Time-Varying Parameters and Endogenous
Learning Algorithms
Eric Gaus∗
July 2015
Abstract
The adaptive learning literature has primarily focused on decreasing gain
learning and constant gain learning. As pointed out theoretically by Marcet
and Nicolini (2003) and empirically by Milani (2007) an endogenous learning
mechanism may explain key economic behaviors, such as recurrent hyperinflation
or time varying volatility. This paper evaluates the mechanism used in those
papers and the adaptive step size algorithm proposed by Kostyshyna (2012)
in addition to proposing an alternative endogenous learning algorithm. The
proposed algorithm outperforms both alternatives in simulations and may result
in exotic dynamics in macroeconomic models and models of hyperinflation.
Keywords: Adaptive Learning, Rational Expectations, Endogenous Learning.
∗
Ursinus College, 601 East Main St., Collegville, PA 19426-1000 (e-mail: egaus@ursinus.edu)
1
Introduction
Macroeconomic empiricists have long been concerned with the potential for time variation in the parameters of their econometric models. This time variation has been
modeled as structural breaks in the Markov switching literature and using a basic
Kalman filter. Macroeconomists are also concerned with accurately capturing how
agents form expectations. Bernanke (2007) suggests that monetary policy research
should use an adaptive learning framework. Adaptive learning is a bounded rationality
exercise that assumes that agents use basic econometric techniques to form forecasts of
the economic variables of interest. Given the concern over time-varying parameters in
the real world, one might be interested in a plausible learning process that addresses
the potential for time-variation.
In learning, agent’s econometric forecasts are updated using a recursive least squares
(RLS) algorithm, which has a critical component called a gain parameter. When a
researcher assumes agents believe the structure of the economy is stable it is common
to use a decreasing gain, since this reduces forecasting errors, but if agents believe the
structure varies over time a constant gain is used, since this type of gain tracks varying
coefficients more efficiently. An alternative notion of the gain is that it relates to how
much data the agent uses. Under a decreasing gain agents increase the amount of data
and a constant gain keeps the size of the data set constant. From a behavioral point
of view, one might expect agents to adjust the size of the data set endogenously with
respect to some statistical or econometric rule.
There have been a few recent papers that consider endogenous gains of this sort.
Marcet and Nicolini (2003) suggest that in the presence occasional structural breaks,
agents might switch between a constant gain, to track the change at a break, and a
decreasing gain, to reduce forecasting error, in stable periods.The switch to a constant
gain occurs when the forecasts errors rise above a certain threshold. According to
Marcet and Nicolini (2003) this type of behavior may help account for recurrent hyperinflations.1 Kostyshyna (2012) uses a similar model, but an alternative endogenous
gain. Drawing on the vast engineering literature on recursive algorithms Kostyshyna
(2012) proposes using the adaptive step size algorithm. Instead of switching between
constant and decreasing gain, the adaptive step size algorithm adjusts the value of the
gain depending upon the series of forecast errors. High forecast errors lead to higher
gains and low forecast errors lead to lower gains. This endogenous algorithm yields
much the same results as Marcet and Nicolini (2003) though with lower mean squared
errors and matches the stylistic facts of the data.
This paper adds to this growing literature on endogenous gains by proposing an
alternative motivated by agent level econometrics. Recall that these endogenous gains
are primarily motivated by situations where agents believe there are occasional structural breaks. When econometricians are concerned about structural breaks they pay
particular attention to the coefficient estimates, not just the forecast errors. This point
is directly evident in Stock and Watson (2007). While forecast errors may be indicative
of structural breaks, in our adaptive learning framework agents update the coefficients
1
This type of algorithm has been used by Milani (2007) as a potential explanation of the Great
Moderation.
1
each period, which suggests that they might be more attentive to large departures from
recent coefficient estimates. This is the basic notion that underlies the proposed gain.
Specifically, if the most recent coefficient estimate is many standard deviations away
from a recent mean of the coefficients then the gain is relatively high, and otherwise it
is relatively low.
We seek to answer two questions. First, would agents choose to use an endogenous
gain over a constant gain? The results of several simulations suggest that in the presence of occasional structural breaks both gain sequences in Marcet and Nicolini (2003)
and Kostyshyna (2012) rarely outperform a constant gain in terms of mean squared
forecast error (MSFE). The proposed algorithm frequently outperforms a constant gain
over a wide range of parameter values with a potential improvement of MSFE of 7 percent. In addition, coefficient estimates of the proposed gain are relatively closer to
the rational expectations solution than the alternatives. Further, the proposed algorithm performs better in the presence of structural breaks as opposed to an AR(1)
time-varying parameter process suggesting that it is in fact responding to structural
breaks and not time variation in general. We also consider a game theoretic scenario
where we test whether a single agent considering the use an endogenous gain might
have better forecasting performance using the endogenous gain in an economy where
the dynamics are driven by agents using a constant gain. We find that such an agent
will always have an incentive (based on MSFE) to switch to the proposed endogenous
gain for the model described below. Second, what economic importance might an
endogenous gain algorithm have? To answer this question we simulate the hyperinflation model presented in Marcet and Nicolini (2003) and a small New Keynesian (NK)
model. We find that the endogenous gain results in qualitatively the same dynamics as
both alternatives in the hyperinflation model. In the NK model we find, under certain
conditions, recurrent episodes of deviations from rational expectations. Whereas the
hyperinflation model studied by Marcet and Nicolini (2003) and Kostyshyna (2012)
impose structural change by way of an exchange rate regime, this NK model has no
inherent structural change yet still results in temporary deviations from rational expectations. This suggests that endogenous gains on their own may result in bubbles
or hyperinflations, without any other structural changes forcing the model back to
the rational expectations equilibrium. This result relies on the E-instability of “large”
constant gain values.
