# A Simple Model of Monetary Policy under Phillips-Curve Causal
Disagreements††thanks: Financial support by ERC Advanced Investigator grant
no. 692995 is gratefully acknowledged. I thank In-Koo Cho, Stephane Dupraz and
Morten Ravn for helpful comments.

Ran Spiegler Tel Aviv University, University College London and CFM. URL:
http://www.tau.ac.il/~rani. E-mail<EMAIL_ADDRESS>
###### Abstract

I study a static textbook model of monetary policy and relax the conventional
assumption that the private sector has rational expectations. Instead, the
private sector forms inflation forecasts according to a misspecified
subjective model that disagrees with the central bank’s (true) model over the
causal underpinnings of the Phillips Curve. Following the AI/Statistics
literature on Bayesian Networks, I represent the private sector’s model by a
direct acyclic graph (DAG). I show that when the private sector’s model
reverses the direction of causality between inflation and output, the central
bank’s optimal policy can exhibit an attenuation effect that is sensitive to
the noisiness of the true inflation-output equations.

##  1 Introduction

Monetary policy is a classic example of an economic interaction in which
policy makers’ ability to achieve their objectives depends on the accuracy of
agents’ forecasts of certain economic variables. Consider a textbook static
model of the type originated by Kydland and Prescott (1977) and Barro and
Gordon (1983). In such a model, the central bank controls a policy variable
that affects inflation. The private sector forms an inflation forecast,
possibly after observing some signal regarding the central bank’s information
or its decision. Private-sector expectations are relevant because they affect
the joint realization of inflation and output, via some kind of an
“expectations-augmented” Phillips curve. It follows that monetary policy
involves “expectations management”. To quote Woodford (2003, p. 15):

> “…successful monetary policy is not so much a matter of effective control of
> overnight interest rates as it is of shaping market expectations of the way
> in which interest rates, inflation and income are likely to evolve…”

Conventional models constrain the central bank’s ability to manage
expectations by assuming that the private sector has “rational expectations” -
i.e., it fully understands the statistical regularities in its environment,
and thus forms an unbiased inflation forecast conditional on its information.
To put it differently, the private sector shares the central bank’s (true)
model of the macroeconomy and uses it to form its beliefs. In Evans and
Honkapohja (2005), Thomas Sargent refers to this feature as “communism of
models”. In this paper, I revisit the textbook model and relax “model
communism”, by assuming that the private sector forms its forecast according
to a different model than the central bank’s (true) model. The key difference
between the two models is in their causal treatment of the Phillips Curve.

The modern history of macroeconomic thought has included a number of forms of
Phillips Curves, which differ in their dependent and R.H.S variables. Popular
descriptions of the Phillips Curve often impose a causal interpretation on
these forms. For instance, when the dependent variable is inflation, the
causal story is about how changes in unemployment affect inflation. When
inflation expectations enter the R.H.S, they are presented as another cause of
inflation. When the dependent variable is unemployment or output, the story is
about how inflation (possibly in addition to inflationary expectations)
affects real variables. Cogley and Sargent (2005), following King and Watson
(1994), describe this as a distinction between two econometric identification
procedures, which they term “Keynesian direction of fit” and “classical
direction of fit”. Taking a broader perspective, Hoover (2001) interprets
debates in the history of monetary theory (and other areas of macroeconomics)
in terms of disagreements over causality.

Therefore, if we wish to model how the private sector forms inflation
forecasts using some kind of a Phillips Curve, it seems reasonable to assume
that it will involve a causal interpretation. Even if the true model is not
fundamentally causal (in the sense that the variables are determined in
equilibrium by some system of simultaneous equations with influences in both
directions), popular conceptions of the Phillips Curve and how it is
integrated with a broader macroeconomic model are likely to be causal in
nature.

Following Spiegler (2016,2020a), I adopt the formalism of Bayesian networks
(Cowell et al. (1999), Pearl (2009), Koller and Friedman (2009)) and represent
the private sector’s causal model with a directed acyclic graph (DAG). Nodes
in the graph represent economic variables and directed links represent
postulated causal influences. The DAG can be interpreted in at least two ways.
First, it may capture intuitive narratives about causal relations between the
relevant variables. Second, it may represent an explicit formal model of the
kind that professional forecasters sometimes employ, consisting of a recursive
system of equations. The private sector fits its subjective model to the
steady-state distribution, and then relies on this “estimated model” to
produce an inflation forecast. When the model is misspecified, the inflation
forecast systematically distorts the true expected inflation conditional on
the private sector’s information.

I incorporate this model of private-sector expectations into a variant on the
familiar static textbook model of central-bank policy. The central bank aims
to minimize the expectation of a quadratic loss function of the deviations of
output and inflation from their exogenous targets. The output target is fixed,
whereas the inflation target is random. The central bank chooses a policy
instrument in response to its inflation target, trading off the deviation of
inflation from its moving target against the deviation of output from its
fixed target. A Phillips Curve links these two deviations. Output deviations
depend on how the private sector’s inflation forecast responds to its
information. The key question is therefore how the private sector’s narrative
regarding the Phillips Curve affects the responsiveness of the private
sector’s inflation forecast, and in turn the responsiveness of the central
bank to the inflation target.

Two qualitative insights emerge from the analysis. First, the private sector’s
wrong causal model can result in rigid inflation forecasts, which in turn
leads to monetary-policy attenuation - that is, the central bank’s equilibrium
response to the inflation target is weaker than in the rational-expectations
benchmark. Second, the variance of the noise terms in the equations for
inflation and output (one of which is a Phillips Curve) can play an important
role in the rigid-forecast effect, when the private sector’s model reverses
the direction of causality between these two variables.

When macroeconomists talk about “flatness” of the Phillips relation, there is
ambiguity about whether this means a weak yet precise relation or a noisy
relation. In a finite sample, a noisy Phillips relation will appear as a
“cloud” of data points, such that the Phillips effect may end up being
statistically insignificant - hence, effectively regarded in much the same way
as a weak, precise relation. However, when the private sector has rational
expectations, the two have very different effects: the variance of the noise
terms is irrelevant for the private sector’s forecast, hence also for the
central bank’s optimal policy. This is no longer the case when the private
sector’s forecasts are based on a misspecified model. Using examples, I show
that the variance of the noise terms in the inflation-output equations has a
subtle effect on the private sector’s inflation forecast and consequently on
central-bank policy.

