Identification of New Keynesian Phillips Curves from a Global Perspective

Dees, Stephane, Pesaran, M. Hashem, Smith, L. Vanessa, Smith, Ron P., Journal of Money, Credit & Banking

NEW KEYNESIAN PHILLIPS CURVES (NKPC) have been widely used in the macroeconomic literature. Yet their empirical implementation raises a number of issues that continue to be of some concern. In this paper, we shall focus on two of these issues--the weak instrument problem and the characterization of the steady states--and propose some solutions from a global perspective.

The first issue relates to the quality of the instruments used to estimate the NKPC model. The problem arises because in a closed economy setting it turns out that there are not many variables that can be used to produce inflation forecasts that improve significantly over a first-order autoregressive model of inflation. Under the NKPC lagged observations are admissible as instruments, in the sense that they are not correlated with the error terms. However, in order to be valid instruments the variables also need to be sufficiently correlated with the endogenous explanatory variables so that the necessary rank condition is satisfied. The solution of the rational expectations (RE) model indicates that this rank condition will not be satisfied unless the lag order of the equation determining the driving variable is greater than that of the NKPC and the extra lags significantly improve the prediction of the driving variable (see, e.g., Mavroeidis 2005, Nason and Smith 2008). In this paper, by taking a global perspective, we suggest some possible routes to resolving the weak instrument problem at least for NKPC models of small open economies.

In addition, the NKPC is typically derived from the first-order optimization conditions of a representative firm, subject to staggered pricing behavior, in the context of a dynamic stochastic general equilibrium (DSGE) framework. Because these first-order conditions are complicated nonlinear stochastic equations, usually they are log-linearized around a steady state. Such an approximation procedure is appropriate if a unique steady state exists, the log-linearization is carried out around the correct steady state, and the approximation errors are relatively small. In cases where the steady states exist and are not time varying the analysis of the DSGE equations as deviations from the steady states does not pose any new difficulties. Inclusion of intercepts in the log-linearized version of the first-order conditions will suffice. Similarly, when the steady state values follow deterministic trends, residuals from regressions on such trend components can be used in the log-linearized DSGE model. The problem arises if the first-order conditions contain variables with stochastic trends that could be cointegrated. In such cases any misspecification of the steady states can seriously bias the estimates of the DSGE equations. In practice, the stochastic trends, for example in the case of output, are often approximated by statistical methods such as the Hodrick-Prescott (HP) filter or a variety of the band pass filters as discussed in Christiano and Fitzgerald (2003). These procedures are purely statistical, in the sense that they are not derived from the assumed DSGE model and need not be consistent with it. In this paper, we present an alternative approach where the derivation of steady states is made consistent with the underlying DSGE model. We propose to measure the steady states by the long-horizon expectations, where expectations are taken consistently with respective to the underlying DSGE model. This is in line with the idea of the model consistent expectations that underpin the NKPC and simply extends it to the long run.

In the empirical section of the paper the steady states are estimated using a global model for 33 countries estimated over the period 1979Q1-2006Q4. Using these estimates NKPC equations are estimated for eight developed economies where it is shown that using global instruments and economic measures of the steady states provide better determined estimates of the NKPC not only for the United States but also for a number of European economies, notably United Kingdom, France, and Spain.

The rest of the paper is set out as follows. Section 1 discusses the identification of the NKPC. Section 2 describes the solution of a multicountry RE DSGE model and shows how the use of global factors as instruments may alleviate the weak instrument problem. Section 3 explains the characterization of steady states as long-horizon expectations. Section 4 discusses how a cointegrating GVAR model based on that of Dees, di Marco, Pesaran, and Smith (2007, DdPS) can be used to provide instruments and theory-consistent estimates of the steady states. It is shown that if the variables in the system are I(1), the long-horizon expectations in a linear system happen to be the same as the permanent or trend component obtained from a multivariate version of the Beveridge and Nelson (1981) decomposition. Section 5 presents estimates of the NKPC for eight countries under a variety of assumptions about available instruments and measures of steady state. Section 6 provides some concluding comments.

1. IDENTIFICATION AND ESTIMATION OF THE PHILLIPS CURVE

Consider a standard closed economy NKPC model. For countries i = 1, 2, ..., N and time periods t = 1, 2, ..., T, the NKPC relates the deviations from steady state of inflation, [[??].sub.it] and a driving output or marginal cost variable, [[??].sub.it], by an equation of the form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (1)

where E([[??].sub.i,t+1]|[T.sub.i,t-1]) denotes expectations formed conditional on information at time t - 1. (1) All variables are measured as deviations from their respective steady states. Steady state values of inflation and output are denoted by [[pi].sup.P.sub.it], and [y.sup.P.sub.it], with deviations from the steady states given by [[??].sub.it] = [[pi].sub.it] - [[pi].sup.P.sub.it] and [[??].sub.it] = [y.sub.it] - [y.sup.P.sub.it].

