Academy of Economic Studies, Bucharest
Doctoral School of Finance and Banking
Long Memory and Structural Breaks in
Romanian Inflation Rates
MSc Student: Virgil Savu
Supervisor: Professor Moisa Altar
- July 2003 -
Contents:
■ Memory
■ Review
■ Unit root tests
■ Chow test of structural
change
■ R/S analysis
■ Spectral analysis
■
GPH
■
Local Whittle
■ Conclusions
Memory
Memory is the series property to depend on its own past realizations. From the
viewpoint of memory characteristics data generating processes can be classified as follow :
■ Processes with no memory:
y t  ε t , ε t - iid(0, σ ε2 ), y t - WN
■ Processes with infinite memory:
y t  a1 y t -1  ε t , y t - ARIMA(1, d,0),
t 1
y t  a y   a1 ε i
t
1 0
i
i 0
- for an AR (1) process a1 is also the first order autocorrelation;
- if a1=1 the process is said to contain an unit root (d=1) and the influence of the
past shocks to present realisations never decreases;
- for a1 =1, the above becomes the random walk model:
Δy t  ε t ,
t 1
yt   εi
i 0
Between those two extremes, processes with finite memory can be found:
■ Processes with short memory:
- if all the roots of AR(L)  1  a1L  a 2 L2  ..... polynomial lie outside the unit circle
then the influence of the shocks decrease exponentially and converge to zero at a
rate depending on the AR coefficients;
- for a1 <1 in the ARMA above and setting y0=0,
y t  a 1t  1 ε t  1  a 1t  2 ε t  2  a 1t  3 ε t  3  .........
-the direct exponential decay of the autocorrelation function is the key feature of
short memory processes when it comes to persistence ;
■ Processes with long memory:
The model with d=1 corresponds to a model with persistence of shocks. However,
persistence need not be infinite so it can be modelled whit d > 0 and < 1. The
autoregressive fractionally integrated moving average (ARFIMA) model aims to capture
the long memory that is apparent in a time series by allowing the difference parameter to
take noninteger values.
A series Xt follows an ARFIMA (p,d,q) process if:
Φ(L) 1  L X t  Θ(l)ε t
d

ε t ~ iid 0, σ ε2

where Φ(L)  1  φ1L  ...  φ p Lp , Θ(L)  1  θ1L  ...  θ q Lq are the AR and MA polynomials. We
assume that all the roots of those polynomials lie outside the unit circle.
The fractionally differencing term (1 - L) d can be written as an infinite order MA process
using the binomial expansion:
1  Ld
 1  dL 

d(d  1) 2 d(d  1)(d  2) 3
k d
L 
L  .....    1  Lk ,
2!
3!
k 0
k
 d  dd  1(d  2)...d  k  1
  
,
k!
k
where
For an ARFIMA (0,d, 0) applying the expansion to Xt yields:
1  Ld X t    1k 

k 0

d k
L X t   A k X t k  ε t
k 0
k
in terms of the gamma function:
ρτ 
.
where the autoregressive
Ak
coefficients are expressed
Γ(k  d)
k d
A k  1   
 k  Γ( d)ΓK  1
Γ(1  d)(τ  d)
 τ 2d1
Γ(d)Γ(τ  1  d)
Hosking 1981
The process is both stationary and invertible if the roots of the AR and MA polynomials
are outside the unit circle and –0.5<d<0. The ARFIMA processes with 0<d<0.5 displays
long memory and is stationary. For 0.5<d<1, the process is invertible but nonstationary.
For d>1 the process is not mean–reverting, and a shock to the process causes it to deviate
away from its starting point.
Review
In the ’80 the persistence of inflation was discussed in the I(0)-I(1) alternative aproach:
■ Nelson and Schwert (1977)
■ Barsky (1987)
■ Ball and Cecchetti (1990)
■ Bruner and Hess (1993)
The possibility that the order of integration could be between 0 and 1 give
rise to an explosion of fractionally integrated models in the ’90. Strictly to
inflation, the most important studies are:
■ Hassler and Wolters (1995)
■ Baillie, Chung and Tieslau (1996)
■ Bos, Frances and Ooms (1999)
■ Hsu and Kuan (2000)
■ Caporale and Gil-Alana (2002)
The data used in the dissertation paper is monthly Romanian
inflation rates taken from December 1990 trough March 2003, computed from
the Consumer Price Index by taking the first difference of the logarithmic
transformed series.
The raw data is show in the figure below:
.28
.24
.20
.16
.12
.08
.04
.00
1992
1994
1996
1998
d L O G (C P I)
2000
2002
Unit root tests
Memory or persistence is closely related to the order of integration. To test for
dependence in the context of nonfractionally integration is equivalent to establish whether
the series is I(0) or I(1). The commonly used tests are KPSS, ADF and PP.
The null of the KPSS test is covariance stationary and is based on the residuals
from the OLS regression of y t on the exogenous x t variables y: t  x t δ  ε t
The LM statistic is defined as:
T
LM   S(t) 2 /(T 2 f̂(0))
t 1 t
ε i and f̂(0) is an estimate of the spectral density of the residuals at zero frequency.
where S(t) = 
i 1
f̂(0) 
T 1
 γ̂(j)K(j/l)
The optimal lag is selected according to:


j  (T 1)
Bartlett kernel
Quadratic
Spectral kernel
j

1  l  1 , j  l
K(j/l)  
0,
jl


  6π j 
 sin  5 l 
25
  cos 6π
 
K(j/l) 

2
6π j
 5
2 j 
12π    5 l
l 


 1.1447(a(1)T) 1/3
for the bartlett kernel
l
1/5
for the quadratic spectral
 1.3221(a(2)T)
- Andrews (1991) assumes that the sample follow a AR(1) process:
4ρ̂ 21
4ρ̂ 21
a(1) 
a(2)

(1  ρ̂ 1 ) 2 (1  ρ̂1 ) 2
(1  ρ̂ 1 ) 4

j 

l 


- Newey and West (1994) is based on a truncated weighted sum of
L0
the cross-moments:
2 i x γ̂ i
i 1
a(x) 
0
L
γ̂ 0  2 γ̂ i
i 1
Null Hypothesis: Y is stationary
Exogenous: Constant
Bandwidth: 9 (Newey-West using Bartlett kernel)
LM-Stat.
Kwiatkowski-Phillips-Schmidt-Shin test statistic
Asymptotic critical values*:
0.928973
1% level
0.739000
5% level
0.463000
10% level
0.347000
Bandwidth: 6.97 (Newey-West using Quadratic Spectral kernel)
LM-Stat.
Kwiatkowski-Phillips-Schmidt-Shin test statistic
Asymptotic critical values*:
0.994218
1% level
0.739000
5% level
0.463000
10% level
0.347000
Bandwidth: 9.12 (Andrews using Bartlett kernel)
LM-Stat.
Kwiatkowski-Phillips-Schmidt-Shin test statistic
Asymptotic critical values*:
0.922180
1% level
0.739000
5% level
0.463000
10% level
0.347000
Bandwidth: 7.96 (Andrews using Quadratic Spectral kernel)
LM-Stat.
Kwiatkowski-Phillips-Schmidt-Shin test statistic
Asymptotic critical values*:
*Kwiatkowski-Phillips-Schmidt-Shin (1992, Table 1)
0.903873
1% level
0.739000
5% level
0.463000
10% level
0.347000
ADF and PP have the null of containing a unit root and are based on tratios. ADF assumes the normality of the errors and when the errors are correlated
the t-statistic does not have the asymptotically t-distribution. So, the critical values are
based on simulations. The most extensive one is that of MacKinnon (1991). PP
correct the t-ratios allowing for a limited degree of serial correlation. The resulting Zstatistics have the asymptotically student distribution.
Null Hypothesis: Y has a unit root
Exogenous: Constant
Lag Length: 1 (Automatic based on SIC, MAXLAG=13)
Augmented Dickey-Fuller test statistic
Test critical values:
t-Statistic
Prob.
-4.470810
0.0003
1% level
-3.475500
5% level
-2.881260
10% level
-2.577365
Null Hypothesis: Y has a unit root
Exogenous: Constant
Bandwidth: 5.86 (Newey-West using Quadratic Spectral kernel)
Phillips-Perron test statistic
Test critical values:
Adj. t-Stat
Prob.
-6.267296
0.0000
1% level
-3.475184
5% level
-2.881123
10% level
-2.577291
As noted in Baillie, Chung and Tieslau (1996) the combined use of ADF, PP and
KPSS test statistics give rise to four possible outcomes:
■ rejection by the ADF and PP and failure to reject by the KPSS is viewed as
strong evidence of covariance stationary I(0) process;
■ failure to reject by the ADF and PP and rejection by the KPSS statistic is
strongly indicative of a unit root I(1) process;
■ failure to reject by all ADF, PP and KPSS is probably due to the data being
insufficiently informative for the long-run characteristics of the process;
■ rejection by all ADF, PP and KPSS indicates that the process is described by
neither I(0) and I(1) processes and that is probable better described by the
fractional integrated alternative.
All tests reject their null at 1% significance level, so a long memory fractionalintegrated alternative is quite appealing for Romanian inflation.
Chow test of structural change
One should be very carefully when testing for long memory if structural breaks
occurred. And there are structural breaks in Romanian inflation. To formally test for the existence
of the breaks I used the Chow breakpoint test. This test necessitates a model specification and a
priori dates for the breaks. I use the test to see where the series experience mean changing and
when the best fitted ARMA representation becomes unstable. The exact dates for the breaks are
taken where the reported F-statistics and Log likelihood ratios are maximized.
Y=C(1)
C(1)
Coefficient
Std. Error
t-Statistic
0.050205
0.003954
12.69572
Prob.
0.0000
Chow Breakpoint Test: 1994:12 1997:06
F-statistic
34.13225
Probability
0.000000
Log likelihood ratio
57.09833
Probability
0.000000
Variable
C
AR(1)
MA(1)
R-squared
Adjusted R-squared
S.E. of regression
Sum squared resid
Log likelihood
Durbin-Watson stat
Coefficient
0.043997
0.889058
-0.529560
0.410372
0.402183
0.037128
0.198506
277.0576
1.908732
Std. Error
0.013408
0.052922
0.100522
Mean dependent var
S.D. dependent var
Akaike info criterion
Schwarz criterion
F-statistic
Prob(F-statistic)
t-Statistic
3.281420
16.79931
-5.268083
Prob.
0.0013
0.0000
0.0000
0.049801
0.048020
-3.728675
-3.667646
50.11086
0.000000
Chow Breakpoint Test: 1993:12 1997:04
F-statistic
25.92426
Probability
0.000000
Log likelihood ratio
110.9525
Probability
0.000000
R/S analysis
To detect long-range dependence in financial markets, Mandelbrot refined Hurst’s R/S
statistic. The R/S statistic is the range of partial sums of deviations of a time series from its
mean, rescaled by its standard deviation.
Let y t  μ  ε t, t = 1,2… where μ = E(y t ) , and ε t are random variables with mean 0 and
Y(t) the partial sum of t observations. Then the rescaled range statistic is:
Q(T) 
where
1
σT