Related literature

The Bayesian-network approach to modeling equilibrium non-rational
expectations was presented in Spiegler (2016). In Spiegler (2020a), I
presented a simple monetary-policy example to illustrate the question of
whether causal misperceptions can lead to systematically biased expectations.
In this paper, I adhere to the standard linear-normal parameterization, such
that the private sector’s expectations are correct on average. Therefore,
systematically biased inflation forecasts are ruled out; the departure from
rational expectations lies in the under-sensitivity of the private sector’s
inflation forecast to its information.

The Game Theory literature contains a number of related approaches to
equilibrium modeling with non-rational expectations, including analogy-based
expectations (Jehiel (2005)), “cursed” beliefs (Eyster and Rabin (2005)) and
Berk-Nash equilibrium (Esponda and Pouzo (2016)). Spiegler (2020b) describes
the relation between these approaches and the Bayesian-network formalism.

Within the macroeconomic theory literature, there are a few precedents for
modeling monetary policy when the rational-expectations assumption is relaxed.
Evans and Honkapohja (2001) and Woodford (2013) review dynamic macroeconomic
models in which agents form non-rational expectations, and explore
implications for monetary policy. Orphanides and Williams (2007) and Garcia-
Schmidt and Woodford (2015) are examples of exercises in this tradition. In
particular, Orphanides and Williams (2007) study the implications of learning-
based private-sector expectations for the structure of central-bank policy.

The most closely related equilibrium concept that is employed in the
macroeconomic literature is known as “restricted perceptions equilibrium”
(Evans and Honkapohja (2001)), which is based on a notion of coarse beliefs in
the same spirit as Piccione and Rubinstein (2003) and Jehiel (2005). Sargent
(2001), Cho et al. (2002) and Esponda and Pouzo (2016) study models in which
it is the central bank that forms non-rational expectations, whereas the
private sector is modeled conventionally.

The idea that real-life monetary policy may be insufficiently responsive to
shocks has been discussed in the literature since at least Rudenbusch (2001).
A number of explanations have been proposed. One of them, originating in
Brainard (1967), is that policy attenuation is a response to uncertainty about
relevant parameters or to Knightian uncertainty regarding the central bank’s
model (see Tetlow and Von zur Muehlen (2001) and Dupraz et al. (2020) for
critical discussions). This paper contributes to this literature by proposing
a new possible motive for monetary-policy distortion: the private sector’s
misperception of the causal relation between inflation and output.

##  2 The Model

The following is a variant on a familiar textbook static model of monetary
policy in the tradition of Kydland and Prescott (1977) and Barro and Gordon
(1983). It most closely resembles Sargent (1999) and Athey et al. (2005).

All variables in the model get real values. A central bank observes an
exogenous, normally distributed variable $\theta$ and chooses an action $a$.
Subsequently, the private sector observes a Gaussian signal $t$ of
$(\theta,a)$ and forms an inflation forecast $e$. Following the realization of
$\theta,a,t,e$, actual inflation $\pi$ and actual output $y$ are realized
according to the following pair of equations:

$\displaystyle\pi$ $\displaystyle=$ $\displaystyle a+\varepsilon$
$\displaystyle y$ $\displaystyle=$ $\displaystyle\pi-\lambda e+\eta$

where $\lambda\in(0,1)$ is a constant that represents the extent to which
anticipated inflation offsets the real effect of actual inflation; and
$\varepsilon$ and $\eta$ are statistically independent noise terms,
$\varepsilon\sim N(0,\sigma_{\varepsilon}^{2})$ and $\eta\sim
N(0,\sigma_{\eta}^{2})$.

The central bank’s objective is to minimize a standard quadratic loss
function:

$L(\pi,y,\theta)=y^{2}+(\pi-\theta)^{2}$

Thus, the exogenous state variable $\theta$ is interpreted as the central
bank’s ideal inflation target. The central bank’s strategy is a function
$(a(\theta))_{\theta}$ that assigns an action to every realization of
$\theta$. The private sector’s strategy is the forecast function $(e(t))_{t}$
that assigns an inflation forecast to every signal $t$.

Given the central bank’s strategy and private sector’s forecast function, we
can define a joint distribution $p$ over all six variables
$\theta,a,t,e,y,\pi$. The exogenous components of $p$ are the distribution
over $\theta$, the conditional distribution over $t$ (given $\theta,a$) and
the conditional distribution of $\pi$ and $y$ (given $\theta,a,t,e$). The
endogenous components are the pair consisting of the central bank’s strategy
$(a(\theta))_{\theta}$ and the private sector’s forecast $(e(t))_{t}$.

We now arrive at the key definition of how the private sector forms its
inflation forecast. The private sector is endowed with a subjective causal
model, which is represented by a directed acyclic graph (DAG) $G=(N,R)$, where
$N$ is a set of nodes and $R$ is a set of directed links. Each node in $N$
represents a variable. Some variables may be omitted from the causal model,
but we assume that $t$, $y$ and $\pi$ are always represented. A link
represents a perceived direct causal influence.

It will be convenient to enumerate the six variables $x_{1},...,x_{6}$, such
that $N$ is some subset of $\\{1,...,6\\}$. Denote $x=(x_{1},...,x_{6})$. For
every $M\subseteq\\{1,...,6\\}$, $x_{M}=(x_{i})_{i\in M}$ is the projection of
$x$ on the variables represented by $M$. Let $R(i)$ be the set of nodes $j$
such that $G$ includes the link $j\rightarrow i$. In other words, $R(i)$
represents the set of variables that are considered to be direct causes of the
variable $x_{i}$. I sometimes use boldface notation of a variable to represent
its label, such that for any variable $z$,its label is $\mathbf{z}$. For
instance, when $R$ postulates that $y$ is a direct cause of $\pi$, we can
rewrite this as $\mathbf{y}\in R(\mathbf{\pi})$.

