Long Memory and Structural Breaks in Romanian Inflation Rates

Long Memory and Structural Breaks in Romanian Inflation

Rates

MSc Student: Virgil Savu

Supervisor: Professor Moisa Altar

Academy of Economic Studies, Bucharest

Doctoral School of Finance and Banking

- July 2003 -

■ Memory

■ Review

■ Unit root tests

■ Chow test of structural change

■ R/S analysis

■ Spectral analysis

■ GPH

■ Local Whittle

■ Conclusions

Contents:

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: WN-y ),σ iid(0, -ε ,εy t

2εttt

■ Processes with infinite memory:

- 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:

1t

0ii

i10

t1ttt1-t1t εayay d,0),ARIMA(1, -y ,εyay

, εΔy tt

1t

0iit εy

Between those two extremes, processes with finite memory can be found:

■ Processes with short memory:

- if all the roots of 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,

-the direct exponential decay of the autocorrelation function is the key feature of

short memory processes when it comes to persistence ;

.....LaLa1AR(L) 221

.........εaεaεay 3t3t

12t2t

11t1t

1t

■ 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: where are the AR and MA

polynomials. We assume that all the roots of those polynomials lie outside the unit circle.

ttd Θ(l)εXL1Φ(L) 2

εt σ0,iid~ε

qq1

pp1 Lθ...Lθ1Θ(L) ,Lφ...Lφ1Φ(L)

The fractionally differencing term can be written as an infinite order MA process using the binomial expansion: where

For an ARFIMA (0,d, 0) applying the expansion to Xt yields:

where the autoregressive coefficients are expressed in terms of the gamma function:

dL)-(1

,Lk

d1 .....L

3!

2)1)(dd(dL

2!

1)d(ddL1L1

0k

kk32d

,

k!

1kd2)...(d1dd

k

d

0ktktkt

0k

kkt

d εXAXLk

d1XL1

.

kA

1Kd)ΓΓ(

d)Γ(k

k

d1A k

k

12dτ τ

d)1Γ(d)Γ(τ

d)(τd)Γ(1ρ

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.

Hosking 1981

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:

.00

.04

.08

.12

.16

.20

.24

.28

1992 1994 1996 1998 2000 2002

d L O G ( C P I )

Unit root testsMemory 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 on the exogenous variables :

ty tx ttt εδxy

The LM statistic is defined as:

where S(t) = and is an estimate of the spectral density of the residuals at zero frequency.

)(0)f/(TS(t)LMT

1t

22

t

1iiε (0)f

(j)K(j/l)γ(0)f1T

1)(Tj

lj 0,

lj ,1l

j1

K(j/l)

l

j

5

6πcos

l

j

5

6πl

j

5

6πsin

l

j12π

25K(j/l)

22

Bartlett kernel

Quadratic Spectral kernel

The optimal lag is selected according to:

spectral quadratic for the )T)1.3221(a(2

kernelbartlett for the )T)1.1447(a(1l

1/5

1/3

- Andrews (1991) assumes that the sample follow a AR(1) process: 2

12

1

12

)ρ(1)ρ(1

ρ4a(1)

41

12

)ρ(1

ρ4a(2)

- Newey and West (1994) is based on a truncated weighted sum of the cross-moments:

0

0

L

1ii0

L

1ii

x

γ2γ

γi2

a(x)

Null Hypothesis: Y is stationary

Exogenous: Constant

Asymptotic critical values*: 1% level 0.739000

5% level 0.463000

Bandwidth: 9 (Newey-West using Bartlett kernel)

LM-Stat.

Kwiatkowski-Phillips-Schmidt-Shin test statistic 0.928973

10% level 0.347000


5% level 0.463000

Bandwidth: 6.97 (Newey-West using Quadratic Spectral kernel)

LM-Stat.


10% level 0.347000


5% level 0.463000

Bandwidth: 9.12 (Andrews using Bartlett kernel)

LM-Stat.


