Rational Bias in Inflation Expectations

Transcription

1 Rational Bias in Inflation Expetations Robert G. Murphy * Boston College Adam Rohde Charles River Assoiates August 2014 Revised August 2015 Abstrat This paper argues that individuals may rationally weight prie inreases for food and energy produts differently from their expenditure shares when forming expetations about prie inflation. We develop a simple dynami model of the eonomy with gradual prie adjustment in the ore (non-food, non-energy setor and flexible pries in the food and energy setors. Serial orrelation of supply shoks to the food and energy setors allows individuals to gain an understanding about future shoks, possibly making it optimal for individuals to plae more weight on the movement of pries in these setors, as past movements of these pries may help predit future inflation. In partiular, if food and energy prie inflation exhibits a suffiient degree of persistene and wage adjustment is not too sluggish, we show that it is rational to put more weight on inflation in these setors than their expenditure shares in the Consumer Prie Index would warrant. We test this predition of the model using data on expeted inflation from the Federal Reserve Bank of Philadelphia s Survey of Professional Foreasters. Our results show the weights implied by the model for onstruting expetations of inflation differ from the expenditure shares of food and energy pries in the CPI for the United States. We find food prie inflation is weighted more heavily and energy prie inflation is weighted less heavily. But importantly, we annot rejet the hypothesis that these weights reflet rational behavior in forming expetations about inflation. Our analysis validates onerns raised by poliymakers as to whether expetations might not be well anhored with respet to some ommodity prie shoks, suh as those to food pries. As a onsequene, poliy may need to be alibrated arefully to prevent suh shoks from beoming embedded in expeted inflation. JEL Classifiation: E30, E31, E52, E58 Keywords: Inflation Expetations, Core Inflation, Food and Energy Pries, Anhored Expetations * Corresponding author: Robert G. Murphy, Department of Eonomis, Boston College, Chestnut Hill, MA 02467, murphyro@b.edu. An earlier version of this paper was presented at the Amerian Eonomi Assoiation Annual Meeting, Boston, MA January 3-5, We thank the referee, Susanto Basu, Brent Bundik, Fabio Shiantarelli, John Seater, and partiipants in the Boston College Maro Lunh seminar for insightful omments on an earlier draft of this paper. The views expressed herein are the views and opinions of the authors and do not reflet or represent the views of Charles River Assoiates or its staff.

2 1. Introdution Expetations about prie inflation play a entral role in modern maroeonomi analysis. They are important for understanding how households and firms make saving, spending, and investing deisions, and are a key input into negotiations for labor ontrats and the priing of finanial instruments. Central banks trak them for omparison with internal foreasts and targets for inflation. The ability of monetary authorities to ahieve prie stability depends on an aurate understanding of inflation expetations. A onern sometimes raised by poliymakers is whether inflation in highly visible produts suh as food and energy might overly influene the publi s pereption of inflation. For example, disussion at the Federal Open Market Committee meeting in June 2008 learly illustrates this onern: Partiipants had beome more onerned about upside risks to the inflation outlook--inluding the possibility that persistent advanes in energy and food pries ould spur inreases in long-run inflation expetations. Some noted that the inrease was greatest for short-term survey measures of households' inflation expetations, whih may be influened disproportionately by onsumers' pereptions of hanges in the pries of food and gasoline 1 This onern is not a reent one, having featured prominently during poliy disussions about supply shoks during the 1970s. 2 1 Federal Open Market Committee, Minutes of the Meeting of June 24-25, 2008, pp See Van Duyne (1982 for several examples.

3 2 In an interesting paper, Van Duyne (1982 onsiders poliymakers onerns that food prie inflation may overly influene the publi s expetations of inflation. He uses a simple model of the inflation proess to illustrate how suh bias an represent rational behavior. 3 Van Duyne is unable to rejet the hypothesis that the weights implied by his model are equal to the atual expenditure shares of food and other items in the onsumer prie index. He onludes that ontrary to the onventional wisdom of poliymakers, onsumers appear not to bias their expetations toward food pries (p Similar to Van Duyne, our paper explores whether pries of items that individuals frequently purhase and that often are quite volatile, suh as food and energy produts, play a larger role in the formation of inflation expetations than their expenditure shares would indiate. We show how it may be rational for individuals to assign relative weights to prie inreases in the food and energy setors that are larger than those setors shares in onsumer expenditure. To illustrate this potential bias in inflation expetations, we develop a simple three-setor dynami aggregate demand-aggregate supply model of the eonomy with gradual prie adjustment in the ore (non-food, nonenergy setor and flexible pries in the food and energy setors. Serial orrelation of shoks to food and energy pries allows individuals to gain an understanding about future shoks, possibly making it optimal for individuals to plae more weight on the movement of pries in these setors, as past movements of these pries may help predit future inflation. 4 3 We adopt Van Duyne s terminology and use the word bias to denote rational overweighting or underweighting of prie index omponents ompared with their expenditure shares. This bias does not represent statistial bias in whih onsistently positive or negative foreast errors arise. 4 We study how expeted inflation responds to movements in ommodity pries and whether this response represents overweighting or underweighting ompared to the expenditure shares of ommodities in the onsumer prie index. A separate but related literature explores the question of whether survey measures

