Rational Expectations Under Knightian Uncertainty — Part 2
How to formalize Knightian uncertainty by opening economic models to unforeseeable structural change
This is the second post in my series that explains the main components of Roman Frydman’s and my new theoretical approach to opening macroeconomic models to unforeseeable structural change and representing rational expectations in the presence of the Knightian uncertainty arising from such change.
We present this approach, which we call the Knight-Muth hypothesis (KMH), in the paper titled Rational Expectations of Inflation Undergoing Unforeseeable Change.
In the first post, I explained that KMH is based on the observation that the economy’s structure changes at times and in ways that cannot be understood in advance. Even if such changes, which we refer to as unforeseeable, occur only rarely, they imply that we cannot have perfect probabilistic knowledge of the future and that we face Knightian uncertainty.
We argue that economists must formalize this in economic models to adequately represent the expectations of rational individuals in real-world markets. That is our motivation for developing KMH.
Opening economic models to unforeseeable change and Knightian uncertainty, however, poses a challenge: how can we characterize economic outcomes with a stochastic process, which enables empirical estimation and testing, and yet acknowledge that future outcomes cannot be reduced to a probability distribution ex ante?
In this post, I will explain how KMH provides a simple answer to this challenge that is tractable and empirically testable. See also this post.
The New Keynesian Phillips curve (NKPC)
In the paper, we present KMH in the context of a simple New Keynesian Phillips curve (NKPC) that specifies inflation, 𝜋ₜ, as a function of an exogenous output gap, 𝑥ₜ, and a representative agent’s discounted expectation of the next period’s inflation, 𝐹ₜ(𝜋ₜ₊₁). Mathematically, this is given by:
for t=1,2,…,T and where 0<𝛽<1.
For simplicity, we assume that the exogenous output gap can be represented in terms of a stationary first-order autoregressive process with constant parameters and mean zero:
where |𝜌|<1.
Our focus is on structural change in the parameter 𝜅ₜ, which captures the time-varying pass-through of the output gap to inflation.
Our Formalization of Structural Change
KMH provides a tractable and empirically testable formalization of unforeseeable change in the pass-through parameter by assuming that
inflation can be characterized with a stochastic process that undergoes nonrepetitive structural changes only intermittently, and that the periods between these changes are stable enough to be modeled as constant.
This can be decomposed into three distinct assumptions to explain their relevance separately.
The three assumptions are:
Assumption 1: The structural change only occurs intermittently, at times 𝜏ᵢ for i=1,2,…, so between these shifts, the pass-through parameter is constant.
Assumption 2: The structural change is nonrepetitive, so 𝜅ₜ shifts between nonrepetitive regimes with parameter values 𝜅⁽ⁱ⁾ for i=1,2,….
Assumption 3: The value of the pass-through parameter in each regime, 𝜅⁽ⁱ⁾ for i=1,2,…, lies within a positive, constant interval.
Based on these three assumptions, we formalize the structural change in the pass-through parameter as:
and so on, where
This formalizes the pass-through parameter to take the value 𝜅⁽¹⁾ during the first period, the value 𝜅⁽²⁾ during the second period, etc. Moreover, in each of these regimes, the parameter lies within a positive, constant interval.
Next, let’s unwrap how each of the three assumptions renders this formalization of structural change empirically testable and yet unforeseeable.
Assumption 1 → Empirically Testable
The first assumption that structural change occurs only intermittently enables empirical estimation of the pass-through parameter based on a historical sample of time-series data.
As we write in the paper, this can be done by estimating the NKPC using survey forecasts to measure the market’s inflation expectations or by estimating the reduced-form expression for inflation that we derive later. That enables empirically testing the model’s formalization of structural change in the NKPC.
It requires, however, a statistical method that identifies the unknown timings of shifts in the pass-through parameter from the time-series data.
I can add here that, relying on different statistical methods, a vast empirical literature has indeed found that the relation between inflation and the output gap (or unemployment), as captured by the pass-through parameter, has changed over time.
For example, the very recent paper by Atsushi Inoue, Barbara Rossi, and Yiru Wang (accepted for publication in Econometric Theory) finds that:
the slope of the Phillips curve weakened since the early 1980s: the slope decreased, in absolute value, by 68 percent in the last two decades. However, we also find that it started reverting back in the most recent pandemic period. We also find that the decrease in the correlation between unemployment and inflation cannot be attributed to monetary policy; rather, to the decrease in the slope of the Phillips curve. (Inoue, Rossi, and Wang, 2025, pp. 2-3)
Assumption 2 → Change is Unforeseeable
The second assumption, that the change is nonrepetitive, is the crucial assumption that renders the structural change unforeseeable and gives rise to Knightian uncertainty.
