecasts with simple underparametrized rules. In our experiment, we also detect cyclical behavior of in(cid:135)ation and the output gap in some treatments. However, we show that these phenomena are not only associated with the parameterization of the rule but also with the design of monetary policy and (the in(cid:135)uence of monetary policy on) the way subjects form expectations. Recently, a setup similar to ours has been used by Assenza et al. (2013), who focus on the analysis of switching between di⁄erent forecasting rules, and by Kryvtsov and Petersen (2013), who quantify the contribution of systematic monetary policy for macroeconomic stabilization. This paper is organized as follows: Section 2 describes the underlying experimental economy and its properties under di⁄erent expectation-formation processes. Section 3 outlines the experimental design. In Section 4 we study the relationship between the monetary policy design and expectation formation; Section 5 provides concluding remarks. 2 A Simple New Keynesian Economy In our experiment, we use a simpli(cid:133)ed version of a forward-looking sticky-price New Keynesian(NK)monetarymodel.4 Themodelconsistsofaforward-lookingPhillipscurve (PC), an IS curve, and a monetary-policy reaction function. In this paper, we focus on the reduced form of the NK model, where we can clearly elicit forecasts and study their relationship with monetary policy. There is a trade-o⁄ between using the model from (cid:147)(cid:133)rst principles(cid:148)and employing a reduced form. The former has the advantage of setting the objectives (payo⁄function) exactly in line with the microfoundations since subjects act as producers and consumers and interact on both the labor and (cid:133)nal product markets (for this approach, see Noussair et al., 2011). However, forecasts are di¢ cult to elicit in such an environment, because subjects do not explicitly provide forecasts. We therefore choose learning-to-forecast design, where incentives are set in order to induce forecasts that are as accurate as possible.5 In this framework, we do not assign the subjects a (2002), who search for hyperin(cid:135)ationary paths in the laboratory. 4Thissmall-scaleNKmodelsuccessfullyreproducesseveralstylizedfactsaboutmajoreconomiesand is also widely used for policy analysis. In an experimental setup, however, it has potential drawbacks. It requiresforecastingtwoperiodsahead. Inaddition,instandardNKmodels,agentshavetoforecastboth in(cid:135)ation and the output gap. We simplify this experiment by asking only for expectations of in(cid:135)ation. 5The argument is similar to that of Marimon and Sunder (1993, 1994). Bao et al. (2013) show that within the same model, convergence to REE occurs much faster in the learning-to-forecast design than in the learning-to-optimize design. 4

particular role in the economy; rather they act as (cid:147)professional(cid:148)forecasters.6 The forecasts for period t+1 are made in period t with the information set consisting of macro variables up to t 1. Mathematically, we denote this as E ((cid:25) ), or simply t t+1 t 1 (cid:0) jI (cid:0) E (cid:25) . In our case, E might not be restricted to just RE. The IS curve is speci(cid:133)ed as t t+1 t follows: y = ’(i E (cid:25) )+y +g ; (1) t t t t+1 t 1 t (cid:0) (cid:0) (cid:0) where the interest rate is i , (cid:25) denotes in(cid:135)ation, y is the output gap, and g is an t t t t exogenous shock.7 The parameter ’ is the intertemporal elasticity of substitution in demand. We set ’ to 0:164.8 One period represents one quarter. Note that we do not include expectations of the output gap in the speci(cid:133)cation. Instead, we have a lagged output gap.9 Compared to purely forward-looking speci(cid:133)cations, our model displays more persistence in the output gap. The supply side of the economy is represented by the Phillips curve: (cid:25) = (cid:12)E (cid:25) +(cid:21)y +u : (2) t t t+1 t t (cid:21) is a parameter that is, among other things, related to price stickiness. McCallum and Nelson (1999) suggest the value 0:3. The parameter (cid:12) is the subjective discount rate and is set to 0:99. The shocks g and u are unobservable to subjects and follow the t t following process: g g g (cid:20) 0 t t 1 t = (cid:10) (cid:0) + ; (cid:10) = ; " u t # " u t 1 # " u t # " 0 (cid:23) # (cid:0) e where 0 < j (cid:20) j < 1 and 0 < j (cid:23) j < 1: g t and u t aree independent white noises, g t v N 0;(cid:27) g 2 and u N (0;(cid:27)2). g could be seen as a government spending shock or a taste shock, t v u t (cid:0) (cid:1) andthestandardinterpretationofue isameark-up(oracost-push)shock. Ineparticular, (cid:20) t and (cid:23)eare set to 0:6, while their standard deviations are 0:08:10 All these shocks are found to be quite persistent in the empirical literature (see, e.g., Cooley and Prescott, 1995, or Ireland, 2004). In the experimental context, it is important to have some exogenous unobservable component in the law of motion for endogenous variables; otherwise all 6One way to think about the relationship between professional forecasters and consumers/(cid:133)rms is that these economic subjects employ professional forecasters to provide them with forecasts of in(cid:135)ation. 7Detailed derivations can be found in, e.g., Walsh (2003) or Woodford (2003). 8We implement McCallum and Nelson(cid:146)s (1999) calibration. 9Onecouldarguethatthisspeci(cid:133)cationoftheISequationcorrespondstothecasewheresubjectshave naive expectations about the output gap or where an extreme case of habit persistence is assumed. The main reason for including a lagged output gap in our speci(cid:133)cation is that we want another endogenous variable to in(cid:135)uence the law of motion for in(cid:135)ation. 10Parameterizationoftheseshocksisquiteimportant. Increasing(cid:20)andvwouldincreasethevariability of in(cid:135)ation and of the output gap. Values of (cid:20) and v higher than 0:6 (and closer to empirical estimates) were avoided as the frequency of the cycles drops and the possibility of having only one big recession (expansion) over the whole experimental time span increases. 5

agents can quickly coordinate on forecasts identical to the in(cid:135)ation target.11 To close the model, we use two alternative forms of Taylor-type interest rate rules in di⁄erent treatments that are explained in Section 3. The forward-looking interest rate rule is speci(cid:133)ed as: i = (cid:13)(E (cid:25) (cid:25))+(cid:25); (3) t t t+1 (cid:0) where the central bank responds to deviations in subjects(cid:146)in(cid:135)ation expectations from the target, (cid:25).12 To ensure positive in(cid:135)ation for most of the periods, we set the in(cid:135)ation target to (cid:25) = 3. We vary (cid:13) in di⁄erent treatments. The second speci(cid:133)cation is the contemporaneous rule, where the monetary authority responds to deviations in current in(cid:135)ation from the in(cid:135)ation target:13 i = (cid:13)((cid:25) (cid:25))+(cid:25): (4) t t (cid:0) 2.1 Rational Expectations In this section, we derive the properties the model (cid:147)should(cid:148)have under REE. When all agents in the economy are rational, their perceived law of motion (PLM) is equal to the actual law of motion (ALM) of the minimum state variable (MSV) form. For a comparison, we solve the model (cid:133)rst as if the agents observe the shocks. Note that (cid:25) t 1 (cid:0) does not enter the REE solution. The corresponding expectations (PLM) of the REE form (representation 1) are: E (cid:25) = (b +b b )+b b y +(b c +c (cid:20))g +(b c +c (cid:23))u : (5) t t+1 (cid:25) (cid:25)y y (cid:25)y yy t 1 (cid:25)y yy (cid:25)y t 1 (cid:25)y y(cid:25) (cid:25)(cid:25) t 1 (cid:0) (cid:0) (cid:0) Parameters b and c represent the REE solution (see Appendix A for details). Note that for the forward-looking rule there exists an alternative representation of the MSV-REE (representation2), whichis moreuseful inourcasewheresubjects donotdirectlyobserve 11Besides that it is more realistic to have AR(1) shocks, without them, this would represent the dominant strategy, as we initialize the model in a REE; at the start of the experiment, we provide 10 data points to the subjects that are generated under RE. 12We assume that the central bank is responding to subjects(cid:146)in(cid:135)ation expectations and not to their own in(cid:135)ation expectations. 13We note that this rule is characterized as non-operational, as at the time of interest rate decision the central bank does not know the realization of (cid:25) : However, theoretical research has to a large extend t focused on these type of instrumental rules. 6

the shocks:14 (cid:13) 1 E (cid:25) = (a +b a ) (cid:25) (cid:0) (’(b c +c (cid:20))+(cid:12)(b c +c (cid:23))) (6) t t+1 (cid:25) (cid:25)y y (cid:25)y yy (cid:25)y (cid:25)y y(cid:25) (cid:25)(cid:25) (cid:0) (cid:13) (cid:18) (cid:19) +(b c +c (cid:23))(cid:25) +(b b +(b c +c (cid:20)) (cid:21)(b c +c (cid:23)))y (cid:25)y y(cid:25) (cid:25)(cid:25) t 1 (cid:25)y yy (cid:25)y yy (cid:25)y (cid:25)y y(cid:25) (cid:25)(cid:25) t 1 (cid:0) (cid:0) (cid:0) (cid:13) 1 1 (b c +c (cid:20))y + (b c +c (cid:20))’ (cid:0) + (cid:12)(b c +c (cid:23)) i (cid:25)y yy (cid:25)y t 2 (cid:25)y yy (cid:25)y (cid:25)y y(cid:25) (cid:25)(cid:25) t 1: (cid:0) (cid:0) (cid:13) (cid:13) (cid:0) (cid:18) (cid:18) (cid:19) (cid:19) In this representation, REE also depends on (cid:25) ; i ; and y . If we used a similar prot 1 t 1 t 2 (cid:0) (cid:0) (cid:0) cedure in the contemporaneous rule treatment we would (cid:133)nd that the REE is dependent on the initial values of the shocks and the whole history of (cid:25) and y. In Table A3, we present the detailed E-stability and determinacy properties of the model, while the summary is in Table 2. E-stability is the asymptotic stability of an equilibrium under least squares learning. By determinacy, we mean the existence of a unique dynamically-stable equilibrium. Our models produce a determinate and E-stable outcome under RE when (cid:13) > 1 (for both representations). When (cid:13) 1, the equilibria (cid:20) are E-unstable and indeterminate. Note that the models we analyze retain these stability properties although we replace the expectations of the output gap by the lagged output gap in the IS equation. 2.2 Restricted Perceptions In this section, we outline ten models of expectation formation that have found support in the empirical literature. As we discuss later on, we will use these rules to describe the behavior of the subjects in our experiment. To be clear, our subjects are not introduced to these forecasting rules; they are asked simply to report their forecast for in(cid:135)ation given the observed data. Based on their observed behavior, we then assign a speci(cid:133)c rule to each subject. This section solves the model assuming agents use expectation-formation mechanisms that are summarized in Table 1. Shocks were not directly observable, so these models do not include them. In model M1, in(cid:135)ation expectations follow a simple AR(1) model, while model M2 represents a weighted-average model similar in formulation to the sticky information model of Carroll (2003).15 We estimate this model stated in terms of observable variables with restrictions on the coe¢ cients, where (cid:17) = b + b b and (cid:17) = b b are REE 0 (cid:25) (cid:25)y y 1 (cid:25)y yy coe¢ cients. We consider two versions of adaptive expectations, where agents revise their expectations according to the last observed error: (cid:133)rst, a constant gain learning (CGL) model 14Inordertoobtainthisrepresentationitiscrucialthattheinstrumentalruleincorporateexpectations of in(cid:135)ation. To derive this representation we replace the g and u in (5) by lagged (1) and (2) and t 1 t 1 then use (3) to substitute E (cid:25) : (cid:0) (cid:0) t 1 t 15AsinCarroll(2003),them(cid:0)odelisaconvexcombinationbetweentherationalforecastandtheforecast made in the previous period. 7

