I asked my colleagues George Evans and Bruce McGough if they would like to respond to a recent post by Simon Wren-Lewis, "Woodford’s reflexive equilibrium" approach to learning:

**The neo-Fisherian view and the macro learning approach **

** George W. Evans and Bruce McGough **

** Economics Department, University of Oregon **

** December 30, 2015**

Cochrane (2015) argues that low interest rates are deflationary — a view that is sometimes called neo-Fisherian. In this paper John Cochrane argues that raising the interest rate and pegging it at a higher level will raise the inflation rate in accordance with the Fisher equation, and works through the details of this in a New Keynesian model.

Garcia-Schmidt and Woodford (2015) argue that the neo-Fisherian claim is incorrect and that low interest rates are both expansionary and inflationary. In making this argument Mariana Garcia-Schmidt and Michael Woodford use an approach that has a lot of common ground with the macro learning literature, which focuses on how economic agents might come to form expectations, and in particular whether coordination on a particular rational expectations equilibrium (REE) is plausible. This literature examines the stability of an REE under learning and has found that interest-rate pegs of the type discussed by Cochrane lead to REE that are not stable under learning. Garcia-Schmidt and Woodford (2015) obtain an analogous instability result using a new bounded-rationality approach that provides specific predictions for monetary policy. There are novel methodological and policy results in the Garcia-Schmidt and Woodford (2015) paper. However, we will here focus on the common ground with other papers in the learning literature that also argue against the neo-Fisherian claim.

The macro learning literature posits that agents start with boundedly rational expectations e.g. based on possibly non-RE forecasting rules. These expectations are incorporated into a “temporary equilibrium” (TE) environment that yields the model’s endogenous outcomes. The TE environment has two essential components: a decision-theoretic framework which specifies the decisions made by agents (households, firms etc.) given their states (values of exogenous and pre-determined endogenous state variables) and expectations;^{1} and a market-clearing framework that coordinates the agents’ decisions and determines the values of the model’s endogenous variables. It is useful to observe that, taken together, the two components of the TE environment yield the “TE-map” that takes expectations and (aggregate and idiosyncratic) states to outcomes.

The adaptive learning framework, which is the most popular formulation of learning in macro, proceeds recursively. Agents revise their forecast rules in light of the data realized in the previous period, e.g. by updating their forecast rules econometrically. The exogenous shocks are then realized, expectations are formed, and a new temporary equilibrium results. The equilibrium path under learning is defined recursively. One can then study whether the economy under adaptive learning converges over time to the REE of interest.^{2}

The essential point of the learning literature is that an REE, to be credible, needs an explanation for how economic agents come to coordinate on it. This point is acute in models in which there are multiple RE solutions, as can arise in a wide range of dynamic macro models. This has been an issue in particular in the New Keynesian model, but it also arises, for example, in overlapping generations models and in RBC models with distortions. The macro learning literature provides a theory for how agents might learn over time to forecast rationally, i.e. to come to have RE (rational expectations). The adaptive learning approach found that agents will over time come to have rational expectations (RE) by updating their econometric forecasting models provided the REE satisfies “expectational stability” (E-stability) conditions. If these conditions are not satisfied then convergence to the REE will not occur and hence it is implausible that agents would be able to coordinate on the REE. E-stability then also acts as a selection device in cases in which there are multiple REE.

The adaptive learning approach has the attractive feature that the degree of rationality of the agents is natural: though agents are boundedly rational, they are still fairly sophisticated, estimating and updating their forecasting models using statistical learning schemes. For a wide range of models this gives plausible results. For example, in the basic Muth cobweb model, the REE is learnable if supply and demand have their usual slopes; however, the REE, though still unique, is not learnable if the demand curve is upward sloping and steeper than the supply curve. In an overlapping generations model, Lucas (1986) used an adaptive learning scheme to show that though the overlapping generations model of money has multiple REE, learning dynamics converge to the monetary steady state, not to the autarky solution. Early analytical adaptive learning results were obtained in Bray and Savin (1986) and the formal framework was greatly extended in Marcet and Sargent (1989). The book by Evans and Honkapohja (2001) develops the E-stability principle and includes many applications. Many more applications of adaptive learning have been published over the last fifteen years.