The rest of the paper is organized as follows. The next section sets up the model
used to test forecasting performance, describes the time varying parameter processes
and presents the endogenous gain algorithms. Section three provides the results of
several head to head comparisons of MSFE of the endogenous gains. The fourth section
describes the stability properties of the endogenous gain algorithms in a model of
hyperinflation and a simple NK model. Section five concludes.
2
Framework
This section presents a simple univariate model that provides an opportunity to examine the performance of each of the endogenous gain algorithms. Two time-varying
processes are used to demonstrate the type of time variation that yields the best results.
2
In addition, this section describes all three endogenous gain algorithms.
2.1
The Time-Varying Parameters Model
Consider the following univariate system,
yt = αt xt−1 + δyte + ηt ,
(1)
where xt is a mean zero random variable, η is normally distributed with mean zero
and variance ση , and the superscript e denotes expectations conditional on information
available at t − 1: yte = E(yt |It−1 ).2 We will consider two potential time varying
parameters scenarios. The first is a model of occasional structural breaks,
(
αt−1 with Prob. (1-ε),
αt =
(2)
υt
with Prob. ε,
where υt is and iid Gaussian error with variance στ2 . Evans and Ramey (2006) use
this particular process because it is straightforward yet econometrically challenging to
estimate. The second scenario we will consider makes the assumption, common in the
time-varying parameters literature, that αt follows an AR(1) process.
αt = ψαt−1 + ωt ,
(3)
where ωt is an iid, mean zero, variance σω white noise process and −1 < ψ < 1. We consider the AR(1) scenario to demonstrate that the endogenous algorithm is particularly
suited to occasional structural breaks not just time variation in general.
We model the expectation formation process using adaptive learning. The adaptive
learning framework considers the conditions under which agents are able to “learn” the
rational expectations solution of a model. These so called E-stability conditions serve
as parameter constraints of the model.3 The E-stability conditions are found by setting
up differential equations based on the relationship between the agents perceived law of
motion (PLM) to the resulting actual law of motion (ALM) when agents expectations
are realized in the model. For the model at hand, the PLM is,
yt = axt−1 + et ,
(4)
yt = (αt + δa)xt−1 + ηt .
(5)
and the resulting ALM is,
If αt = α then we can apply the standard E-stability principle result which, in this
case, implies the equilibrium is E-stable when δ < 1. One might be concerned whether
this model will converge to rational expectations as Bullard (1992) finds that if agents
believe there are time varying parameters they may never learn the rational expectations equilibrium. Instead agents converge to a restricted perception equilibrium since
2
The results found here are robust to setting xt = 1 ∀t. However, the magnitude of the improvements are less.
3
For more on E-stability see Evans and Honkapohja (2001).
3
they are unaware of the underlying time varying process. As pointed out by McGough
(2003) under the right conditions on the stochastic process and the stability parameter
(δ in our case) convergence to the rational expectations equilibrium still obtains.4
In the adaptive learning literature agents recursively updated their estimates of a
using the following algorithm,
at = at−1 + γt Rt−1 xt−1 (xt−1 − xt−1 at−1 ),
Rt = Rt−1 + γt (xt−1 xt−1 − Rt−1 ).
(6)
(7)
where Rt is the variance of xt−1 , and γt is the so called gain parameter. If γt = 1/t
these equations form the standard RLS algorithm and is referred to as decreasing gain
learning. Another common assumption is to set γt equal to some constant. This is
referred to as constant least squares or constant gain learning. In this paper, we assume
that γt will vary endogenously as described below.
2.2
Endogenous Algorithms
Marcet and Nicolini (2003) posit that in a world with occasional structural breaks
agents would prefer to switch between a constant gain and a decreasing gain. They
suggest using an average of past forecast errors to determine when switches occur. In
this context, they propose an algorithm of the following form,
Pt
(
1
i=t−J−1 |yi −ai |
if
< v,
γ̄ −1 +k
J
Pt
(8)
γt =
|y
−a
|
i
i
≥
v
γ̄
if i=t−J−1
J
where k denotes the number of periods since the switch to a decreasing gain, J is
the number of forecast errors used in the average, and v is an arbitrary cutoff point.
Hereafter we refer to this algorithm as MN. Milani (2007) further endogenizes the
algorithm by suggesting that the arbitrary cutoff v be a historical average of forecast
errors. That is, some window of forecast errors larger than J is used to calculate
average forecast errors for the right hand side of the inequalities. In this simple case
such an algorithm, makes considerable sense, should α or δ change value agents would
react if the their forecast errors increase. However, when there are multiple coefficients,
this approach seems rather limited.
By contrast Kostyshyna (2012) draws on the large engineering literature to offer the
adaptive step size algorithm. This algorithm allows the gain parameter to vary based
on probability distributions of the data, and the observation noise. A small gain is
preferred when observation noise is high. The algorithm takes the following form,
Y
e
γt =
(γt−1 + µ(yt−1 − yt−1
)Vt−1 ),
(9)
[γ̄− ,γ̄+ ]
e
Vt = Vt−1 − γt−1 Vt−1 + (yt−1 − yt−1
), V0 = 0.
4
(10)
The specific conditions on the stochastic process are that agents believe that the coefficient of
interest follows a random walk and the associated error decreases over time
4
Q
where µ is the step size parameter, [γ̄− ,γ̄+ ] bounds the value gain, and Vt is the “derivative” of the estimated parameter. The lower bound is not important in applications,
but the upper bound can be influential depending on the stability properties of the
model. The simplicity of our model makes the choice of the upper bound less critical.