Model (Eq.) Speci(cid:133)cation AR(1) process (M1) (cid:25)k = (cid:11) +(cid:11) (cid:25)k t+1t 0 1 tt 1 Sticky information type (M2) (cid:25)k j = (cid:21) (cid:17) +(cid:21) (cid:17)j (cid:0)y +(1 (cid:21) )(cid:25)k Adaptive expectations CGL (M3) (cid:25) t k +1 j t = (cid:25) 1 k 0 + 1 # 1 ((cid:25) t (cid:0) 1 (cid:25)k (cid:0) 1 ) t j t (cid:0) 1 Adaptive expectations DGL (M4) (cid:25) t k +1 j t = (cid:25) t k(cid:0) 1 j t (cid:0) 2 + (cid:19)((cid:25) t (cid:0) 1 (cid:0) (cid:25)k t (cid:0) 1 j t (cid:0) 2 ) Trend extrapolation (M5) (cid:25) t k +1 j t = (cid:28) t (cid:0) + 1 j t (cid:25)(cid:0) 2 + t (cid:28) t (cid:0) ((cid:25) 1 (cid:0) t (cid:0)(cid:25) 1 j t (cid:0) 2 ); (cid:28) 0 General model (M6) (cid:25) t k +1 j t = (cid:11) 0 +(cid:11) t (cid:0) (cid:25) 1 + 1 (cid:11) t (cid:0) y 1 (cid:0) + t (cid:11) (cid:0) 2 y 1 + (cid:21) (cid:11) i t+1t 0 1 t 1 2 t 1 3 t 2 4 t 1 Recursive - lagged in(cid:135)ation (M7) (cid:25)k j = (cid:30) +(cid:30) (cid:0) (cid:25) (cid:0) (cid:0) (cid:0) t+1t 0;t 1 1;t 1 t 1 Recursive - lagged output gap (M8) (cid:25)k j = (cid:30) (cid:0) +(cid:30) (cid:0) y (cid:0) t+1t 0;t 1 1;t 1 t 1 Recursive - trend extrapolation (M9) (cid:25)k j = (cid:30) (cid:0) +(cid:25) (cid:0) +(cid:30) (cid:0) ((cid:25) (cid:25) ) Recursive - AR(1) process (M10) (cid:25) t k +1 j t = (cid:30) 0;t (cid:0) 1 +(cid:30) t (cid:0) 1 (cid:25)k 1;t (cid:0) 1 t (cid:0) 1 (cid:0) t (cid:0) 2 t+1t 0;t 1 1;t 1 tt 1 j (cid:0) (cid:0) j (cid:0) Table 1: Models of in(cid:135)ation expectation formation. Notes: (cid:25) is in(cid:135)ation at time t; y is t t theoutputgap,i istheinterestrate,and(cid:25)k isthekth forecaster(cid:146)sin(cid:135)ationexpectation t t+1t for time t+1 made at time t (with informatijon set t 1). (cid:0) (M3), where # is the constant gain parameter, and second, a decreasing gain learning (DGL), where (cid:19) is the decreasing gain parameter. Next, we evaluate simple trend extrapolationrules(M5). Theseareidenti(cid:133)edinHommesetal.(2005)asparticularlyimportant rules for expectation-formation processes. Simple rules do not capture all the macroeconomic factors that can a⁄ect in(cid:135)ation forecasts. Therefore, we estimate a general model (M6) which coincides with the REE form for the forward-looking rule.16 Wealsoconsiderforecastingproceduresthatallowagentstoreestimateruleswhenever new information becomes available, as postulated in the adaptive learning literature. In the following speci(cid:133)cations, we test whether agents update their coe¢ cients with respect to the last observed error. We use this estimation procedure for models M7(cid:150)M10. When agents estimate their PLM they exploit all the available information up to period t 1. (cid:0) As new data become available, they update their estimates according to a stochastic gradient learning (see Evans et al., 2010) with a constant gain. Let X and (cid:30) be the t t 1 (cid:0) vectors of variables and coe¢ cients, respectively, speci(cid:133)c to each rule; for example, for b model M7, X = 1 (cid:25) and (cid:30) = (cid:30) (cid:30) 0. In this version of CGL, agents t t t 1 0;t 1 1;t 1 (cid:0) (cid:0) (cid:0) update the coe¢ c(cid:16)ients ac(cid:17)cording to the(cid:16)following stoch(cid:17)astic gradient learning rule: b (cid:30) = (cid:30) +(cid:24)X (cid:25) X (cid:30) : (7) t t (cid:0) 2 0t (cid:0) 2 t (cid:0) t (cid:0) 2 t (cid:0) 2 (cid:16) (cid:17) b b b Asabackdropforourempiricalpart, weexaminethestabilitypropertiesoftheserules inAppendixA.17 InTable 2, we summarize the properties of the REEanddi⁄erent RPEs 16The models in groups 19 24 do not have the interest rate as a dependent variable because this (cid:0) would imply multicollinearity due to the design of the monetary policy in our framework. 17Stability properties are presented for the speci(cid:133)c parameterizations of monetary policy rules used across di⁄erent treatments in this experiment. For a detailed description of treatments, see Section 3. 8

Treatment M6, rep. 2 M2, M8, M1, M7, M10 M6; (cid:11) = 0 M5, M9 4 Determinacy yes yes yes (unit root) no no 1 B E-Stability yes yes yes yes (c.e.) no (c.e.) 1 B E-Stability - - - no no (c.e.) 2 Determinacy yes yes yes (unit root) no no 2 B E-Stability yes yes yes yes (c.e.) no (c.e.) 1 B E-Stability - - - no (c.e.) no (c.e.) 2 Determinacy yes yes yes (unit root) no yes 3 B E-Stability yes yes yes yes (c.e.) - 1 B E-Stability - - - no (c.e.) no (c.e.) 2 Determinacy - yes yes no no 4 B E-Stability - yes yes yes (c.e.) no (c.e.) 1 B E-Stability - - - no (c.e.) no (c.e.) 2 Table 2: Properties of solutions in the equilibrium under di⁄erent expectation formation mechanisms. Notes: (c.e.) stands for complex eigenvalues. For a detailed version of this table with speci(cid:133)c values of their respective ALM, determinacy, and E-stability conditions, see Table A3. under both policy rules. Results are also reported in Figure 1. When all agents have RE, a higher (cid:13) leads to less variability in in(cid:135)ation. The general model (M6) produces less variabilityforhigher(cid:13). ItalsoproduceslessvariabilitythantheREE.Thisisasomewhat surprising result because restricted perceptions usually generate more volatility (Evans and Honkapohja, 2001). Trend extrapolation (M5), however, leads to more volatility than the REE. The relationship with (cid:13) is also nonmonotonic for M5: the minimum is at (cid:13) = 1:98. After this threshold, volatility increases with higher (cid:13).18 A comparison between the forward-looking rule and the contemporaneous rule at (cid:13) = 1:5 suggests that the REE for the contemporaneous rule produces about 25% less variability (0:52) than the forward-looking rule.19 As discussed in the Appendix, this result is consistent with a comparison of the eigenvalues of the determinacy condition but not by the eigenvalues of the E-stability condition (see Table A3). A similar di⁄erence is seen for other expectation-formation mechanisms, except for M5, where the di⁄erence is considerably larger: in(cid:135)ation variance that is only 5% of a variance produced by the same expectation-formation mechanism under the contemporaneous rule. In Table 2 we can observe an explanation for this result: under the forward-looking rule only, this equi- 18WeperformanadditionalsimulationinwhichtheagentsuseOLStoestimatethecoe¢ cientsintheir respective rules based on the past data, and compute the standard deviation of in(cid:135)ation while varying (cid:13) between 1 and 2 (see Figure A9). When all the agents employ a sticky information type model, a higher (cid:13) leads to less variability in in(cid:135)ation. Several other expectation formation mechanisms produce a U-shaped in(cid:135)ation variability. In particular, trend extrapolation rules lead to U-shaped behavior and eventually higher variability with increasing (cid:13). The minimum variability of in(cid:135)ation with sticky information and a trend extrapolation rule is achieved at (cid:13) =1:1. Therefore, under certain expectation formation mechanisms, a lower (cid:13) could result in less in(cid:135)ation variability. 19Figure 1 is reproduced for the contemporaneous rule in Figure A10 in Appendix A. 9

pDSMR t 61 41 21 01 8 6 4 2 0 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3 monetary policy reaction coefficient,g M2, M8 equilibrium M1, M7, M10 equilibrium M6 equilibrium M5, M9 equilibrium REE Figure 1: Equilibrium dynamics of in(cid:135)ation under di⁄erent expectation formation rules for the forward-looking rule. Notes: RMSD (cid:25) is root mean squared deviation of in(cid:135)ation t fromitstarget. Figureisbasedonasimulationover1000periods. Simulationisperformed for the equilibrium values of the coe¢ cients of the respective rules, see Appendix A. librium exhibits a unit root. In contrast, under the contemporaneous rule, the variability of M6 is only 3% higher than under the forward-looking rule. Generally, we can conclude that the properties of the system depend crucially on the expectation-formation mechanism. Under RE, a higher value of (cid:13) will result in lower variability of in(cid:135)ation, while under some expectation rules, e.g., trend extrapolation rules (M5), a higher value of (cid:13) leads to more volatile in(cid:135)ation. We label these expectationformation mechanisms as potentially destabilizing. Another type of forecasting rules that we classify as potentially destabilizing are those that do not have a MSV solution. In our case, this holds for adaptive expectations (M3) (see Appendix). Therefore, the relationshipbetweenthevariabilityofin(cid:135)ationanddi⁄erentforecastingrulesisnontrivial. 3 Experimental Design The experimental subjects participate in a simulated economy with 9 agents.20 Each participant is an agent who makes forecasting decisions, and each simulated economy 20Mostlearning-to-forecastexperimentsareconductedwith5to6subjects,e.g.,Hommesetal.(2005), Adam (2007), and Fehr and Tyran (2008). 10

