A Note on the Size of the ADF Test with Additive Outliers and Fractional Errors. A Reapraisal about the (Non) Stationarity of the Latin-American Inflation Series.

34 Pages·2013·0.468 MB·English

by  Gabriel Rodríguez, Dionisio Ramirez

DEPARTAMENTO DE ECONOMÍA
PONTIFICIA UNIVERSIDAD CATÓLICA DEL PERÚ
DEPARTAMENTO DE ECONOMÍA
PONTIFICIA UNIVERSIDAD CATÓLICA DEL PERÚ
DOCUMENTO DE TRABAJO N° 357
DEPARTAMENTO DE ECONOMÍA
A NOTE ON THE SIZE OF THE ADF TEST WITH
ADDITIVE OUTLIERS AND FRACTIONAL ERRORS.
PONTIFICIA UNIVERSIDAD CATÓLICA DE?L PERÚ
A REAPRAISAL ABOUT THE (NON) STATIONARITY
DEPARTAMENTO DE ECONOMÍA
OF THE LATIN-AMERICAN INFLATION SERIES
Gabriel Rodríguez y DionPiOsiNo TRIaFmICiIrAe zUNIVERSIDAD CATÓLICA DEL PERÚ
DEPARTAMENTO DE ECONOMÍA
PONTIFICIA UNIVERSIDAD CATÓLICA DEL PERÚ
DEPARTAMENTO DE ECONOMÍA
PONTIFICIA UNIVERSIDAD CATÓLICA DEL PERÚ
DEPARTAMENTO DE ECONOMÍA
PONTIFICIA UNIVERSIDAD CATÓLICA DEL PERÚ
DEPARTAMENTO DE ECONOMÍA
PONTIFICIA UNIVERSIDAD CATÓLICA DEL PERÚ
DEPARTAMENTO DE ECONOMÍA
PONTIFICIA UNIVERSIDAD CATÓLICA DEL PERÚ
DEPARTAMENTO DE ECONOMÍA
PONTIFICIA UNIVERSIDAD CATÓLICA DEL PERÚ
DEPARTAMENTO DE ECONOMÍA
DEPARTAMENTO
DE ECONOMÍA
DOCUMENTO DE TRABAJO N° 357

A NOTE ON THE SIZE OF THE ADF TEST WITH
ADDITIVE OUTLIERS AND FRACTIONAL ERRORS.
A REAPRAISAL ABOUT THE (NON) STATIONARITY
OF THE LATIN-AMERICAN INFLATION SERIES

Gabriel Rodríguez y Dionisio Ramirez

Julio, 2013

DEPARTAMENTO
DE ECONOMÍA

DOCUMENTO DE TRABAJO 357
http://www.pucp.edu.pe/departamento/economia/images/documentos/DDD357.pdf
© Departamento de Economía – Pontificia Universidad Católica del Perú,
© Gabriel Rodríguez y Dionisio Ramirez

Av. Universitaria 1801, Lima 32 – Perú.
Teléfono: (51-1) 626-2000 anexos 4950 - 4951
Fax: [phone]
[email protected]
www.pucp.edu.pe/departamento/economia/

Encargado de la Serie: Luis García Núñez
Departamento de Economía – Pontificia Universidad Católica del Perú,
[email protected]

Gabriel Rodríguez y Dionisio Ramirez

A Note on the Size of the ADF Test with Additive Outliers and Fractional
Errors. A Reapraisal about the (Non) Stationarity of the Latin-American
Inflation Series.
Lima, Departamento de Economía, 2013
(Documento de Trabajo 357)

PALABRAS CLAVE: Outliers aditivos, Errores ARFIMA, Test ADF.

Las opiniones y recomendaciones vertidas en estos documentos son responsabilidad de sus
autores y no representan necesariamente los puntos de vista del Departamento Economía.

