1 INTRODUCTION
 Top of page
 Abstract
 1 INTRODUCTION
 2 THE UNDERLYING VAR MODEL AND ASSUMPTIONS
 3 BOUNDS TESTS FOR A LEVEL RELATIONSHIPS
 4 THE ASYMPTOTIC POWER OF THE BOUNDS PROCEDURE
 5 AN APPLICATION: UK EARNINGS EQUATION
 6 CONCLUSIONS
 Acknowledgements
 APPENDIX A: PROOFS FOR SECTION 3
 APPENDIX B: PROOFS FOR SECTION 4
 REFERENCES
Over the past decade considerable attention has been paid in empirical economics to testing for the existence of relationships in levels between variables. In the main, this analysis has been based on the use of cointegration techniques. Two principal approaches have been adopted: the twostep residualbased procedure for testing the null of nocointegration (see Engle and Granger, 1987; Phillips and Ouliaris, 1990) and the systembased reduced rank regression approach due to Johansen (1991, 1995). In addition, other procedures such as the variable addition approach of Park (1990), the residualbased 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 pretesting, thus introducing a further degree of uncertainty into the analysis of levels relationships. (See, for example, Cavanagh, Elliott and Stock, 1995.)
This paper proposes a new approach to testing for the existence of a relationship between variables in levels which is applicable irrespective of whether the underlying regressors are purely I(0), purely I(1) or mutually cointegrated. The statistic underlying our procedure is the familiar Wald or Fstatistic in a generalized Dicky–Fuller type regression used to test the significance of lagged levels of the variables under consideration in a conditional unrestricted equilibrium correction model (ECM). It is shown that the asymptotic distributions of both statistics are nonstandard under the null hypothesis that there exists no relationship in levels 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 I(0)/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 Fstatistic 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 Fstatistic 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 tstatistic 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 tstatistic are also detailed.
The empirical relevance of the proposed bounds procedure is demonstrated in a reexamination of the earnings equation included in the UK Treasury macroeconometric model. This is a particularly relevant application because there is considerable doubt concerning the order of integration of variables such as the degree of unionization of the workforce, the replacement 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, did not consider the problem of testing for the existence of a levels relationship between real wages and its 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 predicted by the bargaining theory of wage determination reviewed, for example, in 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 from entering the unemployment equation, or there being other level relationships between the remaining four variables. Our approach accommodates both of these possibilities. 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 of this and later sections is set out in Section 2. This section also addresses the issues involved in testing for the existence of relationships in levels between variables. Section 3 considers the Wald statistic (or the Fstatistic) for testing the hypothesis that there exists no level relationship between the variables under consideration and derives the associated asymptotic theory together with that for the tstatistic 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 following notation is used. The symbol signifies ‘weak convergence in probability measure’, I_{m} ‘an identity matrix of order m’, I(d) ‘integrated of order d’, O_{P} (K) ‘of the same order as K in probability’ and o_{P} (K) ‘of smaller order than K in probability’.
