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JOURNAL OF APPLIED ECONOMETRICS J. Appl. Econ. 16 16:: 289–326 (2001) DOI: 10.1002/jae.616
BOUNDS TESTING APPROACHES TO THE ANALYSIS OF LEVEL RELATIONSHIPS M. HASHEM PESARAN,a * YONGCHEOL SHINb AND RICHARD J. SMITHc a
b
Trinity College, Cambridge CB2 1TQ, UK Department of Economics, University of Edinburgh, 50 George Square, Edinburgh EH8 9JY, UK c Department of Economics, University of Bristol, 8 Woodland Road, Bristol BS8 1TN, UK
SUMMARY This paper develops a new approach to the problem of testing the existence of a level relationship between a dependent variable and a set of regressors, when it is not known with certainty whether the underlying regressors are trend- or first-difference stationary. The proposed tests are based on standard F- and t -statistics used to test test the sig signifi nifican cance ce of the lag lagged ged levels levels of the variab variables les in a univar univariat iatee equ equili ilibri brium um correc correctio tion n mechanism. The asymptotic distributions of these statistics are non-standard under the null hypothesis that there exists no level relationship, irrespective of whether the regressors are I 0 or I1. Two sets of asymptotic critical criti cal values are provided: one when all regr regressors essors are purel purely y I1 and the other if they are all purely I0. These two sets of critical values provide a band covering all possible classifications of the regressors into purely I0, purely I1 or mutually cointegrated. Accordingly, various bounds testing procedures are proposed. It is shown that the proposed tests are consistent, and their asymptotic distribution under the null and suitably defined local alter alternativ natives es are derived. The empir empirical ical relevance relevance of the bound boundss proced procedures ures is demonstrated by a re-examination of the earnings equation included in the UK Treasury macroeconometric model. Copyright © 2001 John Wiley & Sons, Ltd.
1. INTRODU INTRODUCTIO CTION N Over the past decade considerable attention has been paid in empirical economics to testing for the existence existence of relations relationships hips in levels levels between between variables. variables. In the main, this analysis analysis has been based on the use of cointegration techniques. Two principal approaches have been adopted: the two-step residual-based procedure for testing the null of no-cointegration (see Engle and Granger, 1987; Phillips and Ouliaris, 1990) and the system-based reduced rank regression approach due to Johansen (1991, 1995). In addition, other procedures such as the variable addition approach of Park (1990), the residual-based procedure for testing the null of cointegration by Shin (1994), and the stochastic common trends (system) approach of Stock and Watson (1988) have been considered. All of these methods concentrate on cases in which the underlying variables are integrated of order one. This inevitably involves a certain degree of pre-testing, thus introducing a further degree of uncertainty into the analysis of levels relationships. (See, for example, Cavanagh, Elliott and Stock, 1995.) Thiss paper Thi paper propos proposes es a new approach approach to testin testing g for the exi existe stence nce of a relati relations onship hip betwee between n variables in levels which is applicable irrespective of whether the underlying regressors are purely
Ł Corre Corresponde spondence nce to: M. H. Pesaran, Pesaran, Faculty of Econom Economics ics and Politi Politics, cs, University University of Cambridge, Cambridge, Sidgwick Avenu Avenue, e, Cambridge CB3 9DD. E-mail:
[email protected] Contract/grant sponsor: ESRC; Contract/grant numbers: R000233608; R000237334. Contract/gra Contra ct/grant nt sponso sponsor: r: Isaac Newton Trust of Trinity Trinity College, College, Cambridge. Cambridge.
Copyright © 2001 John Wiley & Sons, Ltd.
Received 16 February 1999 Revised 13 February 2001
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M. H. PESAR PESARAN, AN, Y. SHIN AND R. J. SMITH
I(0), purely I(1) or mutually cointegrated. The statistic underlying our procedure is the familiar Wald or F-statistic in a generalized Dicky–Fuller type regression used to test the significance of lag lagged ged level levelss of the variable variabless under under consid considera eratio tion n in a conditional unrestricted equilibrium
correctio correc tion n mod model el (ECM (ECM). ). It is shown shown that that the asymp asymptot totic ic dis distri tribut bution ionss of both both sta statis tisti tics cs are non-standa non-s tandard rd under under the null hypothesis hypothesis that there there exists exists no relations relationship hip in leve levels ls between between the included variables, irrespective of whether the regressors are purely I (0), purely I(1) or mutually cointegrated. We establish that the proposed test is consistent and derive its asymptotic distribution under the null and suitably defined local alternatives, again for a set of regressors which are a mixture of I0/I 1 variables. Two sets of asymptotic critical values are provided for the two polar cases which assume that all the regressors are, on the one hand, purely I (1) and, on the other, purely I (0). Since these two sets of critical values provide critical value bounds for all classifications of the regressors into purely I(1), purely I (0) or mutually cointegrated, we propose a bounds testing procedure. If the computed Wald or F-statistic falls outside the critical value bounds, a conclusive inference can be drawn without needing to know the integration/cointegration status of the underlying regressors. However, if the Wald or F -statistic falls inside these bounds, inference is inconclusive and knowledge of the order of the integration of the underlying variables is required before conclusive inferences can be made. A bounds procedure is also provided for the related cointegration test proposed by Banerjee et al. (1998) which is based on earlier contributions by Banerjee et al. (1986) and Kremers et al. (1992). Their test is based on the t -statistic associated with the coefficient of the lagged dependent variable in an unrestricted conditional ECM. The asymptotic distribution of this statistic is obtained for cases in which all regressors are purely I (1), which is the primary context considered by these authors, as well as when the regressors are purely I(0) or mutually cointegrated. The relevant critical value bounds for this t-statistic are also detailed. The empirical relevance of the proposed bounds procedure is demonstrated in a re-examination of the earnin earnings gs equati equation on inc includ luded ed in the UK Tr Treas easury ury ma macro croeco econom nometr etric ic model. model. Thi Thiss is a particula parti cularly rly relevant relevant application application because there is consi considera derable ble doubt concerning concerning the order of integratio integ ration n of varia variables bles such as the degree degree of unionizat unionization ion of the workforce, workforce, the replacem replacement ent ratio (unemployment benefit–wage ratio) and the wedge between the ‘real product wage’ and the ‘real consumption wage’ that typically enter the earnings equation. There is another consideration in the choice of this application. Under the influence of the seminal contributions of Phillips (1958) and Sargan (1964), econometric analysis of wages and earnings has played an important role in the development of time series econometrics in the UK. Sargan’s work is particularly noteworthy as it is some of the first to articulate and apply an ECM to wage rate determination. Sargan, however, notitsconsider the problem of testing for the existence of a levels relationship between real wagesdid and determinants. The relationship in levels underlying the UK Treasury’s earning equation relates real average earnings of the private sector to labour productivity, the unemployment rate, an index of union density, a wage variable (comprising a tax wedge and an import price wedge) and the replacement ratio (defined as the ratio of the unemployment benefit to the wage rate). These are the variables predic pre dicted ted by the bargai bargainin ning g theory theory of wage wage determ determina inatio tion n rev review iewed, ed, for exa exampl mple, e, in Layard Layard et al. (1991). In order to identify our model as corresponding to the bargaining theory of wage determination, we require that the level of the unemployment rate enters the wage equation, but not vice versa; see Manning (1993). This assumption, of course, does not preclude the rate of change of earnings earnings from enter entering ing the unemploym unemployment ent equation, equation, or there there bein being g othe otherr level level relation relationships ships between betw een the remaining remaining four variables. variables. Our approach approach accommodate accommodatess both of these possibilitie possibilities. s. Copyright © 2001 John Wiley & Sons, Ltd.
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A number of conditional ECMs in these five variables were estimated and we found that, if a sufficiently high order is selected for the lag lengths of the included variables, the hypothesis that there exists no relationship in levels between these variables is rejected, irrespective of whether they are purely I (0), purely I (1) or mutually cointegrated. Given a level relationship between these variables, the autoregressive distributed lag (ARDL) modelling approach (Pesaran and Shin, 1999) is used to estimate our preferred ECM of average earnings. The plan of the paper is as follows. The vector autoregressive (VAR) model which underpins the analysis analysis of this this and later sections sections is set out in Sectio Section n 2. Thi Thiss sectio section n also also add addres resses ses the issues involved in testing for the existence of relationships in levels between variables. Section 3 considers consi ders the Wald statistic statistic (or the F-stat -statistic istic)) for testing the hypothesi hypothesiss that there exists no level relationship between the variables under consideration and derives the associated asymptotic theory together with that for the t -statistic of Banerjee et al. (1998). Section 4 discusses the power properties of these tests. Section 5 describes the empirical application. Section 6 provides some concluding remarks. The Appendices detail proofs of results given in Sections 3 and 4. The follow following ing notati notation on is used. used. The symbol symbol signifies signifies ‘weak ‘weak conve convergen rgence ce in proba probabili bility ty measure’, Im ‘an identity matrix of order m’, Id ‘integrated of order d’, OP K ‘of the same measure’, order as K in probability’ and oP K ‘of smaller order than K in probability’.
)
2. THE UNDERLYING UNDERLYING VAR VAR MODEL AND ASSUMPTIONS ASSUMPTIONS
f g1D is the
Let zt 1 k 1-vector random process. The data-generating process for zt tD1 denote a k VAR model of order p (VAR(p)):
C
f g
t 1
m g tt D e , t D 1, 2, . . . 1 where L is the lag operator, m and g are are unknown k C 1-vectors of intercept and trend coefficients, the k C 1, k C 1 matrix lag polynomial 8L D I C D 8 L with f g D k C 1, k C 1 matrices of unknown coefficients; see Harbo et al. (1998) and Pesaran, Shin and Smith (2000), henceforth HJNR and PSS respectively. The properties of the k C 1-vector error process fe g1 D are given in Assumption 2 below. All the analysis of this paper is conducted given the initial observations Z observations Z z , . . . , z . We assume: Assumption 1. The roo jI C D 8 z j D 0 are either outside the unit circle j zzj D 1 or roots ts of j satisfy z satisfy z 1. D Assumption 2. The vect is IN IN 0, Z , Z positive definite. vector or error error process process fe g1 D is 8Lzt
t
k 1
0
1 p
p i 1
i
p i i 1
i
t t 1
0
p i 1
k 1
i
i
t t 1
Assumption 1 permits the elements of zz t to be purely I (1), purely I (0) or cointegrated but excludes the possibility of seasonal unit roots and explosive roots.1 Assumption 2 may be relaxed somewhat to permit et 1 tD1 to be a conditionally mean zero and homoscedastic process; see, for example, PSS, Assumption 4.1. We may re-e re-expres xpresss the lag polynomia polynomiall 8L in vect vector or equilibri equilibrium um correcti correction on mode modell (ECM) form; i.e. 8L 5 L 0L1 L in which the long-run multiplier matrix is defined by 5
f g
C
1
ff g1tD1 to unity.
Assumptions 5a and 5b below further restrict the maximal order of integration of zt
Copyright © 2001 John Wiley & Sons, Ltd.
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I C 0 D k 1
i
M. H. PESAR PESARAN, AN, Y. SHIN AND R. J. SMITH
p 1 and the short-run response matrix lag polynomial 0L Ik C1 iD1 0i L i , D p jDiC1 j , i D 1, . . . , p 1. Hence, the VAR(p) model (1) may be rewritten in vector p i 1 8i ,
ECM form as
p 1
D a C a t C 5z
zt
0
t 1
1
C
0i zt i
D
i 1
Ce
t
D 1, 2, . . .
t
2
1 L is the difference operator, a 5m C 0 C 5g , a 5g 3 and the sum of the shortshort-run run coeffi coefficie cient nt matric matrices es 0 I D 0 D 5 C D i8 . As detail det ailed ed in PSS PSS,, Se Secti ction on 2, if g 6 D 0, the result resultant ant con constr strain aints ts (3) on the tre trend nd coe coeffi fficie cients nts a 1 in (2) ensure that the deterministic trending behaviour of the level process fz g D is invariant to m D 6 0 and g D 0. the (cointegrating) rank of 5; a similar result holds for the intercept of fz g1 D if m where
0
1
m
p 1 i 1
i
t t 1
p i 1
i
1
t t 1
Consequently, critical regions defined in terms of the Wald and F-statistics suggested below are asymptotically similar .2 The focus of this paper is on the conditional modelling of the scalar variable y t given the k 1 and xt and the past values zti tiD Z 0 , where we have partitioned z partitioned z t y t , x0t 0 . Partitioning vector x vector 1 and Z
f g
D
D y0 , x0 0 as e D ε ω w ZD w
the error term et conformably with with zt
t
t
yy
yx
xy
xx
0 0 and its variance matrix as
yt , e xt
t
we may express ε yt conditionally in terms of e xt as εyt
1 yx Z xx e xt
D w
Cu
4
t
1 w xy and ut is independent of e xt . Substitution of (4) ωyy wyx Z xx where ut IN0, ωuu , ωuu 0 , 50 0 , into (2) together with a similar partitioning of of a0 ay 0 , a x 0 0 0 , a1 ay 1 , a x 0 1 0 , 5 py 0 , 00 0 , i 1, . . . , p 1, provides a conditional model for y t in terms x of 0 g 0y , 0 x 0 0 , 0i g yi xi zt1 , xt , zt1 , . . .; i.e. the conditional ECM
¾
D
D
D
D
D
D
p 1
D c C c t C p
y t
0
1
y.x zt 1
C D
i 1
y0i zti
C w0 x C u t
t
1 w xy , c 0 ay 0 w0 a x 0 , c 1 ay 1 w0 a x 1 , y 0i g yiyi xx where w py.x py w0 x . The deterministic relations (3) are modified to
c0
D p
y.x m
y.x y.x
D 1, 2, . . .
w0 0
C g C p
t
y.x g
xi , i
D 1, . . . , p 1, and
D p
c1
y.x g
g yy w0 0 x . where g yy.x .x We now partition the long-run multiplier matrix 5 conformably with with zt
2
D
yy pyx p xy 5 xx
5
6
D y , x0 0 as t
t
See also Nielsen and Rahbek (1998) for an analysis of similarity issues in cointegrated systems.
Copyright © 2001 John Wiley & Sons, Ltd.
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The next assumption is critical for the analysis of this paper. Assumption 3. The The k -vector p p xy k -vector
0.
D
In the application of Section 6, Assumption 3 is an identifying assumption for the bargaining theory of wage determination. Under Assumption 3,
p 1
D a C a t C 5
xt
x 0
xx xt 1
x 1
C
0 xi zt i
D
i 1
Ce
xt
t
D 1, 2, . . . .
7
1 Thus, we may regard the process xt 1 tD1 as long-run forcing for y t tD1 as there is no feedback 3 from the level of y t in (7); see Granger and Lin (1995). Assumption 3 restricts consideration to xt , irrespective cases in which there exists at most one conditional level relationship between y t and and x 1 4 of the level of integration of the process xt tD1 ; see (10) below. Under Assumption 3, the conditional ECM (5) now becomes
f g
f g
f g
p 1
D c C c t C
y t
0
1
yy y t 1
D 1, 2, . . ., where c D 0 and p p w 5 .
Cp
yx.x xt 1
C D
i 1
y0i zti
C w0x C u t
t
8
t
yy , pyx.x m
0
yx.x
yx
xx
C [g C y.x y.x
D
yy , pyx.x ]g , c1
yy , pyx.x g
9
5
The next assumption together with Assumptions 5a and 5b below which constrain the maximal order of integration of the system (8) and (7) to be unity defines the cointegration properties of the system. Assumption 4. The matrix 5 xx has rank r r , , 0
r k .
0 , where a xx and b xx are both Under Assumption 4, from (7), we may express 5 xx as 5 xx a xx b xx k,r matrices of full column rank; see, for example, Engle and Granger (1987) and Johansen (1991). If the maximal order of integration of the system (8) and (7) is unity, under Assumptions mutually ally cointegrated cointegrated of order order r , 0 r k . Ho Howev wever, er, in 1, 3 an and d 4, the proces processs xt 1 tD1 is mutu contradis contr adistinct tinction ion to, for example, example, Banerjee, Banerjee, Dolado Dolado and Mest Mestre re (1998), (1998), BDM hencefor henceforth, th, who concentrate on the case r 0, we do not wish to impose an a priori specification of r .6 When then xt is weakly exogenous for yy and p yx.x pyx in (8); see, for example, p xy 0 and 5 xx 0, then x
D
D
D
f g D D
D
1 Note that this restriction does not preclude yt t1 D1 being Granger-causal for xt tD1 in the short run. Assump Assumption tion 3 may be straig straightfo htforwa rwardl rdly y ass assess essed ed via a test test for the exclusi exclusion on of the lagged lagged level y t1 in (7). The asymptotic properties of such a test are the subject of current research. 5 PSS and HJNR consider a similar model but where x t is purely I 1; that is, under the additional assumption 5 xx 0. If current and lagged lagged values of a weakl weakly y exogenous purely I 0 vector vector w w t are included as additional explanatory variables 1 in (8), the lagged level vector x t1 should be augmented to include the cumulated sum st D1 ws in order to preserve the asymptotic similarity of the statistics discussed below. See PSS, sub-section 4.3, and Rahbek and Mosconi (1999). 6 BDM, pp. 277–278, also briefly discuss the case when 0 < r k . However, in this circumstance, as will become clear below, the validity of the limiting distributional results for their procedure requires the imposition of further implicit and untested assumptions. 3
4
f g
f g
Copyright © 2001 John Wiley & Sons, Ltd.
