UNIT ROOTS Herman J. Bierens 1 Pennsylvania State University (October 30, 2007)

Transcription

1 UNIT ROOTS Herma J. Bieres Pesylvaia Sae Uiversiy (Ocober 30, 2007). Iroducio I his chaper I will explai he wo mos frequely applied ypes of ui roo ess, amely he Augmeed Dickey-Fuller ess [see Fuller (996), Dickey ad Fuller (979, 98)], ad he Phillips-Perro ess [see Phillips (987) ad Phillips ad Perro (988)]. The saisics ad ecoomerics levels required for udersadig he maerial below are Hogg ad Craig (978) or a similar level for saisics, ad Gree (997) or a similar level for ecoomerics. The fucioal ceral limi heorem [see Billigsley (968)], which plays a key-role i he derivaios ivolved, will be explaied i his chaper by showig is aalogy wih he cocep of covergece i disribuio of radom variables, ad by cofiig he discussio o Gaussia ui roo processes. This chaper is o a review of he vas lieraure o ui roos. Such a review would eail a log lis of descripios of he may differe recipes for ui roo esig proposed i he lieraure, ad would leave o space for moivaio, le aloe proofs. I have chose for deph raher ha breadh, by focusig o he mos iflueial papers o ui roo esig, ad discussig hem i deail, wihou assumig ha he reader has ay previous kowledge abou his opic. As a iroducio of he cocep of a ui roo ad is cosequeces, cosider he Gaussia AR() process y ' β 0 % β y & % u, or equivalely ( & β L)y = β 0 + u, where L is he lag operaor: Ly = y -, ad he u s are i.i.d. N(0,σ 2 ). The lag polyomial & β L has roo equal o /β. If *β * <, he by backwards subsiuio we ca wrie y ' β 0 /(&β ) % ' 4 j'0 βj u &j, so ha y is sricly saioary, i.e., for arbirary aural umbers m < m 2 <...< m k- he joi disribuio of y,y &m,y &m2,...,y &mk& does o deped o, bu oly o he lags or leads m, m 2,..,m k-. Moreover, he disribuio of y, > 0, codiioal o y 0, y -, y -2,..., he coverges o he margial disribuio This is a slighly revised versio of a chaper i Badi Balagi (Ed.), Compaio i Theoreical Ecoomerics, Blackwell Publishers. The useful commes of hree referees are graefully ackowledged.

2 of y if 64. I oher words, y has a vaishig memory: y becomes idepede of is pas, y 0, y -, y -2,..., if 64. If β =, so ha he lag polyomial & β L has a ui roo, he y is called a ui roo process. I his case he AR() process uder review becomes y = y u, which by backwards subsiuio yields for > 0, y ' y 0 % β 0 % ' j' u j. Thus ow he disribuio of y, > 0, codiioal o y 0, y -, y -2,..., is N(y 0 % β 0, σ 2 ), shock i y 0 will have a persise effec o y. The former iercep parameer of he ui roo process ivolved. reasos: β 0 so ha y has o loger a vaishig memory: a ow becomes he drif I is impora o disiguish saioary processes from ui roo processes, for he followig. Regressios ivolvig ui roo processes may give spurious resuls. If y ad x are muually idepede ui roo processes, i.e. y is idepede of x -j for all ad j, he he OLS regressio of y o x for =,..,, wih or wihou a iercep, will yield a sigifica esimae of he slope parameer if is large: he absolue value of he -value of he slope coverges i probabiliy o 4 if 6 4. We he migh coclude ha y depeds o x, while i realiy he y 's are idepede of he x 's. This pheomeo is called spurious regressio. 2 Oe should herefore be very cauious whe coducig sadard ecoomeric aalysis usig ime series. If he ime series ivolved are ui roo processes, aive applicaio of regressio aalysis may yield osese resuls. 2. For wo or more ui roo processes here may exis liear combiaios which are saioary, ad hese liear combiaios may be ierpreed as log-ru relaioships. This pheomeo is called coiegraio 3, ad plays a domia role i moder empirical macroecoomic research. β 0 2 See he chaper o spurious regressio i Badi Balagi (Ed.), Compaio i Theoreical Ecoomerics, Blackwell Publishers. This pheomeo ca easily be demosraed by usig my free sofware package EasyReg, which is dowloadable from websie hp://eco.la.psu.edu/~hbieres/easyreg.htm (Click o "Tools", ad he o "Teachig ools"). 3 See he chaper o coiegraio i Badi Balagi (Ed.), Compaio i Theoreical Ecoomerics, Blackwell Publishers.. 2