10% level 0.347000

Asymptotic critical values*: 1% level

0.739000

5% level 0.463000

*Kwiatkowski-Phillips-Schmidt-Shin (1992, Table 1)

Bandwidth: 7.96 (Andrews using Quadratic Spectral kernel)

LM-Stat.


10% level 0.347000

ADF and PP have the null of containing a unit root and are based on t-ratios. 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 Z-statistics have the asymptotically student distribution. Null Hypothesis: Y has a unit root

Exogenous: Constant

Test critical values: 1% level -3.475500

5% level -2.881260

Lag Length: 1 (Automatic based on SIC, MAXLAG=13)

t-Statistic Prob.

Augmented Dickey-Fuller test statistic -4.470810 0.0003

10% level -2.577365

Null Hypothesis: Y has a unit root

Exogenous: Constant

Test critical values: 1% level -3.475184

5% level -2.881123

Bandwidth: 5.86 (Newey-West using Quadratic Spectral kernel)

Adj. t-Stat Prob.

Phillips-Perron test statistic -6.267296 0.0000

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 fractional- integrated alternative is quite appealing for Romanian inflation.

Chow test of structural changeOne 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)

Coefficient Std. Error t-Statistic Prob.

C(1) 0.050205 0.003954 12.69572 0.0000

F-statistic 34.13225 Probability 0.000000

Log likelihood ratio 57.09833 Probability 0.000000

Chow Breakpoint Test: 1994:12 1997:06

C 0.043997 0.013408 3.281420 0.0013AR(1) 0.889058 0.052922 16.79931 0.0000

R-squared 0.410372 Mean dependent var 0.049801Adjusted R-squared 0.402183 S.D. dependent var 0.048020

S.E. of regression 0.037128 Akaike info criterion -3.728675Sum squared resid 0.198506 Schwarz criterion -3.667646Log likelihood 277.0576 F-statistic 50.11086

Variable Coefficient Std. Error t-Statistic Prob.

MA(1) -0.529560 0.100522 -5.268083 0.0000

Durbin-Watson stat 1.908732 Prob(F-statistic) 0.000000

F-statistic 25.92426 Probability 0.000000

Chow Breakpoint Test: 1993:12 1997:04

Log likelihood ratio 110.9525 Probability 0.000000

R/S analysisTo 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 t = 1,2… where = , and are random variables with mean 0 andY(t) the partial sum of t observations. Then the rescaled range statistic is:

where

t,t εμy μ )E(y t tε

,Y(T)T

tY(t)minY(T)

T

tY(t)max

σ

1Q(T)

Tt1Tt1T

2

1

2

2

2tT Y(T)

T

1y

T

1σ

When the errors are i.i.d. via functional central limit theory it can be shown that: 10 ,)(1 0 WrangeQT

T . Lo (1991) showed that when the errors are correlated(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:

q

1jjjy

2

q

1j

n

1j1jiij

2T

1jjT

2

, γ(q)2σ

μ)μ)(y(y(q)T

2)μ(y

T

1(q)σ

where and are the usual sample variance and autocovariance estimators and:

y2σ jγ

nq ,1q

j1(q)j

0* rangeW1

q)V(T, TQT

In order to choose the optimal truncation lag q the data dependent formula of Andrews (1991) is proposed. Lo showed that when data are short range dependent

is the Bartlett kernel.

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):

1k

2(kv)22W

2

)ev4k(121(v)F

Spectral analysisIn 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.

dωiωexpωfτγdτiωexpτγ

2

1ωf

τcosωτγ

2π

1ωf

τ

ωI4π

1ωf X

ωτcosτγ2πωI

1T

1Tτ

where

is the periodogram function.