Hereafter we refer to this algorithm as K.
Note that both of these methods rely on the forecast errors to adjust the gain
parameter. When an econometrician examines data for potential structural breaks
(s)he looks at the coefficients not the forecast errors. Motivated by this methodology
and the fact that the RLS algorithm and adaptive learning assume that agents have
access to past coefficient estimates, the proposed endogenous gain algorithm uses a
recent coefficient estimate to adjust the gain by the normalized standard deviation.
This results in the following endogenous gain algorithm,
ât −āt σ̄a ,
(11)
γt = γ̄lb + γ̄sf
t
1 + âtσ̄−ā
a
where γ̄lb is the lower bound of the of the endogenous gain, γ̄sf is a scaling factor, ât is
a recent coefficient estimate, āt and σ̄a are the mean and standard deviation of the w
most recent coefficient estimates, respectively. If the recent coefficient is very close to
the mean γt = γ̄lb and as the deviation from the mean increases the value of the gain
approaches γ̄lb + γ̄sf . Therefore, as long as 0 < γ̄lb , γ̄lb < 1 and γ̄lb + γ̄sf < 1 then γt will
be bounded between zero and one. As time progresses agents will increase the value of
the gain in times when their coefficient estimates are different from the recent past and
decrease the value of the gain when their coefficient estimates are similar. Hereafter
we refer to this algorithm as PEG.
3
Forecasting Results
Our results are found via simulation. The simulation procedure starts with an optimization routine to determine the values of the parameters for each of the learning
algorithms. The parameters are optimized to minimize the MSFE of a 50,000 period
simulation. The resulting optimal parameters are then used in 100 independent simulations of 50,000 periods. For MN we optimize over v, j, and γ̄, for the adaptive step
size gain we optimize over µ and the for the proposed algorithm we optimize over w,
γ̄lb , and γ̄sf . We then report the mean and standard deviation of the MSFE for the
last 30,000 periods of the 100 simulations. One could increase the number of periods
or the number of simulations for improved standard deviations, but this would add
a significant amount of computing time. In addition, we calculate the deviations of
the coefficient estimates from the rational expectations solution relative to a constant
gain. The optimal values of the gains are not reported mainly to save space. The
optimal PEG values straddle the optimal constant gain and the gain values increase
as volatility increases. We consider three different expectational environments, no expectational feed back, expectational feedback, and expectational feedback with game
theoretic behavior.
5
Consider (1) with a AR(1) process (3) with ψ = 0.99 and δ = 0. In this case there
will be no expectational feedback and we can focus on the quality of the forecasting.
Table 1 presents the baseline results of several simulations where we have varied the
variance of the AR process. We observe little improvement relative to a constant gain
by all the endogenous gain algorithms. However, the endogenous gains were motivated
by structural breaks and therefore an occasional structural break process (2) may yield
greater improvements. We first demonstrate that these all these algorithms do not
perform well when the breaks are frequent as documented in Table 2. One can find
little difference over a wide range of the variance of the structural break , στ relative
to the variance of the underlying process, ση and over the frequency of the breaks
Table 3 displays the results of simulations of less frequent breaks. The results show
that as the probability of a structural break decreases both MN and PEG improve
performance relative to a constant gain and PEG appears to perform as well or better
than the alternatives. Strikingly, K seems to be equivalent to a constant gain over all
simulations. Performance of PEG increases as the relative variance increases, whereas
MN performance decreases. Table 4 shows the results of the deviations from the RE
solution for the same simulations. Not surprisingly the results mimic the same pattern
found in Table 3. However, one might not expect the same pattern when we add in
expectational feedback.
To that end, we simulate the same model when δ = 0.5 and find a larger improvement
in MSFE for the endogenous gain. Table 5 reports those results for the MSFE. While
we observe a similar pattern for the proposed gain as found with no expectational
feedback, we find the opposite with the alternatives. The MN algorithm decreases
performance as the likelihood of a structural break decreases. Also performance gets
drastically worse as the relative variance increases. The result for the K algorithm is
more subtle. While it does perform worse relative to a constant gain, the performance
is more variable when the relative variances are small. Table 6 gives some insight to
why we observe these striking differences by comparing agents coefficient estimates
to what the RE solution would be at each point in time (that is, for each αt ). The
proposed gain results in coefficient estimates that are much closer to the RE solution
than a constant gain, which is where the improved performance comes from. In contrast
the MN algorithm gets progressively worse. This occurs to a lesser extent with the K
algorithm. These results suggest that with expectational feedback the PEG algorithm
hovers around the closer to the RE solution, while the alternatives wander quite far
away.
These conditions impose that all agents use the same gain process. In order to
determine whether agents would truly choose to use this endogenous gain, consider a
model where all the agents are using some optimal constant gain. Would an agent
be able to improve their forecast if they used and endogenous gain? Figure 1 plots
the values of the optimal response, in terms of MSFE, when the data are generated
with a particular value of constant gain with expectational feedback. It turns out,
that the Nash equilibrium is exactly equal to the value of the optimal constant gain
with expectational feedback. One might also ask whether the proposed gain would
always be a best response over the same range. Figure 2 presents the MSFE within the
model (red line), of the optimal constant gain response (dashed line), and the optimal
proposed gain response (solid line). Just as was found in the previous results the
6
optimal parameters for PEG resulted in a range the encompassed the optimal constant
gain response. Specifically, at the Nash Equilibrium gain value of 0.436 the optimal
endogenous response is γ̄lb = 0.314 and γ̄sf = 0.249. Over all the constant gain values
that generated the data agents will always have an incentive, in terms of MSFE, to
switch to the proposed gain instead of the optimal constant gain. As an example, at
the Nash Equilibrium the MSFE of the constant gain is 5.671 compared with 5.548
with PEG, or about a three percent improvement. This conforms with what one might
perceive about the real world, people who are adaptable, i.e. adjust the information
set relative to past information, tend to predict more accurately than those with more
rigid information sets.
