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Relative Price Shocks and Inflation WP 22-07 Francisco Ruge-Murcia McGill University Alexander L. Wolman Federal Reserve Bank of Richmond Relative Price Shocks and In‡ation Francisco Ruge-Murciay and Alexander L. Wolmanz May 2022 Abstract In‡ation is determined by interaction between real factors and monetary policy. Among the most important real factors are shocks to the supply and demand for di¤erent components of the consumption basket. We use an estimated multi-sector New Keynesian model to decompose the behavior of U.S. in‡ation into contributions from sectoral (or “relative price”) shocks, monetary policy shocks, and aggregate real shocks. The model is estimated by maximum likelihood with U.S. data for the post-1994 period in which in‡ation and the monetary policy regime appeared to be stable. In addition to providing a broad decomposition of in‡ation behavior, we enlist the model to help us understand the in‡ation shortfall from 2012 to 2019, and the dramatic in‡ation movements during the COVID pandemic. JEL classi…cation: E31, E52, E58 Key Words: Monetary policy, sectoral shocks, in‡ation shortfall, COVID-19. This research received …nancial support from the Social Sciences and Humanities Research Council of Canada and the Fonds de recherche du Québec. This paper represents the views of the authors and not of the Federal Reserve Bank of Richmond, the Federal Reserve Board, or the Federal Reserve System. The authors would like to thank Stephanie Schmitt-Grohé for helpful conversations, and Elaine Wissuchek and Zachary Edwards for outstanding research assistance. y Department of Economics, McGill University. Email: francisco.ruge-murcia@mcgill.ca z Federal Reserve Bank of Richmond. Email: alexander.wolman@rich.frb.org 1. Introduction How do factors that drive relative price changes across consumption categories a¤ect in‡ation? In theory, monetary policy can o¤set these factors and perfectly stabilize in‡ation. This does not happen in practice for two broad reasons: …rst, monetary policy may be unable to control in‡ation perfectly; second, monetary policy may choose not to control in‡ation perfectly, instead following a policy that results in equilibrium ‡uctuations in in‡ation. Following the latter branch, we ask what is the contribution of “relative price” shocks to the behavior of in‡ation. We focus on in‡ation determination in a stable monetary policy regime, namely the post-1994 United States. The paper makes three contributions, all based on an estimated multi-sector New Keynesian model. First, we provide summary information on the importance of relative price shocks for the behavior of in‡ation since 1994. Second, we examine the in‡ation shortfall from target that occurred between 2012 and 2019 in the U.S., decomposing that shortfall into policy and real shocks. Third, we apply the model to studying the dramatic change in in‡ation during the COVID pandemic, providing an out-of-sample version of that same decomposition. The relative in‡uence of monetary policy and real factors on in‡ation is one of the most important and long-studied questions in monetary economics. Any such analysis immediately confronts the issue of monetary policy regimes. At one extreme, if there is a stable underlying monetary policy regime, then the tools of rational expectations and local approximation of DSGE models may be appropriate. At another extreme, clearly identi…ed breaks in regime can be the source of facts against which theories are evaluated, but the researcher must decide how to model information sets of private agents and the policymaker. The fundamental assumption of this paper is that from January 1995 to December 2019 the United States was in a stable, well-understood monetary policy regime and that it is appropriate to use a locally approximated DSGE model with rational expectations to study that period. The …rst contribution involves basic analysis of in‡ation and relative price shocks. The model has 15 consumption categories, shocks to productivity whose volatility and dynamics di¤er across categories, and Rotemberg costs of price adjustment that are allowed to vary across categories. We estimate the model on monthly data from 1995-2019 by maximum likelihood. The observable variables are rates of price change for the 15 consumption categories, and the level of the nominal interest rate. We decompose in‡ation into the contribution of monetary policy shocks and shocks to productivity in each of the consumption categories. Over the sample, in‡ation was low and stable, but nonetheless exhibited substantial month-to-month volatility. Our estimates explain that volatility primarily with sectoral shocks, which we label relative price shocks. One way we 1 evaluate the model …t is through its ability to match the historical relationship between the monthly in‡ation rate and the distribution of relative price changes. As discussed in Hornstein, Ruge-Murcia and Wolman (2022), from 1995 to 2019 there was a tight negative relationship between the monthly in‡ation rate and the share of consumption expenditures exhibiting relative price inreases. The model reproduces and explains that relationship. The second contribution is to decompose the shortfall of in‡ation from target in the 20122019 period. According to this decomposition, which uses the smoothed estimates of the model’s structural shocks, the largest contributors to the cumulative shortfall of the price level from the 2% trend implied by the Fed’s target were shocks to gasoline and energy goods and to health care. Note that we are intentionally describing causality; absent a model, when one analyzes in‡ation and category price changes, it is only possible to describe the extent to which a particular category’s price change accounts for in‡ation. The model allows us to provide estimates of the extent to which particular shocks caused the in‡ation shortfall. While nontrivial, the monetary policy contribution is relatively small (about 17%). A leading theoretical explanation for the in‡ation shortfall is that it resulted from interaction between the lower bound on nominal interest rates and the Fed’s implicit policy rule. See for example Bianchi, Melosi and Rottner (2021). Our current results cannot speak directly to that view because we do not impose a lower bound on nominal interest rates. However, during much of the period when the shortfall occurred, the interest rate was at its lower bound, and we do use interest rate data in estimation. The 17% contribution from monetary policy shocks can be interpreted as a lower bound for the e¤ects of the zero bound. The third contribution is an out-of-sample analysis of the high in‡ation episode that began in March 2021 during the COVID-19 pandemic, and is ongoing as of this writing. As with the in‡ation shortfall, we decompose in‡ation into contributions of the sectoral shocks and monetary policy. Here, however, the smoothed estimates of the model’s shocks use out-of-sample data, as our estimation sample ends in December 2019. In conducting local analysis around the model’s (trending) steady state, we assume that the same policy rule remains in place during this high in‡ation episode. But has in‡ation in fact remained anchored? Our approach describes the contributions to in‡ation of policy shocks and real shocks under the assumption that in‡ation remains anchored. If one has outside information about the shocks, then that can be used together with our estimates to evaluate the anchoring question. Summarizing what we …nd: of the 2:7% in‡ation overshoot from March 2021 through November 2021, the largest contributor is shocks to the motor vehicle sector, at 31%. According to the model, monetary policy shocks explain 22% of the in‡ation overshoot. There is a large and varied literature on relative prices and in‡ation. One branch focuses on the 2 role of oil shocks in driving short-run in‡ation ‡uctuations; see for example Kilian and Zhou (2021) and the references therein. Our paper embodies a generalization of that idea: oil shocks are the most volatile relative price shocks, but in any given period some other category might experience an unusually large shock that accounts for a sizable movement in in‡ation— for example, motor vehicles during the COVID period. Some other branches of the literature were concerned mainly with periods when in‡ation was not stable— in particular a set of papers from the 1970s and ’80s that studied causality in the opposite direction, from in‡ation to the variability of relative prices. A key reference is Parks (1978). In principle, our model can have in‡ation (i.e., policy) a¤ect relative