is an independent group. All the participants were undergraduate students recruited at the Universitat Pompeu Fabra and the University of Tilburg. The participants were invited from a database of approximately 1300 students at Pompeu Fabra (in May 2006) and 1200 students at Tilburg (in June 2009). They were predominantly economics and business majors. On average, the participants earned around e15 ( $20), depending on (cid:25) the treatment and individual performance. There are 4 treatments in the experiment, each based on a di⁄erent speci(cid:133)cation of the monetary policy-reaction function. The experiment consists of 24 independent groups of 9 subjects (6 groups per treatment), 216 subjects in total. Each subject was randomly assigned to one group; each group is exposed to only one treatment. The experimentaleconomylastsfor70periods. Wescaledthelengthofeachdecisionsequence and the number of repetitions in such a way that each session lasted approximately 90 to 100 minutes, including the time for reading the instructions and 5 trial periods at the beginning.21 We gathered 15120 point forecasts of in(cid:135)ation from the 216 subjects. The subjects are presented with a simple (cid:133)ctitious economy setup. The economy is described with three macroeconomic variables: in(cid:135)ation, the output gap, and the interest rate. The participants observe time series of these variables in a table up to period t 1. Ten initial values (periods 9;:::;0) are generated by the computer under the (cid:0) (cid:0) assumption of RE. The subjects(cid:146)task is to provide in(cid:135)ation forecasts for period t + 1. Figure 2 provides the timeline of decisions in the experiment. The underlying model of the economy is qualitatively described to them. We explain the meaning of the main macroeconomic variables and inform them that their decisions have an e⁄ect on the realized output, in(cid:135)ation, and interest rate at time t. The parameters of the model are not revealed to subjects. This is the predominant strategy in learning-to-forecast experiments (see Du⁄y, 2012, and Hommes, 2011).22 All the treatments have exactly the same shocks. In every period t, there are two decision variables: i) a prediction of the t+1 period in(cid:135)ation; and ii) the 95% con(cid:133)dence interval of their in(cid:135)ation prediction. In this paper, 21The experimental interface was designed in z-Tree (Fischbacher, 2007). The experimental instructions can be found in the Online Supplementary material of the companion paper, Pfajfar and Z(cid:181)akelj (2014). 22Inlearning-to-forecastexperimentsitisnotpossibletoachievetheREEsimplybyintrospection. This holds even if we provide the subjects with the data generating process because there exists uncertainty as to how other participants forecast, so the subjects have to engage in a number of trial-and-error exercises, or, in other words, adaptive learning. It has been proven by Marcet and Sargent (1989) and further formalized in a series of papers by Evans and Honkapohja (see Evans and Honkapohja, 2001) that agents will achieve the REE if they observe all the relevant variables in the economy and update their forecasts according to the adaptive learning algorithm (their errors). Bao et al. (2013) show that convergence to the REE actually occurs faster in the learning-to-forecast design than in the learning-to-optimize design. For further discussion see Du⁄y (2012) and Hommes (2011). Kelley and Friedman (2008) provide a survey of experiments that support the theoretical result above. Examples of learning-to-forecast experiments are Marimon and Sunder (1993, 1994), Adam (2007), and Hommes et al. (2005). 11

Figure 2: Timeline we focus on in(cid:135)ation expectations, while our companion paper Pfajfar and Z(cid:181)akelj (2011) studies the behavior of con(cid:133)dence intervals. After each period, the subjects receive information about the realized in(cid:135)ation in that period, their in(cid:135)ation expectations, and the payo⁄ they have gained. The subjects(cid:146)payo⁄s depend on the accuracy of their predictions. The accuracy benchmark is the actual in(cid:135)ation rate computed fromthe underlying model on the basis of the predictions made by all the agents in the economy. We replace E (cid:25) in Eqs. (1), (2), and (3) by 1 k(cid:25)k , where (cid:25)k is subject k(cid:146)s point foret t+1 K t+1t t+1t j j cast of in(cid:135)ation (K is the total number of subjects in the economy). In the subsequent P rounds, the subjects are also informed about their past forecasts. They do not observe the forecasts of other individuals or their performance. The payo⁄ function, W, is the sum of two components: 100 W = W +W ; W = max 20; 0 ; f = (cid:25) (cid:25)k : 1 2 1 1+f (cid:0) t (cid:0) t+1t j (cid:26) (cid:27) (cid:12) (cid:12) (cid:12) (cid:12) The (cid:133)rst component, W , depends on the subjects(cid:146)forecast errors and is designed to 1 encourage them to give accurate predictions. It gives subjects a payo⁄ if their forecast errors, f, are less than four.23 The second component, W , represents an independent 2 incentive that refers to their con(cid:133)dence intervals and is not the focus of this paper (see PfajfarandZ(cid:181)akelj,2011). Weaccompaniedthepayo⁄functionwithacarefulexplanation and a payo⁄ matrix on a separate sheet of paper to ensure that all the participants understood the incentives. The participants received detailed instructions, which were read aloud. They also (cid:133)lled in a short questionnaire after they had read the instructions, answering questions about the procedure to demonstrate that they understood it. Thedi⁄erenttreatmentsaresummarizedinTable3: The(cid:133)rstthreetreatments, which are shown in Table 3, deal with the parameterization of the forward-looking rule given in 23Comparedtomorestandardquadraticpayo⁄functions,oursgivesagreaterrewardformoreaccurate predictions and provides an incentive also to think about small variations in in(cid:135)ation, which may be important. Since this experiment can potentially produce quite di⁄erent variations in in(cid:135)ation between di⁄erent sessions, it is important to keep the incentive scheme fairly steep. A similar incentive scheme is used in Adam (2007) and Assenza et al. (2013). 12

Treatment Parameter Forward-looking rule (1) (cid:13) = 1:5 Forward-looking rule (2) (cid:13) = 1:35 Forward-looking rule (3) (cid:13) = 4 Contemporaneous rule (4) (cid:13) = 1:5 Table 3: Treatments Eq. (3). In this setup, the coe¢ cient (cid:13) determines the central bank(cid:146)s aggressiveness in responsetodeviationsofexpectedin(cid:135)ationfromitstarget. Weareparticularlyinterested to see how subjects react to more and less aggressive interest rate policies. We chose (cid:13) = 1:5 as a baseline speci(cid:133)cation in line with the majority of empirical (cid:133)ndings and the initial proposal of Taylor (1993), (cid:13) = 1:35 as a case with a lower stabilization e⁄ect, and (cid:13) = 4asaparameterizationwithahighstabilizinge⁄ect. Initially, weplannedtoperform a treatment with (cid:13) < 1. The (cid:133)ndings from the pilot treatment, however, convinced us that such a low (cid:13) is not a suitable choice, as subjects quickly reached extremely high levels of in(cid:135)ation, leading to explosive behavior of the system.24 Aswepointedoutabove, underRE,higher(cid:13) resultsinlowervariability. Thus, among the (cid:133)rst three treatments, the variability in in(cid:135)ation should be the lowest in treatment 3, where (cid:13) = 4. Comparing treatments 1 and 4, under RE the contemporaneous rule stabilizes in(cid:135)ation better than the forward-looking rule does. These two statements represent testable hypotheses in our experiment. 4 Results Summary statistics of in(cid:135)ation and in(cid:135)ation expectations for each of the 24 independent groups are presented in Table 4. These statistics are used in the analysis below to establish whether the di⁄erences across treatments are signi(cid:133)cant. Unconditionally, the mean in(cid:135)ation forecast for all treatments is around 3:06%, while the mean in(cid:135)ation is 3:02% when the in(cid:135)ation target is set to 3%. The standard deviations of in(cid:135)ation (expectations) vary considerably across the independent groups. The largest standard deviation of in(cid:135)ation expectations is 6:32 and the smallest 0:23, while the largest standard deviation of in(cid:135)ation is 5:87 and the smallest is 0:24. The di⁄erences across treatments are analyzed in the following subsections. Moreover, if we compare the means of the in(cid:135)ation forecasts in treatments 1 and 4, we (cid:133)nd that the median value in the latter treatment is signi(cid:133)cantly higher than in the 24Under these circumstances, in(cid:135)ation never returned to the target in(cid:135)ation but just kept growing. Therefore, the e⁄ect of the output gap on in(cid:135)ation never outweighed the expected in(cid:135)ation e⁄ect. This suggests that under non-rational expectations, the Taylor principle is still required in order to generate stability. Assenza et al. (2013) perform a treatment where (cid:13) = 1: In their economy with i.i.d. shocks this results in a convergence to values of in(cid:135)ation that are di⁄erent from the target value. 13

14

former treatment (at 10% signi(cid:133)cance with the Kruskal-Wallis rank test, see Conover, 1999). Similar results are obtained when comparing treatments 2 and 3: the mean in(cid:135)ation is lower in the latter treatment. If we compute the trend of means of in(cid:135)ation expectationsinin(cid:135)ationforecastingtreatmentsusingJonckheere-Terpstratestforordered alternatives, we (cid:133)nd that the mean is decreasing with higher (cid:13): 4.1 In(cid:135)ation Variability and Monetary Policy Woodford (2003) points out that within a standard NK model, monetary policy should minimizethevariabilityinin(cid:135)ationandtheoutputgaparounditstargets,asthisbehavior correspondstomaximizingtheutilityofconsumers. Inoursetup, themonetaryauthority cares only about in(cid:135)ation, so we focus our analysis on the variability in in(cid:135)ation. We graph the evolution of in(cid:135)ation for all independent groups in Figure 3. Does monetary policy have an in(cid:135)uence on the in(cid:135)ation variability? Theory says that it should: As we demonstrated in Figure 1, simulations under RE show that a forwardlooking rule produces a lower standard deviation of in(cid:135)ation with increasing (cid:13). The (cid:133)rst column of Table 5 summarizes these results. Speci(cid:133)cally, when (cid:13) = 1:35 the standard deviationis0:46,andwhen(cid:13) = 4itreducesto0:15. Table5alsoshowsthatwhen(cid:13) = 1:5, the contemporaneous rule produces a slightly lower standard deviation of in(cid:135)ation than the forward-looking rule. Turning to our experimental results, the standard deviation of in(cid:135)ation is higher than that simulated under RE. The di⁄erence between the average standard deviation and that under RE is signi(cid:133)cant for all treatments (p-value: 0:0110). The average standard deviation among the treatments with the in(cid:135)ation forecasting rule is lowest when (cid:13) = 4 (0:42) and the highest when (cid:13) = 1:5 (2:25). In the treatment with the contemporaneous rule, the average standard deviation is 0:65. Standard Mean Median Comparison deviation standard standard with treat. 1 Treatment Groups under RE deviation deviation (p-value) 1: Fwd-l. rule (cid:13) = 1:5 1 6 0:37 2:25 1:52 (cid:0) (cid:0) 2: Fwd-l. rule (cid:13) = 1:35 7 12 0:46 2:18 1:35 0:6310 (cid:0) 3: Fwd-l. rule (cid:13) = 4 13 18 0:15 0:42 0:29 0:0104 (cid:0) 4: Cont. rule (cid:13) = 1:5 19 24 0:33 0:65 0:50 0:0250 (cid:0) Table 5: Standard deviation of in(cid:135)ation for each treatment and Kruskal-Wallis test of di⁄erences between treatments using group-level standard deviations. Whenwetestfordi⁄erencesinthemedianvariancesofin(cid:135)ationacrossthetreatments, thenullhypothesisthatthemedianvariancesarethesameinallthetreatmentsisrejected at the 1% level with the Kruskal-Wallis test. Table 5 shows a comparison of the median standard deviations of in(cid:135)ation in treatments 2, 3, and 4 with the baseline treatment 15