To arrive at an inflation forecast, the private sector “estimates” its model
by performing measurements that quantify the postulated causal links, and
combining the results of these measurements in accordance with the causal
model. In other words, the private sector fits the causal model $G$ to the
joint distribution $p$. Formally, the private sector’s belief is defined as
follows:

$p_{G}(x_{N})=\mathop{\displaystyle\prod}\limits_{i\in N}p(x_{i}\mid
x_{R(i)})$ (1)

This is a Bayesian-network factorization formula: it factorizes the objective
joint distribution $p$ (defined over $x_{N}$) into a product of conditional-
probability terms. Each term is extracted from $p$, but the terms are put
together in a way that may result in $p_{G}\neq p$. As long as $p$ has full
support, $p_{G}$ is a well-defined distribution, which induces well-defined
marginal and conditional distributions. We say that a distribution $p$ is
consistent with $G$ if $p_{G}=p$. Indeed, the objective joint distribution $p$
is consistent with the following DAG, denoted $G^{\ast}$:

$\begin{array}[]{ccccc}\theta&\rightarrow&a&\rightarrow&\pi\\\
\downarrow&\swarrow&&&\downarrow\\\ t&\rightarrow&e&\rightarrow&y\end{array}$

One interpretation of $p_{G}$ is that it is the limit belief of a discrete-
time process of Bayesian updating, where the prior has full support over all
distributions that are consistent with $G$, and the private sector observes an
independent draw from $p$ at every period (see Spiegler (2020b)). Another
interpretation, which is appropriate when $p$ is multivariate normal, is that
$G$ represents a recursive system of linear regression equations, where every
term $p(x_{i}\mid x_{R(i)})$ in (1) corresponds to a linear regression
equation for $x_{i}$, in which the set of regressors is $R(i)$. In this case,
(1) defines a so-called Gaussian Bayesian network (Koller and Friedman (2009,
Ch. 7)). Each equation is estimated via OLS against an arbitrarily large
sample, and the estimated equations are put together in accordance with the
assumption that the noise terms in the regression equations are independent.
This assumption is wrong when $G$ is a misspecified causal model.

Because the private sector perceives statistical regularities through the
prism of an incorrect model, the subjective belief $p_{G}$ may systematically
distort the correlation structure of the actual distribution $p$.

The private sector uses its belief $p_{G}$ to obtain the inflation forecast:

$e=E_{G}(\pi\mid t)=\mathop{\displaystyle\int}\nolimits_{\pi}\pi
dp_{G}(\pi\mid t)$ (2)

For instance, if $G$ is

$\begin{array}[]{ccccc}\theta&\rightarrow&a&\rightarrow&y\\\
&\searrow&\downarrow&&\downarrow\\\ &&t&&\pi\end{array}$

then

$E_{G}(\pi\mid t)=\int_{\theta,a,y,\pi}p(\theta,a\mid t)p(y\mid a)p(\pi\mid
a)\pi$

As we shall see, because $G$ is a misspecified model, $e$ may be sensitive to
all features of the joint distribution of $p$, including those that would not
matter under rational expectations.

######  Definition 1 (Equilibrium)

The pair $(a(\theta))_{\theta}$ and $(e(t))_{t}$ constitutes an $equilibrium$
if: $(i)$ for every $\theta$, $a(\theta)$ maximizes the central bank’s
expected payoff given $(e(t))_{t}$; and $(ii)$ $(e(t))_{t}$ is given by (2).

######  Definition 2 (Linear equilibrium)

An equilibrium $(a(\theta))_{\theta}$ and $(e(t))_{t}$ is $linear$ if
$a(\theta)$ and $e(t)$ are linear functions.

Throughout the paper, I restrict attention to linear equilibria. This ensures
that the joint distribution $p$ is multivariate normal.

Rational-expectations benchmark

As a benchmark, suppose that the private sector has rational expectations.
That is, its subjective DAG is $G^{\ast}$ itself. Furthermore, suppose it is
fully informed of $\theta,a$. Then, its inflation forecast conditional on
$\theta,a$ is

$e=E(\pi\mid\theta,a)=E(\pi\mid a)=a$

Therefore,

$y=a-\lambda a+\varepsilon+\eta$

The central bank’s expected cost from an action $a$ given the realization of
$\theta$ is:

$(1-\lambda)^{2}a^{2}+(a-\theta)^{2}+2\sigma_{\varepsilon}^{2}+\sigma_{\eta}^{2}$

Therefore, the central bank’s optimal action given $\theta$ is

$a^{\ast}(\theta)=\frac{1}{(1-\lambda)^{2}+1}\theta$

The solution $a^{\ast}(\theta)$ is not fully responsive to changes in
$\theta$. However, as $\lambda$ tends to one - such that only unanticipated
inflation matters for output - the central bank’s policy approaches fully
flexible targeting. Another key observation is that $a^{\ast}(\theta)$ is
invariant to $\sigma_{\varepsilon}^{2}$ and $\sigma_{\eta}^{2}$.

Discussion: Model-based forecasts

The private sector’s forecasting method consists of two steps: if first
estimates a causal model and then uses the estimated model to generate an
inflation forecast. The exact interpretation of this process depends on
whether we think of private-sector agents as lay persons who reason about
macroeconomic processes informally and intuitively, or as professional
forecasters. Under the former interpretation, the idea is that agents have
some qualitative causal perceptions of how macroeconomic variables relate to
one another, and they interpret observational data in light of these
perceptions.

As to the latter interpretation, it is clear that not all professional
forecasting is model-based. However, models do play a role in forecasting. One
virtue of models is that they ground the forecast in a “story” that can be
communicated to other parties (e.g. see Edge et al. (2008)). Another advantage
of a model is that it is a simple vehicle for making a multitude of
conditional predictions. See, for example, the following quote from a recent
speech by Stanley Fisher: “The economy is an extremely complicated mechanism,
and every macroeconomic model is a vast simplification of reality…the large
scale of FRB/US is an advantage in that it can perform a wide variety of
computational “what if” experiments.”111See
https://www.federalreserve.gov/newsevents/speech/fischer20170211a.htm. For a
critical discussion of theory-based forecasting, see Giacomini (2015).