price variability but in practice that channel is weak, as one would expect in a stable policy regime. The papers closest to ours are Ball and Mankiw (1995), Balke and Wynne (2000), Reis and Watson (2010), and Smets, Tielens, and Van Hove (2019). In Ball and Mankiw (1995), menu costs mean that in‡ation tends to move with the price change of sectors hit by the largest shocks to their desired price; prices don’t change in sectors hit by small shocks to their desired price. In our model, Rotemberg costs mean that any …rm hit by a shock will adjust its price; in equilibrium, the interaction of policy and shocks delivers a relationship between in‡ation and the distribution of actual relative prices consistent with the data. Balke and Wynne (2000) conduct an analysis quite similar to ours, but their model has ‡exible prices and monetary policy is characterized by a constant money growth rule. Reis and Watson’s (2010) empirical analysis using a factor model leads them to conclude that “most of the variation in standard aggregate in‡ation indices is associated with relative-price movements.” Our estimated DSGE model leads us to the same conclusion. Like Smets, Tielens, and Van Hove (2019) we use the Kalman …lter to decompose the behavior of in‡ation into aggregate and sectoral shocks. They study network interactions absent from our model, and estimate over a longer sample in which in‡ation was not stable. They do not address the in‡ation-shortfall or COVID episodes. Our paper also relates to the literature that discusses how the Phillips curve can appear to ‡atten when policy is conducted optimally so as to stabilize in‡ation. McLeay and Tenreyro (2020) is a leading example. While we do not study optimal policy, in practice the Federal Reserve did stabilize in‡ation from 1995-2019, and that is embodied in our estimated model. We emphasize that the remaining small ‡uctuations in in‡ation were associated with variation in relative prices across categories. Borio, Disyatat, Xia, and Zakrajšek (2021) also emphasize that in a stable monetary policy regime, sector-speci…c factors can play a major role in driving in‡ation. The paper proceeds as follows. Section 2 describes the model and its balanced growth path. Section 3 describes the empirical analysis and presents the parameter estimates. Section 4 presents 3 the basic analysis of in‡ation and relative price shocks. Sections 5 and 6 cover the in‡ation shortfall and COVID in‡ation, respectively. Section 7 concludes. 2. The Model The economy consists of an in…nitely-lived representative household, continua of …rms in a …nite number of sectors, and a monetary authority. Firms are monopolistic-competitors that each produce a di¤erentiated good using labor as the sole input of production. 2.1 Households: Preferences, Constraints, and Optimality The household maximizes E 1 X t ln (Ct ) + 1 t= where E denotes the conditional expectation as of time , is consumption, Nt is hours worked, and and ! Nt )1 (1 ; (1) 2 (0; 1) is the discount factor, Ct are positive preference parameters. The time endowment is normalized to be one. Consumption is an aggregate of goods produced in di¤erent sectors s = 1; 2; : : : ; S, Ct = S Y s ( s) s (cs;t ) ; (2) s=1 where s 2 (0; 1) are aggregation weights that satisfy S X s = 1 and cs;t is consumption of goods s=1 produced in sector s. Under the Cobb-Douglas speci…cation in (2), the elasticity of substitution between goods produced in di¤erent sectors is one. Within each sector, there is a continuum of monopolistically-competitive …rms that each produce a di¤erentiated good. The household’s preferences for these goods are represented by the CES aggregator cs;t = Z ( 1)= =( ci;s;t 1) ; (3) where ci;s;t is consumption of the good produced by …rm i in sector s and > 1 is the elasticity of substitution between goods produced in the same sector. Since this elasticity is larger than one, goods produced in the same sector (e.g., apples and pears) are closer substitutes than goods produced in di¤erent sectors (e.g., apples and nails). In every period, the household faces the budget constraint Pt Ct + Bt Pt wt Nt + (1 + Rt 4 1 ) Bt 1 + Dt ; (4) where Pt is the aggregate price level, Bt is nominal bonds, wt is the real wage, Rt is the net nominal interest rate, and Dt are pro…ts from …rms, which are transferred to the household in the form of dividends. The price index satis…es Pt = S Y (Ps;t ) s . (5) s=1 The solution to the household’s maximization problem shows that the optimal labor supply satis…es (1 Nt ) 1=Ct = wt ; (6) meaning that the marginal rate of substitution between leisure and consumption equals the real wage; the optimal consumption of good i produced in sector s is ci;s;t = Pi;s;t Ps;t s where Ps;t = Z Ps;t Pt 1=(1 1 Ct ; (7) ) 1 Pi;s;t dl (8) is a sector-speci…c price index; and the optimal (total) consumption satis…es the intertemporal Euler equation 1 = Pt Ct 2.2 (1 + Rt ) Et 1 Pt+1 Ct+1 : (9) Firms: Technology, Market Structure, and Optimality Firm i in sector s produces output using the technology yi;s;t = ezt ezs;t ni;s;t ; (10) where yi;s;t is output, ezt and ezs;t are, respectively, the aggregate and sectoral levels of productivity, and ni;s;t is labor input. Labor is completely mobile across …rms and sectors. Sectoral productivity follows the trend-stationary process, zs;t = as;t = where t is a time index, s s st + as;t ; s as;t 1 (11) + "s;t ; (12) is a constant, as;t represents stochastic deviations from the time trend, 2 ( 1; 1) is a parameter, and "s;t is an independent and identically distributed (i.i.d.) innovation drawn from a normal distribution with mean zero and constant conditional variance 2. s The deviation from the time trend has di¤erent persistence and variance across sectors. The trend itself 5 is also di¤erent across sectors, which is important for our model to account for the trends in relative prices observed in the data. Aggregate productivity follows the process, zt = zt where 1 + "t ; (13) 2 ( 1; 1) is a parameter and "t is an i.i.d. innovation drawn from a normal distribution with mean zero and constant conditional variance 2 .1 Each …rm must incur a cost when it changes its nominal price. The cost is speci…ed in units of the aggregate consumption good, is convex in the size of the adjustment, and is proportional to the quantity produced by the …rm. The per-unit cost, i;s;t where s 0, trend, and = 0 and and s (Pi;s;t ; Pi;s;t 1) = i;s;t , 1 s;t s 2 for …rm i in sector s in period t is e + s s Pi;s;t Pi;s;t 1 0 are parameters that satisfy 0 s + s 2 1 ; (14) 1, s;t is a deterministic are, respectively, the steady-state in‡ation rate and the steady-state sectoral rate of price change. The parameter s controls the degree of price rigidity and takes value zero in the special case where prices are completely ‡exible. Since s is indexed by s (but not by i), price rigidity di¤ers across sectors, but is constant across …rms within a sector. The deterministic trend s;t is speci…ed as s;t S Y = (1=e s )t (e k ) k=1 k !t We require that price-adjustment costs grow at the trend rate for sector s grows at rate e w . s;t because the labor part of costs =e s ; if the price-adjustment part of costs were to grow at a di¤erent rate, then one or the other components of costs would become negligible in the long run. This S Y means that there would be no balanced growth path. We will see below that e w = (e s ) s , so s=1 that the trend can be written as s;t = e( w s )t . Finally, the factors in the denominator of (14) determine the degree and the form of indexation as follows. If = adjustment. If 0 < s = 0, then there is no indexation and …rms incur a cost for any nominal price + s < 1, then there is partial indexation. If = 1 and s = 0, then there is complete indexation to the aggregate in‡ation rate. Thus, price increases equal to the steady-state rate of aggregate price change are costless. If = 0 and s = 1, then there is complete indexation to the sectoral in‡ation rate and price increases at the steady-state rate of sectoral price change 1 Results would be unchanged if we were also to assume a trend in aggregate productivity, for instance, equal to the growth rate of aggregate output and with sectoral trends adjusted accordingly because what matters in this model are relative productivity trends rather than absolute ones. 6 are costless. And, if + s = 1 with < 1 and s < 1, then there is complete indexation to an average of the aggregate and sectoral price changes. The …rm chooses output, labor input, and the price of its good to maximize pro…ts, where costs comprise labor costs and adjustment costs. The maximization is subject to the demand function (7), the demand associated with other …rms’ adjustment costs,2 and the technology (10). The solution to this problem delivers the following optimality condition for …rm i in sector s: 1 Pi;s;t Pi;s;t wt = zt t as;t + s;t s 1 + s sP Pt e e se 2 e i;s;t 1 Pi;s;t Pi;s;t s 1 s;t + + s s s sP Pi;s;t 1 e e i;s;t 1 1 Pt+1 s + s;t+1 Et (1 + Rt ) 1 e + s s Pt Pi;s;t+1 e + s s Pi;s;t 1 Pi;s;t+1 Pi;s;t 1 Ps;t Ps;t+1 2 (15) ys;t+1 : ys;t This equation relates the optimal price selected by …rm i to its marginal cost, including that associated with current and future expected price adjustments. 