D
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M. H. PESAR PESARAN, AN, Y. SHIN AND R. J. SMITH
Johansen (1995, Theorem 8.1, p. 122). In the more general case where 5 xx is non-zero, as yy and pyx.x pyx w0 5 xx are variation-free from the parameters in (7), x t is also weakly exogenous for the parameters of (8).
D
Note No te tha thatt under under Ass Assump umptio tion n 4 the ma maxim ximal al coi cointe ntegra gratin ting g rank rank of the long-r long-run un mul multip tiplie lierr matrix 5 for the system (8) and (7) is r 1 and the minimal cointegrating rank of 5 is r . The next assumptions provide the conditions for the maximal order of integration of the system (8) and (7) to be unity. First, we consider the requisite conditions for the case in which rank5 r . -vector f. In this case, under Assumptions 1, 3 and 4, yy 0 and pyx f0 5 xx 00 for some k -vector 0 Note that p yx.x 0 implies the latter condition. Thus, under Assumptions 1, 3 and 4, 5 has rank r and is given by 0 pyx 0 5 xx
C
D
D
D
D
D
0 , a0 0 and b 0, b0 0 are k 1, r matrices of Hence, we may express 5 ab0 where a ayx xx xx ? , a? full column rank; cf. HJNR, p. 390. Let the columns of the k 1, k r 1 matrices ay ? where a? , b? and a? , b? are respective respectively ly k 1-vec -vectors tors and k 1, k r and b? y y y , b , where matrices, matr ices, denote bases for the orthogona orthogonall compleme complements nts of respectiv respectively ely a and b; in particul particular, ar, ? ? 0 ? ? 0 ay , a a 0 and by , b b 0.
D
C D C C C
D
D
C
D
If rank5 Assumption 5a. If rank
matrix a? , a? 0 0b? , b? is full rank k rank k r C 1 , 0 r D r , the matrix k . y
y
Cf. Johansen (1991, Theorem 4.1, p. 1559). Second, if the long-run multiplier matrix 5 has rank r r 1, then under Assumptions 1, 3 and 4, 0 0 0 0 yy 0 and 5 may be expressed as 5 ay by ab , where ay ˛yy , 00 0 and b y ˇyy , byx are k 1-vectors, the former of which preserves Assumption 3. For this case, the columns of a ? and b ? form respective bases for the orthogonal complements of ay , a and by , b; in particular, a?0 ay , a 0 and b ?0 by , b 0.
6D
C
D
C
C
D
D
D k . Assumption 5b. If rank rank5 D r C 1 , the matrix a a ?0 0b? is full rank k k r , 0 r fx g1D is1a Assumptions 1, 3, 4 and 5a and 5b permit the two polar cases for f x g1 D . First, if f fx g D purely I0 vector process, then 5 , and, hence, a and b , are nonsingular. Second, if f is purely I1, then 5 D 0, and, hence, a and b are also null matrices. Ł D
t t 1
t t 1
xx
xx
t t 1
xx
xx
xx
xx
Us Usin ing g (A (A.1 .1)) in Appe Append ndix ix A, it is easi easily ly se seen en th that at py.x zt
py.x C Let , wh wher eree
g t t
m
a mean zero stationary process. Therefore, under Assumptions fCŁ 6L D1, 3, 4 and 5b, that is, D 0,e git isimmediately follows that there exists a conditional level relationship between y and t
yy
t
xt defined by
D C t C q x C , t D 1, 2, . . . 10 where p m/ , p g / / , q p / and D p CŁ Lε / , also a zero mean D 00, the level relation stationar stati onary y proce process. ss. If p D ˛ b0 C a w a b0 6 relationship ship betw between een y and x is non-degenerate. Hence Hence,, fro from m (10), (10), y ¾ I0 if rankb , b D r and y ¾ I1 if rankb , b D r C 1. In the former case, q is the vector of conditional long-run multipliers and, in this sense, (10) may be interpreted as a conditional long-run level relationship between y and 1 x , whereas, in the latter, because the processes fy g1 D and fx g D are cointegrated , (10) represents 0
y.x
yy
1
yx.x
y t
0
y.x
yy
yy yx
t
1
t
yx.x
yx
vt
v t
yy
xx
yy
t
xx
t
yx
t
y.x
yx
xx
t
xx
t
t
t t 1
t t 1
the conditional long-run level relationship between y t and and xt . Two degenerate cases arise. First,
Copyright © 2001 John Wiley & Sons, Ltd.
J. Appl. Econ. 16: 16 : 289–326 (2001)
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295
if yy 0 and pyx.x 00 , clearly, from (10), y t is (trend) stationary or y t I0 whatever the value of r r . Consequently, the differenced variable y t depends only on its own lagged level y t1 in the conditional ECM (8) and not on the lagged levels levels xt1 of the forcing variables. Second, if
6D
D
¾
0 yy 0,0 that 0is, Assumption 5a holds, and p yx.x ayx w0 a xx b xx 00 , as rank5 r , p yx.x which, fro from m the above, above, yields yields pyx.x xt m x g x t py.x CŁ Let , t 1, 2, . . ., f w a xx b xx which, y t , x0t 0 . Thus, in where m y , m x 0 0 and g yy , g x 0 0 are partitioned conformably with zt (8), y t depends only on the lagged level x level x t1 through the linear combination f w0 a xx of the 0 xt1 for the process xt 1 . Consequently, y t I1 lagged mutually cointegrating relations b xx t D1 whatever the value of r . Finally, if both both yy 0 and pyx.x 00 , there are no level effects in the conditional ECM (8) with no possibility of any level relationship between y t and and xt , degenerate r . or otherwise, and, again, y t I1 whatever the value of r Therefore, in order to test for the absence of level effects in the conditional ECM (8) and, more crucially, the absence of a level relationship between y t and and xt , the emphasis in this paper is a test of the joint hypothesis yy 0 and pyx.x 00 in (8).7,8 In contradistinction, the approach of BDM may be described in terms of (8) using Assumption 5b:
6DD
D
D
D
D
f g
D
D
D
D
DD
¾
¾
D
D
yy ˇyy y t 1
C b0
w0 xt
ut
D c C c t C ˛
y t
0
1
yx xt 1
C a w0 a
0
xx b xx xt 1
yx
p 1
y0i zt
i
11
C C C iD1 D 0, that is, b xx D 0 in (11) or 5 xx D 0 in BDM test for the exclusion of y t1 in (11) when r D
(7) and, thus, xt is purely I1; cf. HJNR and PSS.9 Therefore, BDM consider the hypothesis generally, when 0 < r k , BD BDM M req requir uiree the impos impositi ition on of the ˛yy 0 (or yy 0).10 More generally, 0 0 a w a 0 untested subsidiary hypothesis yx ; that is, the limiting distribution of the BDM test xx is obtained under the joint hypothesis yy 0 and p yx.x 0 in (8). In the following sections of the paper, we focus on (8) and differentiate between five cases of interest delineated according to how the deterministic components are specified:
f g D
D
D
D D
ž Case I (no intercept intercepts; s; no trends) trends) c D 0 and c D 0. That is, m D 0 and g D 0. Hence, the 0
1
Cp
C
ECM (8) becomes
p 1
D
y t
yy y t 1
yx.x xt 1
Case II (restricted (restricted intercept intercepts; s; no trends) trends) c0
ž ECM is D
y t
yy y t 1
Cp y
D
i 1
y0i zti
t
t 1
x
D
i 1
y0i zti
0. The
0. Here Here,, g
D
p 1
12
t
yy , pyx.x m and c1
D x m C
yx.x
C w0 x C u
D
C w0x C u t
t
13
7 This joint hyp hypoth othesi esiss may be jus justifi tified ed by the applic applicati ation on of Roy Roy’s ’s uni unionon-inte interse rsecti ction on prin princip ciple le to tests tests of yy 0 in (8) giv given en pyx.x . Let Wyy pyx.x be the Wald Wald statist statistic ic for testin testing g yy 0 fo forr a gi give ven n va valu luee of pyx.x . The The te test st yy 0 and and p p yx.x 0 in (8). maxyx.x W yy pyx.x is identical to the Wald test of 8 A related approach to that of this paper is Hansen’s (1995) test for a unit root in a univariate time series which, in our context, would require the imposition of the subsidiary hypothesis p yx.x 00 . 9 The BDM test is based on earlier contributions of Kremers et al. (1992), Banerjee et al. (1993), and Boswijk (1994). 10 Partitioning 0 xi g xy,i , 0 xx,i , i 1, . . . , p 1, confor conformab mably ly with zt y t , x0t 0 , BDM BDM al also so se sett g xy,i 0, i 1, . . . , p 1, which implies g xy 0, where 0 x g xy , 0 xx ; that is, y t does not Granger cause cause xt . xy
D
D
D
D
Copyright © 2001 John Wiley & Sons, Ltd.
D D
D
D
D
D
D
D
J. Appl. Econ. 16: 16 : 289–326 (2001)
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M. H. PESAR PESARAN, AN, Y. SHIN AND R. J. SMITH
D 0 and c D 0. ž Case III (unrestric (unrestricted ted intercep intercepts; ts; no trends) trends) c 6 intercept restriction c D , p m is ignored and the ECM is 0
Ag Again ain,, g
1
0
yx.x
yy
D 0.
No Now, w, th thee
14
p 1
D c C
y t
yy y t 1
0
Cp
yx.x xt 1
y0 z C w0 x C u ti t t i iD1
C
ž Case IV (unrestricted IV (unrestricted intercepts; restricted trends) c 6 D 0 and c D 0
yy , pyx.x g .
1
p 1
D c C
y t
t C p
yy y t 1
0
y y
yx.x xt 1
g t x
C D
i 1
y0i zti
C w0 x C u t
t
15
D 0. Here, the deterministic D 0 and ž Case V (unrestricted and c c 6 V (unrestricted intercepts; unrestricted trends) c 6 trend restriction c D , p is ignored and the ECM is 1
0
1
yy
yx.x
p 1
D c C c t C
y t
0
1
yy y t 1
Cp
yx.x xt 1
C iD1
y0i zti
C w0 x C u
t
t
16
It should be emphasized that the DGPs for Cases II and III are treated as identical as are those for Cases IV and V. However, as in the test for a unit root proposed by Dickey and Fuller (1979) compared with that of Dickey and Fuller (1981) for univariate models, estimation and hypothesis testing in Cases III and V proceed ignoring the constraints linking respectively the intercept and trend coefficient, c0 and c1 , to the parameter vector yy , pyx.x whereas Cases II and IV fully incorporate the restrictions in (9). In the following exposition, we concentrate on Case IV, that is, (15), which may be specialized to yield the remainder.
3. BOUNDS TESTS TESTS FOR A LE LEVEL VEL RELA RELATIONSHIP TIONSHIPS S In this section we develop bounds procedures for testing for the existence of a level relationship using (12)– (12)– (16); (16); see (10). The main main approa approach ch tak taken en here, here, cf. Eng Engle le and between y t and xt using Granger (1987) and BDM, is to test for the absence of any level relationship between y t and xt via the exclusion of the lagged level variables y t1 and and xt1 in (12)–(16). Consequently, we yy yx.x define the constituent null hypotheses H0 : yy 0, H0 : p yx.x 00 , and alternative hypotheses : p yx.x 00 . Hence, Hence, the joint joint null null hypoth hypothesi esiss of intere interest st in (12)– (12)– (16 (16)) is H1 yy : yy 0, H1 yx.x given by: yy yx.x H0 H0 17 H0
6 6D
D D \
6 6D
D
and the alternative hypothesis is correspondingly stated as: H1
yy 1
yx.x 1
D H [ H
18
However, as indicated in Section 2, not only does the alternative hypothesis H1 of (17) cover the case of interest in which yy 0 and pyx.x 00 but also permits yy 0, pyx.x 00 and yy 0 degenerate level relationships between y t and x and p yx.x 00 ; cf. (8). That is, the possibility of degenerate and x t is admitted under H 1 of (18). We comment further on these alternatives at the end of this section.
6D
6D
Copyright © 2001 John Wiley & Sons, Ltd.
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6D
D
D
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For ease of exposition, we consider Case IV and rewrite (15) in matrix notation as
D i c C ZŁ pŁ C Zy C u 19 where i is a T-v -vec ecto torr of on ones es,, y y , . . . , y 0 , X x , . . . , x 0 , Z 0 z , . . . , z , i D 1, . . . , p 1, y w0 , y0 , . . . , y0 0 , Z X, Z , . . . , Z , Z Ł t , Z , t 1, . . . , T 0 , Z z , . . . , z 0 , u u , . . . , u 0 and g 0 0 pŁ D p IC y
T 0
1 y.x 1
T
1 i
T i
1 p
1
1
T
T
T 1
0
i
1
p 1
1
T
1
1
T
1
T
yy
y.x
k 1
yx.x
Ł is given by: The least squares (LS) estimator of py.x
˜ ZŁ0 P
Ł ˆ py.x
O Q
˜ Ł where Z 1
P ZŁ ,
1
Z
0 ˜ ZŁ1 1 ˜ ZŁ1 PZ y
P y, P I i
20
1 i 0
0
and PZ IT The Wald and the F-stat -statistic isticss for testing testing the null hypothes hypothesis is H0 of (17) against the alternative hypothesis H1 of (18) are respectively:
1
Z
0 0 Z Z Z 1 Z .
P Z , y
T iT iT
T
T
0 Ł /ωuu , F W Ł0 ˜ py.x ZŁ1 ˆ ZŁ1 PZ ˜ p ˆ y.x 21 k 2 T 1 2 T m k 1p 1 1 is the number of estimated coefficients where ωuu tD1 ut , m and ut , t 1, 2, . . . , T, are the least squares (LS) residuals from (19). The next next theore theorem m pre presen sents ts the asympt asymptoti oticc null null distri distribut bution ion of the Wald sta statis tistic tic;; the lim limit it behaviour beha viour of the F-stat -statist istic ic is a simple simple coroll corollary ary and is not presen presented ted her heree or subseq subsequen uently tly.. 0 0 Let W Let W k r C1 a Wu a, Wk r a denote a k r 1-dimensional standard Brownian motion partitione parti tioned d into the scalar scalar and k r-dim -dimensio ensional nal sub-vecto sub-vectorr independe independent nt stand standard ard Brown Brownian ian motions Wu a and and Wk r a, a [0, 1]. We will also require the corresponding de-meaned k 1 ˜ k r C1 a Wk r C1 a -vector tor standard standard Brownian Brownian motion motion W and d deder 1-vec 0 Wk r C1 ada, an ˜ ˆ meaned and de-trended k r 1-vector standard Brownian motion Wk r C1 a Wk r C1 a 1 ˜ k r C1 a ˜ k r C1 ada, and their 12 a 21 0 a 12 W their respective respective partitioned partitioned coun counterpa terparts rts W ˆ k r a0 0 , a [0, 1]. ˜ k r a0 0 , and W ˆ k r C1 a Wu a, W Wu a, W
W
C
Q D
C
Q
2 C
C OC
C
C
D O
2
D
Theorem 3.1 (Limiting (Limiting distribut distribution ion of of W) If Assumptions 1–4 and 5a hold, then under under H0 : 0 p yx.x 0 of (17), as T yy 0 and p as T , the asymptotic distribution of the Wald statistic statistic W of (21) has the representation
D
D
W ) z0 zr C r
1
0
!1
D
dWu aFk r C1 a0
1
0
Fk r C1 aFk r C1 a0 da
1
1
0
Fk r C1 adWu a 22
¾ N0, I is distributed independently of the second term in (22) and
where z r
r
Fk r C1 a
Wk r C1 a 0 0 Wk r C1 a , 1 ˜ k r C1 a W ˜ k r C1 a0 , a 1 0 W 2 ˆ k r C1 a W
Case I Case II Case III Case IV Case V
D D 0, . . . , k , and Cases I –V are defined in (12)–(16), (12) –(16), a 2 [0, 1].
r
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M. H. PESAR PESARAN, AN, Y. SHIN AND R. J. SMITH
The asymptoti asymptoticc distribut distribution ion of the Wald stat statisti isticc W of (21) (21) dep depend endss on the dimen dimensio sion n and cointegration cointegrat ion rank of the forci forcing ng variable variabless xt , k and r respectivel respectively. y. In Case IV IV,, refe referring rring to (11), the first component in (22), zr 0 zr 2 r, corresponds to testing for the exclusion of the r -
f g
0 xt1 , that is, the hypothesis a yx w0 a xx 00 , whereas the second dimensional stationary vector b xx term in (22), which is a non-standard Dickey–Fuller unit-root distribution, corresponds to testing ? , b? 0 zt1 and, in Cases II and for the exclusion of the k r 1-dimensional I1 vector by IV, the intercept and time-trend respectively or, equivalently, ˛yy 0. We specialize Theorem 3.1 to the two polar cases in which, first, the process for the forcing variables xt is purely integrated of order zero, that is, r k and 5 xx is of full rank, and, second, the xt process is not mutually cointegrated, r 0, and, hence, the xt process is purely integrated of order one.