Thus, we should not think of the private sector as being exclusively
interested in predicting inflation - otherwise, there would be no need to rely
on a model; the private sector could simply measure the steady-state
conditional expectation $E(\pi\mid t)$. Instead, the private sector employs a
model as a multi-purpose vehicle for making sense of statistical regularities
in the macroeconomic environment, and for making conditional predictions as
the occasion arises. The model is a knowing simplification of a complex
environment, and therefore the private sector expects it to get some things
wrong, and would be unfazed if some of the model’s assumptions and predictions
turned out to be incorrect (defending this stance with familiar statements
like “every model is wrong” or “it takes a model to beat a model”).

##  3 The Basic Result

The model assumes that the private sector has a correct causal model of the
joint behavior of the variables $\theta,a,t,e$. The private sector’s error is
that it gets the direct causes of $\pi$ or $y$ wrong. In the true model
$G^{\ast}=(N^{\ast},R^{\ast})$, $\mathbf{y}\notin R^{\ast}(\mathbf{\pi})$ \-
that is, output is not a direct cause of inflation, because causality runs in
the opposite direction. Our first result establishes that if the private
sector’s causal model $G$ shares this property, its inflation forecast in a
linear equilibrium is unaffected by $\sigma_{\varepsilon}^{2}$ and
$\sigma_{\eta}^{2}$. Moreover, the equilibrium can be “rationalized”: it can
be obtained by assuming that $G=G^{\ast}$ while modifying the private sector’s
signal function, given by $(p(t\mid\theta,a))$.

###### Proposition 1

Suppose $G=(N,R)$ satisfies $\mathbf{y}\notin R(\mathbf{\pi})$. Then:
$(i)$ Any linear equilibrium is invariant to
$(\sigma_{\varepsilon}^{2},\sigma_{\eta}^{2})$.
$(ii)$ Any linear equilibrium can be obtained from a specification of the
primitives in which the private sector’s causal model is $G^{\ast}$, and the
private sector is either fully informed or entirely uninformed of
$(\theta,a)$.

Proof. $(i)$ By assumption, $G$ coincides with $G^{\ast}$ over the nodes
$\theta,a,t,e$. Therefore,

$\displaystyle E_{G}(\pi$ $\displaystyle\mid$ $\displaystyle
t)=\sum_{\theta,a,e}p_{G}(\theta,a,e\mid
t)\sum_{\pi}p_{G}(\pi\mid\theta,a,t,e)\pi$ $\displaystyle=$
$\displaystyle\sum_{\theta,a,e}p_{G}(\theta,a,e\mid t)\sum_{\pi}p_{G}(\pi\mid
x_{R(\mathbf{\pi})})\pi$ $\displaystyle=$
$\displaystyle\sum_{\theta,a,e}p(\theta,a,e\mid t)\sum_{\pi}p_{G}(\pi\mid
x_{R(\mathbf{\pi})})\pi$

By assumption,
$R(\mathbf{\pi})\subseteq\\{\mathbf{\theta},\mathbf{a},\mathbf{t},\mathbf{e}\\}$.
Therefore,

$p_{G}(\pi\mid x_{R(\mathbf{\pi})})=\sum_{a^{\prime}}p(a^{\prime}\mid
x_{R(\mathbf{\pi})})p(\pi\mid a^{\prime})$

such that

$E_{G}(\pi\mid t)=\sum_{x_{R(\mathbf{\pi})}}p(x_{R(\mathbf{\pi})}\mid
t)\sum_{a^{\prime}}p(a^{\prime}\mid x_{R(\mathbf{\pi})})\sum_{\pi}p(\pi\mid
a^{\prime})\pi$ (3)

Since

$\sum_{\pi}p(\pi\mid a^{\prime})\pi=E(a^{\prime}+\varepsilon)=a^{\prime}$

we obtain

$E_{G}(\pi\mid t)=\sum_{x_{R(\mathbf{\pi})}}p(x_{R(\mathbf{\pi})}\mid
t)\sum_{a^{\prime}}p(a^{\prime}\mid x_{R(\mathbf{\pi})})a^{\prime}$

The terms $p(x_{R(\mathbf{\pi})}\mid t)$ and $p(a^{\prime}\mid
x_{R(\mathbf{\pi})})$ only derive from $(p(\theta,a,t,e))$, and are therefore
invariant to $(\sigma_{\varepsilon}^{2},\sigma_{\eta}^{2})$.

$(ii)$ Once again, recall that by assumption,
$R(\mathbf{\pi})\subseteq\\{\mathbf{\theta},\mathbf{a},\mathbf{t},\mathbf{e}\\}$.
In a linear equilibrium, $a$ is a deterministic function of $\theta$, and $e$
is a deterministic function of $t$. Therefore, if $R(\mathbf{\pi})$ includes
$\mathbf{a}$ ($\mathbf{\theta}$), it is immaterial whether it also includes
$\mathbf{\theta}$ ($\mathbf{a}$). Likewise, if $R(\mathbf{\pi})$ includes
$\mathbf{t}$ ($\mathbf{e}$), it is immaterial whether it also includes
$\mathbf{e}$ ($\mathbf{t}$).

It follows that we only need to examine three cases. First, suppose that
$\mathbf{a\in}R(\mathbf{\pi})$ (or, equivalently,
$\mathbf{\theta\in}R(\mathbf{\pi})$). Then,

$\displaystyle E_{G}(\pi$ $\displaystyle\mid$ $\displaystyle
t)=\sum_{\theta,a}p(\theta,a,e\mid t)E(\pi\mid x_{R(\mathbf{\pi})})$
$\displaystyle=$ $\displaystyle\sum_{\theta,a}p(\theta,a,e\mid t)E(\pi\mid a)$
$\displaystyle=$ $\displaystyle\sum_{a}p(a\mid t)E(\pi\mid a)$
$\displaystyle=$ $\displaystyle E(\pi\mid t)$

where the second equality follows from the fact that under the objective
distribution, $\pi\perp(\theta,t,e)\mid a$. It follows that if
$\mathbf{a\in}R(\mathbf{\pi})$, the private sector’s inflation forecast is
consistent with rational expectations, even if we do not change its signal
function.