Finally we note that this improvement occurs in a univariate model with only one
coefficient estimate. With a constant gain, MN and K, the gain value will be identical
across all estimated coefficients, however, with PEG there will be a different value for
each coefficient estimate. The agents assess the potential for a structural break in
each of the coefficients separately. Therefore, as the model increases in complexity the
proposed algorithm should have even larger improvements in forecast error.
4
Economic Significance
This section provides two models to demonstrate the economic significance of endogenous gains. The first utilizes the framework of Marcet and Nicolini (2003) to demonstrate that PEG results in similar dynamics as K and MN. This result should not be
surprising since the key feature of Marcet and Nicolini (2003) that results in recurring
hyperinflation episodes is structural change imposed by the exchange rate regime. The
second example demonstrates that changes in the gain parameter over time can, on
their own, result in temporary deviations from the rational expectations equilibrium
in a simple New Keynsian (NK) model.
4.1
Hyperinflation Example
Marcet and Nicolini (2003) develop a model of recurrent hyperinflation that relies
a Cagan-style money demand specification, and a money supply process where the
government either implements an exchange rate regime or seignorage. The reduced
form of the model in terms of inflation and similar notation follows:
πt =
e
1 − γπt+1
1 − γπte − dt /ϕ
(12)
where γ and ϕ are parameters of the money demand equations and dt is as stochastic
process governing seignorage. Under rational expectations there are two deterministic
steady states. Marcet and Nicolini (2003) assume that agents form expectations over
the mean of past inflation rates. The recursive formula for updating the mean βt is as
follows:
βt = βt−1 + gt (πt−1 − βt−1 )
(13)
where gt is the gain parameter.
7
Using the same parameter values as Marcet and Nicolini (2003), specifically γ = 0.4,
φ = 0.37, E(d) = 0.049, and σd = 0.01, one can generate similar results as discussed in
both Marcet and Nicolini (2003) and Kostyshyna (2012) as demonstrated by Figure (3).
The values for the endogenous algorithm were set as γ̄lb = 0.01 and γ̄sf = 0.39, which is
the same range as Kostyshyna (2012). The top panel of the figure displays the time path
of the inflation process and expectations. The bottom panel displays the value of the
gain for each time period. Typically the hyperinflation episodes coincide with a period
where the gain is relatively high. However, a high gain need not precipitate an episode.
Recall that the gain increases when the most recent data causes large changes in the
coefficient estimate. During the hyperinflation episodes the gain increases because
agents make poor predictions of inflation. When the exchange rate regime is enforced
the gain remains high until expectations converge towards the current inflation rate.
Finally, we would like to note that the frequency and severity of the hyperinflation
episodes will be related to the size of the scaling factor (γ̄sf ).
4.2
NK Example
Gaus (2013) shows that MN creates some exotic dynamics when applied to the framework of Duffy and Xiao (2007) and Evans and Honkapohja (2009). In a simple New
Keynesian model, certain parameter sets may be E-stable under a decreasing gain,
but not E-stable under large constant gains. Gaus (2013) demonstrates through simulations that if the constant part of MN is a value that would not be E-stable then
temporary deviations from the rational expectations equilibrium may occur. These
deviations are economically significant, as they exhibit far greater volatility (up to 6
times more) than what would have occurred under rational expectations. A similar
type of dynamics may occur with the proposed gain studied above.
In order to assess the dynamics consider the following NK model, presented in
section 3 of Evans and Honkapohja (2009),5
e
xt = xet+1 − ϕ(it − πt+1
) + gt ,
e
πt = βπt+1 + λxt + ut ,
ϕλ e ϕαx e
π +
x,
it =
αi t
αi t
(14)
(15)
(16)
where ut and gt are AR(1) processes. The following equations govern these processes:
ut = ρut−1 + ũt , and gt = µgt−1 + g˜t ,
where g˜t ∼ iid(0, σg2 ), ũt ∼ iid(0, σu2 ), and 0 < |µ|, |ρ| < 1. Here ϕ is the inter temporal
elasticity of substitution, β is the discount factor, λ is the slope of the Phillips curve, and
αi and αx are relative weights in the monetary policy makers loss function. Substituting
(16) and rewriting the model in reduced form,
e
ξt = M0 ξte + M1 ξt+1
+ P υt
5
See Woodford (2003) for derivation.
8
(17)
where ξt = (xt , yt )0 , υt = (gt , ut )0 , υt = F υt−1 + υ̃t ,
!
ϕ2 αx
ϕ2 λ
1
ϕ
1 0
αi
αi
M0 = ϕ2 λαx ϕ2 λ2
M1 =
P =
λ β + ϕλ
λ 1
α
α
i
F =
µ 0
.
0 ρ
i
Consequently, the MSV solution that serves as the agents PLM is,
yt = a + cυt ,
(18)
with following rational expectations solution,
are = 0
vec(cre ) = (I2 ⊗ M0 + F 0 ⊗ M1 )vec(P )
Note that with the proposed gain there are six coefficients with six different gain value
sequences.
Since PEG requires previous information, all simulations begin with a 40 period
burn-in. During the burn-in each equation receives an additional exogenous error each
period for each equation. This allows for enough variability in the data to generate
variance covariance matrices necessary for the construction of the endogenous-gain.
