Explicit Instrument versus Targeting Rules in the Backward-Looking Model
Richard T. Froyen
Department of Economics
University of North Carolina at Chapel Hill
and
Alfred V. Guender*
Department of Economics
University of Canterbury
Abstract: In the backward-looking model, an explicit instrument rule is almost as efficient
as a target rule. An explicit instrument rule leads to a more stable real rate of interest and
hence an output stabilization bias compared to a target rule.
JEL Classification Code: E5
Key Words: Optimal Monetary Policy; Target Rule; Explicit Instrument Rule; Stabilization
Bias, Interest Rate Volatility
_________________________________________________________________________
* Corresponding Author. Address: Private Bag 4800, University of Canterbury, Christchurch, NEW
ZEALAND. E-mail: [email protected]. Fax: (64)-3-364-2635. The authors wish to thank a
referee for helpful comments.
Introduction
Several contributions, notably McCallum and Nelson (2005) and Svensson (2003,
2005) discuss the relative merits of explicit instrument and targeting rules in the conduct of
monetary policy. The contributions to date focus largely on the standard forward-looking
model. This paper evaluates the performance of an explicit instrument rule in the backwardlooking model where the monetary policy transmission mechanism follows a distinctive lag
pattern.
2. The Backward-Looking Model and the Policymaker’s Preferences
The simple backward-looking model consists of two equations that describe the
dynamic behavior of the output gap and the rate of inflation:
yt  b1 yt 1  b2rt 1   t
(1)
 t   t 1  ayt 1  t
(2)
b2  0 , 0  b1  1
a0
y = the output gap (the difference between real output and its potential)
r = the real rate of interest1
π = the rate of inflation
ε and η = white noise shocks with constant variances  2 and 2 , respectively.
The backward-looking IS relation of equation (1) has two prominent features. First,
the output gap exhibits persistence, with b1 measuring the degree of persistence. Second, the
output gap responds to a change in the real rate of interest with a one-period lag. Persistence
and a lagged response are also critical elements in the Phillips curve. According to equation
(2), the current rate of inflation depends on the previous period’s rate and reacts to the
output gap with a one period lag.
Society is concerned about the variability of the output gap and the rate of inflation
in period t:
E  Lt   V ( yt )  V ( t )
(3)
1
The policymaker is assumed to have full control over the real rate of interest which serves as the policy
instrument. This simplifies the algebra without changing the results of our analysis.
2
The objective is to minimize the above expected loss function where μ is society’s aversion
to inflation relative to output gap variability.
3. Optimal Policy
Given the quadratic objective function, the policymaker follows a linear target rule in
the conduct of policy. As policy works with lags, in the current period the policymaker
chooses the expected output gap next period and treats the expected rate of inflation as
predetermined:
 Et yt 1  Et t 1  0
(4)
The target rule embodied by equation (4) assumes that the target value for the output gap
and the rate of inflation is zero, respectively. The choice parameter θ represents the weight
that the policymaker places on the output gap relative to the rate of inflation when setting
policy.
Combining equations (1) and (2) with equation (4) leads to a reduced form
expression for the output gap. Along with the Phillips curve the reduced form equation for
the output gap can be set up in matrix form to compute the variances of the two target
variables. After substituting the variances of the output gap and the rate of inflation into the
expected loss function, the policymaker minimizes society’s objective function with respect
to  . The resulting optimal value of the parameter is given by: 2
* 
a  a 2  2  4 
2
(5)
Due to the distinctive lag pattern in the transmission process of monetary policy, an increase
in the size of a causes the relative weight on the output gap in the target rule to increase.3
4. Approximating Optimal Policy with an Explicit Instrument Rule
If policy is based on an explicit instrument rule, then the policy instrument responds
in a prescriptive fashion to the target variables of monetary policy. An explicit instrument
rule is best thought of as a mechanical formula that specifies the precise movement in the
policy instrument for observed or expected deviations of the target variables from their
2
See Ball (1999) or Froyen and Guender (2007) for further details on the solution procedures.
