BANCO DE ESPAÑA RECURSIVE AND SEQUENTIAL TEST FOR UNIT ROOTS AND STRUCTURAL BREAKS IN LONG ANNUAL GNP SERIES Anindya Banerjee, Juan J. Dolado and John W. Galbraith SERVICIO DE ESTUDIOS Documento de Trabajo nº 9010 RECURSIVE AND SEQUENTIAL TESTS FOR UNIT ROOTS AND STRUCTURAL BREAKS IN LONG ANNUAL GNP SERIES (*) Anindya Banerjee (**) Juan J. Dolado (***) John W. Galbraith (****) (*) James Stock very kindly provided the data series used. We are grateful to Sieve Ambler, David Hendry and Mark Rush for especially helpful comments. Galbraith thanks FeAR for research support under grant number NC-0Q47. (•• ) Department of Economics College of Bus. Admin. University of Florida Gainesville. FL 32611 USA. ("U) Servicio de Estudios, Banco de Espafia. Alcala, 50. 28014 Madrid. SPAIN. (* .. * ) Department of Economics McGill University. 855 Shcrbrooke St. W. Montreal, QC H3A 217 CANADA. Banco de Espafia. Servicio de Estudios Oocumenlo de Trabajo n." 9010 In publishing this series, the Banco de Espana seeks to diffuse worthwhile studies that help acquaint readers better with the Spanish economy. The analyses. opinions and findings of these studies re· present the views of their authors; they do not necessarily coincide with those of the Banco de Espana. ISBN: 84-779 3"()S7·8 Deposito legal: M. 43464·1990 Imprenta del Banco de Espana Abstract Recur s i ve and sequentia.l estimation procedures are applied to each of two long annual data ser'ies on the logarithm of real per' capi tal U. S, GNP, each of which includes the G,'eat Depre s s ion, The recursively--calculated Augmented Dickey-Fuller statistic for' a unit root, and several sequentially-calculated F-type s tatistics for str'uctural break s. are compared wi th cri tical values appropriate to such sets of statistics, The results for the long data series are consistent with those found on shorter (post"",ar) data sets; we do not find statis tically significant evidence of a trend break , - 5 - A number of recent p&pers, in particular Stock and Watson (1986) ,,,,d Perr'on ..nd Phillips (1987), logarithm of real per' capital test the hypothesis that the gross national product (GNP) in the United StaLes follows a stochastic tn�nd by testing for a unit root in :.:Ieviations fr'om non-par'ametric Phillips deterministic variants, developed (1988) and Perron b�st for a unit rout, (1988, by tr'end, Both Phillips 1989), papers (1987), employ PerTon and of the Dickey--Fuller (1979) Stock and Watson (1986) extract a i, 5% annual tr'end gr'owth to induce cln test that (1987) taking does not allow for u':$e a non-Zero approximately driftless series, more dr'ift. account gener'dl of Lhe a fitted tr'end, Perr'on and tests have bNm proposed version of Non-pararnetr'ic dnd use a for'm of presence the of the Phillips lest which accomodates a as way autocorrelations first-differenced representation of the process, However, in of the> recent work by SChWH('t (1989) suggests that such non·-parametric corrections do not perform ...ell even in large samples, The simulations in Schwert (1989) show that for some (especially MA) err'or processes, the non"·parametric variants of the Dickey-Fuller tests are characterised by low power and incor'r'ect sizes, in samples as large as 1000, and that thQ lJse of the Augmented Dickey-·Fuller (ADF) test is to be preferred to the use of non-parametrica lly corrected Dickey -Fuller test s tati sties. Stock and Watson (1986) consider three separate real per capital eNP series and test for the existence of a unit r'oot in 1 each , The tests are conducted in different sub-·�samples and the results an� var'ied. They conclude that there is little or no evidence against a unit root in the post-1919 sub-samples, - 6- Pen'on "nd Phill ips (198 7 ) , using the lIIore gene,'al version of the test , conclude that using the N-P series does not allow rejection of the null hypothesis 'of a unit r'oot ei ther before or after World n, War pre,,·World war The II F-·S series perlll i ts rejection data but not on the post--World War i� possible for the full sample ) , The evidence purpose of this paper is of II the null on data (rejection to re--examine the previous in the detail allowl�d by recurs ive estimation procedures. A line of argument i n i t iated by Perron (198 9 ) , states that tests for a unit n;,lot \lJhich use lhe full satnple are biased in favour of accepting the un i t root hypothesis i f the series has a structural break at some intermed iate date. Such struc tural bn�aks may take the fonn of changos in the mean level of while Cl. t rend, ser i�s Iilay ( that is, Cl. series or changes in tr'end gr'owth rates , Thus be s tationcH'Y around Cl. broken Il\{�an or a broken the mean or slope coeffic ient takes on two or more diffQt'�rlt values in d i ffen�nt Piu'ts of the sample, and thQ deviation of the series from thi s changing tr'end sub,·-samples) , standar'd unit r'I,)ot i s stationary in each of the te'St which do (It..lt takQ account of these breaks occuring i n sub--samples of the series will be dogged by low power. By standard uni t r'oot tes ts we mean thosa proposed by D i ckey and Fuller (1979) and Fuller (1976 ) , Pen'on ( 1989 ) proposed modifying the standard D ickey-Fuller test by i.ncluding d ummy variables in the Dick ey�-·Fuller regression, i n order to allow for a break i n the trend and mean. He computed cr i t ical values appr'opriate cr' i t ical values , to found this mod i fied regr'ession and, using the new in favour of a structural br'cak in a majori ty of the t ime series inves t i gated earlier by Nelson and Plosser (1974 ) , An impor'l;ant criticism of the Per-ron approach, identi fied ori ginally by Christiano (198 8 ) was that the break date in the series was assum(�d by Per-ron to be known so Lhat thre is Cl the testing pr'ocedure which should