WO R K I N G PA P E R S E R I E S
N O 7 9 4 / A U G U S T 2 0 0 7
(UN)NATURALLY LOW?
SEQUENTIAL MONTE
CARLO TRACKING OF
THE US NATURAL
INTEREST RATE
ISSN 1561081-0
by Marco J. Lombardi
9 7 7 1 5 6 1 0 8 1 0 0 5
and Silvia Sgherri

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W O R K I N G PA P E R S E R I E S
N O 7 9 4 / A U G U S T 2 0 0 7
(UN)NATURALLY LOW?
SEQUENTIAL MONTE
CARLO TRACKING OF
THE US NATURAL
INTEREST RATE 1
by Marco J. Lombardi 2
and Silvia Sgherri 3
In 2007 all ECB
This paper can be downloaded without charge from
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1 To a large extent, the paper was written while both authors were at the European Central Bank. We thank Gianni Amisano and Matteo
Ciccarelli for useful suggestions and Günter Coenen, Ricardo Mestre, Frank Schorfheide, Frank Smets, and Michael Woodford for
insightful feedback on a preliminary version of this paper. We are also grateful for comments to seminars participants at ECB and DNB,
the 2006 Conference on Computation in Economics and Finance, the 17th EC2 Conference, the 2nd Italian Congress of Econometrics
and Empirical Economics, and the Royal Economic Society Annual Conference. The views expressed in this paper
are personal and do not reflect those of any policy institution. The usual disclaimers apply.
2 University of Pisa, Lungarno Pacinotti 43, 56126 Pisa, Italy; e-mail: mjl@ec.unipi.it
3 De Nederlandsche Bank and International Monetary Fund; Postal address: De Nederlandsche Bank, Postbus 98,
1000 AB Amsterdam, Netherlands; e-mail: s.sgherri@dnb.nl

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Working Paper Series is available from
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ISSN 1561-0810 (print)
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CONTENTS
Abstract
4
Non-technical summary
5
1 Introduction
6
2 A generalized “Neo-Wicksellian” framework
8
2.1
Demand
side
9
2.2
Supply
side
11
2.3
Monetary
policy
14
2.4
Model
solution
15
2.5
Prior
specification
16
3 Likelihood evaluation and Bayesian estimation 17

3.1 A sequential Monte Carlo approach
18
3.2
Parameters
estimates
20
3.3
Out-of-sample
forecasts
22
4 Empirical
results
22
4.1
Structural
parameters:
posterior
distributions
22

