Stabilizing Taylor rules and determinacy under unit root supply shocks: A re-examination

Highlights

• I revisit and extend the analysis of unit root cost-push shock in NK models.

• Stabilizing inflation enforces explosive output gap volatility, and vice versa.

• The Taylor principle produces a unique (non-stationary) solution in MSV form.

• It nonetheless fails to replicate the optimal discretionary policy.

Abstract

I revisit the stabilizing and determinacy properties of Taylor-type policy rules in the canonical New Keynesian model when allowing for a unit root in the supply shock process. While able to offset inflationary pressure from non-stationary disturbances, interest-rate feedback rules that are unresponsive to fluctuations in the output gap necessarily produce unstable dynamics and explosive volatility for the latter. Specifically, rules fulfilling the Taylor principle are found to enforce the unique (non-stationary) equilibrium featuring well-anchored inflation expectations and immunity to sunspots; yet there exists no equilibrium predicting stationary behavior for both the inflation and output gap series, irrespective of whether the policy stance induces determinacy or indeterminacy. I show this property survives the adoption of forecast-based instrument rules, and also explore the relationship between Taylor-type rules and optimal discretionary policies in this particular New Keynesian environment.

Introduction

Since Taylor (1993), simple reactive rules stipulating how changing measures of macroeconomic activity feed back into interest rate policy have provided a useful framework for the analysis of monetary policy, from both a positive and a normative perspective. While able to capture historical patterns of policy behavior, especially in the U.S., Taylor rules have also been advocated as a functional guidance in policy decisions, on account of their empirically documented good stabilization performance vis-à-vis alternative proposals (e.g. Taylor, 1999).

The adoption of rule-based interest rate policies embeds a stabilization logic insofar as they conform to the so-called Taylor principle, i.e. when they respond more than one-for-one with inflation. Intuitively, when inflation is expected to rise, abiding by the Taylor principle forces real interest rates to surge, so as to depress aggregate demand and generate disinflationary pressure. When framed in the context of simple general equilibrium models of the monetary business cycle, the Taylor principle has also been shown to impart uniqueness of the dynamically stable rational expectations (RE) equilibrium, i.e. a collection of bounded equilibrium responses of endogenous variables to fundamental macroeconomic disturbances that all converge back to the model’s non-stochastic steady state (e.g. McCallum, 1981, Clarida et al., 2000). This equilibrium is said to be determinate, for it delivers a unique value for each of the endogenous variables of the system conditional on the structure of the economy and the realizations of the fundamental shocks hitting it (e.g. Benhabib and Farmer, 1999, Lubik and Schorfheide, 2003, Lubik and Schorfheide, 2004, Farmer et al., 2015).

How monetary authorities react to such shocks based on their actual (or perceived) persistence is of course key to evaluating the effectiveness of monetary policy against its stabilization goals. Recent evidence suggests that supply disturbances are a key driver of inflation and output dynamics, and that their conventional measures tend to exhibit near unit root behavior. Permanent supply shocks have been documented to largely contribute to variance and autocorrelation patterns for output growth in major European countries and the US (e.g. Hartley and Whitt Jr, 2003). Inspection of high frequency data from the crude oil market reveals that unanticipated exogenous changes in the oil supply are best approximated by a random walk process (e.g. Maslyuk and Smyth, 2008).

System-based inference on the persistence properties of identified supply shocks in structural models of the US economy is, by contrast, somewhat inconclusive. For example, Del Negro and Schorfheide (2008) report evidence of mildly persistent cost-push shocks, whereas both Lubik and Schorfheide (2004) and Benati and Surico (2009) offer maximum likelihood estimates of the persistence parameter ranging from 0.418 to 0.85. Other studies conducted within a Bayesian framework rather point to highly persistent (close to random walk) processes for supply shocks: using data from the postwar United States and different model specifications, Ireland (2004) estimates the persistence of the autoregressive cost-push shock process to be 0.9907 over the post-1980 sample (the Great Moderation period), whereas Smets and Wouters (2007) report a posterior mode for the autocorrelation coefficient of the identified wage markup process of 0.97.¹

