Elsevier

Review of Economic Dynamics

Volume 41, July 2021, Pages 121-173
Review of Economic Dynamics

A toolkit for solving models with a lower bound on interest rates of stochastic duration

https://doi.org/10.1016/j.red.2021.04.001Get rights and content

Abstract

This paper presents a toolkit to solve for equilibrium in a computationally efficient way in economies facing the effective lower bound on the nominal interest rate under a special assumption about the underlying shock process, a two-state Markov process with an absorbing state. We illustrate the algorithm in the canonical New Keynesian model by replicating the optimal monetary policy in Eggertsson and Woodford (2003), and we show how the toolkit can be used to analyse the medium-scale dynamic stochastic general equilibrium model developed by the Federal Reserve Bank of New York. As an application, we show how various policy rules perform relative to the optimal commitment equilibrium. A key conclusion is that previously suggested strategies – such as price level targeting and nominal GDP targeting – do not perform well when there is a small drop in the price level, as observed during the Great Recession, because they do not imply a sufficiently strong commitment to low future interest rates (“make-up strategy”). We propose two new policy rules – the cumulative nominal GDP targeting rule and the symmetric dual-objective targeting rule – that are more robust. Had these policies been in place in 2008, they would have reduced the output contraction by approximately 80 percent. If the Federal Reserve had followed average inflation targeting – which arguably approximates the policy framework announced in August 2020 – the output contraction would have been roughly 25 percent smaller.

Introduction

The effective lower bound (ELB) on nominal interest rates has been widely studied in recent years. The standard way to analyse this problem is with dynamic stochastic general equilibrium (DSGE) models, in which the ELB is an inequality constraint on the nominal interest rate. However, inequality constraints complicate the application of standard solution strategies – for example perturbation methods. These methods approximate the behaviour of a dynamic nonlinear model around a point (usually via linearisation) using differentiability assumptions. Occasionally binding constraints pose a challenge for direct application of these methods.

In this paper, we present a toolkit aiming to facilitate the application of a generalised version of the solution method first used by Eggertsson and Woodford (2003), who analyse the ELB in the context of a two-state Markov process for the exogenous shocks with an absorbing state.1 We illustrate the algorithm in the canonical New Keynesian (NK) model and in the medium-scale DSGE model developed by the Federal Reserve Bank of New York (FRBNY). As an application, we consider various policy rules and study their performance relative to the optimal commitment equilibrium. Previously suggested policy rules – such as price level targeting and nominal GDP targeting – do not perform well when the price level does not fall by a large amount, as observed during the Great Recession, because they do not entail sufficiently strong commitment to a low future interest rate (that is, an adequate make-up strategy). This also applies to a policy rule we term average inflation targeting, which arguably approximates the policy regime of the Federal Reserve as presented by Powell (2020). To address this shortcoming, we propose two new policy rules – what we call cumulative nominal GDP targeting and symmetric dual-objective targeting rule – that are more robust. Had either of these policies been in place in 2008 and believed to be credible, our model simulation suggests the Federal Reserve would have reduced the output contraction (relative to trend) by about 80-90 percent. The comparable number for the average inflation targeting rule is 25 percent (Table 3).

Several strategies have been proposed to deal with inequality constraints in DSGE models. Eggertsson and Woodford (2003) exploit a particular structure for exogenous disturbances: the posited shock process implies that the economy unexpectedly moves to a crisis state and then reverts back to the steady state with a fixed probability. Once back to the steady state, it stays there forever (that is, the steady state is an absorbing state). The idea behind the approach is intuitive: instead of treating a single dynamical system that contains both a set of equality constraints and a set of occasionally binding inequality constraints, we split the system into several parts we call regimes, each of which contains equality constraints exclusively. Once the system is cast in this form, we can apply perturbation methods since each equation is differentiable.

An application to the ELB scenario should make the point clear. We distinguish among four regimes, each of them corresponding to a different combination of the status of the inequality constraint (for example, ELB binding or not) and the exogenous Markov disturbance (crisis or steady state). For the regimes in which the ELB is not binding, we treat the model as if the ELB was not present. In the other two regimes, when the ELB constraint is binding, the equilibrium conditions are characterised by an equality constraint (for example, it=it1=0). Since all four dynamical systems are described by a set of equations, each can be solved using perturbation techniques.