Second, suppose that $\\{\mathbf{a},\mathbf{\theta}\\}\cap R(\mathbf{\pi})$ is
empty, but $\mathbf{t\in}R(\mathbf{\pi})$ (or, equivalently,
$\mathbf{e\in}R(\mathbf{\pi})$). Then, by definition,

$E_{G}(\pi\mid t)=E(\pi\mid t)$

Then, as in the first case, the private sector’s inflation forecast is
consistent with rational expectations, even if we do not change its signal
function.

The only remaining case is that $R(\mathbf{\pi})$ is empty. In this case,

$E_{G}(\pi\mid t)=E(\pi)$

This inflation forecast can be generated in a rational-expectations model, in
which the private sector is entirely uninformed of $\theta,a$ \- i.e., we
modify $p(t\mid\theta,a)$ into $p^{\prime}(t\mid\theta,a)$ that is constant in
$\theta,a$.

This result establishes that nothing “interesting” happens in this model when
the private sector’s model does not invert the causal link between inflation
and output. In fact, the only departure from the rational-expectations
benchmark occurs when the private sector’s model postulates that none of the
other relevant variables are a direct cause of inflation. In this case, the
private sector acts as if it is entirely uninformed of $\theta$ and $a$. It is
easy to show that in the unique linear equilibrium, $e=0$ with certainty,
while the central bank’s policy is

$a(\theta)=\frac{\theta}{2}$

That is, the private sector’s expectations are entirely rigid, as in the
rational-expectations benchmark with $\lambda=0$. This results in an overly
rigid central-bank policy.

##  4 Examples

In this section I analyze in detail two examples in which the private sector’s
causal model violates the condition in Proposition 1. As a result, the noise
terms in the equations for inflation and output play a non-trivial role in the
private sector’s inflation forecast, and therefore also in the central bank’s
equilibrium policy.

In both examples, I assume that $t=a$ \- i.e., the private sector observes $a$
before forming its inflation forecast. (The private sector does not observe
$\theta$, but this is immaterial.) Since $t$ coincides with $a$, we can remove
it from the model, such that $G^{\ast}$ can be written as:

$\begin{array}[]{ccccc}\theta&\rightarrow&a&\rightarrow&\pi\\\
&&\downarrow&&\downarrow\\\ &&e&\rightarrow&y\end{array}$

That is, the central bank’s action $a$ is an immediate consequence of the
exogenous variable $\theta$; actual inflation $\pi$ and the private sector’s
inflation forecast $e$ are conditionally independent consequences of $a$; and
these two variables are the direct causes of real output $y$.

We will now analyze two specifications of the private sector’s DAG $G$.

###  4.1 Case 1: $G:\theta\rightarrow a\rightarrow\pi\leftarrow y$

This DAG tells a causal story, according to which inflation is a consequence
of two independent causes: real output and the central bank’s policy. This
causal model postulates absolute monetary neutrality: it admits no path of
causal links from $a$ to $y$. Thus, the private sector’s causal model distorts
the true model $G^{\ast}$ in two ways: it omits the expectations variable $e$
and inverts the causal link between inflation and output.

The private sector’s inflation forecast after observing the central bank’s
action $a$ is

$E_{G}(\pi\mid a)=\mathop{\displaystyle\int}\nolimits_{\pi}\pi p_{G}(\pi\mid
a)=\mathop{\displaystyle\int}\nolimits_{\pi}\mathop{\displaystyle\int}\nolimits_{y}\pi
p(y)p(\pi\mid a,y)$ (4)

Compare this with the following way to write the rational-expectations
inflation forecast:

$E(\pi\mid
a)=\mathop{\displaystyle\int}\nolimits_{\pi}\mathop{\displaystyle\int}\nolimits_{y}\pi
p(y\mid a)p(\pi\mid a,y)$ (5)

The discrepancy arises because $p_{G}(\pi\mid a)$ involves an implicit
expectation over $y$ without conditioning on $a$. (Note that since
$\pi,y\perp\theta\mid a$ according to both $G^{\ast}$ and $G$, $\theta$ does
not feature in either of the two expressions.) Because the term $p(y)$ in (4)
is actually $not$ independent of $a$, a change in the central bank’s strategy
can lead to a change in $E_{G}(\pi\mid a)$.

###### Proposition 2

There is a unique linear equilibrium, given by

$\displaystyle e(a)$ $\displaystyle=$
$\displaystyle\frac{1-\beta}{1-\beta\lambda}a$ (6) $\displaystyle a(\theta)$
$\displaystyle=$
$\displaystyle\frac{1}{\left(\frac{1-\lambda}{1-\beta\lambda}\right)^{2}+1}\theta$

where

$\beta=\frac{\sigma_{\varepsilon}^{2}}{\sigma_{\varepsilon}^{2}+\sigma_{\eta}^{2}}$
(7)

Proof. Let us begin by deriving a formula for the private sector’s inflation
forecast. Denote

$e(a)=E_{G}(\pi\mid a)=\mathop{\displaystyle\int}\nolimits_{y}p(y)E(\pi\mid
a,y)$

Since $\pi=a+\varepsilon$,

$E(\pi\mid a,y)=a+E(\varepsilon\mid a,y)$

By the Phillips Curve,

$\varepsilon+\eta=y-a+\lambda e(a)$

For given $a$ and $y$, the R.H.S is a constant, whereas the L.H.S is a sum of
two independent variables that are normally distributed with mean zero.
Therefore, to calculate $E(\varepsilon\mid a,y)$, we can apply the standard
formula for $E(X\mid X+Z)$ when $X$ and $Z$ are independent normal variables,
and obtain

$E(\varepsilon\mid a,y)=\beta(y-a+\lambda e(a))$

where

$\beta=\frac{\sigma_{\varepsilon}^{2}}{\sigma_{\varepsilon}^{2}+\sigma_{\eta}^{2}}$

We can now write

$e(a)=\int_{y}p(y)\left[a+\beta y-\beta a+\beta\lambda
e(a)\right]=(1-\beta)a+\beta\lambda e(a)+\beta E(y)$

