Banco Central de Reserva del Per57 Curso de Extensin Universitaria2010EconometraUnit Roots: A Selected SurveyGabriel RodrguezCentral Bank of PeruPontiicia Universidad Catlica del PerUniversity of OttawaUnit Roots: A Selected SurveyGabriel RodrguezBanco Central de Reserva del PerPonticia Universidad Catlica del PerUniversidad del PaccoUniversity of OttawaPonticia Universidad Catlica del PerViernes EconmicoLima, October 24 2008cGabriel Rodrguez, 20081 Motivation (1)-25-20-15-10-5051015250 500 750 1000Random Walk (No Drift)Japn: Exchange Rate Yen for Dollar95.0100.0105.0110.0115.0120.0125.0130.0135.0140.0Ago-01Feb-02Ago-02Feb-03Ago-03Feb-04Ago-04Feb-05Ago-05Feb-06Ago-06Feb-07Ago-07Feb-08Ago-082Motivation (2)0100200300400500250 500 750 1000Random Walk (with Drift)Federal Reserve Board' Industrial Production Index020406080100120Ene-21Ene-23Ene-25Ene-27Ene-29Ene-31Ene-33Ene-35Ene-37Ene-39Ene-41Ene-43Ene-45Ene-47Ene-49Ene-51Ene-53Ene-55Ene-57Ene-59Ene-61Ene-63Ene-65Ene-67Ene-69Ene-71Ene-73Ene-75Ene-77Ene-79Ene-81Ene-83Ene-85Ene-87Ene-89Ene-91Ene-93Ene-95Ene-97Ene-99Ene-01Ene-03Ene-05Ene-073Motivation (3)-50510152025303525 50 75 100Y_AR_0.5 Y_AR_0.97Y_AR_0.98 Y_AR_0.99Y_RANDOM_WALK42 Outline Basic References: Campbell and Perron (1991), Stock (1994), Phillipsand Xiao (1999), Maddala and Kim(2000), Haldrup and Jansson (2006) Data Generating Process Classical Unit Root Statistics Other Unit Root Statistics Recent Unit Root Statistics Some Issues on Unit Roots Structural Change and Unit Roots The Role of the Initial Condition and Unit Roots Covariates and Unit Roots Additive Outliers and Unit Roots Further Issues and/or Limitations of this Survey53 The Data Generating Process (DGP)t= dt + nt. t = 1. .... 1. (1)nt= cnt1 + t. (2) n0 = 0 (initial condition); t = P1i=0ijti with P1i=0i[i[ < and where jt is a martingaledierence sequence; t has a non-normalized spectral density at frequency zero given byo2= o2j(1)2. where o2j = limT!111P1t=11(j2t); Under H0, Functional Central Limit Theorem(FCLT) says: 112P[vT]t=1 t =o\(:). \(:) is a standard Wiener process. dt = 0.t. where .t is a set of deterministic components; H0 : c = 1; H : [c[ < 1; Local-to-unity framework: c = 1 + c,1. Used after.64 Classical Unit Root Statistics4.1 The Dickey-Fuller (DF) Statistic References: Dickey and Fuller (1979, 1981). The regression model ist = 0.t + ct1 + t. (3) Assume that t ~ i.i.d.(0. o2) and .t = O. Then, the asymptoticdistributions are1(b c 1) =R \(:)d\R \(:)2d: = 0.5[\(1)21]R \(:)2d:. (4)tb c =R \(:)d\[R \(:)2d:]12 = 0.5[\(1)21][R \(:)2d:]12. (5) If .t= 1 or .t= 1. t, then \is replaced by \i= \(:) R \2(220)12(:), for i = j. t. \iis the projection of \ onto thespace orthogonal to .. Asymptotic critical values at 5.0% are: -1.94, -2.86, -3.43 for .t = O,.t = 1 and .t = 1. t. respectively.74.2 The Parametric ADF Reference: Said and Dickey (1984). Now, assume that t is 1(0), as in Section 3. In general: t is an1`(j. ) process. Assuming, as before that .t = O, then1(b c 1) =0.5[\(1)21] +`R \(:)2d:. (6)tb c =o0.5[\(1)21] + `o[R \(:)2d:]12. (7)where ` = (o2o2),2o2. Distributions depend of nuissance parameters. The autocorrelation is corrected using the following autoregressiont = 0.t + c0t1 +IXi=1/iti + ct(8)where c0 = c 1. Then, H0 : c0 = 0. If we dene a detrended time series as e t= t b0.t, then (8) isequivalently written ase t = c0e t1 +IXi=1/ie ti + ct(9) If / . /3,1 0. then, 1(b c1) and tb c converge to the expressions(4) and (5). Important empirical application: Nelson and Plosser (1982).84.3 The Semi-Parametric 