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Arturo Estrella and Frederic Mishkin
Federal Reserve Bank of New York
Research Paper No. 9806

April 1998

This paper is being circulated for purposes of discussion and comment.
The views expressed are those of the author and do not necessarily reflect those
of the Federal Reserve Bank of New York of the Federal Reserve System.
Single copies are available on request to:
Public Information Department
Federal Reserve Bank of New York
New York, NY 10045

Rethinking the Role ofNAI RU in Monetary Policy:
Implications of Model Formulation and Uncertainty

Arturo Estrella
Federal Reserve Bank ofNew York
Frederic S. Mishkin
Graduate School of Business, Columbia University,
National Bureau of Economic Research

Prepared for the National Bureau of Economic Research conference on Monetary Policy Rules,
Islamorada, Florida, January 15-17, 1998. We thank participants at the conference and at a
seminars at the Federal Reserve Bank of New York and Columbia University for their helpful
comments, and Elizabeth Reynolds for excellent research assistance. The views expressed in this
paper are those of the authors and do not necessarily represent those of the Federal Reserve Bank
ofNew York, the Federal Reserve System, Columbia University or the National Bureau of
Economic Research.

Rethinking the Role ofNAIRU in Monetary Policy:
Implications of Model Formulation and Uncertainty
In this paper we rethink the NAIRU concept and examine whether it might have a useful
role in monetary policy. We argue that it can, but success depends critically on definin
as a short-run concept and distinguishing it from a long-run concept like the natural
rate of
unemployment. We examine what effect uncertainty has on the use ofNAIRU in policy.
Uncertainty about the level ofNAIRU does not imply that monetary policy should react
less to
the NAIRU gap. However, uncertainty about the effect of the NAIRU gap on inflatio
n does
require adjustments to the policy reaction function. Also, as in Brainard (1967), uncerta
inty about
the effect of the monetary policy instrument on the NAIRU gap reduces the magnitude
of the
policy response. We estimate a simple NAIRU gap model for the United States to obtain
quantitative measures of uncertainty and to assess how these measures affect our view
of the
policy reaction function.

1. IntH"oduction

Because the effects of monetary policy on the aggregate economy have long lags,
monetary policy must necessarily be preemptive, that is, it must act well before inflation starts to

rise. This, of course, is easier said than done. In order to act preemptively, monetary
policymakers must have signals that help them forecast future changes in inflation. One such
signal that has received substantial attention both in the academic literature and in the press is the
gap between unemployment and NAIRU, the non-accelerating inflation rate ofunemployment. 2
In other words, NAIRU is the unemployment rate at which inflation is expected to neither
increase or decrease.
The NAIRU concept has come under quite serious attack in recent years. In the early to
mid 1990s, the common view in the economics profession was that NAIRU in the United States
was around six percent. However, when the unemployment rate began to fall below six percent in
1995 and remained well below that level thereafter without any increase in inflation -- indeed
inflation actually fell -- concern arose that the NAIRU concept might be seriously flawed. In
addition, recent academic research has shown that there is great uncertainty in the estimates of
NAIRU (e.g., Staiger, Stock and Watson (1997a,b)), suggesting that looking at the
unemployment rate relative to NAIRU might not be a very helpful guide for monetary policy.
In this paper, we rethink the NAIRU concept and examine whether NAIRU might have a
useful role in monetary policymaking. We argue that the answer is yes. However, the positive
answer depends critically on redefining NAIRU very carefully and distinguishing it from a longrun concept like the natural rate of unemployment, something that is not typically done in the
literature. Furthermore, as we will see, the view that the NAIRU concept implies that the


monetary authorities should try to move the economy towards the NAIRU, thus to some extent
treating it as a target, is both incorrect and misguided.
The first step in our analysis, in section 2, is to think about defining NAIRU in the context
of setting monetary policy instruments. We adopt a definition that focuses on NAIRU as a
reference point for monetary policy, and show that our definition ofNAIRU is a short-run
concept and is not the same as the natural rate of unemployment. Understanding that short-run
NAIRU and the natural rate of unemployment differ is important, not only for the theoretical
analysis to follow, but also because it suggests that the short-run NAIRU is likely to be highly
variable, in contrast to the natural rate of unemployment. One immediate implication is that
thinking ofNAIRU as a level at which the unemployment rate should settle is not very useful for
policy purposes.
Once we have defined the short-run NAIRU, we then go on to examine how it might be
used in policymaking. We do this in several steps. First we look in section 3 at the certaintyequivalent case, when only inflation enters the policymakers' objective function and then when
unemployment (or equivalently, output) as well as inflation are part of policymakers' objectives.
Although the certainty-equivalent case is useful as a starting point for the analysis, we cannot stop
here because there are several sources of uncertainty that have important implications for how
monetary policy should be conducted. In addition to uncertainty about estimates of the actual
value ofNAIRU, there is uncertainty about the estimated parameters of the model, especially the
parameters that measure the effect of the NAIRU gap on inflation and the impact of monetary
policy instruments on the NAIRU gap. We examine in section 4 what effect these sources of
uncertainty have on how the short-run NAIRU might be used in monetary policymaking, again


under the pure price stability objective and then when unemployment as well as inflation enter the
policymakers' objective function.
Our theoretical analysis shows that uncertainty about the level of the short-run NAIRU
does not necessarily imply that monetary policy should react less to the NAIRU gap. However,
uncertainty about the effect of the NAIRU gap on inflation doe_s require an adjustment to the
reference point for monetary tightening in terms of the level of unemployment and to the weight
applied to the gap between actual and target inflation. Furthermore, as in Brainard ( 1967),
uncertainty about the effect of the monetary policy instrument on the NAIRU gap reduces the
magnitude of the policy response.
There is another sense in which uncertainty about the NAIRU may have an effect on
policy. There may be uncertainty not just about the level ofNAIRU or its effect, but about the
way it is modeled: the exact form of the model specification may be unknown. Errors in model
selection may result in excess uncertainty regarding both inflation forecasts and the parameters of
the model. Thus, model selection has the potential to increase uncertainty about the effect of the
NAIRU gap and to reduce the effectiveness of policy, and the magnitude of this problem may be
more difficult to determine than that of simple parameter uncertainty. In section 5, we focus on
the losses associated with leaving out key information from the model.
Although our theoretical framework shows the qualitative effects of uncertainty on how
monetary policy should be conducted, it cannot tell us whether these effects are economically
important.· To examine this question, we estimate in section 6 a simple NAIRU gap model for the
United States to obtain quantitative measures of uncertainty and to assess how these measures
affect our view of the optimal reaction of monetary policy to movements in unemployment
relative to short-run NAIRU. Using an analogous model based on monthly data, we then examine

how in practice the short-run NAIRU concept could be used in the actual conduct of monetary
policy. The estimated models provide us with measures of short-run NAIRU that indicate that it
is highly variable, suggesting that trying to drive the unemployment rate toward NAIRU, whether
it is a short-run or a long-run concept, would be an inappropriate way to think about how
monetary policy should be conducted. In particular, we use our analysis to evaluate whether the
setting of monetary policy instruments in the face of rapidly falling unemployment rates in recent
years makes sense.

