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### 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
- 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:

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

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.

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:

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 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:
- The preferences vary over time.
- We specify a random walk process:
- For simplicity, we assume that

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)

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.

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.

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

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.

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

- 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

- 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

- 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 model
- Rudebusch and Svensson (1998, NBER)
- Dennis (2005, JAE)

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:
- 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.

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