After the burn-in, the additional exogenous variation shuts down and the simulation
continues without any extraneous noise. Similar to Evans and Honkapohja (2009)
the calibrated parameter values are drawn from Table 6.1 of Woodford (2003), with
αx = 0.048, αi = 0.077, ϕ = 1/0.157, λ = 0.024 and β = 0.99. In addition, µ =
ρ = 0.8 and σg2 = σu2 = 0.2. Evans and Honkapohja (2009) find that this particular
parameterization results in instability if agents use a constant-gain greater than or
equal to 0.024. Therefore we consider γ̄lb = 0.005 and γ̄sf = 0.035 so that the proposed
algorithm may take on a value in the unstable region.
Figure 4 presents the deviations from the rational expectations equilibrium for one
such simulation. Notice the recurrent deviations from the rational expectations, these
continue indefinitely. Whereas Gaus (2013) reports that the deviations occur very
infrequently with MN, these fluctuations occur once every 300 periods on average.
This suggests that PEG implies even greater economic impact of expectations. Also
note there are no structural time varying parameters in this particular model. These
fluctuations occur due to the assumptions on expectation formation. In addition, the
values of PEG remain within the E-stable range, then the deviations do not exist. For
a large enough value of the scaling factor (γ̄sf ) the model will not be stable. For the
intermediate cases as the range between the lower bound, γ̄lb , and the scaling factor,
γ̄sf , increases the departures from RE become more frequent and more severe.
5
Conclusion
This paper systematically explored a proposed endogenous gain that outperforms MN
and K. Specifically we examine a model with occasional structural breaks and find
that the proposed gain is capable of improving forecasting accuracy because it remains
9
closer to the RE equilibrium. We have also demonstrated that agents will have an
incentive, through lower forecasting errors, to switch from a pure constant gain to
an endogenous gain. In addition, simulation of a hyperinflation model and a simple
NK model suggest that the endogenous gain may create additional recurrent dynamics
with significant economic impact. The intuition behind the dynamics found in the NK
model is particularly compelling, agents, wary of structural breaks, may use too little
information to form expectations leading to a period of non-rational economic outcomes. Eventually agents realize their mistakes and reacquire the rational expectation
equilibrium by using more data to form expectations. Endogenous gains seem to be
a natural outgrowth of the adaptive learning literature and have potential to explain
macroeconomic phenomena.
References
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Economics Workshop of the National Bureau of Economic Research Summer Institute July 2007.
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Duffy, John and Wei Xiao, “The value of interest rate stabilization policies when
agents are learning,” Journal of Money, Credit and Banking, 2007, 39 (8), 2041–
2056.
Evans, George W and G Ramey, “Adaptive expectations, underparameterization
and the Lucas critique,” Journal of Monetary Economics, 2006, 53 (2), 249–264.
and Seppo Honkapohja, Learning and expectations in macroeconomics, Princeton: Princeton University Press, January 2001.
and
, “Robust learning stability with operational monetary policy rules,” in
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and Learning, Santiago: Central Bank of Chile, 2009, pp. 145–170.
Gaus, Eric, “Robust Stability of Monetary Policy Rules under Adaptive Learning,”
Southern Economic Journal, January 2013, pp. 1–29.
Kostyshyna, Olena, “Application of an Adaptive Step-Size Algorithm in Models of
Hyperinflation,” Macroeconomic Dynamics, November 2012, 16, 355–375.
Marcet, Albert and Juan P Nicolini, “Recurrent hyperinflations and learning,”
American Economic Review, 2003, 93 (5), 1476–1498.
McGough, Bruce, “Statistical Learning with Time-Varying Parameters,” Macroeconomic Dynamics, 2003, 7 (01), 119–139.
Milani, Fabio, “Learning and time-varying macroeconomic volatility,” Manuscript,
UC-Irvine, 2007.
10
Stock, James H and Mark W Watson, “Why Has U.S. Inflation Become Harder to
Forecast?,” Journal of Money, Credit and Banking, February 2007, 39 (1), 3–33.
Woodford, Michael, Interest and prices: foundations of a theory of monetary policy,
Princeton: Princeton University Press, 2003.
11
Table 1: Relative Performance: AR(1), No Feedback
MSFE (Std.)
Deviation from RE
σω
PEG
MN
K
PEG MN
K
0.1 1.000(0.000) 1.000(0.000) 1.000(0.000)
1.000 1.000
1.000
0.2 1.000(0.000) 1.000(0.000) 1.000(0.000)
0.999 1.000
1.000
0.998 1.000
1.000
0.3 0.999(0.001) 1.000(0.000) 1.000(0.000)
0.993 1.001
1.000
0.4 0.998(0.001) 1.000(0.000) 1.000(0.000)
0.5 0.996(0.002) 1.001(0.001) 1.000(0.000)
0.990 1.003
1.000
0.987 1.005
1.000
0.6 0.995(0.002) 1.002(0.001) 1.000(0.000)
0.7 0.993(0.002) 1.005(0.002) 1.000(0.000)
0.979 1.016
1.000
0.8 0.993(0.003) 1.012(0.003) 1.170(0.021)
0.986 1.032
1.469
0.9 0.992(0.003) 1.021(0.004) 1.000(0.000)
0.981 1.073
1.000
0.969 1.115
1.003
1.0 0.991(0.004) 1.039(0.006) 1.000(0.001)
Values reported in the first panel are the MSFE relative to the optimal constant
gain benchmark with the standard deviations in parenthesis. The second panel
displays the squared deviations from the RE solution relative to a constant gain.
PEG stands for proposed the endogenous gain, MN stands for the Marcet and
Nicolini algorithm, and K represents the Adaptive Step-Size algorithm. Note that
for the MN algorithm is optimized over v, j and the γ̄, the Adaptive Step-Size
algorithm optimized over the µ parameter and PEG is optimized over w, γ̄lb and
γ̄sf .
