The Academy of Economic Studies
Doctoral School of Finance and Banking
Does the Barro-Gordon Model Explain
the Behavior of Inflation in Romania?
MSc Student: Ana Alexe
Supervisor: Professor Moisă Altăr
Bucharest, July 2008
Topics
1.
2.
3.
4.
5.
6.
7.
8.
9.
Introduction
Literature review
Barro-gordon time consistency model
Input data
The natural rate of unemployment
estimations
Inflation and unemployment estimations
according to barro-gordon model
Unit root tests
Cointegration tests
Conclusions
1. Introduction



There are many situations in which one economic agent (the government for
example) has an incentive to deceive another economic agent.
Intuitively, the policymaker, who aims to bring the unemployment rate closer to
its natural rate, is tempted to do so by creating surprise inflation. Households,
being rational agents, perfectly anticipate this temptation to inflate and adjust
their decisions accordingly. As a result, the equilibrium outcome is a situation
with no reduction in unemployment and higher inflation than before (the
`inflation bias' result).
The purpose of this paper is to analyze the time-inconsistency problem
between inflation and unemployment rate series for Romania, using unit root
tests and co-integration tests, Hodrick-Prescott filter and Kalman filter as
estimation techniques, in order to test the Barro-Grodon model’s implications.
2. Literature review




In the literature there are known different types of time-inconsistency:

time-inconsistency due to changes in preferences over time (Strotz (1956))

time-inconsistency of government plans when agents have rational
expectations (Lucas (1976), Kydland & Prescott (1977), Barro & Gordon
(1983)).
The idea that the proper design of monetary policy is crucial to achieve good
inflation outcomes was first proposed by Kydland and Prescott (1977). A key result
is that if policymakers cannot commit to future policies, inflation rates are higher
than if they can commit.
Barro and Gordon (1983) further developed this idea. They argue that an optimal
punishment mechanism is in place and that no intervention is necessary.
Ireland (1999) initially conducted time series tests for the United States based on
the modified Barro-Gordon model. He shows that Barro and Gordon’s (1983) model
of time-consistent monetary policy imposes long-run restrictions on the time series
properties of inflation and unemployment that are not rejected by the data in the
US.
3. Barro-Gordon Time Consistency Model

BG model’s assumption:



inflation varies positively with the natural unemployment rate
inflation will inherit the persistency of the natural rate of
unemployment when the central bank cannot commit to a monetary
policy rule
Ireland (1999):


the actual unemployment rate is non-stationary
control errors for inflation – which permits the model to account for
transitory deviations between the actual unemployment rate and the
natural rate
3. Barro-Gordon Time Consistency Model

The equations of the model

expectations Phillips curve (1)
U t  U tn   ( t   te )

the natural rate (2)
U tn  U tn1   (U tn1  U tn2 )   t

the actual inflation rate (3)
 t   tp   t

minimize a loss function that penalizes variations of unemployment and
inflation around target values: k*Unt and 0
(1 / 2)(U t  kU tn ) 2  (b / 2) t2

(1) and (3) => the policymaker’s problem becomes:


min
Et 1 (1 / 2)[(1  k )U tn   ( tp   t   te )] 2  (b / 2)( tp   t ) 2 ,
p
t
3. Barro-Gordon Time Consistency Model

The solution of the model

the first-order condition (4)
Et 1 L
 Et 1[(1  k )U tn   ( tp   t   te )]  bEt 1 ( tp   t )  0
p
 t
 te   tp
Et 1 t  0

in equilibrium

using (4) => (5) the inflationary bias resulting from the policymaker’s inability to
commit depends positively on the expected natural rate
 tp   te  AEt 1U tn

from (1), (3) and (5) => (6) which shows how the control error for inflation
(ηt) allows the actual unemployment rate to fluctuate, in equilibrium,
around the natural rate
U t  U tn  t
3. Barro-Gordon Time Consistency Model

The solution of the model
U t  U tn1  U tn1   t  t

combining (6) and (2) => (7)

combining (2), (3) and (5) => (8)


 t  AU tn1  AU tn1   t
separately, (7) and (8) indicate that both inflation and unemployment are
non-stationary, inheriting unit roots from the natural rate of
unemployment
together, they imply that a linear combination of inflation and
unemployment is stationary (9)
 t  AU  A t  (1   2 A)t

Equation (9) summarizes the constraint that Barro and Gordon’s theory
imposes on the long-run behavior of inflation and unemployment:
according to the model, these variables should be non-stationary
but co-integrated.
4. Input Data
Variable
Period Analyzed
Source
Observations
Unemployment
rate
Jan 1999 – March
2008 (monthly
observations)
Website of the
National Bank
registered monthly
unemployment rates in %
For the CPI:
National Institute of
Statistics
measured as the first
difference in the logarithm
of the monthly CPI
(Consumer Price Index)
Inflation rate
The relationship between the two variables is a positive one, meaning that an extra one
percentage point of unemployment pushes the inflation rate up.
The relationship between initial inflation and unemployment
rates
3,00
2,50
inflation (%)
2,00
1,50
1,00
0,50
0,00
0,00
-0,50
2,00
4,00
6,00
8,00
10,00
unemployment (%)
12,00
14,00
16,00
5. The natural rate of unemployment
estimations