2.3 Monetary Policy The monetary authority sets the nominal interest rate following the rule, 1 + Rt = (1 + Rt where 2 ( 1; 1), 1) + (1 , and )(1= ) exp( + y + ( t )+ y (ln(Yt ) ln(Y )) + ut ); (16) are parameters representing responsiveness to the lagged interest rate, in‡ation, and output, respectively, rate of output (see below), y t is the gross in‡ation rate (= Pt =Pt 1 ), y is the growth is a policy parameter denoting the in‡ation target, Yt is aggregate output, Y is output in the balanced-growth steady state, and ut is a disturbance that represents movements in the interest rate beyond the control of the central bank.3 This disturbance follows the process ut = ut where 1 + & t; (17) 2 ( 1; 1) is a parameter and & t is an i.i.d. innovation drawn from a normal distribution with mean zero and constant conditional variance 2. & 2 Note that since price adjustment costs require each …rm to purchase the …nal good, this implies that each …rm demands output of all other …rms in the economy. 3 We also estimated the model under a policy rule where the monetary authority responds to average quarterly in‡ation. Parameter estimates are very similar to those reported here and support the same conclusions. These results are available from the authors upon request. 7 2.4 Equilibrium and Balanced Growth Path The equilibrium of the model is symmetric within sectors but asymmetric across sectors. That is, in equilibrium, all …rms within a sector are identical and make exactly the same choices (labor input, price, and output), but …rms in di¤erent sectors make di¤erent choices. In particular, …rms in di¤erent sectors choose di¤erent prices and, hence, rates of price change will di¤er across sectors. In a symmetric equilibrium we can simplify the optimality conditions by dropping the i subscripts, and we impose market clearing in the di¤erent markets. Since we will approximate the model solution around a steady state with long-term growth, it is also helpful to derive some relationships between the steady state growth rates of di¤erent variables. 2.4.1 Sectoral Phillips Curve The fact that …rms in the same sector are identical in equilibrium means that the price charged by each …rm is equal to the sectoral price index, Pi;s;t = Ps;t ; i 2 (0; 1); s = 1; 2; : : : ; S. The …rm’s optimality condition can then be written as the sectoral Phillips curve, 1 + where s;t s;t+1 Ps;t wt = zt t as;t + Pt e e se 1 s Et + s s e = Ps;t =Ps;t 1. s;t s;t s e + 1 1 + Rt 1 s s t+1 s;t+1 1 1 2 e e s;t+1 + s s s;t + s 1 s ys;t+1 ys;t 1 2 ; As in previous literature that derives sectoral Phillips curves in multi- sector economies (e.g., Imbs, Jondeau and Pelgrin, 2011, and Rubbo, 2022), the slope of the linearized Phillips curve is heterogenous across sectors, in our case as a result of heterogeneity in price rigidity. 2.4.2 Goods Market Clearing In each sector, there is a goods market clearing condition, ys;t = cs;t + fs;t , where fs;t denotes the output from sector s used for adjustment costs. Note that this is di¤erent from s;t , the adjustment costs incurred by sector s, and is given by fs;t = s Ps;t Pt 8 1 Ft . Aggregate adjustment costs are Ft = S X s=1 2.4.3 s;t + s s s;t 2 e 2 1 j ys;t : Steady-state Growth Rates The model has an exogenous growth rate for the overall price index in the balanced growth path. That growth rate, , is determined by the in‡ation target in the policy rule. The model also has S exogenous trend growth rates for sectoral productivity, s. Each sector’s output growth rate is equal to its productivity growth rate. Productivity growth rates determine the growth rates of consumption and the real wage. Then, the in‡ation target, along with the productivity growth rates, determine the growth rates of sectoral prices as follows: c = w = S X s s, s=1 s = + w s = + S X k ( k s) . k=1 The trend growth (or decline) in sectoral relative prices is determined entirely by a real factor, namely sectoral relative productivity growth, s = s+ S X k k. k=1 This point bears stressing: while sectoral relative prices can move around temporarily because of monetary policy shocks or the interaction between real shocks and the policy rule, the trend in relative prices is invariant to monetary policy and to parameters representing price stickiness. Finally, note that because we incorporate the exogenous trend output growth ( y) s;t in the adjustment cost speci…cation, is equal to consumption growth and adjustment cost growth on the balanced growth path, both in the aggregate and by sector. The model is solved using a …rst-order perturbation with the rational-expectation solution of the linearized system found using the approach in Klein (2000). 3. Empirical Analysis In this section, we describe the data used for estimation and the estimation method, and we discuss the parameter estimates. 9 3.1 Data The data used to estimate the model are monthly observations of the nominal interest rate and rates of price change for …fteen consumption expenditure categories of the U.S. economy from 1995M1 to 2020M1. The sample starts around the time that the Federal Reserve Board settled on implicitly targeting an in‡ation rate of 2% per year (see the transcripts of the meeting of the Federal Open Market Committee on January 31). The in‡ation target was o¢ cially adopted in January 2012 and applies to the personal consumption expenditures (PCE) price index. The nominal interest rate is the e¤ective federal funds rate and was taken from the FRED website of the Federal Reserve Bank of St. Louis (fred.stlouisfed.org). The original interest rate series quoted as a net annual rate is transformed into a monthly rate and quadratically detrended to account for its secular decline over the sample period. Since in‡ation has been stable over the sample, this secular decline is attributable to the persistent decrease in the natural real rate. This decrease has been widely documented and studied in previous literature (see, for example, Laubach and Williams, 2003, King and Low, 2014, and Rachel and Summers, 2019). Because our model does not contain elements that can capture the decrease in the natural real rate, we follow an econometric strategy that accounts for this decrease by means of a nonlinear time trend, and we focus instead on the cyclical component of the nominal interest rate. The …fteen categories comprise the entirety of PCE. The sectors are motor vehicles and parts, furnishings and household durables, recreational goods, other durable goods, food consumed at home, clothing and footwear, gasoline and other energy goods, other nondurable goods, housing and utilities, health care, transportation services, recreation services, food services and accommodations, …nancial services and insurance, and other services. The raw data used to construct the sectoral price changes are seasonally adjusted price indices available from the website of the Bureau of Economic Analysis (www.bea.gov). To be consistent with the model solution, which describes the dynamics of the variables in deviation from their steady state values, all data series are demeaned prior to the structural estimation of the model. 