Transcript Document

The Academy of Economic Studies
Doctoral School of Finance and Banking
Monetary Policy Rules Evaluation using a Forward
Looking Model for Romania
MSc student Muraraşu Bogdan
Coordinator Professor Moisă Altăr
I motivate the importance of my topic by the following remark of
John Taylor (1998): “researchers first build a
structural model of the economy, consisting of
mathematical equations with estimated numerical
parameter values. They then test out different
rules by simulating the model stochastically with
different policy rules placed in the model. One
monetary policy rule is better than another
monetary policy rule if the simulation results
show better economic performance.”
CONTENTS
• Policy evaluation with a forward
looking model
• Estimation (calibration) of the model
• Klein algorithm (generalized Schur
decomposition)
• Central bank’s loss function and the
optimization problem
• Optimal monetary policy rules
The forward looking model
•Coats, Laxton, and Rose (2003) argued that in order to support the
policy decisions necessary to respect a target for inflation, the framework
had to be forward-looking and capable of dealing with the process of
controlling inflation.
gap
gap
gap
dev
ytgap   Et ytgap

a
y

a
y

b
(
r


1
1 t 1
2 t 2
t 1
t )  ut
dev
dev
 tdev   ytgap


[

E


(1


)

1
1
2 t t 1
2
t 1 ]  vt
(1)
rt gap  1 tdev   2 ytgap  3rt gap
1  wt
•Another specification of the system includes the real effective exchange rate
in the IS curve. Taking into consideration that are no great differences
between the two cases regarding the methodology and even the main results,
I will describe the procedure I follow referring to first model.
•This model introduces two layers of complexity: 1. agents’ actions
depend upon expected future output and inflation which may cause the
existence of zero or many reduced form equations; 2. the system must be
solved for simultaneity.
Estimation vs. Calibration
• Problems:
•
•
data are very limited, both in terms of the coverage and the duration of series
data sample is very short and describes a period of major structural change in
the economy and major change in policy regimes
These are reasons to expect very imprecise identification of the
parameters from any estimation.
•Solutions:
•I chose a full information method of estimation (3SLS) in order to solve
for simultaneity
•after estimation I kept the coefficients that were statistically or
economically significant
•I applied a kind of calibration for the coefficient  from the Phillips
curve which is statistically and economically inconsistent
Data used in estimation
• The model is fitted to quarterly data for the Romanian economy for
1998Q1 – 2006Q2, subject to the restriction that the coefficients of the
policy rule minimize a quadratic loss function.
 tdev deviation
of inflation from its target (inflation is measured as a
percentage change of headline CPI, quarter-over-quarter, at annual rates
and is seasonally adjusted using Demetra (Tramo-Seats))
11.0
lpibrsa95
ytgap
Trend_lpibrsa95
the interest rate gap I
rt gap For
applied a Hodrick-Prescott
10.9
10.8
10.7
10.6
1995
1996
1997
1998
1999
2000
2001
2002
2003
2004
2005
2006
2007
1999
2000
2001
2002
2003
2004
2005
2006
2007
Detrended_lpibrsa95
0.025
0.000
-0.025
1995
1996
1997
1998
filter to the data and I
computed the gap as a
deviation from the trend.
Structural parameters
• quality of the instruments in 3SLS estimation
The stability of the coefficients from the two curves across the
interval of variation of 
Structural system is written in Klein format as
 Xt 
 X t 1 
 ut 


 gap 
 
A  Et ytgap

C
1   B  yt

 vt 
 E  dev 
  dev 
w 
 t t 1 
 t 
 t
'
X t  ( ytgap ,  tdev , rt gap , ytgap
1 )
dev
Et ytgap
1 , Et t 1
1

0
0
A
0
0

0

0
1
0
0
0
0
0
0
1
0
0
0
0
0
0
1
0
0
is a vector of predetermined variables
the forward looking or non-predetermined variables