A disadvantage of this representation is that it includes the unobserved natural rate
of unemployment and there are no direct measures of the natural rate.
Statistical approaches to estimate time-varying natural rate of unemployment:


Hodrick Prescott filter - widely used among macroeconomists to obtain
a smooth estimate of the long-term trend component of a series
Kalman filter - the most commonly used reduced form filtering technique
for estimating the natural rate of unemployment due to its simplicity of
estimation
5. The natural rate of unemployment
estimations

Hodrick-Prescott (HP) filter


HP filter is a two-sided linear filter that computes the smoothed series s of y
by minimizing the variance of y around s, subject to a penalty parameter
The penalty parameter controls the smoothness of the series (λ=14.400 for
monthly data).
Hodrick-Prescott Filter (lambda=14400)
14
12
6
10
4
8
6
2
4
0
2
-2
-4
99
00
01
02
03
UNEMPLOYMENT

04
05
06
Trend
07
08
Cycle
The trend estimated here counts for the natural rate of unemployment.
5. The natural rate of unemployment
estimations

Kalman filter

I use the Kalman filter of Kalman (1960) and Kalman and Bucy (1961), since it
has the major advantage of allowing a time-varying natural rate of
unemployment to be estimated jointly with a Phillips curve.

The general specification:
 t   te   ( t 1   te1 )   (U t  U tn )  xt   t
U tn  U tn1   t
 t ~ N (0,  2 ) (1)
 t ~ N (0,  2 ) (2)
24
Equation (1) is a Phillips curve –
it models unexpected inflation as a
function of: shocks (xt) and the
unemployment gap (Ut - Unt)

The natural rate of unemployment
(Unt) is time varying and its movement is
modeled by Equation (2)

The forecasts from the exponential
smoothing method are used to track
the seasonal movements in the actual
series as it gives good results

20
16
12
8
4
0
99
00
01
02
03
04
UN_KALMAN_SMOOTHED
05
06
07
08
UN_KALMAN
5. The natural rate of unemployment
estimations

Hodric-Prescott filter versus Kalman filter

The key difference between the HP filter and the Kalman filter is
that the HP filter natural rate of unemployment estimates move more
closely with the actual level of unemployment => the size of
unemployment gaps is smaller than those estimates based on the
Kalman filter.
14
12
10
8
6
4
2
99
00
01
02
03
04
05
06
UN_KALMAN_SM
UN_Hodrick_Prescott
UNEMPLOYMENT
07
08
5. The natural rate of unemployment
estimations

Unit root tests for the estimated natural rate of unemployment
6. Inflation and unemployment estimations
according to Barro-Gordon model
Equations (7) and (8) show that according to the model, both inflation and
unemployment rate ought to be unit root processes.

Unit root tests for inflation – HP filter case
 t   'U tn1   ' ' U tn1  t
=>
πt = 0.1343*Unt-1 + 5.6319*ΔUnt-1+ ηt
(15.54)
(6.08)
1.4
1.2
1.0
0.8
0.6
0.4
0.2
0.0
From the table above one can see
that the inflation determined by BG
has a unit root, which is according to the model.


99
00
*Critical points of 1%, 5% and 10% levels of significance: 1.98, 2.63, 3.39
01
02
03
04
05
INFLATION
06
07
08
6. Inflation and unemployment estimations
according to Barro-Gordon model

Unit root tests for inflation – Kalman filter case
 t   'U tn1   ' ' U tn1  t
=>πt = 0.0885*Unt-1 -0.0344*ΔUnt-1 + ηt
(17.31)
(-3.81)
1.0
0.9
0.8
0.7
0.6
0.5
0.4
From the table above one can see
that the inflation determined by BG
has a unit root, which is according to the
model.

0.3
99
00
01
02
03
04
05
INFLATION
06
07
08
6. Inflation and unemployment estimations
according to Barro-Gordon model

Unit root tests for unemployment – HP filter case
U t  U tn1  U tn1   't
=> Ut = Unt-1 +1.7344*( Unt-1- Unt-2) + εt
(2.62)
12
11
10
9
8
7
6
5
4
3
99
From the table above one can see that
the unemployment rate determined by
BG has a unit root, which is according to the model.