4 3 Our framework extends Van Duyne s paper in three important ways. First, we model aggregate demand as a funtion of the real interest rate and inorporate a Taylortype poliy rule for the nominal interest rate, whereas Van Duyne expresses aggregate demand as a funtion of real money balanes and assumes a money supply growth rule. Seond, we allow for a differential optimal response by the monetary authority to inflation aross setors rather than assume a single response to overall inflation. Third, in our empirial analysis we test whether or not any observed bias in the formation of expetations about inflation is rational, whereas Van Duyne tests only for the presene of suh bias but not whether it is rational. We estimate the model using survey data on expeted inflation from the Federal Reserve Bank of Philadelphia. 5 In ontrast to Van Duyne, our results show the weights implied by the model for onstruting expetations of inflation differ from the expenditure shares of food and energy pries in the Consumer Prie Index (CPI for the United States. 6 In partiular, we find food prie inflation is weighted more heavily and energy prie inflation is weighted less heavily. But importantly, we annot rejet the of expeted inflation represent rational foreasts of future inflation. See, for example, the volume by Sinlair (2010 and artiles by Capistran and Timmermann (2009, Mankiw et al (2004, Ehrbek and Waldmann (1996, Noble and Fields (1982, and Mullineaux ( As disussed in Setion 3, we use data on expeted CPI inflation from the Federal Reserve Bank of Philadelphia s Survey of Professional Foreasters (SPF beause it provides measures of expeted inflation one-quarter and two-quarters ahead, whih our methodology requires. Fuhrer (2012 and Mishkin (2007, among other authors, also use the SPF to apture the publi s expetation of inflation at different horizons. See Croushore (1993 for an overview of the SPF. 6 One potential reason why our results differ from Van Duyne s is beause we use a later sample period. Van Duyne estimates his model over the period 1966 to 1977, whereas we estimate our model over a sample period beginning in 1982 determined by the availability of the SPF data for expeted inflation. Van Duyne uses data for expeted inflation from the University of Mihigan s Surveys of Consumers that is available for the earlier time period. His model requires data only for inflation expeted one period in the future whereas our model requires data for inflation expeted one and two periods in the future. Although the Mihigan data also provide measures of inflation expeted over more than one horizon (one-year and five-year, these data do not provide measures of inflation expeted over equal lengths of time one and two periods in the future, as needed for our analysis. Also, the Mihigan five-year horizon is omputed from a question that asks respondents about the next five-to-ten years and so is not very preise. By ontrast, the SPF data exatly math our model s timing.

5 4 hypothesis that these weights reflet rational behavior in forming expetations about inflation. Our analysis validates onerns raised by poliymakers that expetations might not be well anhored with respet to some ommodity prie shoks, suh as those to food pries. As a onsequene, poliy may need to be alibrated arefully to prevent suh shoks from beoming embedded in expeted inflation. Several reent papers have onsidered the response of inflation expetations to ommodity prie movements and reah onfliting onlusions. Counter to our findings, work by Trehan (2011 suggests that households are more sensitive to both food and energy pries in forming inflation expetations than they are to ore measures of inflation that exlude those items. He uses orrelation analysis to show that survey measures of inflation expetations are more losely related to inflation in food and energy items than to ore inflation. Experimental evidene presented by Georganas et al (2014 finds that pereption of the eonomy-wide inflation rate is influened by the frequeny with whih goods pries are observed, onsistent with our results for food pries but not energy. A reent paper by Arora et al (2013 shows that expeted inflation responds strongly to (what they term explosive deviations of overall inflation (often driven by energy pries from ore inflation. On the other hand, work by Verbrugge and Higgins (2015 finds energy prie shoks are muh less important in determining inflation expetations than are other maroeonomi variables, in line with our results and onsistent with evidene in Bernanke (2007 that inflation expetations have beome better anhored with respet to energy pries in reent deades. All of these papers use non-strutural methods in their analyses and none expliitly test whether the observed response reflets

