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Schorfheide, Frank. "DSGE model-based estimation of the New Keynesian Phillips curve.(dynamic stochastic general equilibrium)." Economic Quarterly. Federal Reserve Bank of Richmond. 2008. HighBeam Research. 27 Apr. 2018 <https://www.highbeam.com>.
Schorfheide, Frank. "DSGE model-based estimation of the New Keynesian Phillips curve.(dynamic stochastic general equilibrium)." Economic Quarterly. 2008. HighBeam Research. (April 27, 2018). https://www.highbeam.com/doc/1G1-206162977.html
Schorfheide, Frank. "DSGE model-based estimation of the New Keynesian Phillips curve.(dynamic stochastic general equilibrium)." Economic Quarterly. Federal Reserve Bank of Richmond. 2008. Retrieved April 27, 2018 from HighBeam Research: https://www.highbeam.com/doc/1G1-206162977.html
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An important building block in modern dynamic stochastic general equilibrium (DSGE) models is the price-setting equation for firms. In models in which the adjustment of nominal prices is costly, this equation links inflation to current and future expected real marginal costs and is typically referred to as the New Keynesian Phillips curve (NKPC). Its most popular incarnation can be derived from the assumption that firms face quadratic nominal price adjustment costs (Rotemberg 1982) or that firms are unable to re-optimize their prices with a certain probability in each period (Calvo 1983). The Calvo model has a particular appeal because it generates predictions about the frequency of price changes, which can be measured with microeconomic data (Bils and Klenow 2004, Klenow and Kryvtsov 2008). The slope of the NKPC is important for the propagation of shocks and determines the output-inflation tradeoff faced by policymakers. The Phillips curve relationship can also be used to forecast inflation.
This article reviews estimates of NKPC parameters that have been obtained by fitting fully specified DSGE models to U.S. data. By now, numerous empirical papers estimate DSGE models with essentially the same NKPC specification. In this literature, the Phillips curve implies that inflation can be expressed as the discounted sum of expected future marginal costs, where marginal costs equal the labor share. We document that the identification of the Phillips curve coefficients is tenuous and no consensus about its slope and the importance of lagged inflation has emerged from the empirical studies.
We begin by examining how the NKPC parameters are identified in a DSGE model-based estimation. This is a difficult question. Many estimates are based on a likelihood function, which is the model-implied probability distribution of a set of observables indexed by a parameter vector. The likelihood function peaks at parameter values for which the model-implied autovariance function of a vector of macroeconomic time series matches the sample autocovariance function. Unfortunately, this description is not particularly illuminating. More intuitively, the NKPC parameters are estimated by a regression of inflation on the sum of discounted future expected marginal costs. The likelihood function corrects the bias that arises from the endogeneity of the marginal cost regressor. We show that if one simply uses ordinary least-squares (OLS) to regress inflation on measures of expected marginal costs, the slope coefficient is very close to zero. This finding is quite robust to the choice of detrending method and marginal cost measure. Hence, much of the variation in the estimates reported in the literature is due to the multitude of endogeneity corrections that arise by fitting different DSGE models that embody essentially the same Phillips curve specification.
The review of empirical studies distinguishes between papers in which marginal costs are included in the observations and, hence, are directly used in the estimation and studies that treat marginal costs as a latent variable. In the latter case, NKPC estimates are more sensitive to the specification of the households' behavior, the conduct of monetary policy, and the law of motion of the exogenous disturbances. Estimates of the slope of the Phillips curve lie between 0 and 4. If the list of observables spans the labor share, then the slope estimates fall into a much narrower range of 0.005 to 0.135. No consensus has emerged with respect to the importance of lagged inflation in the Phillips curve. We compare estimates of the relative movement of inflation and output in response to a monetary policy shock, which captures an important tradeoff for monetary policymakers. We find that the estimates in the studies that are surveyed in this article range from 0.07 to 1.4. A value of 0.07 (1.4) implies that a 1 percent increase in output due to a monetary policy shock is accompanied by a quarter-to-quarter inflation rate of 7 (140) basis points.
The remainder of this paper is organized as follows. We discuss the derivation of the NKPC as well as our concept of DSGE model-based estimation in Section 1. In Section 2, a simple DSGE model that can be solved analytically is used to characterize various sources of NKPC parameter identification. Any particular DSGE model-based estimation might exploit some or all of these sources of information. Section 3 provides empirical evidence from least-squares regressions of inflation on the discounted sum of future marginal costs as well as evidence from a vector autoregression (VAR) on the relative movement of output and inflation in response to a monetary policy shock. We thereby characterize some features of the data that are important for understanding the DSGE model-based parameter estimates reviewed in Section 4. Finally, Section 5 concludes.
1. PRELIMINARIES
This section begins with a brief description of the price-setting problem that gives rise to a Phillips curve in New Keynesian DSGE models. We then discuss some of the defining characteristics of DSGE model-based estimation of NKPC parameters.