-2 -1 0 1 2 3 4 5 6 7 8-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5Spectrum of inflation

Real axis

Ima

gina

ry a

xis

0 0.5 1 1.5 2 2.5 3 3.50

10

20

30

40

50

60Periodogram of inflation

Pow

er

Angular frequencies

GPHThe spectral density of a stationary ARFIMA(0,d,0) process ,

integrated of an order d between –0.5 and 0.5 at 0 frequency has the form:

When parameter d is well estimated then is white noise

and when the frequency tends to 0, is a good estimate of the spectral density of Y at frequency 0 and we can write:

tX

ε2didi

x σ)e(1)e(1)(f t

dt XL1Y

2

ωfω/24sinωf)(f)e(1)e(1)(f y

d2xy

didix

Taking the logarithm results:

jx

jx

y

jy

2

jyjx

Y

jY

2

jYjX

2

jjYjX

ωf

ωIln

0f

ωfln

2

ω2sindln0flnωIln

0f

ωfln

2

ω2sindln0flnωfln

2

ω2sindlnωflnωfln

For a given series the periodogram is given by:

1T

1τx τωcosτγ20γ

2π

1ωI ππ,ω

For frequencies close to 0 the term tends to zero and the previous equation can be written as an OLS regression equation:

0fωfln YjY

m1,...,j ,ε2

4sindlna)ln(I( jj2

j

If the OLS estimator of d is consistent then the error term is white noise.

jx

jxj ωf

ωIlnε

The bandwidth parameter m is chosen so that when .

Usually m is taken so that

Robinson (1995a) showed that

0.T

m and m ,T

,TclimT

0,TTclimT

0.TcTlnlim 2

T

.6

π0,N~d)d(m

2

,Tclim ,TmT

c

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 1.033037 0.735025 0.595491 0.595058 0.518525 0.491928)

t-stat (-3.619196) (-2.915859) (-2.899099) (-3.996341) (-5.179419) (-6.221068)

With breaks at 1993 :12 1997 :04

d 0.457745 0.337626 0.26537 0.346781 0.167311 0.298097

t-stat (-1.929224) (-1.354233) (-1.379429) (-2.066868) (-2.43054) (-3.609405)

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 methodWhen is an ARFIMA(p,d,q,) process, Sowell (1992) suggested to

estimate the parameters by the method of maximum likelihood. The likelihood function is:

where , , and is a vector of parameters including ,

ARMA coefficients and unconditional variance. The autcovariance generating function is written as:

The spectral generating function is given by:

leading to the power spectrum which is used extensively in the likelihood function as:

ty

μyβΩ'μy2

1βΩlog

2

1log2π

2

Tβμ,y,L 1

T

T1 y,...,yy jiij γΩ )E(yμ t β d

)Φ(y)Φ(y

)Θ(y)Θ(y))yy)(1((1σβ)γ(y,

1

1d12

ε

2i

2i2di2

ε

)Φ(e

)Θ(ee1σβ)γ(ω,

2π

1β),f( β)γ(ω,

As the autocovariances of an ARFIMA process are complex function of and is a TxT matrix, calculating the exact MLEs is computationally demanding.

β βΩ1

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:

m

1jj

m

1j

2djj ωln

m

2dωωI

m

1lndLW N(0,1).)dd(2m H d

01/2 and that

When have a mean change:ty

, ,...,1k t ημ

k1,..., t ημy

0t2

0t1t T

.Tτ t,Tτμy

,Tτ t,Tτμyη

2t

1tt

where

T

1tt1 ,yTμ

T

1Tτtt.2 y

TτT

1Tτμ

The corresponding periodogram of is then For each hypothetical change point , d can be estimated by minimizing

tη .eτη2π

1τ,I

2T

1t

ittj

j

T

τ

,ωlnm

2dω,ωI

m

1lnd,LW

m

1jj

m

1j

2djj

The change-point estimator is:τ

τ),,τd~

LW(argminττ,ττ

Kuan and Hsu (2000) proved that asymptotic normality of carries over to Thus, can be used as a test statistic for long-range dependence with the asymptotic standard normal distribution. This method can be generalized for multiple breaks.

d ).τ~(d~

)dτ~d~

(2m H~

01/2

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: d H Breaks at:

0 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 For m=T/4


0 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 1991:1 1993:12 1997:1 1997:4 For m=T/8


0 0.47798416 4.05583*

1 0.13071547 1.10915754 1994:1

2 0.0877619 0.74468441 1991:1 1994:2

* indicates significance at 5%

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.