4.2 Unobservable state variables:


A real-time policy assessment
25

4.3 Out-of-sample forecasts and model
evaluation
27
5 Conclusions and prospects
30
References
33
Tables and figures
38
European Central Bank Working Paper Series
48
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Working Paper Series No 794
August 2007
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Abstract
Following the 2000 stockmarket crash, have US interest rates been held "too low" in
relation to their natural level? Most likely, yes. Using a structural neo-Keynesian model,
this paper attempts a real-time evaluation of the US monetary policy stance while ensuring
consistency between the speci…cation of price adjustments and the evolution of the econ-
omy under ‡exible prices. To do this, the model’s likelihood function is evaluated using a
Sequential Monte Carlo algorithm providing inference about the time-varying distribution
of structural parameters and unobservable, nonstationary state variables. Tracking down
the evolution of underlying stochastic processes in real time is found crucial (i) to explain
postwar Fed’s policy and (ii) to replicate salient features of the data.
JEL Classi…cation Numbers: E43, C11, C15
Keywords: Natural Interest Rate; DSGE Models; Bayesian Analysis; Particle Filters
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Non-technical summary
After the 2000 stockmarket bubble burst, the Fed has successfully managed to reduce
business cycle volatility (resulting in the mildest US recession ever!) by reducing short-term
rates to a 45-year low of 1 percent, while e¤ectively convincing the bond markets that rates
would have been kept so low for a considerable period. Yet, it remains an open question whether
or not the Fed has cut interest rates by “too much”and left them low for “too long”in relation
to the current economic conditions. Possibly, the exceptionally low cost of capital–at a time
when the expected worldwide return on capital has likely increased–might have encouraged
excessive borrowing, allowing …nancial imbalances to build up.
Inferring the stance of monetary policy from prevailing interest rates requires some sort of
benchmark. Economists tend to assume that such a benchmark role is played by the equilib-
rium, or “natural,”real rate of interest. Put simply, the natural real rate of interest is the rate
that keeps output at its potential and in‡ation stable, once any shocks to the economy have
played out. According to this concept, one can thus gauge the stance of, say, the US monetary
policy by comparing the actual level of the federal fund rate with the natural rate. Yet, to
make life tricky for both policymakers and market participants, the natural rate of interest is
not observable and may vary over time, in line with changes in the rate of return on capital
or households’rate of time preference. If these movements are su¢ ciently large, any constant
long-term average would be a poor predictor of the natural rate of interest.
By drawing on a structural model featuring explicit theoretical foundations, our paper
proposes an original econometric approach to estimate the evolution of an economy under
equilibrium conditions. The methodology makes it possible to track down and interpret the
di¤erent sources of variation and uncertainty in interest rate setting— such as shifts in labour
productivity, preference shocks, or households’degree of risk aversion— with su¢ cient precision
and in real time. Our estimates show how the Fed— to deal with the economic downturn
following the stock market crash— has been driving the cost of capital signi…cantly below
the natural rate for over four years. The results also provide some support to the belief that
postwar US in‡ation’s rise and fall was due to Fed’s inability to disentangle–in a timely manner–
transitory from permanent shifts in productivity. In fact, over 85 percent of the uncertainty
surrounding future monetary policy stance is found to stem from imprecision surrounding
inferences of the underlying equilibrium position.
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1
Introduction
After the US stockmarket bubble burst, the Fed has successfully managed to reduce business
cycle volatility by reducing short-term rates to a 45-year low of 1 percent, while e¤ectively
convincing the bond markets that rates would have been kept so low for a considerable period.
The question is to determine whether in 2001-03 the Fed has cut interest rates by “too much”
and left them low for “too long” in relation to their underlying equilibrium level. Although
optimal monetary policy is not associated with keeping the actual real rate close to its “normal”
level on a period-by-period basis, a prolonged period of exceptionally low cost of capital–at a
time when the expected return on capital has likely increased–might have encouraged excessive
borrowing, contributing to the buildup of …nancial imbalances.
Inferring the stance of monetary policy from prevailing interest rates requires some sort
of benchmark. Following Wicksell´s seminal idea, economists tend to assume that such a
benchmark role is played by the natural real rate of interest. Put simply, the natural real rate
of interest is the rate that keeps output at its potential and in‡ation stable, once any shocks
to the economy have played out. According to this concept, one can thus gauge the stance of,
say, the Fed’s monetary policy by comparing the actual level of the federal fund rate with the