2 THE UNDERLYING VAR MODEL AND ASSUMPTIONS
 Top of page
 Abstract
 1 INTRODUCTION
 2 THE UNDERLYING VAR MODEL AND ASSUMPTIONS
 3 BOUNDS TESTS FOR A LEVEL RELATIONSHIPS
 4 THE ASYMPTOTIC POWER OF THE BOUNDS PROCEDURE
 5 AN APPLICATION: UK EARNINGS EQUATION
 6 CONCLUSIONS
 Acknowledgements
 APPENDIX A: PROOFS FOR SECTION 3
 APPENDIX B: PROOFS FOR SECTION 4
 REFERENCES
Assumption 1. The roots ofare either outside the unit circle z = 1 or satisfy z = 1.
Assumption 2. The vector error processis IN(0, Ω), Ωpositive definite.
Assumption 1 permits the elements of z_{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 to be a conditionally mean zero and homoscedastic process; see, for example, PSS, Assumption 4.1.
We may reexpress the lag polynomial Φ(L) in vector equilibrium correction model (ECM) form; i.e. Φ(L) ≡ −ΠL + Γ(L)(1 − L) in which the longrun multiplier matrix is defined by , and the shortrun response matrix lag polynomial , , i = 1, …, p − 1. Hence, the VAR(p) model (1) may be rewritten in vector ECM form as
 (2)
where Δ ≡ 1 − L is the difference operator,
 (3)
and the sum of the shortrun coefficient matrices . As detailed in PSS, Section 2, if γ ≠ 0, the resultant constraints (3) on the trend coefficients a_{1} in (2) ensure that the deterministic trending behaviour of the level process is invariant to the (cointegrating) rank of Π; a similar result holds for the intercept of if µ ≠ 0 and γ = 0. Consequently, critical regions defined in terms of the Wald and Fstatistics suggested below are asymptotically similar.2
The focus of this paper is on the conditional modelling of the scalar variable y_{t} given the kvector x_{t} and the past values and Z_{0}, where we have partitioned z_{t} = (y_{t}, x_{t}′)′. Partitioning the error term ϵ_{t} conformably with z_{t} = (y_{t}′, x_{t}′)′ as ϵ_{t} = (ϵ_{yt}, ϵ_{xt}′)′ and its variance matrix as
we may express ϵ_{yt} conditionally in terms of ϵ_{xt} as
 (4)
where u_{t} ∼ IN(0, ω_{uu}), and u_{t} is independent of ϵ_{xt}. Substitution of (4) into (2) together with a similar partitioning of a_{0} = (a_{y0}, a_{x0}′)′, a_{1} = (a_{y1}, a_{x1}′)′, Π = (π_{y}′, Π_{x}′)′, Γ = (γ_{y}′, Γ_{x}′)′, Γ_{i} = (γ_{yi}′, Γ_{xi}′)′, i = 1, …, p − 1, provides a conditional model for Δy_{t} in terms of z_{t − 1}, Δx_{t}, Δz_{t − 1}, …; i.e. the conditional ECM
 (5)
where , c_{0} ≡ a_{y0} − ω′a_{x0}, c_{1} ≡ a_{y1} − ω′a_{x1}, ψ_{i}′ ≡ γ_{yi} − ω′Γ_{xi}, i = 1, …, p − 1, and π_{y.x} ≡ π_{y} − ω′Π_{x}. The deterministic relations (3) are modified to
 (6)
where γ_{y.x} ≡ γ_{y} − ω′Γ_{x}.
We now partition the longrun multiplier matrix Π conformably with z_{t} = (y_{t}, x_{t}′)′ as
The next assumption is critical for the analysis of this paper.
Assumption 3. The kvectorπ_{xy} = 0.
Under Assumption 3, the conditional ECM (5) now becomes
 (8)
t = 1,2, …, where
 (9)
and π_{yx.x} ≡ π_{yx} − ω′Π_{xx}.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 matrixΠ_{xx}has rank r, 0 ≤ r ≤ k.
Under Assumption 4, from (7), we may express Π_{xx} as Π_{xx} = α_{xx}β_{xx}′, where α_{xx} and β_{xx} are both (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 1, 3 and 4, the process is mutually cointegrated of order r, 0 ≤ r ≤ k. However, in contradistinction to, for example, Banerjee, Dolado and Mestre (1998), BDM henceforth, who concentrate on the case r = 0, we do not wish to impose an a priori specification of r.6 When π_{xy} = 0andΠ_{xx} = 0, then x_{t} is weakly exogenous for π_{yy} and π_{yx.x} = π_{yx} in (8); see, for example, Johansen (1995, Theorem 8.1, p. 122). In the more general case where Π_{xx} is nonzero, as π_{yy} and π_{yx.x} = π_{yx} − ω′Π_{xx} are variationfree from the parameters in (7), x_{t} is also weakly exogenous for the parameters of (8).
