The New Keynesian Phillips Curve and lagged inflation: a case of spurious correlation?

• Author: Hall, Stephen G.

1. Introduction

   The New Keynesian Phillips Curve (NKPC) is a key component of much recent theoretical work on inflation. Unlike traditional formulations of the Phillips curve, the NKPC is derivable explicitly from a model of optimizing behavior on the part of price setters, conditional on the assumed economic environment (for example, monopolistic competition, constant elasticity demand curves, and randomly arriving opportunities to adjust prices) (Walsh 2003). In contrast to the traditional specification, in the NKPC framework current expectations of future inflation, rather than past inflation rates, shift the curve (Woodford 2003). Also, the NKPC implies that inflation depends on real marginal cost, and not directly on either the gap between actual output and potential output or the deviation of the current unemployment rate from the natural rate of unemployment, as is typical in traditional Phillips curves (Walsh 2003). A major advantage of the NKPC compared with the traditional Phillips curve is said to be that the latter is a reduced-form relationship; whereas, the NKPC has a clear structural interpretation so that it can be useful for interpreting the impact of structural changes on inflation (Gali and Gertler 1999).

   Although the NKPC is appealing from a theoretical standpoint, empirical estimates of the NKPC have, by and large, not been successful in explaining the stylized facts about the dynamic effects of monetary policy, whereby monetary policy shocks are thought to first have an effect on output, followed by a delayed and gradual effect on inflation (Mankiw 2001; Walsh 2003). To deal with what some authors (for example, McCallum 1999; Mankiw 2001; Dellas 2006a,b) believe to be inflation persistence in the data, (l) a response typically found in the literature is to augment the NKPC with lagged inflation on the supposition that lagged inflation receives weight in these equations because it contains information on the driving variables (that is, the variables driving inflation), thereby yielding a "hybrid" variant of the NKPC. A general result emerging from the empirical literature is that the coefficient on lagged inflation is positive and significant, with some authors (for example, Fuhrer 1997; Rudebusch 2002; Rudd and Whelan 2005) finding that inflation is predominantly backward looking.

   The hybrid NKPC, however, is itself subject to several criticisms. First, derivations of the hybrid specifications typically rely on backward-looking rules of thumb, so that a "more coherent rationale for the role of lagged inflation" has yet to be provided (Gali, Gertler, and Lopez-Salido 2005, p. 1117). In effect we are losing all the supposed advantages of the clear microfoundations. Second, the idea that the important role assigned to lagged inflation derives from its use as a proxy for expected future inflation is contradicted by the large estimates of the effects of lagged inflation obtained even in specifications that include the discounted sums of future inflations (Rudd and Whelan 2005, p. 1179). (2)

   The contention made in this article is that the standard model estimated within the NKPC paradigm is subject to a number of serious econometric problems and that these problems lead not only to ordinary least squares (OLS) being a biased estimator of the true underlying parameters, but that generalized method of moments (GMM) is also subject to these problems in this instance. We will demonstrate that, while GMM and instrumental variables can correctly deal with the standard problem of measurement error and endogeneity, if there are also missing variables and a misspecified functional form, then no valid instruments will exist and GMM becomes inconsistent. Consequently, we argue that the finding of a need for lagged inflation may be a direct result of the biases caused by estimation problems rather than a flaw with the underlying economic theory. We will make this case first at a theoretical level, showing that economic theory clearly suggests both that the standard form of the NKPC is misspecified and that it is subject to omitted variables and misspecified functional form; hence, we will show that GMM is inconsistent. Second, we will apply a time varying coefficient (TVC) estimation procedure that aims to yield consistent estimates under these circumstances, and which finds a coefficient on expected inflation that is essentially unity.

   The remainder of this article is divided into three sections. Section 2 briefly summarizes the theoretical derivation of the NKPC and stresses the simplifying assumptions that imply the misspecification of the model. It then goes on to outline the estimation strategy used in this article, building on the work of Swamy et al. (2008). (3) We contrast our TVC estimation approach with that of the GMM, which has been widely applied in previous empirical studies of NKPCs (e.g., Gali and Gertler 1999; Gali, Gertler, and Lopez-Salido 2005; Linde 2005). Section 3 presents empirical results of NKPCs using U.S. quarterly data. We demonstrate that GMM produces the usual result of significant lagged inflation rates while our estimation approach provides coefficients that are much more closely in line with the microfoundations. Section 4 concludes.