2. Defining Short-Run NAffiU: Why It Differs from the Natural Rate of Unemployment

The concept of the natural rate of unemployment was first developed by Friedman (1968)
and Phelps (I 968) to argue that there would be no long-run tradeoffbetween unemployment and
inflation. The natural rate of unemployment is defined as the level of unemployment to which the
economy would converge in the long run in the absence of structural changes to the labor market.
An implication of this definition is that expansionary monetary policy that leads to higher inflation

would not be able to produce lower unemployment on average. Indeed, as mentioned in
Friedman ( I 968), higher inflation might even have the opposite effect of raising unemployment in
the long run because it would interfere with efficient functioning oflabor markets. The concept
of a natural rate of unemployment leads to the following characterization of an expectationsaugmented Phillips curve:

• p(l)(u,-u,)

7t 1 = 7t 1 +

.. ,

+ u z, + € 1



= inflation rate from t-1 to t


= inflation rate expected at t-1

u1 = unemployment rate at time t

ii, = natural rate of unemployment at time t, which could be a constant, but could shift with
structural changes in the economy


= a vector of variables such as supply shocks, which have zero ex-ante expectation

e, = an unspecified disturbance term.
In order to estimate this expectations-augmented Phillips curve, researchers typically
assume that the expected inflation can be measured as a distributed lag on past inflation and other
variables, and that the inflation rate is integrated of order one, so that A1t, is stationary. The
resulting Phillips curve is then:
A1t 1 = P(L)(u, -ii,)


y(L)A1t 1_ 1






The NAIRU (non-accelerating inflation rate of unemployment) concept was first
developed in a paper by Modigliani and Papademos (1975) and is defined as the rate of
unemployment at which there is no tendency for inflation to increase or decrease. In empirical
work such as Staiger, Stock and Watson (1997a,b) and Gordon (1997), NAIRU is viewed as
being equivalent to the natural rate of unemployment, ii,, in equation (2) and is typically estimated
by assuming that ii, is a constant, a random walk, or a linear transformation of some step function
or spline. 3
For policy purposes, equation (1) indicates that it is perfectly appropriate to think about
the unemployment gap, u, -ii,, as one determinant of changes in the rate of inflation, recognizing
that other factors, represented by the past history of inflation and the z, variables, also affect the
inflation process. However, current unemployment is frequently compared with the estimated

value ofNAIRU, and the resulting NAIRU gap is taken to be an indicator ofinflationary pressure.
Under a strong fonn of this view, if policymakers wish to drive inflation down, they need to raise
the unemployment level above NAIRU, whereas if inflation is at its desired level, monetary policy
needs to keep unemployment from falling below NAIRU.
Policy discussions, therefore, frequently focus on the difference between the current level
of unemployment and the NAIRU as estimated above, in other words, on the variable that enters
the first tenn of equation (1) in a distributed lag. This implicit comparison has the advantage of
simplicity: it focuses the discussion on a single indicator of inflationary pressure, the
unemployment gap, that we know from the model should be zero in long run equilibrium.
However, this advantage is overwhelmed by a number of serious problems associated with this
First, monetary policy does not generally focus only on long run equilibrium, so the gap as
defined above may be of limited usefulness. Second, even if equation ( 1) is viewed as a short-run
forecasting equation, the dependent variable is contemporaneous monthly or quarterly inflation,
which is quite unlikely to be the policy target in practice. Third, the current unemployment gap is
only one of many explanatory variables in the equation, including several lags of the gap itself.
Focusing on only one variable gives an incomplete picture. Fourth, the equation may not even
represent the optimal forecast of inflation, since other potentially important variables may be
Finally, focusing on the unemployment gap may create the impression that the goal of
policy is to drive unemployment towards the NAIRU as a target level. As equation (1)
illustrates, the current unemployment gap, u,-ii,, is only one of many explanatory variables in the
Phillips-curve equation. The presence oflags of A7t in the equation suggests that inflation may

decelerate because expected inflation is falling, even if the unemployment rate is below the natural
rate of unemployment. Similarly, if there have been favorable supply shocks, inflation in the
future may decelerate even though the unemployment rate is well below the natural rate. The
presence oflags of the unemployment gap suggests complicated dynamics in which a current
negative unemployment could also be associated with decelerating inflation. The presence of
many other variables besides the current unemployment gap in the expectations-augmented
Phillips-curve equation therefore implies that the unemployment rate at which there is no tendency
for inflation to rise or fall over the policy horizon can be quite different from the natural rate of
unemployment, ii,. In other words, it can be quite misleading to focus on NAIRU, as an estimate
in equation (1) of the natural rate of unemployment, because it is not clear that the introduction of
policy shocks designed to drive unemployment towards this characterization ofNAIRU will do
anything to control inflation either in the short run or in the long run.
Therefore, we propose an alternative way of thinking about the NAIRU as a reference
point for unemployment that reflects inflationary pressures over the short- or intermediate-run
policy horizon. The key idea is that the reference: point for unemployment at which inflation will
neither increase nor decrease over the relevant policy horizon, which can be thought of as a shortrun NAIRU, not only embodies

ii,, the natural rate of unemployment, but also the other variables

that help predict inflation. In other words, we would like to express the change in inflation over
the relevant policy horizon as a function of u, -n,, where n, is an appropriately constructed shortrun NAIRU.
Thus, suppose that the policy horizon for inflation is from} toj+k months ahead and


lj,k) -


(1200/k) log(p,.1./p,.} - I00log(p/p,_ 12)

as the difference between current annual inflation and inflation over the policy horizon, where p,
is the price level in month t. We then construct equation (2)

which is similar to equation (1 ), save for the dependent variable and the inclusion of a vector x
that contains any predetermined variables that help predict inflation at the targeted horizon. 4
In order to express the change in inflation as a function of the difference between
unemployment and a short-run NAIRU, equation (2) can always be rewritten as

n, = short-nm NA/RU= -(a + (P(L)-P(0))u1 + y(L)'11t 1 + &'x,)IP(O)


where all the predictive power of the equation has been subsumed in the short-run NAIRU n,.
This short-run NAIRU is not an estimate of the long-run equilibrium natural rate, but a reference
rate that represents the level of current unemployment that would correspond to a forecast of no
inflation change over the policy horizon. 5 Another important point that immediately falls out of
this equation is that since the short-run NAIRU is related to past lags of unemployment, inflation
and any other variables that help forecast changes in inflation, the short-run NAIRU may undergo
substantial fluctuations even if the natural rate of unemployment is a constant.