1 (p-values from the Kruskal-Wallis test are reported).25 According to these pairwise comparisons, the standard deviation of in(cid:135)ation in treatment 3 is signi(cid:133)cantly lower than the standard deviation of in(cid:135)ation in both treatments 1 (p-value: 0:0104) and 2 (p-value: 0:0250). However, as can be seen in Figure 3, the frequency of cycles (in terms of number of changes from above to below the in(cid:135)ation target) is higher in treatment 3, where the monetary authority responds more strongly to deviations of in(cid:135)ation expectations from the in(cid:135)ation target. Our results suggest that the median (and mean) standard deviation is lower in treatment 2 compared to treatment 1, although not signi(cid:133)cantly di⁄erent. We can also jointly compare the three in(cid:135)ation forecasting treatments and investigate the behavior in the standard deviation of in(cid:135)ation when changing (cid:13). Using the Jonckheere- Terpstra test, we (cid:133)nd that there is a descending standard deviation of in(cid:135)ation when we increase (cid:13). Thus, we can argue that the size of the policy reaction ((cid:13)) is important. Regarding the form of the policy rule, the contemporaneous rule (treatment 4) produces a signi(cid:133)cantly lower standard deviation of in(cid:135)ation (and in(cid:135)ation forecasts) than the forward-looking rule with the same reaction coe¢ cient (treatment 1); see Table 5. Treatment 1 Treatment 2 0 0 2 2 0 0 1 1 0 0 0 0 1 1 ) 0 10 20 30 40 50 60 70 0 10 20 30 40 50 60 70 % ( n Treatment 3 Treatment 4 o ita 0 0 lfn 2 2 I 0 0 1 1 0 0 0 0 1 1 0 10 20 30 40 50 60 70 0 10 20 30 40 50 60 70 Period Figure 3: Group comparison of in(cid:135)ation realized by treatment. Notes: Each line represents one of the 24 independent groups. Treatment 1 has forward-looking rule (FWR) with (cid:13) = 1:5. Treatment 2 has FWR with (cid:13) = 1:35. Treatment 3 has FWR with (cid:13) = 4. Treatment 4 has contemporaneous rule with (cid:13) = 1:5. Now that we have established that there is a di⁄erence in the variability of in(cid:135)ation between treatments, we further analyze the origins of these di⁄erences. There are two 25Results are identical if we consider only the last 40 periods of our sample. 16

possible explanations: monetary policy and in(cid:135)ation expectations. To proceed with the analysis and disentangle the two e⁄ects, we have to (cid:133)rst establish how the subjects form expectations. 4.2 Formation of Individual Expectations In this subsection, we choose among the ten models introduced in Table 1 to (cid:133)nd the one that (cid:147)best (cid:133)ts(cid:148)the actual expectations of each individual. The models are estimated usingOLS. Weconsideranindividual (cid:147)touse(cid:148)themodel that produces thelowest RMSE among all competing models. In the case of the recursive models (M7(cid:150)M10), we search for the parameter # and initial values that minimize the RMSE between the simulated forecast under adaptive learning and the subjects(cid:146)forecasts (see Pfajfar and Santoro, 2010). We can reject rationality under the assumption of homogeneous expectations for each of 216 subjects.26 In addition, models M4 and M10 describe none of the participants. A detailed discussion on heterogeneity of expectation-formation mechanisms in this experiment can be found in Pfajfar and Z(cid:181)akelj (2014). Model (Eq.) Treatments 1 2 3 4 All n AR(1) process (M1) 0.0 0.0 0.0 1.9 0.5 Sticky information type (M2) 5.6 7.4 11.1 1.9 6.5 Adaptive expectations CGL (M3) 11.1 1.9 7.4 14.8 8.8 Adaptive expectations DGL (M4) 0.0 0.0 0.0 0.0 0.0 Trend extrapolation (M5) 33.3 29.6 13.0 29.6 26.4 General model (M6) 33.3 29.6 55.6 29.6 37.0 Recursive - lagged in(cid:135)ation (M7) 3.7 13.0 3.7 13.0 7.8 Recursive - lagged output gap (M8) 0.0 1.9 1.9 1.9 1.4 Recursive - trend extrapolation (M9) 13.0 16.7 7.4 9.3 11.6 Recursive - AR(1) process (M10) 0.0 0.0 0.0 0.0 0.0 Table 6: In(cid:135)ation expectation formation across treatments (percentage of subjects using a given rule). In Table 6, we compare the empirical models across all the treatments. The behavior of about 37% of the subjects is best described by the general model (M6), using all the relevant information to forecast in(cid:135)ation. About 26% of the subjects simply extrapolate the trend (M5) and another 12% extrapolate the trend while updating their coe¢ cients recursively (M9). About 9% employ adaptive expectations (M3), while the remaining 16% mostly behave in accordance with adaptive learning and sticky-information type models. However, there are considerable di⁄erences across the treatments, especially in the proportion of subjects using the trend extrapolation rule (M5) and subjects using the 26However, in experiments it is possible to go one step further, as we are able to control the subjects(cid:146) information sets. For a detailed assessment of rationality, see Pfajfar and Z(cid:181)akelj (2014). 17

general model. Treatment 3 has the lowest proportion of trend extrapolating subjects and the highest proportion of subjects using the general model (M6). 4.3 In(cid:135)ation Variability and Expectations In the exercise in Section 2, we learned that di⁄erent expectation-formation mechanisms can have di⁄erent implications for the stability of the system. The analysis in the previous sectionshows that several forecasting mechanisms are used, and their structure varies across the treatments. In the present section we analyze these di⁄erences. In particular, we focus on establish the relationship between the observed expectation-formation mechanisms and in(cid:135)ation variability, and the e⁄ect of monetary policy design on in(cid:135)ation variability. TheresultsfromSection4.1demonstratethatthein(cid:135)ationvolatilityineverygroupin our experiment is signi(cid:133)cantly higherthan that simulatedon the basis of Rational Expectations Equilibrium (REE) and Restricted Perceptions Equilibria (RPEs) considered in Section 2.2, possibly with the exception of equilibrium dynamics under M6 in treatments with the forward-looking rule. Possible reasons for this discrepancy are (i) misspeci(cid:133)cation of the Perceived Law of Motion (PLM), (ii) the use of nonoptimal coe¢ cients, and (iii) the use of adaptive learning with a constant gain. In the existing literature, the evidence for these temporary equilibria dynamics is not very abundant. In a forecasting experiment, Adam (2007) argues that subjects rely on simple underparameterized rules to forecast in(cid:135)ation, and thus the equilibrium dynamics resembles the RPE. We observe similar dynamics. In addition, many subjects in our experiment use misspeci(cid:133)ed models as they include in(cid:135)ation in their speci(cid:133)cations of the forecasting rules, e.g., the general model (M6). As discussed above, this has important consequences for in(cid:135)ation dynamics. We (cid:133)rst focus on (i), the role of the speci(cid:133)cation of the PLM. It has already been suggested that the proportion of trend extrapolation subjects plays a particularly important role in the stability of the system. We observe that there is a considerable degree of heterogeneity across the treatments (see Table 6) and that there is a strong correlation between the variability of in(cid:135)ation and the degree of trend extrapolation behavior. We use panel data regressions to test these conjectures regarding the relationship between the variability and the proportions of di⁄erent categories of subjects:27 sd = (cid:17) sd +(cid:17) p +(cid:17) T+" ; (8) s;t 0 s;t 1 1 js;t 2 s;t (cid:0) 27Toobtainthepaneldataforthestandarddeviationofin(cid:135)ationandtheproportionofdi⁄erentrules, we compute for each period t the standard deviation of in(cid:135)ation and determine the best forecasting rule for each individual based on her information set in that period. Note that this is di⁄erent from calculations for Table 6. For details, see Pfajfar and Z(cid:181)akelj (2014). Results for cross-sectional models are reported in the Appendix in Table A1, with both robust and clustered standard errors, as clustered standard errors might not have good properties for small samples. 18

where sd is the standard deviation of in(cid:135)ation in group s up to time t; p is a vector s;t js;t of the proportions of agents in group s that use forecasting rules j (M2(cid:150)M7 and M9 from Table 6) in time t, and T is a vector of treatment dummies. We limit ourselves to models M2(cid:150)M7 and M9 since other rules were selected seldomly or not at all in this exercise. The results are reported in Table 7. sd : (a) (b) (c) (d) s;t sd 1.0065 1.0056 1.0065 1.0033 s;t 1 (cid:3)(cid:3)(cid:3) (cid:3)(cid:3)(cid:3) (cid:3)(cid:3)(cid:3) (cid:3)(cid:3)(cid:3) (cid:0) (0.0065) (0.0073) (0.0072) (0.0054) p (j = M2) -0.0007 -0.0013 -0.0019 -0.0016 js;t (cid:3) (0.0014) (0.0019) (0.0011) (0.0012) p (j = M3) -0.0008 -0.0015 js;t (cid:3)(cid:3) (0.0009) (0.0007) p (j = M4) -0.0015 -0.0017 -0.0027 -0.0021 js;t (cid:3)(cid:3)(cid:3) (cid:3)(cid:3) (0.0009) (0.0011) (0.0010) (0.0009) p (j = M5) 0.0037 0.0033 0.0026 0.0033 js;t (cid:3)(cid:3)(cid:3) (cid:3)(cid:3) (cid:3)(cid:3) (cid:3)(cid:3)(cid:3) (0.0013) (0.0015) (0.0011) (0.0013) p (j = M6) 0.0016 0.0011 js;t (cid:3)(cid:3) (0.0008) (0.0011) p (j = M7) -0.0011 -0.0017 js;t (0.0014) (0.0012) p (j = M9) -0.0011 js;t (0.0011) T2 0.0350 0.0330 0.0363 0.0368 (0.0327) (0.0339) (0.0326) (0.0351) T3 -0.1191 -0.1172 -0.1273 -0.1104 (cid:3)(cid:3) (cid:3)(cid:3) (cid:3)(cid:3) (cid:3)(cid:3) (0.0517) (0.0500) (0.0498) (0.0490) T4 -0.0916 -0.0887 -0.0989 -0.0807 (cid:3)(cid:3) (cid:3)(cid:3) (cid:3)(cid:3) (cid:3) (0.0464) (0.0440) (0.0464) (0.0465) cons -0.0208 0.0301 0.0984 0.0638 (cid:3)(cid:3)(cid:3) (cid:3) (0.0607) (0.1007) (0.0218) (0.0381) N 1560 1560 1560 1560 (cid:31)2 107822.0 216120.0 143881.7 97425.5 Table 7: In(cid:135)uence of the decision model on the standard deviation of in(cid:135)ation. Notes: EstimationsareconductedusingthesystemGMMestimatorofBlundellandBond(1998) for dynamic panels. Arellano-Bond robust standard errors in parentheses. */**/*** denotes signi(cid:133)cance at 10/5/1 percent level. A higher proportion of trend-extrapolation agents increases the standard deviation of in(cid:135)ation. The proportion of these agents probably plays the most important role for the stability of in(cid:135)ation.28 In contrast, having more agents that behave according to the 28It also helps to explain the di⁄erences among groups within the same treatment. Generally, we note that groups with a lower proportion of trend extrapolation rules are more stable than groups with a higher proportion in the same treatment. 19