Hecho el Depósito Legal en la Biblioteca Nacional del Perú Nº 2013-11339.
ISSN 2079-8466 (Impresa)
ISSN 2079-8474 (En línea)

Impreso en Cartolán Editora y Comercializadora E.I.R.L.
Pasaje Atlántida 113, Lima 1, Perú.
Tiraje: 100 ejemplares
A Note on the Size of the ADF Test with
Additive Outliers and Fractional Errors. A
Reapraisal about the (Non)Stationarity of the
Latin-American In(cid:135)ation Series
Gabriel Rodr(cid:237)guez Dionisio Ramirez
Ponti(cid:133)cia Universidad Cat(cid:243)lica del Perœ Universidad Castilla La Mancha
Abstract
ThisnoteanalyzestheempiricalsizeoftheaugmentedDickeyandFuller(ADF)
statistic proposed by Perron and Rodr(cid:237)guez (2003) when the errors are frac-
tional. This ADF is based on a searching procedure for additive outliers based
on (cid:133)rst-di⁄erences of the data named (cid:28) . Simulations show that empirical size
d
of the ADF is not a⁄ected by fractional errors con(cid:133)rming the claim of Perron
and Rodr(cid:237)guez (2003) that the procedure (cid:28) is robust to departures of the unit
d
root framework. In particular the results show low sensitivity of the size of the
ADF statistic respect to the fractional parameter (d). However, as expected,
when there is strong negative moving average autocorrelation or negative au-
toregressive autocorrelation, the ADF statistic is oversized. These di¢ culties
are (cid:133)xed when sample increases (from T =100 to T =200). Empirical applica-
tion to eight quarterly Latin-American in(cid:135)ation series is also provided showing
theimportanceoftakingintoaccountdummyvariablesforthedetectedadditive
outliers.
Keywords: Additive Outliers, ARFIMA Erros, ADF Test
JEL: C2, C3, C5
Resumen
En esta nota se analiza el tamaæo emp(cid:237)rico del estad(cid:237)stico Dickey y Fuller au-
mentado (ADF), propuesto por Perron y Rodr(cid:237)guez (2003), cuando los errores
son fraccionales. Este estad(cid:237)stico se basa en un procedimiento de bœsqueda de
valoresat(cid:237)picosaditivosbasadoenlasprimerasdiferenciasdelosdatosdenomi-
nado(cid:28) . Lassimulacionesmuestranqueeltamaæoemp(cid:237)ricodelestad(cid:237)sticoADF
d
no es afectado por los errores fraccionales con(cid:133)rmando el argumento de Perron
y Rodr(cid:237)guez (2003) que el procedimiento (cid:28) es robusto a las desviaciones del
d
marcodera(cid:237)zunitaria. Enparticular,losresultadosmuestranunabajasensibil-
idad del tamaæo del estad(cid:237)stico ADF respecto al parÆmetro fraccional (d). Sin
embargo, como es de esperar, cuando hay una fuerte autocorrelaci(cid:243)n negativa
de tipo promediom(cid:243)vilo autocorrelaci(cid:243)n autorregresiva negativa, elestad(cid:237)stico
ADFtieneuntamaæoexactomayorqueelnominal. Estasdi(cid:133)cultadesdesapare-
cen cuando aumenta la muestra (a partir de T =100 a T =200). La aplicaci(cid:243)n
emp(cid:237)rica a ocho series de in(cid:135)aci(cid:243)n latinoamericana trimestral proporciona evi-
denciadelaimportanciadetenerencuentalasvariables(cid:133)cticiasparacontrolar
por los outliers aditivos detectados.
Palabras Claves: Outliers Aditivos, Errores ARFIMA, Test ADF.
Classi(cid:133)caci(cid:243)n JEL: C2, C3, C5
A Note on the Size of the ADF Test with Additive
Outliers and Fractional Errors. A Reapraisal
about the (Non)Stationarity of the
Latin-American In(cid:135)ation Series
1
Gabriel Rodr(cid:237)guez2 Dionisio Ramirez
Ponti(cid:133)cia Universidad Cat(cid:243)lica del Perœ Universidad Castilla La Mancha
1 Introduction
Additve outliers a⁄ect inference of parameters in di⁄erent circunstances.
For example they a⁄ect inference of the autoregressive and moving average
estimates of ARMA(p;q) models; see Cheng and Liu (1993), Chan (1992,
1995). They also a⁄ect other topics like causality tests (see Balde and Ro-
dr(cid:237)guez (2005)), fractional estimates (see Fajardo et al. (2009), Chareka
et al. (2006)). In the context of a unit root, additive outliers have been
