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Available online at: www.cepr.org/pubs/dps/DP2848.asp and http://papers.ssrn.com/paper.taf?abstract_id=275026

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No. 2848

THE REAL INTEREST RATE GAP AS AN INFLATION INDICATOR

Katharine S Neiss and Edward Nelson

INTERNATIONAL MACROECONOMICS

ISSN 0265-8003

THE REAL INTEREST RATE GAP AS AN INFLATION INDICATOR

Katharine S Neiss, Bank of England Edward Nelson, Bank of England and CEPR

Discussion Paper No. 2848 June 2001

Centre for Economic Policy Research 90–98 Goswell Rd, London EC1V 7RR, UK

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This Discussion Paper is issued under the auspices of the Centre’s research programme in International Macroeconomics. Any opinions expressed here are those of the author(s) and not those of the Centre for Economic Policy Research. Research disseminated by CEPR may include views on policy, but the Centre itself takes no institutional policy positions.

The Centre for Economic Policy Research was established in 1983 as a private educational charity, to promote independent analysis and public discussion of open economies and the relations among them. It is pluralist and non-partisan, bringing economic research to bear on the analysis of medium- and long-run policy questions. Institutional (core) finance for the Centre has been provided through major grants from the Economic and Social Research Council, under which an ESRC Resource Centre operates within CEPR; the Esmée Fairbairn Charitable Trust; and the Bank of England. These organizations do not give prior review to the Centre’s publications, nor do they necessarily endorse the views expressed therein.

These Discussion Papers often represent preliminary or incomplete work, circulated to encourage discussion and comment. Citation and use of such a paper should take account of its provisional character.

Copyright: Katharine S Neiss and Edward Nelson

CEPR Discussion Paper No. 2848

June 2001

ABSTRACT

The Real Interest Rate Gap as an Inflation Indicator*

A long-standing area of research and policy interest has been the construction of a measure of monetary policy stance. One measure that has been proposed – as an alternative to indices that employ monetary aggregates or exchange rates – is the spread between the actual real interest rate and its flexible-price, or natural-rate, counterpart. We examine the properties of the natural real interest rate and ‘real interest rate gap’ using a dynamic stochastic general equilibrium model. Issues we investigate include: (1) the response of the gap and its components to fundamental economic shocks; and (2) the indicator and forecasting properties of the real interest rate gap for inflation, both in the model and in the data. Our results suggest that the real interest rate gap has value as an inflation indicator, supporting the ‘neo-Wicksellian framework’ advocated by Woodford (2000).

JEL Classification: E31, E43, E52 Keywords: inflation, monetary policy, natural interest rate, output gap and real interest rate gap

Katharine S Neiss Structural Economic Analysis Division Monetary Analysis HO-3 Bank of England Threadneedle Street London EC2R 8AH Tel: (44 20) 7601 4588 Fax: (44 20) 7601 5018 Email: [email protected]

For further Discussion Papers by this author see: www.cepr.org/pubs/new~dps/dplist.asp?authorid=144346

Edward Nelson Research Adviser Monetary Policy Committee, MPC Unit HO-3 Bank of England Threadneedle Street London EC2R 8AH Tel: (44 20) 7601 5692 Fax: (44 20) 7601 3550 Email: [email protected]

For further Discussion Papers by this author see: www.cepr.org/pubs/new~dps/dplist.asp?authorid=144589

* We thank Peter Andrews, Alec Chrystal, Mark Gertler, DeAnne Julius, Robert King, Robert Kollmann, Thomas Laubach, Athanasios Orphanides, Glenn Rudebusch, Chris Salmon, Frank Smets, Paul Tucker, Simon van Norden, Raf Wouters, Michael Woodford and participants at Bank of England seminars and the Canadian Macro Study Group meeting, November 2000, for helpful discussions. The views expressed in this Paper are those of the authors and should not be interpreted as those of the Bank of England, the Monetary Policy Committee, or the Centre for Economic Policy Research.

Submitted 27 March 2001

NON-TECHNICAL SUMMARY

In this Paper, we investigate the business cycle properties of the real interest rate relative to its natural value. Our investigation into the natural real interest rate is motivated by the possibility of constructing a measure of monetary policy stance based exclusively on interest rates. Recent work by Michael Woodford has revived the ideas of Knut Wicksell by focusing on ‘the gap between the current level of the “natural rate” of interest and the interest rate controlled by the central bank’ as the key variable for the analysis of ‘inflationary or deflationary pressures’. In line with this terminology, we describe the spread between actual and natural real interest rates as the real interest rate gap.

This Paper examines a number of questions involving the real interest rate gap, including: • Does the real interest rate gap provide a useful tool for monetary policy

analysis? • Is the real interest rate gap more difficult to measure than the output gap? • How do empirical measures of the real interest rate gap perform in

forecasting UK inflation?

We develop a dynamic stochastic general equilibrium model with sticky prices in order to examine the behaviour of the natural real interest rate and the real interest rate gap. In our model, household spending and asset accumulation, and also the prices that firms set, are based on optimizing behaviour. We build on the existing literature by including capital formation (subject to adjustment costs), habit persistence in consumption, technology and demand shocks, and two alternative models of price stickiness. The baseline model of price stickiness that we use is Calvo price-setting, which can be interpreted as a system of staggered contracts for nominal prices.

We calibrate the model to the UK economy, and examine the response of the natural real interest rate to shocks to both technology and demand. Our focus is mainly on the indicator properties of the real interest rate gap, and so we examine how well the real interest rate gap does in signalling future inflation – both in response to specific shocks (which we examine using impulse response functions) and when all shocks are hitting the economy simultaneously (which we examine using stochastic simulations).

Using our model as a guide, we also construct empirical estimates of the natural real rate and the real interest rate gap from UK data. Our key results include the following:

• The response of the natural real interest rate to a technology shock depends on whether or not capital is included in the model and, if so, whether or not there are capital adjustment costs. We find that with capital adjustment costs, the natural real interest rate can fall in response to a technology shock. For a given actual real interest rate, this leads to a rise in the real interest rate gap.

• Conversely, the natural real interest rate rises in response to a demand shock. For a given actual real interest rate, this leads to a decline in the real interest rate gap.

• The less firms and households are willing to adjust their quantities, the more the natural real rate needs to adjust to maintain equilibrium.

• Stochastic simulations indicate that the real interest rate gap and output gap do equally well in forecasting inflation. In addition, the behaviour of the real interest rate is a reasonable approximation for the behaviour of the real interest rate gap. By contrast, output (or detrended output) is not a good indicator of the behaviour of the output gap. This suggests the value of constructing measures of both gaps instead of concentrating only on output gap measures.

Finally, we test the predictive power of the real interest rate gap for UK inflation. On quarterly UK data, the real interest rate gap is closely related to future inflation, whether the relationship is judged by correlations or by the marginal predictive content of the gap for inflation in regressions. Our results suggest that constructing a real interest rate gap series, using theory as a guide, can have value for evaluating the stance of monetary policy and the prospects for future inflation, in keeping with the neo-Wicksellian framework of Woodford (2000).

3

1 Introduction

In evaluating the consequences of monetary policy actions for the future behaviour of

inflation, it is often useful to construct a measure of monetary policy stance. Typically,

such a measure will use some indicator of monetary or demand conditions, and express

that measure relative to a baseline value (such as the value consistent with price stability).

There are several candidates for variables to include in a measure of monetary stance.

Monetary aggregates (or their rates of growth) have historically been prominent

candidates. A common criticism of their use for this purpose is that they can be distorted

by innovations to the financial system.1 In addition, monetary aggregates have been

viewed as being too far removed from the typical instrument of monetary policy, namely

the short-term nominal interest rate. This is particularly so for broad monetary

aggregates.

One measure of the stance of monetary policy that does not rely on monetary aggregates

is the Monetary Conditions Index (MCI). An MCI consists of a weighted index of the

interest rate and the exchange rate.2 A weakness of this approach is that it relates

changes in the exchange rate to inflation without identifying the shock responsible for the

change in the exchange rate. For example, MCIs tend to interpret exchange rate

depreciations as producing monetary ease, but it is possible for the equilibrium real

exchange rate to depreciate under conditions of price stability (King, 1997). More

generally, the weight given to exchange rates in the index may implicitly assume a much

closer and more mechanical link between exchange rate depreciations and inflation than

can be justified empirically (McCallum and Nelson, 2000; Stock and Watson, 2000).3

In light of the arguments against including monetary aggregates or exchange rates in

measures of monetary policy stance, one potentially fruitful option is to construct a 1 See Svensson (1999) for a recent articulation of this and other criticisms of monetary aggregates. 2 Ericsson and Kerbeshian (1997) provide a detailed bibliography of the MCI literature.