be accounted for. pre·-test bias in In general , i t i s -7- more r'easonable not to assume a priori knowledge of the br'eak date but r'ather to allow its estimation to be part of the empirical exer'cise. This is the view taken by Chdstiano (1988) and is also the pr'emise of the present seems paper, natur'al, min imum To "ndogeneise following values the Christiano, choice to pay of the brnak date it attention to maximum 0 in a sequence of test statistics constructed through the two procedur'es descr'ibed below, Recursive procedur'es for testing for' a unit root or structural break have their most natural uses in this setting, tests date, in their moder'n fon", to Brown, Cl Such Durbin and Evans (1975) and their use of the cumulative sums of squares statistic, and Quandt (1960), Recent Ploberger, paper's by Kramer, Ploberger and Alt (1988) and Kr'amer and Kontrus (1989) have extended the Brown, Durbin and Evans analysis to dynamic models, Chdstiano (1988) and Banerjee, Lumsdaine and Stock (1989) (BLS henceforth) deal with the case where, under the null hypothesis, the data generation process is assumed to have a unit root. Two different clases of statistics are computed to test for the presence class, of of a unit root in N-P and F-S series, The first commonly termed IIrecursiue statistics", consists of a sequence statistics which is constructed starting from some minimum size, Lhe the recursive regression procedure . using each The of by the sample, by one data point at each stage of recursive these incrementing nested algorithm estimates the same samples, given the minimum .ample size required to estimate the first regression i n the sequence, The critical values of the test statistics used for this class of tests are given in BLS (1989), The sequence stage . of second class, statistics termed computed by " sequential using the statistics" , full sample uses at Cl each The regY'essions in this second sequence differ from each other in postulating varying dates for the break, The first regr'ession in -8- break date of k ' where 1 < k < T, o O with sucessive regressions in the sequence postulating k ' k + 1 O O T k as dales for the break, SOllle of the critical values used for O these tests appear in Christiano (1988) and in BLS (1989), We provide the sequence uses a postulated • . . . . • - additional cr'itical values in section 3 of this paper. Section 2 describes some Section detail. the recursive and describes 3 2 provides appropiate critical values .this sequential tests in asymptotic pr'oper'ties Section 4 presents of the tests when applied to the N-P and F-S series, concludes. In viow of the critici�lIls in Schwer't and the n:!sults and section 5 we (1989) limit the discussion to extended ve r s ions of t he Augmented Dickey-..Fuller· tests ' and d o not compute dony non-parametr'ic vers ions. ·A denotes "::;" word on the notation is The symbol "_)11 weak conven3ence of l:he associated probability measur'es, and signifies equality in distribution. .8rownian motion with unit indexed necessary. by wr ' i tt en Stochastic processes such as variance on [0,1] and on its sub-intervals as B and p-d im Browni. an nlotions as B P 1 1 1 Si..i larly, integrals such as I B (r) d r I B(r) dr, etc, I r (B) d r, /i o o 1 1 l are written simply as I B, I r B, I B, etc. All limits given dre O /i 0 as the sample size T -> m. S are , 2, Unit Roots and Recursive Estimation Consider the Cdse where, under the univariate pr ocess is postulated ,t o have a ' root; one possible alter'native hypothesis cl.nd a non-'zero is s tation ary autoregressive process of ordet� p null + lhat hypothesis, a dr'ift and a unit the series is Cl 1 with a non-zero dr'i ft linear trend. ottll;�r albH�native hypoth,�ses consid'�n�d in this paper' are thos{:! of stationar'ity of the autot"egressive pr'ocess about a brok��n trend or Cl broken dri rt or of the series being d riven by a unit root in o n e pa.rt of the sample and by 3 the other' , Cl tr'end-stationar'y pr'ocess in -9 - The estimate null hypothesis can be tested using either the point of the t-statistic. f i rst autor'egressive coefficifmt or its regre s s ion As it is well known the distributions of both statistics, for a given finitQ sample si.ze and asymptotically, are non-nonnal and have been tabulated by Fuller (1976) and Dickey and Fuller ( 1 979 ) . 4 Specifically, the mQ(l�! underlying the test for the unit root is (1) with in capital GNP. this paper, representing the logarithm of r'eal per' is a unit r'oot then cr. ::;;. 0, and this is the 1 . . . 2. null hypothesis. We take ' t a b e 11 . d Wl th varIance a . t , If there Se now turn to a discus sion of the recursive and sequ<-mtial testing procedur'es whi<.:h we shall use to test for a unit r'oot in the N····P and F·-$ series . 2 . 1 . 8ecursive Tests The o-Z T { } • t . 1S o sequence of r'ecurs ive statistics and . computed accord 1n9 to a stand ard a 1 gorithm ( see , e.g . , Harvey ( 1 9 8 1 »; {�1} is the sequence of OLS estimates of the coefficients, {�1} is the estimator t-statistic of the un is standard the smallest sample used for estimation ( i . e . , initialization will give us of the recursive evidence abuut the sequence of an e 's and t is the o t ' the sample 5 ize used for procedure s ) . Examination of {� } 1 the possibility thdt the time se�' i e s is gover'ned by a unit r'oot in s ome par't of the sample while it follows a stationary process in a.nother pdr't of the ::sample. -lU - We will show that although, t-stati stics be a have different for any given sample size, the 5 Dickey-Fuller distribution , there will the usual critical value t-statistics over the sample; probability s tandard that the the maximum of in any giv@n minimum Dickey-Fuller for of cri tic.al thi s values the sequence of t·-ratios the wi 11 sequence is minimum of greater lie below thdn the the nominal size of the test. Thus the use of the Dickey-Ful l e r critical values is prone to l,�ad to exce s s ive rejection of the unit root hypothe sis. BLS ( 1 989) tabulate some critical values by Monte Carlo s imulation, which are repr'oduced in section 3 below. 