00
01
02
03
04
05
UNEMPLOYMENT
06
07
08
6. Inflation and unemployment estimations
according to Barro-Gordon model

Unit root tests for unemployment – Kalman filter case
U t  U tn1  U tn1   't
=> Ut = Unt-1 -3.2745*( Unt-1- Unt-2) + εt
(-3.006)
12
11
10
9
8
7
6
5
4
3
99
00
01
02
03
04
05
06
07
08
UNEMPLOYMENT

As a conclusion, we can not say if the unemployment rate is in accordance with
the model’s hypothesis that the unemployment is non-stationary if the natural rate
of unemployment is non-stationary.
7. Unit root tests



Unit root tests results
In the case when the natural unemployment rate is determined by Hodrick
Prescott filter, both inflation and unemployment seem to be unit root processes,
which is consistent with the model’s implication.
In the case when the natural unemployment rate is determined using Kalman
filter, inflation is a unit root process, but it is difficult to say if unemployment is
stationary or non-stationary, so we can not say whether this case is or not in
accordance with the Barro-Gordon model.
8. Cointegration tests
 t   'U   ' '  t  At
Equation (9)
implies that the linear combination of unemployment rate and inflation is
stationary, even though these two variables are non-stationary
independently.


Phillips-Ouliaris (PO) co-integration test

The tested hypothesis is H0 – no co-integration between inflation and
unemployment rate
8. Cointegration tests



Phillips-Ouliaris (PO) co-integration test
HP filter case: the t-statistic > critical values* at 1%, 2.5% and 5% levels of
significance, we reject H0: there is no co-integration between inflation
and unemployment rate, which means that the data appear to be consistent
with the Barro-Gordon model’s implication that inflation and unemployment are
cointegrated, according to the PO co-integration test.
Kalman filter case: the t-statistic << critical values at 1%, 2.5% and 5%
levels of significance, in this case we can’t reject H0: no co-integration
between inflation and unemployment rate, which means that the linear
combination of inflation and unemployment rate is non-stationary.
*t - critical values at 1%, 2.5% and 5% are -3.39, -3.05 and -2.76 respectively
8. Cointegration tests


Engle Granger (EG) Method
This method involves estimating the long-run Equation (9) by the
standard regression method and then the residuals are recovered and
tested for stationarity by applying the ADF and the PP unit root tests.
8. Cointegration tests






Johansen co-integration test
The analysis below assumes that there is no constant in the co-integration relation
(as implied in the BG model) and that there are no deterministic trends in the data;
under this assumption, no constant terms are included in the preliminary
regression.
There are two statistics to take into account; the trace and maximum eigenvalue.
Given that for both tests, the test statistic exceeds its critical value (5%) when
the null is r = 0, we can conclude that at least one co-integration vector is
present.
For more than one co-integration vector, the test statistic is less than the
critical value so we conclude only a single co-integration vector is present.
Co-integrating Vector: πt=0.085486*ut
8. Cointegration tests


Johansen co-integration test
Given that for both tests, the test statistic is less than its critical value (5%)
when the null is r = 0, we can conclude that no co-integration vector is
present.
8. Cointegration tests




Johansen co-integration test
According to Maximum Eigenvalue statistic, only 1 co-integration vector exists.
We compute the likelihood ratio:
Tested hypothesis: H0 – no co-integration between inflation and
unemployment rate.
As the calculated likelihood ratio=10,2418 < likelihood critical values at 1%,
2.5% and 5% levels of significance, we cannot reject H0: no co-integration
between inflation and unemployment rate.
* Likelihood critical values at 1%, 2.5% and 5% are 15.69, 13.27 and 11.44 respectively.
8. Cointegration tests



Cointegration tests results
The co-integration vectors determined using the three co-integration tests are
similar.
As a conclusion:


I can say that the co-integration implication of the Barro-Gordon model can
only be proved in the case when the natural unemployment rate is estimated
using the Hodrick-Prescott filter.
In the other case, when the natural unemployment rate is estimated using the
Kalman filter, the co-integration implication can not be proved, this is mainly
because we were not able to prove that the unemployment rate is nonstationary or not.
8. Conclusions


The model implies that both inflation and unemployment rate processes depend on
the evolution of the natural rate of unemployment
Under the assumption that the natural rate of unemployment follows a unit root
process, inflation and unemployment rate should be non-stationary while in the
long-run, these two variables are co-integrated.
8. Conclusions

Looking at the results presented and analyzing the Barro-Gordon model’s
implications, I conclude that:



inflation and unemployment are unit root processes – which is in accordance
with the model: both inflation and unemployment are non-stationary in both of
the cases presented for the estimation of the natural rate of unemployment
I could prove that the two variables are co-integrated only in the case when I’ve
made the estimation of natural rate of unemployment using Hodrick Prescott
filter – this is in accordance with the model’s implication of co-integration
The results in this paper support the Barro-Gordon model to explain long-term
inflation behavior in Romania. The results indicate that the policymaker’s inability to
commit in advance to a monetary policy could explain the evolution of inflation.
Thank you!