¾
C
D
f g
f g
D
D D
D D
f g
f g ¾ D
Corollary 3.1 (Limiting (Limiting distributi distribution on of of W if xt I0). If Assu Assump mpti tion onss 1– 4 and and 5a hold and r k , that is, xt and I0 , then under H0 : yy 0 and pyx.x 00 of (17), as T , the asymptotic distribution of the Wald statistic W of (21) has the representation
D D
f g ¾
D Q Q O
W ) z0 zk C k
!1
D
1 2 0 FadWu a 1 0 Fa2 da
23
¾ N0, I is distributed independently of the second term in (23) and
where z k
k
Fa
Wu a 0 Wu a, 1 Wu a Wu a, a, a 12 0
Case I Case II Case III Case IV
Wu a
Case V
D D 0, . . . , k , where Cases I –V are defined in (12)–(16), (12) –(16), a 2 [0, 1]. Assump mpti tion onss 1– 4 and and 5a hold Corollary 3.2 (Limiting (Limiting distributi distribution on of of W if fx g ¾ I1). If Assu 0 D 0 , that is, fx g ¾ I1 , then under H : D 0 and p D 0 of (17), as T ! 1 , the and r D and r
t
0
t
yx.x
yy
asymptotic distribution of the Wald statistic W of (21) has the representation 1
dWu aFk
W
)
0
0 1 a
C
1
Fk 1 aFk
0
0 1 a da
C
C
a is defined in Theorem 3.1 for Cases I–V, a where F k C1 a is
1
1
Fk 1 adWu a
2 0
C
[0, 1].
In practice, however, it is unlikely that one would possess a priori knowledge of the rank r of 5 xx ; that is, the cointegration rank of the forcing variables xt or, more particularly, whether xt I0 or xt I1. Long-run analysis of (12)–(16) predicated on a prior determination of the cointegratio cointegration n rank r in (7) is prone prone to the possibil possibility ity of a pre-te pre-test st specifi specificat cation ion error; error; see, for example, example, Cavanagh et al. (1995) (1995).. Howeve However, r, it may be sho shown wn by sim simula ulatio tion n that that the asymptotic critical values obtained from Corollaries 3.1 (r k and xt I0) and 3.2 (r 0 1 ) pro provid vide e lower low er and upper upp er bounds bou nds res respec pectiv tivel ely y for those tho se cor corres respon pondin ding g to the and xt I genera gen erall case case con consid sidere ered d in Theore Theorem m 3.1 whe when n the cointe cointegra gratio tion n rank rank of the forcin forcing g var variab iables les
f g¾
f g
f g ¾
D D
f g ¾
Copyright © 2001 John Wiley & Sons, Ltd.
f g ¾
D D
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is 0 r k .11 Hence Hence,, these these two two set setss of critic critical al val values ues pro provid videe critical critical value value bounds covering covering all possible possible classifica classifications tions of xt into I0, I1 and mutu mutually ally cointegra cointegrated ted processes. Asymptotic critical value bounds for the F -statistics covering Cases I–V are set out in
fx g proc proces esss t
f g
Tables CI(i)–CI(v) for sizes 0.100, 0.050, 0.025 and 0.010; the lower bound values assume that the forcing variables xt are purely I 0, and the upper bound values assume that xt are purely I1.12 Hence, we suggest a bounds procedure to test H0 : yy 0 and pyx.x 00 of (17) within the conditional ECMs (12)–(16). If the computed Wald or F-statistics fall outside the critical value bounds, a conclusive decision results without needing to know the cointegration rank r of the xt process. If, however, the Wald or F-statistic fall within these bounds, inference would be inconclusive. In such circumstances, knowledge of the cointegration rank r of the forcing variables xt is required to proceed further. The conditional ECMs (12)–(16), derived from the underlying VAR(p) model (2), may also be interpreted as an autoregressive distributed lag model of orders ( p , p , . . . , p) (ARDL(p , . . . , p)). Howeve How ever, r, one could could also also allow allow for differ different ential ial lag length lengthss on the lag lagged ged variab variables les y ti and xti in (2 (2)) to ar arri rive ve at at,, fo forr exam exampl ple, e, an ARDL ARDL((p, p1 , . . . , pk ) model model wi witho thout ut affec affectin ting g the asymptotic results derived in this section. Hence, our approach is quite general in the sense that one can use a flex flexibl iblee choice choice for the dynamic dynamic lag str struct ucture ure in (12)– (12)– (16 (16)) as we well ll as all allow owing ing
f g
f g
D
D
f g f g
for short-run -run r,feedback feed backs s from lagged lagg ed depen dent variable y tisi , isi more 1, . .general . , p, tothan xt the in (7). short Moreove Moreover, within with in the single-eq singlthe e-equati uation on dependent context, context, thevariables, above s,analysis analys cointegration analysis of partial systems carried out by Boswijk (1992, 1995), HJNR, Johansen x t is purely (1992, 1995), PSS, and Urbain (1992), where it is assumed in addition that 5 xx 0 or or x I1 in (7). To conclude this section, we reconsider the approach of BDM. There are three scenarios for the deterministics given by (12), (14) and (16). Note that the restrictions on the deterministics’ coefficients (9) are ignored in Cases II of (13) and IV of (15) and, thus, Cases II and IV are now subsumed by Cases III of (14) and V of (16) respectively. As noted below (11), BDM impose but do not test the implicit hypothesis ayx w0 a xx 00 ; that is, the limiting distributional results given below are also obtained under the joint hypothesis H0 : yy 0 and p yx.x 00 of (17). BDM yy test ˛yy 0 (or H0 : yy 0) via the exclusion of y t1 in Cases I, III and V. For example, in Case V, they consider the t -statistic
D
D
D
D
D
D
D
tyy
ˆy01 PZ , ˆX y 1 1/2 ˆy 1/2 ω ˆy P uu 01 Z ,X ˆ 1 1
D
24
O D O
where ωuu is defined in the line after (21), y PT , T y, yˆ 1 PT , T y1 , y1 0 0 ˆ PT PT , T X1 , X1 PT , T Z , PT , T y 0 , . . . , yT 1 , X1 x0 , . . . , xT1 , Z 0 0 1 ˆ 0 1 0 ˆ ˆ ˆ PZ X1 X1 PZ X1 X1 PZ and PZ PZ PT t T t T PT t T t T PT , PZ , ˆX 1 0 0 1 IT Z Z Z Z .
11
The critical values of the Wald and F -statistics in the general case (not reported here) may be computed via stochastic simulations with different combinations of values for k and 0 r k . 12 The critical values for the Wald version of the bounds test are given by k 1 times the critical values of the F -test in Cases I, III and V, and k 2 times in Cases II and IV.
C
Copyright © 2001 John Wiley & Sons, Ltd.
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M. H. PESAR PESARAN, AN, Y. SHIN AND R. J. SMITH
Tabl ablee CI. CI. As Asymp ymptot totic ic critic critical al value value bou bounds nds for the F-stati -statisti stic. c. Test esting ing for the exi existe stence nce of a levels levels relationshipa Table Table CI(i) Ca Case se I: No intercept intercept and no trend 0.100
0.050
0.025
0.010
Mean
Variance
k
I0
I1
I0
I1
I0
I1
I0
I1
I0
I1
I0
I1
0 1 2 3 4 5 6 7 8 9 10
3.00 2.44 2.17 2.01 1.90 1.81 1.75 1.70 1.66 1.63 1.60
3.00 3.28 3.19 3.10 3.01 2.93 2.87 2.83 2.79 2.75 2.72
4.20 3.15 2.72 2.45 2.26 2.14 2.04 1.97 1.91 1.86 1.82
4.20 4.11 3.83 3.63 3.48 3.34 3.24 3.18 3.11 3.05 2.99
5.47 3.88 3.22 2.87 2.62 2.44 2.32 2.22 2.15 2.08 2.02
5.47 4.92 4.50 4.16 3.90 3.71 3.59 3.49 3.40 3.33 3.27
7.17 4.81 3.88 3.42 3.07 2.82 2.66 2.54 2.45 2.34 2.26
7.17 6.02 5.30 4.84 4.44 4.21 4.05 3.91 3.79 3.68 3.60
1.16 1.08 1.05 1.04 1.03 1.02 1.02 1.02 1.02 1.02 1.02
1.16 1.54 1.69 1.77 1.81 1.84 1.86 1.88 1.89 1.90 1.91
2.32 1.08 0.70 0.52 0.41 0.34 0.29 0.26 0.23 0.20 0.19
2.32 1.73 1.27 0.99 0.80 0.67 0.58 0.51 0.46 0.41 0.37
Table CI(ii) Case II: Restricted intercept and no trend 0.100
0.050
0.025
0.010
Mean
Variance
k
I0
I1
I0
I1
I0
I1
I0
I1
I0
I1
I0
I1
0 1 2 3 4 5 6 7 8 9 10
3.80 3.02 2.63 2.37 2.20 2.08 1.99 1.92 1.85 1.80 1.76
3.80 3.51 3.35 3.20 3.09 3.00 2.94 2.89 2.85 2.80 2.77
4.60 3.62 3.10 2.79 2.56 2.39 2.27 2.17 2.11 2.04 1.98
4.60 4.16 3.87 3.67 3.49 3.38 3.28 3.21 3.15 3.08 3.04
5.39 4.18 3.55 3.15 2.88 2.70 2.55 2.43 2.33 2.24 2.18
5.39 4.79 4.38 4.08 3.87 3.73 3.61 3.51 3.42 3.35 3.28
6.44 4.94 4.13 3.65 3.29 3.06 2.88 2.73 2.62 2.50 2.41
6.44 5.58 5.00 4.66 4.37 4.15 3.99 3.90 3.77 3.68 3.61
2.03 1.69 1.52 1.41 1.34 1.29 1.26 1.23 1.21 1.19 1.17
2.03 2.02 2.02 2.02 2.01 2.00 2.00 2.01 2.01 2.01 2.00
1.77 1.01 0.69 0.52 0.42 0.35 0.30 0.26 0.23 0.21 0.19
1.77 1.25 0.96 0.78 0.65 0.56 0.49 0.44 0.40 0.36 0.33
Table Table CI(iii CI(iii)) Case III: Unrestricte Unrestricted d intercept and no trend 0.100
0.050
0.025
0.010
Mean
Variance
k
I0
I1
I0
I1
I0
I1
I0
I1
I0
I1
I0
I1
0 1 2 3 4 5 6 7 8 9 10
6.58 4.04 3.17 2.72 2.45 2.26 2.12 2.03 1.95 1.88 1.83
6.58 4.78 4.14 3.77 3.52 3.35 3.23 3.13 3.06 2.99 2.94
8.21 4.94 3.79 3.23 2.86 2.62 2.45 2.32 2.22 2.14 2.06
8.21 5.73 4.85 4.35 4.01 3.79 3.61 3.50 3.39 3.30 3.24
9.80 5.77 4.41 3.69 3.25 2.96 2.75 2.60 2.48 2.37 2.28
9.80 6.68 5.52 4.89 4.49 4.18 3.99 3.84 3.70 3.60 3.50
11.79 6.84 5.15 4.29 3.74 3.41 3.15 2.96 2.79 2.65 2.54
11.79 7.84 6.36 5.61 5.06 4.68 4.43 4.26 4.10 3.97 3.86
3.05 2.03 1.69 1.51 1.41 1.34 1.29 1.26 1.23 1.21 1.19
3.05 2.52 2.35 2.26 2.21 2.17 2.14 2.13 2.12 2.10 2.09
7.07 2.28 1.23 0.82 0.60 0.48 0.39 0.33 0.29 0.25 0.23
7.07 2.89 1.77 1.27 0.98 0.79 0.66 0.58 0.51 0.45 0.41
(Continued overleaf ) )
Copyright © 2001 John Wiley & Sons, Ltd.
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Table CI. (Continued ) Table CI(iv CI(iv)) Case IV: Unrestricted Unrestricted intercept intercept and restr restricted icted trend
0.100
0.050
0.025
0.010
Mean
Variance
k
I0
I1
I0
I1
I0
I1
I0
I1
I0
I1
I0
I1
0 1 2 3 4 5 6 7 8 9 10
5.37 4.05 3.38 2.97 2.68 2.49 2.33 2.22 2.13 2.05 1.98
5.37 4.49 4.02 3.74 3.53 3.38 3.25 3.17 3.09 3.02 2.97
6.29 4.68 3.88 3.38 3.05 2.81 2.63 2.50 2.38 2.30 2.21
6.29 5.15 4.61 4.23 3.97 3.76 3.62 3.50 3.41 3.33 3.25
7.14 5.30 4.37 3.80 3.40 3.11 2.90 2.76 2.62 2.52 2.42
7.14 5.83 5.16 4.68 4.36 4.13 3.94 3.81 3.70 3.60 3.52
8.26 6.10 4.99 4.30 3.81 3.50 3.27 3.07 2.93 2.79 2.68
8.26 6.73 5.85 5.23 4.92 4.63 4.39 4.23 4.06 3.93 3.84
3.17 2.45 2.09 1.87 1.72 1.62 1.54 1.48 1.44 1.40 1.36
3.17 2.77 2.57 2.45 2.37 2.31 2.27 2.24 2.22 2.20 2.18
2.68 1.41 0.92 0.67 0.51 0.42 0.35 0.31 0.27 0.24 0.22
2.68 1.65 1.20 0.93 0.76 0.64 0.55 0.49 0.44 0.40 0.36
Table Table CI(v) Case V: Unrestricted Unrestricted intercept and unrestricted unrestricted trend
0.100
0.050
0.025
0.010
Mean
Variance
k
I0
I1
I0
I1
I0
I1
I0
I1
I0
I1
I0
I1
0 1 2 3 4 5 6 7 8 9 10
9.81 5.59 4.19 3.47 3.03 2.75 2.53 2.38 2.26 2.16 2.07
9.81 6.26 5.06 4.45 4.06 3.79 3.59 3.45 3.34 3.24 3.16
11.64 6.56 4.87 4.01 3.47 3.12 2.87 2.69 2.55 2.43 2.33
11.64 7.30 5.85 5.07 4.57 4.25 4.00 3.83 3.68 3.56 3.46
13.36 7.46 5.49 4.52 3.89 3.47 3.19 2.98 2.82 2.67 2.56
13.36 8.27 6.59 5.62 5.07 4.67 4.38 4.16 4.02 3.87 3.76
15.73 8.74 6.34 5.17 4.40 3.93 3.60 3.34 3.15 2.97 2.84
15.73 9.63 7.52 6.36 5.72 5.23 4.90 4.63 4.43 4.24 4.10
5.33 3.17 2.44 2.08 1.86 1.72 1.62 1.54 1.48 1.43 1.40
5.33 3.64 3.09 2.81 2.64 2.53 2.45 2.39 2.35 2.31 2.28
11.35 3.33 1.70 1.08 0.77 0.59 0.48 0.40 0.34 0.30 0.26
11.35 3.91 2.23 1.51 1.14 0.91 0.75 0.64 0.56 0.49 0.44
a The critical values are computed via stochastic simulations using T 1000 and 40,000 replications for the F-statistic for testing testing f 0 in the regression: y t 1, . . . , T, where x where x t x 1t , . . . , x kt 0 and f zt1 a wt t , t
D
D
zt
C
1
C
D
D
0 y t 1 , xt0 1 0 , wt 0 0 0 y t 1 , xt1 , 1 , wt y t1 , xt0 1 0 , wt 1 1 y t1 , xt0 1 , t0 , wt y t1 , xt0 1 0 , wt 1, t0
D
Case I
D D Case II D D D D Case III D D Case IV D D Case V and x t are generated from y t D y t1 C ε1t and x t D Pxt1 C e2t , t D 1, . . . , T, where y 0 D 0, The variables y t and x 0, x x 0 D 0 and et D ε1t , e 02t 0 is drawn as k C 1 independent standard normal variables. If x x t is purely I 1, P D Ik whereas whereas P P D 0 if x if x t is purely I 0. The critical values for k D D 0 correspond to the squares of the critical values of Dickey and Fuller’s (1979)
zt 1 zt1 zt1 zt1
unit root t-statistics for Cases I, III and V, while they match those for Dickey and Fuller’s (1981) unit root F-statistics for Cases II and IV. The columns headed ‘ I0’ refer to the lower critical values bound obtained when x t is purely I 0, while the columns headed ‘ I1’ refer to the upper bound obtained when when xt is purely I 1.