4

measure of the monetary policy stance based exclusively on interest rates. In this vein,

Woodford (1999a, 2000) has advanced the use of the natural or �equilibrium real rate of

interest� (the real interest rate that prevails under price flexibility) in the analysis of price

level determination. Woodford�s analysis revives the ideas of Wicksell (1898, 1906)

within a dynamic stochastic general equilibrium model. He argues that an analysis of

price level determination that is based on the difference between actual and natural real

interest rates is preferable to traditional analysis, which is based on the interaction of

money demand and supply. In models in which the instrument of monetary policy is a

nominal interest rate, this �neo-Wicksellian� analysis of price level determination avoids

the cumbersome procedure of solving for the implied money supply function.

In Woodford�s framework, the key variable for the analysis of �inflationary or

deflationary pressures� is �the gap between the current level of the �natural rate� of

interest and the interest rate controlled by the central bank�(Woodford, 1999a, p. 35). In

line with this terminology, we define the real interest rate gap as the spread between the

actual and �natural� real interest rates, and this paper studies this gap concept.

The position of actual real interest rates relative to their �natural�, �neutral�, or

�equilibrium� value, has been considered by policy makers as well. For example, at the

December 9�10, 1998 meeting of the Bank of England�s Monetary Policy Committee,

�[t]he Committee discussed whether it was helpful to think about the appropriate level of

nominal interest rates by reference to the concept of a �neutral� level, which provides

neither stimulus nor restraint to the economy.� The outcome of this discussion was that

�[w]hile some members of the Committee found the concept of the neutral rate useful in

deciding on interest rate policy, other members found the uncertainty surrounding its

level so large that the concept was of little use as a practical guide to policy.� (Monetary

Policy Committee, 1999, p. 67).

3 One part of the problem is that the relative weights on interest rates and exchange rates in MCIs are typically based on the estimated relative effects of the two variables on aggregate demand, rather than inflation per se. See Eika, Ericsson, and Nymoen (1996) for a discussion of MCI weights.

5

Thus, it is probably accurate to characterise the attitude by many policy makers to the

practical usefulness of the real interest rate gap concept as skeptical. It is also very likely

that the real interest rate gap is regarded as less easy to measure empirically than an

�output gap� concept. Nevertheless, we see three key practical reasons for reconsidering

the importance of looking at real interest gaps:

1. Understanding the behaviour of the natural real rate appears to be important for

understanding the empirical relationship between the real interest rate and output.

Consider the following table, which gives correlations between the level of detrended log

GDP (yt) and the real interest rate (current and lagged) for quarterly US and UK data:

Table 1: Correlations between detrended GDP and the real interest rate

Corr(yt, rt−k)

US 1980 Q1−1999 Q4

UK 1980 Q1−1999 Q4

k = 0 −0.21 0.10 k = 1 −0.28 0.08 k = 2 −0.27 0.07 k = 3 −0.23 0.07 k = 4 −0.21 0.11

Note: For both the US and the UK, yt is obtained from the residuals of a regression of log real GDP (seasonally adjusted) on a quadratic trend, 1976 Q1−1999 Q4, and the nominal interest rate used in the calculation of rt is the quarterly average of the Treasury bill rate. For the US, rt is then measured by the ex-post real interest rate. For the UK, rt is (as in Figure 7 and Section 6 below) the Treasury bill rate minus the forecasts of next-period annualised inflation from a VAR(8). The inflation series used in these calculations is the seasonally adjusted quarterly log change in the CPI (US) or the RPIX (UK).

For the US, the correlation between the two variables is negative and, therefore, perhaps

interpretable as reflecting the negative effects of the real interest rate on aggregate

demand. But for the UK, the correlation is generally positive, indicating that the real

6

interest rate is procyclical.4 To identify the negative effects of the real rate on aggregate

demand suggested by economic theory, it may be necessary to control for the fluctuations

of the natural levels of output and the real rate in response to real shocks.

2. The real interest rate gap potentially has stronger leading indicator properties than the

output gap. In models where the output gap responds to the real interest rate gap with a

lag, and inflation reacts to the output gap gradually, the real interest rate gap should give

advance information about both the output gap and inflation.

3. It is possible that the real interest rate gap may actually be measured with less

uncertainty than the output gap. While measures of the output gap are very commonly

used in policy analysis and forecasting, it is worth remembering that many of these

measures are based on the assumption that potential output evolves according to a

deterministic trend (such as a linear, quadratic, or broken-linear trend). Economic theory

suggests, instead, that potential output, while certainly containing a trend component,

also fluctuates over the business cycle in response to all real shocks. The validity of

many standard measures of the output gap therefore rests on the hypothesis that the

response of potential output to these shocks is relatively flat, so that detrended output

provides a good approximation of the output gap.5 This hypothesis is difficult to justify

in light of the results from many dynamic general equilibrium models (including ours

below) that suggest that potential output fluctuates considerably. On the other hand, it is

possible that the response of the natural real rate to shocks may be relatively flat, which

would make it relatively straightforward to construct reliable measures of the real interest

rate gap even though data are not available on the natural rate.

4 In terms of Boldrin, Christiano, and Fisher (1999), the real interest rate does not exhibit an �inverted leading indicator� property for real GDP in the UK. 5 Similarly, production-function based approaches to measuring potential output typically do not keep track adequately of the distinction between actual and flexible-price values of capital and labour inputs. For example, measuring flexible-price labour supply by the total labour force involves the assumption that labour supply is inelastic�and tends to generate an overly smooth potential output series. A general equilibrium model such as ours provides a way of keeping track of flexible-price values of variables and of their response to real shocks.

7

The above considerations suggest that there are benefits from further study of the real

interest rate gap concept. To this end, this paper develops a sticky-price, dynamic

stochastic general equilibrium (DSGE) model in order to examine the behaviour of the

natural real interest rate and the real interest rate gap. Any general equilibrium model,

whether sticky- or flexible-price, implicitly provides a model of the natural real interest

rate and the real interest rate gap. In flexible-price models, the behaviour of the real

interest rate gap is trivial�it is zero each period by definition, since the real interest rate

is the natural real rate.6 In sticky-price models,7 the real interest rate gap is zero on

average (provided that the Phillips curve is vertical in the long run), but will not be zero

every period (except in the very specific case in which the monetary authority runs a

policy that eliminates completely the real effects of price stickiness). In addition to the

Woodford papers mentioned above, King and Watson (1996), Rotemberg and Woodford

(1997), Clarida, Gali, and Gertler (1999), and Giannoni (2000) have focussed on the

behaviour of real rates relative to their natural values in general equilibrium sticky price

models. Of these, only King and Watson have explicit capital formation.8 Their model,

however, does not include preference shocks, which in theory (and in our own numerical

results below) have a major influence on the natural rate. Our model builds on the

existing literature by including both capital formation and preference shocks, as well as

elements absent from the aforementioned papers, such as non-time-separable preferences.

Unlike the previous work, we also present results for more than one model of price

stickiness.

An advantage of our use of a fully specified general equilibrium model is that we can

trace the determination of the natural real rate�which we denote rt*�back to the

model�s underlying economic shocks. Woodford (1999a, 1999b) assumes that rt* is a

univariate AR(1) exogenous process. By making this high-level assumption, such an

analysis does not permit decomposition of the underlying real shocks that determine rt*. 6 This is so even for models, such as those in Beaudry and Guay (1996), that enrich the dynamics of real business cycle models, but maintain the assumption of price flexibility. 7 Here we presume that the stickiness of prices lasts more than one period. 8 Woodford (2000) adds endogenous capital formation to his model. One difference between his model and ours below is that our calibration of preference parameters, which strongly affect the interest elasticity of

8

And analysis of the underlying shocks affecting rt* could aid in reducing the uncertainty

surrounding its level, which, as noted above, has inhibited the usefulness of the concept

to policy makers.

The framework we adopt also enables us to examine the effect of particular shocks or

model elements on the variance of the natural real rate. This is important in light of the

diverging estimates of the variance of the natural real rate in recent papers. King and

Watson (1996) find that the natural real rate in a calibrated DSGE model has a variance

well below that observed in US data for the actual (ex-post) real rate. The estimates in

Rotemberg and Woodford (1997), however, suggest a more volatile natural real interest

rate series, with a standard deviation of 3.9 percentage points (annualised) on post-1979

US quarterly data9�around 1 percentage point greater than that observed empirically for

the actual real rate in US and UK data over that period.

Our paper is organised as follows. Section 2 describes the model. Section 3 analyses the

response of the natural real interest rate and the real interest rate gap in the model to

shocks. Section 4 compares the model�s dynamics to those in the data, and Section 5

examines the properties of the real interest gap in the model. Our focus is mainly on the

indicator properties of the real interest rate gap, and so we examine how well the real

interest rate gap does in signalling future inflation when all shocks are hitting the

economy simultaneously (which we examine using stochastic simulations). Using the

theoretical model as a guide, Section 6 then constructs estimates of the real interest rate

gap series from UK data and evaluates the series� forecasting ability for inflation.