also analyzed since the contribution of Franses and Haldrup (1994). These
authors show that additive outliers contaminate the limiting distribution of
the unit root statistics; see also Vogelsang (1999) and Perron and Rodr(cid:237)guez
(2003). Vogelsang (1999) suggests to use M-tests based on GLS detrended
data because they are robust to the presence of negative moving average au-
tocorrelation which is induced by the presence of additive outliers. Another
alternative procedure is to estimate an ADF statistic corrected for dummy
variablesrelatedtotheidenti(cid:133)edadditiveoutliersinapreliminarystep. Ro-
dr(cid:237)guez (2004) used four Latin-American in(cid:135)ation series and show that even
the M-tests indicate a rejection of the null hypothesis of a unit root. When
applyinganADFcorrectedfordummyvariables, somecountriesshowanon
rejection of the null hypothesis of a unit root indicating nonstationarity of
the in(cid:135)ation series which is an opposite results obtained from the standard
unit root tests.
1We want to thank Carmen Armas Montalvo and Vanessa Belapatiæo for excellent
researchassistanship. Iacknowledge(cid:133)nancialsupportfromtheDepartmentofEconomics
of the Ponti(cid:133)cia Universidad Cat(cid:243)lica del Perœ and useful comments of Patricia Lengua
Lafosse.
2Address for Correspondence: Gabriel Rodr(cid:237)guez, Department of Economics, Pon-
ti(cid:133)cia Universidad Cat(cid:243)lica del Perœ, Av. Universitaria 1801, Lima 32, Lima,
Perœ, Telephone: [phone] (4998), Fax: [phone]. E-Mail Address:
[email protected].
1
The procedure mentioned above needs the location of the additive out-
liers. Perron and Rodr(cid:237)guez (2003) have suggested a powerful test, denoted
by (cid:28) , which works with (cid:133)rst-di⁄erenced data3. This procedure is more
d
powerful than other based on levels of the data, for example; see Perron and
Rodr(cid:237)guez (2003) for a detailed discussion. These authors claim that (cid:28) is
d
powerful even for departures from the unit root case4. The purpose of this
note is to show that this claim is correct. we do it analyzing the empirical
sizeoftheADFstatistic(using(cid:28) tolocateadditiveoutliers)whentheDGP
d
contains ARFIMA(p;d;q) errors. The experiment deals with di⁄erent val-
ues of the fractional parameter (d) to observe di⁄erent departures from the
unit root hypothesis. Also di⁄erent structure of autocorrelation is analyzed
(moving average and autoregressive).
The Monte-Carlo simulations show that the ADF statistic corrected for
dummyvariablesassociatedtotheadditiveoutlierssu⁄ersofsizedistortions
in only few cases. For example, when the moving average parameter is close
to -1 empirical size is greater than nominal size. Negative autoregressive
autocorrelaci(cid:243)n has impact on the size of the ADF statistic too. However,
most of these issues are (cid:133)xed when sample size increases from T = 100 to
T = 200 in simulations. In general, the ADF test appears to be slightly
undersized. When fractional parameter is higher, distortions appear but at
the same time when correlation is higher. Therefore, fractional parameter
itself does not cause problems or distorions on the size of the ADF test.
After simulations, we present an empirical application using quarterly
in(cid:135)ation series ranging from 1970:1 until 2010:4 of 8 countries. We use a
sample of eigth countries and the spirit of this exercise is very similar to
Rodr(cid:237)guez (2004) where four countries were only used. In this note, we
add more countries and more observations. In particular, the Phillips and
Perron (1988) statistic shows a strong rejection of the null hypothesis of
a unit root which is not rare given the sensitivity of this statistic to the
presence of strong negative moving average correlation which is the case
here because additive outliers are clearly present and literature has shown
that they are related to this type of correlation. Similar results are obtained
