This paper argues that existing empirical models of interest rate rules are too simplistic. The hybrid Phillips curve implies that policymakers should respond to both current and expected future inflation rates, in contrast to existing models. We provide evidence that UK policymakers do this.optimal monetary policy; inflation persistence; Phillips curve

Christopher Martin

Costas Milas

English

Research Papers in Economics

The Rimini Centre for Economic Analysis
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WP 36-07
CHRISTOPHER MARTIN
School of Social Sciences
Brunel University, UK
and
COSTAS MILAS
Department of Economics,
Keele University, UK
and
The Rimini Centre for Economic Analysis, Rimini, Italy
“MONETARY POLICY AND
THE HYBRID PHILLIPS CURVE”
Copyright belongs to the author. Small sections of the text, not exceeding three
paragraphs, can be used provided proper acknowledgement is given.
The Rimini Centre for Economic Analysis (RCEA) was established in March 2007.
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The views expressed in this paper are those of the authors. No responsibility for them
should be attributed to the Rimini Centre for Economic Analysis.  1
 
Monetary Policy and the Hybrid Phillips Curve 
 
 
Christopher Martin  
School of Social Sciences 
Brunel University 
Uxbridge 
Middlesex UB8 3PH 
 
 
Costas Milas
* 
Department of Economics, 
Keele University 
Staffordshire, ST5 5BG 
and 
Rimini Centre for Economic Analysis, Rimini, Italy 
 
September 2007 
 
 
Abstract This paper argues that existing empirical models of interest rate rules are too 
simplistic.  The hybrid Phillips curve implies that policymakers should respond to both current 
and expected future inflation rates, in contrast to existing models.  We provide evidence that 
UK policymakers do this.  
 
 
Keywords: optimal monetary policy; inflation persistence; Phillips curve 
JEL Classification: C51, C52, E52, E58 
 
 
1) Introduction 
This paper argues that existing models of optimal monetary policy are too 
simplistic.  They rely on the assumption that changes in interest rates only 
have a one-off effect on inflation.  But the most influential model of the supply 
                                                           
* Corresponding author: Costas Milas, Department of Economics, Keele University, Keele, ST5 5BG, 
United Kingdom. Phone: +44 (0) 1782 583090. Fax: +44 (0) 1782 717577.  
Email: c.milas@keele.ac.uk 
 
   2
side of the economy, the hybrid Phillips curve introduced by Gali and Gertler 
(1999), features inflation persistence.  Since this implies that changes in 
interest rates will have persistent effects on inflation, the interest rate in the 
optimal monetary policy rule should respond to both current and expected 
future inflation rates.  By neglecting the latter, the existing model is mis-
specified. 
  This paper makes two contributions.  First, we present a simple 
analysis of the optimal monetary policy rule when there is inflation 
persistence, highlighting the weaknesses in the existing conventional model.   
Second, we present empirical evidence.  Using UK data that includes inflation 
forecasts published by the Bank of England, we estimate monetary policy 
rules and present evidence that monetary policymakers do respond to both 
current and expected future inflation rates.  We show that allowing for this 
improves the fit of the model compared to the conventional model. We also 
show that there is a simple parametric restriction under which our model 
simplifies to the existing model; this restriction is rejected by the data.  Section 
2) of the paper develops our model, section 3) presents empirical evidence 
while section 4) summarises and concludes.  
 
 
2) The Model   
The aggregate demand curve is  
 
(1)     11 ()
d
tt t t t t t yi EE y ρ πε ++ = −− + +  
   3
where i is the nominal interest rate, ε
d
t is an i.i.d demand shock and  i ρ  is a 
positive coefficient.  The Phillips curve is  
 
(2)   11 (1 )
s
tt t t t t Em c π θπ θ δ π γ ε −+ = −+ + +  
 
This is the hybrid Phillips curve proposed by Gali and Gertler (1999), in which 
inflation depends on aggregate marginal cost as well as lagged and expected 
future inflation rates.  We assume aggregate demand is proportional to 
marginal cost, so 
 
(3)   tt mc y η =  
 
If policymakers choose the nominal interest rate at the beginning of period t 
on the basis of information available at the end of period t-1, then their 
optimisation problem is  
 
(4)   {}
0
Min  =
t
j
tt j i
j
EL δ
∞
+
=
Λ ∑  
 
subject to (1), (2) and (3) and where  
 
(5)   ()
2 *2 * 2 2
1
1
()( )
22 2 2
tt tt t t L y ii ii
λ μκ
ππ − =− ++ − + −  
   4
is a conventional per-period quadratic loss function, π* is the inflation target, i* 
is the equilibrium interest rate and λ , μ  and κ are positive coefficients.    
  Solving the model under discretion, the first-order condition is 
 