Copyright © 2001 John Wiley & Sons, Ltd.
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M. H. PESAR PESARAN, AN, Y. SHIN AND R. J. SMITH
D
0 , where Theorem 3.2 (Limiting distribution of t t yy ). If Assumptions 1-4 and 5a hold and g xy xy 0 0 x under H0 : yy 0 and pyx.x 0 of (17) g xy , 0 xx , then under (17),, as T , the asymptotic distribution of the t the t-statistic -statistic t t yy of (24) has the representation
D
D
where
Fk r a
1
1
0
!1
D
dWu aFk r a
D Q Q O O Wu a
1 0 Wu aWk r a
Wu a
1 0
˜ k r a0 da Wu aW
Wu a
1 0
ˆ k r a0 da Wu aW
0 da
2
Fk r a da
0
1/2
25
1 0 Wk r a Case I da 1 1 ˜ ˜ k r a Case III ˜ k r a0 da W W a W k r 0 1 1 ˆ k r aW ˆ k r a0 da ˆ k r a Case V W W 1 0 Wk r aWk r a
0
D D 0, . . . , k , and Cases I, III and V are defined in (12), (14) and (16), (16), a a 2 [0, 1].
r
The form of the asymptotic representation (25) is similar to that of a Dickey–Fuller test for a unit root except that the standard Brownian motion Wu a is replaced by the residual from an asymptotic regression of Wu a on the independent ( k r )-vector )-vector standard Brownian motion Wk r a (or their de-meaned and de-meaned and de-trended counterparts). Similarly to the analysis following Theorem 3.1, we detail the limiting distribution of the tstatistic tyy in the two polar cases in which the forcing variables xt are purely integrated of order zero and one respectively.
f g
f g ¾ D
Corollary 3.3 (Limiting (Limiting distribution distribution of of tyy if xt I0). If Assu Assump mpti tion onss 1-4 1-4 and and 5a hold hold , the I0 , then under H0 : yy 0 and pyx.x 00 of (17), as T and r k , that is, xt and asymptotic distribution of the t the t-statistic -statistic t t yy of (24) has the representation
D D
f g ¾
D
1
1
dWu aFa
2
where Fa
1/2
Fa da
0
0
D Q
Wu a Case I Wu a Case III Wu a Case V
O
!1
[0, 1].
and Cases I, III and V are defined in (12), (14) and (16), a
2 f g ¾ f g ¾
I1). If Assum Assumpti ptions ons 1-4 and 5a hol hold, d, Corollary 3.4 (Limiting (Limiting distributi distribution on of of tyy if xt yy I1 , then under under H0 : yy 0 , as and r 0 , that is, xt g xy 0 , where 0 x g xy , 0 xx , and T , the asymptotic distribution of the the t t-statistic -statistic t t yy of (24) has the representation
D !1
D
D D
1
1
dWu aFk a
0
2
D
Fk a da
0
where F where F k a a is is defined in Theorem 3.2 for Cases I, III and V, a
1/2
2 [0, 1].
As above, it may be shown shown by simulation simulation that the asymptoti asymptoticc critical critical values obtained obtained from Corollaries 3.3 (r k and xt is purely I0) and 3.4 (r 0 and xt is purely I1) provide
D D
f g
Copyright © 2001 John Wiley & Sons, Ltd.
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f g
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lower and upper bounds respectively for those corresponding to the general case considered in yy Theorem 3.2. Hence, a bounds procedure procedure for testing H 0 : yy 0 based on these two polar cases may be implemented as described above based on the t-statistic tyy for the exclusion of y t1 in
D
the conditional ECMs (12), (14) and (16) without prior knowledge of the cointegrating rank r .13 These asymptotic critical value bounds are given in Tables CII(i), CII(iii) and CII(v) for Cases I, III and V for sizes 0.100, 0.050, 0.025 and 0.010. As is emphasized in the Proof of Theorem 3.2 given in Appendix A, if the asymptotic analysis yy for the t -statistic t yy of (24) is conducted under H0 : yy 0 only, the resultant limit distribution for tyy depends on the nuisance parameter w f in addition to the cointegrating rank r , where, y t is allowed to Granger-cause xt , that is, under Assumption 5a, a yx f0 a xx 00 . Moreover, if y g xy,i 0 for some i 1, . . . , p 1, then the limit distribution also is dependent on the nuisance f or g xy 0, parameter g xy f0 g xy ; see Appendix A. Consequently, in general, where w xy xy / y yy y
6D
D
D
D
D6
6D
Table CII. Asymptotic critical value bounds of the t -statistic. Testing for the existence of a levels relationship a Table CII(i CII(i): ): Case I: No interc intercept ept and no trend 0.100 I0
k
0 1 2 3 4 5 6 7 8 9 10
1.62 1.62 1.62 1.62 1.62 1.62 1.62 1.62 1.62 1.62 1.62
0.050 I1
1.62 2.28 2.68 3.00 3.26 3.49 3.70 3.90 4.09 4.26 4.42
I0
1.95 1.95 1.95 1.95 1.95 1.95 1.95 1.95 1.95 1.95 1.95
0.025 I1
1.95 2.60 3.02 3.33 3.60 3.83 4.04 4.23 4.43 4.61 4.76
I0
2.24 2.24 2.24 2.24 2.24 2.24 2.24 2.24 2.24 2.24 2.24
0.010 I1
2.24 2.90 3.31 3.64 3.89 4.12 4.34 4.54 4.72 4.89 5.06
I0
Mean I1
2.58 2.58 2.58 2.58 2.58 2.58 2.58 2.58 2.58 2.58 2.58
2.58 3.22 3.66 3.97 4.23 4.44 4.67 4.88 5.07 5.25 5.44
I0
0.42 0.42 0.42 0.42 0.42 0.42 0.42 0.42 0.42 0.42 0.42
Variance I1
I0
I1
0.42 0.98 1.39 1.71 1.98 2.22 2.43 2.63 2.81 2.98 3.15
0.98 0.98 0.98 0.98 0.98 0.98 0.98 0.98 0.98 0.98 0.98
0.98 1.12 1.12 1.09 1.07 1.05 1.04 1.04 1.04 1.04 1.03
Table CII(iii) Case III: Unrestricted intercept and no trend 0.100 I0
k
0 1 2 3 4 5 6 7 8 9 10
2.57 2.57 2.57 2.57 2.57 2.57 2.57 2.57 2.57 2.57 2.57
0.050 I1
2.57 2.91 3.21 3.46 3.66 3.86 4.04 4.23 4.40 4.56 4.69
I0
2.86 2.86 2.86 2.86 2.86 2.86 2.86 2.86 2.86 2.86 2.86
0.025 I1
2.86 3.22 3.53 3.78 3.99 4.19 4.38 4.57 4.72 4.88 5.03
I0
3.13 3.13 3.13 3.13 3.13 3.13 3.13 3.13 3.13 3.13 3.13
0.010 I1
3.13 3.50 3.80 4.05 4.26 4.46 4.66 4.85 5.02 5.18 5.34
I0
3.43 3.43 3.43 3.43 3.43 3.43 3.43 3.43 3.43 3.42 3.43
Mean I1
3.43 3.82 4.10 4.37 4.60 4.79 4.99 5.19 5.37 5.54 5.68
I0
1.53 1.53 1.53 1.53 1.53 1.53 1.53 1.53 1.53 1.53 1.53
Variance I1
I0
I1
1.53 1.80 22..0246 2.47 2.65 2.83 3.00 3.16 3.31 3.46
0.72 0.72 0.72 0.72 0.72 0.72 0.72 0.72 0.72 0.72 0.72
0.71 0.81 0.86 0.89 0.91 0.92 0.93 0.94 0.96 0.96 0.96
(Continued overleaf ) ) yx.x
0 and H 0 : p yx.x 00 is automatically satisfied under the conditions Although Corollary 3.3 does not require g xy xy yx.x of Corollary 3.4, the simulation critical value bounds result requires g xy 0 and H 0 : p yx.x 00 for 0 < r < k . xy
13
Copyright © 2001 John Wiley & Sons, Ltd.
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D
D
D
J. Appl. Econ. 16: 16 : 289–326 (2001)
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M. H. PESAR PESARAN, AN, Y. SHIN AND R. J. SMITH
Table CII. (Continued ) Table Table CII(v) Case V: Unrestricte Unrestricted d interc intercept ept and unrest unrestricted ricted trend 0.100 I0
k
0 1 2 3 4 5 6 7 8 9 10
3.13 3.13 3.13 3.13 3.13 3.13 3.13 3.13 3.13 3.13 3.13
0.050 I1
3.13 3.40 3.63 3.84 4.04 4.21 4.37 4.53 4.68 4.82 4.96
I0
3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41 3.41
0.025 I1
3.41 3.69 3.95 4.16 4.36 4.52 4.69 4.85 5.01 5.15 5.29
I0
3.65 3.65 3.65 3.65 3.65 3.65 3.65 3.65 3.65 3.65 3.65
0.010 I1
3.66 3.96 4.20 4.42 4.62 4.79 4.96 5.14 5.30 5.44 5.59
I0
3.96 3.96 3.96 3.96 3.96 3.96 3.96 3.96 3.96 3.96 3.96
Mean I1
3.97 4.26 4.53 4.73 4.96 5.13 5.31 5.49 5.65 5.79 5.94
I0
2.18 2.18 2.18 2.18 2.18 2.18 2.18 2.18 2.18 2.18 2.18
Variance I1
I0
I1
2.18 2.37 2.55 2.72 2.89 3.04 3.20 3.34 3.49 3.62 3.75
0.57 0.57 0.57 0.57 0.57 0.57 0.57 0.57 0.57 0.57 0.57
0.57 0.67 0.74 0.79 0.82 0.85 0.87 0.88 0.90 0.91 0.92
a
D 1000 and 40 000 replica replications tions for the t -statistic for C C D 1, . . . , T, where x where x t D x 1t , . . . , xkt 0 and wt D 0 Case I 1 Case III w wt Dt D 1, t0 Case V The variables y t and xt are generated from y t D y t1 C ε1t and xt D Pxt1 C e2t , t D 1, . . . , T, where y 0 D 0, x0 D 0 If x t is purely I 1, P D Ik whereas P whereas P D 0 and e t D ε1t , e02t 0 is drawn as k C 1 independent standard normal variables. If x if xt is purely I 0. The critical values for k D D 0 correspond to those of Dickey and Fuller’s (1979) unit root t -statistics.
The critical values are computed computed via stochastic stochastic simulation simulationss using T testing 0 in the regression: y t y t1 d0 xt1 a0 wt t , t
D
D
C
The columns headed ‘ I 0’ refer to the lower critical values bound obtained when x t is purely I0, while the columns headed ‘ I1’ refer to the upper bound obtained when when xt is purely I 1. yy
D
although the t -statistic t yy has a well-defined limiting distribution under H 0 : yy 0, the above yy bounds testing procedure for H0 : yy 0 based on tyy is not asymptotically similar .14 Conseq Con sequen uently tly,, in the light light of the consis consisten tency cy result resultss for the above above sta statis tistic ticss discus discussed sed in Section 4, see Theorems 4.1, 4.2 and 4.4, we suggest the following procedure for ascertaining the existence of a level relationship between y t and x and x t : test H 0 of (17) using the bounds procedure based on the Wald or F-statistic of (21) from Corollaries 3.1 and 3.2: (a) if H0 is not rejected, yy proceed no further; (b) if H H 0 is rejected, test H 0 : yy 0 using the bounds procedure based on yy the t -statistic t yy of (24) from Corollaries 3.3 and 3.4. If H0 : yy 0 is false, a large value of
D
D
D
tyy should result, however, at least asymptotically, confirming the existence of a level relationship between 00 ). y and xt , which, may be degenerate (if pyx.x t and
D
4. THE ASYM ASYMPTO PTOTIC TIC POWER POWER OF THE BOUNDS BOUNDS PROCEDUR PROCEDURE E This section first demonstra demonstrates tes that the proposed bounds testing procedure procedure based on the Wal Wald d statistic of (21) described in Section 3 is consistent. Second, it derives the asymptotic distribution 14
yy
D 0 may be simulated from the limiting representation given in the Proof of Theorem 3.2 of Appendix A after substitution of consistent estimators for f and l xy g xy / yy.x under under 0 H0 : yy D 0, where yy.x yyy y f xy . Although such estimators may be obtained straightforwardly, unfortunately, they necessitate the use of parameter estimators from the marginal ECM (7) for fxt g1 t D1 . In principle, the asymptotic distribution of t t yy under H 0 : yy yy
Copyright © 2001 John Wiley & Sons, Ltd.
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of the Wald sta statis tistic tic of (21 (21)) under under a sequen sequence ce of local local al alter ternat native ives. s. Fin Finall ally, y, we show show tha thatt the bounds procedure based on the t-statistic of (24) is consistent. In the discussion of the consistency of the bounds test procedure based on the Wald statistic 5 of (21), because the rank of theyy long-run or r 1 under the yx.x multiplier matrixyy may be either r yx.x alternative hypothesis H 1 H1 H1 of (18) where H 1 : yy 0 and H 1 : p yx.x 00 , it is yy necessary to deal with these two possibilities. First, under H1 : yy 0, the rank of 5 is r 1 so yy Assumption 5b applies; in particular, ˛ yy 0. Second, under H0 : yy 0, the rank of 5 is r so yx.x Assumption 5a applies; in this case, H1 : p yx.x 00 holds and, in particular, ayx w0 a xx 00 .
D
[
6D 6D
6D
6D
C 6D
D
C 6D
yy
Theorem 4.1 (Consistency of the Wald Wald statistic bounds test pr procedure ocedure und under er H H1 ). If If Assumpt Assumptions ions yy 1-4 and 5b hold, then under H H 1 : yy 0 of (18) the Wald statistic W (21) is consistent against yy : 0 H1 yy in Cases I–V defined in (12)–(16).
6D
6D
yx.x
\
yy
Theorem 4.2 (Consiste (Consistency ncy of the Wald Wald stat statistic istic bounds test proced procedure ure under H1 H0 ). If yx.x yy 0 Assumptions 1 – 4 and 5a hold, then under H H 1 : p yx.x 0 of (18) and H H 0 : yy 0 of (17) the yx.x 0 Wald statistic W statistic W (21) is consistent against H H 1 : p yx.x 0 in Cases I–V defined in (12)–(16).
6D 6D
D
Hence, combining Theorems 4.1 and 4.2, the bounds procedure of Section 3 based on the Wald yy
yx.x
yy
yx.x
H1 against nst H 1 H1 H0 of (17) agai statistic W (21) defines a consistent test of H H 0 H0 of (18). This result holds irrespective of whether the forcing variables xt are purely I 0, purely I1 or mutually cointegrated. We now turn to consider the asymptotic distribution of the Wald statistic (21) under a suitably specified sequence of local alternatives. Recall that under Assumption 5b, py.x [ yy , pyx.x ] 0 ayx w0 a xx b0 . Consequently, we define the sequence of local alternatives ˛yy ˇyy , ˛yy b xy xx
\
D
D
f g
D
H1T : p y.xT [
D
D
C
yyT , pyx.xT ]
1 ˛yy b0
D T ˛ 1
yy ˇyy , T
C T
1 /2
xy
dyx
w0 d
[
0
xx b xx
26
Hence, under Assumption 3, defining 5T
and recalling
yyT pyxT 0 5 xxT
D ab0 , where 1, w0 a D a w0 a D 00 , we have yx
5T
xx
5 D T a b0 C T 1 y y
1/2
dyx d xx
b0
27
In order to detail the limit distribution of the Wald statistic under the sequence of local alternatives H 1T of (26), it is necessary to define the ( k r 1)-dimensional Ornstein–Uhlenbeck process JŁk r C1 a JŁu a, JŁk r a0 0 which obeys the stochastic integral and differential equations, cess a JŁk r C1 a Wk r C1 a ab0 0 JŁk r C1 r a da, r dr and dJk Łr C1 a dWk r C1 a ab0 Jk Łr C1 a ? , a? 0 Za? , where Wk r C1 a is a (k r 1)-dimensional standard Brownian motion, a [ay where y ? 0 ay , b [a? , a? 0 Za? , a? ]1/2 [b? , b? 0 0a? , a? ]1 b? , b? 0 by , togeth tog ether er a a? ]1/2 a? , y y y y y y Ł 0 Ł Ł Jk r a 0 with the de-m de-meane eaned d and de-meane de-meaned d and de-trended de-trended counterparts counterparts J˜ k r C1 a Ju a, ˜ and Jˆ Łk r C1 a JŁu a, ˆJŁk r a0 0 partitioned similarly, a [0, 1]. See, for example, Johanse Johansen n (1995, Chapter 14, pp. 201–210).
D
D
C
C C D
D
D O
Copyright © 2001 John Wiley & Sons, Ltd.