nominal natural rate. If the rate set by the central bank is lower than the natural rate, the
economy and in‡ation would be expected to accelerate, as there will be excessive investment
and borrowing, while households will not save enough. Conversely, if the Fed keeps interest
rates above the natural rate, policy would rein in the economy and in‡ation would eventually
slow.
Yet, to make life tricky for both policymakers and market participants, the natural rate of
interest is not observable and may vary over time, in line with changes in the rate of return
on capital or households’rate of time preference. There are indeed reasons to think that the
Wicksellian rate may have shifted over the last decade. The surge in productivity growth owing
to the information technology revolution is likely to have boosted expectations about future
pro…ts and investment opportunities, moving upwards the demand curve for investment funds,
ceteris paribus. Meanwhile, in industrial countries, households seem to have become more
impatient and decided that they need to save less than they used to, on the belief that rising
asset and house prices will provide them with adequate resources to …nance their retirement.
As a result, the so-called IS (Investment=Saving) curve— the equilibrium line showing the
negative relationship between spending and real interest rate— may have moved rightwards,
implying a higher natural rate of interest for any level of potential output. Structural forces—
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such as enhanced competition from emerging markets, deregulation, and faster productivity
growth— may have also helped to hold down (the equilibrium level of) in‡ation, whilst central
banks’success in taming in‡ation (around such an equilibrium level) has anchored in‡ationary
expectations. This has meant that monetary policy can now be eased more freely to deal with
economic ‡uctuations, provided that in‡ation remains subdued.
From a theoretical perspective, tracking down the evolution of the natural rate of interest
requires to pin down the behavior of the economy under equilibrium conditions. In our paper,
the problem is tackled within Woodford’s (2003) “neo-Wicksellian” framework, which has the
property of ensuring consistency between the speci…cation of price adjustments and the (unob-
servable) behavior of the economy under ‡exible prices.1 Speci…cally, the model entails rational
expectations and neo-Keynesian price rigidities, while guaranteeing a zero output gap in steady
state, in compliance with the natural rate hypothesis. Reconciling short-term and equilibrium
dynamics becomes particularly relevant in system estimation. In this context, each change
to the speci…cation of price adjustments a¤ects the likelihood-maximizing vector of structural
parameters which— together with the dynamics and the distributions of shocks— determine the
(unobservable) behavior of the economy under ‡exible prices.
Originally, our prototypical dynamic optimizing sticky-price model has been generalized
along three important dimensions. First, it accounts for possible nonstationarity in the steady-
state level of in‡ation and real output. Second, it allows for shifts in the underlying stochastic
processes, as well as in the parameterization of the model. Third, it has been estimated in real
time, e.g. by using observable, non-detrended data which are readily available at the time they
are needed.2 To do this, the model’s likelihood function is evaluated using a Sequential Monte
Carlo algorithm, providing joint estimates of the structural parameters and the unobservable,
time-varying, nonstationary state variables. The idea beneath this approach is to represent any
probability law by a large number of random samples, or particles, evolving over time on the
1 Notable examples of dynamic, optimizing models examining the behaviour of the natural levels of output
and interest are McCallum and Nelson (1999), Rotemberg and Woodford (1999), Galí, López-Salido, Vallés
(2003), Woodford (2003), Giammarioli and Valla (2003), Neiss and Nelson (2003), Smets and Wouters (2003),
Amato and Laubach (2004), Andrés, López-Salido, Nelson (2005).
2 The issue of real-time estimation of unobservable equilibrium variables has received increasing attention
in the context of monetary policy analysis. See, among others, Laubach (2001), Larsen and McKeown (2002),
Orphanides and Williams (2002), Laubach and Williams (2003), Basdevant, Björksten, and Karagedikli (2004),
Cuaresma, Gnan, and Ritzberger-Grunenwald (2004), Mésonnier and Renne (2006), Sevillano and Simon (2004),
Garnier and Wilhelmsen (2005). Overall, real-time measures of the time-varying natural rate of interest— like
those of the equilibrium level of output, unemployment, and in‡ation— are found to be highly imprecise, limiting
their relevance for policy.
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basis of a simulation-based updating scheme, so that new observations are incorporated in the
…lter as they become available. By tracking down the whole distribution of the natural rate of
interest in real time, it is possible to interpret the di¤erent sources of variation and uncertainty
in interest rate setting— such as shifts in labour productivity, volatility of preference shocks,
or households’degree of risk aversion— in a timely and helpful manner.
To anticipate our conclusions, our estimates show how the Fed— to deal with the slow-