6 5 rational behavior. 7 By ontrast, we develop a strutural model of the eonomy and use it to test diretly whether inflation expetations respond rationally to food and energy prie movements. The paper proeeds as follows. Setion 2 develops a simple dynami model of the eonomy to show how individuals may optimally overweight or underweight food and energy pries in forming expetations about overall inflation. Setion 3 presents tests of the model s preditions using survey data on expeted prie inflation and provides support for rational bias in inflation expetations. Setion 4 summarizes our findings and offers suggestions for further researh. 2. The Model Our model assumes pries are flexible in the food and energy setors but adjust sluggishly in the ore (non-food and non-energy setor, where firms with market power set prie as a markup over marginal ost. 8 The model onsists of a standard wage-prie Phillips urve augmented to inlude supply shoks and a dynami aggregate demand relationship inorporating a Taylor-type rule for monetary poliy. Our approah builds on work by Van Duyne (1982, but differs from his by inluding a monetary poliy rule for the nominal interest rate (instead of a money growth rule, expressing aggregate demand as a funtion of the real interest rate (instead of the real money supply, and allowing for differential response by the monetary authority to inflation aross setors. 7 Other authors who have analyzed the relationship between expeted inflation and ommodity prie movements using non-strutural methods inlude Wong (2014 and Celasun et al ( This differene in speed of prie adjustment between ommodity and ore setors draws on a distintion long emphasized in the literature between flexprie and fixprie markets. See, for example, Hiks (1974, Okun (1981, and the more reent disussions in De Ceo (2009 and Jespersen (2009. These differenes in prie adjustment have been ited as a possible justifiation for government support programs aimed at limiting volatility of agriultural inomes, as disussed in Congressional Budget Offie (1990, pp

7 6 2.1 Aggregate Supply 9 The omposite prie of goods and servies in the ore setor is given as a onstant markup (µ over marginal ost, whih we proxy by unit labor osts: P t = µ w tl t y t (1 where w is the wage, l is employment, and y is output. 10 For simpliity, we assume a onstant rate of labor produtivity growth in the ore setor so that equation (1 implies ore inflation ( equals wage inflation (ω t minus produtivity growth (g: 11 = ω t g. (2 The equilibrium wage is assumed to rise at the rate of expeted inflation plus produtivity growth, adjusted for the degree of slak in labor markets: ω t * = E t 1 + g a 1 u t (3 where E t 1 is the expetation in period t-1 of inflation for period t and u t is the gap between the atual rate of unemployment and its natural (full-employment level. Atual wage inflation adjusts to its equilibrium rate gradually, either beause overlapping wage ontrats make nominal wages stiky or osts of aquiring information lead to lags in 9 See Whelan (1997 for a derivation of a wage-prie Phillips urve similar to the one we develop here. 10 Profit-maximizing monopolistially ompetitive firms will set prie as a markup over marginal ost, where the markup may vary with demand onditions. For example, Mazumder (2014 finds that the markup in the U.S. manufaturing setor is ounterylial. For simpliity, and following Whelan (1997, we treat the markup as onstant. 11 If real wages rise by less than produtivity growth, we an replae g in equations (2 and (3 by zg where 0 < z < 1, leaving our derivation of equation (7 unhanged. Feldstein (2008, Anderson (2007, and Sherk (2013 find evidene that real ompensation has traked produtivity reasonably losely in line with equation (3.

8 7 updating otherwise flexible wages. 12 We approximate this gradual adjustment of wage inflation with a simple relationship: ω t ω t 1 = (ω t * ω t 1 0 < 1 (4 where parameter determines the speed with whih wage inflation adjusts to its equilibrium value. 13 We ignore differenes in produtivity growth rates aross setors and assume that inflation rates in the food and energy setors relative to the inflation rate in the ore setor are driven by serially orrelated supply shoks (ν t and ε t : f = π t +ν t π e t = π t + ε t (5 ν t = σ ν t 1 + ξ t ε t = δ ε t 1 + η t where σ, δ < 1, and ξ t, η t are mean-zero unorrelated random shoks. In priniple, the supply shoks ould be either positively or negatively serially orrelated and ould be represented by more ompliated time-series proesses. As disussed in Setion 3, we find that a first-order autoregressive proess fits the data well and provides for a parsimonious representation of these shoks. 12 The equilibrium wage an be interpreted as the wage that would prevail under full information. When information is ostly to aquire, firms and workers will update information with some lag, implying a gradual adjustment of wage inflation toward its equilibrium rate. See Mankiw and Reis (2002 for impliations of stiky information in a model of prie setting behavior. 13 Many authors have shown that inflation is persistent. See, for example, Fuhrer and Moore (1995, Nelson (1998, Mankiw (2001, and Roberts (2006. We assume gradual (or stiky adjustment of wage inflation to its equilibrium rate so that prie inflation in the ore setor exhibits persistene in our model. Equation (4 is onsistent with stiky adjustment of the level of the wage when the adjustment equation for the level of the wage is w t w = t 1 * w ( t w t 1.