2
C D
D Q
J. Appl. Econ. 16: 16 : 289–326 (2001)
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M. H. PESAR PESARAN, AN, Y. SHIN AND R. J. SMITH
Theorem 4.3 (Limiting distribution of of W W under H H1T ). If Assumptions 1–4 and 5a hold, then under 0 0 1/2 1 0 as T , the asymptotic distribution of dyx w d xx b of (26), as H1T : y.x T ˛yy by T the Wald statistic W statistic W of (21) has the representation
C
D
W ) z0 zr C r
1
0
dJŁ aFk r C1 a0 u
!1
1
0
Fk r C1 aFk r C1 a0 da
1
1
0
Fk r C1 a a dJŁu a
0 ZŁ1 bŁ , h ZŁ1 PZ ˜ where zr NQ1/2 h, Ir , Q[ Q1/2 0 Q1/2 ] p limT!1 T1 b0Ł ˜ d xx 0 , is distributed independently of the second term in (28) and
D
¾
Fk r C1 a
D
D
d w0 yx
Case I JŁk r C1 a Ł 0 0 Jk r C1 a , 1 Case II ˜ Case III JŁk r C1 a Ł 0 0 ˜ Jk r C1 a , a 1/2 Case IV ˆJŁ Case V k r C1 a
D D 0, . . . , k , and Cases I –V are defined in (12)–(16), (12) –(16), a 2 [0, 1].
r
28
The first compon component ent of (28) (28) zr 0r zr is 0non-centr non-central al chi-squar chi-squaree distr distribute ibuted d with r degrees degrees yx.x of freedom and non-c freedom non-centr entralit ality y paramete parameterr h Qh and corres correspon ponds ds to the local local alt altern ernati ative ve H1T : 0 under Hyy : yy 0. The second term in (28) is a non-standard pyx.xT T1/2 dyx w0 d xx b xx 0 yy T1 ˛yy ˇyy and Dickey– Dick ey– Fuller Fuller unit-root unit-root distributi distribution on under the local local alternat alternative ive H1T : yyT dyx w0 d xx 00 . Note that under H0 of (17), that is, ˛yy 0 and dyx w0 d xx 00 , the limiting representation (28) reduces to (22) as should be expected. The proof for the consiste consistency ncy of the bounds bounds test test proced procedure ure based on the t-sta -statisti tisticc of (24) requires that the rank of the long-run multiplier matrix 5 is r 1 under the alternative hypothesis yy H1 : yy 0. Hence, Assumption 5b applies; in particular, ˛ yy 0.
D
D
D
D
C
6D
D D
6D
yy
Theorem 4.4 (Consiste (Consistency ncy of the the t-statistic bounds test procedure under under H1 ). If Assumptions yy 1–4 and 5b hold, then under H1 : yy 0 of (18) the t-statistic -statistic tyy (24) is consistent against yy H1 : yy 0 in Cases I, III and V defined in (12), (14) and (16).
6D
6D
As no note ted d at th thee end end of Sect Sectio ion n 3, 3, Theo Theore rem m 4.4 4.4 sugg sugges ests ts th thee poss possib ibil ilit ity y of usin using g tyy to yy between H0
yy and H1
yx.x if H0
discriminate 0 0, although, : p yx.x 00 is false, the bounds procedure given via Corollaries 3.3 and 3.4 is not asymptotically similar.
D
: yy
6D
: yy
D
AN APPLICATION: UK EARNINGS EQUATION Following the modelling approach described earlier, this section provides a re-examination of the earnings equation included in the UK Treasury macroeconometric model described in Chan, Savage and Whittaker (1995), CSW hereafter. The theoretical basis of the Treasury’s earnings equation is the bargaining model advanced in Nickell and Andrews (1983) and reviewed, for example, in Layard et al. (1991, Chapter 2). Its theoretical derivation is based on a Nash bargaining framework where firms and unions set wages to maximize a weighted average of firms’ profits and unions’ Copyright © 2001 John Wiley & Sons, Ltd.
J. Appl. Econ. 16: 16 : 289–326 (2001)
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utility. Following Darby and Wren-Lewis (1993), the theoretical real wage equation underlying the Treasury’s earnings equation is given by
Prodt
D 1 C fUR 1 RR /Union
wt
t
t
29
t
where wt is the real wage, Prodt is labour productivity, RRt is the replacement ratio defined as the ratio of unemployment benefit to the wage rate, Uniont is a measure of ‘union power’, and fURt is the probability of a union member becoming unemployed, which is assumed to be an increasing function of the unemployment rate URt . The econometric specification is based on a log-linearized version of (29) after allowing for a wedge effect that takes account of the difference between the ‘real product wage’ which is the focus of the firms’ decision, and the ‘real consumption wage’ which concerns the union.15 The theoretical arguments for a possible long-run wedge effect on real wages is mixed and, as emphasized by CSW, whether such long-run effects are present is an empirical matter. The change in the unemployment rate (URt ) is also included in the Treasury’s wage equation. CSW cite two different theoretical rationales for the inclusion of UR URt in the wage equatio equation: n: the diff differen erential tial moderating moderating effects effects of long- and short short-term -term unempl unemployed oyed on real wages, and the ‘insider–outsi ‘insider– outsider’ der’ theories theories which which argu arguee that only risin rising g unem unemploym ployment ent will be effective in significantly moderating wage demands. See Blanchard and Summers (1986) and Lindbeck and Snower (1989). The ARDL model and its associated unrestricted equilibrium correction formulation used here automatically allow for such lagged effects. We begin our empirical analysis from the maintained assumption that the time series properties of the key variables in the Treasury’s earnings equation can be well approximated by a log-linear VARp model, model, augmente augmented d with appropriate appropriate deterministic deterministicss such as intercep intercepts ts and time trends. To ensure comparability of our results with those of the Treasury, the replacement ratio is not includ inc luded ed in the analysi analysis. s. CSW, CSW, p. 50, 50, report report that ‘.. ‘.... it has not pro proved ved possi possible ble to identi identify fy a significant effect from the replacement ratio, and this had to be omitted from our specification’.16 Also, as in CSW, we include two dummy variables to account for the effects of incomes policies on average earnings. These dummy variables are defined by
D 1, over the period 1974q1 1975q4, 0 elsewhere D7579 D 1, over the period 1975q1 1979q4, 0 elsewhere
D7475t
t
The asympt asymptoti oticc theory theory develo developed ped in the paper paper is not affec affected ted by the inclu inclusio sion n of such such ‘on ‘onee0 0 0 17 off’ dummy variables. Let Let zt wt , Prod Prodt , URt , Wed Wedge get , Union Uniont wt , xt . Then, using the analysis of Section 2, the conditional ECM of interest can be written as
D
D
p 1
D c C c t C c D7475 C c D7579 C
wt
0
1
2
t
3
t
ww wt 1
Cp
wx.x xt 1
C D
i 1
y0i zti
C d0 x C u t
t
30
15
The wedge effect is further decomposed into a tax wedge and an import price wedge in the Treasury model, but this decomposition is not pursued here. 16 It is important, however, that, at a future date, a fresh investigation of the possible effects of the replacement ratio on real wages should be undertaken. 17 However, both the asymptotic theory and associated critical values must be modified if the fraction of periods in which the dummy variables are non-zero does not tend to zero with the sample size T . In the present application, both dummy variables included in the earning equation are zero after 1979, and the fractions of observations where D 7475t and D 7579t are non-zero are only 7.6% and 19.2% respectively. Copyright © 2001 John Wiley & Sons, Ltd.
J. Appl. Econ. 16: 16 : 289–326 (2001)
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M. H. PESAR PESARAN, AN, Y. SHIN AND R. J. SMITH
Under Und er the ass assump umptio tion n that that lag lagged ged real real wag wages, es, wt1 , do not ente enterr th thee subsub-VAR model for xt , the above real wage equation is identified and can be estimated consistently by LS. 18 Notice, however, that this assumption does not rule out the inclusion of lagged changes in real wages in the unemployment or productivity equations, for example. The exclusion of the level of real wages from these equations is an identification requirement for the bargaining theory of wages which permits it to be distinguished from other alternatives, such as the efficiency wage theory which postulates that labour productivity is partly determined by the level of real wages. 19 It is clear that, in our framework, the bargaining theory and the efficiency wage theory cannot be entertained simultaneously, at least not in the long run. The above specification is also based on the assumption that the disturbances ut are serially uncorrela uncor related. ted. It is therefore therefore import important ant that the lag order p of the unde underlyin rlying g VAR is selected appropriately. There is a delicate balance between choosing p sufficiently large to mitigate the residual serial correlation problem and, at the same time, sufficiently small so that the conditional ECM (30) is not unduly over-parameterized, particularly in view of the limited time series data which are available. Finally, a decision must be made concerning the time trend in (30) and whether its coefficient should be restricted.20 This issue can only be settled in light of the particular sample period under consideration. The time series data used are quarterly, cover the period 1970q1-1997q4, and are 21
seasonally adjusted (where relevant). To ensure comparability of results for different choices of p, all estimations use the same sample period, 1972q1–1997q4 ( T 104), with the first eight observations reserved for the construction of lagged variables. The five variables in the earnings equation were constructed from primary sources in the folln1 TEt ln1 TDt lnRPIXt / lnERPRt /PYNONGt , Wedget lowing manner: w t PYNONGt , URt ln100 ILOUt /ILOUt WFEMPt , Prodt lnYPROMt 278.29 lnUDENt , where ERPRt is average private sector YMFt /EMFt ENMFt , and Uniont earnings per employee (£), PYNONGt is the non-oil non-government GDP deflator, YPROMt is outpu outputt in the private, non-oil, non-oil, non-manu non-manufact facturing uring,, and public traded sectors at const constant ant factor cost (£ million, 1990), YMFt is the manufacturing output index adjusted for stock changes (1990 100), EMFt and ENMFt are respectively employment in UK manufacturing and nonmanufactu manuf acturing ring secto sectors rs (thousand (thousands), s), ILOUt is the Int Intern ernati ationa onall Labou Labourr Of Office fice (IL (ILO) O) me measu asure re of unemployment (thousands), WFEMPt is total total employme employment nt (thou (thousands sands), ), TEt is the average average employers’ National Insurance contribution rate, TDt is the average direct tax rate on employment inco incomes, mes, RPIXt is the Retail Price Index excluding excluding mortgage mortgage payments, payments, and UDENt is union density (used to proxy ‘union power’) measured by union membership as a percentage of employment.22 The time series plots of the five variables included in the VAR model are given in Figure Fig uress 1– 3.
D
D D C
ð
D
D
C
C
C D
C
ð
D
18
See Assumption 3 and the following discussion. By construction, the contemporaneous effects x t are uncorrelated with the disturbance term ut and instrumental variable estimation which has been particularly popular in the empirical wage equation literature is not necessary. Indeed, given the unrestricted nature of the lag distribution of the conditional ECM (30), it is difficult to find suitable instruments: namely, variables that are not already included in the model, which are uncorrelate uncorrelated d with u t and also have a reasonable degree of correlation with the included variables in (30). 19 For a discussion of the issues that surround the identification of wage equations, see Manning (1993). 20 See, for example, PSS and the discussion in Section 2. 21 We are grateful to Andrew Gurney and Rod Whittaker for providing us with the data. For further details about the sources and the descriptions of the variables, see CSW, pp. 46–51 and p. 11 of the Annex. 22 The data series for UDEN assumes a constant rate of unionization from 1980q4 onwards.
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(a)
309
4.0
3.5 Real Wages 3.0 e l a c S g o L
2.5
2.0
Productivity
1.5
1.0 1972Q1 1972 Q1 1974 1974Q3 Q3 1977Q1 1977Q1 1979 1979Q3 Q3 1982 1982Q1 Q1 1984 1984Q3 Q3 1987 1987Q1 Q1 1989 1989Q3 Q3 1992 1992Q1 Q1 1994Q3 1994Q3 199 1997Q1 7Q1
Quarters (b)
0.04 0.03 Real Wage 0.02 0.01 0.00
−0.01
−0.02
Productivity −0.03
−0.04
1972Q1 1972Q 1 1974Q 1974Q3 3 1977Q 1977Q1 1 1979Q 1979Q3 3 1982Q 1982Q1 1 1984Q 1984Q3 3 1987Q 1987Q1 1 1989Q 1989Q3 3 1992Q 1992Q1 1 1994Q3 1994Q3 1997Q1 1997Q1
Quarters
Figure 1. (a) Real wages and labour productivity. (b) Rate of change of real wages and labour productivity
It is clear from Figure 1 that real wages (average earnings) and productivity show steadily rising trends with real wages growing at a faster rate than productivity. 23 This suggests, at least initially, that a linear trend should be included in the real wage equation (30). Also the application of unit root tests to the five variables, perhaps not surprisingly, yields mixed results with strong evidence in favour of the unit root hypothesis only in the cases of real wages and productivity. This does not necessarily preclude the other three variables ( UR, Wedge, and Union) having levels impact on real wages. Following the methodology developed in this paper, it is possible to test for the existence of a real wage equation involving the levels of these five variables irrespective of whether they are purely I 0, purely I1, or mutually cointegrated. 23
Over the period 1972q1–97q4, real wages grew by 2.14% per annum as compared to labour productivity that increased by an annual average rate of 1.54% over the same period. Copyright © 2001 John Wiley & Sons, Ltd.
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−0.2
−0.3 UNION
−0.4
−0.5
−0.6
WEDGE
−0.7
−0.8
1972Q1 1974Q3 1977Q1 1979Q3 1982Q1 1984Q3 1987Q1 1989Q3 1992Q1 1994Q3 1997Q1 Quarters
Figure 2. The wedge and the unionization variables
3.0 2.5 2.0 e l a c S g o L
UR 1.5 1.0 0.5 0.0 1972Q1 1974Q3 1977Q1 1979Q3 1982Q1 1984Q3 1987Q1 1989Q3 1992Q1 1994Q3 1997Q1
Quarters
Figure 3. The unemployment rate
To determine the appropriate lag length p and whether a deterministic linear trend is required in addi addition tion to the productivit productivity y variable, variable, we estimate estimated d the conditio conditional nal model (30) by LS, with and without a linear time trend, for p 1, 2, . . . , 7. As pointed out earlier, all regressions were computed over the same period 1972q1–1997q4. We found that lagged changes of the productivity variable, Prodt1 , Prodt2 , . . . , were insignificant (either singly or jointly) in all regressions. Therefore, for the sake of parsimony and to avoid unnecessary over-parameterization, we decided to re-e re-estima stimate te the regressions regressions without without these lagged variables, variables, but including including lagged changes of all other variables. Table I gives Akaike’s and Schwarz’s Bayesian Information Criteria, denoted
D
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respectively by AIC and SBC, and Lagrange multiplier (LM) statistics for testing the hypothesis 2 2 4 respectively. of no residual serial correlation against orders 1 and 4 denoted by SC 1 and SC As might might be expect expected, ed, the lag order selected selected by AIC, AIC, paic 6, irr irresp espect ective ive of whe whethe therr a
D D
deterministic trend term is included or not, is much larger than that selected by SBC. This latter 2 criterion gives estimates psbc statistics also 1 if a trend is included and psbc 4 if not. The SC suggest using a relatively high lag order: 4 or more. In view of the importance of the assumption of serially uncorrelated errors for the validity of the bounds tests, it seems prudent to select p to be either 5 or 6.24 Nevertheless, for completeness, in what follows we report test results for p 4 and 5, as well as for our preferred choice, namely p 6. The results in Table I also indicate that there is little to choose between the conditional ECM with or without a linear deterministic trend. Table II gives the values of the F- and t-statistics for testing the existence of a level earnings equation under three different scenarios for the deterministics, Cases III, IV and V of (14), (15) and (16) respectively; see Sections 2 and 3 for detailed discussions. The various statistics in Table II should be compared with the critical value bounds provided in Tables CI and CII. First, consider the bounds F-statistic. As argued in PSS, the statistic F IV which sets the trend coefficient to zero under the null hypothesis of no level relationship, Case IV of (15), is more appropriate than FV , Case V of (16), which ignores this constraint. Note that,
D
D
D
if the of trend c1 is not subject to this of restriction, a0 ,quadratic in the level realcoefficient wages under the null hypothesis 0 (30) and implies pwx.x 0 which is trend empirically ww implausible. The critical value bounds for the statistics F IV and F V are given in Tables CI(iv) and CI(v). Since k 4, the 0.05 critical value bounds are (3.05, 3.97) and (3.47, 4.57) for F IV and FV , respectively.25 The test outcome depends on the choice of the lag order p. For p 4, the
D
D
D D
D
Table I. Statistics for selecting the lag order of the earnings equation With deterministic trends p
AIC
SBC
2 SC 1
1 2 3 4 5 6 7
319.33 324.25 321.51 334.37 335.84 337.06 336.96
302.14 301.77 293.74 301.31 297.50 293.42 288.04
16.86Ł 2.16 0.52 3.48ŁŁŁ 0.03 0.85 0.17
Without deterministic trends 2 SC 4
35.89Ł 19.71Ł 17.07Ł 7.79ŁŁŁ 2.50 3.58 2.20
AIC
SBC
2 SC 1
317.51 323.77 320.87 3 35.37 33 336.49 337.03 336.85
301.64 302.62 294.43 303.63 299.47 294.72 289.25
18.38Ł 1.98 1.56 3.41ŁŁŁ 0.03 0.99 0.09
2 SC 4
34.88Ł 21.52Ł 19.35Ł 7.13 2.15 3.99 0.64
Notes : p is the lag ord order er of the under underlyi lying ng VAR model for the condition conditional al ECM (30), with zero restricti restrictions ons on the coefficients coeffi cients of lagge lagged d changes changes in the productivity variable. variable. AICp LLp sp and SBCp LLp sp /2 ln T denote Akaike’s and Schwarz’s Bayesian Information Criteria for a given lag order p , where LL p is the maximized log-likelihood 2 2 value of the model, s p is the number of freel freely y estimated coefficients coefficients and T is the sample size. SC 1 and SC 4 are LM Ł ŁŁ ŁŁŁ statistics for testing no residual serial correlation against orders 1 and 4. The symbols , , and denote significance at 0.01, 0.05 and 0.10 levels, respectively.