2.2. Sequential Te s t s A . This tes t u s e s the following T-2k regressions o � Y = � + at + t a 1 Yt-l Y r ( k ) + Y r2 ( k) l 1t 2 t + tJ(t�k), indicator function, for t = + P E a. ] j=2 J ( t)k), k, k+l, , , . , T a nd k = 6 �Y . + e t-] t being J( . ) k ' k + l, o o (2) , , . , T.- k o an ' Model (2) represents ctnd tri.:"!nd and mean discontinuous break at 6=0, time since Under under the value of s" HO' the the 1 imiting The sequential and a =Y =Y 2=0, 1 1 are invariant to hypothe sis, null d i s tribut ions test therefore consists of computing the F-statistics and/or F for F Yl=Y2=O al=a=Yl=Y2=O each regres sion, where each regre s s ion differs from dny other only in Y and/or Y are 1 2 don indication of a s t ructur'al br'eak in the data series (Y for a 2 mean break, Y for a break in the trend). Again, while for any 1 given break at date k (pos tulated independently of the data) the the postulated break date . Non-·zero F-statis tic has the values of the s tandard F-density, in any given sequt:mce of FIs max exceeding the s tandard 5% ( 1 0%) c ritical probabil ity of F values is in excess of 5% ( 10%). - 11 - B . Sequential Test for' Shift in Trend (BL. S ( 1989)) This te.t considers the model as a representing r ( k)=(t-k)l(t�k) and r ( k )=O, Zt 1t trend with a kink at date k . In the spirit of Christiano's test, we estimate the T.-Zk O k=k to o Thus fur fur computed null <I =Y �Y =0 . 1 Z is the T-Zk regressions the F-statistic F o Y 1=YZ=l=O l1lax and F , the maximum of this sequence of F-·statistics is Under the hypothesis, with continous regressions given by mov ing the trend-shift sequentially k=T-k ' O each of but (Z) in 1 compared with the appropiate critical values given below . C . Sequential Test for Jump in Trend (BlS (1989)) This with final class of tests is computed r dS in B above but ( k)=J(t�k) and r (k)=O, since we are testing for a 1t Zt shift in the mean of the series without change of slope in the tr·end . 3 . Asymptotic Distributions and Critical Values 3.1. Recursive Tests The and of test the analysis statistics unit transformation root focuses in under suggested model on (1) the properties of the estimators under the H ' it is ver'y O in Sims, Stock and null <I 1 convenient Watson =�=O . to ( 1990), Because use so the that under H ' (Z) can be rewritten as O (3) where Z! =(/\ yt-1 - it, . . . , /\ Yt-p-ii)', Zt =l . Z � = ( Yt-iit) and zt = t -12 - such that Also l' = P -1 �b with b -(1·- E a..) 1 2 1 E ZI Z , t t let combinations of the unconditional In"an under HO' The transformed P original regressors = O . the regressors with are different linear stochastic orders of convergence. So define the scaling matrix I :.= diag (Tl/2 lp' T 1 3 T /2, T, 1 /2) and 6, Since the partitioned conformably with Z t rec.ursive regressions k(=, ko"'" T), t:.c:"'Ike define the coefficient vector is [15] A 6(5) - where ° < 5 � 0 5 [ E 1 � 1. Z place 5=k T, / Z t·-1 ov�r and t-1 ,]- the 1 [ the samples recursive going OLS fr'om 1 estimator [15] to of ( 4) E 1 rhus (5) where V (5) T = =, Ther'e are analogous computed Wald statistics the the hypothesis q hypoUwsis hypothesis coefficients m�xt and are that are orden�d on restriccions and so F (5) T =, 1 Z t Suppose wher'e, so involve forth, for a the Wald [15] E 1 genar'al n�cursively and for the Dickey-Fuller t-·statistic testing a. "O, 1 R6=r expressions -1 T T (and The (R6(5)-r')' U,,,,t without the test [R( fir's"t perhaps coefficients loss on statistics of 2 Z t' statistic generality, restrictions Z� that Z3 t' 4 (and we Z ) t ' tests the involve perhaps will consider (6) (7) -13- &2 (0) where the is • e stimator er of 2 computed e the residuals the through e stimated [To]-th ij -1 denotes the (i, j ) element o f V (O) . V (o) T T Then using observation and the asymptotic behaviour of the recursive estimators and test-statistics are summarized in the following Theorem (proofs in Appendix) . Theorem 1: Suppose that y t is generated by model (1), then b) Under the null R(e) c) Under the null � 1 -. = r 0 where V(O) and �(o) are partioned conformably with Vll V 22 2 =: (J = e 0 0 p' 0, V 23 V =er Table critical test minimal values (0= 1 ) , ij = 0 (j - TT and 2, 3 , 4 ) O 2 b r Bdr, V = 0 /2 0 24 1, taken for the four form BLS statistics: the reproduce s full Dickey-Fuller min ) and statistic ( t maximal Dickey-Ful ler ( 1 989), sample statistic the spn�ad the 5% ( 10%9 Dickey-Fuller max (t ), the betweQn the -14 - max imal and m i n imal experimentation in BI_S d iff (t The Monte Carlo stat i s t i c s ). d iff show that t i s more powerful but its s ize i s substantially larger than the level, so that care i s needed in the i nterpretat ion of the results with this stat i stic , 3 . Z Sequential Tests The mod e l can 3 Z t The considered be is (2.) which repaf'ameterised defined as under as above, correspond i ng the null that in and scal i ng are, matrices accord i ng to each case: A A) T T B) T B T C C) T T = = = d iag (T d iag (T diag ( r 1/Z 1/Z 1/Z I ' p r I ' T p 1/Z 3/Z 3/Z 1/Z , T , T, T , r ) 1IZ 3/Z 3/Z , T, T ) , T 3/Z 1/Z 1/Z I ' r , r , r, r ) p The estimators and test stat i stics are computed us ing the full r obser'vat i ons, for k=ko'" . r-kO' For k iT -> 0, the proce s s of these sequential stat i stics i s given by (9) where all the sums go from Z to T and 0< 0o� 00'-00 < 1 As in the recur s i ve stati stics case, the asymptotic representat ion of these proce s s e s can be summarized in the follow ing theorem. - 15 - Theorem Suppose 2: that !:l'",- ='Y ='Y =0, then 1 1 2 � a ) In case A, y (e(5)-e) '1'(5) U 11 = 2 6(1), bcr/2[6 (1)-1], 6(1)-I O[Q 0" 2 • Q P ; U 1l . = 0 (j U55 = case b) In '1'(5) = = �/2 -> = 2 (1-5 )/2; U 6 2 = 1-0 2 (1-5 )/2 = 2 (1-5 )/2; U66 6 • B, YT(8(5)-e) O[Q 1/2; U 5 2 3 (1-5 )/3; U 6 4 3 (1-5 )/3; U56 �, 1 6(5)-I 6, 6(1)-6(5)] 6 2,3..., ) = = - 113; U 5 4 with (2) 1 U(5)- '1'(5), wher'e -> � 6, �/26(1)" = model by generated is Y t = 1-5 1 U(S)- '1'(5), where � 1 2 6(1)', 8(1), bcr/2[B (1).