9 8 The overall rate of inflation for the eonomy is measured using a weighted average of prie inflation in the food, energy, and ore setors of the eonomy: = A f + B e + (1 A B (6 where A and B are expenditure shares for food and energy items used to onstrut the prie index for onsumer expenditures. 14 Using equations (2, (3, (4, (5, and (6, we an solve for a Phillips urve relating overall inflation to expeted inflation and lagged values of inflation for the ore, food, and energy setors: = E t 1 a 1 u t + (1 1 f + A{σ ( 1 e 1 + ξ t } + B{δ ( η t } (7 where the last two terms are written as deviations of food and energy inflation from ore inflation and an be interpreted as supply shoks. Equation (7 is similar to the expetations-augmented Phillips urve of Friedman (1968 as generalized by Gordon (1982 to inorporate supply shoks and inertia through lagged ore inflation Aggregate Demand To omplete the model, we speify an IS-type demand equation that relates the unemployment gap to the real interest rate: 14 Equation (6 holds for Laspeyres fixed-weight indexes. The offiial CPI for the United States uses a Laspeyres formulation at its upper level for aggregate items suh as food and energy. See Bureau of Labor Statistis (2007 for details. 15 See also Gordon (1990, Fuhrer (1995, and Murphy (2000 who provide empirial support for standard Phillips urve models of inflation. Bernanke (2008 provides an overview of several important issues for Phillips urve analyses of inflation. Ball and Mazumder (2011 inorporate anhored expetations into an otherwise standard Phillips urve as a possible reason for why the United States did not experiene deflation during and immediately after the Great Reession. Murphy (2014 finds that a Phillips urve modified to aount for unertainty about regional eonomi onditions an explain the behavior of inflation following the Great Reession.

10 9 u t = (i t E t +1 ρ υ t (8 where i t is the nominal interest rate, E t +1 is the expetation of overall inflation in period t+1 as of period t, ρ is the natural or long-run value of the real interest rate at whih the unemployment gap is zero, and υ t is a mean-zero unorrelated shok to demand. 16 The monetary authority is assumed to target the nominal interest rate so as to raise (lower the real interest rate when inflation exeeds (falls short of its target or when the unemployment rate is less than (greater than its natural level. 17 We allow the monetary authority to respond differently to inflation in food, energy, and ore setors: i t = + ρ + (λ f f + λ e e + (1 λ f λ e π * u t 0 <, ; 0 < λ f,λ e,λ f + λ e < 1 (9 where λ f and λ e are the relative weights the monetary authority plaes on food and energy inflation, aptures the response to deviations in overall inflation from its target, π * is the monetary authority s target for inflation, aptures the response to deviations in unemployment from its natural rate, and ρ is the long-run real interest rate. Note that λ f and λ e need not equal the expenditure shares used in equation (6 to onstrut the prie index for onsumer expenditures Romer (2012, Chapter 6, derives a New Keynesian IS urve relationship similar to equation (8 from the maximizing behavior of households. 17 See Taylor (1993 and Bryant, Hooper, and Mann (1993 for a disussion of entral bank interest-rate poliy that appears to be well approximated by suh rules. 18 As shown later in the paper, our solution to the model imposes the ondition that λ f and λ e equal the optimal weights on food and energy inflation used by individuals in forming expetations about overall

11 10 Substituting this Taylor-type rule into equation (8 and using equation (6 to express overall inflation in terms of its setoral omponents yields the following dynami aggregate demand expression: u t = a (A +θ 3 1λ f f (1+ + (B +λ e (1+ π e t + a {(1 A B +θ (1 λ λ } 3 1 e f (1+ (1+ E t+1 (1+ π * 1 (1+ υ t (10 relating unemployment to inflation in the food, energy, and ore setors. Equation (10 shows that when inflation inreases, the unemployment gap also inreases as the monetary authority raises the real interest rate to redue aggregate demand and ontain inflation. Depending on the values of λ f and λ e, the monetary authority may respond differently to inflation in the food, energy, and ore setors. The unemployment gap will respond more to inflation when the monetary authority plaes a larger weight,, on deviations from the inflation target and it will respond less to inflation when the monetary authority plaes a larger weight,, on stabilizing unemployment at its natural rate. An inrease in expeted inflation, given urrent and target inflation, lowers the real interest rate, raises demand and lowers the unemployment gap. Note that when inflation is inflation. This ensures that the monetary authority responds to food and energy inflation in a manner onsistent with their underlying time series dynamis. We do not derive values for and in terms of other model parameters beause this requires speifying the form of the monetary authority s preferene struture and solving a ompliated dynami programming problem that is beyond the sope of this paper. Ball (1994, 1997 argues that solutions for Taylor-type rules are highly sensitive to partiular speifiations of preferenes and model struture. Our empirial analysis aounts for the estimated values of these poliy oeffiients, ensuring our results inorporate poliymakers atual preferenes while allowing differential response to setoral inflation rates.

12 11 onstant and equal to its target level (and the demand shok is zero, equation (10 implies the unemployment gap is zero so that unemployment is equal to its natural rate Equilibrium To solve for the equilibrium value of inflation, we first substitute for period t setoral inflation rates in equation (10 by using equations (2, (3, (4, and (5 to obtain: u t = S π + (1 t S 1 + S + λ f S ν t + λ e ε t S E t 1 S E t+1 S π * 1 S υ t (11 where S = (1+ a 1 +, and where we have used the definition of measured inflation given by equation (6 to ombine setoral inflation rates into overall inflation. Next, we use equation (11 to substitute for the unemployment gap in equation (7 and rearrange to obtain: = a (1+ 3 E t 1 + a a a X X E t+1 + a a a π * + X {(1 (1+ S(σ A + δ B} 1 + Sσ A X X π f t 1 + Sδ B X π e t 1 + SAξ t X + SBη t X a a a λ f ν t a a a λ e ε t + a a 1 2 X X X υ t (12 where X = {1+ + a 1 (1+ } and, as before, S = (1+ a 1 +. Equation (12 shows how inflation in period t depends on inflation expeted for period t and inflation expeted for period t+1. An inrease in expeted inflation for period t as of 19 We do not impose a zero lower bound on the nominal interest rate even though our model is estimated over the reent period when the federal funds rate was zero beause the relevant interest rate in our model is more akin to long-term rates that have the most important effets on aggregate demand. Long-term interest rates remained above zero during and after the Great Reession. One also an view allowing a negative interest rate in our model as a simple way to apture alternative poliy measures like quantitative easing.