24
In the Treasury model, different lag orders are chosen for different variables. The highest lag order selected is 4 applied to the log of the price deflator and the wedge variable. The estimation period of the earnings equation in the Treasury model is 1971q1–1994q3. 25 Following a suggestion from one of the referees we also computed critical value bounds for our sample size, namely T 104. For k 4, the 5% critical value bounds associated with F IV and F V statistics turned out to be (3.19,4.16) and (3.61,4.76), (3.61, 4.76), respectiv respectively, ely, which are only marginally marginally different different from the asympt asymptotic otic critical critical value bounds.
D
DD
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Table II. F - and t-stati -statisti stics cs for tes testin ting g the exi existe stence nce of a levels earnings equation With de dete term rmin inis isti ticc tr treend ndss p
F IV
4 5 6
2.99a 4.42c 4.78c
FV
2.34a 3.96b 3.59b
Without de dete terrmi mini nist stic ic tr treend ndss tV
2.26aa 2.83a 2.44
F III
3.63b 5.23c 5.42c
t III
3.02bc 4.00b 3.48
Notes : See the notes to Table I. F IV is the F-statistic for testing 0 ww 0, pwx.x 0 and c1 0 in (30). FV is the F-statistic for testing ww 0 and pwx.x 0 in (30). F III is the F-statistic for testing ww 0 and pwx.x 0 in (30) with c1 set equal to 0. tV 0 in (30) with and without and t III are the t -ratios for testing ww a deterministic linear trend. a indicates that the statistic lies below the 0.05 lower bound, b that it falls within the 0.05 bounds, and c that it lies above the 0.05 upper bound.
D
D D
D
D D D
D
hypothesis that there exists no level earnings equation is not rejected at the 0.05 level, irrespective
D
of whether the regressors are purely I0, purely I1 or mutually cointegrated. For p 5, the bounds test is inconclusive. For p 6 (selected by AIC), the statistic F V is still inconclusive, but F IV 4.78 lies outside the 0.05 critical value bounds and rejects the null hypothesis that there exists no level earnings equation, irrespective of whether the regressors are purely I0, purely I1 or mutually cointegrated.26 This finding is even more conclusive when the bounds F-test is applied to the earnings equations without a linear trend. The relevant test statistic is F III and the associated 0.05 critical value bounds are (2.86, 4.01).27 For p 4, F III 3.63, and the test result is inconclusive. However, for p 5 and 6, the values of F III are 5.23 and 5.42 respectively and the hypothesis of no levels earnings equation is conclusively rejected. The results from the application of the bounds t -test to the earnings equations are less clear-cut and do not allow the imposition of the trend restrictions discussed above. The 0.05 critical value bounds for t III and tV , when k 4, are ( 2.86, 3.99) and ( 3.41, 4.36).28 Therefore, if a linear trend is included, the bounds t-test does not reject the null even if p 5 or 6. However, when the trend term is excluded, the null is rejected for p 5. Overall, these test results support the exi existe stence nce of a levels levels earnings earnings equati equation on when when a suffic sufficien iently tly high lag order is sel select ected ed and
D
D
D
D
D D
D
D
D
when the statistically insignificant deterministic trend term is excluded from the conditional ECM (30). Such a specification is in accord with the evidence on the performance of the alternative conditional ECMs set out in Table I. In testing the null hypothesis that there are no level effects in (30), namely ( ww 0, pwx.x 0) it is important that the coefficients of lagged changes remain unrestricted, otherwise these tests could be subject to a pre-testing problem. However, for the subsequent estimation of levels effects and short-run dynamics of real wage adjustments, the use of a more parsimonious specification seems advisable. To this end we adopt the ARDL approach to the estimation of the level relations
D
26
The same conclusion is also reached for p See Table CI(iii). 28 See Tables CII(iii) and CII(v).
27
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D 7.
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313
discussed in Pesaran and Shin (1999).29 First, the (estimated) orders of an ARD ARDLp, Lp, p1 , p2 , p3 , p4 model in the five variables wt , Pr Prod odt , URt , Wed Wedge get , Uni Union ont were selected by searching across 5 16, 807 ARDL models, spanned by p 0, 1, . . . , 6, and pi 0, 1, . . . , 6, i 1, . . . , 4, the 7 using the AIC criterion.30 This resulted in the choice of an ARDL6, 0, 5, 4, 5 specification with estimates of the levels relationship given by
D
D
D 1.063
wt
Prodt
D
0.105 UR 0.943 Wedge C1.481 Union C2.701 C O
31
0.242
0.311
0.265
vt
t
t
t
0.034
0.050
D
O
where vt is the equilibrium equilibrium correcti correction on term, term, and the standard standard errors are given in parenthe parenthesis. sis. All levels estimates are highly significant and have the expected signs. The coefficients of the productivity and the wedge variables are insignificantly different from unity. In the Treasury’s earnings equation, the levels coefficient of the productivity variable is imposed as unity and the above estimates can be viewed as providing empirical support for this a priori restriction. Our levels estimates of the effects of the unemployment rate and the union variable on real wages, namely 0.1 0.105 05 and 1.481 1.481,, are also in line line with with the Treasu Treasury ry est estim imate atess of 0.0 0.09 9 and 1.31.31 The main main differ differenc encee betwe between en the two two set setss of estima estimates tes concer concerns ns the lev levels els coeffi coefficie cient nt of the wedge variable. We obtain a much larger estimate, almost twice that obtained by the Treasury. Setting the levels coefficients of the Prodt and Wedget variables to unity provides the alternative
interpretation that the share of wages (net of taxes and computed using RPIX rather than the implicit GDP deflator) has varied negatively with the rate of unemployment and positively with union strength.32 The condit condition ional al EC ECM M reg regres ressio sion n associ associate ated d with with the above above lev level el relati relations onship hip is given given in 33 Table III. These estimates provide further direct evidence on the complicated dynamics that seem to exist between real wage movements and their main determinants.34 All five lagged changes in real wages are stat statistic istically ally significant, significant, further further justifying justifying the choice choice of p 6. The equi equilibri librium um correctio corre ction n coefficie coefficient nt is estimated estimated as 0.2 0.229 29 (0. (0.058 0586) 6) wh which ich is reason reasonabl ably y la large rge and highly highly significant.35 The auxiliary equation of the autoregressive part of the estimated conditional ECM has real roots 0.9231 and 0.9095 and two pairs of complex roots with moduli 0.7589 and 0.6381, which suggests an initially cyclical real wage process that slowly converges towards the equilibrium described by (31).36 The regression fits reasonably well and passes the diagnostic tests against nonnormal errors and heteroscedasticity. However, it fails the functional form misspecification test at
D
29
Note that the ARDL approac approach h advan advanced ced in Pesaran Pesaran and Shin (1999) is applic applicable able irrespective irrespective of whethe whetherr the regre regressors ssors are purely I 0, purely I 1 or mutually cointegrated. 30 For further details, see Section 18.19 and Lesson 16.5 in Pesaran and Pesaran (1997). 31 CSW do not report standard errors for the levels estimates of the Treasury earnings equation. 32 We are grateful to a referee for drawing our attention to this point. 33 Clearly, it is possible to simplify the model further, but this would go beyond the remit of this section which is first to test for the existence of a level relationship using an unrestricted ARDL specification and, second, if we are satisfied that such a levels relationship exists, to select a parsimonious specification. 34 The standard errors of the estimates reported in Table III allow for the uncertainty associated with the estimation of the levels coefficients. This is important in the present application where it is not known with certainty whether the regressors are purely I 0, purely I 1 or mutually cointegrated. It is only in the case when it is known for certain that all regressors are I 1 that it would be reasonable in large samples to treat these estimates as known because of their super-consistency. 35 The equilibrium correction coefficient in the Treasury’s earnings equation is estimated to be 0.1848 (0.0528), which is smaller than our estimate; see p. 11 in Annex of CSW. This seems to be because of the shorter lag lengths used in the Treasury’s specification rather than the shorter time period 1971q1–1994q3. Note also that the t-ratio reported for this coefficient does not have the standard t-distr -distribution ibution;; see Theorem 3.2. 36 The complex roots are 0.34293 0.67703i and 0.17307 0.61386i, where i 1.
š
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D p
š
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M. H. PESAR PESARAN, AN, Y. SHIN AND R. J. SMITH
the 0.05 level which may be linked to the presence of some non-linear effects or asymmetries in the adjustment of the real wage process that our linear specification is incapable of taking into account.37 Recursive estimation of the conditional ECM and the associated cumulative sum and cumulative sum of squares plots also suggest that the regression coefficients are generally stable over the sample period. However, these tests are known to have low power and, thus, may have missed important breaks. Overall, the conditional ECM earnings equation presented in Table III has a number of desirable features and provides a sound basis for further research. Table III. Equilibrium correction form of the ARDL(6, 0, 5, 4, 5) earnings equation Regressor
O 1
vt
wt1 wt2 wt3 wt4 wt5 Prodt URt URt1 URt2 URt3 URt4 Wedget Wedget1 Wedget2 Wedget3 Uniont Uniont1 Uniont2 Uniont3 Uniont4 Intercept D7475t D7579t 2
D
Coefficient
Standard error
0.229 0.418 0.328 0.523 0.133 0.197
0.0586 0.0974 0.1089 0.1043 0.0892 0.0807 0.0954 0.0083 0.0119 0.0118 0.0113 0.0122 0.0534 0.0592 0.0569 0.0560 0.8169 0.8395 0.9023 0.7805 0.7135 0.1554 0.0063 0.0063
0.315 0.003 0.016 0.003 0.028 0.027 0.297 0.048 0.093 0.188 0.969 2.915 0.021 0.101 1.995 0.619 0.029 0.017
O DD
D D
R 0.5589, 0.0083, AIC 339.57, SBC 2 2 1 4.86[0.027] SC 4 8.74[0.068], FF 2 2 0.01[0.993], 2 1 0.66[0.415]. N
D D
H
p-value
N/A 0.000 0.004 0.000 0.140 0.017 0.001 0.683 0.196 0.797 0.014 0.031 0.000 0.417 0.105 0.001 0.239 0.001 0.981 0.897 0.007 0.000 0.000 0.009
D 302.55,
D
Notes : The regression is based on the conditional ECM given by (30) using an ARDL6, 0, 5, 4, 5 specification with dependent variable, wt est estima imated ted ove overr 1972q1– 1972q1– 199 1997q4, 7q4, and the equili equilibri brium um correc correction tion term term 2 given en in (31 (31). ). R is the adjusted squared multiple correlation correlation vt 1 is giv coefficient, is the standard error of the regression, AIC and SBC are 2 2 Akaike’s and Schwarz’s Bayesian Information Criteria, SC 1, 4, FF 2 2 N 2, and H 1 denote denote chi-sq chi-squared uared statistics statistics to test for no residual serial correlation, no functional form mis-specification, normal errors and homoscedasticity respectively with p-values given in [ ]. For details of these diagnostic tests see Pesaran and Pesaran (1997, Ch. 18).
O
O
Ð
37
The conditional ECM regression in Table III also passes the test against residual serial correlation but, as the model was specified to deal with this problem, it should not therefore be given any extra credit! Copyright © 2001 John Wiley & Sons, Ltd.
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315
6. CONCLUSI CONCLUSIONS ONS Empirica Empi ricall analysis analysis of level level relation relationships ships has been an integral integral part of time series econometrics econometrics 38
and pre-dates the recent literature on unit roots and cointegration. However, the emphasis of this earlier literature was on the estimation of level relationships rather than testing for their presence (or otherwise). Cointegration analysis attempts to fill this vacuum, but, typically, under the relatively restrictive assumption that the regressors, x regressors, xt , entering the determination of the dependent variable of interest, y t , are all integrated of order 1 or more. This paper demonstrates that the problem of testing for the existence of a level relationship between y t and x t is non-standard even if all all the regressors under consideration are I 0 because, under the null hypothesis of no level relationship between y t and xt , the process describing the y t process is I1, irrespective of whether the regressors xt are and purely I0, purely I1 or mutually cointegrated. The asymptotic theory developed in this paper provides provi des a simple simple univariat univariatee framewor framework k for testing the existence existence of a singl singlee leve levell relations relationship hip between y t and xt when it is not known with certainty whether the regressors are purely I0, purely I1 or mutually cointegrated. 39 Moreover, it is unnecessary that the order of integration of the underlying regressors be ascertained prior to testing the existence of a level relationship between y t and and xt . Therefore, unlike typical applications of cointegration analysis, this method is not subject to this particular kind of pre-testing problem. The application of the proposed bounds testing procedure to the UK earnings equation highlights this point, where one need not take an a priori position as to whether, for example, the rate of unemployment or the union density variable are I1 or I0. The analysis of this paper is based on a single-equation approach. Consequently, it is inappropriate in situations where there may be more than one level relationship involving y t . An extension of this paper and those of HJNR and PSS to deal with such cases is part of our current research, but the consequent theoretical developments will require the computation of further tables of critical values.
APPENDIX A: PROOFS FOR SECTION 3 We confine the main proof of Theorem 3.1 to that for Case IV and briefly detail the alterations necessary for the other cases. Under Assumptions 1–4 and 5a, the process zt 1 tD1 has the infinite moving-average representation,
f g
zt
m
Cst
g t
CŁ Let
A1
D C C C where whe re the par partia tiall sum s D e , 8zCzŁ D Cz8z D 1 zI C , 8z I C D 1 Johansen (1991 (1991)) and PSS. 8 z , Cz I C C D C z D C C 1 zC z, t D 1, 2 . . .; see Johansen Johansen en (1991, (1991, (4.5), (4.5), p. 1 1559 559). ). Note that C D b? , b? [a? , a? 0 0(b? , b? )] a? , a? 0 ; see Johans Define the k C 2, r and k C 2, k r C 1 matrices b Ł and d by g 0 b and d g 0 b? , b? bŁ
t
i
i
k 1
i 1
y
i
i
t i 1
k 1
i
y
y
1
k 1
p i 1
y
Ik C1
Ik C1
y
38
For an excellent review of this early literature, see Hendry et al. (1984). Of course, the system approach developed by Johansen (1991, 1995) can also be applied to a set of variables containing possibly a mixture of I I 0 and I 1 regressors.
39
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M. H. PESAR PESARAN, AN, Y. SHIN AND R. J. SMITH
? 1, k r 1 matrix matrix whose whose col column umnss are a bas basis is for the orthog orthogona onall where b? y , b is a k ? ? k C1 b . Hence, b, by , b is a basis for R . Let x be the k 2-unit vector 1, 00 0 . complement of b Then, b , x , d is a basis for Rk C2 . It therefore follows that
C
Ł T 1/2 d0 zŁ
D T
1/2
[Ta]
C
C
?0 b? y , b m
C T b? , b? 0 Cs C b? , b?0 T ) b? , b?0CB C a 1/2
[Ta]
y
1/2
y
CŁ Le[Ta]
k 1
y
where z Łt where z t, z0t 0 , B k C1 a is a k 1-vector Brownian motion with variance matrix Z and [Ta] denotes the integer part of Ta Ta, a [0, 1]; see Phillips and Solo (1992, Theorem 3.15, p. 983). Also, 1 1 0 Ł b0 m b0 CŁ Let Similarly, y, noting noting that b0 C 0, we have that bŁ0 zŁt T t a. Similarl T x zt Ł ˜ Ł OP 1. Hence, from Phillips and Solo (1992, Theorem 3.16, p. 983), defining Z 1 Pi Z1 and Pi Z , it follows that Z
D D
2
)
0 Ł Z1 bŁ ZŁ1 ˜ T1 b0Ł ˜ 0 Ł Z1 bŁ ZŁ1 ˜ T1 B0T ˜
C
D
C
D O 1, T b0Ł ˜ ZŁ0 Z D O 1, T Z0Z D O 1 D O 1, T B0 ˜ ZŁ0 Z D O 1 1
P
1
1
P
T
d, T1/2 x . Similarly, defining u ˜
where BT where
D
D
0 u ZŁ1 ˜ T1/2 b0Ł ˜
P
P
1
1
P
A2
Pi u,
D O 1, T P
1/2
0
u Z ˜
D O 1
A3
P
Cf. Johansen (1991, Lemma A.3, p. 1569) and Johansen (1995, Lemma 10.3, p. 146). The next next result result fol follow lowss fro from m Philli Phillips ps and Solo Solo (1992, (1992, The Theore orem m 3.15, 3.15, p. 98 983); 3); cf. Johans Johansen en (1991, Lemma A.3, p. 1569) and Johansen (1995, Lemma 10.3, p. 146) and Phillips and Durlauf (1986).