3Of course, there are many other procedures to identify outliers, for example, those
proposed in Tsay (1986), Chang, Tiao and Chen (1988), Shin, Sharkar and Lee (1996),
Chen and Liu (1993) and G(cid:243)mez and Maravall (1992a, 1992b). Another interesting ap-
proach is proposed by Lucas (1995a, 1995b), and Hoek, Lucas and van Dijk (1995). See
Rodr(cid:237)guez (2004) for a comparison with other approaches.
4However,thisprocedureisnotrobusttodeparturesfromtheassumptionofnormality
in the errors. It is mentioned and discussed by Perron and Rodr(cid:237)guez (2003). See also
Burridge and Taylor (2006) for more evidence about this drawback and the correction
they propose based on the extreme value theory.
2
withtheADFstatisticandevenwiththeM-testsandMP testsusingGLS
T
detrended data as suggested by Elliott, Rothenberg and Stock (1996) and
Ng and Perron (2001), respectively. Only Uruguay and Venezuela show non
rejection of the null. However, when applying the ADF test augmented by
dummy variables related to the location of the addittive outliers identi(cid:133)ed
by the procedure (cid:28) , none of the countries reject the null hypothesis of a
d
unit root.
This note is organized as follows. Section 2 presents the model, dis-
cusses the issue of outlier detection and brie(cid:135)y revises the method proposed
by Perron and Rodr(cid:237)guez (2003). In section 3, I present the results from
the simulations. Section 4 shows the empirical application and Section 5
concludes.
2 The Issue of Outlier Detection and Testing for
Unit Roots with Additive Outliers
The issue of outlier detection in the unit root framework is the approach
taken by Perron and Rodr(cid:237)guez (2003) which is based on Vogelsang (1999)5.
The data-generating process entertained is of the following general form:
m
y = d + (cid:14) D(T ) +u (1)
t t j ao;j t t
j=1
X
where D(T ) = 1 if t = T and 0 otherwise. This permits the presence
ao;j t ao;j
of m additive outliers occurring at dates T (j = 1;:::;m): The term d
ao;j t
speci(cid:133)es the deterministic components. In most cases, d = (cid:22) if the series is
t
non-trending or d = (cid:22)+(cid:12)t if the series is trending. The noise function is
t
integrated of order one, i.e, u = u +v ; where v is a stationary process.
t t 1 t t
(cid:0)
While Perron and Rodr(cid:237)guez (2003), use an ARMA(p;q) for the process v ,
t
in this paper, we assume that v is an ARFIMA(p;d;q) process.
t
5Let t (T ) denote the t-statistic for testing (cid:14) = 0 in (1). Following Chen and Liu
(cid:14) ao
(1993),thepresenceofanadditiveoutliercanbetestedusing(cid:28) =sup t (T ):Assuming
b Tao j (cid:14) ao
that (cid:21) = Tao=T remains (cid:133)xed as T grows, Vogelsang (1999) showed tbhat as T ;
! 1
the limiting distribution of t (T ) is non-standard. More precisely, t (T ) H((cid:21)) =
(cid:14) ao (cid:14) ao )
W(cid:3)((cid:21))=( 01W(cid:3)(r)2dr)1=2; whbere W(cid:3)((cid:21)) denotes a demeaned standardbWiener process: If
(1) also includes a time trend, W(cid:3)((cid:21)) will denote a detrended Wiener process. Further-
R
more,fromthecontinuousmappingtheoremitfollowsthat,(cid:28) sup H((cid:21)) H(cid:3):This
) j j(cid:17)
(cid:21) (0;1)
distributionisinvariantwithrespecttoanynuisanceparameters,i2ncludingthecorrelation
structure of the noise function.
3
As shown in Perron and Rodr(cid:237)guez (2003), the original procedure of
Vogelsang (1999) has severe size distortions when applied in an iterative
fashiontosearchforadditiveoutliers. Thereasonforthisisthatthelimiting
distribution of the statistic is only valid in the (cid:133)rst step of the iterations as
speci(cid:133)edinTheorem1ofPerronandRodr(cid:237)guez(2003). Insubsequentsteps,
the asymptotic critical values used need to be modi(cid:133)ed.
Perron and Rodr(cid:237)guez (2003) have proposed a more powerful iterative
strategy using a test based on (cid:133)rst-di⁄erences of the data. Consider data