0 1 2  
0
0
0
0
(1)
0
0
0
0
0
 0

 0
 0
B
 1
 a1

 
0
0
0
1
0
0
0
0
0
0
3
2
0
0
0
0
b
a2
1
1 (1   2 )
0
0
0
0
0

1
1 

0
b 

1 
Reduced form of the system (Klein(2000) algorithm)
0 0

0 0
0 0
C 
0 0
 1 0

 0 1
0

0
1

0
0

0 
X t  AX t 1  t
Klein algorithm (generalized Schur decomposition)
• solves systems of linear rational expectations
• the system need to be solved distinctly for the predetermined
variables (or backward-looking in the language of Klein) and nonpredetermined ones ( or forward looking variables)
• infinite and finite unstable eigenvalues are treated in a unified way
• preferable from a computational point of view to other similar
numerical methods
• for the pair of square matrices ( A, B) from the equation (1) the Q and Z
orthonormal matrices and the upper triangular matrices S and T
exist such that : A  Q ' SZ ', B  Q 'TZ ' and QQ '  ZZ '  I (2)
• The generalized eigenvalues of the system are the ratios Tii / Sii
where Tii and Sii are the diagonal elements of T and S
• The decomposition matrices can be transformed so that the
generalized eigenvalues are arrayed in ascending modulus
order (stable eigenvalues come first corresponding to backward
looking variables and unstable come next corresponding to
forward looking variables)
Solutions
•)
•)
•)
•)
(3)
(4)
Reduced form
• Now I have the structural system (1) written in the reduced form as:
X t  (Z11S111T11Z111 ) X t 1  Lzt
(5)
'
• X t  ( ytgap ,  tdev , rt gap , ytgap
is a vector of predetermined variables
1 )
•Taking into account equation (5) we can recover the covariance matrix of
structural errors from the covariance matrix of reduced form errors with
1
the relationship:   L111  L ' 
Loss function
• The central bank chooses the values for the coefficients from
the reaction function that minimize the loss function:

Loss  E0   t X t'WX t
t 0
W is
a matrix of policy weights that represent the relative
importance to the central bank of stabilizing inflation,
output and interest rate (stabilization objectives).
•These weights range between zero and one and sum to
one in order to determine whether the performance of the
policies is sensitive to policy objectives (represented by
the weights assigned to stabilize inflation, output and
respectively interest rate).
•By minimizing the loss function I also obtain optimal values
for the coefficients of the reaction function
Computation of the loss function
Because the reduced form errors are linear combinations of the serially
uncorrelated structural errors, they are serially uncorrelated.



Loss  E0   X WX t    trace[WE0 ( X t X )]  trace[W   t E0 ( X t X t' )] 
t
t 0
'
t
t
t 0
'
t
t 0

trace[W   t ( E0 ( X t  E0 X t )( X t  E0 X t )'  ( E0 X t )( E0 X t )' ] trace[W ( M  N )]
t 0
E0 ( X k  E0 X k )( X k  E0 X k )'    AA'  A2( A2 )'  ...  Ak 1( Ak 1 )'

M    t E0 ( X t  E0 X t )( X t  E0 X t ) ' 
t 0
   2 [  AA' ]  ...   k [  AA'  ...  Ak 1( Ak 1 ) ' ]  ... 
(1   ) 1[   2 AA'  ...   k Ak 1( Ak 1 ) '  ...]

N    t ( E0 X t )( E0 X t )'  0.
t 0
Loss  trace[WM ]
Correlograms and serial correlation LM test for the structural errors
•
•
Tests for no autocorrelation of the residual (residual from IS curve)
Tests for no autocorrelation of the residual (residual from Phillips curve)
Alternative policy rules
•The interest rate rules proposed by John Taylor are the most used ones. Taylor
Rule with Interest Rate Smoothing: rt gap  1 tdev  2 ytgap  3rt gap
1
•Original Taylor Rule (Taylor, 1993) assigns exact coefficient values that describe
Federal Reserve policy: 1  1.5; 2  0.5; 3  0.
•Optimal Taylor Rule: 3  0 but chooses the values for
minimize the loss function of the central bank
1
and
2 that
•Taylor Backward-Looking Rule, where lagged values of output and inflation
gap
replace the current values of the two variables: rt gap  1 tdev
1  2 yt 1
•Full State Rule (respond to all, rather than a subset, of the
gap
gap
gap
variables in the state vector): rt gap  1 tdev
1  2 yt 1  3rt 1  4 yt 2
•Woodford (2002) attributes to Goodhart a simple rule where the central bank
responds only to deviations of the inflation rate from its target value: 2  3  0
and choosing an optimal value for 3
•Clarida, Gali and Gertler (1998) suggest that forecast-based rules are optimal for a
central bank with a quadratic objective function: rt gap  1Et [ t 1 ]
 Xt 
 X t 1 
 ut 