3.2 Estimation The model is estimated by the method of maximum likelihood (ML) using the Kalman …lter to evaluate the likelihood function. The state equation of the state-space representation of the model solution is the joint process of exogenous and predetermined variables, Xt+1 = HXt + vt+1 , where Xt = (zt ; a1;t ; : : : ; aS;t ; ut ; Rt 1 ; P1;t 1 ; : : : ; PS;t 1 ) 10 0 and vt = ("t ; "1;t ; : : : ; "S;t ; & t ; 0; 0; : : : ; 0)0 are (2S + 3) 1 vectors, and H is a (2S + 3) (2S + 3) matrix whose elements are the parameters of the exogenous shock processes (in the …rst S + 2 elements of Xt ) and the coe¢ cients of the decision rules of the predetermined variables in the last S + 1 elements of Xt . The observation equation is Qt = GXt , where Qt = (Rt ; 1;t ; : : : ; S;t ) 0 is (S + 1) 1 vector and G is a (S + 1) (S + 1) matrix whose elements are the coe¢ cients of the decision rules of the interest rate and sectoral price changes in Qt . The coe¢ cients of the decision rules are nonlinear functions of the structural parameters. As it is well known, this estimation approach is equivalent to using a Bayesian estimation strategy with di¤use priors. Hansen and Sargent (2013, ch. 8) shows that the ML estimator obtained by applying the Kalman …lter to the state-space representation of dynamic linear models is consistent and asymptotically e¢ cient. Standard errors are estimated by the square root of the diagonal elements of (T I) 1 where T is the sample size and I is the information matrix, which is computed using the outer product of the scores at the maximum. A di¢ culty that we face in estimating this model is that the steady state has to be computed numerically in every iteration of the algorithm that maximizes the likelihood function. Given the relatively large size of our model, this computation is time-consuming. In order to address this challenge, we …rst …x the parameters that determine the steady state and, with these parameters set, we estimate the parameters that determine the dynamics of the model by ML. The parameters that determine the steady state are …xed as follows. The weight of leisure in the utility function ( ) is set to 1:8, such that households work 1=3 of the time in steady state. The parameter that determines the elasticity of substitution between goods from the same sector ( ) is …xed to 10. This value is standard in the literature and implies a mark-up of approximately 10%. We assume complete indexation, meaning that + s = 1. Provided that the latter condition is satis…ed, the steady state is independent of the relative magnitude of and s, and we assume = s = 1=2. The discount rate ( ) is …xed to 0:998. The consumption weights are computed using the consumption expenditure shares in each sector. Recall that the optimal consumption of good i produced in sector s is ci;s;t = s Pi;s;t Ps;t Ps;t Pt 1 Ct : Using the fact that the equilibrium is symmetric within sectors, which implies Pi;s;t = Ps;t , and solving for s delivers s = Ps;t cs;t , P t Ct 11 where Ps;t cs;t are expenditures in sector s and Pt Ct are total consumption expenditures. Estimates of s for each sector are reported in Column 1 of Table 1. Finally, the model suggests a natural strategy to estimate the trends in sectoral productivity based on the result that sectoral price changes in steady state are s for s = 1; 2 : : : ; S. Solving for s, = so that + s = s; c + c (18) s, and using the fact that the model implies that the steady state values of aggregate in‡ation, aggregate consumption growth, and sectoral price changes are equal to their respective unconditional means, delivers estimates of s for each sector. These estimates are reported in the Column 2 of Table 1. 3.3 Parameter Estimates ML estimates of the price rigidity parameter in each sector and the parameters of all shock processes are reported in Table 2. There is substantial heterogeneity in price rigidity across sectors and the null hypothesis that rigidity is the same in all sectors is strongly rejected by the data (see the p-value of the Wald test reported in the last row of Table 2). Heterogeneity in price rigidity across product categories has been documented by previous literature using highly disaggregated components of the consumer price index (CPI) (see, among others, Bils and Klenow, 2004, Klenow and Kryvtsov, 2008, and Nakamura and Steinsson, 2008) and estimated multi-sector dynamic equilibrium models (see, e.g., Bouakez, Cardia, and Ruge-Murcia, 2014). In line with that literature, we …nd that service prices are generally the most rigid in the U.S. economy. The only exceptions in our sample are transportation services and …nancial services and insurance, for which the null hypothesis that prices are ‡exible (i.e., = 0) cannot be rejected at the 5% level. Quantitatively, the most rigid prices are those of food services and accommodations, and housing and utilities. The prices of some categories of durables and nondurable goods are also rigid, meaning that the hypothesis = 0 can be rejected at standard levels (e.g., food at home and motor vehicles). There are, however, other categories like furnishing and clothing for which the hypothesis cannot be rejected. In terms of the expenditure shares, 70% of consumption by U.S. household may be considered to have rigid prices. Sectoral productivity shocks are persistent with autoregressive coe¢ cients ranging from 0:455 to 0:998. The hypothesis that the coe¢ cients are statistically the same in all sectors can be rejected at standard levels and the p-value of the Wald test is less than 0:001. The former estimate (0:455) corresponds to food services and accommodations and all other estimates are concentrated in the interval between 0:962 and 0:998. We explore the possibility that the above rejection is driven by food services and accommodations by performing the Wald test excluding this sector. Since 12 the p-value of this test is also less than 0:001, we conclude that the …nding that the persistence of productivity shocks is heterogenous across sectors is not only due to the moderate estimate for food services and accommodations. There is large heterogeneity in the standard deviation of productivity innovations, and the hypothesis that they are the same in all sectors is strongly rejected by the data. The largest standard deviation is that of gasoline and energy goods, followed by food services and accommodations, which are one order of magnitude larger than that of other sectors. The …nding that the standard deviation of productivity shocks to gasoline and energy goods is large helps explain why even though their price rigidity parameter is of moderate magnitude and statistically signi…cant, the observed frequency of price adjustments for these goods is high. For instance, Bils and Klenow (2004, Table A1 in p. 983) report that the estimated average monthly frequency of price changes for gasoline over 1995–1997 ranges between 0:762 and 0:789 depending on its grade. The estimate of the standard deviation of the aggregate productivity innovation is small and not statistically signi…cant. This has two implications. First, the estimate of the autoregressive coe¢ cient of the productivity shock is poorly identi…ed (see the large standard error in Table 2). Second, as we will see below, aggregate productivity plays a limited role in explaining the aggregate and sectoral in‡ation dynamics. For this reason, the conclusions in this paper are robust to considering a restricted version of the model without an aggregate productivity shock. Table 3 reports the ML estimates of the monetary policy rule under the heading “Benchmark.” The smoothing parameter is large (0:733) and in line with values reported elsewhere in the literature. The coe¢ cients of in‡ation and output are positive, as expected, but imprecisely estimated. It is interesting to compare these estimates, based on the full model, with reduced-form estimates obtained under minimal assumptions and also reported in Table 3 under the heading “Unrestricted.” The latter are estimates of the …rst-order linearization of (16) computed by ML but without imposing the restrictions of the model. Estimates are quantitatively similar suggesting that the restrictions of the model do not limit its ability to correctly represent the dynamics of the monetary policy rule. (The last two columns of Table 3 will be discussed later). 3.4 Predicted Moments Table 4 reports the standard deviations and autocorrelations of the nominal interest rate, aggregate in‡ation, and sectoral price changes predicted by the model and compares them with U.S. data. In contrast to method of moments estimators (e.g., GMM or SMM), our ML estimation procedure does not explicitly target these moments. The table shows that the model quantitatively captures the high persistence of the interest rate, the low autocorrelation of month-to-month in‡ation and 13 sectoral price changes, and the volatility of all series, notably the large standard deviation of price changes in gasoline and other energy goods. The correlation between the two sets of moments is high: 0:994. We, therefore, conclude that our model does a good job reproducing the key moments of the data. 4. Relative Price Shocks and In‡ation over the Entire Sample With parameter estimates in hand, we can now turn to the model’s implications regarding the role of relative price shocks in driving in‡ation dynamics. We begin with impulse response analysis and variance decompositions. Then we focus on a summary statistic for the distribution of relative price changes that has not— to our knowledge— previously been examined in the literature. We conclude this section with a discussion of the (non)importance of sticky prices for some of our qualitative …ndings. 