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Nason, James M.; Gregor Smith,. "The New Keynesian Phillips curve: lessons from single-equation econometric estimation." Economic Quarterly. Federal Reserve Bank of Richmond. 2008. HighBeam Research. 20 Jul. 2018 <https://www.highbeam.com>.
Nason, James M.; Gregor Smith,. "The New Keynesian Phillips curve: lessons from single-equation econometric estimation." Economic Quarterly. 2008. HighBeam Research. (July 20, 2018). https://www.highbeam.com/doc/1G1-206162976.html
Nason, James M.; Gregor Smith,. "The New Keynesian Phillips curve: lessons from single-equation econometric estimation." Economic Quarterly. Federal Reserve Bank of Richmond. 2008. Retrieved July 20, 2018 from HighBeam Research: https://www.highbeam.com/doc/1G1-206162976.html
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The last decade has seen a renewed interest in the Phillips curve that might be an odd awakening for a macroeconomic Rip van Winkle from the 1980s or even the 1990s. Wasn't the Phillips curve tradition discredited by the oil prices shocks of the 1970s or by theoretical critiques of Friedman, Phelps, Lucas, and Sargent? It turns out that the New Keynesian Phillips curve (NKPC) is consistent with both the theoretical demands of modern macroeconomics and some key statistical properties of inflation. In fact, the NKPC can take a sufficient number of guises to accommodate a wide range of perspectives on inflation.
The NKPC originated in descriptions of price setting by firms that possess market power. For example, Rotemberg (1982) describes how a monopolist sets prices if it faces a cost of adjustment that rises with the scale of the price change. He shows that prices then gradually track a target price and also depend on expected, future price targets. Calvo (1983) instead describes firms that are monopolistic competitors. They change their prices periodically. Knowing that some time may pass before they next set prices, firms anticipate future cost and demand conditions, as well as current ones, in setting their price. Also, the staggering or nonsynchronization of price setting by firms creates an aggregate stickiness: The aggregate price level will react only partially on impact to an economy-wide shock, such as an unexpected change in monetary policy.
These theoretical models link prices to a targeted real variable such as a markup on the costs faced by the price-setting firm. Therefore, they also relate the change in prices over time (i.e., the inflation rate) to real variables. So it is natural to label them as Phillips curves. In fact, there are a range of setups called NKPCs that vary depending on (a) the information and price-setting behavior attributed to firms and (b) the measure of costs or demand that firms are assumed to target. Whether a specific version of the NKPC fits inflation data has implications for our understanding of recent macroeconomic history and for the design of good policy. For example, the parameters of the NKPC influence how monetary policy ideally should respond to external shocks. Schmitt-Grohe and Uribe, in this issue, make this connection clear. (King, also in this issue, describes the uses of an older Phillips curve tradition in policymaking.)
Yet, putting the NKPC to use for policy analysis requires that it has a good econometric track record in describing actual inflation dynamics. In this article we review this record using single-equation statistical methods that study the NKPC on its own. These methods stand in contrast to approaches that place the NKPC in larger economic models, sometimes referred to as systems methods, which are reviewed by Schorfheide in this issue. A disadvantage of single-equation methods is that they do not make use of everything known about the economy (e.g., the monetary policy regime), so they generally do not provide the greatest statistical precision. Their advantage is that they allow us to be agnostic about the rest of the economy, and so their findings remain valid and will not be affected by misspecification of other parts of a larger macroeconomic model.
This article asks the following questions: How can we estimate the NKPC and what do we find when we do so for the United States? Are its parameters stable over time and well-identified? Is there a relation between inflation and real activity? Do we reach similar conclusions about the NKPC regardless of the way in which we measure inflation, forecast future inflation, or model the costs or output gap that inflation tracks?
We focus on marginal cost as the real activity variable in the NKPC. We find that the single-equation statistical evidence for this relationship is mixed. Since 1955 there does seem to be a stable NKPC for the United States with positive parameter values as we would expect from economic theory. But our confidence intervals for these parameter values are somewhat wide, the findings depend on how we model expected future inflation, and further research is needed on the best way to represent the marginal cost variable to which price changes react. Before outlining the methods and findings, though, we begin by introducing the specific NKPC that we will estimate and the inflation history it aims to explain.
1. FOUNDATIONS AND INTERPRETATIONS
The New Keynesian Phillips curve arises from a description of staggered price setting, which is then linearized for ease of study. The result is an equation in which the inflation rate, [[pi].sub.t], depends on the expected inflation rate next period, [E.sub.t][[pi].sub.[t+1]], and a measure of marginal costs, denoted [x.sub.t]:
[[pi].sub.t] = [[gamma].sub.[Florin]][E.sub.t][[pi].sub.[t+1]] + [lambda][x.sub.t].