Equation (3) has several important advantages over equation (1). In contrast to the
conventional equation, the dependent variable in equation (3) is the change in inflation over the
target horizon. Second, the current NAIRU gap, u, -n,, is the only explanatory variable in the
equation and it subsumes all the predictive power of the equation. Third, the equation provides
an optimal forecast of targeted inflation, given current information.
The analysis of this paper will focus on equations (2) and (3) and on our corresponding
definition of short-run NAIRU. For the purposes of theoretical analysis, we use a simplified
version of these equations with a limited lag structure. We return to the more general
specification, however, when we consider empirical estimates using monthly data in section 6.

3. The Role of the NAIRU in Policy-Making: the Certainty-Equivalence Case

3. l Objective Function with Inflation Only
For the theoretical analysis, we start with a simple joint model of unemployment and
inflation that is isomorphic to the one employed by Svensson (1997) with an output gap. In
addition to inflation 1t and an unemployment gap ii, the model contains an exogenous variable x
and a monetary policy control variable r. This model will be the basis for the next few sections of
the paper. However, some specific assumptions will be adjusted in subsequent sections in order
to address particular issues. Assume for the purposes of this section that the parameters of the
model are known with certainty.




where ii,= u, -ii and r, = the monetary policy variable. Equation (5) is a dynamic Phillips curve in
which both unemployment and x are predictors of inflation one-period ahead, say a year.
Equation (6) is an IS curve and equation (7) defines the dynamics of the exogenous variable, x.
The equilibrium level of all the variables is zero. Note, therefore, that the policy variable r might
be more similar to a change in the interest rate rather than the level.
The reduced form expression for inflation two periods ahead based on current values of
the variables is


Assume now that the policy objective is to minimize

Although this assumption seems simplistic, Svensson (1997) has shown that the solution obtained
in this manner is equivalent to the dynamic solution of a model in which the target is a weighted
sum of all future squared deviations ofinflation from the target level. Note also that equation (8)
is analogous to equation (2) above in that it corresponds to an optimal forecast of inflation
acceleration over the policy horizon, which is given by

The conditional variance of inflation is

Since the variance of inflation does not depend on the policy variable, the result is determined by
certainty equivalence, that is, the optimal rule may be obtained by setting expected inflation equal
to the target,

1t • ,

and solving for the value of the policy variable. The optimal value of the policy

variable is given by


where the short-run NAIRU (defined as a deviation from ii) is:



Equation (9) is a variant of the Taylor (1993) rule, but which differs in that it is expressed
in terms of unemployment rather than output. In addition, it allows for the reference point for
monetary tightening in terms of the level of unemployment to be a short-run NAIRU rather than a
fixed natural rate. In effect what this variation on the Taylor rules does is that it brings in
additional information that helps forecast inflation in deriving an optimal setting of the policy
Even in this relatively simple setting, the short-run NAIRU n, is not a constant, but is
instead a function of the exogenous variable x. lflags of inflation, unemployment and the policy
variable appear in equations (5) and (6), their role in the policy rule -- and therefore in the
definition of short-run NAIRU -- would be like that of x in the model. Of course, if the only
variable that helps predict inflation over the policy horizon, other than the unemployment rate, is a
constant, then the NAIRU will be constant as in a more standard formulation. Note also that, like
ii, the short-run NAIRU of our theoretical model is measured in relation to ii. In empirical

applications, we would want to focus on the equivalent of n, +ii as a measure of short-run
Equation (9) also helps to clarify the proper use of the NAIRU for policy purposes. The
policy objective is not to drive unemployment to the NAIRU, which is a temporary and variable
reference point, but to use the NAIRU unemployment gap as one indicator of the direction to
move the policy variable, by an amount dictated by the coefficients of the model. Also, the
NAIRU gap indicator is not to be interpreted in isolation, but must be weighed against the effect
on the optimal setting of the policy variable suggested by the other indicator which is also
included in the reaction function, the gap between actual and target inflation.


It is also important to recognize that our equation (9) variant of the Taylor rule is
completely consistent with the result ofSvensson (1997). Setting the policy instrument according
to (9) is equivalent to setting expected inflation over the policy horizon equal to the inflation
target 1t*, which is the Svensson (1997) optimality condition if only inflation is in the objective
We can also draw some conclusions about the sign of the coefficient ofx in the definition
ofNAIRU, based on whether x represents a supply or a demand effect. For example, ifx is a
supply effect such as an oil price shock, then a and b would have the same sign. Since the
other parameters in equation (l 0) were chosen to have positive values, the two terms in the
coefficient would be offsetting and the net effect ofx on short-run NAIRU would be
indeterminate. In contrast, if x represents a demand effect, then a and b would have opposite
signs and the two terms would be reinforcing. The sign of the effect is positive if the demand
variable x increases inflation and vice versa. In other words, a demand shock that raises inflation
would lead to a higher value of short-run NAIRU, which implies more tightening given the same
value of unemployment.
Supply and demand shocks also have differential effects on the overall implication about
the optimal setting of the policy variable. The cumulation of supply effects would tend to drive
both unemployment and inflation in the same direction, producing offsetting effects in equation
(9). Cumulated demand effects, however, would drive inflation and unemployment in different
directions, providing an unambiguous policy reaction. Therefore, demand effects that raise
inflation should provoke a policy tightening.


3.2 Output As Well As Inflation in the Objective Function
Even when inflation is the only concern of policymakers, as in section 3.1, the optimal
policy assigns a significant role to the level of unemployment or to the unemployment gap, as seen
in equation (9). In this section, we explore how policy should be conducted when policymakers
include both inflation and output in their objectives. We do this by including a second term in the
objective function, which now becomes

The economic significance of this change is that the policy objective assigns some weight to
reducing the variability of unemployment around wro, which is the equilibrium level. 6
The optimal value of the policy variable in this case is

The modification of the objective function to reflect an unemployment target changes the weights
on u, x and

1t1 - 1t'

the weight on

in the optimal policy rule, but does not affect its general form. Specifically,

u1 relative to the weight on 1t1 -1t'

rises with A. In the extreme, if the weight on

unemployment becomes infinitely large (A approaches infinity), the optimal rule simplifies to

b, - - -x
b2 '





in which the inflation gap has disappeared and only an unemployment gap remains. This result
may also be obtained by certainty equivalence, setting expected unemployment equal to its
equilibrium level and solving for the value of the policy variable.