adaptive expectations models (M3 and M4) (and potentially M2) decreases the standard deviationof in(cid:135)ationandthushasastabilizinge⁄ectontheexperimental economy. From the treatment dummies, we learn that treatments 3 and 4 both produce e⁄ects that are signi(cid:133)cant even when controlling for the subjects(cid:146)alternative forecasting rules. These e⁄ects are negative, which con(cid:133)rms that, compared to treatment 1, the monetary policies in treatments 3 and 4 have a stabilizing e⁄ect on the in(cid:135)ation variability. Thesecondreason(ii) fortheincreasedvolatilityinin(cid:135)ationisnon-optimalparameter estimates of certain rules. In the Appendix we present simulations that demonstrate this point (Figures A1 and A2). Higher updating coe¢ cients are related to higher in(cid:135)ation variability, especially for trend extrapolation and adaptive expectations. Hommes et al. (2005)showthatcoe¢ cientsinthetrendextrapolationrulesthatareabove1canseverely compromise the dynamic stability of the model. The coe¢ cients of individuals that use a given rule in our experiment are quite di⁄erent across treatments. We observe that the average coe¢ cient of the trend extrapolation rule ((cid:28) ) in M5 is higher in the treatments where in(cid:135)ation is more volatile, on average. 1 It is the highest in treatment 1 (0:53) and the lowest in treatment 3 (0:38). Sticky information type rules (M2) also exhibit signi(cid:133)cant di⁄erences across the treatments. The subjects in treatment 3 have the highest average (cid:21) (0:37), while those in treatment 2 1 have the lowest (0:11). Therefore, these expectation rules produce a less destabilizing e⁄ect in treatment 3 than in treatment 2. Similar evidence is also found for the adaptive expectation rule (M3), where rules with a coe¢ cient ((cid:28) or #) larger than 1 represent 1 another threat to stability. As can be seen in Figure A7, updating coe¢ cients of the trend-extrapolation rule that are higher than 0:6 could induce severe instability.29 It is possible to evaluate those e⁄ects more formally by estimating the e⁄ects of the averagecoe¢ cientofthetrendextrapolationruleineachgrouponthestandarddeviation of in(cid:135)ation (see Table A1). The coe¢ cient is positive and signi(cid:133)cant; the higher it is, the higher is in(cid:135)ation variability. Furthermore, we also investigate the joint e⁄ect of the proportion of agents using the trend extrapolation rule and their average coe¢ cients, and we (cid:133)nd the same results. Compared to the previous two regressions for the trend extrapolation rule, this regression explains the most variability of the standard deviation of in(cid:135)ation. In all of these regressions, the treatment dummies have a signi(cid:133)cant e⁄ect, emphasizing the importance of the monetary policy (see Table A3). The third issue (iii) we investigate is the relationship between the gain parameter in adaptive learning PLMs and the stability of the system: constant gain learning produces greater variability of the underlying series than does decreasing gain learning. Marcet and Nicolini (2003) show that this relationship could explain the evolution of in(cid:135)ation 29Results in this paragraph are based on estimations of all models in Table 2 for each individual. For further details see Figures A3 (cid:150)A8, where we plot these results for di⁄erent expectation formation mechanisms. 20

in Latin America. Furthermore, the variability increases with the level of the (constant) gainparameter. Ifthismechanismrepresentedanimportantsourceofvolatility, wewould expect higher average gains in more volatile treatments. However, we (cid:133)nd higher average (andmedian)gainsformorestabletreatments(3and4)thanformorevolatiletreatments (1 and 2). This result suggests that constant gain learning cannot explain the di⁄erences in volatility across the treatments. In addition to the e⁄ect of the monetary policy that was evident from the signi(cid:133)cance of the treatment dummies in regressions (8) (see Table 7), it seems plausible that the monetary policy also, at least partly, in(cid:135)uences the choice of the expectation-formation mechanism. The relationship between the underlying model and the expectation formation has recently been studied by Heemeijer et al. (2009) and Bao et al. (2012). They compare experimental results from positive and negative expectation feedback models.30 In a positive expectation system, e.g., an asset pricing model, they observe a cyclical behavior of prices similar to our behavior of in(cid:135)ation, and they note that when there is stronger positive feedback more agents resort to trend following rules. This result is also evident in Assenza et al. (2013). The link between the realized in(cid:135)ation and the expectation-formation mechanism can be represented by the expectational feedback, which is determined by the underlying model (monetary policy). The expectational feedback is the e⁄ect of a change in the average expectations in period t for period t + 1, E (cid:25) , on the change in the realization of in(cid:135)ation in period t, (cid:25) , formally @(cid:25)t . It t t+1 t @Et(cid:25)t+1 can be calculated by substituting the monetary policy rule into the IS equation (1) and then substituting the resulting equation into the PC equation (2). The expectational feedback for the forward-looking rule is (cid:12) + (cid:21)’(1 (cid:13)), while for the contemporaneous (cid:0) rule it is (cid:12)+(cid:21)’ . We see that this derivative is decreasing in (cid:13) for both rules. Comparing (cid:21)(cid:13)’+1 treatments 1 and 4, we see that the derivative is higher for the contemporaneous rule than for the forward-looking rule. By changing the monetary policy, we augment the degree of positive feedback from in(cid:135)ation expectations to current in(cid:135)ation. In an environment with higher expectational feedback, in(cid:135)ation expectations have a higher importance relative to the output gap for the realization of in(cid:135)ation. This makes in(cid:135)ation more vulnerable to the presence of potentially destabilizing expectation-formation mechanisms, such as the trend extrapolation rule. When at least one subject extrapolates the trend, the (cid:133)rst and second lags of in(cid:135)ation also enter the ALM for in(cid:135)ation. This has at least two e⁄ects: In(cid:135)ation variability increases, and it becomes optimal for others to use the two lags of in(cid:135)ation as well (to have the PLM of the same form as the ALM), which results in a further increase in the in(cid:135)ation variability. If we compare systems with higher and lower expectationalfeedbacks, theformerwillrequirefewersubjectsthatusepotentiallydestabilizing expectation-formation mechanisms (with given coe¢ cients) to produce the same in(cid:135)ation 30Fehr and Tyran (2008) also compare the two environments, although in a di⁄erent context. 21

variability. Alternatively, if the number of subjects using these rules is the same, the coe¢ cients must be higher to achieve the same e⁄ect. Therefore, the design of monetary policy is important for the expectation-formation mechanism and vice versa. We found that both the percentage of potentially destabilizing expectation-formation mechanisms (e.g., trend extrapolation rules or adaptive expectations) and the variability of in(cid:135)ation are the lowest in treatment 3, where the expectational feedback is the lowest. 5 Conclusion In a macroeconomic experiment where the subjects are asked to forecast in(cid:135)ation, we study the e⁄ectiveness of alternative monetary policy designs. The underlying model of the economy is a simpli(cid:133)ed version of the standard New Keynesian model that is commonly used for the analysis of monetary policy. In di⁄erent treatments, we employ various modi(cid:133)cations of Taylor-type instrumental rules. We compare two forms of the Taylor-type rules responding to either deviations of in(cid:135)ation expectations or current in(cid:135)ation from the target, and study the e⁄ects of varying the degree of responsiveness to deviations of the in(cid:135)ation expectations from the target level. Under rational expectations, we expect the contemporaneous rule to result in a lower variability in in(cid:135)ation than under the forward-looking rule. We also expect lower variability in in(cid:135)ation when the reaction coe¢ cient attached to deviations of the in(cid:135)ation expectations from the target level ((cid:13)) is higher. However, these policy prescriptions are altered under certain potentially destabilizing expectations formation mechanisms, especially trend extrapolation and adaptive expectations. Under these mechanisms, a higher (cid:13) may result in a higher volatility of in(cid:135)ation. The degree of expectational feedback also plays an important role in reducing the likelihood of ending up in the self-enforcing e⁄ect of potentially destabilizing expectations. In all treatments of our experiment, we observe the cyclical behavior of in(cid:135)ation and the output gap around their steady states. The variance of in(cid:135)ation in all the groups in the experiment is higher than that under rational expectations. We (cid:133)nd that monetary policy matters in our environment and that there are sizeable di⁄erences in in(cid:135)ation variability across the alternative designs under scrutiny. Among the monetary policy rules that react to deviations of the in(cid:135)ation expectations from in(cid:135)ation target, the one with a reaction coe¢ cient 4 results in a lower in(cid:135)ation variability compared to those with reaction coe¢ cients 1:35 and 1:5. Between the latter two there is no statistical di⁄erence. We (cid:133)nd that instrumental rules that are less aggressive are more vulnerable to the emergence of potentially destabilizing forecasting mechanisms. Wealsoexplorethecontemporaneousrule,aninstrumentalrulethatreactstoin(cid:135)ation rather than in(cid:135)ation expectations. The results show that the in(cid:135)ation variance under the contemporaneous rule is signi(cid:133)cantly lower than under the forward-looking rule at the 22

samelevel of sensitivityof theinterest ratetothedeviationof thein(cid:135)ation(expectations) from the target. Bernanke and Woodford (1997) also suggest that forward-looking rules may entail undesirable properties. It is noteworthy that the lower in(cid:135)ation variance is not accompanied by a signi(cid:133)cantly smaller proportion of subjects using potentially destabilizing expectation-formation mechanisms. Under the contemporaneous rule, both thevariabilityofinterestratesandtheexpectationalfeedbackarelower, resultinginlower in(cid:135)ation variability. Our analysis suggests that both the design of the monetary policy and the expectation-formation mechanisms are important for the dynamic stability of the model. Therefore, it is imperative to understand the interplay between the two. 23