D )
? , b? 0 d, T1/2 x and define Ga G1 a0 , G2 a0 , where G1 a by Lemma A.1 Let B BT 1 ˜ k C1 a[ B1 a0 , B ˜ k a0 0 ] Bk C1 a C ˜ Bk C1 a , B Bk C1 ada , and G and G2 a a 21 , a [0,1]. 0 Then 1 1 0 1 0 ˜ Ł0 2 0 ˜ Ł0 ˜ Ł Gad BŁu a u GaGa da da,, T BT Z1 ˜ T BT Z1 Z1 BT
D Q
D
0
0
Q
where BŁu a
)
2
Q
˜ a D B Q a, B˜ a0 0 , a Q B a w0 ˜ B a and a and B , a 2 [0, 1] 1
k
k
1
k
Proof of Theorem 3.1 Under H H 0 of (17), the Wald statistic W statistic W of (21) can be written as
D C D
O
ωuu W
Ł ˜ Ł 1 ˜ Ł0 P ˜ Ł0 P ˜ ˜ Z Z Z Z 1 Z u 1 1 1 Z Z 1 0 Ł0 0 u AT ˜ Z1 PZ ˜ ZŁ1 AT ZŁ1 AT A0T ˜ u˜ 0 PZ ˜ ZŁ1 PZ ˜
D u˜ 0P
0 ZŁ A . It follows from (A2) ZŁ1 PZ ˜ where AT T1/2 bŁ , T1/2 BT . Consider the matrix A0T ˜ 1 T and Lemma A.1 that 0 ˜ Ł bŁ 00 ZŁ1 PZ Z T1 b0Ł ˜ 0 1 Ł 0 Ł ˜ ˜ oP 1 A4 AT Z1 PZ Z1 AT 0 Ł Z1 BT ZŁ1 ˜ 0 T2 B0T ˜
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BOUNDS TESTING FOR LEVEL RELATIONSHIPS
0 u. From (A3) and Lemma A.1, ZŁ1 PZ ˜ Next, consider A A 0T ˜ 0 ZŁ 1 P ˜ u T1/2 b0 ˜ 0 0 Z Ł Ł 0 u Z1 PZ ˜ AT ˜ u ZŁ ˜ T1 B0 ˜
317
D C O D C D C D D C C ) ¾ ) ¾ Q Q oP 1
1
T
A5
Finally, the estimator for the error variance ω variance ω uu (defined in the line after (21)), ωuu
T
m1 ˜ u0 ˜ u
T
u0 ˜ u m1 ˜
0 0 0 ˜ u ZŁ1 AT 1 AT ZŁ1 PZ ˜ u˜ 0 PZ ˜ ZŁ1 AT A0T ˜ ZŁ1 PZ ˜
oP 1
oP 1
ωuu
A6
From (A4)–(A6) and Lemma A.1,
1 0 Ł 0 0 u/ωuu bŁ ˜ Z1 PZ ˜ ZŁ1 bŁ ZŁ1 PZ ˜ ZŁ1 bŁ T1 b0Ł ˜ u0 PZ ˜ T1 ˜ 1 0 Ł 0 0 Ł u/ωuu oP 1 BT ˜ Z1 ˜ u0 ˜ Z1 BT ZŁ1 BT T2 B0T ˜ ZŁ1 ˜ T2 ˜
W
A7
We consider each of the terms in the representation (A7) in turn. A central limit theorem allows us
1/2 1/2 0 Ł0 1/2 bŁ ˜ u/ωuu zr N0, Ir T Z1 PZ ˜ chi-square random variable with Hence, the first term in (A7) converges converges in distribution to z0 zr , a chi-square to state
0 ZŁ b T1 b0Ł ˜ ZŁ1 PZ ˜ 1 Ł
r
r degrees degrees of freedom; that is,
0 ZŁ1 bŁ T1 b0Ł ˜ u0 PZ ˜ ZŁ b ZŁ1 PZ ˜ T1 ˜ 1 Ł
1 0 Ł 0 u/ωuu bŁ ˜ Z1 PZ ˜
zr 0 zr
2 r
A8
From Fro m Lemma A.1, the second second term in (A7) weakly converge convergess to 1
0
which, as C
u
GaGa0 dr
1
1
0
0
Gk C1 ad BuŁ a/ωuu
D b? , b?[a? , a? 0 0ˇ? , b? )] a? , a?0 , may be expressed as y
y
d BuŁ a
Q
a
1 2
1
y
0
? 0 ˜ a? y , a Bk 1 a
1 0
1
d BŁ aGa0
C
y
? 0 ˜ a ? y , a Bk 1 a
1
a
0
1
ð
1 2
C
? , a? 0 ˜ Bk 1 a ay
Q
? 0 ˜ a? y , a Bk C1 a a
0
1 2
a
1 2
C
d BŁu a/ωuu
1
0
da
? 1, w0 0 and a? 0, a? 0 0 where Now, noting that under H0 of (17) we may express ay xx ? 0 a xx 0 , we define the k r 1-vec -vector tor of independen independentt de-me de-meaned aned stand standard ard Bro Brownia wnian n a xx motions,
D
˜ k r C1 a[ W
D
D
C
D QW a, W˜ a0 0] [a? , a?0Za? , a? ] D a? 0 Z ωa? BQ a B a a? 0 ˜
u
k r
1/2 uu
xx
xx xx
Copyright © 2001 John Wiley & Sons, Ltd.
y
u 1/2
xx
k
y
1/2
? , a? 0 ˜ ay Bk C1 a
J. Appl. Econ. 16: 16 : 289–326 (2001)
318
Q
M. H. PESAR PESARAN, AN, Y. SHIN AND R. J. SMITH
D Q
Q
˜ k C1 a B1 a, B ˜ k a0 0 is par˜ k a and B where BuŁ a B1 a w0 ˜ Bk a is independent of B titioned according to zt y t , x0t 0 , a [0, 1]. Hence, the second term in (A7) has the following asymptotic representation: representation:
D
Q 1
dWu a
0
2
Q ð 0
W ˜ k r C1 a a 12 1
W ˜ k r C1 a a 21
0
W ˜ k r C1 a a 12
W ˜ k r C1 a a 21
0
da
1 A9
dWu a
0
Q
1
2
Note that d d Wu a a in in (A9) may be replaced by dW u a , a , a [0, 1]. Combining (A8) and (A9) gives the result of Theorem 3.1. For the re remai mainin ning g cases, cases, we need need only only make make minor minor mod modific ificati ations ons to the proof proof for Cas Casee IV IV.. ? ? ? ? k C1 d. For Cas In Case I, d by , b with b, by , b a ba basi siss for for R and BT Casee II II,, whe where re
D
ZŁ1 D iT , Z01 0 , we have
bŁ
D
D m0
Ik C1
b
and, consequently, we define x as in Case IV, d
D I Cm0
b? y y , , b? and B T
D d, x ..
k 1
Case III is similar to Case I as is Case V.
D D k . Proof of Corollary 3.2 Follows immediate D 0. immediately ly from Theorem Theorem 3.1 by setting setting r r D
Proof of Corollary 3.1 Follows immediate immediately ly from Theorem Theorem 3.1 by setting setting r r
Proof of Theorem 3.2 We provide provide a proof for Case V which which may be simply simply adapted for Cases I and III. To emphasize the potential dependence of the limit distribution on nuisance parameters, the proof is initially conducted under Assumptions 1-4 together with Assumption 5a which implies yy pyx.x ? H0 : yy 0 but not necessarily necessarily H0 : p yx.x 00 ; in particular, note that we may write ay yy -vector f. The The t-statistic for H0 : yy 0 may be expressed as the square 1, f0 0 for some k -vector root of 1 0 0 0 ˆ ˆ Z0 P Z 1 AT A0 ˆ Z 1 AT A ˆ Z P y P y/ωuu A10
D
D
Z , ˆ X
T
1
1
D
D T
Z
1
ˆ Z ,X
1
? ? by , b . Note that only the diagonal element of the T1/2 b, T1/2 BT and BT where AT inverse in (A10) corresponding to b? y is relevant, which implies that we only need to consider 2 0 ˆ 0 Z B and T T 1 B0T ˆ Z01 PZ , ˆX y in (A10). Therefore, using (A2) and the blocks T blocks T BT Z1 PZ ˆ 1 1 T (A3), (A10) is asymptotically equivalent to
D O
Z1 BT Z01 ˆ Z1 BT T2 B0T ˆ T1 ˆu0 PX ˆ 1 b? ˆ xx where P ˆX1 b xx ?
1 1 0 ˆ 0 u/ωuu T BT Z1 PX ˆ b? ˆ 1 xx
A11
I Xˆ b? b? 0 ˆX0 ˆX b? b? 0 ˆX0 . Now, B C a T b? 0 ˆx ) 0, b? 0 b? [a? , a? 0 0(b? , b? )] a? , a? 0 ˆ D b? 0b? [a? 0 0 l g b? ] a? 0 ˆB a 1 xx
T
1/2
xx
xx
1
1
1 xx
1
xx
1
[Ta]
xx
xx xx
Copyright © 2001 John Wiley & Sons, Ltd.
xx
xx
y
xx
f f xy yx.x
k 1
y
y
1
xx
xx
f k
J. Appl. Econ. 16: 16 : 289–326 (2001)
BOUNDS TESTING FOR LEVEL RELATIONSHIPS
319
? ˇ? , 00 0 and b? where whe re,, for conven convenien ience, ce, but wit withou houtt loss loss of gen genera eralit lity, y, we hav havee set by yy ? 0 0 0 0 ˆ k a l B a , ˆ B , yy.x yyy f g xy , g yx.x g yx f 0 xx and Bk a 0, b xx , l xy g xy / yy.x , y yx xy u ˆ , a [0, 1]. Hence, (A11) weakly converges to Bu a B1 a 0 Bk a , a
D
O 2 O adW a BO a ˆB a0 da a? a? 0 B
O
1
1
u
0
u
0
ð 1
ð a? 0 xx
1
a? 0 xx
ł O O O k
u
2
0
1
xx
xx
0
1
ˆ Bk adWu a
0
O
0
? ˆ Bk a0 da a xx Bk a ˆ
1
1
Bu a2 da 1
a? 0 xx
0
? ˆ Bk a ˆ Bk a0 da a xx
D
1
? Bk a0 da a xx Bu a ˆ
f ˆ Bk a Bu ada
0
O
D O
Under Und er the condit condition ionss of the theor theorem, em, f w and l xy 0 and, therefo therefore, re, Bu a[ BuŁ a] ˆ k r a , a Bk a] a? 0 Z xx a? 1/2 W B a[ a? 0 ˆ , a [0, 1]. ω1/2 Wu a a and and a ? 0 ˆ uu
O
xx
D
k
D
D
xx
D
xx
2
xx
D
D D k .
Proof of Corollary 3.3 Follows immediate immediately ly from Theorem Theorem 3.2 by setting setting r r
immediately ly from Theorem Theorem 3.2 by setting setting r r Proof of Corollary 3.4 Follows immediate
D D 0.
APPENDIX B: PROOFS FOR SECTION 4 Proof of Theorem 4.1 Aga Again, in, we consider consider Case IV; IV; the remaini remaining ng Cases I–III and V may be yy 0 ab0 where dealt with similarly. Under H H1 : yy 0 , Assumption 5b holds and, thus, thus, ay by 0 0 ; see above Assumption 5b. Under Assumptions 1–4 and 5b, ay ˛yy , 00 0 and by ˇyy , byx m g t Cst CŁ Let , the process zt 1 tD1 has the infinite moving-average representation, zt the k where now C b? [a?0 0b? ]1 a?0 . We redefine b Ł and d as the k 2, r 1 and k 2, k r matrices, g 0 bŁ by , b Ik C1
D
f g
D C D C C C C C C
6D
D
and
g 0
d
b? ,
IC
k 1
matrix whose columns are a basis for the orthogonal complement of where b ? is a k 1, k r r matrix ? thus, bŁ , x , d a basis for Rk C2 , where again Hence, by , b, b is a basis for Rk C1 and, thus, by , b. Hence, x is is the k the k 2-unit vector 1, 00 0 . It therefore follows that
C
C
D T b?0 m C T b?0 Cs C b?0 T CŁLe ) b?0 CB C a Also, as above, T above, T x 0 zŁ D T t ) a and b 0Ł zŁ D b , b0 m C b , b0 CŁ Le D O 1. The Wald statistic (21) multiplied by Oω may be written as 0 Ł0 0 0 0 ˜ ˜ ˜ ZŁ l , ˜ u C l0Ł ˜ ZŁ P ˜ u C 2l0Ł ˜ ZŁ P A ˜ Z P ZŁ A ZŁ A A0 ˜ ˜ uP ZŁ P Ł T1/2 d0 zŁ[Ta]
1/2
1 /2
1
1
t
1 /2
[Ta]
y
t
[Ta]
k 1
t
y
P
uu
Z
1
T
T
1
Z
1
Copyright © 2001 John Wiley & Sons, Ltd.
T
1
T
1
Z
1
Z
1
Z
1
B1
J. Appl. Econ. 16: 16 : 289–326 (2001)
320
M. H. PESAR PESARAN, AN, Y. SHIN AND R. J. SMITH
bŁ ay , a0 1, w0 0 , AT d, T1/2 x . Note that (A6) T1/2 bŁ , T1/2 BT and BT where lŁ yy continues to hold under under H1 : yy 0. A similar argument to that in the Proof of Theorem 3.1 demonstrates that the first term in (B1) divided by ω by ω uu has the limiting representation
z0
C
1
r 1 zr 1
C C
0
6D
1
dWu aFk r a0
0
Fk r aFk r a0 da
1
1
0
Fk r adWu a
B2
˜ k r a0 , a 1 0 and W ˜ k r a a? 0 Z xx a? 1/2 a? 0 ˜ where z r C1 N0, Ir C1 , F k r a W xx Bk a xx xx 2 is a k r-vec r-vector tor of de-me de-meaned aned independent independent stand standard ard Brownian Brownian motio motions ns inde independe pendent nt of the 1 standard Brownian motion Wu a , a [0, 1]; cf. (22). Now, 0 Fk r adWu a is mixed normal 1 with condition conditional al variance variance matrix matrix 0 Fk r aFk r a0 da There refor fore, e, the seco second nd te term rm in (B2) (B2) is da.. The unconditionally distributed as a a 2 k r r random random variable and is independent of the first term; cf. (A4). Hence, the first term in (B1) divided by ω uu has a limiting limiting 2 k 1 distribution. The second term in (B1) may be written as
D
¾
2
0 21, w0 a , ab0Ł ˜ ZŁ P y
1
and the third term as
C
1/2
˜ u D 2T
Z
1, w0 a , a y
D
0 u T1/2 b0Ł ˜ ZŁ1 PZ ˜
OP T1/2 ,
w0a , ab0Ł ˜ ZŁ0 P ˜ ZŁ bŁa , a0 1, w0 0 DT1, w0 a , a T b0Ł ˜ ZŁ0 P Z ˜ Ł bŁ a , a0 1, w00 D O T
1,
y
1
y
1
Z
1
y
B3
1
1
Z
y
P
B4
0 ZŁ b converg ZŁ1 PZ ˜ converges es in probabi probability lity to a posit positive ive definite definite matrix. matrix. Mor Moreove eover, r, as as T1 b0Ł ˜ 1 Ł yy H 1 : yy 0 , the Theorem is proved. 1, w0 ay , a 00 under H
6D
6D
Proof of Theorem 4.2 A similar decom decomposit position ion to (B1) for the Wald Wald statistic statistic (21) holds under yy yx.x H0 except except that that bŁ and d are now as defi H1 defined ned in the Proof Proof of Theor Theorem em 3 3.1. .1. Altho Although ugh yx.x yy 0 have H1 : p yx.x 0 . Therefore, as in Theorem 3.2, note that we may H0 : yy 0 holds, we have 0 0 ? 1, f for some k some k -vector -vector f f w. Consequently, the first term divided by ω by ω uu may be write a y written as
\
6D 6D
D D
1 0 Ł0 0 bŁ ˜ ZŁ1 bŁ u0 PZ ˜ T1 ˜ ZŁ1 PZ ˜ ZŁ1 bŁ T1 b0Ł ˜ u/ωuu Z1 PZ ˜ 1 0 ˜ Ł0 0 Ł u/ωuu oP 1 Z1 BT BT Z1 ˜ ZŁ1 ˜ u0 ˜ ZŁ1 BT T2 B0T ˜ T2 ˜
C
C
B5
cf. (A7). As in the Proof of Theorem 3.1, the first term of (B5) has the limiting representation z r 0 zr where z r N0, Ir ; cf. (22). The second term of (B5) has the limiting representation
¾
Q Q Q Q Q Q ð D 1
0
d BŁu a 1
0
Bu a ? 0 ˜ Bk a a xx a 12
Bu a ? 0 ˜ a xx Bk a a 12
0
1
0
Bu a ? Bk a a xx 0 ˜ a 12
d BuŁ a/ωuu
Bu a ? 0 ˜ Bk a a xx a 21
0
1
da
OP 1
Copyright © 2001 John Wiley & Sons, Ltd.