Price Setting in DSGE Models
New Keynesian DSGE models typically assume that production is carried out by two types of firms: final good producers and intermediate goods producers. The latter hire labor and capital services from the households to produce a continuum of intermediate goods. The final good producers purchase the intermediate goods and bundle them into a single aggregate good that can be used for consumption or investment. The intermediate goods are imperfect substitutes and, hence, each producer faces a downward-sloping demand curve. Price stickiness is introduced by assuming that it is costly to change nominal prices. Rotemberg (1982) assumed that the price adjustment costs are quadratic, whereas Calvo (1983) set forth a model of staggered price setting in which the costs are either zero or infinite with fixed probabilities, i.e., only a fraction of firms is able to change or, more precisely, re-optimize prices.
Aggregating the optimal price-setting decisions of firms leads to the following expression for inflation in the price of the final good, referred to as the New Keynesian Phillips curve:
[[~.[pi]].sub.t] = [[gamma].sub.b] [[~.[pi]].sub.(t-1)] + [[gamma].sub.[Florin]] = [E.sub.t] [[[~.[pi]].sub.(t+1)]] + [gamma] [[~[MC.sub.t] + [[xi].sub.t]. (1)
Here [[~.[pi]].sub.t] represents inflation, [[~.MC].sub.t] is real marginal costs, and [[~.[xi]].sub.t] is an exogenous disturbance that is often called a mark-up shock. We use [[~.z].sub.t] to denote percentage deviations of a variable, [z.sub.t], from its steady state. The coefficients [[gamma].sub.b],[[gamma].sub.[Florin]], and X are functions of model-specific taste and technology parameters. For instance, in Calvo's (1983) model of price stickiness
[[gamma].sub.b] = [[omega]/[1 + [beta][omega]]], [[gamma].sub.[Florin]] = [[beta]/[1 + [beta][omega]], and [gamma] = [[(1 - [zeta]) (1 - [zeta][beta]]/[[zeta] (1 + [beta][omega]]],
where [beta] is the households' discount factor and [zeta], is the probability that an intermediate goods producer is unable to re-optimize its price in the current period. In the derivation of (1), it was assumed that those firms that are unable to re-optimize their prices either adjust their past price by the steady-state inflation rate or by lagged inflation. The parameter [omega] represents the fraction of firms that indexes their prices to lagged inflation.
Assuming that [beta]= 0.99, the sum of [[gamma].sub.b]and [[gamma].sub.[Florin]] is slightly less than 1 and the coefficient of lagged inflation lies between 0 (no dynamic indexation, ([omega] = 0) and 0.5 (full dynamic indexation, [omega]) = 1). If [omega] = 0 and steady-state inflation is 0, then 1/(1 - [zeta]) can be interpreted as the expected duration between price changes. For instance, [zeta] = 2/3 implies that the expected duration of a price set by an intermediate goods producer is three quarters, which leads to a slope coefficient of [gamma] = 0.167. On the other hand, if [zeta] = 7/8, which means that the duration of a price is eight quarters, then the NKPC is much flatter: [lambda] = 0.018.
Our survey of the empirical literature will focus on coefficient estimates for [[gamma].sub.b], [[gamma].sub.[Florin]], and [lambda] rather than the model-specific preference-and-technology parameters. The slope, [lambda], determines the output-inflation tradeoff faced by central banks and affects, for instance, the relative response of output and inflation in response to an unanticipated monetary policy shock. A detailed exposition of the role that the NKPC plays in the analysis of monetary policy is provided in an article by Stephanie Schmitt-Grohe and Martin Uribe in this issue. The coefficient [[gamma].sub.b] affects the persistence of inflation and, for instance, the rate at which inflation effects of shocks to marginal costs die out. This is an important parameter, particularly for central banks that pursue a policy of inflation targeting. If we rearrange the terms in (1), such that expected inflation appears on the left-hand side and all other terms on the right-hand side, then the Phillips curve delivers a forecasting equation for inflation.
DSGE Model-Based Estimation
This article focuses on estimates of [[gamma].sub.b], [[gamma].sub.[Florin]], and [lambda] that are obtained by exploiting the full structure of a model economy. Thus, we consider approaches in which the researcher solves not only the decision problems of the firms but also those of the other agents in the economy and imposes an equilibrium concept. If the economy is subject to exogenous stochastic shocks, the DSGE model generates a joint probability distribution for time series such as aggregate output, inflation, and interest rates. Suppose we generically denote the vector of time, t, observables by [x.sub.t], and assume that the DSGE model has been solved by log-linear approximation techniques. Then the equilibrium law of motion takes the form of a vector autoregressive moving average (VARMA) process of the form (omitting deterministic trend components)
[x.sub.t] = [[PHI].sub.1][x.sub.[t-1]] + ... [[PHI].sub.p][x.sub.[t-p]] + R[[member of].sub.t] + [[PSI].sub.1] R[[member of].sub.[t-1]] + ... [[PSI].sub.q]R[[member of].sub.[t-q]]. (2)
The matrices [[phi].sub.i], [[psi].sub.j], and R are complicated functions of the Phillips curve parameters [[gamma].sub.b], [[gamma].sub.[Florin]], and [lambda], as well as the remaining DSGE model parameters, which we will summarize by the vector[theta]. The vector [[member of].sub.t], stacks the innovations to all exogenous stochastic disturbances and is often assumed to be normally and independently distributed.