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A Properties of the Model under Di⁄erent Expectation- Formation Mechanisms The actual dynamics of endogenous variables in the model is a result of the interaction between the underlying model and the expectation-formation mechanism. Several recent papers, using both experimental and survey data, have shown that the expectations of individuals are heterogeneous.31 In this section we outline the properties of the underlying model under di⁄erent expectation-formation mechanisms in order to compare these properties with the observed aggregate behavior in the experiment. A.1 Rational Expectations When all agents in the economy are rational, their perceived law of motion (PLM) is equal to the actual law of motion (ALM) of the minimum state variable (MSV) form. If agents would observe the shocks there would exist a unique evolutionary stable REE with the following form: y 1 g g b b c c t = B +C t 1 +D t ; B = y yy ;C = yy y(cid:25) : (cid:0) (cid:25) y u u b b c c t t 1 t 1 t (cid:25) (cid:25)y (cid:25)y (cid:25)(cid:25) (cid:20) (cid:21) (cid:20) (cid:0) (cid:21) (cid:20) (cid:0) (cid:21) (cid:20) (cid:21) (cid:20) (cid:21) (cid:20) (cid:21) e B is the matrix of coe¢ cients speci(cid:133)c to each treatment. It is presented in the (cid:133)rst e column of Table 2 below along with the other properties of possible equilibria in this framework. C and D are matrices of coe¢ cient values for the exogenous variables. D is speci(cid:133)c to the form of the Taylor rule employed. Note that (cid:25) does not enter the REE t 1 (cid:0) solution. To solve this model for RE we use the method of undetermined coe¢ cients. The corresponding expectations (PLM) of the REE form (representation 1) are: E (cid:25) = b +b y +c g +c u ; t t (cid:25) (cid:25)y t 1 (cid:25)y t 1 (cid:25)(cid:25) t 1 (cid:0) (cid:0) (cid:0) E (cid:25) = b +b E y +c E g +c E u ; t t+1 (cid:25) (cid:25)y t t (cid:25)y t t (cid:25)(cid:25) t t = (b +b b )+b b y +(b c +c (cid:20))g +(b c +c (cid:23))u : (9) (cid:25) (cid:25)y y (cid:25)y yy t 1 (cid:25)y yy (cid:25)y t 1 (cid:25)y y(cid:25) (cid:25)(cid:25) t 1 (cid:0) (cid:0) (cid:0) We insert (10) into the IS equation (1), where we substitute in the monetary policy rule andthePCequation(2). WethusobtaintheALM.BycomparingthePLMandtheALM we solve this model for the MSV-REE. The parameters of the RE forecasting rule (B and C) can be found in Table A3 in the Appendix. Note that for the forward-looking rule treatments there exists an alternative representation of the MSV-REE (representation 2), which is actually more useful in our case where subjects do not directly observe the shocks: (cid:13) 1 E (cid:25) = (a +b a ) (cid:25) (cid:0) (’(b c +c (cid:20))+(cid:12)(b c +c (cid:23))) (10) t t+1 (cid:25) (cid:25)y y (cid:25)y yy (cid:25)y (cid:25)y y(cid:25) (cid:25)(cid:25) (cid:0) (cid:13) (cid:18) (cid:19) +(b c +c (cid:23))(cid:25) +(b b +(b c +c (cid:20)) (cid:21)(b c +c (cid:23)))y (cid:25)y y(cid:25) (cid:25)(cid:25) t 1 (cid:25)y yy (cid:25)y yy (cid:25)y (cid:25)y y(cid:25) (cid:25)(cid:25) t 1 (cid:0) (cid:0) (cid:0) (cid:13) 1 1 (b c +c (cid:20))y + (b c +c (cid:20))’ (cid:0) + (cid:12)(b c +c (cid:23)) i (cid:25)y yy (cid:25)y t 2 (cid:25)y yy (cid:25)y (cid:25)y y(cid:25) (cid:25)(cid:25) t 1: (cid:0) (cid:0) (cid:13) (cid:13) (cid:0) (cid:18) (cid:18) (cid:19) (cid:19) 31Supportinsurveydataisfoundin,e.g.,Branch(2004)andPfajfarandSantoro(2010). Forasurvey of experimental support see Hommes (2011). Fehr and Tyran (2008) and Arifovic and Sargent (2003) also suggest that the expectations of individuals are heterogeneous. 27

In this representation REE also depends on (cid:25) ; i ; and y . If we used a similar prot 1 t 1 t 2 (cid:0) (cid:0) (cid:0) cedure in the contemporaneous rule treatment we would (cid:133)nd that the REE is dependent on the initial values of the shocks and the whole history of (cid:25) and y. A.2 Other models A.2.1 Stability Properties of Restricted Perceptions It is important to analyze the stability properties of the equilibria in all four underlying models under di⁄erent expectation-formation mechanisms.32 It is not possible to use the undetermined coe¢ cients technique to calculate the optimal coe¢ cients in adaptive expectation models (M3 and M4): in our setting there are no solutions for the coe¢ cients # and (cid:19). Therefore, only temporary equilibria exist.33 In the case of the sticky information type model (M2), this technique shows that the optimal coe¢ cient is (cid:21) = 1; and is studied in the second column of Table 2. Also, the AR(1) 1 process model (M1) in equilibrium has a coe¢ cient (cid:11) = 0 and thus reduces to forecast- 1 ing the steady state. Of course, recursive representations of the models have optimal coe¢ cients equal to the static counterparts. In general, we can write all the remaining forecasting models using (cid:25) t k +1t = (cid:30)X t , where X t = 1 y t (cid:25) t 1 (cid:25) t 2 (cid:25) t k t 1 0. But (cid:133)rst we de(cid:133)ne the RPE, which exi j sts for all models except M3 and (cid:0) M4:3 (cid:0) 4 j (cid:0) (cid:2) (cid:3) De(cid:133)nition 1 RestrictedPerceptionEquilibriainModelsM (M M1;M2;M5;:::;M10 ) (cid:3) (cid:3) 2 f g are stationary sequences y ;(cid:25) generated by (1), (2) and either (3) or (4) dependf t t g 1 t=0 ing on the treatment where agents use Model M (cid:25)k = (cid:30)X with parameters (cid:30) to (cid:3) t+1t t (cid:3)M j forecast in(cid:135)ation at time t for time t+1 where (cid:30) (cid:3)M (cid:16) is the ortho(cid:17)gonal projection of (cid:25) t on X : t De(cid:133)nition 2 There exist four classes of Restricted Perception Equilibria in Model M : (cid:3) 1. I⁄ M M2;M8 , (cid:30) is the orthogonal projection of (cid:25) on 1 y , the dy- (cid:3) 2 f g (cid:3)M t t (cid:0) 1 namics are characterized as a Underparameterized Perception Equilibrium level 1 (cid:2) (cid:3) (UPE1).35 2. I⁄ M M1;M7;M10 , (cid:30) is the orthogonal projection of (cid:25) on 1 , the dy- (cid:3) 2 f g (cid:3)M t namics are characterized as a Underparameterized Perception Equilibrium level 2 (cid:2) (cid:3) (UPE2). 3. I⁄ M = M6 and (cid:11) = 0, (cid:30) is the orthogonal projection of (cid:25) on 1 y (cid:25) , (cid:3) 3 (cid:3)M t t 1 t 1 (cid:0) (cid:0) the dynamics are characterized as a Misspeci(cid:133)ed Perception Equilibrium level 1 (cid:2) (cid:3) (MPE1). 32Stabilityanalysisoftheeconomywithasingleforecastingruleis,ofcourse,notdirectlyapplicableto the environment of heterogeneous agents as observed in our experiment (see Berardi (2007) for analysis of such an environment). Given the number of rules considered in our case, too many combinations are possible to make an informed conclusion. Thus, separate analysis of each rule is more indicative of the possible outcomes. 33Strictlyspeaking, theremightexistanequilibriumwithadi⁄erent(nonfundamental)representation using alternative methods to the undetermined coe¢ cients, e.g., common factor representation. 34It is worth pointing out that in general our stochastic gradient models M7(cid:150)M10 converge to a path around the RPE. 35M2ismisspeci(cid:133)ed,buttheinclusionofpastforecastsdoesnotalterthepropertiesoftheequilibrium. 28

4. I⁄ M M5;M9 , (cid:30) is the orthogonal projection of (cid:25) on 1 (cid:25) (cid:25) , (cid:3) 2 f g (cid:3)M t t (cid:0) 1 t (cid:0) 2 the dynamics are characterized as a Misspeci(cid:133)ed Perception Equilibrium level 2 (cid:2) (cid:3) (MPE2).36 In Table 2 we present the REE and di⁄erent RPEs and the summary of their determinacy and E-stability properties across all treatments. For the parameter of the ALM, B, under each expectation-formation mechanism, the corresponding eigenvalues of the determinacy condition, and the values of the eigenvalues of the T-map, see Table A3 in the Appendix.37 The second column in the table presents a UPE1, which has the same form as the REE (9) except that we omit shocks from the representation because they were not directly observable by the subjects in our experiment. UPE1(cid:146)s determinacy and E-stability propertiesarethesameasthoseoftheRE. ThethirdcolumnofTable2representsUPE2. In this case only a constant (equal to in(cid:135)ation target) is used for the forecasting. The models in these two columns are determinate and E-stable. The fourth column of Table 2 contains the stability results for a MPE1. As in the previous case, the optimal coe¢ cient on the lagged in(cid:135)ation is always zero (see Table A3 in Appendix). Note that the di⁄erence between UPE1 and MPE1 is a result of the inclusion of (cid:25) in M6. Comparing these results with those for the UPE1 in the (cid:133)rst t 1 (cid:0) column, it can be observed that the inclusion of a lagged in(cid:135)ation causes indeterminacy and di⁄erent values for the ALM. Furthermore, this inclusion causes the eigenvalues of the T-map to be complex in all treatments, and only the B solutions are E-stable. As 1 MarimonandSunder(1995)observe, iftheeigenvaluesarecomplex, thentheconvergence is cyclical. The MPE2 in the last column yields a determinate outcome only in treatment 3. The other treatments have two evolutionary stable solutions (thus indeterminacy), which could result in higher in(cid:135)ation volatility. Furthermore, solutions in all treatments are E-unstable. The trend extrapolation rule (M5) is restricted to positive coe¢ cients (cid:28) , 1 so only solution B is sensible in treatments 1, 2, and 4, while no evolutionary stable 1 solution with positive (cid:28) exists in treatment 3 (they exist only for (cid:13) < 2:99). 1 Generally, we can conclude that the stability and determinacy of the system crucially depend on the expectation-formation mechanism. A system that is E-stable and determinate under RE might not be so under di⁄erent expectation rules. In E-stable models under RE, a higher value of (cid:13) will result in lower eigenvalues of both the determinacy and E-stability conditions.38 On the contrary, under some expectation rules, e.g., trend extrapolation rules (M5), a higher value of (cid:13) can produce higher eigenvalues of the determinacy and E-stability conditions and thus more volatile in(cid:135)ation. We label these expectation-formation mechanisms as potentially destabilizing. Another type of forecasting rules that we classify as potentially destabilizing are those that do not have a MSV 36This equilibria is similar to the Behavioral Learning Equilibria of Hommes and Zhu (2014). 37Table A3 reports numerical values for di⁄erent treatments. In the case of indeterminacy we report both solutions and their corresponding eigenvalues of the E-stability condition. The analytical solutions canbeobtaineduponrequestfromtheauthors. WealsoomittheeigenvaluesoftheE-stabilitycondition corresponding to the shocks because they are always less than one and speci(cid:133)c only to treatments (thus C and D are omitted as well) and not to the expectation formation rules for the cases under scrutiny. 38Increasing(cid:13) hastwoe⁄ectsonthedynamicbehaviorofin(cid:135)ation: i) italwaysincreasesthefrequency ofcyclesregardlessoftheexpectationformationmechanism,andii) ita⁄ectstheamplitudeofthecycle, dependingontheexpectationformationmechanism. FormodelsthathaveadecreasingpatterninFigure 1, theamplitudeislowerwhen(cid:13) ishigher, whileintheothercases, mostnotablyforthelaggedin(cid:135)ation model, the relationship is not monotonic. 29

solution, i.e., adaptive expectations (M3), as seen in the simulations in Figures A3 and A4. Therefore, the relationship between the variability of in(cid:135)ation and di⁄erent forecasting rules is nontrivial. We con(cid:133)rm the results of Marimon and Sunder (1995), that the stability properties of the system, especially the eigenvalues of the determinacy condition, provide a good explanation for in(cid:135)ation volatility, but only with respect to stable expectation-formation mechanisms (mechanisms that always produce less variability of in(cid:135)ation when we increase (cid:13)). B Additional Tables and Figures 30