Section 7 concludes, and an Appendix describes how we measure output and real interest

rate gaps in the model.

2 The model

This section describes the DSGE model employed in the paper.

aggregate demand, is more in keeping with those suggested by econometric studies such as Fuhrer (2000) and Ireland (2000). 9 See Woodford (1999b, fn. 50).

9

2.1 Households

The economy is inhabited by a large number of households, each of which has

preferences defined over a composite consumption good (denoted Ct in period t), leisure

(lt), and real money balances (Mt / Pt). Money enters the utility function directly to

capture the idea that real balances provide a transactions-facilitating service. The

representative household chooses a sequence of consumption, leisure, nominal money

and one-period bond (Bt+1) holdings, and capital (Kt+1), to maximise lifetime utility:

σ 1 1 ε

σ

0 1

σ γ E β λ

σ 1 1 ε t j t jj

t t t jh j t j t j

C M bl

C P

− −

∞ + +

+ = + − +

� � � � � �� �+ +� � � �� �� � � �− − � �� �

� (1)

subject to a series of real period budget constraints:

1 1 1 φ , 0,1,...,

t j t j t j t j t jG t j t j t j t j t j t j t j t j

t j t j t j t j t j

M B B M C X w N z K R X j

P P P P P ητ+ + + + + − +

+ + + + + + + − + + + + + +

+ + + = + + + − − ∀ = ∞

(2)

where b > 0, γ > 0, ε > 0, and β∈ (0,1). Xt in equation (2) is related to Kt by:

( )1 1t j t j t jX K Kδ+ + + += − − , (3)

where δ∈ [0,1). In equation (2), RtG denotes the gross nominal interest rate, tN denotes

labour supplied, τt denotes government transfers, and zt the return on capital. In the

numerical results of Section 5, we consider two versions of the model: a simplified

stripped-down version of the model with no capital variation (Kt = Kss and Xt = δ Kss for

all t), and one with capital accumulation but with adjustment costs operative. If the

model had no capital adjustment costs, the variable Xt could be interpreted as investment

expenditure; with capital adjustment costs, we refer to Xt as quasi-investment. The size

10

of capital adjustment costs is determined by the parameters ϕ (ϕ > 0) and η (1 < η < ∞).10

When capital accumulation is subject to adjustment costs there is a wedge between the

return on capital and the real interest rate, beyond the constant wedge implied by the rate

of depreciation of capital (δ).

The parameter σ in the consumer�s utility function indexes the curvature of the utility

function: the larger σ, the more the household is willing to shift consumption across

periods in response to interest rate changes. 1/(σε) is the steady-state consumption

elasticity of money demand, and γ determines the steady-state consumption/money ratio.

Preferences over consumption take on a non-time-separable form to capture the idea that

households may exhibit habit formation in their consumption patterns. The parameter h

∈ [0,1) indexes the degree of habit formation. If h = 0, the household exhibits no habit

formation in consumption, and preferences are time-separable. For 0 < h < 1, utility from

current consumption depends partially on consumption in adjacent periods. We work

with strictly positive h in this paper in light of evidence that doing so reduces some of the

empirical shortcomings of quantitative business cycle models (Boldrin, Christiano, and

Fisher, 1999; Fuhrer, 2000). Finally, the utility function is augmented by a disturbance

(λt) to consumption preferences, which we interpret as an �IS� or �real demand� shock.

The time allocation constraint of the household is normalised so that labour and leisure

sum to one:

1t tN l+ = . (4)

The composite consumption good tC is a Dixit-Stiglitz aggregate of a multiplicity of

differentiated goods, indexed by i∈ [0,1]. Under this scheme, the composite consumption

good and price index are defined as:

10 Our specification of the capital adjustment cost function follows Abel (1983) and Casares and McCallum (2000).

11

( ) 1

1

0 t tC C i di

ρ ρ� �

= � � � � � (5)

( ) 1

1

1

0 t tP p i di

ρ ρρ

ρ

−

− � �

= � � � � � , (6)

which can be derived from two-stage budgeting. The parameter ρ∈ (0,1) determines the

degree of substitutability of differentiated goods in consumption. For example, if ρ is

close to 1, goods are perfectly substitutable in consumption and firms are perfectly

competitive. The inverse of ρ is therefore the (gross) steady-state markup of price over

marginal cost.

Letting ψt denote the Lagrange multiplier on (2), optimal household behaviour yields the

following conditions:

Ct:

σ 1 σ 1 σ σ

1 1

1

1 1 λ β E λ ψt tt t t th h

t t t t

C C h

C C C C

− −

+ +

−

� � � � − =� � � �

� � � � (7)

lt: ψt tb w= (8)

Mt: 1 1

ψ ψ βEt tt t

tt t

M PP

P

ε

γ −

+

+

� � = −� �

� � (9)

Bt+1: 1 1

βψ 0 ψ EG tt t t

t t

R P P

+

+

= − (10)

Kt+1: ( ) ( ) ( )1 11 1 10 ψ 1 φ βE ψ 1 δ 1 φηt t t t t tX X zη ηη − −+ + +� �= + − − + +� � (11)

12

ψt : 1 1

1 φ Gt t t t t

t t t t t t t t t t t t t

M B B M C X w N z K R X

P P P P P ητ+ −

−+ + + = + + + − − . (12)

2.2 Firms

There is a continuum of monopolistically competitive firms, indexed by [ ]0,1j ∈ . Each firm j chooses price (pjt), labour (Njt), and capital (Kjt+1) to maximise its profits:

jt jt t jt t jt

t

Y p w N z K

P − − , (13)

subject to a demand function for its good, written as:

( )1 jt jt

t t

p Y P Y

ρ− − � �

= � � � �

, (14)

and a production function: α 1 α

jt t jt jtY A N K −= , (15)

where Pt and Yt are the aggregate price and output levels, At is a technology shock, and

α∈ (0,1). In a symmetric equilibrium (under which we can drop the subscript j), the first

order conditions for the representative firm are given by:

lt: α µGt t t t

Y w

N = , (16)

Kt: (1 α) µGt t t t

Y z

K − = . (17)

Here, µtG is the gross markup of price over marginal cost; in steady state this variable

equals 1/ρ, as noted above.

2.3 Market Clearing

Finally, the economy is subject to the following resource constraint:

13

φt t t tY C X X η= + + , (18)

which differs from the usual closed-economy constraint due to the presence of capital

adjustment costs.

2.4 Equilibrium

In order to investigate the dynamics of the model, we log-linearise the above optimality

conditions and technological constraints around the steady state. The resulting equations

are as follows (with the superscript ss denoting steady-state values):

Consumption 2 2

λ 1 1

β (σ 1) β σ β β σ 1 (σ 1) 1 β ρ E ψ λ

σ(1 β ) σ(1 β ) σ(1 β ) 1 βt t t t t t h h h h h h

c c c h h h h+ −

− − + − − − = − − +

− − − − (19)

Labour market equilibrium

0 ψ µt t t ty n= − + − (20)

Money demand

1 1 1 0 ψ

ε εt t tss rm R

R = − − − (21)

Euler equation

1ψ ψt t t tE r+ = − (22)

Fisher equation

1πt t t tE R r+ = − (23)

Quasi-investment

14

( ) ( )

( ) ( ) ( )

( ) ( )1 1 1 11 1 1 α 1

1 δ E E µ 1 φ 1 φ

ss ss

t t t t t t t t ss ss

Y Kx y k x r

X X η η

ρ

η η η η + + + +− −

− − + − − = +

− − (24)

Law of motion for capital

1 δ (1 δ)t t tk x k+ = + − (25)

Resource constraint

η φηss ss ss

t t tss ss C X X

y c x Y Y

+ = + (26)

Production function

( )α 1 αt t t ty a n k= + + − (27)

where yt, ct, kt, nt, rmt, µt, and at are the log deviations of Yt, Ct, Kt, Nt, (Mt / Pt), µGt and At,

respectively, from their steady-state values. Similarly, the Lagrange multipliers should

now be interpreted as log-deviations from the steady state of the corresponding variables

in the original nonlinear model. πt is the demeaned quarterly net inflation rate, and Rt and

rt denote the demeaned net nominal and real interest rates, respectively.

Thus we have a log-linear model describing the path of eleven endogenous variables:

output (yt), capital (kt), consumption (ct), quasi-investment (xt), the nominal interest rate

(Rt), the real interest rate (rt), the marginal utility of consumption (ψt), real balances (rmt),

labour input (nt), the markup (µt), and inflation (πt). The system so far consists of nine

equations. To complete the model, we need two more equations: a price setting equation

and a monetary policy rule. We turn now to the first of these.