This stands in marked contrast to the forward-looking model where an increase in the size of the coefficient
on the output gap in the Phillips curve causes the relative weight on the output gap in the target rule to
decrease.
3
3
respective target levels. For the problem at hand, we let the policy instrument react to
deviations of the linear target rule from its target level (which is zero):
rt  (  Et yt 1  Et t 1 )
0 
(6)
 is set by the policymaker. A large (small) value of  indicates that the policy instrument
reacts vigorously (languidly) to any observed deviation of the target rule from its target
level. Instrument rules such as equation (6) are at the heart of the debate between McCallum
& Nelson and Svensson. As specified, the above instrument rule is consistent with
McCallum and Nelson (2005, p. 603): “Thus, in a sense, one can accomplish with an
instrument rule anything that can be accomplished with a specific targeting rule, according
to our argument.” Svensson (2005, p. 616) in turn criticizes instrument rules because “[a]
large response coefficient [of the policy parameter  ] does indeed make the instrument rate
very volatile”. Svensson’s claim is disputed by McCallum and Nelson.
To get the reduced form equation for the output gap, we proceed as follows. First,
update and take conditional expectations of both equations (1) and (2). Second, substitute
both expectations into equation (6). Finally, substitute the lag of this explicit instrument rule
into the IS equation. The output gap evolves then in the following way:
yt 
1
1   b2
[( b1  b2  a )yt 1  b2  t 1 ]   t
(7)
The size of the policy parameter  is critical in determining the sign of the coefficient on
the lagged output gap in the above equation. Ceteris paribus, the greater the strength with
which the policymaker responds to deviations of the target rule, the more likely it is that the
coefficient on yt 1 is negative.4
Along with the Phillips curve, equation (7) can be set up in a two-equation
framework to calculate the variances of the output gap and the rate of inflation:5
V( yt ) 
2aA 2  b2  B2
aD
2
3
2
a 2 AB 2 1  b1( 1  b1( 1  b1 ))  b2 C  ( b2  ) E  ( b2  ) F  
V(  t ) 

ab2  D
ab2  D
(8)
(9)
In the limit, when    the policymaker enforces the target rule. In this case equation (7) reduces to the
a
1
following expression: yt   yt 1   t 1   t . This is an unlikely scenario, however, as central banks
4


generally adjust interest rates in small steps.
5
It can be verified that the coefficients A,….F are positive. Simply insert the limiting values of  , i.e.   a
and    and evaluate each expression.
4
A  ( b2  1)2  0
B  1  b1  b2  0 C  2a( 1  b1( 1  b1 ))   ( 4  ( 1  b1 )2 )  0
D  ( 1  b1  b2 )( 2B  ab2 )  0 E  ( 1  ( b1  2 )) 2  2a( 2  b1 )  a 2 ( 1  b1 )  0
F   ( 2 2  (   a )2 )  0
Equations (7) – (9) illustrate that policymaking becomes far more complex under an
explicit instrument rule compared to a target rule. The IS parameters b1 and b2 now appear in
the policy-augmented IS equation and hence in the expressions for the variances of the
target variables. Ideally, the policymaker would want to set  so as to minimize the
expected loss function (equation (3)). Due to the complexity of the policy problem, no
analytical solution exists.
However, the policymaker can still calculate the variances of the target variables and
the policy instrument by assigning different values to the policy parameter  . Table 1
reports the variances of the output gap, inflation, and the policy instrument under an explicit
instrument rule for 0.5    50 .6 The last column of the table lists the outcomes for the
target variables under the target rule approach. Two cases are considered. Table 1A contains
the results for the case where μ=1 while Table 1B reports the results for the case where
μ=10.
A few observations are noteworthy. The target rule and the explicit instrument rule
produce approximately the same loss score in the backward-looking framework if the policy
parameter  is set unrealistically high at 50. This observation is similar to what McCallum
and Nelson (2005) report for the forward-looking model. Notice, however, that an explicit
instrument rule introduces a stabilization bias. Irrespective of the size of the policy
parameter  , the variance of the output gap is always lower under an explicit instrument rule
compared to a target rule. By contrast, the variance of the rate of inflation is always higher
under an instrument rule compared to a target rule. Closer inspection of the variance of the
policy instrument reveals why this is the case. For 0.5    50 , an explicit instrument rule
leads to consistently lower variability of the policy instrument and hence a more stable
output gap than a target rule.