The assumptions about the shock structure allow us to solve the model recursively in regimes. Starting from the last regime, in which the ELB is not relevant, we work backwards to the period when the shock hits the system and we obtain a piecewise solution. Since outcomes in later regimes influence behaviour in earlier ones through expectations, the strategy is not based on a simple merger of separate models.

Our approach has two key advantages: first, its relative simplicity allows us to handle models with many state variables; second, unlike local-solution techniques, our strategy allows us to assume the basic stochastic structure, making it attractive for simple estimations.2 Note that it is not just the interest-rate constraint, but any model with a constraint that is temporarily binding can be solved using our toolkit.

As a new feature, the toolkit uses an algorithm that generalises the solution method in Eggertsson and Woodford (2003). In particular, it allows for the case of a regime in which the two-state Markov process is in the crisis state but the ELB is not binding. This feature is of particular importance when we analyse policy rules, as a common property of policy rules is that they imply an inertial response of the interest rate. An example is a Taylor-type rule with lagged terms for the nominal interest rates. Rules of this kind often do not imply an immediate reduction of the interest rate to the ELB once the two-state Markov disturbance switches to the crisis state. The new feature is thus a meaningful addition and facilitates the analysis of different types of policy rules in the presence of an ELB, which is the main application in this paper.

The idea of attacking the problem by constructing a piecewise solution is not new, nor is the idea of a toolkit for applying the solution. In fact, Guerrieri and Iacoviello (2015), henceforth OccBin, provide a toolkit for solving dynamic models with occasionally binding constraints in a similar fashion. The main difference is that we do not assume perfect foresight – that is a deterministic setting. This feature also differentiates our approach from several other strategies, such as the extended path (EP) algorithm. To achieve this, we rely on the specific shock structure implied by a two-state Markov process with an absorbing state. Expectations about the future path of variables are a crucial component of models related to the ELB (for example, uncertainty about whether the economy will hit the ELB and uncertain timing of lift-off), and hence allowing for uncertainty is a useful feature of the toolkit.

Adding a two-state Markov process with an absorbing state usually implies the following timing for the models analysed with the toolkit: initially, a shock hits the economy and the response of the central bank might be to immediately lower the interest rate to zero. In every period the disturbance reverts to its initial absorbing condition with some probability. There is often a transition period, lasting from the point when the shock reverts to its initial level until all other variables of the model return to their steady-state values. One benefit of our setup is that one can separately calibrate the expected time during which the constraint is binding from the actual realised time. Empirical evidence on the Great Recession, for instance the Blue Chip financial forecasts (Aspen Publishers 2008-12), suggests that market participants were expecting the ELB to be binding for a much shorter time than turned out to be the case. We account for this evidence, and we analyse several questions related to it, such as what the output gains would have been had the Federal Reserve adopted alternative policy regimes to that in place during the financial crisis of 2008.

The expected duration of the ELB episode is not necessarily exogenously determined simply by the transition probability of the shock; in the case of a central bank that has commitment power, the duration of the binding ELB will typically be longer than the persistence of the disturbance in its crisis state. The periods in which the inequality constraint is binding therefore do not coincide with the periods in which the shock is in the low state. This means that the duration at the ELB is endogenously determined in the model, as it depends on the optimal decisions taken by the monetary authority, which are a function of, among other things, the realisation of the shock. This is a key challenge in solving the model and is discussed in detail in the paper.

Our main application is monetary policy when the interest rate reaches the ELB. Since the standard policy tool of affecting nominal interest rates is not available at that point, influencing expectations about their future path becomes the main lever through which the monetary authority can affect present variables. In this environment, policy rules that are able to mimic some form of commitment from the central bank are believed to perform relatively well. For example, Eggertsson and Woodford (2003), who predict strong deflation, argue that rules that commit to bringing the price level back to pre-crisis levels and to inflate in the future are very effective. A key finding is that price level targeting and nominal GDP targeting do not do well if there is little fall in inflation, as was the case during the financial crisis of 2008 in the United States. The policy of price level targeting we consider is arguably equivalent to the policy of average inflation targeting recently adopted by the Federal Reserve when it amended its policy framework in August 2020, if the average is taken over a sufficiently long period. We also consider an average inflation targeting regime for which the average time period is shorter. This policy provides even less stimulus at the ELB.