15

2.5 Price setting

This paper is concerned with the real interest rate gap, defined as the spread between the

actual and �natural� real interest rates. We therefore need to specify the equilibria

associated with each of the two interest rate concepts. The natural real interest rate is the

real rate that prevails in the case of fully flexible prices, whereas the actual real interest

rate is the real rate that prevails under sticky prices.11

Our procedure for obtaining a sticky-price equilibrium consists of two steps:

(a) First, we solve for natural output (in logs, yt*) and the natural real interest rate (rt*) by

obtaining the flexible-price equilibrium of the log-linear model above.12 This flexible-

price equilibrium is characterised by a constant markup, so that µt = 0 (see for example

Ireland, 1997). The flexible-price equilibrium can therefore be obtained by imposing this

condition.

(b) We specify a model of gradual price adjustment, describing how the price level

responds to the output gap (yt � yt*). Our baseline specification of gradual price

adjustment will be the Rotemberg (1982)�Calvo (1983) model, which, following Roberts

(1995), we write as the New Keynesian Phillips curve:

1 µβE π π α µt t t t+ = + (28)

where µα > 0. To examine the sensitivity of our results to the price-setting specification,

we report supplementary results using a different model of price setting, that of Fuhrer

and Moore (1995):

11 As in other sticky-price DSGE models, the assumption of monopolistic competition among firms provides groundwork for the assumption of gradual price setting. 12 A monetary policy rule needs to be appended to this system to complete the model, but due to monetary neutrality under price flexibility, the solution for potential output will be invariant to the rule selected.

16

1 1 µ0.5E π π 0.5π +α µt t t t t+ −= − . (29)

In both models, the size of µα governs the degree to which prices are sticky. The larger

is µα , the more flexible are goods prices. 13

2.6 Shocks

There are two types of real shocks in this model: a technology and a demand (IS) shock.

Each of the two shocks is assumed to follow an AR(1) process:

1ρt a t ata a e−= + , (30)

λ 1 λλ ρ λt t te−= + , (31)

where ρa and ρλ lie in [0,1], and ate and eλt are white noise innovations.

2.7 Policy rule

In an environment of flexible prices, the actual real interest rate will always equal the

natural rate regardless of the monetary authority�s policy rule. On the other hand,

monetary policy has real effects in an environment of sticky prices. Therefore, the

specification of monetary policy rule will have implications for the real interest rate gap.

Taking this into account, we have examined the properties of the real interest rate gap

under a variety of different rules. Our results below will focus on a baseline policy rule

estimated on UK data and described in Section 3.

13 For the Calvo model, the flexible-price equilibrium is reached as αµ approaches infinity.

17

2.8 Open economy considerations

Our model has no explicit open-economy elements. It is therefore worthwhile to discuss

in what sense the model�s explanation of real interest rate determination may be useful

for small open economies in a world of integrated capital markets.

One interpretation of our model is that it refers to the determination of the world interest

rate, or the interest rate in a large country. We believe, however, that our analysis has

some value as a description of interest rate determination in a small open economy. The

conditions under which a small open economy�s real interest rate is determined

completely by global factors are actually quite stringent. One consequence of this may

be that in small open economies, domestic rather than solely global factors play a

significant role in the determination of real interest rates.

This point is particularly relevant for a model intended for monetary policy analysis.

Inflation-targeting central banks typically operate with a short-term nominal interest rate

instrument, which, combined with some inertia in inflation, implies considerable short-

run influence by the domestic monetary authority over the domestic short-term real

interest rate, even though these economies are highly open and are part of a global capital

market. Indeed, many observers would consider the following model features essential

for a realistic analysis of monetary policy: (i) central bank control of nominal interest

rates and short-run influence over real rates; (ii) a considerable amount of persistence in

inflation; and (iii) investment in physical capital being very important for cyclical

fluctuations and being a major channel through which monetary policy affects aggregate

demand. Our closed-economy model can (under certain settings) satisfy all three criteria;

yet very few existing open-economy DSGE models can. In the Obstfeld-Rogoff (1995)

model, for example, real interest rates are not affected by domestic monetary policy and

are equal to the foreign real rate every period. The failure of standard open-economy

models to meet criterion (ii) is documented in McCallum and Nelson (2000); and

introducing endogenous physical capital is often problematic in open-economy models.

18

We therefore proceed with our use of a closed-economy model, but take the openness of

the UK economy into account in our calibration of the model.

In addition to their effect on capital markets, open-economy elements alter goods market

behaviour by adding net export demand, which is a function of foreign and domestic

demand, to total domestic aggregate demand. We can take these factors into account in

our closed-economy model by modifying the calibration of the aggregate demand side of

the economy. Consider the Euler equation for consumption in our model, for simplicity

abstracting from habit formation (h = 0). In this case, ct = Etct+1 �σrt + + σ(1−ρλ)λt.

Iterations on this condition produce:

ct = −σEtΣΣΣΣj=0∞rt+j + σλt, (32)

so consumption depends on current and expected future real short rates. Now suppose

that net foreign demand for domestic output is given by the log-linear equation:

nxt = b1 qt + κt, (33)

where qt is the real exchange rate (an increase signifying a depreciation), κt is a shock to

foreign demand, and b1 > 0. Supplementing the model with the real interest parity

condition:

qt = Etqt+1 − rt + ut, (34)

where the shock ut includes both the foreign real interest rate and the foreign exchange

risk premium, one can write (33) as:

nxt = − b1 EtΣΣΣΣj=0∞rt+j + b1EtΣΣΣΣj=0∞ut+j + κt. (35)

Thus, aggregate non-investment demand is given by:

19

scct + sNX nxt = −(scσ + sNXb1) EtΣΣΣΣj=0∞rt+j + exogenous shocks, (36)

where sc and sNX are the steady-state shares of consumption and net exports in GDP,

respectively. The effect of adding open-economy influences is therefore to raise the

interest elasticity of (non-investment) aggregate demand from scσ to (scσ + sNXb1). In our

calibration, we take these influences into account by calibrating σ to a higher value than

would be suggested by the interest responsiveness of consumption alone.

Finally, we discuss the relevance for our analysis of two other aspects of open-economy

analysis. First, the consumption Euler equation (19) in our model would still hold if we

made the model open. However, some open-economy models assume either finite

horizons for consumers or an endogenous discount factor, with the effect of making the

external asset position relevant for consumption behaviour.14 The evidence suggests,

however, that the business cycle frequency dynamics of endogenous variables�the

frequency with which the present paper is concerned�are little changed by the

introduction of these features (Kollmann, 1991; Kim and Kose, 2000).

Secondly, openness makes imports a component of the consumer price index; this may

create an extra channel through which shocks are transmitted to inflation. However, as

we discussed in the introduction, the empirical relationship between exchange rates and

inflation is weak. There is little empirical case for including exchange rate terms in the

Phillips curve (Stock and Watson, 2000), possibly because of slow or incomplete pass-

through of exchange rate movements to import prices. This suggests that our use of a

Phillips curve with no explicit open-economy terms is a reasonable approximation.

3 Model calibration and properties

In this section we describe the responses of the natural real rate and the real interest rate

gap in a calibrated version of our model. We first turn to our calibration.

20

3.1 Calibration

The parameter values assigned to our model correspond to those commonly found in the

literature and are similar to those found in earlier work on sticky-price general

equilibrium models, including King and Watson (1996) and McCallum and Nelson

(1999). These are presented in Table 2, and we now discuss some of the key choices.

The parameter σ indexes the curvature of the utility function, and also determines the

interest-elasticity of the non-investment component of aggregate demand. Due to the

habit formation in our utility function, our calibrated value for this parameter must be

between 0 and 1. While many business cycle studies select values of σ near unity, the

estimates of Euler and optimising IS equations on US data in Hall (1988), McCallum and

Nelson (1999), Fuhrer (2000), and Ireland (2000) suggest a lower value, of around 0.2.15

On the other hand, as we discussed in Section 2.8, the openness of the UK economy

justifies a higher value. We therefore choose σ = 0.6.16

The habit formation parameter is set to h = 0.8, in line with Fuhrer�s (2000) estimate.

The capital adjustment cost parameters are calibrated so that (quasi-)investment is

considerably more interest elastic than consumption, but not implausibly large. The

capital adjustment cost settings in Table 2 imply a semi-elasticity of investment with

respect to the short-term real interest rate of about 3.2%.17

14 See Kollmann (1991), Kim and Kose (2000), and Smets and Wouters (2000) for applications. 15 Higher values of σ have been reported by Rotemberg and Woodford (1997) and Amato and Laubach (1999), but, unlike the studies mentioned in the text, these are not based on conventional econometric estimation procedures such as instrumental variables or maximum likelihood. 16 By comparison, Boldrin, Christiano, and Fisher take logarithmic preferences (the limit as σ→1) and set h = 0.9. Beaudry and Guay (1996) impose logarithmic preferences and estimate h to be in the 0.3�0.5 range. Woodford (2000) sets σ→1 and h = 0. All of these parameter settings seem to us to impose an unrealistically high interest elasticity of consumption, compared to the estimates cited above. 17 Capital adjustment costs are also important for generating realistic output behaviour under sticky prices. Without capital adjustment costs, output exhibits an extremely elastic response to monetary policy shocks (see Casares and McCallum, 2000).

