Transcript Estimating Time Varying Preferences of the FED

Estimating Time Varying
Preferences of the FED
Ümit Özlale
Bilkent University,
Department of Economics
OUTLINE: Introduction

INTRODUCTION
Change in the conduct of monetary policy
 Estimated policy rules vs. Optimal policy
rules
 What’s missing?
 What is the contribution of this paper?

The U.S. economy since late 1970’s

General consensus: Favorable economic
outcomes in the U.S. economy since the late
1970’s.

Little consensus: Role of monetary policy

Several papers, including Clarida et al (2000, QJE)
report a change in the conduct of monetary policy,
which contributes to overall improvement in the
economy
Why is there a change in the
conduct of monetary policy?

Fed’s preferences have changed over time

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Variance and nature of shocks changed.
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References: Romer and Romer(1989, NBER), Favero and
Rovelli (2003, JMCB), Ozlale (2003, JEDC), Dennis (2005,
JAE)
References: Hamilton (1983, JPE), Sims and Zha (2006,
AER)
Learning and changing beliefs about the economy

References: Sargent (1999), Taylor (1998), Romer and Romer
(2002)
Estimated Policy Rules vs. Optimal
Policy Rules

To understand the changes in the monetary
policy, two main approaches:

Estimate interest rate rules, which started with the
celebrated Taylor Rule

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Some references: Taylor (1993, Carnegie-Rochester CS),
Boivin (2007, JMCB)
Derive optimization based policy rules

Some references: Rotemberg and Woodford (1997,
NBER), Rudebusch and Svensson (1998, NBER)
Estimated Policy Rules
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Advantages:
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Capturing the systematic relationship between
interest rates and macroeconomic variables
Empirical support
Disadvantages:

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Do not satisfy a structural understanding of
monetary policy
Unable to address questions about policy
formulation process or policy regime change
Optimal Policy Rules
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Advantages:

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Optimization based policy rules
Theoretical strength
Disadvantages:

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Cannot adequately explain how interest rates move
over time.
Estimate more aggressive responses to shocks
than typically observed.
Combining optimal rule with the data
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Combine the two areas by:
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

Assuming that monetary policy is set optimally
Estimating the policy function along with the
parameters that characterize the economy
References:
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Salemi (1995, JBES) uses inverse control
Favero and Rovelli (2003, JMCB) uses GMM
Ozlale (2003, JEDC) uses optimal linear regulator
Dennis (2004, OXBES and 2005, JAE) uses optimal linear
regulator
Combining optimal rule with the data
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Advantages:
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Assess whether observed outcomes can be
reconciled within optimal policy framework
Assess whether the objective function has changed
over time
Allows key parameters to be estimated
Disadvantages:

None!
A general framework

Specify a quadratic loss function and AS-AD
system such as:

Lt =E t   j [ ( t  j   * )2 y ( yt  j ) 2  i (it  j  it  j 1 ) 2 ]
j 0
subject to the following linear constraints:
 t 1  f ( t , yt )
yt 1  g ( yt , it   t )
A general framework

Each period, the central bank attempts to minimize
a loss function

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Which depends on the deviations from inflation,
output gap and interest rate targets
The preferences of the central bank are  , y , i
The linear constraints are inflation and output gap
equations.
Inflation is expected to have an inertia and it is
affected from the output gap.
The output gap is affected from the real interest rate
Solving via Optimal Linear Regulator

When the loss function is quadratic and the
constraints are linear, the problem can be regarded as
a stochastic optimal linear regulator problem, for
which the solution takes the form:
it   fX t

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which means that the control variable, which is the
interest rate, is a function of the state variables in the
model
The vector f contains both the loss function
(preference) and the system parameters to be
estimated.
Estimation
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One way to estimate the parameters is to
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Cast the model in state space form
Developing a MLE for the problem
Under certain conditions, executing the
Kalman filter provide consistent and efficient
estimates
Main findings
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A substantial change in the Fed’s response to
inflation and output gap

The response of Fed to inflation has become
more aggressive since the late 1970’s.
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There is an incentive for the Fed to smooth the
interest rates
What’s missing?

The preferences that characterize the loss
function are assumed to stay constant over
time.
 In technical terms, previous studies did not
allow for a continual drift in the policy objective
function.
 Thus, these studies could not identify
preference shocks of the Federal Reserve.
What to do?