The parameters are nonlinear functions of underlying structural parameters. For instance, following Gali, Gertler, and Lopez-Salido (2005) suppose that there is staggered price setting, with a proportion of firms, (1 - [[theta].sub.i]), resetting prices in any period, and a proportion [[theta].sub.i] keeping prices unchanged. Of those firms able to adjust prices only a fraction (1 - [[omega].sub.i]) set prices optimally on the basis of expected marginal costs. A fraction [[omega].sub.i] use a rule of thumb based on lagged inflation. Then for a subjective discount factor, [[lambda].sub.i], we have

[[beta].sub.fi] = [[lambda].sub.i][[theta].sub.i][[phi].sup.-1.sub.i], [[beta].sub.bi] = [[omega].sub.i][[phi].sup.- 1.sub.i],

[[gamma].sub.i] = (1 - [[omega].sub.i])(1 - [[theta].sub.i])(1 - [[lambda].sub.i][[theta].sub.i])[[phi].sup.-1.sub.i], (2)

where [[phi].sub.i] = [[theta].sub.i] + [[omega].sub.i][1 -0i(1 - [[theta].sub.i])]. If [[omega].sub.i] = 0, all those who adjust prices do so optimally, then [[beta].sub.fi] = [[lambda].sub.i], and [[beta].sub.bi] = 0. If the discount factor, [[lambda].sub.i] = 1, then [[beta].sub.fi] + [[beta].sub.bi] = 1 in either case. Because the discount factor is likely to be very close to unity, this case is worth attention. Note that there is no reason for these parameters to be the same across countries with very different market institutions and property rights (which will influence [[lambda].sub.i]), so we allow them to be heterogeneous from the start.

Traditionally, the driving variable has been a measure of unemployment or the output gap. More recently, measures of marginal cost and the share of labor have been used. We will use the output gap because it is the variable that is relevant to policy, which relates inflation to aggregate demand not marginal cost; it is the variable that appears in the standard three equation macro model; and it is the variable that is available for all the countries in our sample. We will compare the performance of two measures of the output gap obtained using either the HP filter or the GVAR measure of the steady states, but there are various issues of identification and estimation to be considered first.

It is common to assume that inflation is stationary, and that its steady state is a constant, say [[bar.[pi]].sub.i], then equation (1) becomes the special case

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)

The solution of the model depends on the process generating [[??].sub.it] and [[epsilon].sub.it]. It is typically assumed that [[epsilon].sub.it] is a martingale difference process, and [[??].sub.it] follows a stationary time series process. Consistent estimation of the NKPC critically depends on the nature of the [[??].sub.it] process. The empirical literature typically assumes that suitable instruments (or moment conditions) exist and uses GMM to estimate the following version of the NKPC

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (4)

or

[[??].sub.it] = [[theta]'.sub.i][[chi].sub.i,t+1] + [[xi].sub.i,t+1],

where

[[xi].sub.i,t+1] = [[epsilon].sub.it] - [[beta].sub.fi][[upsilon].sub.i,t+1],

and [v.sub.i,t+1] is the expectations error of inflation, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

The estimation of (4) requires at least three instruments that are

(i) not correlated with [[xi].sub.i,t+1], namely,

E([z.sub.i,t-1] [[xi].sub.i,t+1] | [T.sub.i,t-1]) = 0,

where [z.sub.i,t-1] denotes the s x 1 vector of instruments, and at the same time are

(ii) sufficiently correlated with [[chi].sub.i,t+1], such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Given the nature of the RE hypothesis there are no difficulties finding instruments that satisfy condition (i). Condition (ii) is more problematic and whether it holds critically depends on the nature of the [[??].sub.it] process. To determine if the NKPC is identified requires solving the RE model.

1.1 Alternative Solutions

In the case where there are no feedbacks from inflation to output gap, [[beta].sub.bi], [[beta].sub.fi] [greater than or equal to] 0, [[beta].sub.fi][[beta].sub.bi] [less than or equal to] 1/4, and [[beta].sub.bi] +[[beta].sub.fi] [less than or equal to] 1, the NKPC (1) has the unique solution,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (5)

where [[lambda].sub.bi] and [[lambda].sub.fi] are roots of …

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Publication information: Article title: Identification of New Keynesian Phillips Curves from a Global Perspective. Contributors: Dees, Stephane - Author, Pesaran, M. Hashem - Author, Smith, L. Vanessa - Author, Smith, Ron P. - Author. Journal title: Journal of Money, Credit & Banking. Volume: 41. Issue: 7 Publication date: October 2009. Page number: 1481+. © 1999 Ohio State University Press. COPYRIGHT 2009 Gale Group.

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