2. Theoretical Considerations and Empirical Methodology

   The NKPC Is a Misspecified Model

   There are a number of ways of deriving the NKPC. A standard way of doing so is based on a model of price setting by monopolistically competitive firms (Gali and Gertler 1999). (4) Following Calvo (1983), firms are allowed to reset their price at each date with a given probability (1-[theta]), implying that firms adjust their price taking into account expectations about future demand conditions and costs, and that a fraction [theta] of firms keep their prices unchanged in any given period. Aggregation of all firms produces the following NKPC equation in log-linearized form

   [[??].sub.t] = [beta][E.sub.t][[??].sub.t+1] + [[lambda].sub.1] [s.sub.t] + [[eta].sub.0t] (1)

   where [[??].sub.t] is the inflation rate, [E.sub.t] [[??].sub.t+1] is the expected inflation in period t+1 as it is formulated in period t, [s.sub.t] is the (logarithm of) average real marginal cost in percent deviation from its steady state level, and [[eta].sub.0t] is a random error term. The coefficient, [beta], is a discount factor for profits that is on average between 0 and 1, [[lambda].sub.1] = [(1 - [theta])(1 - [beta][theta])]/[theta] is a parameter that is positive, where [theta] is the probability that the firm will change its price in any quarter; [[??].sub.t] increases when real marginal cost, which is a measure of excess demand, increases (as there is a tendency for inflation to increase). Since marginal cost is unobserved, in empirical applications real unit labor cost (ul[c.sub.t]) is often used as its proxy. (5)

   If we look a little deeper into the microfoundations, however, we find a number of serious simplifications that underline this equation. Batini, Jackson, and Nickell (2005) emphasize the underpinnings of the NKPC. They begin their derivation with a Cobb-Douglas production function in which capital is replaced by a variable labor-productivity rate. They then assume a representative firm with a simple quadratic cost-minimization objective function and derive a standard NKPC, which even then includes terms in employment. Later, in the same article, they generalize the NKPC to an open economy case, at which point a number of extra variables play an important part, including foreign prices, exchange rates, and oil prices. Given this derivation, it is clear that the standard NKPC involves the following simplifications:

   * The basic functional form is misspecified. In the standard derivations the NKPC is a linearization of a theoretical formulation based on quadratic costs and Cobb-Douglas technology. In fact, both of these assumptions are unrealistic. Cobb-Douglas technology is almost always rejected wherever it is tested, and so the real production function must be more complex. Similarly, quadratic objective functions are convenient, but far from realistic. Clearly, according to the theory, the NKPC is a linear version of a much more complex nonlinear model.

   * The basic NKPC is subject to the omission of a potentially large number of omitted variables. Batini, Jackson, and Nickell (2005) emphasize the need to include exchange rates, foreign prices, oil prices, employment, and a labor productivity variable. The representative firm assumption could well mean that variables capturing firm heterogeneity are important.

   * The variables used in the NKPC are almost certainly measured with error. For example, unit labor costs can only be modeled as the labor share under Cobb-Douglas technology. A constant elasticity of substitution (CES) function would involve a much richer set of variables to properly capture the real wage, but even this function would be only an approximation, as empirical support for CES technology is not overwhelming. Clearly, the representative-firm assumption also suggests that average or total measures of labor share may not be the correct measure. Additionally, there are well-known problems in measuring inflation itself.

   Thus, the case is very strong, from a theoretical perspective, that any of the standard NKPC models would be subject to measurement error, omitted variable bias, and a misspecified functional form.

   The response of many authors to the poor estimation results often produced from the NKPC is to start to find largely "ad hoc" reasons for augmenting the NKPC with lags. Many authors assume that firms can save costs if prices are changed between price adjustment periods according to a rule of thumb. For example, Gali and Gertler (1999) assume that only a portion (1 - [rho]) of firms is forward-looking and the rest are backward-looking. This implies that only a fraction (1 - [rho]) of firms set their prices optimally, and the rest...