Selective bibliographyAndrews, D.W.K. (1991), “Heteroskedasticity and Autocorrelation Consistent Covariance Matrix Estimation”, Econometrica, 59, 817-858. Baillie, R. T., C.-F. Chung, and M. A. Tieslau (1996), “Analysing inflation by the fractionally integrated ARFIMA-GARCH model”, Journal of Applied Econometrics11, 23-40.

Ball, L. and S.G. Cecchetti (1990), “ Inflation and uncertainty at short and long horizons”, Brookings Papers on Economic Activity, 215-254. Barsky, R.B. (1987), “ The Fisher hypothesis and the forecastability and persistence of inflation”, Journal of Monetary Economics 19, 3-24.

Bos C.S., P.H. Franses and M. Ooms (1998), “Long Memory and Level Shifts: Re-Analizing Inflation Rates”, Tinbergen Institute Discussion Paper TI98-039/4.

Brunner, A.D. and G.D. Hess (1993), “ Are higher level of inflation less predictable? A state-dependent conditional heteroskedasticity approach”, Journal of Business and Economic Statistics, 11, 187-197

Caporale G.M. and L.A. Gil-Alana (2002), “Long memory and structural breaks in hyperinflation countries”, working paper, South Bank University, London

Franses P.H. and N .Hyung (2001), “ Structural breaks and Long Memory in US inflation rates: Do they matter for forecasting?”, Econometric Institute (Erasmus University Rotterdam) Research Report EI2001-13.Granger, C.W.J. (1980), " Long memory relationships and the aggregation of dynamic models", Journal of Econometrics, 14, 227-238. Granger, C.W.J. and R. Joyeaux (1981), " An introduction to long memory time series models and fractional differencing", Journal of Time Series Analysis, 1, 15-39.

Geweke, J. and S. Porter-Hudak (1983), "The estimation and application of long memory time series models", Journal of Time Series Analysis, 4, 221-238.

Hassler, U. and J. Wolter (1995), " Long Memory in Inflation Rates: International Evidence", Journal of Bussines and Ecinomic Statistics, 13, 37-45. Hosking, J.R.M. (1981), " Fractional Differencing", Biometrica, 68, 165-176.

Hsu C.C. and C. M. Kuan (2000), “Long Memory or Structural Change: Testing Method and Empirical Examination” , working paper, National Central University, Taiwan Kuan, C.M. and C.C. Hsu (1998), "Change-point estimation of fractionally integrated processes", Journal of Time Series Analysis, 19, 693-708.

Kwiatkowski, D., Philips, P.C.B., Schmidt, P., and Y. Shin (1992), " Testing the null hypothesis of stationatity against the alternative of a unit root: How sure are we that the economic time series have a unit root?", Journal of Econometrics 54, 1559-178.Lo A.W. (1991), “Long-Term Memory in Stock Market Prices”, Econometrica 59, 1279-1313.

Mandelbrot, B. (1972), " Statistical Methodology of Non-Periodic Cycles: From the Covariance to R/S Analysis", Annals of Economic and Social Measurement, 1, 259 - 290.

Nelson, C.R. and G.W. Schwert (1977), “ Short term interest rates as predictors of inflation: on testing the hypothesis that the real rate of interest is constant”, American Economic Review, 67, 478-486.

Newey, W.K and K.D. West (1994), " Automatic Lag Selection in Covariance Matrix Estimation", Review of Economic Studies, 61, 631-653.

Robinson, P.M. (1995a), " Log Periodogram Regression of Time Series with Long Range Dependence", The Annals of Statistics, 23, 1048-1072. (1995b), "Gaussian Semiparametric Estimation of Long Range Dependence", The Annals of Statistics, 23, 1630- 1661.

Sowell, F.B. (1992), " Maximum likelihood estimation of stationary univariate fractionally integrated time series models", Journal of Econometrics, 53, 165-188.

Documents

Long Memory and Structural Breaks in Romanian Inflation Rates