Figure 1: The Nash equilibrium constant-gain value. στ = 5, ση = 2, and ε = 0.01.
Note: The red line is the 45-degree line and the black line is the optimal constant
gain response (based on MSFE) to the data generating gain value.
12
Table 2: Relative MSFE Performance: Structural Break, Without Feedback
ε
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
PEG
(0.000) (0.000) (0.000) (0.000) (0.000) (0.000) (0.000) (0.000)
1.000
0.999
0.999
0.999
1.000
0.999
1.033
1.154
1.25 MN
(0.000) (0.000) (0.001) (0.000) (0.000) (0.002) (0.009) (0.016)
1.268
1.005
1.153
1.108
1.209
1.157
1.000
1.006
K
(0.013) (0.001) (0.010) (0.009) (0.013) (0.012) (0.000) (0.002)
1.008
1.000
1.000
1.000
1.000
1.000
1.000
1.000
PEG
(0.001) (0.000) (0.000) (0.000) (0.000) (0.000) (0.000) (0.000)
1.009
1.000
1.000
0.999
0.999
1.015
1.181
1.381
2.5 MN
(0.001) (0.000) (0.000) (0.001) (0.001) (0.014) (0.025) (0.023)
1.010
1.198
1.197
1.195
1.115
1.060
1.000
1.000
K
(0.004) (0.013) (0.012) (0.014) (0.010) (0.008) (0.001) (0.000)
1.000
1.000
1.000
1.000
1.000
1.000
1.000
0.999
PEG
(0.000) (0.000) (0.000) (0.000) (0.000) (0.000) (0.000) (0.001)
0.999
1.000
0.999
0.999
1.000
1.084
1.291
1.402
5 MN
(0.000) (0.000) (0.001) (0.001) (0.001) (0.021) (0.024) (0.025)
1.179
1.171
1.154
1.162
1.087
1.000
1.002
1.000
K
(0.013) (0.011) (0.012) (0.012) (0.010) (0.000) (0.002) (0.001)
1.000
1.000
1.000
1.000
1.000
1.000
0.999
0.996
PEG
(0.000) (0.000) (0.000) (0.000) (0.000) (0.000) (0.001) (0.003)
0.998
0.996
1.000
1.000
1.047
1.153
1.238
1.337
10 MN
(0.001) (0.001) (0.000) (0.001) (0.020) (0.017) (0.021) (0.026)
1.026
1.171
1.182
1.035
1.001
1.000
1.001
1.000
K
(0.004) (0.012) (0.013) (0.005) (0.001) (0.001) (0.001) (0.001)
1.000
1.001
1.000
1.000
1.000
1.000
0.998
0.995
PEG
(0.000) (0.000) (0.000) (0.000) (0.000) (0.001) (0.005) (0.004)
1.000
0.996
1.000
0.999
1.039
1.115
1.150
1.252
20 MN
(0.000) (0.001) (0.000) (0.001) (0.013) (0.015) (0.015) (0.021)
1.124
1.183
1.116
1.101
1.000
1.000
1.015
1.001
K
(0.010) (0.012) (0.010) (0.010) (0.000) (0.000) (0.005) (0.002)
Values reported are the MSFE relative to the optimal constant gain benchmark with the
standard deviations in parenthesis. PEG stands for proposed the endogenous gain, MN
stands for the Marcet and Nicolini algorithm, and K represents the Adaptive Step-Size
algorithm. Note that for the MN algorithm is optimized over v, j and the γ̄, the Adaptive
Step-Size algorithm optimized over the µ parameter and PEG is optimized over w, γ̄lb and
γ̄sf .
στ /ση
13
Table 3: Relative MSFE Performance: Structural Break, Without Feedback
ε
0.1
0.09
0.08
0.07
0.06
0.05
0.04
0.03
0.02
0.01
στ /ση
1.000
1.000
1.000
1.000
1.000
1.000
1.000
0.999
0.998
0.996
PEG
(0.000) (0.000) (0.000) (0.000) (0.000) (0.000) (0.000) (0.001) (0.002) (0.002)
1.226
1.216
1.169
1.165
1.117
1.083
1.044
1.016
0.998
0.997
1.25 MN
(0.013) (0.012) (0.010) (0.011) (0.009) (0.008) (0.006) (0.005) (0.003) (0.001)
1.000
1.000
1.000
1.025
1.006
1.000
1.000
1.000
1.000
1.014
K
(0.000) (0.000) (0.000) (0.005) (0.002) (0.000) (0.000) (0.000) (0.000) (0.004)
0.997
0.996
0.996
0.995
0.995
0.994
0.993
0.990
0.987
0.983
PEG
(0.003) (0.002) (0.003) (0.003) (0.002) (0.003) (0.002) (0.005) (0.004) (0.005)
1.360
1.331
1.327
1.282
1.253
1.177
1.121
1.060
1.005
0.993
2.5 MN
(0.018) (0.021) (0.020) (0.019) (0.017) (0.017) (0.013) (0.013) (0.008) (0.004)
1.000
1.000
1.000
1.002
1.000
1.000
1.000
1.002
1.000
1.000
K
(0.000) (0.000) (0.001) (0.002) (0.000) (0.000) (0.000) (0.002) (0.001) (0.001)
0.993
0.992
0.987
0.986
0.983
0.981
0.979
0.976
0.971
0.966
PEG