There are other approaches to learning in macro that have a related theoretical motivation, e.g. the “eductive” approach of Guesnerie asks whether mental reasoning by hyper-rational agents, with common knowledge of the structure and of the rationality of other agents, will lead to coordination on an REE. A fair amount is known about the connections between the stability conditions of the alternative adaptive and eductive learning approaches.^{3} The Garcia-Schmidt and Woodford (2015) “reflective equilibrium” concept provides a new approach that draws on both the adaptive and eductive strands as well as on the “calculation equilibrium” learning model of Evans and Ramey (1992, 1995, 1998). These connections are outlined in Section 2 of Garcia-Schmidt and Woodford (2015).^{4}

The key insight of these various learning approaches is that one cannot simply take RE (which in the nonstochastic case reduces to PF, i.e. perfect foresight) as given. An REE is an equilibrium that begs an explanation for how it can be attained. The various learning approaches rely on a temporary equilibrium framework, outlined above, which goes back to Hicks (1946). A big advantage of the TE framework, when developed at the agent level and aggregated, is that in conjunction with the learning model an explicit causal story can be developed for how the economy evolves over time.

The lack of a TE or learning framework in Cochrane (2011, 2015) is a critical omission. Cochrane (2009) criticized the Taylor principle in NK models as requiring implausible assumptions on what the Fed would do to enforce its desired equilibrium path; however, this view simply reflects the lack of a learning perspective. McCallum (2009) argued that for a monetary rule satisfying the Taylor principle the usual RE solution used by NK modelers is stable under adaptive learning, while the non-fundamental solution bubble solution is not. Cochrane (2009, 2011) claimed that these results hinged on the observability of shocks. In our paper “Observability and Equilibrium Selection,” Evans and McGough (2015b), we develop the theory of adaptive learning when fundamental shocks are unobservable, and then, as a central application, we consider the flexible-price NK model used by Cochrane and McCallum in their debate. We carefully develop this application using an agent-level temporary equilibrium approach and closing the model under adaptive learning. We find that if the Taylor principle is satisfied, then the usual solution is robustly stable under learning, while the non-fundamental price-level bubble solution is not. Adaptive learning thus operates as a selection criterion and it singles out the usual RE solution adopted by proponents of the NK model. Furthermore, when monetary policy does not obey the Taylor principle then neither of the solutions is robustly stable under learning; an interest-rate peg is an extreme form of such a policy, and the adaptive learning perspective cautions that this will lead to instability. We discuss this further below.

The agent-level/adaptive learning approach used in Evans and McGough (2015b) allows us to specifically address several points raised by Cochrane. He is concerned that there is no causal mechanism that pins down prices. The TE map provides this, in the usual way, through market clearing given expectations of future variables. Cochrane also states that the lack of a mechanism means that the NK paradigm requires that the policymakers be interpreted as threatening to “blow up” the economy if the standard solution is not selected by agents.5 This is not the case. As we say in our paper (p. 24-5), “inflation is determined in temporary equilibrium, based on expectations that are revised over time in response to observed data. Threats by the Fed are neither made nor needed ... [agents simply] make forecasts the same way that time-series econometricians typically forecast: by estimating least-squares projections of the variables being forecasted on the relevant observables.”

Let us now return to the issue of interest rate pegs and the impact of changing the level of an interest rate peg. The central adaptive learning result is that interest rate pegs give REE that are unstable under learning. This result was first given in Howitt (1992). A complementary result was given in Evans and Honkapohja (2003) for time-varying interest rate pegs designed to optimally respond to fundamental shocks. As discussed above, Evans and McGough (2015b) show that the instability result also obtains when the fundamental shocks are not observable and the Taylor principle is not satisfied. The economic intuition in the NK model is very strong and is essentially as follows. Suppose we are at an REE (or PFE) at a fixed interest rate and with expected inflation at the level dictated by the Fisher equation. Suppose that there is a small increase in expected inflation. With a fixed nominal interest rate this leads to a lower real interest rate, which increases aggregate demand and output. This in turn leads to higher inflation, which under adaptive learning leads to higher expected inflation, destabilizing the system. (The details of the evolution of expectations and the model dynamics depend, of course, on the precise decision rules and econometric forecasting model used by agents). In an analogous way, expected inflation slightly lower than the REE/PFE level leads to cumulatively lower levels of inflation, output and expected inflation.