-I], B(I)- I B, (1-5) B(1)- I B] Clnd U i s a( 5x5) matrix with elements as i n case A except that U 5 3 1 ,," [(1-5) B(I)-I B]; U 5 6 4 . = C • c) In case C, Y (8(6)-e) T '1'(5) dnd = U o[Q is _.> �/2 B(I)', B(1), dS in case A , matr'ix owe eliminated. 3 3 (1-5 )/3 - (0·-6 )/2 = U(5) -1 '1'(5), wher·e � 2 bcr/2[B (1)-I], B(I)-I B, (1) - B(o)] exct:!pt l:hC'tt the 5 lh column clnd row of the - The result provides 16 - joint uniform convergence of all estimators and test statistics, including Wald te'3ts as in Theorem 1 . Accordindly we use simulations to compute the cr'itical v alues of the max distribution of F in the sequence of regressions given by model (2), The critical values, in case A, are giv\!n in the second and third columns in Table 2, for sample sizes of 50, 100 and -150 in their 2 an 4 d. f vers ions and are cons is b!rI t wl th those repor'ted by Chri 5 tiano (1988) for- a sample si.ze of 1 5 2 , using a bootstrap method, This table agedn reflects lhe relative insensibility of the critical values to changes in the sample size. The cr'itieal values. in cases B and C, an� presented in the third and foud.:h columns in Table 2 respectively and again we n:!ly upon the insensitivity of these critical values in applying lhem to both F-S a.nd N--P series, Not� ar'e stated \='Y2=O finally, for in lhe Theor'em that although null model the 2, the results in both theor-ems in results "' �O d"d Il�-o (also ( sufficiently gen�I"al to which are handle the case where '''' '<1, eto ('Y to, 'Y tO) as noted by B LS (1989) , 1 1 2 In this case natur'ally the result on t (6) does no longer hold, OF since its distribution is standard. 4. Empirical Results Figures Fr'iedHlan--Schwar-!;z inspection of and 1 r'(�al these P(H' 2 plot capi Lal figures, the GNP two the Nelson-Plosser s,!ries, series As are is cl(�d.r well, but and from not perfectly. corTeldl�d over the periods in which they overlap, We begin our analysis of these ser'ie. by repor-ting the full-sample estimates of the Augmented Oicke Y--Fuller regression applied to each (wi th one lag): Nelson--Plosser (19 11---1970) IlY t =- 1.278 (3 , 05) + 0:0035t - 0,182Y t _l + 0 , 410 IlYt-l (3 , 02) (3 , 05) (3 , 3 9) s = 0 , 059; LM(5) = 2 , 52; N(2) = 3,61; ARCH (5) _ (11) 1,94; H(6) = 10_8 1 -17 - F r i edman-Schwar'l;z ( 1 8 71-1975) II Y t s = = + 1. 474 (3 .93 ) 0.0037 t - 0;226 Yt_ 1 (3.8 9 ) (3.8 1 ) = 0.0 6 1 ; LM(5) = 8.50; N ( 2 ) + 0.247I1 Y t._ 1 (2.55) 24.35; ARCH (5) - 6. 79; H ( 6 ) Absolute values o f t-rat ios are i n parentheses; error of the regr'es s ion; LM ( q ) (12) 5 = 12.15 denotes the s tandard i s the Lagrange Mult iplier test against an AR ( q ) or MA ( q ) i n the d i s turbance ( see Godfrey (1978»; N(2) i s the Jar·que·-Ber·a (1980) normality test; ARCH (g) i s th aga inst 8 order ARCH d i strubances; H(.) Engle's is (1982) White's test (1980) hetero s k eda s t i c i ty test. In Qach case the s tat i s t i c i s asymptotically 2 d i s t r i buted x w i th the number of degrees of freedom b racketed. U s i ng Fuller's c r i t ic"l values of ·-3.50 and ·-3.45 for the on it t-ratios in (4) and (5) respec t i vely , we can reject the null 1 hypothe s i s of a un i t root on the full sample for the F-S series , but not for the s hor'ter N -P series. There are, however, several featur'es of these results w h i c h lead us to consider a further exam ination of th.. data. F i rst, sUbstantially, as F i gures parti cularly s o that the inference w i th change Lhe sample. over 3 in and 4 s how, the t-ratios change the v i c inity of lhe Great Depress ion , respect t o the null o f a un i t root would One could pos s i bly argue, following Perron, that a s tructur'al break could account .for the low absolute values of these s ta t i s t i c s the in the v i c in i ty of case of the F-S series res iduals values w i ll computed errors . exc<>ed is rejected. not using Since the 5" be is very Th i s sens i t ive prec i s ely real i sed Moreover, to a in suggests that the D i ckey-Fuller c r i t ical appropriate, process stat i s t i c for theSe! which the c r i t ical value by a large marg i n , deemed conclus i ve. Second, the null hypothe s i s of normal ity of the a data gener'at i on the 1 93 0 in each series . values hav i ng been incorporatQd F--S series nor'mal does not lhe test cannot be as t h i s skewness-kurto s i s normality test few especially large dev i ations. it is -18 - conceivOl.blc a str'uctur'al that also account for the n or 'ma l i t y for str'uctur'bl.l BLS take we the ('Titical lninimlJm DickGY dnd Fuller � tat istl c s sampl e s . SHt N-·P and size The samj:)ll� may also se r-ies, of n�aliably th (� since is 100 values for' to conduc;t a r'e t(�!St succ��ss ive of th(� F·-S data thH :> ho r' b�r' inFer'ence in not \.\J(� maximum and th(� o bse t 'v � d ov��r' a :>et uf values cr-itical Lhe � )(� n�dsons, r-ur' close to the si.ze us (� d be se r'i ,�� could shift in the result. te,:;t br'eaks bel ow . From tr'end bn.