RDP 2013-11: Issues in Estimating New-Keynesian Phillips Curves in the Presence of Unknown Structural Change 1. Introduction

Many papers that estimate models with forward-looking expectations report that the magnitude of the coefficient attached to the forward expectations term is very large compared with that attached to the term which captures past dynamics. This has often been regarded as implausible, leading to the conjecture that the estimator of the former is biased upwards. Exactly why there should be an upward bias is less clear. One possibility is that weak instruments can result in an estimator bias in small samples (e.g. Mavroeidis 2004), although there is no reason to think that it is an upwards bias. Another possibility is specification error in the structural equation containing the expectations. While there is little one can say about this in general, as the nature of the specification error will be crucial, a specific argument has been that the bias could be due to changes in the means of the variables entering the structural equation. Russell et al (2010) give this explanation. Using Bai-Perron tests they find that there were eight breaks in the mean of the inflation rate in the United States over 1960–2007. Assuming that the timing of these breaks coincides with changes in the intercept of the New-Keynesian Phillips curve (NKPC), they then augment the NKPC with dummy variables to capture intercept breaks. Re-estimating with such dummies, they show that the coefficient of the expectations term is greatly reduced. This leads them to conclude that ‘Once the shifts in the mean rate of inflation have been accounted for in the estimation of the United States Phillips curves we find that … there is no significant role for expected inflation … in the NK and hybrid models of inflation’ (p 5).

Castle et al (2010) provide an explanation of what Russell et al (2010) found. It revolves around the fact that a standard way of estimating an equation with forward-looking expectations, like the NKPC, involves replacing the expectations term with future inflation and then applying an instrumental variables estimator to the resulting equation in observable variables. If the intercept breaks are unaccounted for when estimating the NKPC, a specification error exists and this will be correlated with future values of inflation. Thus the argument in Castle et al (2010) is that the explanatory power of the future observable variables is due to the breaks and not to forward-looking expectations. Technically, one gets an inconsistency in the instrumental variables (IV) estimator of the coefficient of the future expectations term. In some simple experiments they show that this effect can be substantial.

In this paper we investigate the issue of upward bias in the estimated coefficients of the expectations variable based on a model where we can see what causes the breaks and how to control for them. Since many of the applications involve a NKPC, we work with that as the structural equation, embedding it in a standard New-Keynesian (NK) model that also has equations for real marginal cost and an interest rate rule. In each case the agent may know of the breaks but the econometrician is assumed to be ignorant of when they occur. We discuss how to solve this model in the presence of breaks, both when agents know exactly where the breaks occur and also when they get the timing of the break wrong. The method of solution does not depend on the simple model we use for experiments but can be used for any model with forward-looking expectations. The method is set out in detail in Kulish and Pagan (2012).

Because the model is simple, we are able to perform an experiment in which there are breaks in the means of the target inflation rate and real marginal cost that offset one another exactly so as to produce no breaks in the intercept of the NKPC. This experiment just makes the point that breaks in the means of variables such as inflation may not cause breaks in the intercepts of the NKPC. Yet in this experiment we find a bias in some commonly used estimators. Since the equation is correctly specified, due to the intercept being constant (and in this experiment we assume that agents know exactly the timing of the mean shifts), the only reason that this bias can arise is the presence of weak instruments. This leads us to make a distinction between large-sample biases due to specification errors and those arising in smaller samples owing to weak instruments. The latter can sometimes be resolved by using different estimators whereas the former cannot. We find that breaks in the means of the series can often change the properties of instruments a great deal, and may well be a bigger source of small-sample bias than that coming from specification error. Moreover, we also find that the direction of the specification bias is not predictable. With some estimators and breaks the coefficient of the expectations variable is over-estimated, but with others it is under-estimated. This leads to the conclusion that it is necessary to check for factors such as the presence of weak instruments before deciding that the magnitude of any estimator bias reflects specification errors coming from breaking means.

The next section sets out our simple model and distinguishes three estimators of the NKPC. One of these cannot be implemented in practice but gives a useful benchmark. Section 3 then provides a simplified account of how the NK model can be solved in the presence of structural change. Section 4 looks at a range of simulations, beginning with no breaks, moving on to breaks in the reduced form but not the structure, and finishing with breaks in both the structure and reduced form. Breaks in the reduced form correspond to breaks in the means of the inflation rate while breaks in the structure come from changes in the intercept in the NKPC. The estimators introduced in Section 2 are examined, and we assess which one performs best in the presence of breaks. In this section we also investigate the robustness of our results to agents not knowing the timing of breaks precisely when they form expectations.

Section 5 looks at the empirical work in Castle et al (2010) on the NKPC in the euro area. They argue that the structural equation requires the addition of a number of indicator variables which, when added, reduce the estimated expectations coefficient by a large amount. Of course indicators are only very short-lived breaks, whereas the breaks we look at in this paper are rather longer-lived. Nevertheless, even short-lived breaks can cause specification bias, and their presence in the reduced form can lead to weak instruments. We assess whether the smaller coefficient on expectations that Castle et al note when dummy variables are present is due to weak instruments or to specification issues. Our finding suggests that it is probably a consequence of weak instruments.