Returning to the NK model, additional insight is obtained by considering a nonlinear NK model with a global Taylor rule that leads to two steady states. This model was studied by Benhabib, Schmidt-Grohe and Uribe in a series of papers, e.g. Benhabib, Schmitt-Grohe, and Uribe (2001), which show that with an interest-rate rule following the Taylor principle at the target inflation rate, the zero-lower bound (ZLB) on interest rates implies the existence of an unintended PFE low inflation or deflation steady state (and indeed a continuum of PFE paths to it) at which the Taylor principle does not hold (a special case of which is a local interest rate peg at the ZLB). From a PF/RE viewpoint these are all valid solutions. From the adaptive learning perspective, however, they differ in terms of stability. Evans, Guse, and Honkapohja (2008) and Benhabib, Evans, and Honkapohja (2014) show that the targeted steady state is locally stable under learning with a large basin of attraction, while the unintended low inflation/deflation steady state is not locally stable under learning: small deviations from it lead either back to the targeted steady state or into a deflation trap, in which inflation and output fall over time. From a learning viewpoint this deflation trap should be a major concern for policy.^{6,7}

Finally, let us return to Cochrane (2015). Cochrane points out that at the ZLB peg there has been low but relatively steady (or gently declining) inflation in the US, rather than a serious deflationary spiral. This point echoes Jim Bullard’s concern in Bullard (2010) about the adaptive learning instability result: we effectively have an interest rate peg at the ZLB but we seem to have a fairly stable inflation rate, so does this indicate that the learning literature may here be on the wrong track?

This issue is addressed by Evans, Honkapohja, and Mitra (2015) (EHM2015). They first point out that from a policy viewpoint the major concern at the ZLB has not been low inflation or deflation per se. Instead it is its association with low levels of aggregate output, high levels of unemployment and a more general stagnation. However, the deflation steady state at the ZLB in the NK model has virtually the same level of aggregate output as the targeted steady state. The PFE at the ZLB interest rate peg is not a low level output equilibrium, and if we were in that equilibrium there would not be the concern that policy-makers have shown. (Temporary discount rate or credit market shocks of course can lead to recession at the ZLB but their low output effects vanish as soon as the shocks vanish).

In EHM2015 steady mild deflation is consistent with low output and stagnation at the ZLB.^{8} They note that many commentators have remarked that the behavior of the NK Phillips relation is different from standard theory at very low output levels. EHM2015 therefore imposes lower bounds on inflation and consumption, which can become relevant when agents become sufficiently pessimistic. If the inflation lower bound is below the unintended low steady state inflation rate, a third “stagnation” steady state is created at the ZLB. The stagnation steady state, like the targeted steady state is locally stable under learning, and arises under learning if output and inflation expectations are too pessimistic. A large temporary fiscal stimulus can dislodge the economy from the stagnation trap, and a smaller stimulus can be sufficient if applied earlier. Raising interest rates does not help in the stagnation state and at an early stage it can push the economy into the stagnation trap.

In summary, the learning approach argues forcefully against the neo- Fisherian view.

**Footnotes**

^{1}With infinitely-lived agents there are several natural implementations of optimizing decision rules, including short-horizon Euler-equation or shadow-price learning approaches(see, e.g., Evans and Honkapohja (2006) and Evans and McGough (2015a)) and the anticipated utility or infinte-horizon approaches of Preston (2005) and Eusepi and Preston (2010).

^{2}An additional advantage of using learning is that learning dynamics give expanded scope for fitting the data as well as explaining experimental findings.

^{3}The TE map is the basis for the map at the core of any specified learning scheme, which in turn determines the associated stability conditions.

^{4}There are also connections to both the infinite-horizon learning approach to anticipated policy developed in Evans, Honkapohja, and Mitra (2009) and the eductive stability framework in Evans, Guesnerie, and McGough (2015).

^{5}This point is repeated in Section 6.4 of Cochrane (2015): “The main point: such models presume that the Fed induces instability in an otherwise stable economy, a non-credible off-equilibrium threat to hyperinflate the economy for all but one chosen equilibrium.”

^{6}And the risk of sinking into deflation clearly has been a major concern for policymakers in the US, during and following both the 2001 recession and the 2007 - 2009 recession. It has remained a concern in Europe and Japan as well as in Japan during the 1990s.

^{7}Experimnetal work with stylized NK economies has found that entering deflation traps is a real possibility. See Hommes and Salle (2015).

^{8}See also Evans (2013) for a partial and less general version of this argument.

**References**

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_______ (2015): “Do Higher Interest Rates Raise or Lower Inflation?, "Working paper, University of Chicago Booth School of Business.

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Evans, G.W. (2013): “The Stagnation Regime of the New KeynesianModel and Recent US Policy,” in Sargent and Vilmunen (2013), chap. 4.

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_______ (2015b): “Observability and Equilibrium Selection,” mimeo, University of Oregon.

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