4.1 Impulse Responses Figure 1 reports the responses of relative prices (Ps;t =Pt ) to a negative productivity shock in each sector. For example, panel A reports the response of the relative price of motor vehicles (thick line) and other relative prices (thin lines) to a negative productivity shock in the motor vehicles sector. The size of the shock is one standard deviation of the productivity innovation, s (see (11)). In all panels, the productivity shock leads to a large increase in the relative price of the good produced in that sector. This increase is one or two orders of magnitude larger than the muted decrease in the relative price of goods produced by the other 14 sectors.4 These 14 impulse responses are quantitatively similar, and for this reason they sometimes appear as a single black line in Figure 1. The result that a sectoral productivity shock yields a large response in the relative price of its own good and a mild response in other relative prices motivates our interpretation of sectoral productivity shocks as relative price shocks. Figure 2 reports the response of in‡ation to aggregate and sectoral shocks. The horizontal axis is months, and the vertical axis is the in‡ation deviation from in steady state value, expressed in percentage points at an annual rate. Panel A reports the in‡ation response to a monetary policy shock that reduces the nominal interest by & (see eq. (17)). The other panels report the in‡ation response to a negative relative price shock with size equal to one standard deviation of the productivity innovation, 5 s. Note that to allow comparisons across panels, the scale of the vertical 4 Recall that the aggregate price index, Pt , is a geometric average of sectoral prices (see eq. (5)) and hence the sum of the weighted responses of relative prices must add up to zero. 5 We abstract from reporting the response to the aggregate productivity shock because its small standard deviation (see Table 2) implies that the in‡ation response to a productivity shock of a plausible magnitude is basically zero. 14 axis is the same in all panels except for panels A, H, and O, which respectively report the responses to the monetary policy shock and the shocks to gasoline and …nance and insurance. This …gure shows that di¤erent shock sizes across sectors and, to some extent, di¤erent price stickiness across sectors, imply heterogeneity in the e¤ects of relative price shocks on in‡ation. Quantitatively, the largest e¤ects are due to relative price shocks to gasoline (panel H), …nance and insurance (panel O), housing and utilities (panel J), other services (panel P), and food at home (panel F). 4.2 Accounting for the Variance of In‡ation This section reports the variance decomposition for the in‡ation rate at di¤erent horizons. (Recall that this is the proportion of the mean squared error of the forecast of in‡ation at di¤erent horizons that is accounted for by each of the shocks.) Panel A in Figure 3 shows that the aggregate productivity shock accounts for a negligible proportion— less than 0:001%— of the variance of the in‡ation forecast error at all horizons. Instead, monetary policy and relative price shocks account for basically all the variance of the in‡ation forecast error. Panels B and C show that they respectively account for 23:7% and 76:3% of the variance of the in‡ation forecast error one-month ahead and 24:7% and 75:3% of the unconditional variance of in‡ation. The …nding that relative price shocks account for a large proportion of the in‡ation forecasts error at all horizons is consistent with results in Reis and Watson (2010, p. 146) who report that 76% of the movements in aggregate in‡ation are accounted for by a relative-price index.6 A similar result is reported by Smets, Tielens, and Van Hove (2019) who …nd that sectoral shocks, by ways of pipeline pressures, are an important contributor to the variance and persistence of headline in‡ation. The remaining panels in Figure 3 (D through R) report the contribution of relative price shocks in each sector to the variance decomposition of in‡ation. Panel J shows that relative price shocks to gasoline account for around 43% of the variance of the in‡ation forecast error one-month ahead and for 42% of the unconditional variance of in‡ation. Panels H, L, M, P, and Q show that relative price shocks to food at home, housing and utilities, health, food services, and …nancial services substantially contribute to the variance of the in‡ation forecast error one-month ahead (3%, 4:7%, 4:4%, 2:8%, and 8:2%, respectively) and to its unconditional variance (2:9%, 4:6%, 4:3%, 3:1%, and 7:8%, respectively). Research based on dynamic factor models (e.g., see Boivin, Giannoni, and Mihov, 2009) typically …nds that aggregate factors are the main driver of aggregate in‡ation.7 6 Note, however, that the structural interpretation of their index, which is based on a model of price setting under imperfect information, is di¤erent from our relative-price shocks because their index includes, for instance, the unanticipated component of the rate of money growth, which is an aggregate variable. 7 Boivin, Giannoni and Mihov (2009) also emphasize di¤erential responses of sectoral prices to aggregate and sectoral shocks. Carvalho, Lee and Park (2021) show how adding input-output linkages and labor market segmentation can help a multi-sector New Keynesian model match these patterns. 15 Using a structural model with input-output interactions, Onatski and Ruge-Murcia (2013) show that macroeconomics shocks can indeed be considered factors in that they nontrivially a¤ect most variables in the model. However, principal components analysis has a hard time replicating the macroeconomic factor space because sectoral shocks can act as aggregate shocks. Table 5 reports the contribution of the “own” relative price shock (Column 1) and of the monetary policy shock (Column 2) to the unconditional variance of each sectoral price change under the benchmark model. (Columns 3 and 4 will be discussed below.) The table shows that the “own” relative price shock accounts for most of the variance of all sectoral price changes. The contribution of the monetary policy shock is also substantial. In contrast, the contribution of the aggregate productivity shock and relative price shock in other sectors is negligible.8 The …nding that the sector-speci…c shocks account for most of the variance of sectoral price changes is consistent with results reported in Boivin et al. (2009) and Mackowiak, Moench, and Wiederholt (2009) based on estimated dynamic factor models. Boivin et al. (p. 356) report that 85% percent of the monthly disaggregated in‡ation ‡uctuations are attributable to sector-speci…c shocks, while Mackowiak et al. (p. 82) report that the proportion of the variance in sectoral price changes due to sector-speci…c shocks in their median sector is 90% with a 90% con…dence interval ranging from 79% to 95%. In our sample, sector-speci…c shocks account for 80% of the variance of sectoral price changes in the median sector, and 87% of our estimates (that is, 13 out of 15 sectors) fall in the interval from 62:4% to 94:8%. It is interesting to note that there is basically no relationship between price rigidity and the proportion of the variance that is accounted for by the monetary policy shock. The correlation between the price rigidity parameters in Table 2 and the proportions in Column 2 is 0:287 and not statistically signi…cant. In contrast, the correlation between the standard deviation of sectoral productivity shocks in Table 2 and the proportions in Column 1 is 0:584 and statistically signi…cant at the 5% level. 4.3 In‡ation and the Distribution of Relative Price Changes Panel A in Figure 4 displays the monthly PCE in‡ation rate rate on the vertical axis against the share of relative price increases— or equivalently, the share of price increases greater than the in‡ation rate— from 1995 through 2019. A striking feature of this plot is the close empirical relationship between the two variables.9 Panel B in the same …gure plots the same relationship but 8 These …gures are not reported to save space, but their sum can be computed by substracting the contributions in Table 5 from 100% for each sector. 9 This empirical relationship is discussed in Wolman (2022) in relation to the high in‡ation observed since in March 2021, and it is examined in more detail by Hornstein, Ruge-Murcia, and Wolman (2022). 