4. The NAIRU and Policy-Making: Implications of Parameter Uncertainty

4. I Objective Function with Inflation Only
4. I. I Uncertainty about the natural rate of unemployment

We begin to examine the consequences of uncertainty in the model of section 3 by looking
at the effects of uncertainty regarding the natural rate of unemployment or, equivalently, the longrun NAIRU. We start with this particular question for two reasons. First, it seems that in the
policy discussion on the use of the NAIRU, it is this question that is most frequently in the minds
of the policymaker, although it is not always precisely formulated. Second, the examination of
this narrower issue provides helpful intuition for the more general results that follow in the rest of
this section.
Thus, consider a more focused version of the model of section 3 in which the traditional
long-run NAIRU is the appropriate reference point for monetary policy in terms of the
unemployment rate
7t 1 = 7t 1 _1

-a 1 (u,


= 7t 1_1 - a 1 u, 1 •



+ e






where a0 = a1 ii and, as in section 3, ii is the natural rate and r1 is the monetary policy variable.
We write these equations explicitly in terms of ii in order to focus on uncertainty with regard to
this parameter. For the same reason, we assume that the parameters b 1 and b2 in equation (6a)
are known.
The second expression for equation (Sa), under the natural stochastic assumptions, may be
estimated using least squares. It is straightforward then to calculate the asymptotic distribution of
the parameter estimates, which are consistent. In particular, we can derive that T V(ii1 , ii0), the
asymptotic variance of the vector of estimates (ii1 , ii0) multiplied by the number of observations
T, is

where ii and o! are the unconditional asymptotic mean and variance of u 1 and o; is the variance
of e 1 . Now, if J is the Jacobian of the transformation (ai, a0 ) I- (ai, ii)= (ai, a/a 1), then
asymptotically, TV(iii, ~) = T JV(iii, ii0 )J 1, which equals

where we have made use of the fact that the unconditional mean of equation (Sa) is

an = 0.

The foregoing derivations may now be incorporated into the optimization problem of
section 3, again with the objective function £,(n,, 2 - 7t*)2 , but now



In the expression for the variance, the terms that include a~ do not depend on the policy variable.
Since the estimators of ii and a1 are orthogonal, the optimal rule will not depend on the
uncertainty with regard to ii, as shown in the expression

n __
+ _1_ ,_;,,
r, = --b-(u,-ii)



Thus, uncertainty about the natural rate, in and of itself, does not affect the solution to the
policymaker' s optimization problem, as defined in this section and in section 3. However, the
uncertainty about the natural rate does increase the cost function because, as seen above, it
increases the conditional variance of n,. 2 . The uncertainty about the parameter a1 , the effect on
inflation acceleration of the gap between unemployment and the natural rate, does figure in the
optimal policy through the term (I +1:;


2 1

which is a essentially a function of the t-statistic on a 1 .

Its effect, however, is not on the term containing the unemployment gap, but rather on the term
containing the gap between current and target inflation. The greater the uncertainty about a1 , the
lower is

1: 1

and therefore (1 +1:;


2 1


so the less weight the policymaker should place on the

current inflation gap. This result is very robust, as it obtains in the models of subsequent sections,
in which we introduce more complex specifications with fairly general parameter uncertainty.

4.1.2 General Parameter Uncertainty
Consider again the model defined by equations (5)-(7) of section 3.1, but assume now that
there is uncertainty at time t about the allthe coefficients of the model (a" a3 , b" b2 , b3 , c3 )
and about the disturbance of the reduced form (~), but that the uncertainty in all of these variables
is pairwise orthogonal. Although these uncertainty assumptions are not entirely general -- on
account of the assumed orthogonality -- they are more extensive than those that the previous
literature has examined. 7 The orthogonality assumptions are easily relaxed for coefficients
belonging to the same equation, but the inclusion of the corresponding covariances does not
provide greater intuition and is therefore not pursuc,d here. Thus at time t, the expectation and
variance of inflation at time t+ 2 are given by







+ (a3 ac3 +a0 3 (I +c3) +a0 3ac3 +a1 ab3 +a0 I b3 +a0 1ab3 )x,

where the values of the coefficients denote their expected values. 8
As in section 3.1, the policy objective is to c:hoose r1 so as to minimize the objective


In this case, the optimal value of the policy variable is given by




t 1 =a1 la.

= -1_2

1 +t2


t 2 =b/ab.

Equation (11) can be rewritten as

( ---(ii-(n+<f>))+-'--·--·(1t-1t*)
, ,
1 +t; 2 a1





<I>, =




1 +t~ a, (1 +b,) ,

Comparison of equations (9) and (12) indicates that the presence of uncertainty introduces
two multiplicative terms of the form (1 +t;-





These terms are essentially functions of the t

statistics corresponding to the parameters a 1 and b2 , respectively, which correspond to the oneperiod-ahead effects of unemployment on inflation and of the policy variable on unemployment.
All other variance-related terms in the objective function drop out of the calculation. When there
is no uncertainty about a 1 and b2 , the two multiplicative terms become 1, reverting to the
certainty equivalent case of section 3.1.
One of the two uncertainty effects, the one related to b2 , the coefficient on the policy
variable in equation (6), takes a form that is predictable from the analysis by Brainard (1967).
Specifically, as abl rises, the term (1 +t;




falls so that uncertainty about the magnitude of the

effect of the policy variable leads to a partial policy reaction -- a reaction that is less than that in
the certainty equivalence case.

In contrast, uncertainty about a1 , the effect of unemployment on the change in inflation in
equation (5), has an effect not on the scale of the policy reaction, but rather on the weight applied

1t1 -1t'

and on the reference point in terms of unemployment at which that reaction occurs.

Specifically, as o01 rises, the term (I +i:;




falls so that the weight on

1t1 - 1t'

falls. A rise in o


causes the term (I +,:~f 1 and the absolute value of the adjustment term <I>, to rise. If x has a
positive impact on inflation (i.e., aye, is positive), then <I>, is negative and so the reference point
for monetary tightening in terms of unemployment, n1 + <I>,, falls.
The effect of uncertainty about a1 on how the reference point responds to change in x is
somewhat more complex. The net effect on the reference point n, + <I>, depends on whether xis a
supply or demand variable, as discussed in section 3. I. Consider the combined expression

Ifx is a supply variable, the direction of the effect of uncertainty on the magnitude of the
reference point is unclear. It is clear, however, that as uncertainty about a 1 approaches infinity,
the sign of the coefficient is the same as the sign of -b3 • Ifx is a demand variable, uncertainty
reduces the absolute magnitude of the reference point unambiguously.