Since $\pi=a+\varepsilon$ and $E(\varepsilon)=0$, $E(\pi)=E(a)$. Plugging the
Phillips curve, we obtain

$E(y)=E(a)-\lambda E(e(a))$

and therefore,

$e(a)=(1-\beta)a+\beta\lambda e(a)+\beta[E(a)-\lambda E(e(a))]$

This functional equation defines $e(a)$. Taking expectations, we obtain

$\displaystyle E(e(a))$ $\displaystyle=$
$\displaystyle(1-\beta)E(a)+\beta\lambda E(e(a))+\beta E(a)-\beta\lambda
E(e(a))$ $\displaystyle=$ $\displaystyle E(a)=E(\pi)$

Plug this into the expression for $e(a)$, and get:

$e(a)=\frac{1-\beta}{1-\beta\lambda}a+\frac{\beta-\beta\lambda}{1-\beta\lambda}E(a)$

Plugging the expression for $\beta$, we obtain

$e(a)=\frac{\sigma_{\eta}^{2}}{\sigma_{\eta}^{2}+(1-\lambda)\sigma_{\varepsilon}^{2}}a+\frac{(1-\lambda)\sigma_{\varepsilon}^{2}}{\sigma_{\eta}^{2}+(1-\lambda)\sigma_{\varepsilon}^{2}}E(a)$

Plugging the expression we obtained for $e(a)$, the expected output
conditional on $a$ is

$E(y\mid
a)=\frac{1-\lambda}{1-\beta\lambda}a-\lambda\frac{\beta-\beta\lambda}{1-\beta\lambda}E(a)$

We can now write down the central bank’s expected payoff from $a$ given
$\theta$ as follows:

$\left[\frac{1-\lambda}{1-\beta\lambda}a-\lambda\frac{\beta-\beta\lambda}{1-\beta\lambda}E(a)\right]^{2}+(a-\theta)^{2}$

Our definition of linear equilibrium adopts the interim approach - i.e., when
finding the optimal $a$ given $\theta$, we need to take $E(a)$ as given. (This
does not matter in fact; we could also take the ex-ante approach and the
result would be the same, although this is an artifact of the linearity of the
model.) The first-order condition with respect to $a$ is

$\left(\frac{1-\lambda}{1-\beta\lambda}\right)\left[\frac{1-\lambda}{1-\beta\lambda}a-\lambda\frac{\beta-\beta\lambda}{1-\beta\lambda}E(a)\right]+a-\theta=0$

Taking the expectation with respect to $a$ and recalling that $E(\theta)=0$,
it is immediate that $E(a)=0$. Therefore,

$a(\theta)=\frac{1}{\left(\frac{1-\lambda}{1-\beta\lambda}\right)^{2}+1}\theta$

We have now obtained the precise formulas for the unique linear equilibrium,
given by (6).

Thus, when $\lambda=1$, $e(a)=a=E(\pi\mid a)$ \- i.e., the private sector’s
inflation forecast coincides with the rational-expectations benchmark. Despite
the private sector’s misspecified Phillips Curve, it ends up having correct
expectations and therefore the central bank’s strategy coincides with the
rational-expectations benchmark.

However, when $\lambda<1$ \- i.e., when anticipated inflation has real effects
- the inflation forecast is “rigid” in the sense of being a convex combination
of the correct conditional expected inflation $E(\pi\mid a)$ and the ex-ante,
unconditional expected inflation $E(\pi)$. As a result, the central bank’s
equilibrium strategy is less responsive to $\theta$ than in the rational-
expectations benchmark.

To understand the reason for the private sector’s partially responsive
expectations, recall that according to the DAG $G:\theta\rightarrow
a\rightarrow\pi\leftarrow y$, the private sector erroneously regards $y$ as an
exogenous variable that affects $\pi$, and therefore assigns some weight to
the ex-ante expected value of $y$ when forming its inflation forecast. Because
$y$ is in fact a consequence of $a$, the private sector ends up assigning
weight to the ex-ante expectation of $a$, thus failing to fully condition on
the actual realization of $a$.

The extent of this failure depends on the relative magnitudes of
$\sigma_{\varepsilon}^{2}$ and $\sigma_{\eta}^{2}$. As the Phillips relation
becomes less noisy (relative to the relation between the central bank’s action
and inflation), the erroneous weight on $E(a)$ increases and the deviation
from rational expectations is exacerbated. This “expectational rigidity” leads
to a policy-attenuation effect that increases with
$\sigma_{\varepsilon}^{2}/\sigma_{\eta}^{2}$.

The intuition behind the comparative statics is as follows. The private sector
tries to account for fluctuations in $\pi$ (conditional on $a$) by the
variation in $y$, mistakenly treating the latter as exogenous. The variables
$\pi$ and $y$ are normally distributed. When
$\sigma_{\eta}^{2}/\sigma_{\varepsilon}^{2}$ is low, the definition of
conditional expectation of normal variables implies that $y$ gets a large
weight in the private sector’s inflation forecast, and therefore the model
misspecification error generates a larger forecast error - which in turn
impels the central bank to adopt a more rigid policy.

###  4.2 Case 2: $G:\theta\rightarrow a\rightarrow y\rightarrow\pi$

This DAG tells a causal story, according to which the central bank’s action
has a causal effect on real output, which in turn is the sole direct cause of
inflation. As in Case 1, this specification of $G$ inverts the true causal
link between output and inflation. However, unlike Case 1, this causal model
regards $y$ as an endogenous variable that mediates the effect of monetary
policy on inflation.

The private sector’s inflation forecast after observing the central bank’s
action $a$ is

$E_{G}(\pi\mid a)=\mathop{\displaystyle\int}\nolimits_{\pi}\pi p_{G}(\pi\mid
a)=\mathop{\displaystyle\int}\nolimits_{\pi}\mathop{\displaystyle\int}\nolimits_{y}\pi
p(y\mid a)p(\pi\mid y)$ (8)

Compare this expression with (5). The error in (8) lies in the term $p(\pi\mid
y)$, which fails to condition $\pi$ on $a$. This failure reflects the
assumption, embedded in $G$, that $\pi$ is independent of $a$ conditional on
$y$. This assumption is inconsistent with the true model $G^{\ast}$.