21

Table 2: Model calibration

Parameter Description Quarterly

value

α Labour share 0.64 β Discount factor 0.99 σ Parameter indexing the curvature of

the utility function 0.6

h Habit formation parameter 0.8 δ Rate of depreciation 0.025

1/σε Scale elasticity of money demand 1 ϕ Capital adjustment cost parameter 0.75 η Capital adjustment cost parameter 2

1/ρ, µ Steady state gross markup 1.25 ρλ AR(1) parameter, IS shock 0.33 ρa AR(1) parameter, technology shock 0.95

σ2ea Variance of technology innovations (0.007) 2

σ2eλ Variance of IS innovations (0.01) 2

Nss Steady-state fraction of time in employment

0.33

µα Degree of nominal rigidity under sticky prices. Calibrated value corresponds to a 75% probability that firms will not be able to change their price.

0.086

3.2 Model properties

3.2.1 Response of the natural real rate to shocks

Figures 1a and 1b plot the responses of the natural real interest rate to technology and real

demand shocks for the calibrated model.18 Three cases are considered: a setting of the

model with no capital; the model with capital formation but with adjustment costs; and

the model with capital that can be adjusted costlessly. The no-capital and costless-

18 Details on how we computed these impulse responses are presented in the Appendix.

22

capital-adjustment cases are presented to indicate the effect our specification of capital

formation has on the behaviour of the natural real rate.

Figure 1a indicates that a temporary 1% shock to technology raises the natural real rate

by about 5 basis points when capital adjustment is costless, but that it reduces the rate in

the cases of no capital and capital with adjustment costs. In the no-capital case, a

technology shock raises output and consumption today by more than in future periods.

Output and consumption therefore jump today and return later to their original levels.

Households would like to smooth their consumption (especially given the habit formation

in their preferences), and attempt to shift resources away from the current period to future

periods. But, in equilibrium, all output must be consumed today. The natural real

interest rate declines to ensure that this occurs.19

When capital formation is present, investment demand goes up because the technology

shock has raised the profitability of production. By itself, this would push up the real

interest rate. Offsetting pressure comes from households� desire to save some of their

increased income, and the disincentive to rapid investment provided by adjustment costs.

The net effect of these pressures is again to reduce the real interest rate, though by less

than in the no-capital case.

Figure 1b depicts the effect of a temporary 1% shock to real demand (a shock to

households� preferences). An �IS� shock of this type, unlike a monetary policy shock,

affect the values of real variables such as the real interest rate, even when prices are

flexible. This shock raises households� consumption demand on impact. With fully

flexible capital, firms are willing to adjust their investment in response to higher

aggregate demand without a large change in the real interest rate; hence the limited

response of the natural rate depicted for that case in Figure 1b. If there is no capital or if

firms face large adjustment costs, then production does not meet the higher demand, and

19 Figure 1a matches Woodford�s (2001) result that temporary productivity shocks (i.e. 0 ≤ ρa < 1) lead to a counter-cyclical response of the equilibrium real interest rate. If the technology shock were completely persistent (i.e., had a unit root), and there were no habit formation in preferences, the technology shock would leave the equilibrium real rate unchanged, as in Clarida, Gali, and Gertler (1999).

23

so real interest rates must rise by a relatively large amount to dampen the rise in

consumption.

A common feature of Figures 1a and 1b is that the natural real interest rate responds by

more when the capital stock cannot adjust costlessly.20 A flexible-price, flexible-capital

model would imply almost no variation in the natural real interest rate since quantities

bear the bulk of the adjustment.

The responses for potential output (Figures 2a and 2b) are the mirror image of the real

interest rate responses. Although the capital-adjustment-cost case continues to hold an

intermediate position, potential output responds less to a technology shock when there is

no capital in the model and more in response to a real demand shock. Impediments to

adjusting capital inhibit the rise in investment that a technology shock would normally

20 This is true even if preferences do not exhibit habit formation, although habit formation magnifies the response of the real rate to both technology and real demand shocks.

24

induce. And when the capital stock cannot be varied at all, it is not possible for higher

consumption demand to be satisfied partly by a rise in consumption�s share of income

(which a rise in real rates would normally induce, as investment is more interest-elastic

than consumption). So the real demand shock leads to sharp increases in both real

interest rates and output.

In Figure 3, we present the natural real rate response to a domestic demand shock for a

different model, namely McCallum and Nelson�s (2000) open-economy model. We

argued in Section 2.8 above that our choice of σ = 0.6 in the closed-economy model

studied in the present paper serves to approximate the effects of openness on real interest

rate dynamics. The figure gives the natural rate response in the McCallum-Nelson (MN)

model to a shock to households� preferences in the domestic economy, where preferences

are characterised by equation (1) but with σ = 0.2.21 As the MN model has no capital,

21 The model is identical to that in McCallum and Nelson (2000, Figures 2�4) except that: (i) flexible prices are assumed, to isolate the natural rate response; (ii) the steady-state share of exports in GDP has been

25

Figure 3 is an open-economy analogue of the no-capital case in Figure 1a. The size and

shape of the response in Figure 3 closely resembles that in Figure 1a. Thus, our

approximation of open-economy effects appears to be a good one.

3.2.2 Response of the real interest rate gap to shocks

We now turn to the sticky-price case, and examine the response of the real interest rate

gap�the spread between actual and natural real interest rates�to shocks.

Examination of the sticky-price case requires that we choose a price-adjustment

specification and policy rule to close the model. For price adjustment, we use the Calvo

model, although in Section 4 we will also present results with the Fuhrer-Moore model.

For the monetary policy rule, an estimated policy rule for the UK for 1980�99 would be

raised from 0.11 to 0.25, a more appropriate value for the UK; and (iii) habit formation (with parameter h = 0.8) has been included to make preferences identical (other than regarding the σ value) to those in our model.

26

desirable, as we will later be comparing our model with data for that sample period. But

regime changes in UK monetary policy over 1980�99 mean that the full-sample estimates

of a policy rule are not very reliable, and exhibit some problems of interpretation (for

example, they fail to satisfy the condition that the nominal interest rate responds by more

than one-for-one to expected inflation). Instead, we use the following policy rule

estimated by Nelson (2000) for the UK for the sub-sample 1992 Q4−1997 Q1:

4*Rt = ρR 4*Rt−1 + (1 − ρR)1.267Et ∆4pt+1 + (1 − ρR)0.47yt + 4*eRt (37)

where ρR = 0.29, ∆4pt = ΣΣΣΣj=03πt−j is the annual inflation rate, and the standard deviation of

the policy shock eRt is 0.001. The policy rule is similar in specification to the Taylor

(1993) rules estimated by Clarida, Gali, and Gertler (1999). According to equation (37),

monetary policy adjusts the nominal interest rate in response to expected future annual

inflation (with a greater than one-for-one long-run response), and to detrended output (yt).

Detrended output in the policy rule is meant to proxy for the output gap.

Figures 4 to 6 display the response of the real interest rate gap, detrended output, and the

difference between output and potential output to technology, demand, and policy shocks,

respectively. The policy rule operates according to equation (37), and capital formation

is subject to adjustment costs.

Figure 4 depicts the effect of a technology shock. The shock generates an increase in the

real interest rate gap�an effective policy tightening�for two reasons. First, the natural

real rate falls, so for a given actual real interest rate monetary policy becomes tighter.

Second, if policy responds to the level of output, a productivity shock induces policy

makers to raise the nominal interest rate, raising the actual real interest rate in the

process.

In the case of a real demand shock, however, both the actual and natural real rate increase

(Figure 5). The rise in the actual real rate, due to the monetary policy tightening in

27

response to higher output, is less than the rise in the natural rate. The overall policy

stance is therefore looser, once the natural rate movement is taken into account.

Figure 6 simply illustrates that a monetary policy tightening affects only the actual real

interest rate. The real rate gap therefore rises one-for-one with the increase in the actual

real interest rate. The output gap and output responses are identical because monetary

policy cannot affect potential output.

The response of the real rate gap to different shocks illustrates how a policy rule like

(37), which responds to the level of output, can yield perverse results. Effectively, the

rule responds symmetrically to positive supply and demand shocks. This is because it

does not take into account that the natural values of both the real interest rate and output

have been affected by the real shocks. The natural real interest rate falls in response to

the technology shock, and therefore the monetary policy response should be to lower, not

raise, interest rates. In the case of a real demand shock, on the other hand, the natural rate

increases. A policy aimed at minimising output gap and inflation variations would raise

28

29

the interest rate in line with the increase in the natural rate. But in fact the policy rule

(37) generates a smaller increase in the real rate than is necessary to keep the interest rate

gap zero. One reason why responding to output appears to lead, for both technology and

real demand shocks, to counterproductive results is that detrended output is not a good

indicator of output gap behaviour in our model. Figures 4�6 also demonstrate the mirror-

image relationship between the output gap and the real interest rate gap. We discuss

these issues further in Section 5 below.