4.2 Output As Well As Inflation in the Objective Function
We now modify the results of the previous subsection by assuming that the policy
objective function includes both inflation and unemployment. As in section 3.2, the objective
function becomes


The optimal value under parameter uncertainty is

The effect of including a target for unemployment, as represented by A, is analogous to the
effect of uncertainty about a1 • In the above equation, these two terms occur additively in the
same expression in the terms corresponding to the exogenous variable and the inflation gap. Only
in the unemployment term does A appear separately. Intuitively, the reason for this is that
uncertainty about a 1 makes the relationship expressed in equation (5) less reliable, so policy
becomes more concerned with affecting the "intermediate target" of equilibrium unemployment.
If the weight on unemployment becomes infinitely large, the optimal rule simplifies to

in which, as in the certainty equivalence case, the inflation gap has disappeared and only an
unemployment gap remains. Here, the only effect of uncertainty is of the rescaling type, as
identified by Brainard (1967).


5. The NAIRU and Policy-Making: tlhe Implications of Model Selection

In this section, we discuss another type of uncertainty that affects the definition of shortrun NAIRU, its computation, and the policy rule that results from inflation targeting. Specifically,
we focus on uncertainty regarding the correct form of the basic model and the associated problem
of model selection. Whereas in section 4 we assumed that the form of the model was known, but
that the parameters were estimated with uncertainty, we now suppose that the policymaker
ignores some key information variable in the optimization problem. 9
In general, if inflation two periods ahead is the policy target, and if a variable helps predict
inflation at that horizon, it is inefficient not to include the information in the model. For example,
the models of sections 3 and 4 define the policy rule in terms of a short-run NAIRU, which in tum
is a function of the exogenous variable x. What is the result ofignoring the predictive content of

x? Alternatively, what is the cost ofrelying on a long-run equilibrium NAIRU (zero in this case)
when a short-run informative NAIRU is available?
Thus, suppose that the policymaker ignores the presence ofx in the basic model (5)-(7).
The values of al and bl are implicitly set to zero, while the third equation is dropped altogether.
Under these conditions, the constrained optimal rule for inflation targeting becomes

We know, of course, that the value of the objective function has to be higher (i.e., worse) when
evaluated at this constrained optimum that when evaluated at the unconstrained optimum r,' as in
section 4.1. In fact, we can calculate the difference between the constrained and unconstrained
values as


Somewhat surprisingly, uncertainty about b2 ameliorates the left-out-variable problem. 10
Uncertainty about a 1 , in contrast, can make matters worse.
The left out variable problem can also increase uncertainty regarding the estimates of the
included coefficients, with consequences for the size of the policy response or the reference point
for monetary tightening in terms of unemployment. To see this, suppose the inflation equation (5)
is estimated by ordinary least squares, leaving out the variable x, after rewriting it in the following

One implication ofleaving out x, well known from econometrics textbooks, is that the estimate of
a 1 may be biased. This occurs unless x and u are contemporaneously uncorrelated. 11 However,

even if the two regressors are indeed uncorrelated so that the estimate of a is unbiased,
uncertainty in the estimate is greater by the amount

~ -2



where the numerator of the last term is the difference between the R 2 s of the unconstrained and
the constrained models. Thus, excluding the variable x from the model, in addition to producing a
policy rule that improperly excludes x, increases uncertainty about a • One possible consequence
is that, for the reasons provided in section 4, the policymaker may react to the higher level of aa,

by adjusting the weight on

1r1 -1t'

downward and by increasing the absolute size of the NAIRU

adjustment <!>,.

6. Empirical Estimates of the short-run NAIRU

6.1 Empirical Evidence on the Importance ofUncertainty
Although our theoretical framework shows qualitatively the effects of uncertainty on how
monetary policy should be conducted, it cannot tell us whether these effects are economically
important. To examine this question, we estimate in this section a simple NAIRU gap model for
the United States to obtain measures of uncertainty and to assess how these measures affect our
view of the optimal reaction of monetary policy to movements in unemployment relative to shortrun NAIRU. In order to have in the model a simple lag structure that mimics that of the
theoretical model of equations (5)-(7), we start by estimating a model with annual U.S. data over
the period 1956 to 1996. The model is



where 1t is the log change in the CPI from December of year t-1 to December of year t, u is the
unemployment rate in December of year t and r is the average monthly three-month Treasury bill
during year t. Note that o: 1 ,
uncertainty ratios


correspond to a 1 , h2 in the theoretical model, and that the key

1/, -c;2will be based on the fonner.

The results are presented in table 1.

These estimates provide some guidelines regarding the importance of uncertainty for
monetary policy in this context. First, the adjustments to the unemployment reference point and
to the policy reaction as a result of parameter uncertainty are not large. The key parameters are
estimated with some precision, and the implied multiplicative adjustment factors are both close to
1. The Brainard-type adjustment -- a 2.5 percent reduction -- is particularly small, suggesting that
the magnitude of the policy reaction should only be shaded down slightly to reflect parameter
uncertainty. However, the unemployment effect adjustment is also less than 5 percent.
These results are confirmed by looking at the implicit optimal policy that corresponds to
the two-year ahead inflation target of the theoretical model in which only inflation is included in
the objective function. The rule that results is very similar to the simple Taylor (1993} rule when
adjustments are made for the fact that Taylor's rule was defined in terms of quarterly data and an
output gap. The annual and quarterly results are presented in table 2. If 6 is the weight on the
lagged interest rate in the annual model, the corresponding quarterly lag is assigned a weight of
6 114 and the weights on the inflation and unemployment lags are divided by I +6 114 +6 214 +6 314 . A
rule based on the output gap is obtained by applying a simple Okun' s law adjustment, dividing the
unemployment weight by 2.
The table confirms that the practical significance of parameter uncertainty is quite small.
Furthermore, the quarterly results with the output gap are remarkably similar, even numerically, to
the parameters suggested by Taylor (1993). The only key difference is that the interest rate is
assumed to be much more persistent here, since Taylor did not include a lagged interest rate in the
form of his rule. 12


6.2 Empirical Estimates of the Short-Run NAIRU
In this section, we present estimates of the short-run NAIRU. For these purposes, we
return to the more general model of equations (2)-(4) and estimate the equations with monthly
data from January 1954 to November 1997, using ll 12-month-ahead, 12-month horizon (i=k=12)
and 12 lags of both the change in inflation and unemployment. 13 Figure 1 shows the estimated
short-run NAIRU together with the contemporaneous unemployment rate, as well as the shortrun NAIRU gap. This figure demonstrates the high variability of the short-run NAIRU, in
contrast with long run measures designed to estimate a natural rate as in Gordon ( 1997) and
Staiger, Stock and Watson (1997a,b). For exampfo, consider a version of our equation (1), which
may be used to estimate a constant ii that is comparable to the long-run measure of those papers:
A1t 1 = P(L)(u, -ii,) + y(L)A1t1_ 1 + e,