13 12 period t-1 leads to higher wage inflation, whih in turn is partly passed through into higher ore inflation and, hene, overall inflation. An inrease in expeted inflation for period t+1 as of period t lowers the real interest rate and raises aggregate demand (i.e., redues the unemployment gap, thereby inreasing inflation. The equation exhibits a neutrality property: if expeted inflation rates, target inflation, and lagged setoral inflation rates all inrease by the same proportion (and demand and supply shoks are zero, then overall inflation also will inrease by the same proportion, and from equation (10 the unemployment gap will be unaffeted. To obtain a relationship desribing equilibrium expeted inflation, we take expetations of both sides of equation (12 as of time t-1, and rearrange to yield: E t 1 = µπ * f +ψ E t ζ [α 1 e + β 1 + (1 α β 1 ] (13 where: α = σ (SA a 1 λ f (1 (1+ β = δ (SB a 1 λ e (1 (1+ ζ = (1 (1+ Z µ = a 1 Z ψ = a 1 Z S = (1+ a 1 + Z = (1 (1+ + [a 1 (1+ ]. Equation (13 relates inflation expeted one period in the future to inflation expeted two periods in the future, the monetary authority s inflation target, and a weighted average of

14 13 lagged setoral inflation rates. The monetary authority is assumed to use the same optimal relative weights that individuals use to form expetations about inflation. This implies that! λ f = α and! λ e = β. Applying these restritions and solving gives the following expressions for! λ f and! λ e in terms of the parameters of the model: σ A(1+ a λ f = 1 + (1 (1+ +σ a 1 δ B(1+ a λ e = 1 +. (1 (1+ + δ a 1 (14 Substituting these expressions for! λ f and! λ e into equation (13 provides the omplete solution for expeted inflation: E t 1 = µπ * +ψ E t ζ [α * f 1 + β * e 1 + (1 α * β * 1 ] (15 where: α * = β * = σ A(1+ a 1 + {(1 (1+ +σ a 1 } δ B(1+ a 1 + {(1 (1+ + δ a 1 } ζ = (1 (1+ Z µ = a 1 Z ψ = a 1 Z Z = (1 (1+ + [a 1 (1+ ]. Similar to equation (13, equation (15 relates inflation expeted one period in the future to inflation expeted two periods in the future, the monetary authority s inflation target,

15 14 and a weighted average of lagged setoral inflation rates. But now the relative weights on setoral inflation rates assume the monetary authority responds in a manner onsistent with the underlying time series dynamis of setoral inflation. Note that the relative weights on lagged food, energy, and ore inflation sum to one but in general are not equal to the expenditure shares used to ompute the overall rate of inflation. In partiular, the relative weights on food and energy inflation (α * and β * will exeed their respetive expenditure shares (A and B when σ + > 1 and δ + > 1. These onditions are more likely to hold when the rate of inflation in the food and energy setors is persistent (supply shoks to these setors exhibit suffiient serial orrelation so that σ and δ are large and the speed of wage adjustment is not too sluggish (so that is large. Conversely, the relative weights will fall short of their respetive expenditure shares when σ + < 1 and δ + < 1. The intuition for these onditions is that serial orrelation of shoks to food and energy setors allows individuals to gain an understanding about future shoks. Higher serial orrelation of shoks and the assoiated greater persistene in food and energy inflation, other things equal, leads individuals to optimally overweight food and energy inflation (and underweight ore inflation in forming expetations. But if the speed of wage adjustment is slow enough, ore inflation, whih depends diretly on wage inflation, will exhibit suffiient persistene, leading individuals to overweight ore inflation and underweight food and energy inflation in forming expetations. Equation (15 imposes following restritions on the relative weights individuals should optimally plae on inflation in the food and energy setors when forming expetations about overall inflation:

16 15 α * = β * = σ A(1+ a 1 + {(1 (1+ +σ a 1 } δ B(1+ a 1 + {(1 (1+ + δ a 1 }. (16 In the next setion, we desribe how these restritions an be tested using survey data on inflation expetations to estimate the parameters of our model. 3. Estimation Results We estimate relationships of the following form that math equation (15: t 1π E t = γ 0 + γ 1 t 1 π E t+1 f + γ 2 1 e 1 1 (17 where t 1 π E E t is expeted CPI inflation for period t as of period t-1, t 1 +1 is expeted CPI inflation for period t+1 as of period t-1, and π f, π e, and π are CPI inflation rates for food, energy and ore setors. Beause the monetary authority s target for inflation is assumed fixed (and equal to two perent, we inlude a onstant term in the equation. To estimate equation (17, we need data for inflation expeted one and two periods in the future. The Federal Reserve Bank of Philadelphia s Survey of Professional Foreasters (SPF provides quarterly measures of CPI inflation in the United States expeted for multiple time periods in the future, allowing diret estimation of equation ( Although the University of Mihigan s Survey of Consumers is more representative of the population at large than the SPF, the Mihigan survey reports monthly measures of 20 The Survey of Professional Foreasters provides median inflation expeted one, two, three, and four quarters ahead, along with an estimate for the next ten years. See Federal Reserve Bank of Philadelphia (2014 for details. The Philadelphia Fed also oversees the Livingston survey of foreasters, but that survey is onduted only every six months and provides foreasts for inflation over the next 6 and 12 months. See Murphy (1986 for an analysis of the term struture of inflation foreasts using the Livingston expeted inflation data.

17 16 inflation expeted over the next year and the next 5 to 10 years, not for one and two periods ahead. 21 In addition, the Mihigan survey asks only about generi inflation, whereas the SPF asks about speifi measures inluding the CPI. Beause the SPF mathes the timing of the expeted inflation variables in our model and asks speifially about the CPI, we use those data in our analysis. 22 The parameters α *, β *, ζ, µ, and ψ in equation (15 are related to the estimated oeffiients of equation (17 by the following expressions: γ 2 α * γ 2 γ 3 β * γ 2 ζ γ 2 µ γ 0 / π * (18 ψ γ 1 where α * and β * represent the relative weights plaed on food and energy pries in forming expetations about inflation. Under the null hypothesis that individuals form expetations about future inflation using our simple model, equations (16 and (18 imply the following restritions: (1 γ 2 {σγ 0 + (γ 2 π * } σ (γ 2 {(1 γ 0 + (γ 2 π * } = A (1 γ 3 {δγ 0 + (γ 2 π * } δ (γ 2 {(1 γ 0 + (γ 2 π * } = B (19 21 See Curtin (1996 for details about the Mihigan expeted inflation series. 22 See Fuhrer (2012 and Mishkin (2007 who also use the SPF median to measure the publi s inflation expetations at various horizons.

18 17 where we interpret A and B as the expenditure shares for food and energy used to ompute the CPI. To test these restritions, we need estimates of,σ, δ, and π * in addition to the parameters of equation (17. This requires estimation of equation (17 in ombination with the wage adjustment equation and the autoregressive relationships for the supply shoks. To reover the wage adjustment parameter,, we estimate a Phillips urve for wage inflation derived by substituting equation (3 into equation (4 and rearranging: ω t = φ 0 φ 1 u t + φ 2 t 1 E + (1 φ 2 ω t 1 (20 where φ 2. We impose the onstraint that the oeffiients on expeted inflation and lagged wage inflation sum to one when estimating equation (20, as implied by equations (3 and (4. 23 To obtain estimates of the autoregressive parameters, σ and δ, we use equations (5 to derive the following relationships: f π f t = ϑ 1 ( 1 1 π e t π e t = ϑ 2 ( 1 1 (21 where ϑ 1 σ and ϑ 1 δ. Finally, we assume that target inflation,! π *, equals 2 perent. 24 Using equations (20 and (21, the restritions given by equation (19 an be written in terms of the estimated parameters: 23 This restrition ensures that wage inflation will approah its equilibrium value in the long run. 24 Estimates of our model using a time-varying impliit inflation target from Leigh (2005 were similar to those we present below using a onstant inflation target of two perent.

19 18 (1 φ 2 γ 2 {ϑ 1 γ 0 + (γ 2 2} ϑ 1 (γ 2 {(1 φ 2 γ 0 + (γ 2 2} = A (1 φ 2 γ 3 {ϑ 2 γ 0 + (γ 2 2} ϑ 2 (γ 2 {(1 φ 2 γ 0 + (γ 2 2} = B (22 where, again, A and B are the expenditure shares of food and energy in the CPI. We employ generalized method of moments (GMM to jointly estimate equations (17, (20, and (21 using quarterly data for the United States. 25 For the inflation variables, we use the quarterly perentage hange expressed at an annual rate for the food, energy, and ore omponents of the CPI for All Urban Consumers (CPI-U. We measure the unemployment gap using the differene between the quarterly ivilian unemployment rate for workers unemployed less than 27 weeks and the quarterly value of the natural rate estimated by the Congressional Budget Offie ( For wage inflation, we use the quarterly perentage hange in the average hourly earnings expressed at an annual rate. And as disussed earlier, we measure expeted inflation using the SPF, whih provides data for median expeted inflation at an annual rate one and two quarters into the future. We start our sample period in 1982, the first full year that the SPF provides onsistent data for multi-horizon expetations of inflation. In our 25 The GMM estimation proedure allows for a generalized variane-ovariane struture of regression error terms for these equations, whih we use in our hypothesis tests. All data, exept as noted, are from the U.S. Bureau of Labor Statistis. 26 Gap measures using the short-term unemployment rate have been shown by Ball and Mazumder (2015 and Krueger et al (2014 to be a better indiator of eonomi slak in Phillips urve models than gap measures using the overall unemployment rate. In appendix Table A-1, we report results using the overall unemployment rate, whih are very similar to those disussed here exept that the oeffiient on the gap term is smaller and not statistially signifiant at standard levels.