3 3. Tess of parameer resricios i (auo)regressios ivolvig ui roo processes have i geeral differe ull disribuios ha i he case of saioary processes. I paricular, if oe would es he ull hypohesis β = i he above AR() model usig he usual -es, he ull disribuio ivolved is o-ormal. Therefore, aive applicaio of classical iferece may give icorrecly resuls. We will demosrae he laer firs, ad i he process derive he Dickey-Fuller es [see Fuller (996), Dickey ad Fuller (979, 98)], by rewriig he AR() model as say, esimaig he parameer y ' y &y & ' β 0 % (β &)y & % u ' α 0 % α y & % u, () α by OLS o he basis of observaios y 0,y,...,y, ad he esig he ui roo hypohesis α = 0 agais he saioariy hypohesis -2 < α < 0, usig he -value of α. I Secio 2 we cosider he case where saioariy hypohesis. I Secio 3 we cosider he case where bu o uder he saioariy hypohesis. α 0 = 0 uder boh he ui roo hypohesis ad he α 0 = 0 uder he ui roo hypohesis The assumpio ha he error process u is idepede is quie urealisic for macroecoomic ime series. Therefore, i Secios 4 ad 5 his assumpio will be relaxed, ad wo ypes of appropriae ui roo ess will be discussed: he Augmeed Dickey-Fuller (ADF) ess, ad he Phillips-Perro (PP) ess. I Secio 6 we cosider he ui roo wih drif case, ad we discuss he ADF ad PP ess of he ui roo wih drif hypohesis, agais he aleraive of red saioariy. Fially, Secio 7 coais some cocludig remarks. 2. The Gaussia AR() case wihou iercep: Par 2. Iroducio Cosider he AR() model wihou iercep, rewrie as 4 y ' α 0 y & % u, where u is i.i.d. N(0,σ 2 ), (2) ad y is observed for =,2,..,. For coveiece I will assume ha 4 The reaso for chagig he subscrip of α from i () o 0 is o idicae he umber of oher parameers a he righ-had side of he equaio. See also (39). 3

4 y ' 0 for # 0. (3) This assumpio is, of course, quie urealisic, bu is made for he sake of rasparecy of he argume, ad will appear o be ioce. The OLS esimaor of α 0 is: ˆα 0 ' j y & y j y 2 & ' α 0 % j y & u j y 2 &. (4) If -2 < α 0 < 0, so ha y is saioary, he i is a sadard exercise o verify ha (ˆα 0 &α 0 ) 6 N(0,&(%α 0 ) 2 ) i disribuio. O he oher had, if α 0 = 0, so ha y is a ui roo process, his resul reads: ˆα 0 6 N(0,0) i disr., hece plim 64 ˆα 0 = 0. However, we show ow ha a much sroger resul holds, amely ha ˆρ 0 / ˆα 0 coverges i disribuio, bu he limiig disribuio ivolved is o-ormal. Thus, he presece of a ui roo is acually advaageous for he efficiecy of he OLS esimaor ˆα 0. The mai problem is ha he -es of he ull hypohesis ha α 0 = 0 has o loger a sadard ormal asympoic ull disribuio, so ha we cao es for a ui roo usig sadard mehods. The same applies o more geeral ui roo processes. I he ui roo case uder review we have y ' y & % u ' y 0 % ' j' u j ' ' j' u j for > 0, where he las equaliy ivolved is due o assumpio (3). Deoig ad ˆσ 2 ' (/)' u 2, i follows ha S ' 0 for # 0, S ' j j' u j for $. (5) j u y & ' 2 j (u %y & ) 2 & y 2 & & u 2 ' 2 j y 2 & j y 2 & & j u 2 (6) ' 2 ( y 2 / & y 2 0 / & ˆσ2 ) ' 2 ( S 2 / & ˆσ2 ), ad similarly, 2j y 2 & ' j 4 (S & / ) 2. (7)

5 Nex, le W (x) ' S [x] /(σ ) for x 0 [0,], (8) where [z] meas rucaio o he eares ieger # z. The we have 5 : j u y & ' 2 (σ2 W () 2 & ˆσ 2 ) ' 2 (σ2 W () 2 & σ 2 & O p (/ )) ' σ 2 2 (W ()2 & ) % o p (), (9) ad 2j y 2 & ' j σ 2 W ((&)/) 2 ' σ 2 m W (x)2 dx, (0) where he iegral i (0) ad below, uless oherwise idicaed, is ake over he ui ierval [0,]. The las equaliy i (9) follows from he law of large umbers, by which las equaliy i (0) follows from he fac ha for ay power m, m W (x)m dx ' W m (x) m dx ' W m (z/) m dz ' j ' 0 %m/2j m & 0 (S [z] /σ ) m dz ' Moreover, observe from (), wih m =, ha %m/2j *W (x)dx & ˆσ 2 ' σ 2 % O p (/ ). The W m (z/) m dz (S & /σ ) m. () is a liear combiaio of i.i.d. sadard ormal radom variables, ad herefore ormal iself, wih zero mea ad variace E W m (x)dx 2 mi [x],[y] ' E W mm (x)w (y) dxdy ' dxdy 6 mi(x,y)dxdy ' mm mm 3. (2) Thus, *W (x)dx 6 N(0,/3) i disribuio. Sice *W (x) 2 dx $ (*W (x)dx ) 2, i follows herefore ha *W (x) 2 dx is bouded away from zero: 5 Recall ha he oaio o p (a ), wih a a deermiisic sequece, sads for a sequece of radom variables or vecors x, say, such ha plim 64 x /a = 0, ad ha he oaio O p (a ) sads for a sequece of radom variables or vecors x such ha x /a is sochasically bouded: œg 0 (0,) M 0 (0,4): sup $ P(*x /a * > M) <g. Also, recall ha covergece i disribuio implies sochasic boudedess. 5