When estimated over the same period as equations (2)-(4), the estimate of ii is 6.1 percent, as
shown in figure 1.
Staiger, Stock and Watson (1997a) have pointed out that such estimates of a constant
long-run NAIRU tend to be quite imprecise. Using the delta method in an equation similar to (1'),
they obtain an estimate of ii=6. 2 percent, with a standard error of about O. 6. Our estimate of
ii=6.l has a standard error of0.43, which is somewhat smaller -- perhaps partly because of our
larger sample -- but is of the same order of magnitude. Estimates of the short-run NAIRU n, are
more precise. The standard error of n, is a time-varying function of the values of the variables in
expression (4). Over the sample period, the standard errors range from 0.11 to 0.42, with a mean
of0.20, less than half of the standard error of ii. 14


Thus, the short-run NAIRU is estimated with more than twice the precision than the
standard long-run NAIRU. The practical significance of this result, however, is limited, since we
have shown in the theoretical sections that this type of uncertainty plays no role in the
determination of the policy rule. Nevertheless, a reduction in the uncertainty may produce a
reduction is the value of the cost function, as shown in section 5, even if the policy rule remains

6.3 A case study: recent signals from a short-run NAIRU
Using the estimates of the NAIRU gap from section 6.2, we now examine the hypothetical
results of using the methodology ofthis paper in the of conduct monetary policy in the U.S. since
June 1992, when the unemployment rate began a prolonged decline. The results will of course be
somewhat simplistic, but they may provide some general support for the concepts developed in
this paper.
If we refer to one of the policy rules in the theoretical part of the paper, say to equation

(9), we note that the appropriate interest rate is determined essentially by two gaps: the difference
between unemployment and short-run NAIRU and the difference between actual and target
inflation. We present in figure 2a the gap between short-run NAIRU and unemployment (signed
so that a positive value indicates that monetary policy should be tightened) and the level of
inflation (12 previous months) since 1992.
From June 1992 to the end of 1993, declining unemployment brought the NAIRU gap
from levels suggesting, if anything, the need for ease to relatively neutral levels. Meanwhile
inflation declined over the period and, in fact, continued to decline into the beginning of 1994.
Beginning in 1994, however, the NAIRU gap became positive and remained so until early 1995,

suggesting a need for tightening. In addition, inflation stopped declining, remaining around the 3
percent level. These two factors combined are consistent with the monetary tightening
undertaken by the Federal Reserve throughout 1994 and into early 1995.
Since then, the NAIRU gap has indicated some pressure to tighten twice, in 1996 and
1997. In the first case, the pressure from the NAIRU gap was accompanied by a rise in inflation.
Even though inflation subsided toward the end of the year, this episode may seem somewhat
inconsistent with the absence of further tightening. Figure 2b suggests one reason for this result.
Figure 2b presents the results ofrepeating the analysis of figure 2a, but using core inflation
(excluding food and energy prices) instead of total inflation. Core inflation tends to be a better
signal of persistent changes in inflation than total inflation.
Figure 2b shows both the level of core inflation as well as the gap between unemployment
and the short-run NAIRU computed using core inflation in equations (2)-(4). Comparisons of the
two panels of figure 2 suggests that the effect of using core inflation in the calculation of the
NAIRU gap is very slight. But core inflation was falling in 1996, in contrast to the rising total
inflation, and this fall may have offset the tightening signals from the NAIRU gap.
In 1997, the pressure arising from the unemployment gap seems stronger than in the
previous year. Inflation, however, both total and core, moved downward again, offsetting at least
partially the signals from the NAIRU gap indicator. Arguably, only during 1994 and early 1995
were there consistent signals for tightening and this is when the Federal Reserve engaged in most
of its monetary tightening.
In order to evaluate the net effect of the unemployment and inflation indicators, it would
be helpful to summarize the information in a single measure, as in the policy rules of table 2. We
would like to do this, not to explain actual policy, but to suggest how the theoretical constructs of


this paper could be used in practice. However, this is a problem for two reasons. First, we would
have to construct a full optimization model in the context of the monthly equations, which is
beyond the scope of the present paper. 15 Second, we would have to know or make an assumption
about the target level of inflation. Thus, we present only a limited version of a policy rule in
which we deal with those problems as follows.
First, we take the weights for the NAIRU and inflation gaps from the annual results of
table 2 allowing for uncertainty, making allowance also for the monthly frequency of our data.
Since the coefficient of the lagged interest rate, 61112 =0.98, is very close to 1, we further simplify
by assuming that the weights are used to calculate a monthly change in the interest rate. We then
divide the annual weights by 1 + 6 1112 + ... + 611/12 to obtain weights of -0.23 for the NAIRU gap
and 0.15 for the inflation gap with total inflation, and -0.25 and 0.19, respectively, using core
inflation. 16 To deal with the second problem, the fact that the inflation target is unknown, we
scale the results so that the policy rule with total inflation is neutral, on average, over the period
since June 1992. This assumption is equivalent to an inflation target of 3 percent.
The results are presented as the solid line in the two panels of figure 3. Note that the
weighted results are consistent with our earlier discussion of the individual components. In panel
3a, which contains the results using the total CPI, the strongest signal for tightening comes during
1994. Note also, however, thatthere were distinct signals for tightening in 1992-93 and 1996-97,
and that there were fairly strong signals for easing at the beginning and towards the end of the
sample period. In panel 3b, which contains results using the core CPI, there are also strong
signals to tighten in 1994, but because the core inflation rate was higher than total CPI in late
1992 and early 1993, there are also strong signals to tighten in this period. In contrast to panel
3a, the results with the core CPI do not suggest any need to tighten in 1996.

We may contrast these results with a rule based on the standard unemployment gap -- the
gap between unemployment and a constant long-run NAIRU. The results are presented as the
dashed line in the two panels of figure 3. To obtain weights that are consistent with the
assumption of a constant NAIRU, we estimated equations (12) and (13) without the second lag of
unemployment, which produces an estimate ofNAIRU that is constant. These new weights are
-0.35 for the NAIRU gap and 0.34 for the inflation gap using total inflation, and -0.36 and 0.37,
respectively, using core inflation. Note, however, that ifwe use of the same weights as before,
the qualitative results are the same as with these weights.
The results for the long-run NAIRU gap, which are driven by the large steady decline in
unemployment over this period, are fairly robust. The main feature of the alternative rule is that it
argues for easing throughout the first part of the period, and then for tightening throughout the
second part of the period. What this rule misses is that a long-run natural rate is not the best
reference point for unemployment if the goal is to target inflation in the short run.

