Estimating Time Varying Preferences of the FED

Ümit ÖzlaleBilkent University, Department of Economics

OUTLINE: Introduction

INTRODUCTIONChange in the conduct of monetary policyEstimated policy rules vs. Optimal policy

rulesWhat’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 References: Romer and Romer(1989, NBER), Favero and

Rovelli (2003, JMCB), Ozlale (2003, JEDC), Dennis (2005, JAE)

Variance and nature of shocks changed. 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 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

Advantages: Capturing the systematic relationship between

interest rates and macroeconomic variables Empirical support

Disadvantages: Do not satisfy a structural understanding of

monetary policy Unable to address questions about policy

formulation process or policy regime change

Optimal Policy Rules

Advantages: Optimization based policy rules Theoretical strength

Disadvantages: Cannot adequately explain how interest rates move

over time. Estimate more aggressive responses to shocks

than typically observed.

Combining optimal rule with the data

Combine the two areas by: Assuming that monetary policy is set optimally Estimating the policy function along with the

parameters that characterize the economy References:

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

Advantages: 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:

subject to the following linear constraints:

* 2 2 2t 1

0

=E [ ( ) ( ) ( ) ]jt t j y t j i t j t j

j

L y i i

1

1

( , )

( , )t t t

t t t t

f y

y g y i

A general framework

Each period, the central bank attempts to minimize a loss function Which depends on the deviations from inflation,

output gap and interest rate targets The preferences of the central bank are 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

, ,y i

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:

which means that the control variable, which is the interest rate, is a function of the state variables in the model

The vector contains both the loss function (preference) and the system parameters to be estimated.

t ti fX

f

Estimation

One way to estimate the parameters is to 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

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.

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:

* 2 2t , ,

0

, , 1 ,

, , 1 ,

=E [ ( ) ( ) ]jt t t j y t t j

j

t t t

y t y t y t

L y

Estimation method

We use a two-step procedure: 1st step: Estimate the linear optimization

constraints, which are the parameters in the inflation and the output gap equation.

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.

Identify Fed’s preference shocks from the data.

In technical terms: Given the linear constraints and the state variables, estimate the time-varying parameters in a quadratic objective function.

Related work

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 modelTheoretical support for the loss functionEmpirical support for the backward-looking

modelEstimating the optimization constraintsEstimating time-varying preferences

The Model: Loss Function

We assume that the loss function is:

The preferences vary over time. We specify a random walk process:

For simplicity, we assume that

2 2t , ,

0

=E [ ( ) ( ) ]jt t t j y t t j

j

L y

, , 1 ,t t t

, , 1t y t

Theoretical Support: Loss Function

A quadratic loss function, although hypothetical, is convenient set-up for solving and analyzing linear-quadratic stochastic dynamic optimization problems

Supporting references: Svensson (1997) and Woodford (2002)

Since inflation data is constructed as deviation from the mean, we did not specify any inflation target.

Theoretical Support: Loss Function

The assumption of random walk: 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

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)

3

1 10

1 3

1 10 0

1( )4

t j t j y t tj

t j t j r t k t k tj k

y

y y i

Empirical Support: Backward Looking Model

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)

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

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

1 1 2 3 1

2

3

1 1 10

2

0.38 0.17 0.30 0.15 0.08

1.35

11.21 0.28 0.02( )

4

1.40

t t t t t t t

t t t t k t k tk

y

y y y i

Estimating Time Varying Preferences: Method

Step 1: The solution for the optimal linear regulator is:

Step 2: Let be the difference (control

error) between the observed control variable and the optimal control variable.

opt observedt t ti i e

opttt ti f x

Some Boring Stuff!

In the Kalman filtering algorithm, the estimate for the state vector is:

which can also be written as:

Since the optimal feedback rule for the linear regulator is

1 1 11 1 1ˆ ˆt t tt t t tx x B e

1 1 1 11 1 1ˆ ˆ ( )opt observed

t t t tt t t tx x B i i

1 1 1ˆt t t ti f x

Still Boring!

The new state vector is

For simplicity, let

Then, the problem reduces down to obtaining the elements of at each step .

Keep in mind that the matrix includes the parameters of the model.

1 1 1 1 11 1 1ˆ ˆ( ) obs

t t t t tt t t tx B f x B i

1 1 1 1( )t t t tA B F

1tA

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 seriesIdentifying preference shocksComparing observed and optimal

interest ratesRobustness checks

Time varying preferences

Time varying preferences

Regardless of the starting values, the preference parameter for output stability goes down to zero.

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

-.006

-.004

-.002

.000

.002

.004

.006

.008

1975 1980 1985 1990 1995 2000

LAMBDAPI Residuals

Preference shocks

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.

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 modelRudebusch 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

-.008

-.004

.000

.004

.008

.012

1975 1980 1985 1990 1995 2000

LPIRS Residuals

Using the estimated coefficients from Dennis

Using the estimated coefficients from Dennis

Using the estimated coefficients from Dennis

-.008

-.004

.000

.004

.008

.012

1975 1980 1985 1990 1995 2000

LPID Residuals

Correlation between preference shocks

Corr (RS, DE)=0.98Corr (RS, OZ)=0.90Corr (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: 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.