Note that under Assumption 4 the maximal cointegrating rank of the longrun multiplier matrix Π for the system (8) and (7) is r + 1 and the minimal cointegrating rank of Π 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 rank(Π) = r. In this case, under Assumptions 1, 3 and 4, π_{yy} = 0 and π_{yx} − ϕ′Π_{xx} = 0′ for some kvector ϕ. Note that π_{yx.x} = 0′ implies the latter condition. Thus, under Assumptions 1, 3 and 4, Π has rank r and is given by
Hence, we may express Π = αβ′ where α = (α_{yx}′, α_{xx}′)′ and β = (0, β_{xx}′)′ are (k + 1, r) matrices of full column rank; cf. HJNR, p. 390. Let the columns of the (k + 1, k − r + 1) matrices and , where , and α^{⟂}, β^{⟂} are respectively (k + 1)vectors and (k + 1, k − r) matrices, denote bases for the orthogonal complements of respectively α and β; in particular, and .
Assumption 5a. If rank(Π) = r, the matrixis full rank k − r + 1, 0 ≤ r ≤ k.
Cf. Johansen (1991, Theorem 4.1, p. 1559).
Second, if the longrun multiplier matrix Π has rank r + 1, then under Assumptions 1, 3 and 4, π_{yy} ≠ 0 and Π may be expressed as Π = α_{y}β_{y}′ + αβ′, where α_{y} = (α_{yy}, 0′)′ and β_{y} = (β_{yy}, β_{yx}′)′ are (k + 1)vectors, the former of which preserves Assumption 3. For this case, the columns of α^{⟂} and β^{⟂} form respective bases for the orthogonal complements of (α_{y}, α) and (β_{y}, β); in particular, α^{⟂}′(α_{y}, α) = 0 and β^{⟂}′(β_{y}, β) = 0.
Assumption 5b. If rank(Π) = r + 1, the matrixα^{⟂}′Γβ^{⟂}is full rank k − r, 0 ≤ r ≤ k.
Assumptions 1, 3, 4 and 5a and 5b permit the two polar cases for . First, if is a purely I(0) vector process, then Π_{xx}, and, hence, α_{xx} and β_{xx}, are nonsingular. Second, if is purely I(1), then Π_{xx} = 0, and, hence, α_{xx} and β_{xx} are also null matrices.
Using (A.1) in Appendix A, it is easily seen that π_{y.x} (z_{t} − µ − γt) = π_{y.x}C*(L)ϵ_{t}, where {C*(L)ϵ_{t}} is a mean zero stationary process. Therefore, under Assumptions 1, 3, 4 and 5b, that is, π_{yy} ≠ 0, it immediately follows that there exists a conditional level relationship between y_{t} and x_{t} defined by
 (10)
where θ_{0} ≡ π_{y.x}µ/π_{yy}, θ_{1} ≡ π_{y.x}γ/π_{yy}, θ ≡ −π_{yx.x}/π_{yy} and v_{t} = π_{y.x}C*(L)ϵ_{t}/π_{yy}, also a zero mean stationary process. If π_{yx.x} = α_{yy}β_{yx}′ + (α_{yx} − ω′α_{xx})β_{xx}′ ≠ 0′, the level relationship between y_{t} and x_{t} is nondegenerate. Hence, from (10), y_{t} ∼ I(0) if rank(β_{yx}, β_{xx}) = r and y_{t} ∼ I(1) if rank(β_{yx}, β_{xx}) = r + 1. In the former case, θ is the vector of conditional longrun multipliers and, in this sense, (10) may be interpreted as a conditional longrun level relationship between y_{t} and x_{t}, whereas, in the latter, because the processes and are cointegrated, (10) represents the conditional longrun level relationship between y_{t} and x_{t}. Two degenerate cases arise. First, if π_{yy} ≠ 0 and π_{yx.x} = 0′, clearly, from (10), y_{t} is (trend) stationary or y_{t} ∼ I(0) whatever the value of r. Consequently, the differenced variable Δy_{t} depends only on its own lagged level y_{t − 1} in the conditional ECM (8) and not on the lagged levels x_{t − 1} of the forcing variables. Second, if π_{yy} = 0, that is, Assumption 5a holds, and π_{yx.x} = (α_{yx} − ω′α_{xx})β_{xx}′ ≠ 0′, as rank(Π) = r, π_{yx.x} = (ϕ − ω)′α_{xx}β_{xx}′ which, from the above, yields π_{yx.x} (x_{t} − µ_{x} − γ_{x}t) = π_{y.x}C*(L)ϵ_{t}, t = 1,2, …, where µ = (µ_{y}, µ_{x}′)′ and γ = (γ_{y}, γ_{x}′)′ are partitioned