7. Summary and Conclusions
In this paper, we examine how a variant of the NAIRU concept can be usefully employed
in the conduct of monetary policy. By thinking ofNAIRU in this way, we obtain insights that
might be quite useful to monetary policymakers. Because there are quite a few results sprinkled
throughout the paper, we list the main ones here.


The NAIRU concept that is useful for the conduct of monetary policy differs from the
estimate of the natural rate of unemployment, the long-run concept used previously by
many researchers. Instead, NAIRU can be viewed as a short-run construct, which is

related to past levels of unemployment and inflation as well as other economic variables,
that helps forecast future accelerations or decelerations ofinflation.


The short-run NAIRU should be viewed not as a target for policy, but as helping to define
the reference point which policymakers can compare to the current rate of unemployment
to derive a signal for the appropriate stance of policy. Furthermore, as long as inflation is
an element in the policymakers' objective function, the NAIRU gap is not the only signal
that should affect the setting of policy instruments: the deviation of inflation from its target
level also has an important role in the determination of the appropriate stance of policy.


The policy rule that comes out of our analysis is a variant of a Taylor rule using an
unemployment gap rather than an output gap, but has one major difference from more
standard formulations. The standard Taylor rule implicitly assumes that the reference
point to which unemployment should be compared in the unemployment gap term is
constant, while in our formulation, the reference point is related to the short-run NAIRU
which can have substantial short-run fluctuations over time.


Uncertainty about the level of the NAIRU has no influence on the setting of policy
instruments, although it does affect the value of the objective function. This type of
uncertainty makes the economy worse off, but does not alter policy behavior.


Uncertainty about the effect of unemployment on inflation leads to an additive adjustment
to the short-run NAIRU to calculate the reference point for monetary tightening in terms

of the level of unemployment. In addition, uncertainty about the unemployment effect on
inflation changes the weight on the inflation gap in the policy rule.


Uncertainty about effect of the policy variable leads to a scaling down of the reaction of
the policy variable, the well-known Brainard (1967) result.


Uncertainty about model selection can have important effects on the form of the policy
rule. In particular, if a constant NAIRU is used -- as occurs ifNAIRU is viewed as a
long-run concept -- so that information about the state of the economy that could be used
to forecast inflation is ignored, the performance of the policy rule can be substantially
worse. In addition, leaving out relevant variables that help forecast inflation increases the
uncertainty about the effect of unemployment on inflation, with the resulting implications
described above.


Although parameter uncertainty has potentially large effects on how policy should be
conducted, our empirical results suggest that parameter uncertainty may not be all that
important for the setting of policy. We find some evidence of changes in the policy rule
resulting from the parameter uncertainty we explored in our theoretical model, but these
effects are very modest. They affect the weights in the policy rule by less than five percent
in both the case of uncertainty about the impact of unemployment and the case of
uncertainty about the effect of the policy variable.



Estimates of the short-run NAIRU are highly variable over time. However, there is a fair
degree of precision in these estimates.


Substantial positive NAIRU gap estimates arose throughout 1994 and early 1995 and in
parts of 1996 and 1997. However, core inflation was substantially lower in 1996 and
1997 than in 1994. Thus the one period since June 1992 during which there were
consistent signals for tightening occurred during 1994 and early 1995, which is when the
Fed engaged in most ofits monetary tightening.

These results suggest that a short-run NAIRU is indeed a useful concept and that it can be
used by policymakers, particularly in deciding on how monetary policy should be conducted.
However, there are some subtle issues in how the short-run NAIRU concept might be used
correctly. First, because our view ofNAIRU sees it as a short-run construct, it is dangerous to
think ofNAIRU as a potential target for unemployment which stays around a particular value,
such as 6 percent, for any period of time. Second, deviations of inflation from its target are every
bit as important a factor in thinking about setting policy as is the NAIRU gap. Third, uncertainty
about parameter values and model selection do have effects on the optimal setting of policy
instruments, but do not appear to be a barrier to a useful role for the NAIRU concept in policy
We hope that this paper helps resurrect NAIRU as a useful concept, but only if it is used
properly. As we have shown, a short-run NAIRU is a useful construct because it helps tell
policymakers what might happen to inflation in the future. Furthermore, the model of this paper
suggests that policymakers may want to avoid the impression that an objective of policy is to raise


unemployment when it falls below NAIRU or to lower it when it is above NAIRU. To think of
policy in this way might lead the public to think that policymakers are against low unemployment,
an outcome that can reduce support for central bank efforts to control inflation.


1. If price stability has already been achieved, then inflation falling below its target is every bit as
damaging as a rise in inflation above the target. Thus, in this situation monetary policy must also
be just as preemptive against declines in inflation below target levels.
2. See, for example, Stiglitz (1997), Gordon (1997), Staiger, Stock and Watson (1997a,b),
Economic Report of the President (1997). For a history ofNAIRU, see Espinosa-Vega and
Russell (1997). The NAIRU acronym would better be expressed as NIIRU (the non-increasing
inflation rate of unemployment) because it is the unemployment rate at which inflation is expected
to neither increase or decrease.
3. See, for example, Stiglitz (1997), Gordon (1997), Staiger, Stock and Watson (1997a,b),
Economic Report of the President (1997). For a history ofNAIRU, see Espinosa-Vega and
Russell (1997). The NAIRU acronym would better be expressed as NIIRU (the non-increasing
inflation rate of unemployment) because it is the unemployment rate at which inflation is expected
to neither increase or decrease.
4. The variables x differ from z in the Gordon (1997) and Staiger, Stock and Watson (1997a,b)
equations in that z represents primarily supply shocks that are contemporaneous with the
dependent variable, whereas x is more general in that it includes any predetermined variables other
than unemployment and inflation (and their lags) that help predict future inflation.
5. Equation (4) is a generalization of the model of short-run NAIRU in Estrella (1997). After
writing this paper, we discovered that Layard and Bean (1988) also have a similar definition of
short-run NAIRU in the context of a one-period change in inflation.
6. Once again, this is a relatively simple objective function designed to highlight the key points of
this paper. A more complex dynamic solution of a similar model may be found in Svensson

(1997), which exhibits properties that are qualitatively analogous to those of the simpler model of
this paper.
7. Other papers that look at the effect of parameter uncertainty in a similar context are Svensson
(1997), Clarida, Gali and Gertler (1997), and Wieland (1997).
8. This convention economizes on notation and is correct by definition if the coefficient estimates
are unbiased.
9. The complementary problem ofincluding too many variables in the model is in principle less
serious, since consistent parameter estimates should assign zero weight to the superfluous
10. The intuition is that as uncertainty about b2 grows, the optimal response of the policy
variable r is reduced so that there is less loss from using the incorrect model.
11. See, for instance, Theil (1971), section 11.2. This problem may be bypassed formally by
thinking ofx as the component of the additional variable that is uncorrelated with u.
12. Recent estimates of the Taylor rule by Rudebusch and Svensson (1998) and Rotemberg and
Woodford (1998), among others, suggest that the persistence parameter is close to 1. Fuhrer and
Moore (1995) assume that it equals 1.
13. Somewhat surprisingly, extending the horizon toj+k=60 months or even longer does not
materially affect the point estimates of the short-run NAIRU. Of course, the fit of the equation
deteriorates with longer horizons.
14. All our standard errors are estimated consistently using the Newey-West (1987) technique
with a 24-lag window.
15. A model along those lines has been developed for the U.S. in Clarida, Gali and Gertler
(Forthcoming). See also the references in that paper.