generated by (1) with d = (cid:22), and a single outlier occurring at date T with
t ao
magnitude (cid:14). Then,
(cid:1)y = (cid:14)[D(T ) D(T ) ]+v ; (2)
t ao t ao t 1 t
(cid:0) (cid:0)
whereD(T ) = 1,ift = T (0,otherwise)andD(T ) = 1;ift = T 1
ao t ao ao t 1 ao
(cid:0) (cid:0)
(0, otherwise). If the data are trending, a constant should be included.
In this case, we are interested in (cid:28) = sup t (T ) ; where t (T ) =
d Taoj (cid:14) ao j (cid:14) ao
^(cid:14)=(2(R^ (0) R^ (1)) and R (j) is the autocovariance function of v at delay
u u u t
j.6 (cid:0) b b
To detect for multiple outliers, we can follow a strategy similar to that
suggested by Vogelsang (1999), by dropping the observation labelled as an
outlier before proceeding to the next step. The important feature is that,
unlike for the case of the test based on levels, the limit distribution of the
test (cid:28) is the same as each step of the iterations when dealing with multiple
d
outliers. Thedisadvantageofthisprocedure, comparedtothatbasedonthe
level of the data, is that the limiting distribution depends on the speci(cid:133)c
distribution of the errors v , though not on the presence of serial correlation
t
andheteroskedasticity7. Thisproblemisexactlythesameasthatfor(cid:133)nding
outliers in stationary time series.
In this note, we analyze the empirical size of the ADF test corrected for
detected additive outliers when errors v are ARFIMA(p;d;q) process. It
t
is equivalent to using the t-statistic for testing that (cid:11) = 1 in the following
regression:
k+1 k
y = (cid:22)+(cid:11)y + (cid:14) D(T ) + d (cid:1)y +v ; (3)
t t 1 j ao;j t j i t i t
(cid:0) (cid:0) (cid:0)
j=0 i=0
X X
(2).6RT^uh(ejn),=R^T((cid:0)j1) isTta=(cid:0)c1jov^ntsv^ist(cid:0)tejnwtietshtiv^mtattheeolefaRst-(sjq)u.aresresidualsobtainedfromregression
u P u
7The dependence of the distribution or departures of the normality of v has been
t
mentioned by Perron and Rodr(cid:237)guez (2003). However, Burridge and Taylor (2006) deals
with this issue using extreme value theory.
4
where D(T ) = 1 if t = T and 0 otherwise, with T (j = 1;2;:::;m)
ao;j t ao;j ao;j
being the dates of the outliers identi(cid:133)ed using the statistic (cid:28) . Notice that
d
k +2 one-time dummy variables have to be included in (3) to remove all
possible in(cid:135)uences of the additive outliers.
3 Monte Carlo Results
In order to analyze the empirical size of the ADF statistic, we consider the
following experiment. Let y follow (1) where u = u +v (a unit root
t t t 1 t
process) and v is an ARFIMA(p;d;q) process, that (cid:0)is (cid:26)(L)(1 L)dv =
t t
(cid:0)
(cid:18)(L)(cid:15) , where (cid:15) is an i:i:d. N(0;1). More exactly, in one case we consider
t t
p = 1 ((cid:26)(L) = 1 (cid:26)L) and q = 0, that is (cid:26)(L)(1 L)dv = (cid:15) , while in the
t t
(cid:0) (cid:0)
other p = 0 case and q = 1 ((cid:18)(L) = 1+(cid:18)L); that is (1 L)dv = (cid:18)(L)(cid:15) .
t t
(cid:0)
The fractional parameter d [ 0:48 to 0:48] with a step of 0:12.
2 (cid:0)
EachTableandeachvalueof(cid:18) or(cid:26)presentthreerowsnamed(cid:147)without(cid:148),
(cid:147)with(cid:148), and (cid:147)total(cid:148). The row named (cid:147)without(cid:148) indicates the size of the
ADF statistic when no additive outliers has been found. The word (cid:147)with(cid:148)
indicates the size of the ADF statistic when additive outliers have been
identi(cid:133)ed. Therefore, the row entitle (cid:147)total(cid:148)means simply the sum of the
two previous rows. If size is correct we expect that this row should be close
to the nominal size of 5.0%.
In order to save space we present only selected Tables. Each experiment
is performed using 10,000 replications, nominal size at 5.0% and we use
tabulated critical values (Table 1 of Perron and Rodr(cid:237)guez (2003)) for T =
100 and T = 200. Other extensive Tables are available upon request. In