6 Combiig (9), (0), ad (3), we ow have: m W (x)2 dx & ' O p (). (3) ˆρ 0 / ˆα 0 ' (/)' u y & (/ 2 )' y 2 & ' (/2)(W ()2 & ) % o p () *W (x) 2 dx ' 2 W () 2 & *W (x) 2 dx % o p (). (4) This resul does o deped o assumpio (3). 2.2 Weak covergece of radom fucios I order o esablish he limiig disribuio of (4), ad oher asympoic resuls, we eed o exed he well-kow cocep of covergece i disribuio of radom variables o covergece i disribuio of a sequece of radom fucios. Recall ha for radom variables X, X, X 6 X i disribuio if he disribuio fucio F (x) of X coverges poiwise o he disribuio fucio F(x) of X i he coiuiy pois of F(x). Moreover, recall ha disribuio fucios are uiquely associaed o probabiliy measures o he Borel ses 6, i.e., here exiss oe ad oly oe probabiliy measure F (B) o he Borel ses B such ha F (x) = F ((-4,x]), ad similarly, F(x) is uiquely associaed o a probabiliy measure F o he Borel ses, such ha F(x) = F((-4,x]). The saeme X 6 X i disribuio ca ow be expressed i erms of he probabiliy measures F ad F: F (B) 6 F(B) for all Borel ses B wih boudary δb saisfyig F(δB) = 0. I order o exed he laer o radom fucios, we eed o defie Borel ses of fucios. For our purpose i suffices o defie Borel ses of coiuous fucios o [0,]. Le C[0,] be he se of all coiuous fucios o he ui ierval [0,]. Defie he disace bewee wo fucios f ad g i C[0,] by he sup-orm: ρ(f,g) ' sup 0#x# *f(x)&g(x)*. Edowed wih his orm, he se 6 The Borel ses i ú are he members of he smalles σ-algebra coaiig he collecio Œ, say, of all half-ope iervals (-4,x], x 0ú. Equivalely, we may also defie he Borel ses as he members of he smalles σ-algebra coaiig he collecio of ope subses of ú. A collecio ö of subses of a se Ω is called a σ-algebra if he followig hree codiios hold: Ω0ö; A0ö implies ha is compleme also belogs o ö: Ω\A0ö (hece, he empy se i belogs o ö); A 0ö, =,2,3,.., implies ^4 ' A 0ö. The smalles σ-algebra coaiig a collecio Œ of ses is he iersecio of all σ-algebras coaiig he collecio Œ. 6

7 C[0,] becomes a meric space, for which we ca defie ope subses, similarly o he cocep of a ope subse of ú: A se B i C[0,] is ope if for each fucio f i B we ca fid a g > 0 such ha {g 0 C[0,]: ρ(g,f) <g} d B. Now he smalles σ-algebra of subses of C[0,] coaiig he collecio of all ope subses of C[0,] is jus he collecio of Borel ses of fucios i C[0,]. A radom eleme of C[0,] is a radom fucio W(x), say, o [0,], which is coiuous wih probabiliy. For such a radom eleme W, say, we ca defie a probabiliy measure F o he Borel ses B i C[0,] by F(B) = P(W 0 B). Now a sequece W * of radom elemes of C[0,], wih correspodig probabiliy measures F, is said o coverge weakly o a radom eleme W of C[0,], wih correspodig probabiliy measure F, if for each Borel se B i C[0,] wih boudary δb saisfyig F(δB) = 0, we have F (B) 6 F(B). This is usually deoed by: W * Y W (o [0,]). Thus, weak covergece is he exesio o radom fucios of he cocep of covergece i disribuio. I order o verify ha W * Y W o [0,], we have o verify wo codiios. See Billigsley (968). Firs, we have o verify ha he fiie disribuios of W * coverge o he correspodig fiie disribuios of W, i.e., for arbirary pois x,..,x m i [0,], (W * (x ),...,W * (x m )) Y (W(x ),...,W(x m )) i disribuio. Secod, we have o verify ha W * is igh. Tighess is he exesio of he cocep of sochasic boudedess o radom fucios: for each g i (0,) here exiss a compac (Borel) se K i C[0,] such ha F (K) > -g for =,2.,... Sice covergece i disribuio implies sochasic boudedess, we cao have covergece i disribuio wihou sochasic boudedess, ad he same applies o weak covergece: ighess is a ecessary codiio for weak covergece. As is well-kow, if X 6 X i disribuio, ad Φ is a coiuous mappig from he suppor of X io a Euclidea space, he by Slusky's heorem, Φ(X ) 6 Φ(X) i disribuio. A similar resul holds for weak covergece, which is kow as he coiuous mappig heorem: If Φ is a coiuous mappig from C[0,] io a Euclidea space, he W * Y W implies Φ(W * ) 6 Φ(W) i disribuio. For example, he iegral Φ(f) ' *f(x) 2 dx wih f 0 C[0,] is a coiuous mappig from C[0,] io he real lie, hece W * Y W implies ha *W ( (x)2 dx 6 *W(x) 2 dx i disribuio. The radom fucio W defied by (8) is a sep fucio o [0,], ad herefore o a radom eleme of C[0,]. However, he seps ivolved ca be smoohed by piecewise liear ierpolaio, * yieldig a radom eleme W of C[0,] such ha sup 0#x# *W ( (x) & W (x)* = o p (). The fiie 7