 gap 
 
A  Et ytgap
1   B  yt
  C  vt 
 E  dev 
  dev 
w 
 t t 1 
 t 
 t
1

0
0
A
0
0

0

0
1
0
0
0
0
0
0
1
0
0
0
0
0
0
1
0
0









0 1 2  
0
0
0
0
0
0
0
0
0
'
X t  ( ytgap ,  tdev , rt gap , ytgap
)
1
 0

 0
 0
B
 1
 a1

 
0
0
0
1
0
0
0
0
0
0
3
2
0
0
0
0
b
a2
1
1 (1   2 )
0
0
0
0
0

1
1 

0
b 

1 
Results
•
Table 1 reports the policy rule that achieved the lowest loss level for each set of
policy objective weights considered.
Taylor Rule with Interest Rate Smoothing rt gap  1 tdev  2 ytgap  3rt gap
1
Goodhart Rule rt gap  1 t
gap
Expected Inflation Rule rt  1Et t 1
In the case where NBR
gives an important weight
to inflation stabilization,
as this is its primary
objective
and
output
represents an important
but secondary objective,
the Taylor Rule with
Interest Rate Smoothing is
the best rule to adopt.
W y gap
W dev
Relative performance of the rules
The figure shows that the Taylor Rule with Interest Rate Smoothing
performs at all times better than the Taylor Backward Looking Rule.
When the NBR
is preoccupied
by the
stability of
output then it
has to
respond
currently to
output gap
and not with a
lag.
Taylor with Interest Rate Smoothing vs. Full State Rule and Goodhart
Rule
The figure shows the superiority
of the Taylor rule against the rule
which takes into consideration
the entire state vector. This rule
performs better than the Taylor
type rule only when the stability
of inflation is the only objective
of the central bank.
The figure shows that this simple rule
can perform better than the Interest
Rate Smoothing Rule when the output
weight is small and also that the
performance of this rule is not
sensitive to weight assigned to
interest rate stabilization.
Full State Rule vs. Expected Inflation and
Taylor Backward Looking vs. Optimal Taylor
• the central bank should not
adopt a policy rule in which
the nominal rate of interest
responds only to changes in
the current expectation of
future inflation
• the conclusion is that the
central bank performs better if
it conditions its policy on
current rather than lagged
economic variables
Impulse responses to positive demand shock for four policy rules, namely:
Taylor Rule with Interest Rate Smoothing; Full State Rule; Backward Looking
Rule and Goodhart Rule
•
•
Taylor Rule with Interest
Rate Smoothing
Backward Looking Rule
•
•
Full State Rule
Goodhart(interest conditioned on current inflation)
Impulse responses to positive demand shock of expected inflation and output
•
Taylor Rule with Interest Rate Smoothing;
•
•
•
Backward Looking Rule;
Full State Rule
Goodhart Rule
Conclusions
•It is clear that the Taylor Rule with Interest Rate Smoothing achieves a
much more stable output gap and inflation, in spite of a relatively small
increase in the nominal interest rate. This is achieved by credibly
committing to a fixed coefficient rule that conditions the short-term
interest rate to current economic variables and to lagged interest rate.
•Taylor Rule with Interest Rate Smoothing responds better to economic
conditions in Romania
•A central bank like ours, which takes care mostly about stabilizing
inflation and is concerned about the economic stability, should control
the interest rate using a Taylor Rule with Interest Rate Smoothing.
•Paper provides evidence on the practical importance to a central bank
of analyzing the performance of the commitment mechanism
•In future work, I intend to compare the performance of fixed coefficients
rules to unconstrained optimal commitment policy and discretionary
policy, two alternatives proposed by Clarida, Gali and Gertler (1999).
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