(6)       () ( )
() ()
** *
11
22 * 33 *
23
{( ) ( ) ( 1 )
(1 ) (1 ) ....} 0
tt t t t t t
t
tt
d
Ey i ii i
di
ρλμ κ ρ γ η π πρ γ η δθ ππ
ργηδ θ π π ργηδ θ π π
−+
++
Λ
=− + − + − − −− − −
−−− −−− =
 
 
which implies 
 
(7)   ()
**
1
0
(( 1 ) )
j
tt t t t t j
j
ii iE yE
κ μ ρλ ργη
δ θπ π
μκ μκ μκ μκ
∞
− +
=
=+ + + −−
++ + + ∑  
 
Existing models of optimal monetary policy uses the purely forward-looking 
New Keynesian Phillips curve; this is obtained when  1 θ =  in (2), in which case 
the policy rule in (7) simplifies to  
 
(8)   ()
**
1 tt t t t t ii iE yE
κ μ ρλ ργη
π π
μκ μκ μκ μκ
− =+ + + −
++ + +
 
 
Comparison of (7) and (8) shows the implications of the hybrid Phillips curve 
for optimal monetary policy: interest rates respond to the discounted sum of 
current and expected future inflation rates, rather than just to the current 
inflation rate, as in the conventional model.  The policy rule in (7) simply 
reflects the fact that interest rate changes at time t affect the inflation rate at 
time t, but also at times  (t+1), (t+2), (t+3), etc.  This analysis suggests that   5
existing models of monetary policy may be incorrect because they ignore the 
impact of the sequence of expected future inflation rates on interest rates.  
Inclusion of this in empirical models should lead to improved estimates.  We 
next consider whether this is the case.  
 
3) Evidence 
Expressing the model in terms of identifiable parameters, we estimate  
 
(9)   ()
*
1
0
(1 ){ * ( ) }
j
tt y t t t f t j
j
ii i E y E π φ φβ β β π π
∞
−+
=
=+ − + + − ∑  
 
where 
κ
φ
μ κ
=
+
,  y
ρλ
β
μ
= ,  π
ργη
β
μ
=  and  (1 ) f β δθ = − ; we have followed the 
literature (eg Clarida et al, 1999) in expressing the weights on output and 
inflation as the long-run responses, net of interest rate smoothing.  If there is 
no inflation persistence, then  0 f β =  and our model simplifies to the existing 
model.  We could of course obtain more structural estimates by joint 
estimation of (1), (2), (3) and (9), but this is beyond the scope of this short 
paper. 
We use UK data for 1992Q4-2007Q1, covering the period of inflation 
targeting.  We use the 3-month Treasury bill rate as the nominal interest rate 
(this has a close relationship with the various interest rate instruments used 
over this period; see Nelson, 2003 and Adam et al., 2005).  For inflation we 
use the rate targeted by the Bank of England: the annual change in the retail 
price index excluding mortgage interest payment (RPIX) until 2003q4, 
thereafter the annual change in the Consumer Price Index.   We use two   6
alternative published forecasts of this variable for up to eight quarters ahead 
published by the Bank of England; the first assumes constant interest rates 
over the forecast period, while the second (which is only available from 
1998Q1) assumes interest rates follow market expectations (see Britton et al, 
1998).  We use two measures of the output gap series: the difference 
between final output and a HP trend and the residuals from a regression of 
final output on a quadratic trend.   
Estimates of (9) are presented in columns (i)-(iii) of table 1) (where we 
truncate the infinite sum after 8 periods).   Column (i) uses a constant interest 
rate forecast of inflation and uses the Hodrick-Prescott filter (1997) in 
constructing the output gap.   Column (ii) uses the alternative measure of the 
output gap while column (iii) uses the alternative forecast of inflation.  The 
estimates in each column are quite similar, the best fit being obtained in 
column (iii).  The estimates of  y β  and  π β  are significant and exceed unity.  
The estimates of  f β  are all significant and vary between 0.11 and 0.17.   The 
model in (9) simplifies to the existing model if the restriction H0: f β =0 is 
accepted.  It is clearly rejected.  We also note that there is no evidence of a 
break during the sample period, which is important as the target shifted from 
RPIX to CPI inflation in 2003Q4 (we also note that some other papers (Favero 
and Rovelli, 2003, Aguiar and Martins, 2005) have estimated policy rules in 
which interest rates respond to more than one inflation rate; however these 
papers do not test this against the conventional model and do not relate this 
issue to the hybrid Phillips curve).  Assuming δ  is close to unity, our 
estimates suggest a relatively low, but nevertheless significant, degree of 
persistence in the Phillips curve, which is consistent with the estimates in Gali   7
and Gertler (1999).   Taking the model in column (iii), our estimates imply the 
weight on current inflation is 1.29, the weight on inflation in period (t+1) is 0.15 
and that the weight on periods further ahead is small.  Columns (iv) and (v) 
present estimates of the conventional model in (8) using our alternative 
measures of the output gap. The estimates of the included parameters are 
similar to those in earlier columns, but the fit of the model is worse, providing 
further empirical support for our model. 
  We estimated other versions of the model (not reported but available 
form the authors on request).  These included models that used actual rather 
than forecast future inflation rates, models that used expected future rather 
than current output gaps and models that allowed for a delay in the impact of 
interest rates on inflation.  Estimates of these models were broadly similar but 
inferior to those reported in table 1).  
 