Iterating this NKPC difference equation forward gives inflation as the present value of future marginal costs:
[[pi].sub.t] = [lambda][[infinity].summation over (i=0)] [[gamma].sub.[Florin].sup.i] [E.sub.t][x.sub.[t+i]].
This present-value relation shows that firms consider both their current marginal costs, [x.sub.t], and their expectations or forecasts of future costs when adjusting prices.
Lacker and Weinberg (2007) describe the history and derivation of this key building block of New Keynesian macroeconomic models. Dennis (2007) outlines a range of environments that can underpin a hybrid NKPC. Calvo's (1983) specific price-setting model is only one of several possible microfoundations for the NKPC. In a Calvo-based NKPC, a fraction, [theta], of firms cannot change prices in a given period. Firms also have a discount factor, [beta]. The reduced-form parameters of the NKPC, [[gamma].sub.[Florin]] and [lambda], are related to these two underlying pricing parameters according to
[[gamma].sub.[Florin]] = [beta], [lambda] = (1 - [theta])(1 - [beta][theta])/[theta].
Because [beta] is a discount factor, both it and [[gamma].sub.[Florin]] must range between zero and one. The same holds for [theta] because it represents the fraction of firms unable to move prices at any moment.
Many estimates of the NKPC find that lagged inflation helps to explain current inflation. We report much the same in this article. This has suggested to some economists that a better fit to inflation history can be obtained with this equation:
[[pi].sub.t] = [[gamma].sub.b][[pi].sub.[t-1]] + [[gamma].sub.[Florin]][E.sub.t][[pi].sub.[t+1]] + [[lambda][x.sub.t].
which Gali and Gertler (1999) call the hybrid NKPC. They develop their NKPC by modifying Calvo's (1983) description of price-setting decisions. In this case, a fraction, [omega], of firms can change prices, but do not choose this option. Define [phi] = [theta] + [omega][l - [theta](1 - [beta])]. Then the mapping between these structural parameters and the reduced-form parameters is
[[gamma].sub.b] = [omega]/[phi], [[gamma].sub.[Florin]] = [beta][theta]/[phi], [lambda] = (1 - [omega])(1 - [theta])(1 - [beta][theta])/[phi].
Gali and Gertler (1999) note that this mapping between the structural, price-setting parameters [omega], [theta], and [beta] and the reduced-form hybrid NKPC parameters [[gamma].sub.b], [[gamma].sub.[Florin]] and [lambda] is unique if the former set of parameters lie between zero and one.
This form of the NKPC also is consistent with the incomplete indexing model of Woodford (2003). He assumes that those firms that cannot optimally alter their prices instead index to a fraction of the lagged inflation rate. This feature makes current inflation depend on lagged inflation, and so provides an alternative interpretation of the hybrid NKPC. Christiano, Eichenbaum, and Evans (2005), among others, study the implications of full indexation, which is equivalent to the restriction [[gamma].sub.b] + [[gamma].sub.[Florin]] = 1 in the Galf-Gertler hybrid NKPC.
Like the original NKPC, the hybrid version can be rewritten in present-value form:
[[pi].sub.t] = [[delta].sub.1][[pi].sub.[t-1]] + ([[lambda]/[[delta].sub.2][[gamma].sub.[Florin]]]) [[infinity].summation over (k=0)] [(1/[[delta].sub.2]).sup.k] [E.sub.t][x.sub.[t+k]],
where [[delta].sub.1] and [[delta].sub.2] are stable and unstable roots, respectively, of the characteristic equation in the lag operator L:
[-L.sup.-1] + [1/[[gamma].sub.[Florin]]] - [[[gamma].sub.b]L]/[[gamma].sub.[Florin]]] = 0.
The present-value version of the hybrid NKPC shows that inflation persistence can arise from the influence of lagged inflation or the slow evolution of the present value of marginal costs.
The NKPC is often derived by log-linearizing a typical firm's price-setting rule around a mean zero inflation rate. Ascari (2004) shows that non-zero mean inflation can affect the response of inflation to current and future marginal cost in the NKPC. However, we follow much of the empirical NKPC literature and demean the data.
