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Parameter Uncertainty
References
• Brainard, W. (1967), Uncertainty and effectiveness of policy,
AER (papers and proceedings), 57 (2), pp. 411-425.
• Sack, B. (1998) Does the Fed Acts Gradually? A VAR
Analysis, Fed Working Paper, No. 17.
• Batini, Martin, Salmon (1999) Monetary Policy and
Uncertainty, BoE Quarterly Bulletin.
• Primiceri, G. Why Inflation Rose and Fell: Policymakers'
Beliefs and US Postwar Stabilization Policy, The Quarterly
Journal of Economics, 121, August 2006, pp. 867-901.
Rudebusch, G. (2001), Is the Fed Too Timid? Monetary Policy
in an Uncertain World, Review of Economics and Statistics
83(2), May 2001, 203-217
• Sargent, T., N. Williams, T. Zha (2006) Shocks and
Government Beliefs: The Rise and Fall of American Inflation,
American Economic Review, 96(4): 1193-1224.
• Carboni, G., Ellison, M., (2007), Learning and the Great
Inflation
Preliminaries
• In most of the literature central bank
knows the ‘true’ model of the economy
• BC fluctuations occur due to shocks
(additive errors)
• CB acts based on its expectations not
based on the uncertainty in its
expectations
• Poole (1970) instrument choice under
certainty equivalence
Certainty Equivalence
• If the uncertainty faced by the central bank
takes a particularly simple form then the
optimal policy of the central bank is to
behave as if everything was known with
certainty.
• This will typically be the case if the only
source of uncertainty is an additive error
term.
Certainty Equivalence
• When uncertainty is additive, the central
bank can ignore the uncertainty and set
policy as if everything was known with
certainty.
• Theil (1958) and Tinbergen (1952)
• not really applicable to most real-world
situations of interest
Certainty Equivalence
• Only shocks are unknown
• Simple illustration
– Phillips Curve
 t 1  a t  yt 1
– IS Curve
yt 1  bit   t 1
reduced
form
 t 1  a t  bit   t 1
Certainty Equivalence
• Objective function
min U=-( t+1   )
* 2
t 1
• CB knows with certainty
– Parameter values (a,b)
– State of the economy
– True model
Certainty Equivalence
• Optimal rule
a
it   t
b
• The optimal rule is certainty equivalent
• This rule will completely offset the effects to inflation
• Calls for very active monetary policymaking
However: Interest Rate Smoothing
Interest Rate Smoothing
• CBs change interest rates in small steps
and often in the same direction for
consecutive periods!
Possible explanations (Sack and
Wieland (1999))
• Forward looking variables
• Data uncertainty
• Parameter uncertainty
Forward looking expectations
• Estimated policy rules are more efficient in
stabilizing inflation and output for a given
rate of interest rate volatility than rules
without partial adjustment
• If policy exhibits large degree of partial
adjustment forward looking agents will
expect a gradual adjustment (in the same
direction) of the policymakers w.r.t.
changes in the fundamentals
Data uncertainty
• Large measurement errors call for caution
• If interest rate movements are large, it
may trigger large mistakes (as measured
by the distance of TRUE actual values
from desired values of real output and
inflation)
Parameter Uncertainty
• Policymakers are not only uncertain about
the state of the economy but also on the
structural parameters of the economy
• Aggressive policy moves which might
otherwise (under certainty or certainty
equivalent) offset excess inflation can
trigger further uncertainty via policy
changes
A Research director at the US Fed
FOMC meeting in 1987 (cited in
Rudebusch (2001))
Alan Blinder
• the then vice-chairman of the Board of
Governors of the US Fed
“a little stodginess at the central bank is
entirely appropriate”,
“central banks should calculate the change
in policy required to get it right and then do
less”.
Parameter Uncertainty: Brainard (1967)
• Multiplicative uncertainty
• Uncertainty about the impact of the policy
instrument
– CB doesn’t know exactly the value of the
parameters but knows the distribution from
which they were drawn
– the more a policy is used the more that the
uncertainty is multiplied into the system.
Parameter Uncertainty: Brainard (1967)
• Suppose a, b have a , b mean and s a2 ,s b2
variance
• Optimal policy rule becomes
 ab 
it   2

2 t
b  sb 
• Suggests caution: as uncertainty increases
about how inflation will respond to changes in
interest rates (s2b becomes larger), interest rate
response to deviations from target becomes
smaller
Brainard Conservatism
• Coefficient of variation
sb
b
• Trade-off between returning inflation to target
and increasing uncertainty about inflation
depends on s2b relative to its average level b
Brainard Conservatism
• A large coefficient of variation means for a small
reduction in the inflation bias CB inserts large
variance into future inflation.
• Once parameter uncertainty is taken into
account inflation variance depends on the
interest rate reactions.
• Policymakers decisions affect uncertainty of
future inflation.
• Thus rather than cold turkey, gradualism
(sustained policy reaction) is advised.
Brainard
• Vertical axis: Expected value y (ybar)
• Horizontal axis: standard deviation of y
• Indifference curves around target y (based
on (y*-ybar)2+sy2
Brainard
Sack (1998)
• Optimal funds rate response (minimizing the
weighted sum of) deviations of unemployment
from ‘natural level’ and deviations of inflation
from desired one given the dynamic response of
these variables obtained from a VAR.