A linear equilibrium is strongly linear if $e(a)$ is proportional to $a$ and
$a(\theta)$ is proportional to $\theta$. The next result restricts attention
to strongly linear equilibria, in order to avoid the minor technical
distraction of proving that every linear equilibrium must be strongly linear
(this could be demonstrated transparently in Case 1, but unfortunately not in
the present case).

###### Proposition 3

There is a unique strongly linear equilibrium, given by

$\displaystyle e(a)$ $\displaystyle=$ $\displaystyle\gamma a$ (9)
$\displaystyle a(\theta)$ $\displaystyle=$ $\displaystyle\delta\theta$

where $\gamma$ and $\delta$ are uniquely given by the pair of equations:

$\displaystyle\gamma$ $\displaystyle=$
$\displaystyle\frac{(1-\gamma\lambda)^{2}\delta^{2}\sigma_{\theta}^{2}+(1-\gamma\lambda)\sigma_{\varepsilon}^{2}}{(1-\gamma\lambda)^{2}\delta^{2}\sigma_{\theta}^{2}+\sigma_{\varepsilon}^{2}+\sigma_{\eta}^{2}}$
(10) $\displaystyle\delta$ $\displaystyle=$
$\displaystyle\frac{1}{(1-\gamma\lambda)^{2}+1}$

Proof. Let us begin by deriving a formula for the private sector’s inflation
forecast. Denote

$e(a)=E_{G}(\pi\mid a)=\mathop{\displaystyle\int}\nolimits_{y}p(y\mid
a)E(\pi\mid y)$ (11)

Guessing a strongly linear equilibrium, suppose $e(a)=\gamma a$ and
$a(\theta)=\delta\theta$. Therefore,

$y=a-\lambda\gamma a+\varepsilon+\eta$

Given $\theta$, the central bank effectively chooses $a$ to minimize

$E(y^{2}\mid a)+(a-\theta)^{2}=(1-\gamma\lambda)^{2}a^{2}+(a-\theta)^{2}$

The first-order condition with respect to $a$ pins down the expression for
$\delta$ given by (10).

Our remaining task is to derive the expression for $\gamma$. To do so, let us
first obtain an expression for $E(\pi\mid y)$. By the true equations for
inflation and output,

$\displaystyle E(\pi$ $\displaystyle\mid$ $\displaystyle
y)=E(a+\varepsilon\mid a-\lambda e(a)+\varepsilon+\eta=y)$ $\displaystyle=$
$\displaystyle E(a+\varepsilon\mid a-\lambda\gamma a+\varepsilon+\eta=y)$

which is equal to

$\frac{1}{1-\gamma\lambda}E[(1-\gamma\lambda)a\mid(1-\gamma\lambda)a+\varepsilon+\eta=y]+E[\varepsilon\mid(1-\gamma\lambda)a+\varepsilon+\eta=y]$

The expression $(1-\gamma\lambda)a+\varepsilon+\eta$ is a sum of independent,
normally distributed variables with mean zero. Denote

$\sigma_{a}^{2}=\delta^{2}\sigma_{\theta}^{2}$

Then, using the conditional-expectation rule for such variables,

$E[a\mid(1-\gamma\lambda)a+\varepsilon+\eta=y]=\frac{(1-\gamma\lambda)\delta^{2}\sigma_{\theta}^{2}}{(1-\gamma\lambda)^{2}\delta^{2}\sigma_{\theta}^{2}+\sigma_{\varepsilon}^{2}+\sigma_{\eta}^{2}}\cdot
y$

and

$E[\varepsilon\mid(1-\gamma\lambda)a+\varepsilon+\eta=y]=\frac{\sigma_{\varepsilon}^{2}}{(1-\gamma\lambda)^{2}\delta^{2}\sigma_{\theta}^{2}+\sigma_{\varepsilon}^{2}+\sigma_{\eta}^{2}}\cdot
y$

Therefore,

$E(\pi\mid
y)=\frac{(1-\gamma\lambda)\delta^{2}\sigma_{\theta}^{2}+\sigma_{\varepsilon}^{2}}{(1-\gamma\lambda)^{2}\delta^{2}\sigma_{\theta}^{2}+\sigma_{\varepsilon}^{2}+\sigma_{\eta}^{2}}\cdot
y$

Now, plug this expression in (11) and obtain

$e(a)=\frac{(1-\gamma\lambda)\delta^{2}\sigma_{\theta}^{2}+\sigma_{\varepsilon}^{2}}{(1-\gamma\lambda)^{2}\delta^{2}\sigma_{\theta}^{2}+\sigma_{\varepsilon}^{2}+\sigma_{\eta}^{2}}\cdot
E(y\mid a)$

By the inflation and output equations,

$\displaystyle E(y$ $\displaystyle\mid$ $\displaystyle a)=E(\pi-\lambda
e(a)+\eta\mid a)$ $\displaystyle=$ $\displaystyle E(a-\lambda\gamma
a+\varepsilon+\eta\mid a)$ $\displaystyle=$ $\displaystyle(1-\gamma\lambda)a$

Therefore,

$e(a)=\frac{(1-\gamma\lambda)\delta^{2}\sigma_{\theta}^{2}+\sigma_{\varepsilon}^{2}}{(1-\gamma\lambda)^{2}\delta^{2}\sigma_{\theta}^{2}+\sigma_{\varepsilon}^{2}+\sigma_{\eta}^{2}}\cdot(1-\gamma\lambda)a=\gamma
a$

where $\gamma$ is given by (10).

It remains to establish uniqueness of $(\gamma,\delta)$. Clearly, $\delta$ is
a monotone function of $\gamma$ in the relevant range. Therefore, it suffices
to show that $\gamma$ is unique. Plugging the equation for $\delta$ in the
equation for $\gamma$, we obtain

$\gamma=\frac{\frac{(1-\gamma\lambda)^{2}}{[(1-\gamma\lambda)^{2}+1]^{2}}\sigma_{\theta}^{2}+(1-\gamma\lambda)\sigma_{\varepsilon}^{2}}{\frac{(1-\gamma\lambda)^{2}}{[(1-\gamma\lambda)^{2}+1]^{2}}\sigma_{\theta}^{2}+\sigma_{\varepsilon}^{2}+\sigma_{\eta}^{2}}$

It can be checked that the R.H.S of this equation is continuously and
monotonically decreasing in $\gamma$ in the range $[0,1]$. Moreover, it is
strictly positive when $\gamma=0$ and strictly below $1$ when $\gamma=1$.
Therefore, there is a unique intersection between the curve given by the R.H.S
and the line given by the L.H.S.