t
t




Y(t)  Y(T) ,
 Y(t)  Y(T)   min

max
T
T
 1t T 

 1t T 
1
1
2
1
2
σ T    y t  2 Y(T) 2 
T
T

When the errors are i.i.d. via functional central limit theory it can be shown that:
1
QT  rangeW 0 ( ), 0    1. Lo (1991) showed that when the errors are correlated
T
(strong mixing) the R/S do not have the above limiting distribution so the null
of short memory is rejected to often. To allow for a limited degree of serial
correlation Lo modified the R/S by letting the denominator to take the form of
the “long-run variance”, specifically:
 n

1 T
2 q
2
σ̂ T (q)   (y j  μ )   j (q)   (y i  μ)(y i  j  μ) 
T j1
T j1
1 j1

2
q
 σ̂ 2 y 2 j (q)γ̂ j ,
j1
where σ̂ 2 y and γ̂ j are the usual sample variance and autocovariance estimators and:
 j (q)  1 
j
, qn
q 1
is the Bartlett kernel.
In order to choose the optimal truncation lag q the data dependent formula of Andrews (1991)
1 *
Q T  rangeW 0
is proposed. Lo showed that when data are short range dependentV(T, q) 
T
The Mandelbrot statistic is the coresponding V(T,0). The optimal truncation lag q* is 9 for the
Bartlett Kernel and 8 for the quadratic spectral.
With no structural change
T
V(T,0)
V(T,q)
V(T,10)
V(T,20)
Kernel
148
3.3663879400
1.5478684650
1.4961427840
1.2311548840
BT
148
3.3213233010
1.5400482670
1.4221796900
1.1572671420
QS
With beaks at 1993: 12 and 1997 :04
T
V(T,0)
V(T,q)
V(T,10)
V(T,20)
Kernel
148
1.7295304890
1.2863583570
1.2690816140
1.2676390590
BT
148
1.7295304890
1.2662254040
1.2268272460
1.2638703410
QS
Critical values for 1%, 5%, 10% are 2.001, 1.747, 1.620
The critical values are computed as fractiles of the distribution of the range of a standard
Brownian bridge on the unit interval. The cdf is given in Feller (1951):