down following the stock market crash— has been cutting interest rates aggressively, driving
the cost of capital signi…cantly below the natural rate for over four years. In the most recent
quarters, however, repeated monetary policy tightening has managed to correct this disequi-
librium while slowing down growth. Yet, the in‡ation gap is found to be still positive at the
end of 2006. From a positive perspective, our …ndings are broadly in line with Cogley and
Sbordone’s (2005) general argument that ignoring time variation in the underlying evolution of
the economy under ‡exible prices may alter the estimates of structural parameters. While the
volatility of preference shocks is likely to have fallen over time, structural pricing parameters
are essentially time-invariant once shifts in trend growth and in‡ation are also accounted for.
From a normative perspective, our results provide some support to the hypothesis that–over
the postwar period–Fed’s inability to promptly disentangle permanent from transitory shifts
in productivity may have translated into persistent movements in the in‡ation rate itself. By
the same token, uncertainty about future interest rate settings is found to be primarily due to
vagueness in the estimate of the underlying stochastic equilibrium rather than to imprecision
about the structural parameterization of the model economy or the policy rule.
The rest of the paper is organized as follows. Section 2 lays down a dynamic stochastic model
consistent with the neo-Wicksellian approach to price determination. Section 3 presents the
methodology adopted to evaluate the likelihood of such a model, estimate posterior densities for
its parameters, and track down the time-varying distribution of its unobservable state variables.
Section 4 discusses estimation and forecasting results, while Section 5 concludes the paper by
providing an outlook on future work.
2
A Generalized “Neo-Wicksellian” Framework
Our structural model has a prototypical rational expectations speci…cation, similar to the one
used by Boivin and Giannoni (2003), Giannoni and Woodford (2005), and described in Wood-
ford (2003). On the demand side, we take into account habit persistence in the level of aggregate
expenditure, assuming that for each household i in period t, utility depends not only on current
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expenditure, but also on the level of expenditure in the previous period. On the supply side,
we allow for price indexation for …rms that are not allowed to set their price optimally in a
given period, in order to generate more realistic level of in‡ation inertia.3 The setup is then
originally enriched by letting in exogenous non-stationary processes— describing (labor) pro-
ductivity and target in‡ation dynamics, respectively— as well as idiosyncratic demand, supply,
and policy shocks.
2.1
Demand Side
We assume a model economy producing a continuum of goods indexed by j and populated
by a continuum of households indexed by i, uniformely distributed over the [0; 1] interval.
By considering the limiting case of a cashless economy— and thereby abstracting from real
balances— each household i seeks to maximize the following in…nite discounted sum of future
utilities:
1
1
1
E
t
t
P1 "u
Ci
Ci
Hi (j)1+%
(1)
=t
1
1
1 + %
where Ci is an index of the household’s consumption of the di¤erentiated goods supplied at
t and Hi (j) denotes the amount of hours supplied by each household i for the production of
each good j. In addition,
= 1=(1 + r) 2 (0;1) is the households’discount factor, is the
coe¢ cient of relative risk aversion or the inverse of the intertemporal elasticity of substitution,
and % represents the inverse of the elasticity of work e¤ort with respect to the real wage.
The parameter 0
1 measures the degree of habit formation in consumption. Consumers’
utility hence depends positively on deviations of consumption Ci from an existing stock Ci 1,
and negatively on the total labor supplied.4 Equation (1) also contains a shock to the discount
rate that a¤ects the intertemporal preferences of households, "u, distributed as a log-normal
with mean equals to 1.
Households maximize their objective function (1) subject to an intertemporal budget con-
straint that is given by:
Bi
Bi
b
=
1 + (W iHi + Ai )
Ci
(2)
P
P
3 This extension has been recently proposed by Christian, Eichenbaum, and Evans (2005). In‡ation indexa-
tion has been also used, for instance, in Smets and Wouters (2005), Milani (2005), and Andrés et al. (2005).
4 The consumption habit is assumed here to depend on the household’s own past level of expenditure, and
not on that of other households.
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Document Outline

• (Un)naturally low? Sequential Monte Carlo tracking of the US natural interest rate
• Contents
• Abstract
• Non-technical summary
• 1 Introduction
• 2 A Generalized "Neo-Wicksellian" Framework
  • 2.1 Demand Side
  • 2.2 Supply Side
  • 2.3 Monetary Policy
  • 2.4 Model Solution
  • 2.5 Prior Speci•cation
• 3 Likelihood Evaluation and Bayesian Estimation
  • 3.1 A Sequential Monte Carlo Approach
  • 3.2 Parameters Estimates
  • 3.3 Out-of-Sample Forecasts
• 4 Empirical Results
  • 4.1 Structural Parameters: Posterior Distributions
  • 4.2 Unobservable State Variables: A Real-Time Policy Assessment
  • 4.3 Out-of-Sample Forecasts and Model Evaluation
• 5 Conclusions and Prospects
• References
• Tables and figures
• European Central Bank Working Paper Series

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