In light of such mixed evidence on the time series properties of supply shocks, and given their role in shaping fundamental policy trade-offs in New Keynesian (NK) environments (e.g. Galí, 2008), the question of how monetary authorities should design policies in response to the destabilizing effects on macroeconomic volatility of non-stationary supply-side disturbances is a sensible one. The main contribution of the present study is to re-expose and further explore the stabilizing and determinacy properties of Taylor rules in the prototypical NK model, with an explicit focus on unit root behavior for the supply (here, cost-push) shock process. Specifically, I analyze what NK theory has to say about the ability of rule-based interest rate policies to insulate the economy from a non-stationary cost-push shock as well as from non-fundamental (sunspot) noise. I show that Taylor-type rules fail to prevent emergence of stationary equilibrium representations for the endogenous variables, no matter whether the policy stance on inflation stabilization induces determinacy or indeterminacy, or whether a forward-looking rather than contemporaneous rule specification is adopted. What is more, a Taylor-type rule that is unresponsive to output gap fluctuations and yet satisfies the Taylor principle is unable to replicate the optimal discretionary policy targeting micro-founded objective functions, i.e. those encompassing welfare-relevant measures of economic activity other than inflation. These core findings qualify previous results in the literature, while also extending them along several dimensions, all of which appear to be relevant for the economic insights of the model and its policy implications.

Yao (2014) is the first to explore the consequences of permanent cost-push shocks for inflation and output gap stabilization, when a Taylor-type instrument rule describes monetary policy behavior. He establishes that (i) stabilizing inflation requires the Taylor rule not to depend on the contemporaneous output-gap measure, no matter whether the model features backward dependence (inertia) or not; and that (ii) the Taylor principle suffices to warrant a determinate RE equilibrium, in the sense of being the unique dynamically stable solution to the NK model.

I first argue that Yao (2014)’s findings need a qualification. While the Taylor principle enforces a uniquely determined stable path for inflation as a function of fundamental shocks, stabilizing inflation requires that the output gap follow a non-stationary process. In fact, the model under scrutiny fails to admit a (stable) saddle-path representation. As a result, invoking a zero response of the nominal interest rate to changes in the output gap necessarily forces a unit root in the output gap dynamics. More specifically, I establish that a muted response of the interest-rate rule to output gap fluctuations and fulfillment of the Taylor principle jointly manage to stabilize inflation dynamics, and yet produce unbounded growth in the unconditional variance of the output gap series (Proposition 1).

I then go a step further and show that interest-rate feedback rules that violate the Taylor principle (and thus open room to equilibrium indeterminacy) cannot safeguard against time-varying volatility originating from non-stationary shocks, no matter whether output gap fluctuations are targeted or not; that is to say, even when considering model’s parameter configurations that support existence of multiple (indeterminate) equilibria, there exists no equilibrium in which both inflation and output gap exhibit stationary behavior (Proposition 2). Being grounded in the theory of linear stochastic recursions with martingale difference errors (e.g. Broze and Szafarz, 1991), this finding holds true irrespective of the method used to represent and/or estimate the full set of equilibria under indeterminacy (e.g. Lubik and Schorfheide, 2003, Fanelli, 2012, Farmer et al., 2015, Bianchi and Nicolò, 2017).

As mentioned, I also investigate whether these insights extend to NK environments where the interest-rate feedback rule reacts to forecasts of future rather than contemporaneous output and inflation deviations, and thus is forward-looking in nature (e.g. Bullard and Mitra, 2002); and explore the relationship between Taylor rules that fully stabilize inflation in the presence of unit root supply shocks and optimal policies derived under discretion when monetary policy objectives reflect society’s welfare losses. I find that (i) similarly to the contemporaneous data specification, forecast-based rules are unable to impart stationary dynamics for equilibrium inflation and output gap in the presence of a permanent supply shock process; and that (ii) the optimal discretionary policy targeting a welfare-theoretic loss function cannot be implemented via a Taylor-type rule that aggressively targets the inflation rate and stipulates no response to fluctuations in real economic activity.

The remainder of the paper is as follows. Section 2 lays out the model, while Section 3 is devoted to the analysis of the model’s equilibrium. Section 4 discusses two variants of the benchmark model under investigation. Section 5 offers concluding remarks.