16 based on arti…cial data simulated from our estimated model. Comparing both panels shows that the model can account for this feature of the U.S. data. This result is due to the fact that our estimated speci…cation matches well the behavior of category price changes and, importantly, that it does so by assigning a large role to sectoral, versus aggregate, shocks. We now provide some intuition for Figure 4 from the perspective of the model. To begin, note that there is a simple relationship between (i) the shares of relative price increases and decreases and (ii) the average sizes of relative price increases and decreases. Because the average relative price change is zero, the ratio of the share of relative price increases to the share of relative price decreases is identical to the ratio of the average relative price decrease to the average relative price increase, ! r+ (1 !) r = 0; which implies !=(1 !) = ( r )=( r+ ); where ! is the share of relative price increases, and r+ and r are the average relative price increase and decrease, respectively. Note that the way we have written these equations, the average relative price decrease is a positive number. Consider now a point on the far right of panel A; because the share of relative price increases is close to one, the ratio of the share of relative price increases to the share of relative price decreases is very high, and therefore the average relative price increase is very small compared to the average relative price decrease. In principle, this can happen in many ways— we’ll consider two. One possibility is that in‡ation is much higher than target. This does not occur in the data or in the model, but it’s instructive to work through what it would involve. A large share of relative price increases and in‡ation far above target would mean that the relative price increases correspond to nominal price increases that are much higher than target. This could happen from a combination of a large in‡ationary aggregate shock and a small share of sectors experiencing a large increase in productivity, so that they choose to reduce their relative price by a large amount. But according to our estimates, large aggregate shocks are unlikely. Or, it could happen without an aggregate shock, if most sectors experienced negative sectoral productivity shocks. But this is also unlikely, because the sectoral shocks are uncorrelated. A second possibility when there is a large share of relative price increases is that in‡ation is much lower than target, which is the typical outcome in the …gure. A large share of relative price increases and in‡ation far below target means that the relative price decreases— few in number, correspond to nominal price decreases that are much lower than target. This could happen from a combination of a large de‡ationary aggregate shock and a small share of sectors experiencing a 17 large (possibly additional) increase in productivity, so that they choose to reduce their relative price by a large amount. But again, large aggregate shocks are unlikely. Or, it could happen without an aggregate shock, if a small share of sectors experienced large positive productivity shocks, and in response chose large nominal price decreases. While this scenario is unconditionally unlikely, conditional on a high share of relative price increases, and conditional on the model estimates, it is in fact likely. The discussion thus far has not touched on the systematic behavior of monetary policy, but that behavior is of course central to the relationship in Figure 4. The key point here is that monetary policy responds— in the model and generally in the data— to economywide aggregates, as opposed to sectoral price changes. When a relative price shock hits a particular sector, the desired relative price change is accomplished mainly by a nominal price change of the same sign for that sector, so that in‡ation moves in the same direction (Figure 2). It’s theoretically possible for monetary policy to perfectly stabilize in‡ation (a ‡at line in Figure 4), or even to generate an upward sloping relationship between the share of relative price increases and the in‡ation rate. But either of these cases would require that in the face of a large relative price shock for just one sector, monetary policy generates large nominal price changes for all other sectors in the opposite direction. This seems implausible, and is not what we see in the data.10 We summarize as follows: most of the time, the share of relative price increases is close to one half, and in‡ation is close to target. When the share of relative price increases is large, it is associated with a small share of sectors experiencing large positive productivity shocks, and choosing large nominal price decreases, which results in low in‡ation. When the share of relative price increases is small, it is associated with a small share of sectors experiencing large negative productivity shocks, and choosing large nominal price increases, which results in high in‡ation. The systematic behavior of monetary policy delivers these relationships as equilibrium outcomes: it is optimal for sectors experiencing changes in their desired relative price to move their nominal price in the same direction, and for other sectors to respond very little. 4.4 A Flexible-Price Version of the Model As discussed above and reported in Table 2, there is substantial heterogeneity in our estimates of price stickiness across sectors. The price-stickiness parameter s is statistically signi…cant for more than half of sectors that account for 70% of the expenditures by U.S. households. Most notably, the 10 Additionally, in New Keynesian models such as the one we employ, it is generally not optimal to stabilize the price level in response to large relative price shocks, unless those shocks hit sectors in which nominal rigidities are especially large. Goodfriend and King (1997) …rst made this general point; Aoki (2001) provided analytical results in a two-sector model; and Eusepi, Hobijn and Tambalotti (2011) conducted quantitative analysis in a model similar to ours. 18 joint hypothesis that the s are all equal is rejected by the data. It may be tempting to conclude then that the relationship between relative price changes and in‡ation displayed in Figure 4 is also closely related to price stickiness. We …nd this not to be the case. We estimate a ‡exible-price version of the model, meaning a version of the model in section 2 subject to the constraint that the price rigidity parameters are zero in all sectors. ML estimates of the parameters are reported in Table 6 and the last two columns of Table 3. Note in Table 6 that the estimates of the autoregressive coe¢ cients of sectoral productivity shocks are extremely high. The reason is simply that without price rigidity, the persistence of sectoral price changes can only be accounted for by a high autocorrelation of the sectoral shocks. Estimates of the standard deviation of productivity innovations and Taylor rule parameters obtained under this model are roughly consistent with those obtained under the benchmark model with sticky prices. Note that this model is strongly rejected by the data: the likelihood ratio statistic of the test of the restriction s = 0 for all s (that is, that all prices are ‡exible) is 2 (25612:7 25540:8) = 143:8 which is larger than the 5% critical value of 25. The point that we want to make here, however, is that price rigidity is not essential for our result that the dynamics of aggregate in‡ation and sectoral price changes are driven by relative price shocks. To see this, consider the following. First, regarding aggregate in‡ation, monetary policy and relative price shocks respectively account for 33:7% and 66:3% of the unconditional variance of in‡ation under the ‡exible-price model, compared to the values of 23:7% and 76:3% reported above for the benchmark model. Second, the last two columns in Table 5 report the contribution of the “own”relative price shock (Column 3) and of the monetary policy shock (Column 4) to the unconditional variance of each sectoral price change under the ‡exible-price model. These estimates are quantitatively similar to those in columns 1 and 2 obtained under the benchmark model and lead to the same conclusions. Finally, panel C in Figure 4 shows that strong negative relationship between the monthly in‡ation rate on the vertical axis and the share of relative price increases predicted by our model is robust to assuming that all prices are completely ‡exible. These results relate to work by Balke and Wynne (2000). Compared with Ball and Mankiw (1995), whose explanation for the relationship between relative price changes and in‡ation relies on price stickiness, Balke and Wynne consider a ‡exible-price model. There are three central elements to their analysis. First, they point out that in a ‡exible price model, properties of the distribution of sectoral productivity growth would be re‡ected in the distribution of relative price changes. Second, they provide evidence that the distribution of productivity changes in fact had much in common with the distribution of sectoral price changes. These two elements are also present in our model in a more general setup where the data are allowed to determine the extent of price rigidity in each 19 sector (see Figure 1). Finally, they show that in a calibrated multi-sector model with ‡exible prices, basic properties of the empirical relationship between the in‡ation and the distribution of relative price changes are replicated when the model is driven by an estimated sectoral productivity process. In contrast to their work, we focus on the heterogeneity in the variance of sectoral productivity innovations and the limited role of aggregate shocks in accounting for the empirical relationship between relative price changes and in‡ation. 