2b and 2t Statistics References: Phillips (1987, 1988), Phillips and Perron (1988). The coecient c is estimated from equation (3). Residuals b t are usedin constructing an estimator of o2. Therefore, the autocorrelation istaken into account in a non-parametric way:b o2= :2= 11Xb 2t + 211IXt=1n(t. /)TXt=t+1b tb tt. (10)n(t. /) = 1 t/ + 1. (11)9 Using (10) and (11), we have that2b c= 1(b c 1) 0.5(:2b :2)12PTt=22t1=0.5[\(1)21]R \(:)2d:. (12)2t= (b :: )tb c0.5(:2b :2)[12PTt=22t1]12=0.5[\(1)21][R \(:)2d:]12. (13)which are the same as in (4) and (5), respectively. Asymptotic critical values of 2b c at 5.0% are -8.0, -14.1, and -21.7 for.t = O, .t = 1 and .t = 1. t. respectively. Asymptotic critical values of 2t at 5.0% are -1.94, -2.86, -3.43 for .t =O, .t = 1 and .t = 1. t. respectively.104.4 The M-Statistics References: Stock (1999), Perron and Ng (1996). Denitions:`2c=11~ 2T :2212PTt=1 ~ 2t1. (14)`o1 = [12PTt=1 ~ 2t1:2]12. (15)`2t=11~ 2T :2[4:212PTt=1 ~ 2t1]12. (16)where: ~ t = t ^0.t, :2= :2cI,[1 ^/(1)]2, :2cI = PTt=I+1 ^ c2tI, ^/(1) =PI)=1^/), obtained from the autoregression (9). The limiting distributions of `2c ans `2t are the expressions (4) and(5), respectively. The `o1 = [R \(:)2d:]12. Asymptotically: `2t = (`2c) (`o1). Asymptotic critical values: see Stock (1999). Simulation Monte-Carlo evidence.11PJACTSc4/t'e rSi'EE14Rt4 1,4F4ijD 1200o`it?!oUo. oo9o135i2Ii oo5sJoOLif/6o60LCSOntcOL,19o.o103J10.0331o oS?031011LQ00q2cO725ouCflOLLzt=0.5=fizooz 0!o.019o oo $101; tOo.023o0i5=0.04A1 AJ /T9&/5 Recent Unit Root Statistics5.1 The 11GLS References: Elliott, Rothenberg and Stock (ERS, 1996), Ng and Perron(2001). Under local-to-unity framework: c = 1 + c,1. Then, 112n[Tv] =o\c(:). where \c(:) is an Ornstein-Uhlenbeck process. It bridges the gap between I(0) and I(1) asymptotics. If c ,1(b c 1) and tb c have I(0) distributions. If c +, 1(b c 1) and tb chave a Cauchy and Normal distributions, respectively. Particular characteristic: use of GLS detrended data with c = 1+c,1. Construction of GLS detrended Data: ct = [1. (1 c1)t]. t = 2. ..... 1. (17). ct = [.1,(1 c1).t]. t = 2. ...... 1. (18) Let ^ be the estimator that minimizes:o() = ( ct 0. ct )0( ct 0. ct ). (19)12 Detrended series: e t = t ^0G1S.t. All unit root statistics may be used with e t. For the ADF, see ERS(1996) and for the M-statistics, see Ng and Perron (2001). When .t = 1 and .t = 1. t, the limiting distributions are:11G1Sj=0.5[\c(1)21][R \c(:)2d:]12. (20)11G1St=0.5[\c,c(1)21][R \c,c(:)2d:]12. (21)where \c,c(:. c) = \c(:) :/, / = `\c(1) + 3(1 `)R :\c(:)d:, ` =(1 c),(1 c + c2,3). Asymptotic critical values: see ERS (1996), Ng and Perron (2001).135.2 A Feasible Point Optimal Test References: Dufour and King (1991), ERS (1996). This test is denoted by 1G1STand dened by:1G1ST(c.c) = o( c) co(1):2. (22) where o( c) and o(1) are the sums of squared errors from GLS regres-sions with c =c and c = 1, respectively. Limiting distributions:1G1ST,j (c.c) =c2Z10\c(:)2d: c\c(1)2. (23)1G1ST,t(c.c) =c2Z10\c,c(:. c)2d: + (1 c)\c,c(1. c)2. (24)for .t = 1, and .t = 1. t, respectively. Selection ofc. Asymptotic critical values: see ERS (1996), Ng and Perron (2001).146 Some Issues on Unit Root Tests6.1 The Asymptotic Gaussian Power Envelope There is no uniform most powerful (UMP) or uniform most powerfulinvariant (UMPI) statistic in unit root framework. With c = 1 +c,1, derivation of the asymptotic Gaussian power enve-lope. Power envelope allows to judge between dierent alternative statistics. The asymptotic Gaussian power envelope is dened by::