�bl.k or' sensitive vet'y to cha.nges in the �a11lple size . sample, On the N--P on the in 1924; mini.mum in t hese dr<� 1.930 ver-y non-nol"mality the minimum of the t--statistics is --3.80 series the l1l inimum F-··S Fr 'om th(� of -3,76, close to Lhe in c se r i es F-·f, the 10% va lw� s in the hy pothes is that However, Lhe maximum whereas th(� bo th valUt�s of cases non rejecting the null f, fo r the trond N-P series, sign i f i ca n t model with and 6 in at eVl-!n the no plot 1930 break, 10% and for for- br ea k . In bn�ak test trend is trend ) . a we o r'd(� r' bet te r miqht nut wdnt an� d , f. test under' the null local in the mind application d. not finlt ar'e of n!jec;tion. and ·--1,36 2..44 and 2.,21 ·-1.68 r'esp(�ctively, F-statistics in The maxima an� 8.19 ttle F-S irl jo int alternative the 1939 for' the uF of a data the trend and of str-uc;t\Jr-cd thdt Lhe stochd.stic than statiol"lOl.r'y is mt!an d. de t e r'm ini st i (; compute the critical values for the hy pothe � i s 2. r'Cmdl)m walk r'esult of this to be cer' tain of with series. Neither' of these under the null poss i ble to but Christiano--type chanl1.ctet"isation al:w cl it is ch!ar- that liter-al t '-S Ldt i stic s to b(� confident in ( s i nc e it is lh hd-ve sOllle suggestive Qvidence root, level, a wi hy poth esis. c\nd m(�an break, and 6.07 I, Tabh� makes lIlay lhe differ'ence t-·statistic� Figur'es 5 d, uni . t Cl of in 19.15, r i ti c al values, flear'ing these critical valu\�s hdzardous, \oJe against is ,-3.89 d(�vi.at.i.(Hls fr'olll 2 a -19 - deterministic trend (again following Christiano). We do so in Table 3. The for'm of nu 11 hypothes is chosen is based upon the reg,'ess ions (3) (4) : and to these = Y t results a fair approximation is 1 +0.003t + 0.8Y _ +e t t l and to those of models which drop the term in 8Y _ ' on both da.ta sets. t l to be a fixed constant. Again, the The initial observation is again chosen test statistics just given are lower than the critical values. For neilh�r of the null models, then, Cdn we conclude that there is a structural break. tests r'epor'tad The t-statistics in the unit root Q"arlier Cdn the�'efore be interpreted in the usual way; a structural break is not a persuasive explanation for any failures to reject the unit root model. Similarly, computed, Christiano-·type F-statistics with 4 d.f. are with maximal values of 13.75 "nd 14.10 for the l\I-P ,md F-S series, both values being highly non significant. S�quential statistics for the altEH'native of a tr'end br'eak against the null of no break in each of the series are plotted in Figur'es 7 and 8. The maxima of lhe F-statistics are 5.73 in 1932 for N-P, and 6.61, also in 1932., for F-S. Again, neither is significant at even the 2 above). 10% level, Finally, hypotyhesis of a using lhe BLS sequential Ill��a.n bt'(,�ak an� cd tical values (reproduced in Table statistics repot'ted for in the Figu?'e 9 alternative and 10. The noaxima a,"e 6.87 in 1938 (N-P) and 6.63 in 1938 (F-S). Once again, the statistics offer no statistically significant Qvidence of such a break. 5. Conclusion In summary, we ar'e able to reject the nu 11 hypothesi s of a unit ,'<lot only for Lhe full F-S sample, although we give r·ea.ons for - 20 - in ter'preting this rejection cautiousl Y ; the recursive test statistics cOlne close to rejecting this null as well. There is much less eviden<:::e against the null hypothesis of no structut�O\l break, whether' or' not the also null incorporab�s process for GNP; a diFference-stationary although the their la"gest values have reach even our 10% Monte finding us allows to or trend-stationary statistics used for testing the b,'"eak en the neighbourhood Car'lo cr'itical values interpr'ct Lhe uf 1930's, they do not for these tests. This recursive t-statistics face dt value, putting aside the possi.bility of a break, In classical statistics, the �nvesti9ator is vicwed as using data to test an hypothesis formed independently of the data on which it is to be tested. bas,!d on the data, will it is clearly relatively easy to find on(� which not be rejp-cb�d on the given sample. In per capita eX When the hypoth esis undHr test is chosen the GNP, case of at least some hypotheses F'(�lating to real par'ticularly those involving structur'al breaks, it is little difficult to imagine an investigator in the purely classical position; the gener'al hypothesis such as HO: f��atures of the dclLa a structu,'al b,'eak i. n are GNP well known, took place c, An 1930 seems very likely to have been influenced by knowledge of the data, It is with this in mind lhat sequential tests are applied here, Clearly an investigator in the classical position, with the a priori hypothesis that there was a structur'al break in GNP c, 19 30, would reject a null of no break; the corr'esponding F-·statistics exceed S% critical values by a comfortable mar'gin. mo"e ,'ealistic test, that However, for' the p��rhaps of the hypothesis that a br'eak took place somewhere on the sample ... - so thctt we do not use our' knowl(�dge of the data C·pr·e .... test·· of the spit(� examination) to choose the br'eak point - ... rejection null of no bn�ak is not possible dot conventional levels, of the app(�ar'(il,neE:! of game fair'ly large statistie g. 1hus relatively (1 988) and long BLS dclLa ( 1 989) sets fo,' conf .l,r-m the results post--war dat.. sets, found by I in these Christiano - 21 - Appendix for convenience ! is iid(O,,, ), being drawn martingale Both fr'om it is assumed assumptions a (MSO) sequence wi th yo �O and that generalised, be di.str'ibution stationary difference can that at and to to e being a t bounded fourth least moments, at the cost of complicating the algebra, Proof of Theorem 1 a) Without loss of generality it is assumed that r and define the sequence of par'ti"l s<,ems S (O)= E T 1 Central Umit Theorem implies that S (o) T C(l) � - (1 P i -1 E '''; l ) and write Y t 2 has all its Lemma: with in Let yt -2 2 E Y T t ii) T -1 E Y to C (l), ,/e e t-1 f t t + Then C�'(l) l C( _.