5. In‡ation Shortfall 2012-2019 We estimated the model over post-1994 U.S. data. Over that period, in‡ation averaged around 2% and was quite stable by historical standards. Since 2012, the Federal Reserve has had a formal 2% target for PCE in‡ation, but from 2012 to 2019, PCE in‡ation averaged only 1:4%. We focus here on the post-2012 period and use the model to decompose the in‡ation shortfall into contributions from the various estimated shocks. Because mean in‡ation is 2%, if the model state variables were at their steady states and there were no shocks, then in‡ation would be constant at 2%. The model explains an episode of deviations from 2% in‡ation by a combination of initial state variables being away from steady state and subsequent shocks. Figure 5 plots annual in‡ation during the undershoot period, along with the contributions to the cumulative price level undershoot from the gasoline shock, the health care shock and the monetary policy shock. The cumulative undershoot is measured starting from January 2012. The contributions are based on the smoothed inferences of each shock from the Kalman …lter with our estimated parameters. The early part of the undershoot was driven mainly by health care shocks, and then somewhat later by gasoline shocks, whereas monetary policy became more important starting in late 2016. Note however that the period where monetary policy began to account for more of the undershoot coincided with the Fed’s increase in interest rates. With the caveat that our estimation procedure does not impose the zero bound, the narrative that seems most consistent with Figure 5 is that health care and gasoline shocks drove in‡ation below target initially, with policy at the lower bound, then in‡ation stayed below target in part because of the Fed’s raising the Funds rate starting late in 2015. While that rate increase was gradual by historical standards, according to our estimated model, it represented contractionary policy shocks. 6. COVID In‡ation We now use our estimated model to interpret the behavior of U.S. in‡ation in the period immediately after our estimation sample; that period corresponds to the COVID-19 pandemic, when in‡ation 20 was at …rst volatile and then consistently far above the Fed’s 2% target.11 Because it interprets the data through the lens of the estimated model, this analysis assumes that the U.S. economy has remained in a rational expectations equilibrium involving local ‡uctuations around a steady state with 2% in‡ation. One might respond sceptically that we are assuming the answer to the most important question: has the U.S. economy remained in that equilbrium, or has the Fed’s behavior deviated from its previous rule to such an extent that private agents no longer perceive that rule to be in place? We do not dispute the importance of that question, and in fact see our work as contributing to an answer: under the assumption that in‡ation has remained anchored, we provide estimates of the contributions to observed in‡ation from monetary policy and sectoral shocks. An evaluation of those estimates based on independent information can then help in assessing whether in‡ation has in fact remained anchored. As above, we decompose observed in‡ation into the contributions of the various shocks. The data is now outside our estimation sample, but the procedure is otherwise identical. Note that this procedure also yields a counterfactual policy analysis: in the absence of monetary policy shocks, how would in‡ation have behaved? Figure 6 presents selected elements of the COVID in‡ation decomposition. In this case, we plot the price level instead of in‡ation, along with the 2% trend line and the price level paths implied by the shocks to gasoline, motor vehicles, food at home, and monetary policy. The …gure uses data only though November 2021, so it misses some of the high in‡ation months. (The …gure will be updated in the next revision of the paper). According to our estimated model, expansionary monetary policy and (to a lesser extent) food at home were signi…cant contributors to in‡ation in this period. However, their positive contributions came early, when the Fed lowered its interest rate target, and they were compensated by a large drop in the price of gasoline and energy goods. A few shocks (especially motor vehicles) were the main culprits in the subsequent increase in in‡ation (along with dissipation of other shocks). It will be interesting to see how this decomposition looks with more recent data, as Wolman (2022) shows that at the end of 2021 and the beginning of 2022, the share of relative price increases has been consistent with relatively low in‡ation, according to the historical relationship in Figure 4. The fact that in‡ation has in fact been quite high suggests that expansionary policy is playing a large role. 11 Recent work that studies the macroeconomic implications of COVID-19 include Baqaee and Farhi (2022), Guerrieri, Lorenzoni, Straub, and Werning (2022), and Woodford (2022). However, contrary to our analysis, which focuses on in‡ation, their work is concerned with the transmission of an asymmetric (COVID) shock in economies with complementarity in production (Baqaee and Farhi, and Guerrieri et al.) or a network payments structure (Woodford). 21 7. Conclusions We set out to evaluate the contribution of sectoral shocks— relative price shocks — to the behavior of in‡ation, and we found their contribution to be very large indeed. For instance, we …nd that the early part of the in‡ation undershooting after 2012 was driven mainly by shocks to health care and to gasoline and other energy goods. We …nd that expansionary monetary policy has been a signi…cant contributor to the high in‡ation outcomes following the COVID pandemic, but sectoral shocks (especially motor vehicles) were the main culprits in the burst of in‡ation from May through November 2021. To be clear, we are not claiming that in‡ation is always and everywhere a relativeprice-shock phenomenon. Rather, the lesson is simply that if monetary policy has crediblility and behaves in a way that delivers stable in‡ation, then the in‡ation ‡uctuations that remain seem to be driven by sectoral shocks. Our model-based analysis assumes credibility and stability of the policy regime, and the data we use to estimate the model comes from a period when in‡ation was stable, so the assumption is plausible. In future work, we plan to address some of the caveats in the analysis reported in this paper. First, we focus primarily on nominal variables but any subsquent research that also goes after the real-nominal relationship and seeks to explain in‡ation should also be consistent with the fact that there is a systematic relationship between the distribution of relative price changes and in‡ation. Second, in order to simplify the analysis as much as possible, we have considered only sectoral supply shocks. A natural and perhaps necessary extension is to add sectoral demand shocks. We expect that the qualitative results would be unchanged, but there would be an additional factor driving relative price changes. 