In addition, we discuss our two proposed rules – the cumulative nominal GDP targeting rule and the symmetric dual-objective targeting rule – that imply a commitment from the central bank to make up for past deviations from the target price level and output. We study the two rules' performance in the standard NK model and in the NYFRB DSGE model and show that they generally perform better than the standard rules in the literature. We show this in an environment with low inflation and small movement in the price level, as experienced during the Great Recession. Since both rules imply an aggressive reaction to past output misses, they communicate that the longer the crisis is, the more accommodative monetary policy will be. This in turn generates enough stimulus to prevent a large recession in the first place.

Among previously proposed policy rules, the ones that perform best are the superinertial rule described in Rotemberg and Woodford (1999) and the augmented Taylor rule detailed by Reifschneider and Williams (2000). Policy rules that do not perform as well include price level targeting, nominal GDP targeting and average inflation targeting, a result in line with Reifschneider and Wilcox (2019). The key problem with these rules is that they do not prescribe strong-enough stimulus when inflation is not falling.

The most important advantage of the stochastic structure our toolkit embodies is that it allows for a clear distinction between the expected duration of the shock and the realised duration of it, though the two of course coincide under perfect foresight. This allows us to clearly show the advantage of policy commitments, such as those exemplified by our targeting rules, relative to optimal time-dependent policy – a strategy that resembles the interventions of several central banks during the crisis of 2008. Under optimal time-dependence, the duration of the ELB is tied to calendar time. In contrast, the targeting rule we consider implies a duration at the ELB that depends upon economic conditions. We highlight that a properly chosen state-contingent policy rule vastly outperforms optimal time-dependent policy, a distinction that is not as transparent in a deterministic setting.

In an additional application, we utilise our toolkit to contribute to a recent debate on the economic effects of forward guidance policy. We distinguish two cases: in the standard theory, forward guidance creates additional stimulus by a fully credible announcement that the central bank will keep the interest rate at the ELB for additional periods; this is an expansionary policy. The second case is what Campbell et al. (2012) call Delphic forward guidance and Nakamura and Steinsson (2018) refer to as information effects. Here, the expected duration of the ELB episode lengthens as well, but this time solely because the revelation of information leads agents to update their beliefs about economic fundamentals, which entails a contraction. We show that both scenarios can match the same increase in the expected duration at the ELB but that they lead to vastly different outcomes.

The paper is structured as follows: section 2 outlines how the solution method relates to the literature; section 3 presents the solution algorithm; section 4 provides a few applications in the context previously defined; section 5 applies our toolkit to the medium-scale FRBNY DSGE model; section 6 concludes.

Section snippets

Related literature

There exists a sizeable literature on solution methods for DSGE models.3 Solution strategies can be classified into local and global. The former include perturbation methods, while the latter includes projection methods. Projection methods can handle occasionally binding constraints in a direct way, but they are associated with a considerable computational burden and suffer from the curse of

Basic idea

In this section we outline our approach of applying perturbation methods to models with inequality constraints.7 Technically, the use of the implicit function theorem (IFT), on which perturbation methods rely, requires that the function approximated

Applications

As an application, we revisit Eggertsson and Woodford (2003), henceforth EW2003, and analyse the optimal monetary policy at the ELB in the standard NK model. We then ask what kind of policy rule can implement it. Our key finding is that EW2003 suggests a simplified price level targeting rule, which performs poorly in replicating the optimal commitment policy in numerical experiments in which the price level does not drop much at the ELB. We consider this scenario because of its similarity to

Medium-scale DSGE model

The results discussed so far were derived using a simple two-equation NK model. In this section we show that the findings generalise to a medium-scale DSGE model and are not an artefact of the simple structure of the baseline exercise. Rules that imply substantial make-up behaviour or feature inertia in the path of interest rates have the best outcome. Our two proposed policies, HD-NGDPT and SDTR, outperform all alternative candidates by implying substantial stimulus compared to a simple Taylor

Conclusions

We provide a toolkit to solve DSGE models that involve occasionally binding constraints. The solution method generalises that of Eggertsson and Woodford (2003) and exploits the properties of a two-state Markov process for the exogenous disturbances. The toolkit performs well even in the presence of a large number of state variables and features a tractable stochastic structure. This modelling assumption is particularly relevant in analysing macroeconomic problems in which uncertainty is

Acknowledgements

This research was partly conducted when Luca Riva was visiting Danmarks Nationalbank; its financial support and kind hospitality are gratefully acknowledged.

This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.

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