> Because uniform be an 1(1) -> -3/2 E t iii) T is (0,1) , given by Y =C (l) S t t i) = b there V T and T (1987): C(l) S t roots outside the unit circle, -> o'bB(I) Y r k (k:;::ko'" TL � � -> " / the the e" let The functional 1 + C*(L) e where C*(l) t -1/2 It follow th",t T sample of following size each � 1 and 1 lemma in to of' Stock ti.me series with a Wold representation et' then 2 I B 2 2 C(l) 0 12 [B (1)-1] e -) <1 Vll' is element e 1 [B (1) - I B) 0 obtain all the stochastic limiting distributions in except r=[T5] B (o), Then let convergence apply I'�O, intet-changing the (0,1) integt-al the text, limits by - 22- 1 4 ) ( .., it follows £, is a MSD wi th sUP(z it_..1 et 1 0 2 B (I». i. e a p-dim. Bt�ownian motion with a that � lT(l» -) e P 1 2 0 2 1 P z. ' :;;:; a deterministic Finally the cova,'iance (j E Z t t e e P V emu ar'e obtained a. J d ., components such as V , V 24 22 44 2 J .d. and J • d. with integral limits between 0 and I). (0,1». Because b & t-1 2 c) Given the conve"\lence re.U 1 t. in (a) ",nd that � (I» limiting dist,·ibution. Stock ",nd Watson (1990). follow directly f,'om 1heorem 2 -) (J both • 2 of Sims, a) The pr'oof is s imi.lar to the proof of Theor'em 1 except that now the siz e sdomple f2t (k) T such goes a. -·3/2 T E t J(t�k) £t 1 Fr'om '¥5 - 1 to and -3/2 T T Et k T. "'6 £t The tar'IllS follow involving fl (k) and t di,'ectly. For example wich implies integr'al 1 imits from I) to 1. The dete'"Illinbtic component. such as U55 (o)'=T • T tend in limit to b &. J! . 2 ··-3 d. etc. c) The argument fot� both cases is similar to that for' case 1. going ,. E[tJ (t�k) ]2 1 - 23 - Footnotes 1 , These are the ( 1 869-1975) Nel son Friedlll..n and ( a .,odif ied and Plos ser (constructed by annual serles from the US Opt . annual series of Corl1mer'ce) and fr'om 1 909 to the National 1 9 70 Income Only the f i r'St two, wh i c h we re fer to and N- ,P , are used her'e; the NIPA s e" i e s star't. only in 1929 , F-S �nd ( 1982) ver'sion of the Kuznets ( 1 9 6 1 ) serie s , the ( 1982) and Product Account series , as Schwar'l;z is thereflJ r'e Depession rlut when long a. l lowance enoLlgh is made Fur eXCl.l1lination of the Gn':!d.t for i n i t i a l i zation of the recurs i ve procedures. 2 , Th i s section re l i es upon results in BLS ( 1989) ",her'e weaker cond i t ions are as sumed for the d i sturbance e ' t 3 , Th i s i s , of course the a l ternat lve con s idered by Christ iano ( 1 988 ) and PerTon ( 1 9 89 ) , 4 , Which may d i ffer' from the data gener'ation proces s i t se l f , ? , Baner'jee et a ! . critical Values ( 1 990) provide an appr'oxilllate a l gorithm to compute for the D i c key-Ful ler' stat i s t i c in recur s i ve sd.mples . 6 , Where T is estimating the total sample minu s the " l:wo lags. This sample is ohser'vat ions at each end . two obser'vations lost further by tri mmed in two - 24 - References Banerjee, A., J. Do l ado and J.W. Galbraith (1990) "Orthogonality Tests De-Trended with Dat a : Interpreting Monte Carlo Results Using Nagar Expansions" Economics Letters. 3 2 , 19-24. Banerjee, A., R. Lumsdaine and J.H. Stock (1989) " Recurs ive and Sequential Tests for a Unit Root : Theory and International E v i dence " , Manuscript, University of Florida. Brown, R.L. , J . Durbin and J.M. Evans (1975) "Techniques for Testing l:he Constancy of Regn"!ssion Re lationships over Time (with Journ�."�f_the discussion) ! ! , Series B, Chow, G.C . Royal Stat i stical Society, 37, 149-192 . (1960) "Tests of Equal i t y Between Sets of Coefficients in Two L i near Regress i ons " , Econometr ica 49, 1593-1596. Chdsti ano, L. (1988) "Searching for a Break in GNP". NBER D i scussion Paper No. 2695. Dickey , D.A. and W . A. Fuller (1979) " D i stribution of the Estimator for Autoregressive T ime Series with a U n i t Rootll, "�ournal of !.b.!L. �!"er i can Stati shcal Society 74, 427-A31. . Engle, R.F. (1982) " Autoregressive Conditional Heteroskedasticity with Estimates of the Vari ance of United Kingdom Inflationll, Econometri c a 50, 987-1008 . Fri edman, M. and A. J. Schwartz (1982) t'l0netary Trends i n the UDited _ States and Price and the United K i ngdom: Their Interest Rates, 1867-1975 . Press, Chicago . R e l ation to Income Uni verstiy of Ch icago - 25 - Fuller, W . A . (1976) Introg�ct�on to Statistical Time ��ries . Wiley, New York . Godfrey, L . G. ( 1 978) " Testing Against General Autoregressive and Moving Average Error Models when the Regressors Include Lagg.ed Dependent Variables", Econometrica 46, 1293-1301 . Harvey, A.C . ( 1 981) The Econom�tri� Ana l�s of Time Series. Philip A l lan, Oxford. Jarque, C.M. a�d A.K . Sera (1980) "Efficient Tests for Normality, Homoskedastici ty dnd Serial Independence of Regression Residuals " , Economics Letters 6 . 255-259 . Kralfler, W . , W. Ploberger and R'. A l t (1988) : "Tes ting for Structural Change in Dynamic Model s " I Econometrica, 56 1355--1369 . Kuznets, S. (1961) capital in the Economy : Its Formation and_ ·Financing. Princeton University Press/NBER, Princeton, New Jers y , Nelson, C . R. c.r. P l osser (1982) "Trends and Random Walks in Macro-economic Time Serie s : Some Evidence and Implicationsll, Journal of Monetary Economics 10, 139-162 . Newey, W.K. and K. D. West (1987) " A Simple Positive Definite Heteroskedasticity and Autocorrelation Consistent Covariance Matrix", Econometrica 5 5 . 703-708. Perron. P. (1988) "Trends and Random Walks in Macro-economic Time Serie s : Further Evidence from a New Approachll • Economic Dynamics and Control 12, 297-332. Journal of - 26- Perron, P. ( 1 9 8 9 ) "The Great Crash, the Oil Shock and the Unit Root Hypothes is " . Econometric .. 