FW (v) 1  2 (1  4k 2 v 2 )e  2(kv)
k 1
2
Spectral analysis
In ordinary time series analysis we are mainly focused on serial dependencies
(autocorrelations) in the time series. This is usually called “time domain analysis”. In
spectral analysis or “frequency domain analysis” we are interested in cyclical regularities
of the underlying generating process of the observed time series. Conversion from time
domain to frequency domain is done by Fourier transformation. The goal of the analysis
is to determine how important cycles of different frequencies are in accounting for the
behavior of the series. This is achieved by estimating the spectral density.

1 
1 
f ω  
f ω 
γτ exp  iωdτ  γτ    f ωexp iωdω
 γτ cosωτ

2π τ  
2 
T 1
1
f̂ ω  
I X ω  where Iω  2π  γ̂τ cosωτ is the periodogram function.
4π
τ   T 1
Periodogram of inflation
Spectrum of inflation
2.5
60
2
50
1.5
1
40
Power
Imaginary axis
0.5
0
30
-0.5
20
-1
-1.5
10
-2
-2.5
-2
-1
0
1
2
3
Real axis
4
5
6
7
8
0
0
0.5
1
1.5
2
Angular frequencies
2.5
3
3.5
GPH
The spectral density of a stationary ARFIMA(0,d,0) process X t , integrated of an order
d between –0.5 and 0.5 at 0 frequency has the form:
f x ( )  (1  e i ) d (1  e i ) d σ 2 ε
When parameter d is well estimated then Y  1  L X is white noise and when the
frequency tends to 0,  2  is a good estimate of the spectral density of Y at frequency 0 and
we can write: f ( )  (1  e i ) d (1  e i ) d f ( )  f ω  4sin 2 ω/2d f ω
d
t
x
y
Taking the logarithm results:


ln f X ω j 

x
t
y


 ω j 
 ω j 
 f Y ω j 
 ln f Y ω j   dln 2sin    ln f X ω j   ln f Y 0  dln 2sin    ln 

 2 
 2 
 f Y 0 




 ln I x ω j 
For a given series

2


2
 f y ω j 
 I x ω j 

 ω j 
 ln f y 0  dln 2sin    ln 
  ln 

 2 

 f y 0 
 f x ω j 
T 1
the periodogram is given by: I x ω  1 γ0  2 γτcosτω 
2π 
τ 1



2
ω   π, π
For frequencies close to 0 the term ln f Y ω j  f Y 0 tends to zero and the previous equation can
be written as an OLS regression equation:

 j 
ln(I(  j )  a  dln  4sin 2     ε j ,
 2 

j  1,..., m
 I x ω j 

 f x ω j 
If the OLS estimator of d is consistent then the error term ε j  ln 
is white noise.
The bandwidth parameter m is chosen so that when
Usually m is taken so that
m  Tc ,
T  , m   and
m
 0..
T
 