20 19 estimation, we use as instruments four lags eah of overall inflation, wage inflation, the unemployment gap, and inflation expeted two quarters ahead. 27 To test the hypotheses given by equation (22, we use the average relative importane weights of food and energy in the CPI-U as reported by the Bureau of Labor Statistis for our values of A and B. Figure 1 plots these relative importane weights over our sample period. Exept for an initial deline from 1982 through 1986, both shares are reasonably stable, flutuating in a band of only a ouple of perentage points around their average values. The food share gradually trends down after 1986 while the energy share moves up slightly during reent years. We report in Table 1 estimates for the oeffiients of equations (17, (20, and (21 along with their standard errors. 28 For the period 1982 to 2014, all oeffiients are of the orret positive sign and all are statistially different from zero at high levels of onfidene. Hansen s J-statisti shows that we annot rejet at standard levels of onfidene the set of overidentifying restritions on the instruments. Our point estimate of the persistene of inflation in the food setor (ϑ 1 is higher than for the energy setor (ϑ 2, onsistent the view that underlying shoks to raw energy pries feed through more rapidly into final-stage produts and thus dissipate more quikly than do shoks to raw food pries. 29 Wage inflation adjusts to its equilibrium value at a rate that eliminates roughly 20 perent of the gap in one quarter, as aptured by the point estimate of φ Our instrument set uses lagged values of only two-quarter-ahead expeted inflation, sine the onequarter-ahead and two-quarter-ahead series are highly orrelated. Estimates when both series are inluded in the instrument set and when only the one-quarter-ahead series is inluded give results similar to those reported in the text. 28 Estimates of the onstant terms in equations (17 and (20 (not reported were small in magnitude and generally not statistially signifiant. 29 See Pedersen (2011 for analysis of the relative speed with whih food and energy prie shoks dissipate.

21 20 Table 1 also presents test results for the hypothesis that individuals form expetations rationally so that the parameters jointly satisfy the two restritions given by equation ( The p-value of 0.41 indiates that we annot rejet the hypothesis that individuals form expetations rationally in aord with these restritions. We an, however, rejet the hypothesis that the estimated relative weights, given by equation (18 and reported in Table 1, are equal to the atual expenditure weights, as seen by the p- value of 0.00 for the test of relative weights. The estimated relative weight for food (0.275 is larger than its atual value (0.155 but the estimated relative weight for energy (0.047 is smaller than its atual value ( This suggests rational overweighting for food but rational underweighting for energy in forming expetations about inflation. 31 The underweighting of energy inflation is onsistent with evidene that monetary poliy was tightened sharply in response to energy-related supply shoks during the 1970s and early 1980s so as to dampen inflation, leading expetations to beome better anhored and less responsive to these shoks in subsequent deades. 32 As shown in Figure 1, the relative weights of food and energy in the CPI were higher in the early 1980s before falling sharply and beoming more stable after To assess whether our findings are robust to exluding the early 1980s, Table 1 also provides results for the period from 1986 to The estimates and test results are similar to 30 The reported test statisti is distributed as χ 2 and is omputed using the method desribed in Greene (2012, p. 528, for both linear and nonlinear hypotheses. We use this method when alulating test statistis for all hypothesis tests reported in our paper. 31 Consistent with these results, we annot rejet the simple hypothesis that φ 2 + ϑ 1 1 (p-value = 0.35 but an rejet the hypothesis that φ 2 + ϑ 2 1 (p-value = Similar results also hold for the 1986 to 2014 sample period. 32 Hooker (2002 finds that monetary poliy has responded less forefully to energy prie shoks sine around 1980 beause expetations of inflation may have beome less sensitive to suh shoks. The underweighting of energy inflation in expetations formation is onsistent with evidene disussed in Bernanke (2007 that inflation expetations have beome better anhored with respet to energy prie shoks in reent deades.