We allow for the preference parameters in the
loss function to vary over time, while keeping
the linear constraints:

Lt =E t   j [ ,t ( t  j   * ) 2  y ,t ( yt  j ) 2 ]
j 0
 ,t   ,t 1    ,t
 y ,t   y ,t 1   y ,t
Estimation method

We use a two-step procedure:
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1st step: Estimate the linear optimization
constraints, which are the parameters in the
inflation and the output gap equation.
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2nd step: Conditional upon the estimated
constraints, estimate the time-varying preferences
of the Fed.
Main contribution of the paper

Generate a time series that will reflect the
preferences of the Fed.
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Identify Fed’s preference shocks from the
data.
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In technical terms: Given the linear constraints
and the state variables, estimate the timevarying parameters in a quadratic objective
function.
Related work
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Sargent, Williams and Zha (2006, AER) find that Fed’s
optimal policy is changing because of a change in the
parameters of the Phillips curve (not because of a
change in the parameters of the objective function)

Boivin (2007, JMCB) uses a time-varying set-up to
investigate the changes in the parameters of a
forward-looking Taylor-type rule. However, he does not
consider a change in the preferences of the objective
function.
OUTLINE: The Model

The Model
Introducing the model
 Theoretical support for the loss function
 Empirical support for the backward-looking
model
 Estimating the optimization constraints
 Estimating time-varying preferences

The Model: Loss Function

We assume that the loss function is:

Lt =E t   j [ ,t ( t  j )2 y ,t ( yt  j )2 ]
j 0

The preferences vary over time.
 We specify a random walk process:
 ,t   ,t 1   ,t

For simplicity, we assume that
 ,t  y ,t  1
Theoretical Support: Loss Function
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A quadratic loss function, although hypothetical, is
convenient set-up for solving and analyzing linearquadratic stochastic dynamic optimization problems
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Supporting references: Svensson (1997) and
Woodford (2002)
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Since inflation data is constructed as deviation from
the mean, we did not specify any inflation target.
Theoretical Support: Loss Function
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The assumption of random walk:
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Cooley and Prescott (1976, Ecta) state that a
random walk assumption is the best way to account
for the Lucas’ critique.

A TVP specification has the ability to uncover
changes of a general and potentially permanent
nature for each parameter separately.
Linear Constraints

The linear constraints of the model are
3
 t 1    j t  j   y yt   t 1
j 0
1
yt 1    j yt  j
j 0


1 3
  r (  it  k   t  k )  t 1
4 k 0
To satisfy the long-run Phillips curve, coefficients of
the lagged inflation terms sum up to unity.
This backward looking model is adopted from
Rudebusch and Svensson and it is used in several
studies, including Dennis (2005, JAE)
Empirical Support: Backward
Looking Model
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Forward looking models tend not to fit the data
as well as the Rudebusch-Svensson model,
which is also reported in Estrella and Fuhrer
(2002)
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There is no evidence of parameter instability in
this version of the backward-looking model, as
stated in Ozlale (2003)
Estimating the optimization
constraints: Data
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We use monthly data from 1970:2 to 2004:10, where
the output gap is derived by using a linear quadratic
trend.

For robustness purposes, we also use quarterly data,
where inflation is derived from GDP chain weighted
price index, the output gap series is taken from CBO.

In each case, we use federal funds rate as the policy
(control) variable.
Estimating the optimization
constraints: SUR

We estimate the parameters in the
backward looking model by using the
Seemingly Unrelated Regression.

Estimating each equation by OLS returns
similar results, implying weak/no
correlation between the residuals.
Estimated Parameters
 t 1  0.38 t  0.17 t 1  0.30 t 2  0.15 t 3  0.08 yt   t 1
 2  1.35
1 3
yt 1  1.21yt  0.28 yt 1  0.02(  it k   t k )  t 1
4 k 0
 2  1.40
Estimating Time Varying
Preferences: Method
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Step 1:


The solution for the optimal linear regulator is:
Step 2:

itopt   ft xt
opt
observed
i

i
 et be the difference (control
Let t
t
error) between the observed control variable
and the optimal control variable.
Some Boring Stuff!
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In the Kalman filtering algorithm, the
estimate for the state vector is:
xˆt t 1  t 1xˆt 1t 1  Bt 1et 1
which can also be written as:
observed
xˆt t 1  t 1 xˆt 1 t 1  Bt 1 (itopt

i
)
1
t 1

Since the optimal feedback rule for the
linear regulator is
it   ft 1xˆt 1t 1
Still Boring!
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The new state vector is
xˆt t 1  (t 1  Bt 1 ft 1 ) xˆt 1 t 1  Bt 1itobs
1

For simplicity, let At 1  (t 1  Bt 1Ft 1 )

Then, the problem reduces down to obtaining the
elements of At 1 at each step .
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Keep in mind that the matrix includes the parameters
of the model.
How to estimate the loop

The model can be cast in a non-linear state
space model.