conformably with z_{t} = (y_{t}, x_{t}′)′. Thus, in (8), Δy_{t} depends only on the lagged level x_{t − 1} through the linear combination (ϕ − ω)′α_{xx} of the lagged mutually cointegrating relations β_{xx}′x_{t − 1} for the process . Consequently, y_{t} ∼ I(1) whatever the value of r. Finally, if both π_{yy} = 0 andπ_{yx.x} = 0′, there are no level effects in the conditional ECM (8) with no possibility of any level relationship between y_{t} and x_{t}, degenerate or otherwise, and, again, y_{t} ∼ I(1) 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 x_{t}, the emphasis in this paper is a test of the joint hypothesis π_{yy} = 0 and π_{yx.x} = 0′ in (8).7, 8 In contradistinction, the approach of BDM may be described in terms of (8) using Assumption 5b:
 (11)
BDM test for the exclusion of y_{t − 1} in (11) when r = 0, that is, β_{xx} = 0 in (11) or Π_{xx} = 0 in (7) and, thus, {x_{t}} is purely I(1); cf. HJNR and PSS.9 Therefore, BDM consider the hypothesis α_{yy} = 0 (or π_{yy} = 0).10 More generally, when 0 < r ≤ k, BDM require the imposition of the untested subsidiary hypothesis α_{yx} − ω′α_{xx} = 0′; that is, the limiting distribution of the BDM test is obtained under the joint hypothesis π_{yy} = 0 and π_{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:
Case I (no intercepts; no trends) c_{0} = 0 and c_{1} = 0. That is, µ = 0 and γ = 0. Hence, the ECM (8) becomes
 (12)
Case II (restricted intercepts; no trends) c_{0} = −(π_{yy}, π_{yx.x})µ and c_{1} = 0. Here, γ = 0. The ECM is
 (13)
Case III (unrestricted intercepts; no trends) c_{0} ≠ 0 and c_{1} = 0. Again, γ = 0. Now, the intercept restriction c_{0} = −(π_{yy}, π_{yx.x})µ is ignored and the ECM is
 (14)
Case IV (unrestricted intercepts; restricted trends) c_{0} ≠ 0 and c_{1} = −(π_{yy}, π_{yx.x})γ.
 (15)
Case V (unrestricted intercepts; unrestricted trends) c_{0} ≠ 0 and c_{1} ≠ 0. Here, the deterministic trend restriction c_{1} = −(π_{yy}, π_{yx.x})γ is ignored and the ECM is
 (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, c_{0} and c_{1}, to the parameter vector (π_{yy}, π_{yx.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 FOR A LEVEL RELATIONSHIPS
 Top of page
 Abstract
 1 INTRODUCTION
 2 THE UNDERLYING VAR MODEL AND ASSUMPTIONS
 3 BOUNDS TESTS FOR A LEVEL RELATIONSHIPS
 4 THE ASYMPTOTIC POWER OF THE BOUNDS PROCEDURE
 5 AN APPLICATION: UK EARNINGS EQUATION
 6 CONCLUSIONS
 Acknowledgements
 APPENDIX A: PROOFS FOR SECTION 3
 APPENDIX B: PROOFS FOR SECTION 4
 REFERENCES
In this section we develop bounds procedures for testing for the existence of a level relationship between y_{t} and x_{t} using (12–16); see (10). The main approach taken here, cf. Engle and Granger (1987) and BDM, is to test for the absence of any level relationship between y_{t} and x_{t} via the exclusion of the lagged level variables y_{t − 1} and x_{t − 1} in (12–16). Consequently, we define the constituent null hypotheses , , and alternative hypotheses , . Hence, the joint null hypothesis of interest in (12–16) is given by:
 (17)
and the alternative hypothesis is correspondingly stated as:
 (18)
However, as indicated in Section 2, not only does the alternative hypothesis H_{1} of (17) cover the case of interest in which π_{yy} ≠ 0 and π_{yx.x} ≠ 0′ but also permits π_{yy} ≠ 0, π_{yx.x} = 0′ and π_{yy} = 0 and π_{yx.x} ≠ 0′; cf. (8). That is, the possibility of degenerate level relationships between y_{t} and x_{t} is admitted under H_{1} of (18). We comment further on these alternatives at the end of this section.
For ease of exposition, we consider Case IV and rewrite (15) in matrix notation as
 (19)