It is interesting to note that the output stabilization bias associated with an explicit
instrument rule ( for   50 ) increases as the relative aversion to inflation variability
increases. For   1 the output stabilization bias amounts to 0.7 percent while for   10 it
6
This is the same range chosen by McCallum and Nelson (2005).
5
hovers around 3.5 percent.7 We also observe that for   1 , the variance of the output gap
does not increase monotonically as the size of the policy parameter increases while it does
so for   10.
5. Conclusion
If minimizing a standard expected loss function is the criterion by which the
performance of monetary policy rules is measured in the backward-looking model, then an
explicit instrument rule is almost as efficient as a target rule, albeit for rather high values of
the policy parameter  . At the same time, an explicit instrument rule is associated with a
more stable real rate of interest which in turn leads to a less volatile output gap and hence an
output stabilization bias compared to a target rule. Hence, following an instrument rule does
not invariably cause extreme interest rate volatility.
7
Calculated as 100*(V(yt)IR- V(yt)TR)/ V(yt)TR
6
References:
Ball, Laurence, "Efficient Rules for Monetary Policy,” International Finance, 2, April 1999,
63-83.
Froyen, Richard T. and Alfred V. Guender, Optimal Monetary Policy under Uncertainty,
Edward Elgar, Cheltenham, UK, 2007.
McCallum, Bennett T. and Edward Nelson, “Targeting versus Instrument Rules for
Monetary Policy,” Federal Reserve Bank of St. Louis Review 87(5),
September/October 2005, 597-611.
-------- “Commentary,” Federal Reserve Bank of St. Louis Review 87(5), September/October
2005, 627-631.
Svensson, Lars E.O., “What is Wrong with Taylor Rules? Using Judgment in Monetary
Policy Through Targeting Rules,” Journal of Economic Literature, 41(2), 2003, 426477.
-------- “Targeting versus Instrument Rules for Monetary Policy: What’s Wrong with
McCallum and Nelson?” Federal Reserve Bank of St. Louis Review 87(5),
September/October 2005, 613-625.
7
Table 1: Comparison of Explicit Instrument Rule with Target Rule
 1
A
Instrument Rule
Target Rule
  0.5
 1
 5
  10
  25
  50
V( yt )
2.19
2.17
2.30
2.35
2.39
2.40452
2.42185
V(  t )
5.14
4.09
3.30
3.20
3.15
3.13367
3.11558
V( yt ) +  V(  t )
7.33
6.26
5.60
5.55
5.54
5.53819
5.53743
V( rt )
0.45
0.72
1.46
1.68
1.84
1.90122
1.96756
Relative Loss( %)
32.31
13.07
1.05
0.30
0.05
0.01375
  10
B
Instrument Rule
Target Rule
  0.5
 1
 5
  10
  25
  50
V( yt )
3.06
3.23
3.92
4.42
4.56
4.70552
4.87597
V(  t )
4.77
3.50
2.50
2.38
2.32
2.29679
2.27748
V( yt ) +  V(  t )
50.77
38.26
28.92
28.06
27.76
27.6735
27.6508
V( rt )
0.58
1.06
3.17
4.20
5.22
5.68131
6.23071
Relative Loss( %)
83.63
38.42
4.58
1.52
0.30
0.0822
Note:
E[ L ]  E[ L ]
IR
a. Relative Loss is calculated as 100x
E[ L ]
TR
TR
where IR= Instrument Rule and TR=Target Rule.
b. The following values were chosen for the parameters of the model: a=0.4, b1=0.8, b2=1;       1
2
c. V ( rt )  c [ e V ( yt )  V (  t )  2eCov( yt ,  t )]
2
2
c

1   b2
d. Slight differences in the calculations are due to rounding errors.
2
e   b1  a
8