4 Model evaluation

We now present some evidence on the match between our model and the data by

examining data and model vector autocorrelations. The version of the model we consider

includes Calvo price setting, capital formation subject to adjustment costs, and the policy

rule (37) described in the previous section. The model correlations are averages of the

output of 100 stochastic simulations of the model (for 200 periods each time). The

corresponding data correlations are based on quarterly UK data for the 1980 Q1−1999 Q4

sample period. The data consist of yt (quadratically detrended log real GDP), Rt (the

quarterly average of the nominal Treasury bill rate), rt (Rt minus the expected value of

next quarter�s seasonally adjusted RPIX inflation, πt+1),22 and annual inflation, ∆4pt.

Figure 7 plots the correlations for the model (solid line) and data (broken line). The

model succeeds in reproducing the autocorrelations of the four series and, to a lesser

extent, matching several other dynamic relationships, including the real / nominal interest

rate relationship, and the close correlation between four-quarter inflation and the nominal

interest rate. The correlation between the level of output and the lags of the real interest

rate is generally positive in the model, which is qualitatively the relationship observed in

UK data, as noted earlier in Table 1. The main exception is the contemporaneous

correlation between the two series, which is slightly negative in the model.

22 Expected inflation is calculated from the VAR(8) described in Section 6 below.

30

The model�s most serious weaknesses are its ability to generate negative correlations

between output and future interest rates (both nominal and real). These are at variance

with the positive correlations observed in the UK for 1980−99. Part of this problem may

reflect the use in the model of a policy reaction function that is estimated over only a sub-

sample of 1980−99: the reaction of nominal interest rates to output movements may have

differed substantially in that sub-sample (1992�1997) from that observed over the full

sample. But the discrepancy between the data and the model also reflects a property of

the structure of the model: the model�s forward-looking aggregate demand specification

implies that aggregate demand (yt) depends negatively on expected future real interest

rates.

31

The model also does poorly in accounting for the correlation between output and annual

inflation. The divergences between data and model in Figure 7 suggest that future work

could focus on modifying the aggregate demand specification.23

5 Indicator properties of the real interest rate gap

We now examine some dynamic properties of our model, focusing on statistics that

describe the relationship between the real interest rate gap, aggregate demand, and

inflation. Table 3 gives selected correlations and standard deviations for four settings of

the model (constant or fluctuating capital, with Calvo or Fuhrer-Moore price setting).

The variables focused upon are log output (yt), the real interest rate (rt), log of natural

output (yt*), the natural real interest rate (rt*), the output gap (yt − yt*), and the real

interest rate gap (rt − rt*). A notable feature of Table 3 is that it reveals that the real

interest rate gap and the output gap have quite different properties. The two series have

the expected strong negative relationship (the correlation ranges from −0.63 to −0.92

depending on model setting), but differences between the two series emerge when we

analyse the contribution of individual components of each gap.

In particular, the behaviour of the real interest rate (rt) is a reasonable approximation for

the behaviour of the real interest rate gap. The correlation between these two series is

relatively high, and the two series are of roughly the same degree of volatility.

Correspondingly, the natural real rate varies much less than the actual real rate. This

contrasts with Rotemberg and Woodford�s (1997) dynamic stochastic general equilibrium

model, which generates a standard deviation for the natural real rate, of 3.7 per cent, that

is higher than that observed empirically for the actual real rate. But it is consistent with

King and Watson (1996), whose DSGE model produced low variability in the real

interest rate under price flexibility. Rotemberg and Woodford�s finding may have arisen

partly from larger shock variances than are typically found in other studies.

23 Other model specifications we analyse, including the no-capital and Fuhrer and Moore pricing specification, retain many of the weaknesses discussed here.

32

In addition, the actual and natural real interest rates are not very closely related to each

other (the correlation between the two series ranges from +0.02 to −0.17). In other

words, the variation in the real interest rate gap is dominated by variation in the

�observable� component of the gap, the real interest rate, rather than the �unobservable�

natural real rate component.

By contrast, the level of yt is not a good indication of the behaviour of the output gap

yt − yt*. In fact, the two series have an inverse relationship, with correlations ranging

from −0.06 to −0.68, and the output gap has a standard deviation that is less than half that

of output. Due to the absence of steady-state growth in our model, output in our model

corresponds to detrended output in the data. Our results therefore question the

widespread practice of measuring the output gap by detrended actual output. Rather, in

our model variations in detrended output are dominated by variations in potential

output�the two series have a correlation ranging from 0.96 to 0.99�so the

�unobservable� potential output variable should not be treated as constant (or growing

steadily) when generating an output gap series.

The last statistics in Table 3 indicate that the real interest rate gap seems to have a

reasonably strong negative correlation with quarterly inflation, ∆pt. Fuhrer-Moore price

setting appears to make this relationship a leading relationship. To examine the

forecasting properties of the real interest rate gap in a different way, we now report

averages across model simulations of estimates of the regression:

∆4pt = b10 + b11(rt−1 − rt−1*) + b12∆4pt−1,

which can be thought of as the inflation equation in a one-lag, bivariate VAR system that

consists of the annual inflation rate, ∆4pt, and the real interest rate gap. For comparison,

we also report results using the output gap:

∆4pt = b20 + b21(yt−1 − yt−1*) + b22∆4pt−1.

33

Table 3: Model statistics

Model setting

No capital;

Calvo

Capital with

adjustment

costs; Calvo

No capital;

Fuhrer-

Moore

Capital with

adjustment

costs;

Fuhrer-

Moore

σ(yt) 1.59 2.22 1.76 2.70

σ(yt*) 1.81 2.82 1.81 2.82

σ(yt − yt*) 0.30 0.80 0.19 0.69

σ(rt) 1.05 1.06 1.25 1.32

σ(rt*) 0.96 0.56 0.96 0.57

σ(rt − rt*) 1.41 1.21 1.68 1.53

Corr(yt, yt*) 0.99 0.98 0.99 0.97

Corr(yt, yt − yt*) −0.68 −0.68 −0.20 −0.06

Corr(rt, rt*) 0.02 −0.03 −0.14 -0.17

Corr(rt, rt − rt*) 0.73 0.88 0.82 0.93

Corr(yt, rt) −0.02 −0.048 −0.04 −0.03

Corr(yt, rt−1) −0.02 0.06 −0.004 0.09

Corr(yt − yt*, rt − rt*) −0.58 −0.74 −0.93 −0.97

Corr(yt − yt*, rt−1 − rt−1*) −0.54 −0.40 −0.72 −0.45

Corr(∆pt, rt − rt*) −0.50 −0.41 −0.38 −0.33

Corr(∆pt, rt−1 − rt−1*) −0.42 −0.34 −0.44 −0.38

Corr(∆pt, rt−2 − rt−2*) −0.39 −0.31 −0.43 −0.37

Note: σ(• ) denotes standard deviation; Corr(• ,• ) the simple correlation coefficient. Standard

deviations of the interest rate variables are annualised. Statistics reported are computed from

analytical formulae for the model moments.

34

We supplement these results by reporting regressions that replace the �gap� variables with

the actual real interest rate and actual output.

∆4pt = b30 + b31rt−1 + b32∆4pt−1,

∆4pt = b40 + b41yt−1 + b42∆4pt−1.

These last two regressions can be thought of as what a researcher might estimate if they

approximated the natural real interest rate by a constant in constructing a real interest rate

gap series, or potential output by a trend in constructing an output gap series.

Regressions on the artificial data sets are summarised in Table 4. Comparison of the

regressions that use the gap measures indicates that the lagged real interest rate gap and

the lagged output gap each is highly significant when added to an autoregression for

inflation. Moreover, the signs of the estimated coefficients have economic

interpretations�the positive output gap coefficient reflect the effect of �excess demand�

on inflation; and the real interest rate gap coefficient indicates the negative relationship

between real interest rates (relative to their natural value) and excess demand.24 The

regressions with the output gap fit better than the regressions with the real interest rate

gap, though the difference is large in only one case (Fuhrer-Moore price setting with no

capital).