8 disribuios of W * are herefore asympoically he same as he fiie disribuios of W. I order o aalyze he laer, redefie W as W (x) ' [x] j e for x 0 [ &,], W (x) ' 0 for x 0 [0, & ), e is i.i.d. N(0,). (5) (Thus, e = u /σ), ad le The W ( (x) ' W & ' W (x) % x & (&) e % x & (&) W for x 0 sup *W ( (x) & W (x)* # max ## *e * 0#x# & W & &,, ',...,, W ( (0) ' 0. (6) ' o p (). (7) The laer coclusio is o oo hard a exercise. 7 I is easy o verify ha for fixed 0 # x < y # we have W (x) W (y) & W (x) ' ' [x] e ' [y] '[x]% e - N 2 0 0, [x] 0 0 [y]&[x] (8) 6 W(x) W(y)&W(x) i disr., where W(x) is a radom fucio o [0,] such ha for 0 # x < y #, W(x) W(y) & W(x) - N 2 0 0, x 0 0 y&x. (9) 7 Uder he assumpio ha e is i.i.d. N(0,), for arbirary g > 0. P max ## *e *#g ' & 2 m 4 8 g exp(&x 2 /2) dx 2π 6

9 This radom fucio W(x) is called a sadard Wieer process, or Browia moio. Similarly, for arbirary fixed x,y i [0,], W (x) W (y) 6 W(x) W(y) 0 - N 2 0, x mi(x,y) mi(x,y) y i disr. (20) * ad i follows from (7) ha he same applies o W *. Therefore, he fiie disribuios of W * coverge o he correspodig fiie disribuios of W. Also, i ca be show ha W is igh [see * Billigsley (968)]. Hece, W Y W, ad by he coiuous mappig heorem, (W ( (), m W ( (x)dx, m W ( (x)2 dx, xw ( m (x)dx)t 6 (W(), W(x)dx, W(x) 2 dx, xw(x)dx) T (2) m m m i disr. This resul, ogeher wih (7), implies ha: LEMMA. For W defied by (5), (W (),*W (x)dx,*w (x) 2 dx,*xw (x)dx) T coverges joily i disribuio o (W(),*W(x)dx,*W(x) 2 dx,*xw(x)dx) T. 2.3 Asympoic ull disribuios Usig Lemma, i follows ow sraighforwardly from (4) ha: ˆρ 0 / ˆα 0 6 ρ 0 / 2 W() 2 & m W(x)2 dx i disr. (22) The desiy 8 of he disribuio of is displayed i Figure, which clearly shows ha he disribuio ivolved is o-ormal ad asymmeric, wih a fa lef ail. ρ 0 8 This desiy is acually a kerel esimae of he desiy of ˆρ 0 o he basis of 0,000 replicaios of a Gaussia radom walk y = y - + e, = 0,,...,000, y = 0 for < 0. The kerel ivolved is he sadard ormal desiy, ad he badwidh h = c.s0000 -/5, where s is he sample sadard error, ad c =. The scale facor c has bee chose by experimeig wih various values. The value c = is abou he smalles oe for which he kerel esimae remais a smooh curve; for smaller values of c he kerel esimae becomes wobbly. The desiies of ρ, τ, ρ 2, ad τ 2 i Figures 2-6 have bee cosruced i he same way, wih c =. 9

10 Figure : Desiy of ρ 0 Also he limiig disribuio of he usual -es saisic of he ull hypohesis = 0 is oα 0 ormal. Firs, observe ha due o (0), (22), ad Lemma, he residual sum of squares (RSS) of he regressio (2) uder he ui roo hypohesis is: RSS ' j ( y & ˆα 0 y & ) 2 ' j u 2 & (ˆα 0 ) 2 (/ 2 ) j y 2 & ' j u 2 % O p (). (23) Hece RSS/(&) ' σ 2 + O p (/). Therefore, similarly o (4) ad (22), he Dickey-Fuller -saisic ˆτ 0 ivolved saisfies: ˆτ 0 / ˆα 0 (/ 2 )' y 2 & RSS/(&) ' (W ()2 & )/2 *W (x) 2 dx % o p () 6 τ 0 / (W()2 & )/2 *W(x) 2 dx i disr. (24) Noe ha he ui roo ess based o he saisics ˆρ 0 / ˆα 0 ad ˆτ 0 are lef-sided: uder he aleraive of saioariy, -2 < α 0 < 0, we have plim 64ˆα 0 ' α 0 <0, hece ˆρ 0 6 &4 i probabiliy a rae, ad ˆτ 0 6 &4 i probabiliy a rae. The o-ormaliy of he limiig disribuios ρ 0 ad τ 0 is o problem, hough, as log oe is aware of i. The disribuios ivolved are free of uisace parameers, ad asympoic criical values of he ui roo ess ˆρ 0 ad ˆτ 0 ca easily be abulaed, usig Moe Carlo simulaio. I paricular, 0

11 P(τ 0 # &.95) ' 0.05, P(τ 0 # &.62) ' 0.0, (25) (see Fuller 996, p. 642), whereas for a sadard ormal radom variable e, P(e # &.64) ' 0.05, P(e # &.28) ' 0.0 (26) Figure 2: Desiy of τ 0 compared wih he sadard ormal desiy (dashed curve) desiy of I Figure 2 he desiy of τ 0 τ 0 is compared wih he sadard ormal desiy. We see ha he is shifed o lef of he sadard ormal desiy, which causes he differece bewee (25) ad (26). Usig he lef-sided sadard ormal es would resul i a ype error of abou wice he size: compare (26) wih P(τ 0 # &.64). 0.09, P(τ 0 # &.28). 0.8 (27) 3. The Gaussia AR() case wih iercep uder he aleraive of saioariy If uder he saioariy hypohesis he AR() process has a iercep, bu o uder he ui roo hypohesis, he AR() model ha covers boh he ull ad he aleraive is: y ' α 0 % α y & % u, where α 0 '&cα. (28) If -2 < α < 0, he he process y is saioary aroud he cosa c:

12 y '&cα % (%α )y & % u ' j 4 j'0 (%α ) j (&cα % u &j ) ' c % j 4 j'0 (%α ) j u &j, (29) hece E(y 2 ) ' c 2 % (&(%α ) 2 ) & σ 2, E(y y & ) ' c 2 % (%α )(&(%α ) 2 ) & σ 2, ad he OLS esimaor (4) of α 0 i model (2) saisfies plim 64 ˆα 0 ' E(y y & ) E(y 2 & ) & ' α % (c/σ) 2 (&(%α ) 2 ), (30) which approaches zero if c 2 /σ Therefore, he power of he es ˆρ 0 will be low if he variace of u is small relaive o [E(y )] 2. The same applies o he -es. We should herefore use he OLS ˆτ 0 esimaor of α ad he correspodig -value i he regressio of y o y - wih iercep. Deoig ȳ & ' (/)' y &, ū ' (/)' u, he OLS esimaor of α is: ˆα ' α % ' u y & & ū ȳ &. ' y 2 & & ȳ (3) 2 & Sice by (8), ū ' σw (), ad uder he ull hypohesis α = 0 ad he maiaied hypohesis (3), ȳ & / ' j S & ' σ W m (x)dx, (32) where he las equaliy follows from () wih m =, i follows from Lemma, similarly o (4) ad (22) ha ˆρ / ˆα ' (/2)(W ()2 & ) & W ()*W (x)dx *W (x) 2 dx & (*W (x)dx ) 2 % o p () 6 ρ / (/2)(W()2 & ) & W()*W(x)dx *W(x) 2 dx & (*W(x)dx ) 2 i disr. (33) The desiy of ρ is displayed i Figure 3. Comparig Figures ad 3, we see ha he desiy of ρ is farher lef of zero ha he desiy of, ad has a faer lef ail. ρ 0 2

13 Figure 3: Desiy of ρ As o he -value ˆτ of α i his case, i follows similarly o (24) ad (33) ha uder he ui roo hypohesis, ˆτ 6 τ / (/2)(W()2 & ) & W()*W(x)dx *W(x) 2 dx & (*W(x)dx ) 2 i disr. (34) Agai, he resuls (33) ad (34) do o hige o assumpio (3). of The disribuio of τ is eve farher away from he ormal disribuio ha he disribuio τ 0, as follows from compariso of (26) wih P(τ # &2.86) ' 0.05, P(τ # &2.57) ' 0. (35) See agai Fuller (996, p. 642). This is corroboraed by Figure 4, where he desiy of compared wih he sadard ormal desiy. τ is 3

14 Figure 4: Desiy of τ compared wih he sadard ormal desiy (dashed curve) We see ha he desiy of τ is shifed eve more o he lef of he sadard ormal desiy ha i Figure 2, hece he lef-sided sadard ormal es would resul i a dramaically higher ype error ha i he case wihou a iercep: compare P(τ # &.64). 0.46, P(τ # &.28) (36) wih (26) ad (27). 4. Geeral AR processes wih a ui roo, ad he Augmeed Dickey-Fuller es The assumpio made i Secios 2 ad 3 ha he daa-geeraig process is a AR() process, is o very realisic for macroecoomic ime series, because eve afer differecig mos of hese ime series will sill display a fair amou of depedece. Therefore we ow cosider a AR(p) process: y ' β 0 % ' p j' β j y &j % u, u - i.i.d. N(0,σ 2 ) (37) By recursively replacig y &j by y &j % y &&j for j = 0,,..,p-, his model ca be wrie as where y ' α 0 % ' p& j' α j y &j % α p y &p % u, u - i.i.d. N(0,σ 2 ), (38) α 0 ' β 0, α j ' ' j i' β i &, j ',..,p.. Aleraively ad equivalely, by recursively replacig y &p%j by y &p%j% & y &p%j% for j = 0,,..,p-, model (37) ca also be wrie as y ' α 0 % ' p& j' α j y &j % α p y & % u, u - i.i.d. N(0,σ 2 ), (39) 4