• Interest rate responds to past interest rate!
• Compare the results with actual policy response
• Compare the results with uncertainty about the
parameters (as captured by the variance
covariance matrix of the VAR coefficients)
Sack
• the optimal policy rule taking parameter
uncertainty into account is closer to the
actual behavior of the federal funds rate
than an optimal policy disregarding
parameter uncertainty.
Interest Rate Smoothing
Empirical Evidence
• Sack (1998): US policy fits better with
Brainard type of conservatism.
• Batini et al. (1999) similar findings for the
UK.
Rudebusch (REStat, 2001)
• Question: How close was the US monetary
policy to a behaviour recommended by optimal
policy rule?
• Assume single true model exists
• Two observations
– Taylor rule (optimal or not) fits well observed interest
rate movements as a function of inflation and output
gap.
– Taylor rule estimated with recent data indicates low
response coefficients for output and inflation.
• Cautious adjustment of interest rates than recommended by
an optimal rule!
• Brainard again!
Optimal Policy Rules a la
Rudebusch and Svensson (1999)
• We will see more details later in the lecture
series.
• Consists of an IS Curve and a Phillips Curve
(backward looking therefore open to Lucas
critique)
• Estimate coefficients for the IS and PC Curves
• Optimal response of interest rates are calculated
within the dynamic programming framework
IS and PC Curves
Rudebusch (REStat, 2001)
(1961:1-1996:4)
Optimal Taylor Rule
• Minimize the loss given by
E ( Lt )  var  t   *     yt   (it )
No Uncertainty
With Parameter Uncertainty
• Simulate the model assuming that model
coefficients of the economy is like the IS
and PC Curves on average
• Two possibilities
– Uncertainty about a single parameter
– Uncertainty about all parameters
• Policymaker chooses the values of Taylor
Rule (gp, gy)
Learning
• Recently huge literature (for references check
for instance Wieland (2000))
• Main argument
– cautious policy is suboptimal (very poor from a
learning point of view)
– If the CB is cautious in its use of policy then it will be
very difficult to learn what the effects of monetary
policy are
– CB should be more aggressive in policy since that
way it learns the key parameters about how the policy
works
Sargent (1999)
• explaining the Great Inflation
as resulting from changes in
the conduct of monetary policy
itself, which occurred as the
monetary authority learned and
revised its view of the
monetary transmission
mechanism.
• Sargent (1999): American
inflation dynamics can be
explained by the Federal
Reserve discovering and
subsequently abandoning the
Phillips curve.
Three stories about US Inflation in
the 70s and 80s
– Data uncertainty (DU)
– ‘triumph of the natural rate theory’
• run up of inflation: MonPol tempts to exploit a nonexpectational PC
• Volcker learns correct rational expectations version of the
NAIRU hypothesis; somehow managing to commit itself to
the Ramsey policy.
- ‘vindication of econometric policy evaluation’
• the Fed never learned rational expectations version of
NAIRU.
• Instead, sequentially refitting the misspecified nonexpectational Phillips curve led the Fed to discover inflationunemployment dynamics to stabilize inflation by the early
1980s.
• allow the data to speak continuously; even through estimates
of a misspecified Phillips curve model, will do a good enough
job to stabilize.
A fourth one: Primiceri (2006)
• Convex combination of data uncertanity and learning
• a model in which rational policymakers learn about the
behavior of the economy in real time and set stabilization
policy optimally, conditional on their current beliefs.
• The steady state associated with the self-confirming
equilibrium of the model is characterized by low inflation.
• Prolonged episodes of high inflation ending with rapid
disinflations can occur when policymakers underestimate
both the natural rate of unemployment and the
persistence of inflation in the Phillips curve.
Sargent,
Williams and Zha (2006)
• The underlying structure of the economy is
described by a Lucas natural-rate Phillips
curve and a true inflation process:
SWZ
• Agents understand the policy reaction function,
therefore
• the monetary authority is assumed to be
unaware of the underlying structure determining
unemployment in the economy.
• it has an approximating model of unemployment
- inflation dynamics not the ‘true’ model (1)
SWZ
• Φt is a vector of current inflation, lags of inflation,
lags of unemployment and a constant.
• Compared to the true Phillips curve (1), the
approximating model is misspecified; it fails to
recognise the role of inflation expectations in
determining unemployment.
• the monetary authority believes (incorrectly) that
the coefficients in the approximating model
follow a simple drifting process
at = at−1 + Λt
i.e. policymaker’s problem
• Subject to misspecified Phillips Curve and
beliefs on the coefficients
• at any point in history the government updates
its beliefs as it learns
• the government believes that the true economy
‘drifts’ over time.
• Learns via Kalman filter or RLS
policymaker
• The mean estimate of at for the econometric
model
• With
• Given the government’s model, the mean
estimates are optimally updated via the special
case of Bayes rule known as the Kalman filter.
Carboni and Ellison (2007, WP),
(2009, JME, Appendix E)
• SWZ results ignore the Brainard problem!
Carboni&Ellison
• policy internalising uncertainty generates
greater persistence in intended inflation
and a rise in the degree of inflation
persistence in the model.
In sum
• Parameter uncertainty and certainty
equivalence
• Parameter uncertainty and caution
• learning