A more practical question, however, is how much of the fit of these regressions is due to

the explanatory power provided by the �unobservable� components of the gaps�the

natural values of the real interest rate and output. The answer to this question is provided

by the regressions in Table 4 that use the actual values of rt and yt as regressors rather

than the corresponding gap variables. For the regressions with rt, the omission of the rt*

term does cost some explanatory power (confirming that rt − rt* is the more appropriate 24 Beyond this sign interpretation, it would not be sensible to interpret the estimated coefficients further, as they are reduced-form. In particular, if interpreted as structural equations, the estimated inflation regressions appear to suggest that long-run non-zero output and real interest rate gaps can be obtained by

35

Table 4: Regressions on simulated data: Dependent variable ∆∆∆∆4pt (annual inflation rate)

Model Setting Coefficient on: SEE σ(∆4pt)

rt−1 − rt−1* ∆4pt−1

No Capital, Calvo −0.90 (0.08) 0.96 (0.01) 0.40 2.44

Capital with Adj. Costs, Calvo

−1.16 (0.11) 0.97 (0.01) 0.47 3.03

No Capital, FM −0.95 (0.05) 0.97 (0.01) 0.32 2.80

Capital with Adj. Costs, FM −1.19 (0.07) 0.98 (0.01) 0.38 3.59

yt−1 − yt−1* ∆4pt−1

No Capital, Calvo 2.80 (0.19) 0.72 (0.02) 0.35 2.44

Capital with Adj. Costs, Calvo

0.70 (0.06) 0.87 (0.01) 0.45 3.03

No Capital, FM 2.49 (0.09) 0.95 (0.01) 0.23 2.80

Capital with Adj. Costs, FM 0.67 (0.04) 0.98 (0.01) 0.39 3.59

rt−1 ∆4pt−1

No Capital, Calvo −1.00 (0.13) 1.01 (0.01) 0.44 2.44

Capital with Adj. Costs, Calvo

−1.20 (0.14) 0.99 (0.01) 0.49 3.03

No Capital, FM −1.17 (0.09) 1.02 (0.01) 0.36 2.80

Capital with Adj. Costs, FM −1.34 (0.09) 1.00 (0.01) 0.40 3.59

yt−1 ∆4pt−1

No Capital, Calvo 0.00 (0.04) 0.98 (0.02) 0.40 2.44

Capital with Adj. Costs, Calvo

−0.05 (0.05) 0.95 (0.03) 0.47 3.03

No Capital, FM −0.02 (0.04) 0.97 (0.02) 0.32 2.80

Capital with Adj. Costs, FM −0.08 (0.06) 0.94 (0.03) 0.38 3.59

Note: Numbers are reported in Columns 2�3 are the means across simulations of parameter

estimates and their corresponding standard errors.

altering the long-run inflation rate. But the underlying structural model that generated the data used for the regressions does not have this property.

36

index of inflationary pressure) but�because of the low variability of rt* in our model�

the loss of fit is minor. In other words, the real interest rate again appears to be a

reasonable proxy for the real rate gap for the purpose of forecasting inflation.

On the other hand, replacing yt−1 − yt−1* by yt−1 leads to a drastic loss of fit and to a

�wrongly� signed parameter estimate (if the coefficient on output is to be interpreted as an

excess demand coefficient). Thus, in this model, both the real interest rate gap and the

output gap are significant leading indicators of inflation�but the leading indicator

properties of the real rate gap largely come from the observable component, while the

leading indicator properties of the output gap are attributable to the unobservable

potential output component. Our results suggest that over the business cycle, fluctuations

in potential output are great enough to make detrended output an unreliable indicator of

the output gap, but that the accompanying fluctuations in the natural real rate are quite

small. Under these circumstances, the real interest rate can be quite useful as an indicator

of demand pressure and of future inflation.

6 Empirical properties of the real interest rate gap

In this section we construct, for the UK, empirical counterparts to the natural real interest

rate and the real interest rate gap series that appear in our model, and investigate the

relationship between the resulting gap series and inflation.

In our model, the natural real rate interest is a function of IS (real demand) and

technology shocks. Technology shocks are typically measured by �Solow residuals�,

which are obtained by subtracting log values of labour and capital inputs (each

respectively weighted by their coefficients in the production function) from the log of

total output. We use a Solow residual series constructed from quarterly UK data.25

25 The Solow residual is defined empirically by log (Yt)− (1−α)log (Kt) � α log (Nt), as suggested by equation (15). Yt is measured by the level of quarterly UK real GDP. Kt is a capital stock series obtained by formula (3), with Xt measured by the investment series in Bank of England (2000, p. 62), and the initial (1962 Q1) value of K given by our model�s steady-state (K/X) value multiplied by the 1962 average level of Xt. Labour input is measured by employment (British Labour Statistics data up to 1978, spliced into the series defined in Batini, Jackson, and Nickell (2000) from 1978). As our model does not allow for secular

37

We measure IS shocks as follows: Combining the log-linearised Euler equations for

consumption and bonds, as well as the Fisher equation, the IS shock λt is given by:

λt = (−βg1Et∆ct+2 + g2Et∆ct+1 + g3Et∆pt+1 − g1∆ct − g3Rt)/ g4, (38)

where g1 = (h − σh), g2 = (1+ βh2 − σβh2 − σβh), g3 = σ(1 − βh), and g4 = −σ(1 − ρλ

+ βhρλ2 − βhρλ). We use formula (38) to generate IS shocks from quarterly UK data for

1980 Q1−1999 Q4. In constructing the shock series, we measure Rt by the Treasury bill

rate (quarterly average), ct by the log of quarterly private consumption (less durables),

and πt by the seasonally adjusted log-change in the quarterly average of the RPIX index.

We measure expected values of series by forecasts from an eight-lag VAR estimated over

1976 Q1−1999 Q4. The variables in the VAR are Rt, ∆ct, and πt, and each equation is

supplemented by two intercept-dummy variables, DERMt and D924t. These dummy

variables capture changes in UK monetary policy regime, and take non-zero values

during the UK�s membership of the Exchange Rate Mechanism (1990 Q4−1992 Q3) and

its period of inflation targeting (1992 Q4 onward). Once expected values of variables

have been constructed, the IS shock series is then generated using our calibrated values of

β, σ, ρλ, and h.

In the version of our model with capital formation (subject to adjustment costs), the

natural real rate is related to IS and technology shocks according to:

rt* = 0.0450λt + 0.0581 λt−1 + 0.0283 λt−2 + 0.0139 λt−3

+ 0.007 λt−4 + 0.0036 λt−5 + 0.0019 λt−6 �0.0783 at

+ 0.0264 at−1 + 0.0124 at−2 + 0.0057 at−3 + 0.0023 at−4 + 0.0007 at−5,

growth, the at series in our model corresponds to detrended total factor productivity. We quadratically detrend our empirical Solow residual series to construct an empirical counterpart to the at series.

38

Table 5: Correlations between inflation and real interest rate gap, UK data

Sample period: 1980 Q1–1999 Q4

Simple correlations

k=0 k=1 k=2 k=3 k=4 k=5 k=6 k=7 k=8

Corr (∆4pt, rt−k − rt−k*) −0.14 −0.20 −0.27 −0.41 −0.47 −0.41 −0.42 −0.46 −0.48

Corr (∆4pt, rt−k) 0.01 −0.10 −0.18 −0.35 −0.44 −0.38 −0.42 −0.47 −0.52

Partial correlations, conditional on policy regime breaks

k=0 k=1 k=2 k=3 k=4 k=5 k=6 k=7 k=8

Corr (∆4pt, rt−k − rt−k*) −0.51 −0.55 −0.59 −0.67 −0.73 −0.63 −0.60 −0.60 −0.59

Corr (∆4pt, rt−k) −0.37 −0.45 −0.52 −0.61 −0.70 −0.61 −0.59 −0.61 −0.61

Note: Construction of rt and rt* series is as described in the text.

where we neglect the low coefficients on further lags of λt and at. We use this formula to

generate an rt* series for the UK, and a real interest rate gap series (rt − rt*).

Table 5 gives the simple correlations on UK data between annual inflation (∆4pt) and lags

of the real interest rate gap series. For comparison, the correlations between inflation and

the real interest rate are also reported. The results indicate that the correlation between

the real interest rate gap and inflation is negative, and tends to be more negative than the

correlations between inflation and the real rate. These results are in keeping with our

model�s predictions.

On the other hand, there is a limitation in studying simple correlations calculated over the

1980�99 sample period. The model we have used in this paper was one that described

fluctuations around a fixed steady state for a given monetary policy rule. By contrast, the

period 1980−99 is characterised by several changes in the UK�s monetary policy

regime�most notably the movements into the Exchange Rate Mechanism in 1990 and to

inflation targeting in 1992, described above�and by different unconditional means of the

real interest rate across regimes (with higher means in the pre-inflation targeting period).

39

Figure 8: Annual inflation and real interest rate gap two years earlier,

UK data 1980�99, six-quarter moving averages

In evaluating the quality of our real rate gap series as an inflation indicator, it is desirable

to account for these shifts. Therefore, we also include in Table 5 the partial correlation

coefficients, which condition on the regime shifts in 1990 and 1992. Essentially these are

correlation coefficients between the real rate, real rate gap series and inflation, once each

series has been purged of its correlation with the dummy variables DERMt and D924t described above. The partial correlations are uniformly more negative than the simple

correlations, confirming that controlling for regime change is important.