4 3 2 5.3 3 5.2 5 0 5 0 01 5 0 6 4 2 0 01 0 01 4 3 2 5.3 3 5.2 6 4 2 0 5 4 3 2 6 4 2 0 6 4 2 0 01 5 0 5 Rational expectations Rational expectations 0 10 20 30 40 50 60 70 0 10 20 30 40 50 60 70 PLM of RPE,x = 0.05 PLM of RPE,x = 0.05 0 10 20 30 40 50 60 70 0 10 20 30 40 50 60 70 Naive expectations Naive expectations 0 10 20 30 40 50 60 70 0 10 20 30 40 50 60 70 Adaptive expectations,J = 0.75 Adaptive expectations,J = 0.75 0 10 20 30 40 50 60 70 0 10 20 30 40 50 60 70 Adaptive expectations,J = 1.4 Adaptive expectations,J = 1.4 0 10 20 30 40 50 60 70 0 10 20 30 40 50 60 70 Trend extrapolation, t = 0.3 Trend extrapolation, t = 0.3 1 1 0 10 20 30 40 50 60 70 0 10 20 30 40 50 60 70 Trend extrapolation, t = 0.7 Trend extrapolation, t = 0.7 1 1 0 10 20 30 40 50 60 70 0 10 20 30 40 50 60 70 Inflation (t) Group's inflation expectation for (t) )%( noitalfnI Treatment 1 Treatment 2 period Figure A1: Simulation of in(cid:135)ation under alternative expectation formation rules (treatments 1 and 2). 31

4.3 2.3 3 8.2 2.3 3 8.2 4 3 2 4 3 2 001 0 001 5 0 004 002 0 002 4 3 2 5.3 3 5.2 4 3 2 4 3 2 6 4 2 4 3 2 6 4 2 0 Rational expectations Rational expectations 0 10 20 30 40 50 60 70 0 10 20 30 40 50 60 70 PLM of RPE,x = 0.05 PLM of RPE,x = 0.05 0 10 20 30 40 50 60 70 0 10 20 30 40 50 60 70 Naive expectations Naive expectations 0 10 20 30 40 50 60 70 0 10 20 30 40 50 60 70 Adaptive expectations,J = 0.75 Adaptive expectations,J = 0.75 0 10 20 30 40 50 60 70 0 10 20 30 40 50 60 70 Adaptive expectations,J = 1.4 Adaptive expectations,J = 1.4 0 10 20 30 40 50 60 70 0 10 20 30 40 50 60 70 Trend extrapolation, t = 0.3 Trend extrapolation, t = 0.3 1 1 0 10 20 30 40 50 60 70 0 10 20 30 40 50 60 70 Trend extrapolation, t = 0.7 Trend extrapolation, t = 0.7 1 1 0 10 20 30 40 50 60 70 0 10 20 30 40 50 60 70 Inflation (t) Group's inflation expectation for (t) )%( noitalfnI Treatment 3 Treatment 4 period Figure A2: Simulation of in(cid:135)ation under alternative expectation formation rules (treatments 3 and 4). 32

0 0 1 0 5 0 2 n o ita 0 1 lfn i fo 5 n o ita2 iv e d d1 ra d n a tS 5 . 0 .2 .4 .6 .8 1 a 1 coefficient Treatment 1 Treatment 2 Treatment 3 Treatment 4 Simulated values Subjects' estimates Figure A3: Standard deviation of in(cid:135)ation, subjects estimates of the AR(1) process (M1), and simulated values across the values of (cid:11) parameter. 1 0 1 5 n o ita 2 lfn i fo n1 o ita iv e d d 5 . ra d n a tS5 2 . 0 .2 .4 .6 .8 1 l 1 coefficient Treatment 1 Treatment 2 Treatment 3 Treatment 4 Simulated values Subjects' estimates Figure A4: Standard deviation of in(cid:135)ation, subjects estimates of the Sticky information process (M2), and simulated values across the values of (cid:21) parameter. 1 33

0 1 5 n o ita 2 lfn i fo n1 o ita iv e d d 5 . ra d n a tS5 2 . 0 .5 1 1.5 J coefficient Treatment 1 Treatment 2 Treatment 3 Treatment 4 Simulated values Subjects' estimates Figure A5: Standard deviation of in(cid:135)ation, subjects estimates of the Adaptive expectations CGL (M3), and simulated values across the values of # parameter. 0 1 5 n o ita 2 lfn i fo n1 o ita iv e d d 5 . ra d n a tS5 2 . 0 .5 1 1.5 i coefficient Treatment 1 Treatment 2 Treatment 3 Treatment 4 Simulated values Subjects' estimates Figure A6: Standard deviation of in(cid:135)ation, subjects estimates of the Adaptive expectations DGL (M4), and simulated values across the values of (cid:19) parameter. 34

0 0 1 0 5 0 n2 o ita lfn i fo 0 1 n o5 ita iv e d d ra 2 d n a tS1 5 . 0 .5 1 1.5 t 1 coefficient Treatment 1 Treatment 2 Treatment 3 Treatment 4 Simulated values Subjects' estimates Figure A7: Standard deviation of in(cid:135)ation, subjects estimates of the Trend extrapolation (M5), and simulated values across the values of (cid:28) parameter. 1 0 0 1 0 5 0 2 n o0 ita 1 lfn i fo 5 n o ita iv2 e d d ra 1 d n a5 tS. 0 .5 1 1.5 2 a 1 coefficient Treatment 1 Treatment 2 Treatment 3 Treatment 4 Simulated values Subjects' estimates Figure A8: Standard deviation of in(cid:135)ation, subjects estimates of the General model (M6), and simulated values across the values of (cid:11) parameter. 1 35

noitaived dradnats 9 8 7 6 5 4 3 2 1 0 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 gamma Sticky information (M2) General model (M6) Trend extrapolation (M5) Adaptive expectations (M3) Figure A9: Variability of in(cid:135)ation and alternative expectation formation rules (forward-looking rule). Notes: Figure is based on real-time OLS estimations of a particular rule for 1000 periods. pDSMR t 5 4 3 2 1 0 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3 monetary policy reaction coefficient,g M2, M8 equilibrium M1, M7, M10 equilibrium M6 equilibrium M5, M9 equilibrium REE Figure A10: Equilibrium dynamics of in(cid:135)ation under di⁄erent expectation formation rules for the contemporaneous rule. Notes: RMSD (cid:25) is root mean squared deviation of t in(cid:135)ation from its target. Figure is based on a simulation over 1000 periods. 36

dnert evisruceR EPR evisruceR .fni gal evisruceR ledom lareneG .partxe dnerT .pxe evitpadA .ofni ykcitS ssecorp )1(RA )9M( )8M( )7M( )6M( )5M( )3M( )2M( )1M( : ds s tsubor retsulc tsubor retsulc tsubor retsulc tsubor retsulc tsubor retsulc tsubor retsulc tsubor retsulc tsubor retsulc 0950.0- 8431.0 1958.0- 7010.1- 3515.0- 3363.0- 8031.0- 9252.0- 5933.0 1883.0 5452.0- 4333.0- 5354.0- 2884.0- 9341.0 0285.0p (cid:3) (cid:3)(cid:3) (cid:3)(cid:3)(cid:3) (cid:3)(cid:3) sj )182.0( )162.0( )775.0( )814.0( )922.0( )761.0( )451.0( )541.0( )101.0( )301.0( )102.0( )352.0( )653.0( )082.0( )302.0( )605.0( 4526.0 8167.0 5913.0 9216.0 4253.0 9172.0 3536.0 0095.0 2T )121.1( )941.1( )350.1( )911.1( )498.0( )102.1( )990.1( )651.1( 1044.1- 3772.1- 4828.1- 7901.1- 8769.0- 9865.1- 7391.1- 4024.1- 3T (cid:3) (cid:3) (cid:3)(cid:3) (cid:3)(cid:3) (cid:3) (cid:3) )047.0( )396.0( )157.0( )729.0( )865.0( )596.0( )116.0( )086.0( 2341.1- 4179.0- 7314.1- 1490.1- 542.1- 9551.1- 3972.1- 0271.1- 4T (cid:3) (cid:3) (cid:3) )786.0( )107.0( )757.0( )047.0( )376.0( )376.0( )827.0( )007.0( 7498.1 8422.1 7538.1 7194.1 4514.2 6226.1 1971.2 4802.2 1789.0 5344.0 8351.2 2346.1 4260.2 1056.1 7538.1 6983.1 snoc (cid:3)(cid:3) (cid:3)(cid:3) (cid:3) (cid:3)(cid:3)(cid:3) (cid:3) (cid:3)(cid:3)(cid:3) (cid:3) (cid:3)(cid:3) (cid:3)(cid:3)(cid:3) (cid:3)(cid:3) (cid:3)(cid:3) (cid:3) (cid:3)(cid:3) (cid:3) )768.0( )086.0( )076.0( )325.0( )508.0( )236.0( )826.0( )957.0( )315.0( )432.0( )647.0( )015.0( )957.0( )646.0( )076.0( )605.0( 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 N 72.0 10.0 03.0 50.0 04.0 70.0 92.0 01.0 94.0 23.0 92.0 40.0 13.0 60.0 72.0 10.0 2R dradnatS .setamitse SLO :setoN .1 elbaT ni detoned sa sepyt laroivaheb niatrec ot noita(cid:135)ni fo noitaived dradnats fo noitaleR :1A elbaT .detaluclac era srorre dradnats tsubor ,tsubor nmuloc rednU .level tnecrep 1/5/01 ta ecnac(cid:133)ingis setoned ***/**/* .sesehtnerap ni srorre .tnemtaert nihtiw noitalerroc rof wolla srorre dradnats ,retsulc nmuloc rednU 37

Trend extrapolation (M5) sd : cluster robust cluster robust s (cid:28) 1.6490 1.8727 1;s (cid:3)(cid:3) (1.016) (0.730) (cid:28) p 0.4539 0.4565 1;s s (cid:3) (cid:3)(cid:3)(cid:3) (0.186) (0.137) T2 0.6676 0.2027 (0.849) (0.817) T3 -0.9487 -1.0316 (cid:3) (cid:3) (0.541) (0.519) T4 -1.6194 -1.6396 (cid:3) (cid:3)(cid:3) (0.799) (0.748) cons 0.5515 0.8765 0.4929 1.0452 (cid:3) (cid:3) (cid:3) (cid:3)(cid:3) (0.1810) (0.461) (0.203) (0.461) N 24 24 24 24 R2 0.21 0.49 0.33 0.56 Table A2: Relation of standard deviation of in(cid:135)ation to the average coe¢ cient (cid:28) from 1 equation(M5)ofsubjectsthatusetrendextrapolatingrule. Notes: OLSestimates. Standard errors in parentheses. */**/*** denotes signi(cid:133)cance at 10/5/1 percent level. Under column robust, robust standard errors are calculated. Under column cluster, standard errors allow for correlation within treatment. 38