where ι_{T} is a Tvector of ones, Δy ≡ (Δy_{1}, …, Δy_{T})′, ΔX ≡ (Δx_{1}, …, Δx_{T})′, ΔZ_{ − i} ≡ (Δz_{1 − i}, …, Δz_{T − i})′, i = 1, …, p − 1, ψ ≡ (ω′, ψ_{1}′, …, ψ_{p − 1}′)′, ΔZ_{−} ≡ (ΔX, ΔZ_{−1}, …, ΔZ_{1 − p}), , τ_{T} ≡ (1, …, T)′, Z_{−1} ≡ (z_{0}, …, z_{T−1})′, u ≡ (u_{1}, …, u_{T})′ and
The least squares (LS) estimator of is given by:
 (20)
where , , , P_{ι} ≡ I_{T} − ι_{T} (ι_{T}′ι_{T})^{−1}ι_{T}′ and . The Wald and the Fstatistics for testing the null hypothesis H_{0} of (17) against the alternative hypothesis H_{1} of (18) are respectively:
 (21)
where , m ≡ (k + 1)(p + 1) + 1 is the number of estimated coefficients and ũ_{t}, t = 1,2, …, T, are the least squares (LS) residuals from (19).
The next theorem presents the asymptotic null distribution of the Wald statistic; the limit behaviour of the Fstatistic is a simple corollary and is not presented here or subsequently. Let W_{k − r + 1} (a) ≡ (W_{u} (a), W_{k − r} (a)′)′ denote a (k − r + 1)dimensional standard Brownian motion partitioned into the scalar and (k − r)dimensional subvector independent standard Brownian motions W_{u} (a) and W_{k − r} (a), a ∈ [0,1]. We will also require the corresponding demeaned (k − r + 1)vector standard Brownian motion , and demeaned and detrended (k − r + 1)vector standard Brownian motion , and their respective partitioned counterparts W̃_{k − r + 1} (a) = (W̃_{u} (a), W̃_{k − r} (a)′)′, and Ŵ_{k − r + 1} (a) = (Ŵ_{u} (a), Ŵ_{k − r} (a)′)′, a ∈ [0,1].
Theorem 3.1 (Limiting distribution of W) If Assumptions 1–4 and 5a hold, then under H_{0}:π_{yy} = 0 andπ_{yx.x} = 0′ of (17), as T ∞, the asymptotic distribution of the Wald statistic W of (21) has the representation
 (22)
wherez_{r} ∼ N(0, I_{r}) is distributed independently of the second term in (22) and
r = 0, …, k, and Cases I–V are defined in (12–16), a ∈ [0,1].
The asymptotic distribution of the Wald statistic W of (21) depends on the dimension and cointegration rank of the forcing variables {x_{t}}, k and r respectively. In Case IV, referring to (11), the first component in (22), z_{r}′z_{r} ∼ χ^{2} (r), corresponds to testing for the exclusion of the rdimensional stationary vector β_{xx}′x_{t − 1}, that is, the hypothesis α_{yx} − ω′α_{xx} = 0′, whereas the second term in (22), which is a nonstandard Dickey–Fuller unitroot distribution, corresponds to testing for the exclusion of the (k − r + 1)dimensional I(1) vector and, in Cases II and IV, the intercept and timetrend respectively or, equivalently, α_{yy} = 0.
We specialize Theorem 3.1 to the two polar cases in which, first, the process for the forcing variables {x_{t}} is purely integrated of order zero, that is, r = k and Π_{xx} is of full rank, and, second, the {x_{t}} process is not mutually cointegrated, r = 0, and, hence, the {x_{t}} process is purely integrated of order one.
Corollary 3.1(Limiting distribution of W if {x_{t}} ∼ I(0)). If Assumptions 1–4 and 5a hold and r = k, that is, {x_{t}} ∼ I(0), then under H_{0}:π_{yy} = 0 andπ_{yx.x} = 0′ of (17), as T ∞, the asymptotic distribution of the Wald statistic W of (21) has the representation
 (23)
wherez_{k} ∼ N(0, I_{k}) is distributed independently of the second term in (23) and
r = 0, …, k, where Cases I–V are defined in (12–16), a ∈ [0,1].
Corollary 3.2(Limiting distribution of W if {x_{t}} ∼ I(1)). If Assumptions 1–4 and 5a hold and r = 0, that is, {x_{t}} ∼ I(1), then under H_{0}:π_{yy} = 0 andπ_{yx.x} = 0′ of (17), as T ∞, the asymptotic distribution of the Wald statistic W of (21) has the representation
whereF_{k + 1} (a) is defined in Theorem 3.1 for Cases I–V, a ∈ [0,1].
Table CI. Asymptotic critical value bounds for the Fstatistic. Testing for the existence of a levels relationshipaTable CI(i) Case I: No intercept and no trend 