(c) = Pr[H1GLST(c. c) < /1GLST

(c)]. (25)where /1T

(c) is such thatPr[H1GLST(0. c) < /1GLST

(c)] = . (26)with the size of the test. Selection ofc (-7.0 for .t = 1 and -13.5 for .t = 1. t).156.2 Asymptotic Power Functions The asymptotic power functions of the tests are dened by::

J(c. c) = Pr[HJGLS(c.c) < /JGLS( c)].where J() = `2c, `o1, `2t, and 11, and the constant /JGLS( c)is such that Pr[HJGLS(0.c) < /JGLS( c)] = , the size of the tests.164S 2 & 2t 24-corewLIte,ogisu:828 32OoaO`oE',.owEitHoO!oooof.n:,c 36.3 Selection of the Lag length Information Criteria: AIC, BIC/oic= arg minfIg log(:2cI) + 2/1 . (27)/bic= arg minfIg log(:2cI) + log(1)/1. (28) Recursive t-sig method Modied Information Criteria: MAIC, MBIC (Ng and Perron, 2001):/nic = arg minfIg log(:2cI) + CT[^ tT(/) + /]1(29)where^ tT(/) = (:2cI)1b c20TXt=1e 2t1. (30) The MAIC uses CT = 2 and the MBIC uses CT = log(1). Ng and Perron (2001), based on theoretical considerations and simula-tions, recommended MAIC. The advantage of the MIC is that it takes into account the possibledependence of b c0 on /.176.4 Summary of Monte-Carlo Evidence All asymptotic valid tests exhibit nite-sample size distortions for mod-els close to 1(0) model. Importance of data dependent methods to select lag length. Presence of non-normality or conditional heteroskedasticity increasessize distortions. Including additional trend terms reduce the power of the unit root testif the trends are unnecessary. Span is important, not the frequency. Power of the unit root depends of the initial condition n0. If trend is underspecied, unit root tests and estimators are inconsis-tent.18Sinl 77 bUiJY03 / 0O0 `7 oto &i/o+00oOOTI5791QVLiGOQ&'QZ5C060LOOjQcli vYW03QOOjObooo Y o01! ?vh&QooVocoECOhOzoooGoi wny5go080Yoo800oo00/OQZ2?09O0o o9001bAOozoA/ooQ001?`O-QY29j?JJIYJ7U.SiLLS`4 viiouWd5r1e.1/-9 =` *oo =17SV VV3fj7&Qj/2l vv ?7 Structural Change and Unit Root Tests7.1 Introduction References: Perron (1989), Christiano (1992), Banerjee et al.(1992),Zivot and Andrews (1992), Perron (1997), Perron and Rodrguez (2003a). Basic idea: misspecication of the trend function is responsible for thenonrejection of the null hypothesis of a unit root in Nelson and Plosser(1982). Models (I, II, III):.t= 1. 1(t11). t..t= 1. t. 1(t11)(t 11)..t= 1. 1(t11). t. 1(t11)(t 11).where 1(.) is the indicator function and 11 is the break point. Assumethat 11 = o1, for some o (0. 1). Perron (1989) Christiano (1992) Zivot and Andrews( 1992)196276747268664626 -----------`---90'3 &3Note. flw broLen ttai&hihite ji a taedtrendDL-O:f :l929and DL it:'929.Kl Lajr.}n .f Nott:nt1 V.Lgc.75-7'-i-/7372`16995C .955 r65 9.' 975.e4eesNote Tse br,kcn `L:a;gZ; l.nc ji al ,J trtnd `o' 01.5 of sic Sim. , --- DT;crcD70:!973:i.o, lE'-. Tifr>H!.!-*-tu Kj 2.-i_.gar:lt o -,asr Qualteri kca1/A//25Ir .SCE80 7228 3r; 34O "t,. 973.;; lict:,,i, .,::a.ht.;ocis Sic tren. 01.5 o! cc 5 tt- - Oh,r;cL%D7Or7ctaruL rL-.DT-I:r,,:97t