4) Conclusions 
This paper has argued that existing empirical models of optimal monetary 
policy rules are over-simplified.  Persistence of inflation in the Phillips curve, 
for which there is strong empirical support suggests interest rates should 
respond to the discounted sum of current and expected future inflation rates, 
rather than just to the current inflation rate, as in the conventional model.  We 
have provided empirical evidence that suggests this is the case. 
  Our work is clearly preliminary.  A more detailed study would estimate 
the policy rule, Phillips curve, aggregate demand equation and the 
relationship between marginal cost and the output gap as a system, allowing   8
the structural parameters to the identified.  However we would not expect this 
extension to affect our main conclusion.    9
 
Table 1 
Estimates of (9) 
   (i)  (ii)  (iii)  (iv)  (v) 
Sample    1992Q4-2007Q1 1992Q4-2007Q1 1998Q1-2007Q1 1992Q4-2007Q1 1992Q4-2007Q1 
Inflation 
forecast  
 Constant  interest 
rates 
Constant interest 
rates 
Expected 
interest rates  
  
Output  gap    HP QT HP HP 
 
QT 
 
         
φ     0.87 (0.05)  0.86 (0.06)  0.83 (0.05)  0.87 (0.05)  0.89 (0.05) 
y β     1.81 (0.60)  1.32 (0.60)  2.40 (0.68)  2.45 (1.16)  1.53 (1.02) 
π β     1.33 (0.60)  1.54 (0.62)  1.29 (0.60)  1.74 (1.12)  1.29 (1.40) 
f β     0.11 (0.04)  0.17 (0.05)  0.12 (0.04)     
Regression 
standard error 
  0.37 0.39 0.34 0.40 0.39 
Parameter 
stability 
  0.30 0.31 0.31 0.30 0.28 
Notes:  
1.  Standard errors in parentheses. 
2.  Estimation by GMM; instruments comprise a constant, and 3 lags of the interest rate, 
inflation and the output gap. 
3.  Inflation forecast: “Constant interest rates” denotes use of inflation forecasts that 
assume constant interest rates, while “expected interest rates” denotes forecasts that 
use market expectations of interest rates.   
4.  Output gap:  “HP” denotes proportional deviation of GDP from its Hodrick-Prescott 
trend; “QT” denotes deviation from fitted values from a regression of GDP on a 
quadratic trend. 
5.  Parameter stability test: (p-value; see Lin and Teräsvirta, 1994).  
6.  All models include an estimated intercept (not reported). 
   10
References 
Adam, C., D. Cobham and E. Girardin (2005). Monetary frameworks and 
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Oxford Bulletin of Economics and Statistics, 67, pp. 497-516. 
 
Aguiar, A. and M. Martins (2005). Testing for asymmetries in the preferences 
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Britton, E., P. Fisher and J. Whitley (1998). The Inflation Report Projections: 
Understanding the Fan Chart, Bank of England Quarterly Bulletin, February, 
pp. 30-37. 
 
Clarida, R, J. Gali, and M. Gertler (1999). The Science of Monetary Policy: A 
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Favero, C. and R. Rovelli (2003). Macroeconomic stability and the 
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Inflation dynamics: a structural econometric analysis,

Macroeconomic stability and the preferences of the Fed. A formal analysis,

Monetary frameworks and institutional constraints: UK monetary policy reaction functions, 1985-2003,

Postwar U.S. business cycles: An empirical investigation,

Testing for asymmetries in the preferences of the euro-area monetary policymaker, mimeo,

Testing the Constancy of Regression Parameters Against Continuous Structural Change,

Monetary Policy and the Hybrid Phillips Curve

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