15 where ow α j '&' j i' β i, j ',..,p&, α p ' 'p i' β i &. If he AP(p) process (37) has a ui roo, he clearly = 0 i (38) ad (39). If he process (37) is saioary, i.e., all he roos of he lag polyomial - ' p j' β j L j lie ouside he complex ui circle, he α = - < 0 i (38) ad (39). 9 p ' p j' β j The ui roo hypohesis ca herefore be esed by esig he ull hypohesis α p = 0 agais he aleraive hypohesis α p < 0, usig he -value ˆ p of α p i model (38) or model (39). This es is kow as he Augmeed Dickey-Fuller (ADF) ess. We will show ow for he case p = 2, wih iercep uder he aleraive, i.e., y ' α 0 %α y & % α 2 y & % u, u - i.i.d. N(0,σ 2 ), ',...,. (40) ha uder he ui roo (wihou drif 0 ) hypohesis he limiig disribuio of α p ˆα p is proporioal o he limiig disribuio i (33), ad he limiig disribuio of ˆ p is he same as i (34). Uder he ui roo hypohesis, i.e., α 0 ' α 2 ' 0, *α * <, we have y ' α y & % u ' (&α L) & u ' (&α ) & u % [(&α L) & &(&α ) & ]u ' (&α ) & u & α (&α ) & (&α L) & (&L)u ' (&α ) & u % v & v &, (4) say, where v '&α (&α ) & (&α L) & u '&α (&α ) & ' 4 j'0 αj u &j is a saioary process. Hece: y / ' y 0 / % v / & v 0 / % (&α ) & (/ )' j' u j ' y 0 / % v / & v 0 / % σ(&α ) & W (/) (42) ad herefore, similarly o (6), (7), ad (32), i follows ha 9 To see his, wrie - ' p j' β j L j = Π p j' (&ρ j L), so ha - 'p j' β j = Π p j' (&ρ j ), where he /ρ j s are he roos of he lag polyomial ivolved. If roo /ρ j is real valued, he he saioariy codiio implies - < ρ j <, so ha - ρ j > 0. If some roos are complex-valued, he hese roos come i complex-cojugae pairs, say /ρ = a+i.b ad /ρ 2 = a-i.b, hece (&ρ )(&ρ 2 ) = (/ρ &)(/ρ 2 &)ρ ρ 2 ' ((a&) 2 %b 2 )/(a 2 %b 2 ) > 0. 0 I he sequel we shall suppress he saeme "wihou drif". A ui roo process is from ow o by defaul a ui roo wihou drif process, excep if oherwise idicaed. 5

16 (/) j y & / ' σ(&α ) & m W (x)dx % o p (), (43) (/ 2 ) j y 2 & ' σ2 (&α ) &2 m W (x)2 dx % o p (), (44) (/) j u y & ' (/) j u & (&α ) & j u j % y 0 % v & & v 0 j' ' (&α ) & (/) j & u j u j % (y 0 &v 0 )(/) j u % (/) j u v & j' (45) Moreover, ' (&α )& σ 2 2 W () 2 & % o p () ad plim(/) j y & ' E( y ) ' 0, 64 plim(/) j ( y & ) 2 ' E( y ) 2 ' σ 2 /(&α 2 ) (46) 64 (/) j y & y & ' (/) j ( y & ) 2 % (/) j y &2 y & ' (/) j ( y & ) 2 % 2 (/) j y 2 & & (/) j y 2 &2 & (/) j ( y & ) 2 (47) ' 2 (/) j hece ( y & ) 2 % y 2 & / & y 2 & / ' 2 σ2 /(&α 2 ) % σ2 (&α ) &2 W () 2 % o p () (/) j y & y & / ' O p (/ ). (48) Nex, le ˆα ' (ˆα 0,ˆα,ˆα 2 ) T be he OLS esimaor of α ' (α 0,α,α 2 ) T. Uder he ui roo hypohesis we have ˆα 0 (ˆα &α ) ' D ˆΣ & ˆΣ xx xu ' D & ˆα 2 ˆΣ xx D & & D & ˆΣ xu, (49) 6

17 where D ' , (50) ˆΣ xx ' (/)' y & (/)' y & (/)' y & (/)' ( y & )2 (/)' y & y & (/)' y & (/)' y & y & (/)' y 2 &, (5) ad ˆΣ xu ' (/)' u (/)' & u y &. (52) (/)' u y & I follows from (43) hrough (48) ha D & ˆΣ xx D & ' 0 σ(&α ) & *W (x)dx 0 σ 2 /(&α 2 ) 0 σ(&α ) & *W (x)dx 0 σ 2 (&α ) &2 *W (x) 2 dx % o p (), (53) hece, usig he easy equaliy 0 a 0 b 0 a 0 c & ' c&a 2 c 0 &a 0 b & (c&a 2 ) 0 &a 0, i follows ha 7

18 ˆΣ xx D & & σ &2 (&α ' ) 2 *W (x) 2 dx&(*w (x)dx ) 2 D & σ 2 (&α ) &2 *W (x) 2 dx 0 &σ(&α ) & *W (x)dx 0 *W (x) 2 dx&(*w (x)dx) (&α 2 )(&α )2 0 % o p (). (54) &σ(&α ) & *W (x)dx 0 Moreover, i follows from (8) ad (45) ha D & ˆΣ xu ' σw () (/ )' u y & σ 2 (&α ) &2 (W () 2 &)/2 % o p (). (55) Combiig (49), (54) ad (55), ad usig Lemma, i follows ow easily ha ˆα 2 &α ' 2 (W ()2 & ) & W ()*W (x)dx % o *W (x) 2 dx & (*W (x)dx) 2 p () 6 ρ i disr., (56) where ρ is defied i (33). Alog he same lies i ca be show: THEOREM. Le y be geeraed by (39), ad le ˆα p be he OLS esimaor of α p. Uder he ui roo hypohesis, i.e., α p = 0 ad α 0 = 0, he followig hold: If model (39) is esimaed wihou iercep, he ˆα p 6 (&' p& j' α j )ρ 0 i disr., where ρ 0 is defied i (22). If model (39) is esimaed wih iercep, he ˆα p 6 (&' p& j' α j )ρ i disr., where ρ is defied i (33). Moreover, uder he saioariy hypohesis, plim 64ˆα p ' α p < 0, hece plim 64 ˆα p '&4, provided ha i he case where he model is esimaed wihou iercep his iercep,, is ideed zero. α 0 Due o he facor & ' p& j' α j i he limiig disribuio of ˆα p uder he ui roo 8