k  0.100  0.050  0.025  0.010  Mean  Variance 

I(0)  I(1)  I(0)  I(1)  I(0)  I(1)  I(0)  I(1)  I(0)  I(1)  I(0)  I(1) 

0  3.00  3.00  4.20  4.20  5.47  5.47  7.17  7.17  1.16  1.16  2.32  2.32 
1  2.44  3.28  3.15  4.11  3.88  4.92  4.81  6.02  1.08  1.54  1.08  1.73 
2  2.17  3.19  2.72  3.83  3.22  4.50  3.88  5.30  1.05  1.69  0.70  1.27 
3  2.01  3.10  2.45  3.63  2.87  4.16  3.42  4.84  1.04  1.77  0.52  0.99 
4  1.90  3.01  2.26  3.48  2.62  3.90  3.07  4.44  1.03  1.81  0.41  0.80 
5  1.81  2.93  2.14  3.34  2.44  3.71  2.82  4.21  1.02  1.84  0.34  0.67 
6  1.75  2.87  2.04  3.24  2.32  3.59  2.66  4.05  1.02  1.86  0.29  0.58 
7  1.70  2.83  1.97  3.18  2.22  3.49  2.54  3.91  1.02  1.88  0.26  0.51 
8  1.66  2.79  1.91  3.11  2.15  3.40  2.45  3.79  1.02  1.89  0.23  0.46 
9  1.63  2.75  1.86  3.05  2.08  3.33  2.34  3.68  1.02  1.90  0.20  0.41 
10  1.60  2.72  1.82  2.99  2.02  3.27  2.26  3.60  1.02  1.91  0.19  0.37 
Table CI(ii) Case II: Restricted intercept and no trend 

k  0.100  0.050  0.025  0.010  Mean  Variance 

I(0)  I(1)  I(0)  I(1)  I(0)  I(1)  I(0)  I(1)  I(0)  I(1)  I(0)  I(1) 

0  3.80  3.80  4.60  4.60  5.39  5.39  6.44  6.44  2.03  2.03  1.77  1.77 
1  3.02  3.51  3.62  4.16  4.18  4.79  4.94  5.58  1.69  2.02  1.01  1.25 
2  2.63  3.35  3.10  3.87  3.55  4.38  4.13  5.00  1.52  2.02  0.69  0.96 
3  2.37  3.20  2.79  3.67  3.15  4.08  3.65  4.66  1.41  2.02  0.52  0.78 
4  2.20  3.09  2.56  3.49  2.88  3.87  3.29  4.37  1.34  2.01  0.42  0.65 
5  2.08  3.00  2.39  3.38  2.70  3.73  3.06  4.15  1.29  2.00  0.35  0.56 
6  1.99  2.94  2.27  3.28  2.55  3.61  2.88  3.99  1.26  2.00  0.30  0.49 
7  1.92  2.89  2.17  3.21  2.43  3.51  2.73  3.90  1.23  2.01  0.26  0.44 
8  1.85  2.85  2.11  3.15  2.33  3.42  2.62  3.77  1.21  2.01  0.23  0.40 
9  1.80  2.80  2.04  3.08  2.24  3.35  2.50  3.68  1.19  2.01  0.21  0.36 
10  1.76  2.77  1.98  3.04  2.18  3.28  2.41  3.61  1.17  2.00  0.19  0.33 
Table CI(iii) Case III: Unrestricted intercept and no trend 

k  0.100  0.050  0.025  0.010  Mean  Variance 

I(0)  I(1)  I(0)  I(1)  I(0)  I(1)  I(0)  I(1)  I(0)  I(1)  I(0)  I(1) 