57, 1 3 6 1-1402. Perron, P. and P.C . B . Ph illips ( 1 9 8 7 ) "Does GNP have a Unit Root? a Re-evaluation" , Economics Letters 2 3 , 1 3 9-145 . Perron, P . and P.C . B . Ph illips ( 1 9 8 8 ) "Testing for a Unit Root in T ime Series Regress ion". �iometri:..ka 75, 3 3 5-246 , Ph illips , P.C.B. ( 19 8 7 ) IIT ime Series Regress ion with a Unit Rootl! Econometrica 55, 277-· 3 02. Plober'ger, W . , W. Kramer and K . Kontr'us ( 1 9 8 9 ) , Structural Stability in the "A New Test for L i near Regress ion Model'· , .Journ.",l of Econometrics, 40, 307·-3 1 8 . Quandt, R.E . ( 1 960) "Tests of the Hypothesis that a Linear Regress ion Obeys Two S{�par'ate Regimes " , Statist ical Association, 55, Journal of the Amer::ican 3 2 4-·3 30 . Schwert ( 1 9 8 9 ) "Tests for Unit Roots: A Monte Carlo Inve s t igation", ;r.ournal of Business and Economic Statistics 7, 147-·159. S ims, C . , Stock, J. and M. Watson ( 19 9 0 ) " Inference in Linear Time Series Models with Some Unit Roots" gconometr ica, 58 . Stock, J.H . ( 1 9 8 7 ) " Asymptotic Pr'operties of Least-Squares Est imators of Cointegrating Vectors . Econometric .. 55, 1035-·1056. Stock, J . H . and M.W. Watson ( 1 9 8 6 ) "Does GNP have a Unit Root?" Economics L�tlers 2 2 , 1 47-1 51 . :'o t b . U. S. Oepar·tment of Commerce ( 1 9 7 3 ) Long Ter..m Eco'!2.'!!.i.£.J?.!w Wash ington D. C. - 27 - Wh i te , H . ( 1 980) " A Heteroskedast i c i ty -Cons i stent Covariance Md trix E s t i mator and a D i rect Econome t r i ca 48 , 8 1 7-8 3 9 . Test for Heteroskedas t i c i ty " . - 28- TABLE 1 Recursive U n it Root Tes t s : Cr i t i c al Values ( 2 . 000 R e p l i cations) 5� ( IOX) C r i t i c a l Values T t DF min t (I) t " ma)( diff t lOO - 3 . 4 5(-3 . 1 5 ) ) - I . 9 3 (--2 2 1 ) 3 . 3 7 ( 2 . 95 ) 250 -3 . 43 (-3 . 1 3 ) -4 . 07 ( -3 . '0) -I . 88 ( - 2 . 1 4 ) 3 . 3 6 ( 2 . 9' ) 500 - 3 . 42 (-3 . 1 3 ) - 4 . 1 0 (·· 3 . ' 2 ) - I . 8 8 (- 2. . 1 4 ) 3 . 45 ( 3 . 0 1 ) Note : Critical values by values for - 4 . 13(-3 . tor { l ) f o r t h e rema i n i ng OYt the nu l l model = takcm fr'um Table t-·stat i s t i c s tt, w i th 0 . 25 taken $ 6 � 8.5.2 i n F u l ler from lable 1. 1 ( 19 76 ) ; the in B L S ( 1 989 ) , (; r i t i c a l generated ,ABLE 2 Sequent i a l Tests for Trene and Mean Break (Case s A. 8, Cl: E s t in�ted C r i t i c a l values (Nu l l incorporat e s SI st o c h ast ic tre� ( 2 . 000 R p- p l i c a t i o n s ) 5 X ( 10�) Critical Values C( 3 ) A(2) A(4) B(3) 50 1 2 . 1 1 ( 10 8 1 ) 20 3 ' ( 1 ' . 1 3 ) 1 7 6 3 ( 15 . 16 ) 18 7 3 ( 1 6 . 0 2 ) lOO 1 2 . 32(10.91) 20 . 3 1 ( 1 8 . 1 1 ) 1 6 . 74 ( 1 4 . 30 ) 1 B . 40 ( 1 5 . 9 2 ) ISO 1 2 . 1 5 ( 1 0 . 90) 20 . 29 ( 1 8 . 1 7 ) 1 6 . 3 9 ( 1 3 . 8 5) 1 ' . 50 ( 1 6 . 06 ) Note : The rur �ntries 0 . 10 � respe c t i v e l y , W� o � computed 0 . 90 , whereas B ( . ) u s ing dala gnne ,·att!d A(Z) and A(4) refltr and C( . ) have 3 d . f . by to lhe te � t s nu l l wilh modQl 2 OY t �nd � Et d . f. - 29 - lABLE 3 Sequential Tests for Trend and Mean Break (eas e A): Est imated Critical Values ( N u l l incorporates a deterministic t rend) ( 2 . 000 RepliCations) T/size Note : The Yt-l entries were 9. lOO 8.4 computed for O . lO,8� O . 90 10� 50 51 I� 10 . 7 13.7 '.8 11.7 u s i ng data gene,"ated by the n u l l model Y t::=lrO. OO3t+O . 8 - 30 - Figure 1 N-P: 1909-1970 8.5 .------, 8 "- z " !! '0. • 0 7.5 � c- o; e � 7 '" ': t 1 860 1 880 1900 1 920 1940 1 960 1980 year F-S: 1869-1975 Figure 2 8.5 ,------, 8 "- z " 7.5 !! '0. • 0 � c- o; e � 7 '" 6.5 ---L---__ __ __ __ __ __ � __ __ � � � __ __ __ � 6 � 1860 1880 1900 1920 1 940 1 960 1980 year - 31 - Figure 3 N-P: RECURSIVE ADF T-STATlSTlCS 4 3.5 " .� 3 � '0 • , .. 2.5 > • '5 '0 • � • 2 1.5 t 1 1 860 1880 1900 1920 1 940 1 960 1 980 year Figure 4 F-S: RECURSIVE ADF T-STATlSTlCS 4 3.5· " 't; .� ! '0 • , .. > S 3 2.5 2 '0 • � • 1.5 1 1 860 1880 1 900 ' i920 year 1940 1960 - 32 - Figure 5 N-P: F-TESTS FOR TREND AND MEAN BREAK 10 I> 8 0 :� ;; 6 " U. 4 � 2 o L-----------�-- � __ __ 1 860 1880 1 900 1 920 ������� __ __ 1940 1960 1 980 year Figure 6 F-S: F-TESTS FOR TREND AND MEAN BREAK 10 r-------, 8 0 � �U. 6 4 2 o 1860 1 900 1 920 year 1980 - 33 - Figure 7 N·P: F·TESTS FOR TREND BREAK 5.8 � 5.6 5.4 u 5.2 .� 5 .� •• � 4.8 4.6 4.4 f 4.2 4 1 860 1880 1 900 1 920 1 940 1 960 1 980 year Figure 8 F·S: F·TESTS FOR TREND BREAK 5.8 5.6 f 5.4 u � .� ;; "- 5.2 5 4.8 •. 6 4.4 4.2 4 1 860 1880 1 900 1920 year 1940 1960 1980 - 34 - Figure 9 N-P: F-TESTS FOR MEAN BREAK 7.5 . 7 u 6.5 .� .u.� 6 � I r f-L 5.5 II 5 rr ; 4. 5 4 f- L '860 '880 , 900 '920 '940 ' 960 ' 980 year Figure 10 F-S : F-TESTS FOR MEAN BREAK 7.5 r 7 u .� ." � u. 6.5 � I r , 6 � 5.5 � 5 , , r i r � 4.5 i rf 4 ' 860 '880 '900 '920 year ' 940 1960 1980 - 35 - DOCUMENTOS DE TRABAJO (1): 850 1 Agustin Maravall: Prediccion con modelos de series temporales. 8503 Ignacio Mauleon: Prediccion multivariante de 105 tipos interbancarios. 8502 Agustin Maravall: On structural time series models and the cha racterization of cempo· nents. 8504 Jose Vinals: El deficit publico y sus efectos macroecon6micos: algunas reconsideraciones. 8506 Jose Vifials: Gasto publico, estructu ra impositiva y actividad macroeconornica en una 8505 8507 Jose Luis Malo de Molina y Eloisa Ortega : Estructuras de ponderacion y de precios relativos entre 105 deflactores de la Contabilidad Nacional. economfa abierta. Ignacio Mauleon: Una tuncion de exportaciones para la economia espanola. 8508 J. J. Dolado, J. L. Malo de Molina y A. Zabalza : El dese'llpleo en el sector industrial 8509 Ignacio MauleDn: Stability testing in regression models. 85 1 1 J. J. Dolado and J. L. Malode Molina: An expectational model of labour demand in Spanish 8512 J. Albarracin y A. Yago: Agregacion de la Encuesta Industrial en 105 1 5 sectores de la 8513 Juan J. Dolado, Jose Luis Malo de Molina y Eloisa Ortega : Respuestas en el deflactor del 8514 Ricardo Sanz: Trimestralizacion del PIS por ra-nas de actividad, 1 964-1 984. 