22 21 those for the full sample period, with the exeption that the oeffiient on the unemployment gap in equation (20 is signifiantly different from zero at only slightly above the ten-perent level, possibly refleting greater importane for the shorter sample period of downward nominal wage rigidity during the prolonged episode of high unemployment assoiated with the Great Reession. 33 One again, we annot rejet the hypothesis of rational bias in expetations formation. We also onsider a version of the model in whih food and energy setors are ombined. Here we use data on pries in the food and energy setors along with relative expenditure shares to ompute a prie index for the ombined setors. 34 We estimate a model of the same struture as before exept that now only one non-ore inflation rate enters the model: t 1π E t = γ 0 + γ 1 t 1 π E t+1 ef + γ ω t = φ 0 φ 1 u t + φ 2 t 1 E + (1 φ 2 ω t 1 (17a (20a ef π ef t = ϑ 1 ( 1 1 (21a where ef is the omposite rate of prie inflation in the food and energy setors. We use generalized method of moments to jointly estimate equations (17a, (20a, and (21a and, as before, employ as instruments four lags eah of overall inflation, wage inflation, the 33 Gali (2011 also obtains an insignifiant oeffiient on unemployment in wage Phillips urves and argues that the estimate is overly influened by downward nominal wage rigidity during the period of prolonged high unemployment aompanying the Great Reession. 34 See Bureau of Labor Statistis (2007, Chapter 17, pp , for details on how to onstrut omposite prie indexes using relative expenditure shares and prie indexes for setors.

23 22 unemployment gap, and inflation expeted two quarters ahead. The restrition in equation (22 now takes the form: (1 φ 2 γ 2 {ϑ 1 γ 0 + (γ 2 2} ϑ 1 (γ 2 {(1 φ 2 γ 0 + (γ 2 2} = C (22a where C is the ombined expenditure share of food and energy in the CPI. Table 2 provides results for the omposite version of our model. All oeffiients are of the orret positive sign and all are statistially signifiant at the one-perent level. For the period 1982 to 2014, we annot rejet the hypothesis that expetations are formed rationally aording to the struture of our model, as indiated by the probability value of 0.87 for the test of the restrition given in equation (22a. We an, however, rejet the hypothesis that the estimated relative weight for the food-energy omposite setor is equal to its atual value, as shown by the p-value of The estimated relative weight of is smaller than its atual value of 0.237, indiating rational underweighting of omposite food-energy inflation in the formation of expetations. Given the results of Table 1 showing underweighting of energy inflation and overweighting of food inflation, this underweighting of omposite food-energy inflation suggests that the influene of energy prie inflation dominates the influene of food prie inflation in forming expetations of overall inflation. As disussed above, this underweighting may reflet better anhoring of inflation expetations with respet to energy prie shoks sine the early 1980s. We again provide a robustness hek on our results by estimating the

24 23 omposite model over the period 1986 to The seond olumn of Table 2 shows that estimates over this shorter sample period are similar to those for the full sample Summary This paper has argued that individuals may rationally weight prie inreases in the food and energy setors differently from the expenditure shares of these setors in the CPI when forming expetations about overall prie inflation. We developed a simple dynami model of the eonomy to illustrate this finding. Serial orrelation of shoks to food and energy pries allows individuals to gain an understanding about future shoks, possibly making it optimal for individuals to plae more weight on the movement of pries in these setors, as past movements of these pries may help predit future inflation. In partiular, if the degree of persistene for shoks to food and energy inflation is high enough and the speed of wage adjustment is not too sluggish, the model predits individuals will overweight prie movements for food and energy ompared with their expenditure shares in the CPI. Using data on expeted inflation from the Federal Reserve Bank of Philadelphia s Survey of Professional Foreasters, we show the weights implied by the model for onstruting expetations of inflation differ from the expenditure shares of food and energy in the CPI. Speifially, we find food prie inflation is weighted more heavily and energy prie inflation is weighted less heavily. But we annot rejet the hypothesis that these weights reflet rational formation of expetations about inflation. These results indiate that expetations might not be well anhored with respet to some ommodity 35 In appendix Table A-2, we report estimates for the omposite version of our model using the overall unemployment rate in the gap measure. These results are very similar to those reported in Table 2.

25 24 prie shoks, suh as those to food pries. It follows that poliymakers may need to tailor responses to suh shoks to aount for the disparity between expenditure shares and the weights individuals plae on setoral inflation in forming expetations. This may mean more foreful poliy is neessary in response to some shoks to prevent them from beoming embedded in expeted inflation. Our goal in this paper was to develop a simple dynami model of the maroeonomy to illustrate the possibility of rational bias in expetations formation and to provide some preliminary results using survey data on inflation expetations. The analysis showed how this bias depends on persistene in ommodity prie inflation and the degree of inertia in wage adjustment. Future researh should extend our work by exploring the links between rational bias and models of rational inattentiveness, where differenes in the ost of aquiring information aross setors (or differenes in willingness to pay attention to news aross setors ould lead to over or underweighting of inflation aross setors. 36 Our approah ould also be applied to ountries other than the United States to asertain similarities and differenes aross ountries in the response of inflation expetations to movements in food and energy pries. Finally, our simple model ould be modified to inorporate a greater degree of forward-looking behavior similar to New Keynesian stiky-prie models. 36 See, for example, Mankiw and Reis (2002 and Carroll (2003 for models where information is updated only periodially.

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