lim cT   , lim cT   , lim cT  T  0, lim ln T 2 cT  0.
T 
T 
T 
T
 π2 
m (d̂  d) ~ N 0, .
 6 
Robinson (1995a) showed that
The band of frequencies used to compute GPH is the one where c=0.45, 0.5, 0.55,
0.65, 0.75. Usually c=0.5 is taken.
c=0.45
c=0.5
c=0.55
c=0.65
c=0.75
m=T/2
With no structural breaks
d
t-stat
1.033037
0.735025
0.595491
0.595058
0.518525
0.491928)
(-3.619196)
(-2.915859)
(-2.899099)
(-3.996341)
(-5.179419)
(-6.221068)
0.457745
0.337626
0.26537
0.346781
0.167311
0.298097
(-1.929224)
(-1.354233)
(-1.379429)
(-2.066868)
(-2.43054)
(-3.609405)
With breaks at 1993 :12 1997 :04
d
t-stat
With breaks at 1991 :01 1993: 12
1997 :01
1997 :04
d
0.75594
0.32851
0.222216
0.189945
0.132862
0.061987
t-stat
(-1.853001)
(-0.941894)
(-0.818308)
(-1.152486)
(-1.243071)
(-0.784212)
The local Whittle method
When y t is an ARFIMA(p,d,q,) process, Sowell (1992) suggested to estimate the parameters
by the method of maximum likelihood. The likelihood function is:
L T y, μ, β   
T
1
1
log2 π  log Ωβ   y  μ ' Ω 1 β y  μ 
2
2
2
where y  y1 ,..., y T , Ωij  γ i j , μ  E(y t ) and β is a vector of parameters including d ,
ARMA coefficients and unconditional variance.
The autcovariance generating function is written as:
γ(y, β)  σ ε ((1  y)(1  y 1 )) d
2
Θ(y)Θ(y 1 )
Φ(y)Φ(y 1 )
The spectral generating function is given by:
γ(ω, β)  σ ε 1  e
2
i  2d
Θ(e i )
2
Φ(e i )
2
leading to the power spectrum which is used extensively in the likelihood function as:
f( , β) 
1
γ(ω, β)
2π
As the autocovariances of an ARFIMA process are complex function of β and
Ω1 β  is a TxT matrix, calculating the exact MLEs is computationally demanding.
Following a approximation proposed by Whittle and focusing only on the spectral
density in a neighborhood of zero, Robinson (1995b) proved that maximizing the
exact likelihood function is equivalent to minimizing:
1 m
 2d m
LW d   ln   Iω j ω 2d
ln ω j 

j 
m
m
j

1
j

1


and that Ĥ  2m1/2 (d̂  d0 ) d N(0,1).
When y t have a mean change:
μ  η t
yt   1
μ 2  η t
where
T 
μ̂1 T  y t ,
t 1
t  1,..., k 0
t  k 0  1,..., T
,
y  μ̂ 1 Tτ, t  Tτ,
η̂ t     t
 y t  μ̂ 2 Tτ, t  Tτ.
T
1
μ̂ 2 Tτ 
 y t.
T  Tτ t Tτ 1
The corresponding periodogram of η̂ t is then I j , τ   1
2πT
T
 η̂ τe
t 1
t
2
it j
.
For each hypothetical change point τ , d can be estimated by minimizing
1 m
 2d m
2d


LW d,   ln   I ω j , ω j    ln ω j ,
 m j1
 m j1
The change-point estimator τ̂ is:
~
τ̂  argmin ττ,τ LW( d τ , τ),
~
Kuan and Hsu (2000) proved that asymptotic normality d̂ of carries over to d (~τ ).
~
~
Thus, H
can be used as a test statistic for long-range dependence with the
 2m1/2 (d ~τ   d 0 )
asymptotic standard normal distribution. This method can be generalized for multiple breaks.
The local Whittle method allows testing jointly for the difference parameter and
the breaks. The bandwidth parameter m is usually set at T/4. The test is also
computed with m=T/3 and m=T/8. For m=T/8 the number of the frequencies
in the band is small. For m=T/3 results approaches those with m=T/4.
For m=T/3
No. of breaks:
0
d
H
0.42814512
5.9940316*
1
0.27677194
3.8748071*
1993:11
2
0.26270424
3.6779593*
1991:1 1993:12
3
0.26748676
3.7448146*
1991:1 1993:12 1997:4
4
0.02542706
0.3559788
1991:1 1993:12 1997:1 1997:4
No. of breaks:
0
d
H
Breaks at:
0.49781192
6.0561433*
1
0.31209051
3.7967449*
1993:11
2
0.28287419
3.4413130*
1991:1 1993:12
3
0.27888657
3.3928015*
1991:1 1993:12 1997:4
4
0.08615833
1.0481613
No. of breaks:
d
H
0
0.47798416
4.05583*
1
0.13071547
1.10915754
1994:1
2
0.0877619
0.74468441
1991:1 1994:2
Breaks at:
For m=T/4
1991:1 1993:12 1997:1 1997:4
For m=T/8
* indicates significance at 5%
Breaks at:
Conclusions
Neglecting structural breaks leads to a spurious finding of long memory
properties in Romanian inflation rates. When structural change is present
traditional tests are biased towards a nonrejection of the null hypothesis of long
memory. Without the spike in 1997 there is no evidence of long memory in
inflation series. Therefore, a short-memory process can better explain the inertia in
the Romanian inflation rates.
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