16. Adjusting fully for coefficient uncertainty would require, in addition to the adjusted weights,
an adjustment to short-run NAIRU corresponding to the term cl>, defined in section 4.1.2. We do
not make this adjustment here because our equation for monthly NAIRU is essentially a reduced
form and the components are difficult to disentangle, and also because the coefficient of the
adjustment factor (l +i;~f 1s0.065 is empirically small.


Brainard, William. 1967. "Uncertainty and the Effectiveness of Policy." American Economic

Review, 411-425.
Clarida, Richard, Jordi Gali and Mark Gertler. Forthcoming. "The Science of Monetary Policy."

Journal ofEconomic Literature.
Council of Economic Advisers. 1997. Economic Report of the President, 45-54.
Espinosa-Vega, Marco A. and Steven Russell. 1997. "History and Theory of the NAlRU: A
Critical Review." Economic Review, Federal Reserve Bank of Atlanta, 4-25.
Estrella, Arturo. 1997. "Aggregate Supply and Demand Shocks: A Natural Rate Approach."
Federal Reserve Bank of New York Research Paper No. 9737.
Friedman, Milton. 1968. "The Role of Monetary Policy." American Economic Review, 58:1-21.
Fuhrer, Jeffi:ey C. and George R. Moore. 1995. "Monetary Policy Trade-Offs and the
Correlation between Nominal Interest Rates and Real Output." American Economic Review,
Gordon, Robert J. 1997. "The Time-Varying NAlRU and its Implications for Economic Policy."

Journal ofEconomic Perspectives, 11: 11-32.
Layard, Richard and Charles R. 1988. "Why Does Unemployment Persist?" Centre for Labour
Economics, London School of Economics, Discussion Paper No. 321 (August).
Modigliani, Franco and Lucas Papademos. 1975. "Targets for Monetary Policy in the Coming
Year." Brookings Papers on Economic Activity, 1: 141-63.
Newey, Whitney K. and Kenneth D. West. 1987. "A Simple-Positive Semi-Definite,
Heteroskedasticity and Autocorrelation Consistent Covariance Matrix," Econometrica,

Phelps, Edmund. 1968. "Money-Wage Dynamics and Labor-Market Equilibrium."
Journal of

Political Economy, 76:678-711.
Rotemberg, Julio J. and Michael Woodford. 1998. "Interest Rate Rules in an Estima
ted StickPrice Model." This conference.
Rudebusch, Glenn D. and Lars E.O. Svensson. 1998. "Policy Rules for Inflation Targeti
This conference.
Staiger, Douglas, James H. Stock and Mark W. Watson. 1997a. "How Precise are Estima
tes of
the Natural Rate of Unemployment?" In Reducing Inflation: Motivation and Strateg
Christina D. Romer and David H. Romer, eds., Chicago: University of Chicago Press.
Staiger, Douglas, James H. Stock and Mark W. Watson. 1997b. "The NAIRU, Unemp
and Monetary Policy." Journal ofEconomic Perspectives, 11 :33-49.
Stiglitz, Joseph. 1997. "Reflections on the Natural Rate Hypothesis." Journal ofEconom

Perspectives, 11 :3-10.
Svensson, Lars E.O .. 1997. "Inflation Forecast Targeting: Implementing and Monito
Inflation Targets." European Economic Review, 41: 1111-1146.
Taylor, John B. 1993. "Discretion Versus Policy in Practice." Carnegie-Rochester

Series on Public Policy, 39:195-214.
Theil, Henri. 1971. Principles ofEconometrics. New York: Wiley.
Wieland, Volker. 1997. "Monetary Policy and Uncertainty about the Natural Unemp
Rate." Manuscript, Board of Governors of the Federal Reserve System.


Table I. Estimates of annual U.S. model (1956 to 1996)



































Table 2. Implicit interest rate rules
Weight on


Lagged interest rate

Inflation gap

or output gap























With output gap
Uncertainty adjusted:

With output gap


Figure 1. Short-run NAIRU, unemployment and the short-run NAIRU gap
Jan. 1954 to Nov. 1997
11.2 - , - - - - - - - r - - - - - - - - - - - - - - - - - ,

9 . 6 + -""




















•) I

''"' 1~1 I


1, :,~



~ I \



















Figure 2. NAIRU gap and inflation, June 1992 to Nov. 1997
2a. Inflation = Total CPI










. .,,



., 00



- -~..,









2b. Inflation= Core CPI





'''I ",/ '~'




















Figure 3. Simple policy rules based on short-run NAIR.U (solid line)
and long-run NAIR.U (dashe,s), June 1992 to Nov. 1997



~----------="'::·:::'•:::llatioo::·=-•:.;Tola=l..:C::_Plc__ _ _ _ _ _ _ _ _ __


' ''',,,,.,

11' ... ' ,







._._.__,~~I \

,~ ~'' "














3b. Inflation •= Core CPI








I \





.... ..L-.,.,.-------~--,,--~-.,.,--~---,.,----J.,.--',":.;'":.;-;;;::;;_








The following papers were written by economists at the Federal Reserve Bank of
New York either alone or in collaboration with outside economists. Single copies of up
to six papers are available upon request from the Public Information Department,
Federal Reserve Bank of New York, 33 Liberty Street, New York, NY 10045-0001
(212) 720-6134.

9801. Higgins, Matthew, and Carol Osler. "Asset Market Hangovers and Economic Growth:
U.S. Housing Markets." January 1998.
9802. Lopez, Jose. "Methods for Evaluating Value-at-Risk Estimates." March 1998.
9803. Malz, Allan. "Interbank Interest Rates as Term Structure Indicators." March 1998.
9804. Peristiani, Stavros. "Modelling the Instability of Mortgage-Backed Prepayments."
March 1998.
9805. Morgan, Donald. "Judging the Risk of Banks: What Makes Banks Opaque?"
March 1998.

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