0  6.58  6.58  8.21  8.21  9.80  9.80  11.79  11.79  3.05  3.05  7.07  7.07 
1  4.04  4.78  4.94  5.73  5.77  6.68  6.84  7.84  2.03  2.52  2.28  2.89 
2  3.17  4.14  3.79  4.85  4.41  5.52  5.15  6.36  1.69  2.35  1.23  1.77 
3  2.72  3.77  3.23  4.35  3.69  4.89  4.29  5.61  1.51  2.26  0.82  1.27 
4  2.45  3.52  2.86  4.01  3.25  4.49  3.74  5.06  1.41  2.21  0.60  0.98 
5  2.26  3.35  2.62  3.79  2.96  4.18  3.41  4.68  1.34  2.17  0.48  0.79 
6  2.12  3.23  2.45  3.61  2.75  3.99  3.15  4.43  1.29  2.14  0.39  0.66 
7  2.03  3.13  2.32  3.50  2.60  3.84  2.96  4.26  1.26  2.13  0.33  0.58 
8  1.95  3.06  2.22  3.39  2.48  3.70  2.79  4.10  1.23  2.12  0.29  0.51 
9  1.88  2.99  2.14  3.30  2.37  3.60  2.65  3.97  1.21  2.10  0.25  0.45 
10  1.83  2.94  2.06  3.24  2.28  3.50  2.54  3.86  1.19  2.09  0.23  0.41 
Table CI(iv) Case IV: Unrestricted intercept and restricted trend 

k  0.100  0.050  0.025  0.010  Mean  Variance 

I(0)  I(1)  I(0)  I(1)  I(0)  I(1)  I(0)  I(1)  I(0)  I(1)  I(0)  I(1) 

0  5.37  5.37  6.29  6.29  7.14  7.14  8.26  8.26  3.17  3.17  2.68  2.68 
1  4.05  4.49  4.68  5.15  5.30  5.83  6.10  6.73  2.45  2.77  1.41  1.65 
2  3.38  4.02  3.88  4.61  4.37  5.16  4.99  5.85  2.09  2.57  0.92  1.20 
3  2.97  3.74  3.38  4.23  3.80  4.68  4.30  5.23  1.87  2.45  0.67  0.93 
4  2.68  3.53  3.05  3.97  3.40  4.36  3.81  4.92  1.72  2.37  0.51  0.76 
5  2.49  3.38  2.81  3.76  3.11  4.13  3.50  4.63  1.62  2.31  0.42  0.64 
6  2.33  3.25  2.63  3.62  2.90  3.94  3.27  4.39  1.54  2.27  0.35  0.55 
7  2.22  3.17  2.50  3.50  2.76  3.81  3.07  4.23  1.48  2.24  0.31  0.49 
8  2.13  3.09  2.38  3.41  2.62  3.70  2.93  4.06  1.44  2.22  0.27  0.44 
9  2.05  3.02  2.30  3.33  2.52  3.60  2.79  3.93  1.40  2.20  0.24  0.40 
10  1.98  2.97  2.21  3.25  2.42  3.52  2.68  3.84  1.36  2.18  0.22  0.36 
Table CI(v) Case V: Unrestricted intercept and unrestricted trend 

k  0.100  0.050  0.025  0.010  Mean  Variance 

I(0)  I(1)  I(0)  I(1)  I(0)  I(1)  I(0)  I(1)  I(0)  I(1)  I(0)  I(1) 