(0.003) (0.005) (0.005) (0.005) (0.006) (0.006) (0.006) (0.009) (0.009) (0.012)
1.432
1.387
1.380
1.363
1.329
1.285
1.233
1.143
1.082
1.001
5 MN
(0.028) (0.030) (0.025) (0.027) (0.029) (0.025) (0.024) (0.022) (0.023) (0.011)
1.000
1.000
1.001
1.039
1.000
1.000
1.000
1.001
1.001
1.001
K
(0.000) (0.000) (0.002) (0.011) (0.000) (0.000) (0.001) (0.001) (0.001) (0.003)
0.984
0.983
0.982
0.981
0.977
0.976
0.971
0.972
0.960
0.952
PEG
(0.006) (0.007) (0.007) (0.007) (0.008) (0.010) (0.010) (0.015) (0.015) (0.015)
1.436
1.423
1.424
1.420
1.411
1.410
1.355
1.315
1.214
1.045
10 MN
(0.032) (0.034) (0.034) (0.037) (0.036) (0.039) (0.040) (0.041) (0.040) (0.024)
1.000
1.000
1.000
1.000
1.001
1.000
1.000
1.000
1.000
1.002
K
(0.000) (0.001) (0.001) (0.000) (0.002) (0.001) (0.000) (0.001) (0.000) (0.004)
0.987
0.986
0.983
0.983
0.979
0.976
0.972
0.972
0.959
0.950
PEG
(0.007) (0.008) (0.008) (0.009) (0.010) (0.010) (0.017) (0.013) (0.017) (0.018)
1.416
1.446
1.426
1.436
1.436
1.472
1.478
1.437
1.353
1.107
20 MN
(0.035) (0.040) (0.041) (0.040) (0.047) (0.048) (0.061) (0.057) (0.060) (0.043)
1.000
1.007
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
K
(0.000) (0.005) (0.000) (0.000) (0.001) (0.001) (0.000) (0.001) (0.002) (0.001)
Values reported are the MSFE relative to the optimal constant gain benchmark with the standard deviations in
parenthesis. PEG stands for proposed the endogenous gain, MN stands for the Marcet and Nicolini algorithm,
and K represents the Adaptive Step-Size algorithm. Note that for the MN algorithm is optimized over v, j and
the γ̄, the Adaptive Step-Size algorithm optimized over the µ parameter and PEG is optimized over w, γ̄lb and
γ̄sf .
14
Table 4: Relative deviations from the RE solution: Structural Break, Without Feedback
ε
0.1
0.09
0.08
0.07
0.06
0.05
0.04
0.03
0.02
0.01
PEG 1.000 1.000 1.000 0.999 0.998 0.998 1.000 0.992 0.988 0.972
1.25 MN 1.874 1.851 1.648 1.677 1.469 1.337 1.195 1.060 0.970 0.983
K
1.000 1.000 1.000 1.113 1.018 1.000 1.000 1.000 1.000 1.072
PEG 0.983 0.978 0.982 0.977 0.975 0.971 0.983 0.977 0.952 0.904
2.5 MN 2.086 2.006 1.953 1.940 1.777 1.584 1.362 1.158 0.974 0.963
K
1.000 1.000 1.002 1.001 1.000 1.001 1.000 1.000 0.999 1.007
PEG 0.951 0.952 0.955 0.937 0.943 0.935 0.959 0.942 0.965 0.923
5 MN 2.438 2.324 2.209 2.016 1.973 1.793 1.701 1.462 1.262 1.018
K
1.000 1.000 1.003 1.124 1.000 1.000 1.000 0.998 1.001 1.009
PEG 0.902 0.882 0.903 0.863 0.897 0.869 0.854 0.921 0.906 0.945
10 MN 2.944 2.958 2.917 2.787 2.605 2.572 2.141 2.054 1.553 1.097
K
1.000 0.998 1.000 1.000 1.002 1.000 1.000 0.999 1.000 1.005
PEG 0.872 0.878 0.915 0.873 0.873 0.845 0.837 0.769 0.766 0.937
20 MN 4.010 4.033 3.861 3.610 3.323 3.467 3.487 3.119 2.493 1.368
K
1.000 1.050 0.999 1.000 1.001 0.999 1.000 1.003 0.998 1.000
Values reported are the squared deviations from the RE solution relative to a constant gain.
PEG stands for proposed the endogenous gain, MN stands for the Marcet and Nicolini algorithm, and K represents the Adaptive Step-Size algorithm. Note that for the MN algorithm
is optimized over v, j and the γ̄, the Adaptive Step-Size algorithm optimized over the µ
parameter and PEG is optimized over w, γ̄lb and γ̄sf .
στ /ση
Figure 2: The MSFE of best responses of constant and endogenous gains to particular
constant gain values. στ = 5, ση = 2, and ε = 0.01.
Note: The red line indicates the errors within the data generated by the gain
value on the x-axis. The dashed line represents the MSFE of the optimal constant
gain response and the solid black line represents the MSFE of the optimal PEG
response.