As we write:
Although the values of 𝜅ₜ in past regimes can be empirically estimated, the formalization of the structural change as nonrepetitive renders future structural changes in the inflation process unforeseeable. Because the values of the objective parameter 𝜅ₜ in future regimes differ from those in past regimes, those future values cannot be estimated based on historical time-series data. Thus, at any time t, there can be no objective assessment of the values of 𝜅ₜ₊ₕ for h>0, which cannot even be characterized probabilistically. Such unforeseeable change gives rise to Knightian uncertainty about future inflation, in addition to the probabilistic risk represented in the model by future shocks to the output gap.
Let’s unwrap this.
First, the formalization of structural change as nonrepetitive renders it unforeseeable.
What we mean by this is that, at any time t, the future values of the pass-through parameter cannot be objectively assessed. We can estimate the past pass-through parameters from the past data. But there is no way we can learn the value of the future parameters today, as they differ from the past ones.
So we simply cannot know today what the future pass-through parameter will be. Tomorrow’s pass-through parameter might be identical to today’s. But it might also shift to a new value that we have not yet observed. There is no way we can assess today when the changes will occur in the future, and there is no way we can assess today the magnitude of these changes. And this is the case even if we knew exactly what the pass-through parameter is today and what it has been in the past.
The important implication of this is that, although we specify inflation to unfold according to a stochastic process (which also depends on our representation of inflation expectations), we have to acknowledge that the future pass-through parameters are always inherently unknowable.
Consequently, by opening the model to unforeseeable change, we acknowledge that our knowledge of how inflation will unfold in the future is inherently imperfect—we simply cannot have perfect probabilistic foresight.
Note that the imperfection of our knowledge arises from the possibility that an unforeseeable change occurs. Thus, our knowledge of the future is inherently imperfect even if these changes occur only very rarely.
Second, the unforeseeable change gives rise to Knightian uncertainty about future inflation.
As in other economic models, the random shocks to the output gap represents standard probabilistic risk.
The unforeseeable change, however, adds an additional layer of uncertainty. This arises from the possibility that the pass-through parameter shifts in the future. Because these shifts are unforeseeable, the defining feature of this additional layer of uncertainty is that it cannot be reduced to a probability distribution. Thereby, KMH provides a tractable formalization of Knight’s (1921) argument that “true” uncertainty—nowadays referred to as Knightian uncertainty—arises from unforeseeable structural change.
Within the model, the crucial implication of unforeseeable change is that the conditional expectation of future inflation is inherently unknowable ex ante. Sure, the model specifies a conditional expectation of future inflation. But this depends on the future pass-through parameters, which are inherently unknowable ex ante. Thus, viewed from any point in time, the model’s conditional expectation of future inflation is inherently unknowable.
As a side comment, it is worth mentioning that representing structural change as nonrepetitive is not sufficient to render it unforeseeable. In other words, nonrepetitive change is not necessarily unforeseeable. You could, for example, assume that the pass-through parameters in the nonrepetitive regimes are random draws from a fixed distribution with constant parameters. Although the change would still be nonrepetitive (you continue to shift to a new regime without going back to past regimes), it would not be unforeseeable: the constant parameters of the regime parameters’ fixed distribution could be learned from historical data, such that the future pass-through parameters could be characterized probabilistically ex ante. This illustrates that the defining feature of unforeseeable change is that it is formalized in terms of different parameters during different periods. Our formalization of nonrepetitive change here ensures that.
It is also worth mentioning that it is not necessary to specify the nonrepetitive change as deterministic, as we do, to render it unforeseeable. As we discuss in the paper, you could, for example, formalize the pass-through parameter to shift probabilistically according to a Markov process with nonrepetitive regimes. In that case, the parameters in future regimes would still be inherently unknowable ex ante, and the change would be unforeseeable.
Assumption 3 → Bounds on Knightian Uncertainty
The third assumption—that the pass-through parameter lies within a positive interval during each of the nonrepetitive regimes—imposes a regularity over time: although the pass-through parameter shifts over time, we assume that it is positive in all the regimes.
This bounds the Knightian uncertainty in the model. Although the future pass-through parameters are inherently unknowable in advance, they lie within the interval. Consequently, a set of conditional distributions characterizes the Knightian uncertainty about future inflation.
Thus, KMH’s formalization of Knightian uncertainty corresponds to unresolvable ambiguity. As we write in the paper, it
corresponds explicitly to the specific type of ambiguity that Epstein and Schneider (2007, p. 2726) characterize as “not vanish[ing] in the long run, since the time-varying features remain impossible to know even after many observations.
As I will discuss in the next post, this is crucial for how we represent rational expectations in the presence of Knightian uncertainty.
References
Epstein and Schneider (2007), “Learning Under Ambiguity,” Review of Economic Studies, 74. Link.
Knight (1921), Risk, Uncertainty, and Profit, Houghton Mifflin.
Inoue, Rossi, and Wang (2025), “Has the Phillips Curve Flattened and Why?,” working paper. Link.