19 hypohesis, we cao use ˆα p direcly as a ui roo es. However, i ca be show ha uder he ui roo hypohesis his facor ca be cosisely esimaed by & ' p& j' ˆα j, hece we ca use ˆα p /* & ' p& j' ˆα j * as a ui roo es saisic, wih limiig disribuio give by (22) or (33). The reaso for he absolue value is ha uder he aleraive of saioariy he probabiliy limi of & ' p& j' ˆα j may be egaive. The acual ADF es is based o he -value of, because he facor & ' p& will cacel ou i he limiig disribuio ivolved. We will show his for he AR(2) case. α p Firs, i is o oo hard o verify from (43) hrough (48), ad (54), ha he residual sum of squares RSS of he regressio (40) saisfies: RSS ' j j' α j u 2 % O p (). (57) This resul carries over o he geeral AR(p) case, ad also holds uder he saioariy hypohesis. Moreover, uder he ui roo hypohesis i follows easily from (54) ad (57) ha he OLS sadard error, s 2, say, of ˆα 2 i model (40) saisfies: s 2 ' RSS/(&3) σ &2 (&α ) 2 *W (x) 2 dx&(*w (x)dx) 2 % o p () ' &α *W (x) 2 dx&(*w (x)dx) 2 % o p (), (58) hece i follows from (56) ha he -value ˆ 2 of ˆα 2 i model (40) saisfies (34). Agai, his resul carries over o he geeral AR(p) case: THEOREM 2. Le y be geeraed by (39), ad le ˆ p be -value of he OLS esimaor of α p. Uder he ui roo hypohesis, i.e., α p = 0 ad α 0 = 0, he followig hold: If model (39) is esimaed wihou iercep, he ˆ p 6 τ 0 i disr., where τ 0 is defied i (24). If model (39) is esimaed wih iercep, he ˆ p 6 τ i disr., where τ is defied i (34). Moreover, uder he saioariy For example, le p = 2 i (37) ad (39). The α '&β, hece if β < & he & α <0. I order o show ha β < & ca be compaible wih saioariy, assume ha β 2 ' 4β 2, so ha he lag polyomial & β L & β 2 L 2 has wo commo roos &2/*β *. The he AR(2) process ivolved is saioary for &2 <β < &. 9

20 hypohesis, plim 64 ˆ p / < 0, hece plim 64 ˆ p '&4, provided ha i he case where he model is esimaed wihou iercep his iercep,, is ideed zero. α 0 5. ARIMA processes, ad he Phillips-Perro es The ADF es requires ha he order p of he AR model ivolved is fiie, ad correcly specified, i.e., he specified order should o be smaller ha he acual order. I order o aalyze wha happes if p is misspecified, suppose ha he acual daa-geeraig process is give by (39) wih α 0 ' α 2 = 0 ad p >, ad ha he ui roo hypohesis is esed o he basis of he assumpio ha p =. Deoig e = u /σ, model (39) wih α 0 ' α 2 = 0 ca be rewrie as where y ' (' 4 j'0 γ j L j )e ' γ(l)e, e - i.i.d. N(0,), (59) γ(l) ' α(l) &, wih α(l) ' & ' p& j' α j L j. This daa-geeraig process ca be esed i he auxiliary model y ' α 0 % α y & % u, u ' γ(l)e, e - i.i.d. N(0,). (60) We will ow deermie he limiig disribuio of he OLS esimae of he parameer α while i realiy (59) holds. ˆα ad correspodig -value ˆ i he regressio (60), derived uder he assumpio ha he u 's are idepede, Similarly o (4) we ca wrie y ' γ()e % v & v &, where v ' [(γ(l)&γ())/(&l)]e is a saioary process. The laer follows from he fac ha by cosrucio he lag polyomial γ(l)&γ() has a ui roo, ad herefore coais a facor -L. Nex, redefiig W (x) as [x] W (x) ' (/ ) j e if x 0 [ &,], W (x) ' 0 if x 0 [0, & ), (6) i follows similarly o (42) ha hece y / ' y 0 / % v / & v 0 / % γ()w (/), (62) y / ' γ()w () % O p (/ ), (63) ad similarly o (43) ad (44) ha 20