A natural approach of exploiting (2) is likelihood-based estimation. Maximum likelihood (ML) estimation of optimization-based rational expectations models in macroeconomics dates back at least to Sargent (1989) and has been widely applied in the DOGE model literature (e.g., Altug [1989], Leeper and Sims [1994], and many of the papers reviewed in Section 4). The likelihood function is defined as the joint density of the observables conditional on the parameters, which can be derived from (2). Let [X.sup.t] = {[x.sub.1], ..., [x.sub.t]}, then
p([[X.sup.T]|[[gamma].sub.b], [[gamma].sub.[Florin]], [lambda], [theta]]) = p([[x.sub.1]|[gamma].sub.b], [[gamma].sub.[Florin]], [lambda], [theta]]) [T.[product](t=2)] p([x.sub.t]|[X.sup.[t-1]], [[gamma].sub.b], [[gamma].sub.[Florin]], [lambda], [theta]). (3)
The evaluation of the likelihood function typically requires the use of numerical methods to solve for the equilibrium dynamics and to integrate out unobserved elements from the joint distribution of the model variables (see, for instance, An and Schorfheide [2007]). A numerical optimization routine can then be used to find the maximum of the (log-)likelihood function. The potential drawback of the ML approach is that identification problems can make it difficult to find the maximum of the likelihood function and render standard large sample approximations to the sampling distribution of the ML estimator and likelihood ratio statistics inaccurate.
A popular alternative to the frequentist ML approach is Bayesian inference. Bayesian analysis tends to interpret the likelihood function as a density function for the parameters given the data. Let p([[gamma].sub.b], [[gamma].sub.[Florin]], [lambda], [theta]) denote a prior density for the DSGE model parameters. Bayesian inference is based on the posterior distribution characterized by the density.
p([[gamma].sub.b],[[gamma].sub.[Florin]], [lambda], [theta]|[X.sup.T]) = [p([X.sup.T]|[[gamma].sub.b], [[gamma].sub.[Florin]], [lambda], [theta]) p([[gamma].sun.b],[[gamma].sub.[Florin]], [lambda], [theta])/[integral]p([X.sup.T]|[[gamma].sub.b],[[gamma].sub.[Florin]],[lambda],[theta]) p([[gamma].sub.b], [[gamma].sub.[Florin]], [lambda],[theta])d([[gamma].sub.b], [[gamma].sub.[Florin]], [lambda], [theta])] (4)
Notice that the denominator does not depend on the parameters and simply normalizes the posterior density so that it integrates to one. The controversial ingredient in Bayesian inference is the prior density as it alters the shape of the posterior, in particular if the likelihood function does not exhibit much curvature. On the upside, the prior allows the researcher to incorporate additional information in the time series analysis that can help sharpen inference. Many of the advantages of Bayesian inference in the context of DSGE model estimation are discussed in Lubik and Schorfheide (2006) and An and Schorfheide (2007). The implementation of Bayesian inference typically relies on Markov-chain Monte Carlo methods that allow the researcher to generate random draws of the model parameters from their posterior distribution. These draws can then be transformed--one by one--into statistics of interest. Sample moments computed from these draws provide good approximations to the corresponding population moments of the posterior distribution.
Notwithstanding all the desirable statistical properties of likelihood-based estimators, the mapping of particular features of the data into parameter estimates is not particularly transparent. Superficially, the likelihood function peaks at parameter values for which a weighted discrepancy between DSGE model-implied autocovariances of [x.sub.t] and sample autocovariances is minimized. The goal of the next section is to explore the extent to which this matching of autocovariances can identify the parameters of the New Keynesian Phillips curve.
2. IDENTIFYING THE NKPC PARAMETERS
The identification of DSGE model parameters through likelihood-based methods tends to be a black box because the relationship between structural parameters and autocovariances or other reduced-form representations is highly nonlinear. This section takes a look inside this black box to develop some understanding about particular features of the DSGE model that contribute to the identifiability of NKPC parameters. Rather than asking whether there is enough variation in postwar data to estimate the NKPC parameters reliably, for now we focus on sources of identification in infinite samples. In practice, the estimation of a particular model might exploit several of these sources of information simultaneously. …
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