Table 5 indicates that both the simple and partial correlations between inflation and the

real interest rate gap tend to be more negative at lower lags than those between inflation

and the real interest rate. Correspondingly, a smoothed version of the real interest rate

gap series appears to have a reasonably strong inverse leading relationship with smoothed

UK inflation (Figure 8). Finally, we use our interest rate gap series for the UK to

estimate, on actual data for 1979 Q1 to 1999 Q4, the equation that we estimated on

artificial data sets in Table 4: a regression of ∆4pt on ∆4pt−1 and (rt−1 � rt−1*). A constant

1980 1985 1990 1995 2000

-.05

0

.05

.1

.15 infl_ma rgap_ma_8

40

and the DERMt and D924t intercept dummies are also included in the equation. The

estimated coefficient on the lagged real interest rate gap is of the expected, negative, sign;

and a formal test for the exclusion of the gap from the regression leads to a rejection with

a probability higher than 0.001, reinforcing the visual and correlation evidence. Thus,

there is some tentative evidence that our attempt to control for movements in the natural

interest rate has enhanced the value of the real interest rate as an indicator of inflation.

7 Conclusions

This paper has examined what a dynamic stochastic general equilibrium model has to say

about the value of the real interest rate gap as an indicator of inflation. A shortcoming of

the concept is that it requires the construction of an �unobservable� natural real rate

series. However, the same is true of the output gap concept, and our results suggest that

output gap concepts as often constructed (which are based on a deterministic model of

potential output) can be seriously misleading. Under these circumstances, a real interest

rate gap series could provide useful auxiliary information in evaluating the monetary

policy stance and the prospects for future inflation, in keeping with the neo-Wicksellian

framework of Woodford (1999a, 2000).

Our model is capable of reproducing several empirical regularities in UK data, notably

the procyclical behaviour of real interest rates. In other respects, there is room for

improvement in the dynamic specification of the model. In particular, the model predicts

that output is negatively correlated with future inflation and interest rates, which is the

opposite of the pattern observed in UK data. It would also be desirable to obtain a more

gradual inflation response to movements in the real interest rate gap.

Our empirical results in Section 6 suggested that there are some benefits to inflation

forecasting from controlling for movements in the natural real rate, and basing the

construction of the natural rate series on general equilibrium theory. The quantitative

benefits may not be great, however, because the empirical variation in the natural rate

appears to be quite small.

41

Appendix: Calculation of output and interest rate gaps

In this appendix, we describe some of the complications involved in calculating (log)

capacity output (yt*) and the natural real interest rate (rt*) in our model, and our

procedure for overcoming the complications. This allows us then to compute the impulse

responses for the natural real rate in Section 3, and to analyse the output gap (yt − yt*) and

the real interest rate gap (rt − rt*) in Section 5.

The model outlined above features a time-varying, endogenous capital stock, as well as

habit formation in consumption. Consequently, both the period t capital stock, and the

previous period�s consumption level, are elements of the state vector. Thus, there are two

endogenous variables in the state vector, in addition to the exogenous technology (at),

real demand (λt), and monetary policy (eRt) shock variables. As a result, the model�s

solutions for yt* and rt* cannot be written as simple functions of the exogenous variables

alone.

To see this, consider output under sticky prices. From the production function,

yt = at + (1 − α) kt + α nt. (39)

Similarly, output under flexible prices, or capacity output, may be written as:26

yt* = at + (1 − α) kt* + α nt*, (40)

where asterisks denote flexible-price values. Had we assumed inelastic labour supply

and an exogenous capital stock, then yt* would simply be a linear combination of two

current-dated exogenous variables, at and kt*, and calculation of yt - yt* would be

straightforward. But for the more general case the solution to kt* and nt* are functions of

the whole state vector:

26 The technology shock at, being exogenous, behaves identically under price flexibility and price stickiness, and therefore requires no asterisk in (40).

42

kt* = φ1st−1* (41)

nt* = φ2st−1* (42)

where the φi are 1 x 4 coefficient vectors, and st*= [kt* ct−1* at λt]� is the state vector

under flexible prices.27

To calculate yt*, one procedure that would appear illegitimate is to condition on actual

capital and consumption kt and ct−1�in effect replacing the unobserved flexible-price

variables in st* with their sticky-price counterparts, and using equations (40), (41) and

(42) to compute yt*. This procedure seems illegitimate because the behaviour of the two

endogenous state variables, capital and consumption, will be a function of monetary

shocks and the monetary policy rule under sticky prices. However, Woodford (2000) has

made a case for this method of defining the natural output and interest rate, and we

discuss it further below.

In principle, another method would be to enlarge the model by supplementing the list of

endogenous variables in the model with their flexible-price counterparts, and including a

flexible-price and a sticky-price version of each behavioural equation. But by effectively

doubling the size of the model to be solved, this procedure would be quite cumbersome.

Instead, we employ a more economical method for calculating yt* and rt*. In essence, we

want to express yt* and rt* as functions of the exogenous variables only�effectively

�substituting out� kt* and ct−1* of the solution equations. To this end, we note that since

we are assuming (vector) autoregressive processes for the exogenous elements of st*, st*

follows the law of motion st* = Rst−1* + ΨΨΨΨεεεεεεεεt, where εεεεt is a vector of innovations. st* thus

has a vector moving average representation giving each element of st* as a (possibly

infinite) distributed lag of the εεεεt. Specifically, kt* and ct* may be written as:

27 The monetary policy shock eRt is omitted from this list due to the neutrality of real variables to monetary shocks under flexible prices.

43

kt* = f1(εεεεt−1, ε ε ε εt−2, ε ε ε εt−3, ...), (43)

ct* = f2(εεεεt, ε ε ε εt−1, ε ε ε εt−2, ...), (44)

where the fi(• ) are linear functions. It follows from (43) and (44) that

yt* = at + f3(εεεεt, ε ε ε εt−1, ε ε ε εt−2, ...). (45)

Since the exogenous variables at and λt each are infinite moving averages of the ε ε ε εt, it

follows that:

yt* = f4(at, at−1, ..., λt, λt−1, ...). (46)

By the same argument, rt* may be written as:

rt* = f5(at, at−1, ..., λt, λt−1, ...). (47)

To obtain the impulse responses and simulation results reported in the text, we

approximate the right-hand sides of (46) and (47) by a long but finite distributed lag of

the exogenous variables. Our complete procedure for calculating the gap measures (yt −

yt*) and (rt − rt*) is as follows:

1. Solve the model of Section 2 under flexible prices.

2. Simulate the model. Using the data generated from these simulations, run a

regression on generated data of the form:

yt* = c1at + ... + cj at−j + d1λt + ... + dj λt−j,

rt*= g1at + ... + gj at−j + h1λt + ... + hj λt−j,

for a finite j that is high enough to generate a good fit.

44

4. Store the ci, di, gi and hi coefficients (actually, averages of each of these coefficients

across simulations).

5. Solve the model of Section 2 incorporating sticky prices. Augment the exogenous

variable vector with yt* and rt*, where these are defined as the indicated linear

combinations of current and lagged at and λt.

6. Define the output gap as yt − [yt* as defined in Step 4]; real interest rate gap as

rt − [rt* as defined in Step 4].

We have found that this procedure generates fitted measures of rt* and yt* that are near-

perfect approximations for the correct measures given by equations (46) and (47).

Woodford (2000) has argued instead for defining the natural rate conditional on the

realised (sticky-price) values of endogenous state variables such as the capital stock. He

argues that this is preferable to our procedure given above because �[i]t seems odd to

define the economy�s �natural� level of activity� in a way that makes irrelevant the

capital stock that actually exists and the effects of this upon the economy�s productive

capacity� (Woodford, 2000, p. 67). But our procedure does not make the actual capital

stock irrelevant: while the flexible-price capital stock concept appears in our definitions

of natural levels of variables, it is the actual, existing capital stock that appears in the

production function, capital law of motion, capital adjustment cost function, and resource

constraint of the sticky-price economy.

One consideration is that conditioning on the actual capital stock in defining potential

output, as Woodford recommends, can lead to some curious policy prescriptions, if the

monetary policy reaction function is characterised by occasional unavoidable mistakes

that can only be corrected in later periods. For then, a policy mistake last period that

reduced investment demand reduces the capital stock today; this automatically reduces

potential output today, so need not create an output gap, and therefore no compensating

policy response today to correct last period�s mistake. Under our alternative definition,

45

potential output (but not actual output) is invariant to monetary policy, so last period�s

mistake creates an output gap today and justifies an easing today to correct the mistake.

Most importantly, our definition readily extends to the case of endogenous state variables

beside the capital stock. The physical constraint evoked by Woodford to justify his

definition is applicable only to state variables that enter the production function, whereas

our model contains an endogenous state variable, lagged consumption, that does not enter

the production function.

Nonetheless, we would not over-emphasise the differences between Woodford�s natural-

rate concept and our own. The impulse response functions for the natural real rate and

potential output in Section 3.2.1, for example, would be identical under either definition,

since those responses start from a steady state common to both the flexible and sticky-

price economies.

46

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