21 ȳ & / ' j y & / ' γ() m W (x)dx % o p (), (64) ad 2j y 2 & ' γ()2 m W (x)2 dx % o p (). (65) Moreover, similarly o (6) we have where j ' 2 γ()2 W () 2 & j ( y )y & ' 2 y 2 / & y 2 0 / & j ( y ) 2 (γ(l)e ) 2 % o p () ' γ() 2 2 (W ()&λ) % o p (), (66) λ ' E(γ(L)e )2 ' ' 4 j'0 γ2 j. γ() 2 (' 4 j'0 γ (67) j )2 Therefore, (33) ow becomes: ˆα ' (/2)(W ()2 & λ) & W ()*W (x)dx *W (x) 2 dx & (*W (x)dx) 2 % o p () 6 ρ % 0.5(&λ) *W(x) 2 dx&(*w(x)dx) 2 (68) i disr., ad (34) becomes: ˆ ' (/2)(W ()2 & λ) & W ()*W (x)dx *W (x) 2 dx & (*W (x)dx) 2 % o p () 6 τ % 0.5(&λ) *W(x) 2 dx&(*w(x)dx) 2 (69) i disr. These resuls carry sraighforwardly over o he case where he acual daa-geeraig process is a ARIMA process α(l) y ' β(l)e, simply by redefiig γ(l) ' β(l)/α(l). The parameer γ() 2 is kow as he log-ru variace of u = γ(l) e : σ 2 L ' lim var[(/ )' u ] ' γ()2 (70) 64 which i geeral is differe from he variace of u iself: σ 2 u ' var(u ) ' E(u 2 ) ' E('4 j'0 γ j e &j )2 ' ' 4 j'0 γ2 j. (7) 2

22 If we would kow σ 2 L ad σ 2 u, ad hus λ ' σ 2 u /σ2 L, he i follows from (64), (65), ad Lemma, ha σ 2 L & σ2 u (/ 2 )' (y & &ȳ & )2 6 & λ *W(x) 2 dx & (*W(x)dx) 2 i disr. (72) I is a easy exercise o verify ha his resul also holds if we replace y - by y ad. Therefore i follows from (68) ad (72) ha, ȳ ' (/)' y ȳ & by THEOREM 3. (Phillips-Perro es ) Uder he ui roo hypohesis, ad give cosise esimaors ˆσ 2 L ad ˆσ 2 u of σ 2 L ad σ 2 u, respecively, we have ˆ Z ' ˆα & (ˆσ 2 L &ˆσ2 u )/2 6 ρ (/)' (y &ȳ i disr. (73) )2 This correcio of (68) has bee proposed by Phillips ad Perro (988) for paricular esimaors ˆσ 2 L ad ˆσ 2 u, followig he approach of Phillips (987) for he case where he iercep α 0 i (60) is assumed o be zero. I is desirable o choose he esimaors ˆσ 2 L ad ˆσ 2 u such ha uder he saioariy aleraive, plim 64 Ẑ '&4. We show ow ha his is he case if we choose ˆσ 2 u ' j û 2, where û ' y & ˆα 0 & ˆα y &, (74) ad ˆσ 2 L such ha σ 2 L ' plim64ˆσ2 L $ 0 uder he aleraive of saioariy. Firs, i is easy o verify ha ˆσ 2 u is cosise uder he ull hypohesis, by verifyig ha (57) sill holds. Uder saioariy we have plim 64ˆα ' cov(y,y & )/var(y ) & ' α (, say, plim 64ˆα 0 '&α ( E(y ) ' α( 0, say, ad plim64ˆσ2 u ' ( & (α( %)2 )var(y ) ' σ 2 (, say. Therefore, plim 64 Ẑ / '&0.5(α ( 2 % σ 2 L /var(y )) < 0. (75) Phillips ad Perro (988) propose o esimae he log-ru variace by he Newey-Wes (987) esimaor 22

23 where û ˆσ 2 m L ' ˆσ2 u % 2 j [ & i/(m%)](/) j û û &i, (76) i' 'i% is defied i (74), ad m coverges o ifiiy wih a rae o( /4 ). Adrews (99) has show (ad we will show i agai alog he lies i Bieres (994)) ha he rae o( /4 ) ca be relaxed o o( /2 ). The weighs - j/(m+) guaraee ha his esimaor is always posiive. The reaso for he laer is he followig. Le = u for =,..,, ad = 0 for < ad >. The, ˆσ (2 L / j %m m m% j j'0 u ( &j u ( u ( 2 ' m m% j j'0 %m j u ( 2 &j % 2 m& m&j m% j j j'0 i' %m j u ( &j u ( &j&i ' m m% j j'0 %m&j j &j u ( 2 % 2 m% j m& m&j j j'0 i' %m&j j &j u ( u ( &i (77) ' j u 2 % 2 m& m&j m% j j j'0 i' j u u &i ' 'i% j u 2 % 2 m m% j (m%&i) i' j u u &i 'i% is posiive, ad so is. Nex, observe from (62) ad (74) ha ˆσ 2 L Sice û ' u & ˆα γ()w (/) & ˆα v % ˆα (v 0 &y 0 ) & ˆα 0. (78) E*(/)' %i u W ((&i)/)* # (/)' %i E(u 2 ) (/)' %i E(W ((&i)/)2 ) = O(), i follows ha (/)' %i u W ((&i)/) = O p (). Similarly, (/)' %i u &i W (/) = O p (). Moreover, ˆα = O p (/), ad similarly, i ca be show ha ˆα 0 ' O p (/ ). Therefore, i follows from (77) ad (78) ha m ˆσ 2 L & ˆσ(2 L ' O p (/) % O p j [&i/(m%)]/ ' O p (/) % O p (m/ ). (79) i' A similar resul holds uder he saioariy hypohesis. Moreover, subsiuig u = σ 2 Le + v - v -, ad deoig = e, = v for =,..,, = = 0 for < ad >, i is easy o verify ha e ( v ( v ( e ( uder he ui roo hypohesis, 23