15
Table 5: Relative MSFE Performance: Structural Break, With Feedback
ε
0.1
0.09
0.08
0.07
0.06
0.05
0.04
0.03
0.02
0.01
στ /ση
0.999
0.999
0.998
0.997
0.996
0.996
0.995
0.994
0.992
0.992
PEG
(0.001) (0.001) (0.001) (0.001) (0.002) (0.001) (0.002) (0.002) (0.002) (0.002)
1.398
1.434
1.476
1.528
1.587
1.646
1.727
1.816
1.945
2.122
1.25 MN
(0.030) (0.029) (0.033) (0.039) (0.045) (0.048) (0.054) (0.064) (0.089) (0.138)
1.529
1.588
1.695
1.722
1.851
1.781
1.672
1.889
1.000
1.873
K
(0.148) (0.213) (0.173) (0.231) (1.461) (0.198) (0.186) (0.313) (0.000) (0.303)
0.988
0.987
0.987
0.983
0.982
0.980
0.978
0.975
0.974
0.973
PEG
(0.003) (0.003) (0.003) (0.004) (0.004) (0.004) (0.004) (0.005) (0.006) (0.006)
2.045
2.149
2.270
2.417
2.602
2.804
3.093
3.458
3.993
4.835
2.5 MN
(0.056) (0.068) (0.072) (0.085) (0.093) (0.117) (0.157) (0.172) (0.259) (0.488)
1.000
1.485
1.000
1.483
1.501
1.608
1.610
1.589
1.725
1.826
K
(0.000) (0.105) (0.000) (0.086) (0.094) (0.549) (0.194) (0.144) (0.386) (0.206)
0.978
0.976
0.973
0.971
0.967
0.965
0.958
0.953
0.947
0.945
PEG
(0.006) (0.006) (0.005) (0.006) (0.007) (0.008) (0.007) (0.009) (0.009) (0.012)
2.779
2.977
3.222
3.518
3.874
4.389
5.064
6.076
7.606
10.750
5 MN
(0.106) (0.118) (0.128) (0.166) (0.208) (0.215) (0.295) (0.353) (0.564) (1.109)
1.457
1.430
1.438
1.467
1.462
1.533
1.518
1.558
1.595
1.611
K
(0.048) (0.067) (0.067) (0.072) (0.051) (0.054) (0.074) (0.072) (0.140) (0.184)
0.979
0.975
0.973
0.970
0.967
0.961
0.958
0.952
0.940
0.925
PEG
(0.005) (0.006) (0.008) (0.010) (0.008) (0.010) (0.009) (0.011) (0.012) (0.013)
3.322
3.590
3.961
4.396
4.959
5.732
6.945
8.568
11.780 19.304
10 MN
(0.141) (0.158) (0.174) (0.213) (0.248) (0.338) (0.468) (0.625) (1.106) (2.195)
1.562
1.464
1.606
1.499
1.491
1.499
1.687
1.667
1.656
1.652
K
(0.040) (0.043) (0.044) (0.041) (0.040) (0.049) (0.061) (0.089) (0.088) (0.572)
0.981
0.979
0.980
0.976
0.973
0.971
0.966
0.967
0.949
0.932
PEG
(0.018) (0.013) (0.007) (0.010) (0.030) (0.009) (0.014) (0.012) (0.012) (0.017)
3.661
3.996
4.406
4.928
5.677
6.661
8.034
10.376 14.719 26.505
20 MN
(0.171) (0.198) (0.211) (0.257) (0.351) (0.418) (0.586) (0.885) (1.275) (3.320)
1.612
1.563
1.553
1.592
1.697
1.798
1.631
1.627
1.639
1.764
K
(0.045) (0.043) (0.046) (0.044) (0.074) (0.068) (0.064) (0.066) (0.079) (0.127)
Values reported are the MSFE relative to the optimal constant gain benchmark with the standard deviations in
parenthesis. PEG stands for proposed the endogenous gain, MN stands for the Marcet and Nicolini algorithm,
and K represents the Adaptive Step-Size algorithm. Note that for the MN algorithm is optimized over v, j and
the γ̄, the Adaptive Step-Size algorithm optimized over the µ parameter and PEG is optimized over w, γ̄lb and
γ̄sf .
16
Table 6: Relative deviations from the RE solution: Structural Break, With Feedback
ε
0.1
0.09
0.08
0.07
0.06
0.05
0.04
0.03
0.02
0.01
PEG 1.000 1.000 1.000 0.999 0.998 0.998 1.000 0.992 0.988 0.972
1.25 MN 1.874 1.851 1.648 1.677 1.469 1.337 1.195 1.060 0.970 0.983
K
1.000 1.000 1.000 1.113 1.018 1.000 1.000 1.000 1.000 1.072
PEG 0.951 0.957 0.955 0.941 0.951 0.939 0.944 0.930 0.936 0.884
2.5 MN 4.125 4.344 4.597 4.618 5.644 6.228 7.014 7.786 9.973 13.72
K
1.000 2.046 1.000 2.539 2.404 2.453 3.489 2.403 3.203 3.392
PEG 0.921 0.925 0.926 0.909 0.926 0.915 0.907 0.903 0.911 0.874
5 MN 6.093 6.403 7.163 7.627 7.917 9.347 10.96 12.77 17.62 25.19
K
2.234 2.104 2.110 2.011 2.209 2.261 2.239 2.141 2.257 2.581
PEG 0.917 0.903 0.915 0.909 0.908 0.895 0.894 0.881 0.876 0.853
10 MN 7.868 8.946 10.32 10.34 11.83 13.97 14.92 18.03 26.09 33.61
K
2.783 2.527 2.820 2.442 2.326 2.318 2.522 2.454 2.494 2.489
PEG 0.927 0.948 0.936 0.926 0.899 0.898 0.911 0.883 0.846 0.899
20 MN 10.91 11.66 12.46 14.26 14.11 18.14 22.50 28.37 38.51 65.68
K
3.196 2.981 3.040 2.798 2.825 3.264 3.059 3.012 2.384 2.841
Values reported are the squared deviations from the RE solution relative to a constant gain.
PEG stands for proposed the endogenous gain, MN stands for the Marcet and Nicolini algorithm, and K represents the Adaptive Step-Size algorithm. Note that for the MN algorithm
is optimized over v, j and the γ̄, the Adaptive Step-Size algorithm optimized over the µ
parameter and PEG is optimized over w, γ̄lb and γ̄sf .
στ /ση
Figure 3: Recurring hyperinflation episodes
17
Figure 4: Economic fluctuations of an endogenous gain.
18