Cogley and Sbordone (forthcoming) build on Ascari's work, among others, by log-linearizing the NKPC around time-varying trend inflation. This procedure assumes that inflation is nonstationary to obtain a NKPC with time-varying coefficients even though the underlying Calvo-pricing parameters are constant. The resulting NKPC is purely forward-looking, assigning no role to lagged inflation. Cogley and Sbordone estimate the structural coefficients of their NKPC using a vector autoregression. Hornstein (2007) assesses the implications of this approach for the stability of the NKPC. Since these studies use system estimators, we omit them from our review.
In sum, the hybrid NKPC is consistent with various pricing or information schemes. This suggests a focus on the reduced-form coefficients [lambda], [[gamma].sub.b], and [[gamma].sub.[Florin]] (rather than the structural price-setting parameters [omega], [theta], and [beta]), which is our emphasis in this article. We also use single-equation estimators to explore the fit of the hybrid NKPC to U.S. data. After all, obtaining a good fit for the hybrid NKPC is a necessary first step in attributing a monetary transmission mechanism to staggered price setting by firms.
Economists have not yet reached a consensus on two key questions concerning the NKPC parameters. First, what is the mixture of forward ([[gamma].sub.[Florin]]) and backward ([[gamma].sub.b]) weights? If [[gamma].sub.[Florin]] is large, events in the future (including changes in monetary policy) can influence the current inflation rate. If, instead, [[gamma].sub.b] is large, inflation has considerable inertia independent of any slow movements in the cost variable. Such inertia affects the design of monetary policy (again, see Uribe and Schmitt-Grohe). Woodford (2007) reviews several explanations for inflation inertia and also discusses whether it could be stable over time.
Second, can we identify a significant [^.[gamma]], the coefficient on marginal costs? In this case, identification simply refers to measuring a partial correlation coefficient in historical data rather than the possibility of misspecification (i.e., whether this coefficient necessarily measures the theoretical parameter studied in New Keynesian models.) Finding a significant value is a sine qua non for empirical work with the NKPC. If we cannot find a way to represent a price-setting target, we cannot hope to identify the adjustment process of inflation to its target. Consequently, much of the research on estimating the NKPC involves exploring the x-variable or how to measure marginal costs.
Before looking at formal econometric methods, let us look at the data. Figure 1 plots the U.S. inflation rate (the black line) and a measure of marginal cost (the gray line) from 1955:1 to 2007:4. We measure the inflation rate, [[pi].sub.t], as the quarter-to-quarter, annualized growth rate in the U.S. implicit GDP deflator (GDPDEF from FRED at the Federal Reserve Bank of St. Louis). Marginal cost, [x.sub.t], is the update of the series on real unit labor costs used by Gali and Gertler (1999) and Sbordone (2002). It is given by 1.0765 times the logarithm of nominal unit labor costs in the nonfarm business sector (the ratio of COMPNFB to OPHNFB from FRED) divided by the implicit GDP deflator. Both series have fallen since 1980, which in itself provides some statistical support for the idea that inflation tracks marginal cost. There also are some obvious divergences, for example around 2000-2001. However, it is possible that these occurred because inflation was tracking expected future marginal cost (as in the present-value model) or because it was linked to lagged inflation (as in the hybrid NKPC). We next describe the statistical tools economists have used to see if either of these explanations fits the facts.
[FIGURE 1 OMITTED]
2. ESTIMATION
The fundamental challenge with estimating the parameters of the hybrid NKPC is that expected inflation cannot be directly observed. The most popular econometric method for dealing with this issue begins from the properties of forecasts. To see how this works, let us label the information used by price-setters to forecast inflation by [I.sub.t], so their forecast is E[[[pi].sub.[t+1]]|[I.sub.t]]. Economists do not observe this forecast, but it enters the NKPC and influences the current inflation rate. Denote by E[[[pi].sub.[t+1]]|[Z.sub.t]] an econometric forecast that uses some variables, [Z.sub.t], to predict next period's inflation rate, [[pi].sub.[t+1]]. Suppose that [Z.sub.t] is a subset of the information available to price setters. To construct our econometric forecast, we simply regress actual inflation on our set of variables (sometimes called instruments), [Z.sub.t], like this:
[[pi].sub.[t+1]] = b[Z.sub.t] + [[member of].sub.[t+1]],
so that our forecast is simply the fitted value
E[[[pi].sub.[t+1]]|[Z.sub.t]] = [[^.b]z.sub.t].
By construction it is uncorrected with the residual term [[^.[member of]].sub.[t+1]].
A key principle of forecasts (or rational expectations) is the law of iterated expectations. According to that law, our econometric prediction of price-setters' forecast of inflation is simply our forecast. Symbolically,
E[E[[[pi].sub.[t+1]]|[I.sub.t]|[Z.sub.t]] = E[[[pi]. …
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