9M,5M 0=4(cid:11);6M 01M,7M,1M 8M,2M 6M tnemtaerT noitpecreP de(cid:133)icepssiM noitpecreP de(cid:133)icepssiM noitpecreP deziretemaraprednU noitpecreP deziretemaraprednU snoitatcepxElanoitaR )2level( muirbiliuqe )1level( muirbiliuqe )2level( muirbiliuqe )1level( muirbiliuqe )2 .per( muirbiliuqe )89:0;89:0:21:0;69:0;69:0;23:0( on )0;89:0;0;89:0( on )0;1( sey )0;78:0( sey )42:0;77:0( sey ).vnegiE( ycanimreteD 20:0 01:0 0 1 62:0 0 0 0 89:0 30:0 0 0 0 1 70:0 0 0 0 78:0 310:0 0 030:0 73:0 93:0 130:0 92:0 12:1(cid:0) 0 3:0 70:0 0 0 0 25:0 16:2 0 0 0 3:0 31:2 0 0 0 58:1 18:2 0 53:0(cid:0) 14:4 34:7 64:2 1B noituloS (cid:21) (cid:0) (cid:0)(cid:20) (cid:21) (cid:20) (cid:21) (cid:20) (cid:21) (cid:20) (cid:21) (cid:0) (cid:0)(cid:20) 10:0 70:0 0 1 62:0 0 0 0 89:0 60:0 11:0(cid:0) 38:0(cid:0) 0 3:0 01:0 0 0 0 25:0 36:3(cid:0) - - - 2B noituloS (cid:21) (cid:0)(cid:20) (cid:21) (cid:0) (cid:20) 18:1 0 0 i20:0+29:0 0 49:0 0 18:0 0 96:0 0 )a(1B .vnegiE 1 i71:0 (cid:6) 67:1 71:1(cid:3) 0 ::: 0(cid:2) i30:0+19:0 i20:0(cid:3)+40:0 (cid:0) 0 0(cid:2) 49:0 20:0 (cid:0)(cid:3) 0 0(cid:2) 96:0 51:0 (cid:0)(cid:3) 0 0(cid:2) 64:0 03:0 (cid:0) 72:0 (cid:0) 0(cid:3) ::: 0(cid:2) )b(1B .vnegiE (cid:3) 58:1 0 0 (cid:2) (cid:3) i20:0 (cid:0) 10:1 0 (cid:2) (cid:3) (cid:2) - (cid:3) (cid:2) - (cid:3) - (cid:2) )a(2B .vnegiE i41:0 (cid:6) 38:1 77:0 (cid:3) 0 ::: 0 (cid:2) i00:0 (cid:0) 99:0 i20:0 (cid:3)(cid:0) 40:0 (cid:0) 0 0 (cid:2) - - - )b(2B .vnegiE )(cid:3)99:0;99:0;11:0;79:0:79:0;72:0( on(cid:2) (cid:3) )0;99:0;0;99:0( on(cid:2) )0;1( sey )0;98:0( sey )10:0;67:0( sey ).vnegiE( ycanimreteD 20:0 01:0 0 1 81:0 0 0 0 99:0 30:0 0 0 0 1 60:0 0 0 0 98:0 110:0 0 120:0 73:0 04:0 140:0 52:0 81:1(cid:0) 0 3:0 50:0 0 0 0 15:0 84:2 0 0 0 3:0 09:1 0 0 0 02:2 87:2 0 53:0(cid:0) 32:6 5:01 81:2 1B noituloS (cid:21) (cid:0) (cid:0)(cid:20) (cid:21) (cid:20) (cid:21) (cid:20) (cid:21) (cid:20) (cid:21) (cid:0) (cid:0)(cid:20) 10:0 70:0 0 1 81:0 0 0 0 99:0 30:0 11:0(cid:0) 38:0(cid:0) 0 3:0 60:0 0 0 0 15:0 14:3(cid:0) - - - 2B noituloS (cid:21) (cid:0)(cid:20) (cid:21) (cid:0) (cid:20) 68:1 0 0 i10:0+49:0 0 69:0 0 58:0 0 17:0 0 )a(1B .vnegiE 2 i41:0 (cid:6) 28:1 51:1(cid:3) 0 ::: 0(cid:2) i20:0+39:0 i10:0(cid:3)+30:0 (cid:0) 0 0(cid:2) 69:0 20:0 (cid:0)(cid:3) 0 0(cid:2) 47:0 31:0 (cid:0)(cid:3) 0 0(cid:2) 84:0 82:0 (cid:0) 62:0 (cid:0) 0(cid:3) ::: 0(cid:2) )b(1B .vnegiE (cid:3) 98:1 0 0 (cid:2) (cid:3) i10:0 (cid:0) 200:1 0 (cid:2) (cid:3) (cid:2) - (cid:3) (cid:2) - (cid:3) - (cid:2) )a(2B .vnegiE i21:0 (cid:6) 78:1 08:0 (cid:3) 0 ::: 0 (cid:2) i00:0+99:0 i10 (cid:3) :0 (cid:0) 30:0 0 0 (cid:2) - - - )b(2B .vnegiE (cid:3) )69:0;69:0;61:0( sey(cid:2) (cid:3) )0;38:0;0;38:0( on(cid:2) )0;1( sey )0;37:0( sey )51:0;97:0( sey ).vnegiE( ycanimreteD 0 0 0 38:0 30:0 0 0 0 1 90:0 0 0 0 37:0 720:0 0 81:0 84:0 43:0 22:1 - 0 0 0 95:0 19:2 0 0 0 3:0 18:2 0 0 0 77:0 29:2 0 03:0(cid:0) 28:0 34:1 81:4(cid:0) 1B noituloS (cid:21) (cid:20) (cid:21) (cid:20) (cid:21) (cid:20) (cid:21) (cid:0) (cid:0)(cid:20) 70:0 83:0 0 1 04:2 0 0 0 38:0 01:0 21:0(cid:0) 56:0(cid:0) 0 3:0 31:1 0 0 0 95:0 51:3(cid:0) - - 2B noituloS (cid:21) (cid:0)(cid:20) (cid:21) (cid:0) (cid:20) - 55:0 0 96:0 0 64:0 0 93:0 0 )a(1B .vnegiE 3 i60:0+14:0 i30:0+03:0 (cid:0)(cid:3) 0 0(cid:2) 96:0 51:0 (cid:0)(cid:3) 0 0(cid:2) 83:0 (cid:0) 32:0(cid:3) 0 0(cid:2) 85:0 (cid:0) 54:0 (cid:0) 812:0 0(cid:3) ::: 0(cid:2) )b(1B .vnegiE 73:1 0 0 (cid:3) 31:1 0 (cid:2) (cid:3) (cid:2) - (cid:3) (cid:2) - (cid:3) - (cid:2) )a(2B .vnegiE i82:0 (cid:6) 92:1 93:0 (cid:3) 0 ::: 0 (cid:2) i00:0 (cid:0) 99:0 i30:0 (cid:0) 92:0 (cid:3) 0 0 (cid:2) - - - )b(2B .vnegiE )(cid:3)59:0;59:0;21:0;39:0;39:0;13:0( on(cid:2) (cid:3) )0;19:0;0;19:0( on(cid:2) )0;39:0( sey )0;48:0( sey - ).vnegiE( ycanimreteD 20:0 90:0 0 39:0 42:0 0 0 0 19:0 30:0 0 0 0 39:0 70:0 0 0 0 48:0 610:0 82:0 02:1(cid:0) 0 82:0 60:0 0 0 0 25:0 56:2 0 0 0 82:0 31:2 0 0 0 94:1 58:2 - 1B noituloS (cid:21) (cid:0) (cid:0)(cid:20) (cid:21) (cid:20) (cid:21) (cid:20) (cid:21) (cid:20) 10:0 60:0 0 39:0 42:0 0 0 0 19:0 90:0 11:0(cid:0) 38:0(cid:0) 0 82:0 90:0 0 0 0 25:0 22:4(cid:0) - - - 2B noituloS (cid:21) (cid:0)(cid:20) (cid:21) (cid:0) (cid:20) 38:1 i0:0 (cid:6) 0 i10:0+39:0 0 59:0 0 68:0 0 - )a(1B .vnegiE 4 i61:0 (cid:6) 47:1 61 (cid:3) :1 0 ::: 0 (cid:2) i20:0+58:0 i10:0 (cid:3) +40:0 (cid:0) 0 0 (cid:2) 88:0 20:0 (cid:0)(cid:3) 0 0 (cid:2) 07:0 11:0 (cid:0)(cid:3) 0 0 (cid:2) - )b(1B .vnegiE (cid:3) 78:1 i0:0 (cid:6) 0 (cid:2) (cid:3) i10:0 (cid:0) 600:1 0 (cid:2) (cid:3) (cid:2) - (cid:3) (cid:2) - - )a(2B .vnegiE i31:0 (cid:6) 18:1 87 (cid:3) :0 0 ::: 0 (cid:2) i00:0+29:0 i10 (cid:3) :0 (cid:0) 40:0 0 0 (cid:2) - - - )b(2B .vnegiE (cid:3) (cid:21)y y (cid:25) yc c (cid:2) (cid:25) (cid:25) (cid:25) yc c (cid:3) (cid:20) =C , (cid:21)2 2 (cid:25) (cid:25) (cid:25) yb b (cid:25) (cid:25) (cid:25) yb b 2 2 y y (cid:25) yb b y y (cid:25) yb b (cid:2) (cid:25) yb b (cid:20) =B , (cid:21)1 1 (cid:0) (cid:0) t t u g (cid:20) =1 (cid:0) tZ,0 (cid:3) 2 (cid:0) t(cid:25) 1 (cid:0) t(cid:25) 2 (cid:0) ty 1 (cid:0) ty 1 (cid:2) =1 (cid:0) tW , (cid:21)1 1 (cid:0) (cid:0) t t (cid:25) y (cid:20) =tX erehw;1 (cid:0) tZC+1 (cid:0) tWB=tX :mrof noituloS EER eht osla stneserper nmuloc dnoces ehT :setoN .smsinahcem noitamrof noitatcepxe tnere⁄id rednu snoitulos fo seitreporP :3A elbaT rehto htiw detaicossa era )b( elihw ,tnatsnoc eht htiw detaicossa era )a( htiw dellebal seulavnegiE .)skcohs eht rof tpecxe( 1 .per rednu .B xirtam ni detneserper sa ledom eht ni selbairav suonegodne 39