As in Case 1, the variance of the noise terms in the inflation and output
equations plays a role in the strongly linear equilibrium in the present case.
However, this role is different in a number of ways.

First, the effect of the noisiness of the Phillips Curve (i.e.
$\sigma_{\eta}^{2}$) on expectational rigidity goes in the opposite direction.
Expectations become more rigid as the Phillips relation becomes less precise.
The intuition is that in Case 2, $G$ posits that the effect of $a$ on $\pi$ is
mediated by $y$. As the statistical relation between $\pi$ and $y$ becomes
noisier, the perceived mediation effect is attenuated, and this means that the
perceived indirect effect of $a$ on $\pi$ becomes weaker.

Second, unlike Case 1, the private sector’s expectations depart from the
rational-expectations benchmark even when $\lambda=1$. That is, even when
anticipated inflation fully offsets the effect of inflation on output, the
private sector’s causal model leads to incorrect, rigid inflation forecasts.
In particular, when $\lambda=1$ and $\sigma_{\varepsilon}^{2}$ (which measures
the noisiness of the mapping from $a$ to $\pi$) gets large, $\gamma$
approaches $\frac{1}{2}$ \- i.e., the inflation forecast is only “half-
responsive” to changes in $a$.

Finally, unlike Case 1, the variance of the exogenous inflation target
$\theta$ plays a role in the characterization of the linear equilibrium. As
$\sigma_{\theta}^{2}$ becomes larger, the private sector’s expectations become
less rigid, and as a result the central bank’s policy approaches the rational-
expectations benchmark. The intuition is that larger fluctuations in $\theta$
translate to larger fluctuations in $a$. This in turn implies that the private
sector’s interpretation of fluctuations in inflation and output becomes
dominated by the fluctuations in $a$, which is the correct interpretation; the
misperception of the role of $\sigma_{\eta}^{2}$ and
$\sigma_{\varepsilon}^{2}$ becomes less important for the private sector’s
expectations.

##  5 Discussion

The comparison between Cases 1 and 2 shows that the details of the private
sector’s causal misinterpretation of the Phillips Curve matter for the
qualitative properties of the equilibrium in the interaction between the
central bank and the private sector. However, the broad lesson that
$\sigma_{\eta}^{2}$ and $\sigma_{\varepsilon}^{2}$ matter for the equilibrium
holds in both cases because they share the reverse-causality error that
misperceives the direction of causality between output and inflation.

Note, however, that not every $G$ that exhibits this reverse causality will
give rise to this effect. For example, suppose
$R(\mathbf{\pi})=\\{\mathbf{y}\\}$ and $R(\mathbf{y})=\emptyset$. Then, $G$
admits no causal path from $a$ or $e$ to $\pi$. As in earlier examples that
exhibited this feature, the private sector’s equilibrium expectations will be
entirely rigid: $e=0$ with probability one, leading to the most rigid central-
bank policy that is possible in this model, without any sensitivity to
$\sigma_{\eta}^{2}$ and $\sigma_{\varepsilon}^{2}$. Another example is
$R(\mathbf{\pi})=\\{\mathbf{a},\mathbf{y}\\}$ and
$R(\mathbf{y})=\\{\mathbf{a}\\}$. In this case, $G$ induces rational
expectations. This can be shown algebraically, or using basic equivalence
results from the Bayesian-network literature (see Spiegler (2016,2020b) for
details). Thus, reverse causality is a necessary condition for the noise-
sensitivity of central-bank policy, but not a sufficient one.

The noise-sensitivity of linear equilibria is of interest in light of the
“caution principle” attributed to Brainard (1967), which has been suggested as
an explanation of monetary-policy attenuation. A high
$\sigma_{\varepsilon}^{2}$ captures a situation in which the central bank is
highly uncertain about the effect of monetary policy on inflation. However,
given our conventional linear-quadratic parameterization of the model, this
uncertainty has no effect on the central bank’s policy under rational
expectations. However, when the private sector’s model reverses the causal
link between inflation and output, high $\sigma_{\varepsilon}^{2}$ can lead to
an attenuated policy - although, as we saw, the extent of this effect depends
on details of the private sector’s model. The reason is that
$\sigma_{\varepsilon}^{2}$ affects the private sector’s interpretation of
inflation fluctuations, and therefore its inflation forecast conditional on
its information. The lesson is that Phillips-Curve disagreements between the
central bank and the private sector can rationalize policy distortions due to
uncertainty about the consequences of monetary policy - even in settings that
otherwise could not give rise to policy distortions.

The policy distortion that emerged in this paper takes the form of attenuation
\- i.e., the policy is less responsive to exogenous shocks than in the
rational-expectations benchmark. However, I do not want to stress this point
too strongly. The important effect that our analysis has highlighted is the
attenuated response of the private sector’s inflation forecast, as a result of
its causally misperceived Phillips Curve. The fact that this effect translated
to a policy-attenuation effect is an artifact of how the model treated the
central bank’s private information $\theta$ \- namely, as an exogenous
inflation target. Under alternative specifications of the central bank’s
preferences, rigid private-sector forecasts would lead to over-responsive
central-bank policy. One could also imagine a model in which $\theta$ has no
direct payoff relevance, and instead plays a role as a parameter in the
inflation/output equations.222I thank Stephane Dupraz for making these
observations. Depending on how the private sector incorporates this dependency
into its causal model, such a specification could lead to a more complicated,
less transparent relation between the private sector’s forecast errors and the
central bank’s policy distortion. Since paper is a first attempt to
incorporate causal disagreements into familiar models of monetary policy, I
prioritized pedagogical considerations and refrained from adding these
complications. However, I believe they merit further study.

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