22 Table 1. Sectoral Consumption Weights and Productivity Trends Sector Motor vehicles and parts Furnishings and household durables Recreational goods Other durable goods Food at home Clothing and footwear Gasoline and other energy goods Other nondurable goods Housing and utilities Health care Transportation services Recreation services Food services and accommodations Financial services and insurance Other services 23 Consumption Weight (1) 0:0488 0:0296 0:0311 0:0166 0:0878 0:0414 0:0308 0:0801 0:1877 0:1509 0:0343 0:0371 0:0661 0:0738 0:0840 Productivity Trend 102 (2) 0:2605 0:3936 0:7428 0:3076 0:1710 0:3487 0:1343 0:1564 0:0891 0:0464 0:1586 0:0983 0:1003 0:0978 0:1041 Table 2. Estimates of Sectoral Parameters and Aggregate Productivity Process Autoregressive Coe¢ cient Price Rigidity Sector Motor vehicles and parts Furnishings and household durables Recreational goods Other durable goods Food at home Clothing and footwear Gasoline and other energy goods Other nondurable goods Housing and utilities Health care Transportation services Recreation services Food services and accommodations Financial services and insurance Other services Aggregate productivity Wald test (p-value) Estimate (1) 7:316 0:186 0:676 < 0:001 4:229 0:516 3:673 0:030 8:485 4:090 0:255 2:094 111:121 < 0:001 12:883 s.e. (2) 2:362 0:625 0:660 0:023 1:572 0:584 1:159 0:345 1:752 1:591 0:527 0:994 24:342 0:348 3:966 < 0:001 Estimate (3) 0:990 0:991 0:998 0:995 0:997 0:995 0:962 0:990 0:997 0:997 0:975 0:997 0:455 0:991 0:998 0:819 < 0:001 Note: s.e. stands for standard error. The superscripts s.e. (4) 0:012 0:011 0:002 0:009 0:002 0:006 0:015 0:012 0:002 0:002 0:021 0:004 0:062 0:007 0:002 216:745 Standard Deviation Estimate s.e. 102 102 (5) (6) 0:460 0:041 0:386 0:028 0:376 0:028 0:584 0:026 0:324 0:029 0:547 0:039 6:144 0:301 0:315 0:015 0:192 0:016 0:184 0:018 0:530 0:039 0:243 0:039 1:989 0:456 0:710 0:040 0:272 0:031 < 0:001 0:386 < 0:001 and y denote statistical signi…cance at the …ve and ten percent levels, respectively. The value of the log likelihood function at the optimum (excluding the constant term) is 25540:8. 24 Table 3. Estimates of Taylor Rule Parameter Smoothing parameter In‡ation coe¢ cient 102 Output coe¢ cient 102 AR coe¢ cient Standard deviation 102 Benchmark Estimate s.e. (1) (2) 0:733 0:146 4:312 2:725 0:310 0:227 0:673 0:176 0:049y 0:027 Note: see notes to Table 2. 25 Unrestricted Estimate s.e. (3) (4) 0:989 0:261 1:007 0:249 0:146y 0:079 0:456 0:038 0:011 < 0:001 Flexible-Price Estimate s.e. (5) (6) 0:721 0:095 5:122 2:481 0:037 0:111 0:621 0:130 0:052 0:018 Table 4. Empirical and Theoretical Second Moments Variable Nominal interest rate Aggregate in‡ation Sectoral price changes: Motor vehicles and parts Furnishings and household durables Recreational goods Other durable goods Food at home Clothing and footwear Gasoline and other energy goods Other nondurable goods Housing and utilities Health care Transportation services Recreation services Food services and accommodations Financial services and insurance Other services Standard Deviation Data Model (1) (2) 0:110 0:043 0:187 0:252 0:315 0:377 0:355 0:597 0:263 0:517 4:986 0:303 0:140 0:148 0:511 0:208 0:169 0:722 0:171 0:348 0:432 0:415 0:628 0:307 0:566 4:828 0:382 0:197 0:224 0:558 0:273 0:211 0:762 0:221 Autocorrelation Data Model (3) (4) 0:992 0:939 0:387 0:139 0:307 0:019 0:026 0:170 0:266 0:016 0:349 0:140 0:338 0:116 0:045 0:097 0:056 0:236 0:356 0:325 0:000 0:047 0:013 0:228 0:039 0:211 0:024 0:324 0:207 0:005 0:127 0:288 0:012 0:410 Note: The predicted moments are the sample average of the moments computed using 1000 simulations with number of observations equal to the sample size T = 301. 26 Table 5. Variance Decomposition Variable Sectoral price changes: Motor vehicles and parts Furnishings and household durables Recreational goods Other durable goods Food at home Clothing and footwear Gasoline and other energy goods Other nondurable goods Housing and utilities Health care Transportation services Recreation services Food services and accommodations Financial services and insurance Other services Benchmark Own Monetary Shock Policy (1) (2) Flexible-Price Own Monetary Shock Policy (3) (4) 86:481 78:458 79:421 89:216 77:909 88:414 99:904 71:308 62:428 58:413 87:256 62:771 94:805 92:735 77:021 81:676 86:024 83:966 93:401 73:178 91:765 99:900 79:135 52:783 53:192 91:800 64:388 55:789 95:489 57:717 27 9:988 15:438 14:980 7:711 16:674 8:372 0:072 20:537 28:985 31:883 9:139 27:274 3:060 5:240 17:813 15:474 11:865 13:429 5:537 22:582 6:908 0:084 17:451 41:515 40:677 6:961 29:905 36:976 3:782 35:958 Table 6. Estimates of Sectoral Parameters and Aggregate Productivity Process under Flexible-Price Model Motor vehicles and parts Furnishings and household durables Recreational goods Other durable goods Food at home Clothing and footwear Gasoline and other energy goods Other nondurable goods Housing and utilities Health care Transportation services Recreation services Food services and accommodations Financial services and insurance Other services Aggregate productivity Autoregressive Coe¢ cient s.e. Estimate 103 (1) (2) 0:999 0:010 0:999 0:006 0:999 0:021 0:999 0:012 0:999 0:017 0:999 0:025 0:982 0:135 0:983 0:152 0:999 0:008 0:999 0:008 0:999 0:010 0:999 0:010 0:967 0:180 0:997 0:064 0:999 0:008 0:903 979:108 Standard Deviation Estimate s.e. 102 102 (3) (4) 0:316 0:012 0:369 0:015 0:354 0:013 0:583 0:025 0:247 0:009 0:517 0:017 4:948 0:115 0:305 0:007 0:128 0:005 0:134 0:006 0:504 0:018 0:202 0:009 0:175 0:007 0:716 0:010 0:162 0:007 < 0:001 0:034 Wald test (p-value) < 0:001 < 0:001 Sector Note: see notes to Table 2. The value of the log likelihood function at the optimum (excluding the constant term) is 25612:7. 28 References [1] Aoki, Kosuke. 2001. “Optimal monetary policy responses to relative-price changes.” Journal of Monetary Economics 48. 55–80. [2] Balke, Nathan S. and Mark A. Wynne. 2000. “An Equilibrium Analysis of Relative Price changes and Aggregate In‡ation.” Journal of Monetary Economics 45. 269–292. [3] Ball, Laurence and N. 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[31] Woodford, Michael. 2020. “E¤ective Demand Failures and the Limits of Monetary Stabilization Policy.” American Economic Review 112(5). 1475–1521. 31 Figure 1: Responses of Relative Prices to Negative Productive Shock in Own Sector A. Motor Vehicles B. Furnishings C. Recreational Goods 0.4 0.4 0.3 0.3 D. Other Durables E. Food at Home F. Clothing and Footwear H. Other Nondurables G. Gasoline 0.6 0.3 0.6 6 0.3 0.4 0.2 0.4 4 0.2 0.2 0.1 0.2 2 0.1 0 0 0 0.4 0.3 0.2 0.2 0.1 0.2 0.1 0.1 0 0 0 -0.1 1 48 96 1 I. Housing 0.2 48 96 J. Health 1 48 96 -0.1 1 48 96 1 0.6 0.3 96 -0.1 1 M. Food Services K. Transportation L. Recreation 0.2 48 0 0 48 96 1 0.8 96 O. Other Services N. Finance 0.2 48 0.3 0.6 0.2 0.4 0.1 0.2 0.1 0.1 0.1 0.2 0 0.4 0.1 0.2 0 0 0 1 48 96 1 48 96 0 0 1 48 96 1 48 96 1 48 96 0 1 48 96 1 48 96 1 48 96 Figure 2: Inflation Responses A. Monetary Policy B. Motor Vehicles C. Furnishings D. Recreational Goods E. Other Durables F. Food at Home G. Clothing and Footwear 1.5 H. Gasoline 2 0.6 0.6 0.6 0.6 0.6 0.6 1.5 1 0.4 0.4 0.4 0.4 0.4 0.4 1 0.5 0 1 6 12 0.2 0.2 0.2 0.2 0.2 0.2 0 0 0 0 0 0 18 1 I. Other Nondurables 6 12 18 1 J. Housing 6 12 18 K. Health 1 6 12 18 1 6 12 18 1 6 12 18 0.5 0 1 N. Food Services L. Transportation M. Recreation 6 12 18 1 6 12 18 P. Other Services O. Finance 1 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.8 0.4 0.4 0.4 0.4 0.4 0.6 0.4 0.4 0.4 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0 0 1 6 12 18 0 1 6 12 18 0 1 6 12 18 0 1 6 12 18 0 1 6 12 18 0 0 1 6 12 18 1 6 12 18 1 6 12 18 Figure 3: Variance Decomposition of the Inflation Rate A. Aggregate Productivity 0.01 B. Monetary Policy C. All Relative D. Motor Vehicles 25 E. Furnishings 0.32 F. Recreational Goods 1.12 G. Other Durables H. Food at Home I. Clothing and Footwear 0.38 1.34 3.05 0.7 1.1 1.32 0.37 76 24 0.31 3 0.69 1.3 1.08 0.36 2.95 0 1 12 24 36 J. Gasoline 23 1 12 24 36 75 K. Other Nondurables 1 12 24 36 0.68 1 12 24 36 0.3 1 12 24 36 1.06 1 12 24 36 M. Health N. TransportationO. Recreation L. Housing 43.5 1.28 1 12 24 36 1 12 24 36 1 12 24 36 P. Food Services Q. Finance R. Other Services 0.66 8.2 3.55 1.72 0.5 4.4 3 3.5 8 43 4.7 0.64 1.68 0.49 4.3 42.5 3.45 2.8 1.64 1 12 24 36 1 12 24 36 4.6 1 12 24 36 1 12 24 36 0.62 1 12 24 36 0.48 1 12 24 36 7.8 1 12 24 36 1 12 24 36 1 12 24 36 Figure 4: Inflation and the Share of Price Changes Larger than Inflation B. Benchmark Model C. Flexible-Price Model 1 1 0.5 0.5 0.5 0 -0.5 Monthly Inflation 1 Monthly Inflation Monthly Inflation A. U.S. Data 0 -0.5 -0.5 Corr = -0.55 Corr = -0.60 Corr = -0.75 -1 -1 0 0.5 Share 0 1 -1 0 0.5 Share 1 0 0.5 Share 1 Figure 5: Undershooting: Inflation and Contributions from Gasoline, Health Care, and Monetary Policy 0.01 0 -0.01 -0.02 Fitted Inflation Gasoline Health Care Monetary Policy Jan 2012 Jan 2013 Jan 2014 Jan 2015 Jan 2016 Jan 2017 Jan 2018 Jan 2019 Dec 2019 Figure 6: COVID Period: Price Level and Contribution from Selected Shocks 0.07 0.06 Actual PCE 2% Trend Motor Vehicles Food at Home Gasoline Monetary Policy 0.05 0.04 0.03 0.02 0.01 0 -0.01 Jan. 2020 Jun. 2020 Dec. 2020 Jun. 2021 Nov. 2021