all Tables, the total iterative procedure is applied, that is, we search for all
outliers and procedure (cid:133)nish when no outliers are found. Two sets of Tables
are presented. In one case, the lag lenght of (3) is (cid:133)xed to be k = 1 while
in the other case, we use the procedure t-sig proposed by Campbell and
Perron (1991) for k [0;5]. In each Table, three cases are presented. In the
2
(cid:133)rst case, no outliers are in the process, that is, (cid:14) = 0 for i = 1;2;3;4. In
i
the second case, we consider medium sized additive outliers: (cid:14) = 5;3;2;2.
i
The (cid:133)nal case is for high sized addtive outliers, that is, (cid:14) = 10;5;5;5. In
i
summary, the design of the experiment follow closely Perron and Rodr(cid:237)guez
(2003). When there are outliers a maximum of four additive outliers is
considered and they are located at positions 0:20T, 0:40T, 0:60T and 0:80T,
respectively.
Table1showstheresultsforthecasewhereerrorsareARFIMA(0;d;0).
The (cid:133)rst set of columns are the case where no outliers are present in the
5
data. The other columns shows medium and high sized additive outliers,
respectively. TheresultsshowthatthesizeoftheADFisoversizedforevery
d < 0. More negative values of d imply more oversized ADF tests. This is
true for the case where no outliers are found and when they are present in
the data. For other values of d, the ADF is slighthly undersized but close
to the nominal size of 5%. Given these results, in what follows, we do not
consider cases where d < 0.
Tables2a-2cshowsizeoftheADFtestforARFIMA(0;d;1)errors, that
is when there exists moving average correlation. In order to save space, we
only show results for d = 0:00; 0:24; and 0:48. Table 2a indicates that ADF
test is oversized for (cid:18) = 0:8 and for (cid:18) = 0:4. Small distortion is also
(cid:0) (cid:0)
found for (cid:18) = 0:8. In all other cases of (cid:18) and for cases where there are or
not additive outliers, exact size is close to 5%. Table 2b shows the case
for d = 0:24. Again, ADF test is oversized for (cid:18) = 0:8 but distortions
(cid:0)
are smaller than before. In all other cases, size is better although slightly
undersized. When d = 0:48 (Table 2c), that is, when memory of the errors
is large the size of the ADF test is very close to the nominal size of 5%. It
is true when there are or not additive outliers and for both sample sizes. In
summary, there is some di¢ culties when (cid:18) goes to -1 but the performance
is better when d goes to 0.5.
Tables3a-3cshowsizeoftheADFtestforARFIMA(1;d;0)errors, that
is, when there exists autoregressive autocorrelation. Again, in order to save
space, we only show results for d = 0:00;0:24; and 0:48. Table 3a indicates
that ADF test has good exact size except for the case where (cid:26) = 0:8
(cid:0)
and when the process is contaminated for medium and high sized additive
outliers. It is worth to mention that the distortions are smaller compared to
the previous Tables and we observe that size is better when sample size is
higher. Table 3b shows the case for d = 0:24. In this case, the ADF test has
exact size close to the 5% although we observe small oversized results when
(cid:26) = 0:8. This results is more evident when d = 0:48 (Table 3c) even when
there is no outliers in the process. This problem is not (cid:133)xed when sample
size is higher. It is more evident for extreme values of (cid:26) ( 0:8 and 0:8).
(cid:0)
Previous results (undersized or oversized results) may be due to the se-
lection of the lag length which has been (cid:133)xed to unity. In order to observe if
this issue is important, we present similar simulations as in the previous Ta-
blesbutnowthelaglengthisselectedusingtheproceduret-sigassuggested
by Campbell and Perron (1991) considering a k [0;5]. Table 4 presents
2
resultsforARFIMA(0;d;0)errorsandford 0. ThemessageisthatADF
(cid:21)
test has exact size close to the 5%. In some cases, it presents slight smaller
exact size.
6