The linear Kalman filter is inappropriate for the
non-linear cases.

Thus, we use the extended Kalman filter and
estimate both the optimal control sequence
and the time-varying parameters in the model.
Outline: Estimation Results
Time varying preference series
 Identifying preference shocks
 Comparing observed and optimal
interest rates
 Robustness checks

Time varying preferences
Time varying preferences
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Regardless of the starting values, the preference
parameter for output stability goes down to zero.
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Such a finding is consistent with Dennis (2005, JAE),
which states that output gap enters the policymaking
process only because its indirect effect on inflation.

The estimated series follow random walk, which is
consistent with our initial assumptions.
Preference Shocks
.008
.006
.004
.002
.000
-.002
-.004
-.006
1975
1980
1985
1990
1995
LAMBDAPI Residuals
2000
Preference shocks

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Beginning with the second half of 1980’s we do not
observe any significant shocks in the policy
preferences. Thus, the Greenspan period is silent in
terms of preference changes.
The significantly positive shocks, which indicate an
increased emphasis on price stability occur in the
Volcker period.
Such a finding supports the view that Volcker period is
a one-time discrete change in the policy.
These shocks are found to be normally distributed and
autocorrelated.
Actual vs. optimal interest rates
Actual vs. optimal interest rates

The estimated interest rate is slightly sharper than the
observed interest rate, which may be related to the
absence of interest rate smoothing in the loss function.
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The correlation between the two series is found to be
0.93.

Such a finding implies that the observed control
sequence (interest rate) can be generated by putting
increasingly more emphasis on price stability.
Robustness Checks

In order to see whether the estimated results
are robust, we set the optimization constraints
according to the findings of two studies, which
use the same model
 Rudebusch and Svensson (1998, NBER)
 Dennis (2005, JAE)
Using the estimated coefficients
from Rudebusch and Svensson
Using the estimated coefficients
from Rudebusch and Svensson
Using the estimated coefficients
from Rudebusch and Svensson
.012
.008
.004
.000
-.004
-.008
1975
1980
1985
1990
1995
LPIRS Residuals
2000
Using the estimated coefficients
from Dennis
Using the estimated coefficients
from Dennis
Using the estimated coefficients
from Dennis
.012
.008
.004
.000
-.004
-.008
1975
1980
1985
1990
1995
LPID Residuals
2000
Correlation between preference
shocks
Corr (RS, DE)=0.98
 Corr (RS, OZ)=0.90
 Corr (OZ, DE)=0.91


These findings provide robustness for
the estimation methodology and the
results.
Interest rate smoothing

Several studies, including mine!, except Rudebusch
(2002, JME) have found that interest rate smoothing is
an important criteria for the Fed.

Rudebusch (2002) states that lagged interest rates
soak up the persistence implied by serially correlated
policy shocks.

Given that, we find a serial correlation in preference
shocks, Rudebush (2002) argument seems to be valid.
Results

In this paper, we showed that, given the state
of the economy, it is possible to estimate the
“hidden” time-varying preferences of the Fed.

Such a methodology also allows us to
generate the preference shocks of the Fed.
Results

The results are consistent with the literature:


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The weight of the output gap in the loss function
goes down to zero, implying that output gap is
important as long as it affects inflation
There is a one-time discrete change in policy in the
Volcker period. The Greenspan period is silent.
It is possible to generate almost identical interest
rates, even without imposing interest rate
smoothing incentive to the loss function.
Further research

The paper can be significantly improved if the
parameters in the constraints and the preferences are
simultaneously estimated.

Estimating time-varying preferences for inflation
targeting and non-inflation targeting countries will
provide important clues about whether the overall
decrease in inflation rates for IT countries can be
explained by a preference change.