8516 A. Espasa y R. Galian: Parquedad en la parametrizacion y o'llisiones de factores: el modelo 8510 8515 85 17 espanol: algunos factores explicativos. (Publicada una edicion en ingles con el mismo numeral. Ascension Molina y Ricardo Sanz: Un indicador mensual del consumo de energia electrica para usos industriales, 1976-1 984. industry. Contabilidad Nacional de 1970. valor anadido en la industria ante variaciones en los cestes laborales unitarios. Ignacio Maule6n: La inversion en bienes de equipo: determinantes y estabilidad. de las lineas aereas y las h ipotesis del census X-l 1 . (Publ icada una edicion en ingles con el mismo numeral. Ignacio Mauleon: A stabil ity test for simu ltaneous equation models. 8518 Jose Viiials: iAumenta la apertura financiera exterior las fluctuaciones del tipo de cambio? 8519 Jose Viiials: Deuda exterior y objetivos de balanza de pagos en Espana: Un analisis de largo plazo. 8520 (publicada una edicion en ingles con el ."ismo mjrnera). Jose Marin Areas: Algunos indices de progresividad de la imposicion estata l sobre la renta en Espana y otros paises de la OCDE. 860 1 Agustin Marayall: Revisions in ARI'v'IA signal extraction. 8603 Agustin Maravall: On minimum mean squared error estimation of the noise in unobserved 8602 8604 8605 8606 Agustin Marayall and Dayid A. Pierce: ... prototypical seasonal adjustment model. c:Jmponent models. Ignacio MauleDn: Testing the rational expectations model. Ricardo Sanz: Efectos de variaciones en los precios energeticos sobre los precios sectoria­ les y de la demanda final de nuestra economia. F. Martin Bourgon: Indices anuales de valor unitario de las exportaciones: 1972-1980. 8607 Jose Viiials: La politica fiscal y la restriccion exterior. (Publicada una edicion en inglE�s con 8608 Jose Viiials and John Cuddington: Fiscal policy and the current account: what do capital 8609 Gonzalo Gil: Politica agricola de la Comunidad Economica Europea y montantes campen­ el '11 ismo numeral. controls do? satorios monetarios. 8610 Jose Viiials: iHacia una menor flexibilidad de los tipos de cambio en el sistema monetario 870 1 Agustin Marayall: The use of ARIMA models in unobserved components estimation: an 8702 Agustin Maravall: Descomposicion de series temporales: especificacion, estimacion e internacional? appl ication to spanish monetary control. inferencia (Con una aplicacion a la oferta monetaria en Espana). - 36 - 8703 Jose Vhials y Lorenzo Domingo: la peseta y el sistema monetario europeo: un rnodelo de tipo de cambio peseta-marco. 8704 Gonzalo Gil: The functions ofthe Bank of Spain. 8705 Agustin Maravall: Descomposicion de series temporales, con una apl icacion a la oferta 8706 P. L'Hotellerie y J. Viilals: Tendencias del comercio exterior espanol. t>.pendice estadistico. monetaria en Esparia: Comentarios y contestacion. 8707 Anindya Banerjee and Juan Dolado : Tests ofthe Life Cycle-Permanent Income Hypothesis 8708 8709 Juan J. Dolado and TIm Jenkinson: Cointegration: A survey of recent developments. in the Presence of Random Walks: Asymptotic Theory and Small-Sample Interpretations. Ignacio Maul"';n: La demanda de dinero reconsiderada. 880 1 8802 Agustin Maravall: Two papers on arimB signal extraction. 8803 Agustin Maravall and Daniel Pena: l\/Ii55in9 observations in time series and the «dual» 8804 Jose Viiials: El 5istema Monetario Europeo. Espana y la politica macroeconomica. (Publi- 8805 Antoni Espasa : Metodos cua ntitativos y analisis de la coyuntura economica. 8806 8807 8808 890 1 8902 8903 8904 9001 9002 9003 9004 9005 9006 9007 Juan Jose Camio y Jose Rodriguez de Pablo: El conSU'TlO de ali mentos no elaborados en Esparia: Analisis de la inforrnaci6n de Mercasa. autocorrelation function. cada una edicion en ingles con el mismo numero). Antoni Espasa: El pertil de crecimiento de un fenomeno economico. Pablo Martin Acena: Una estimaci6n de los principales agregados monetarios en Espaiia: 1 940-1962. Rafael Repullo: Los efectos econo'T1icos de los coeficientes bancarios: u n analisis te6rico. M.a de los Llanos Matea Rosa : Funciones de transferencia simultaneas del indice de precios al consumo de bienes elaborados no energeticos. Juan J. Dolado: COintegraci6n: una panoramica. Agustin Maravall: La extracci6n de senales y el analisis de coyuntura. E. Morales, A. Espasa y M. L. Rojo : Metodos cuantitativos para el analisis de la actividad industrial espaiiola. (Publicada una edici6n en ingles con el mismo numeral. Jesus Albarracin y Concha Artola: El crecimiento de los salarios y el deslizamiento salarial en el periodo 1981 a 1 988. Antoni Espasa. Rosa G6mez-Churruca y Javier Jareno: Un analisis econometrico de los ingresos porturismo en la economia espaiiola. Antoni Espasa : Metodologia para realizar el analisis de la coyuntura de un fenomeno economico. (Publicada una edicion en ingles con el mismo numeral: Paloma Gomez Pastor y Jose Luis Pellicer Miret: Informaci6n y documentacion de las Comunidades Europeas. Juan J. Dolado. Tim Jenkinson and Simon Sosvilla-Rivero: Cointegration and unit roots: a survey. Samuel Bentolila and Juan J. Dolado : Mismatch and Internal Migration in Spain, 1962-1 986. Juan J. Dolado, John W. Galbraith and Anindya Banerjee: Estimating euler equations with integ rated series. 9008 Antoni Espasa y Daniel Peila: Los modelos ARIMA, el estado de equilibrio en variables 9009 Juan J. Dolado and Jose Vinals: lIJIacroeconomic policy, external targets and constraints: . the case of Spain. 9010 (1) econ6micos y su estirnacion. (Publicada una edicion en ingles con el mismo numeral. Anindya Banerjee, Juan J. Dolado and John W. Galbraith: Recursive and sequential tests for unit roots and structural br.eaks i n long annual GNP series. Los Docurnentos d e Trabajo anteriores a 1985 figuran en el cat.illogo de publicaciones del Banco de Esparia. Informaci6n: Banco de Esparia Secci6n de Publicaciones. Negociado de Distribuci6n y Gesti6n TelE'dono: 338 51 80 Alcalcfl, 50. 28014 Madrid