J. Appl. Econ. 16: 16 : 289–326 (2001)
BOUNDS TESTING FOR LEVEL RELATIONSHIPS
Q
where Buf a becomes
321
Q B a f0 ˜ B a , a 2 [0, 1]; cf. Proof of Theorem 3.2. The second term of (B1)
21,
1
k
w0ab0Ł ˜ ZŁ0 P 1
and the third term
u Z ˜
1/2 1,
D 2T
w0 a
T1/2 b0Ł ˜ u ZŁ0 1 PZ ˜
D
OP T1/2
w0 ab0Ł ˜ ZŁ0 P ˜ ZŁ bŁa0 1, w0 0 D T1, w0a ð T b0Ł ˜ ZŁ0 P ˜ ZŁ bŁ a01, w00 D O T D 00 under H D 00. The Theorem follows as as 1, w0 a 6 H : D 0 and and H H : p 6 1,
1
1
1
Z
Z
1
P
1
yy 0
pyx.x 1
yy
yx.x
Pr Proof oof of The Theor orem em 4 4.3 .3 We conc concen entr trat atee on Case Case IV; IV; th thee remai emaini ning ng Ca Case sess I– II IIII and and V are are T denote the proce process ss und under er H1T of (26). (26). Hen Hence, ce, proved by a similar argument. Let ztT tD1 denote m g tt x tT t 1] et and 5T 5 is 8LztT 5T 5[zt1T m g t tT , where x tT tT Ł C 1 zCŁ z and given give n in (27). Therefo Therefore, re, ztT g tt Cx tT C Lx tT tT , Cz ? ? ? 1 ? ? ? ? 0 ? 0 C by , b [ay , a 0(by , b )] ay , a , and thus,
f g D
D
D
C
C D C
I C C T Ca b0 L]z m g tt D Ce C CŁLx
[Ik C1
1
k 1
y y
tT
tT
B6
tT tT
where
T
etT
1/2
dyx d xx
b0 [zt1T
m g tt 1] C e , t D 1, . . . , T , T D 1, 2, . . . t
Inverting (B6) yields
s 1
ztT D Ik C1 C T1 Cay b0y s zsT m g ss C m C g t C
D
i 0
Ik C1
C T Ca b0 1
i
y y
ð[Ce C CŁLx ] m g tt 1] C e . It therefor Note that that x D 5 5[z thereforee follows that that T T d0 zŁ ) b? , b? 0 CJ C a , where d is defined above Lemma A.1 and zŁ D t, z0 0 , J C a exp r is an Ornstein-Uhlenbeck process and B a is is a k C 1-vector BrowB C a fa b0 Ca rgdB C r is nian motion with variance matrix Z , a , a 2 [0, 1]; cf. Johansen (1995, Theorem 14.1, p. 202). t iT
tT tT
y
y y y
t t iT
t 1T
T
1/2
t
k 1
tT
k 1
tT
k 1
k 1
Similarly to (A4),
A0T ˜ Z01 PZ ˜ Z A D 1 T
0 ZŁ b ZŁ1 PZ ˜ T1 b0Ł ˜ 1 Ł 0
00 0 Ł Z1 BT ZŁ1 ˜ T2 B0T ˜
C
[Ta]T a 0
oP 1
O
Therefore, expression (B1) for the Wald statistic (21) multiplied by ωuu is revised to
C
C
Ł0 P y Ł bŁ 1 b0 ˜ Ł bŁ T1 b0 ˜ Ł0 P ˜ ˜ Z Z Z Z 1 Z Ł Ł 1 1 1 Z Z 1 0 Ł0 0 Ł BT ˜ Z1 PZ y oP 1 Z1 BT ZŁ1 BT T2 B0T ˜ T2 0 yPZ ˜ ZŁ1 ˜
O uu W D T1 0yP ω
Copyright © 2001 John Wiley & Sons, Ltd.
B7
J. Appl. Econ. 16: 16 : 289–326 (2001)
322
M. H. PESAR PESARAN, AN, Y. SHIN AND R. J. SMITH
The first term in (B7) may be written as
1 0 Ł0 Z 1P b ˜ ˜ u Z Ł Z Ł Z Ł Ł 1 C 2T1 ˜ u0 PZ ˜ ZŁ1bŁ T1b0Ł ˜ ZŁ01PZ ˜ ZŁ1bŁ b0Ł ˜ ZŁ01PZ ˜ ZŁ1pyTŁ0 1 C T1pyTŁ ˜ ZŁ0 1PZ ˜ ZŁ1bŁ T1b0Ł ˜ ZŁ01PZ ˜ ZŁ1bŁ b0Ł ˜ ZŁ01PZ ˜ ZŁ1pyTŁ0 B8 Ł T1 ˛yy b0 C T1/2 dyx w0 d xx b0 . Defining h dyx w0 d xx 0 , consider where p yT Ł y Ł 0 0 Ł0 D T1/2 b0 ˜ ZŁ1 pyT ZŁ b ˛ T1 C bŁ hT1/2 T1/2 b0Ł ˜ ZŁ1 PZ ˜ ZŁ1 PZ ˜ Ł 1 y Ł yy D T1b0Ł ˜ ZŁ01PZ ˜ ZŁ1bŁh C oP 1 B9 where we have made use of T a. Therefore, (B8) divided by ω uu may be T 1/2 b0y Ł z[ŁTa]T ) b0y CJk C1 a. ˜ ZŁ 1 b
re-expressed as
0 u T1/2 b0Ł ˜ ZŁ1 PZ ˜
0 ZŁ 1 P T1 b0 ˜
˜ ZŁ 1 b
u0 P T1 ˜
0 1 Q
C Qh
0 u T1/2 b0Ł ˜ ZŁ1 PZ ˜
C C ) ) DQ D
C Qh
/ωuu
C o 1 D z0 z C o 1
¾ D
P
P
r r
B9 0 ZŁ b and zz r NQ1/2 h, Ir . ZŁ1 PZ ˜ T1 b0Ł ˜ 1 Ł 0 Ł0 u˜ . Ł0 Ł0 u˜ , T1 B0 ˜ ZŁ1 pyT As Z ZŁ1 PZ ˜ T1 B0T ˜ ˜ ZŁ1 pyT T Z1 PZ y Z 0 Ł0 which after substitution ZŁ1 pyT ZŁ1 PZ ˜ Consider the second term in (B7), in particular, T1 B0T ˜ Ł p yT becomes for p where Q
p lim !1 P y D P T
0 ZŁ1 bŁ h ZŁ1 PZ ˜ T3/2 B0T ˜ ? 0 ? 1 Jk C1 a b , b C ˜
0 ZŁ1 by Ł ˛yy ZŁ1 PZ ˜ T2 B0T ˜
y
a
0
Therefore,
0 ZŁ1 PZ y T1 B0T ˜
Consider
˜ JŁk r C1 a[
1
1 2
? 0 ˜ b? y , b CJk C1 a a
0
JŁu a, ˜ JŁk r a0 0 ]
1 2
D T B0 ˜ ZŁ0 P 2
1
T
Z
D Q
˜ ZŁ1 by Ł ˛yy
C o 1 P
˜ Jk C1 a0 C0 by ˛yy da
Q
1 /2 ωuu dWu a
C ˜ J C a0 C0 b ˛
y yy da
k 1
? 0 ? ? 1/2 a? , a? 0 ˜ Jk C1 a [a? y , a Zay , a ] y
Q
1/2 Ju a ωuu
? 0 xx a? 1/2 a? 0 ˜ Jk a a xx xx xx
Q
C
Q
Jk a Jk a0 0 , a is independent of J˜ k a a and and J˜ k C1 a J1 a, ˜ , a [0, 1]. where Ju a J1 a w0 ˜ Ł Ł ˜ ˜ k r C1 ˜ Now, Jk r C1 a satisfie satisfiess the stochasti stochasticc inte integral gral and differe differentia ntiall equations equations,, Jk r C1 a W a ˜ Ł 0 0 ? Ł Ł ˜ k r C1 a ab ˜ a da , where a [ay , a? 0 a ab 0 Jk r C1 rdr and d Jk r C1 a and d ˜ Jk r C1 a dW ? 1/2 a? , a? 0 ay and bb [a? , a? 0 Za? , a? ]1/2 [b? , b? 0 0a? , a? ]1 b? , b? 0 a? y y y y y y y , a ] Ł ˜ by ; cf. Johansen (1995, Theorem 14.4, p. 207). Note that the first element of Jk r C1 a satisfies 1/ 2 ˛yy b0 a ˜ JŁ rdr and d 1/ 2 ˛yy b0 JŁ a JŁu a Wu a ωuu d JŁu a dWu a ωuu da.. k r C1 a da 0 k r C1
C
Q
D Q
D
C
Copyright © 2001 John Wiley & Sons, Ltd.
C
D
ð D Q
Q
D
C
D
2
Q
J. Appl. Econ. 16: 16 : 289–326 (2001)
BOUNDS TESTING FOR LEVEL RELATIONSHIPS
Therefore, 0 ZŁ1 PZ Y T1 B0T ˜
? 0 ˜ b? y , b CJk C1 a
1
)
0
Hence, the second term in (B7) weakly converges to
1
ωuu
0
1
d JQ Ł aFk r C1 a0 u
0
1 /2 ωuu d JŁu a
Q
1
a
2
Fk r C1 aFk r C1 a0 da
1
1
0
Fk r C1 ad JŁu a
Q
where F k r C1 a ˜ JŁk r C1 a0 , a 21 0 . Combining (B9) and (B10) gives the result stated in Theorem 4.3 as ωuu a.. a may may be replaced by dJ by dJ Łu a H1T of (26) and noting d noting d JŁu a
D
323
B10
O ω D O 1 under
Q
uu
P
Proof of Theorem 4.4 We consider Case V; the remainin remaining g Cases I and III may be dealt with yy ˆ similarly. simil arly. Under H1 : yy 0 , from (10), yˆ 1 X1 q ˆv1 , where vˆ 1 PZ ,X v and ˆ 1 1 y vˆ 01 PZ ,X v1 0, v1 , . . . , vT1 0 . Ther Therefor efore, e, yˆ 01 PZ ,X ˆy Y and yˆ 01 PZ ,X ˆ 1 ˆ 1 ˆ 1 1 0 ˆv1 PZ , ˆX ˆv1 . 1 As in Appendix A,
6D
D
D
C
D
D
D T b? 0 m C T b? 0g t C T b? 0 b? a?0 0b? a?0 s C 0, b? 0T CŁLe b? D 0 , b 0 x D T b0 m C T b0 g t C 0, b0 CŁ Le . Consequently,
? 0 x[Ta] T1/2 b xx
1/2
1/2
x
xx
1/2
A0
xx
1/2
xx t
ˆ ˆ 0 xT X1 PZ X1 A xT D
xx
xx
[Ta]
[Ta]
xx
0 b xx and noting that b
1
1/2
xx x
1/2
xx x
xx x
xx
t
C D D
0 ˆX0 P X 00 T1 b xx 1 Z ˆ 1 b xx ? ? 0 ˆX0 P X 0 T2 b xx 1 Z ˆ 1 b xx
oP 1
? . b xx , T1/2 b xx 0 0 ˆX0 Z OP 1 , 0 ˆX0 ˆv1 OP 1 , , T1 Z Z , T1 b xx Now, because because T1 b xx OP 1 and 1 1 0 0 ˆX0 P ˆv1 OP 1. Also because T1 b? 0 ˆX0 ˆv1 OP 1 OP 1 , hence T1 b xx T1 Z ˆv1 1 Z xx 1 1 ? 0 ˆ 0 1 ? 0 ˆ 0 hence T and T b xx X1 Z OP 1 , hence T b xx X1 PZ ˆv1 OP 1; cf. (A3). Hence, noting that 0 ? 0 ? 1 0 ˆ 0 2 ˆ b ˆ b X1 PZ X T b xx X1 PZ X OP 1 and and T T b xx ˆ OP 1 , 1 xx 1 xx
T
where A xT
1/2
D D
D
D D D D D D C D C D D D C D T1 ˆy01 PZ , ˆX ˆy1 1
T1 ˆv01 PZ
ˆv , ˆX1 b xx 1
T1 ˆv01 PZ
T1 ˆv01 PZ
ˆv , ˆX1 b xx 1
oP 1
v1 ˆ 1 b? ˆ ,X xx
oP 1
0 ˆ b 1 b0 ˆ ˆ b b0 ˆ X X0 P PZ X where PZ ,X PZ ˆ 1 b? xx X1 PZ and PZ ,X 1 xx xx 1 Z 1 xx ˆ 1 b xx xx ? 0 ? 0 ? 1 0 1 ? 0 ˆ 0 ˆ ˆ ˆ as T ˆv1 ˆv1 PZ X1 b xx b xx X1 PZ X1 b xx b xx X1 PZ . Therefore, as T OP 1 , T1 ˆy01 PZ
y1 ˆ 1 ˆ ,X
OP 1
B11
The numerator of t t yy of (24) may be written as yˆ 01 PZ , ˆX y vˆ 01 PZ , ˆX ˆu ˆv01 PZ , ˆX 1 1 0 1 0 0 0 1 /2 1/2 0 ˆ 0 ˆ b xx X1 ˆu OP 1 and Z1 l , where l by , bay , a 1, w . Beca and T Because use T Z ˆu Copyright © 2001 John Wiley & Sons, Ltd.
D
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324
M. H. PESAR PESARAN, AN, Y. SHIN AND R. J. SMITH
0 ˆX0 P ˆu OP 1 , , T T 1/2 b xx 1 Z Therefore,
D O 1 , and, as X0 P , T b? 0 ˆ X0 ˆu D O 1 , T as T T b? 0 ˆ 1
P
xx
1
1
P
xx
1
Z
ˆu
D O 1. P
01 PZ ,X ˆ b ˆu T1/2 ˆv01 PZ,X ˆ b? ˆu C oP 1 1 /2 0 ˆv1 PZ ,X ˆu C oP 1 D OP 1 ˆ b 6 00 , T1l0 ˆZ01Z D OP 1 , T1l0 ˆZ01 noting T1/2 ˆv01 ˆu D OP 1. Similarly, as 1, w0 ay , a D ? D OP 1. Therefore, ˆ 1 b xx D OP 1 and Z01 ˆ and T T 1 l0 ˆ X1 b xx X ˆ ˆ Z l C oP 1 Z l T1 ˆv01 PZ ,X Z l D T1 ˆv01 PZ ,X T1 ˆv01 PZ , ˆX ˆ ˆ b? 1 ˆ b 1 1 D T1 ˆv01 PZ,X ˆ b ˆZ1l C oP 1 D OP1 noting T noting T 1 ˆv01 ˆ Z1 l D OP 1. Thus, ˆ l D OP T1/2 . B12 T1/2 ˆv01 PZ , ˆX Z 1 T1/2 ˆv01 PZ ,X u ˆ 1 ˆ
D T D T
1
1/2 ˆ v
1 xx
1 xx
1 xx
1 xx
1 xx
1 xx
1
O ω D o 1 , combining (B11) and (B12) yields the desired result.
Because ωuu
uu
P
ACKNOWLEDGEMENTS
We are gratef grateful ul to the Editor Editor (Da (David vid Hendry Hendry)) and three three ano anonym nymous ous refere referees es for their their helpfu helpfull comments on an earlier version of this paper. Our thanks are also owed to Michael Binder, Peter Burridge, Clive Granger, Brian Henry, Joon-Yong Park, Ron Smith, Rod Whittaker and seminar participants at the University of Birmingham. Partial financial support from the ESRC (grant Nos R000233608 R0002 33608 and R00023733 R000237334) 4) and the Isaac Newton Newton Tru Trust st of Tri Trinity nity College, Cambridge, Cambridge, is gratefully acknowledged. Previous versions of this paper appeared as DAE Working Paper Series, Nos. 9622 and 9907, University of Cambridge.
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