0  9.81  9.81  11.64  11.64  13.36  13.36  15.73  15.73  5.33  5.33  11.35  11.35 
1  5.59  6.26  6.56  7.30  7.46  8.27  8.74  9.63  3.17  3.64  3.33  3.91 
2  4.19  5.06  4.87  5.85  5.49  6.59  6.34  7.52  2.44  3.09  1.70  2.23 
3  3.47  4.45  4.01  5.07  4.52  5.62  5.17  6.36  2.08  2.81  1.08  1.51 
4  3.03  4.06  3.47  4.57  3.89  5.07  4.40  5.72  1.86  2.64  0.77  1.14 
5  2.75  3.79  3.12  4.25  3.47  4.67  3.93  5.23  1.72  2.53  0.59  0.91 
6  2.53  3.59  2.87  4.00  3.19  4.38  3.60  4.90  1.62  2.45  0.48  0.75 
7  2.38  3.45  2.69  3.83  2.98  4.16  3.34  4.63  1.54  2.39  0.40  0.64 
8  2.26  3.34  2.55  3.68  2.82  4.02  3.15  4.43  1.48  2.35  0.34  0.56 
9  2.16  3.24  2.43  3.56  2.67  3.87  2.97  4.24  1.43  2.31  0.30  0.49 
10  2.07  3.16  2.33  3.46  2.56  3.76  2.84  4.10  1.40  2.28  0.26  0.44 
In practice, however, it is unlikely that one would possess a priori knowledge of the rank r of Π_{xx}; that is, the cointegration rank of the forcing variables {x_{t}} or, more particularly, whether {x_{t}} ∼ I(0) or {x_{t}} ∼ I(1). Longrun analysis of (12–16) predicated on a prior determination of the cointegration rank r in (7) is prone to the possibility of a pretest specification error; see, for example, Cavanagh et al. (1995). However, it may be shown by simulation that the asymptotic critical values obtained from Corollaries 3.1 (r = k and {x_{t}} ∼ I(0)) and 3.2 (r = 0 and {x_{t}} ∼ I(1)) provide lower and upper bounds respectively for those corresponding to the general case considered in Theorem 3.1 when the cointegration rank of the forcing variables {x_{t}} process is 0 ≤ r ≤ k.11 Hence, these two sets of critical values provide critical value bounds covering all possible classifications of {x_{t}} into I(0), I(1) and mutually cointegrated processes. Asymptotic critical value bounds for the Fstatistics covering Cases I–V are set out in 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 {x_{t}} are purely I(0), and the upper bound values assume that {x_{t}} are purely I(1).12
Hence, we suggest a bounds procedure to test H_{0}:π_{yy} = 0 and π_{yx.x} = 0′ of (17) within the conditional ECMs (12–16). If the computed Wald or Fstatistics fall outside the critical value bounds, a conclusive decision results without needing to know the cointegration rank r of the {x_{t}} process. If, however, the Wald or Fstatistic fall within these bounds, inference would be inconclusive. In such circumstances, knowledge of the cointegration rank r of the forcing variables {x_{t}} 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)). However, one could also allow for differential lag lengths on the lagged variables y_{t−i} and x_{t−i} in (2) to arrive at, for example, an ARDL(p, p_{1}, …, p_{k}) model without affecting the asymptotic results derived in this section. Hence, our approach is quite general in the sense that one can use a flexible choice for the dynamic lag structure in (12–16) as well as allowing for shortrun feedbacks from the lagged dependent variables, Δy_{t−i}, i = 1, …, p, to Δx_{t} in (7). Moreover, within the singleequation context, the above analysis is more general than the cointegration analysis of partial systems carried out by Boswijk (1992, 1995), HJNR, Johansen (1992, 1995), PSS, and Urbain (1992), where it is assumed in addition that Π_{xx} = 0 or x_{t} is purely I(1) 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 α_{yx} − ω′α_{xx} = 0′; that is, the limiting distributional results given below are also obtained under the joint hypothesis H_{0}:π_{yy} = 0 and π_{yx.x} = 0′ of (17). BDM test α_{yy} = 0 (or ) via the exclusion of y_{t − 1} in Cases I, III and V. For example, in Case V, they consider the tstatistic
 (24)
where is defined in the line after (21), , ŷ_{−1} ≡ Py_{−1}, y_{−1} ≡ (y_{0}, …, y_{T−1})′, X̂_{−1} ≡ PX_{−1}, X_{−1} ≡ (x_{0}, …, x_{T−1})′, , P ≡ P − Pτ_{T} (τ_{T}′Pτ_{T})^{−1}τ_{T}′P, and .
Theorem 3.2(Limiting distribution of t). If Assumptions 14 and 5a hold andγ_{xy} = 0, whereΓ_{x} = (γ_{xy}, Γ_{xx}), then under H_{0} : π_{yy} = 0 andπ_{yx.x} = 0′ of (17), as T ∞, the asymptotic distribution of the tstatistic tof (24) has the representation
 (25)
where
r = 0, …, k, and Cases I, III and V are defined in (12), (14) and (16), a ∈ [0,1].
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 W_{u} (a) is replaced by the residual from an asymptotic regression of W_{u} (a) on the independent (k − r)vector standard Brownian motion W_{k − r} (a) (or their demeaned and demeaned and detrended counterparts).
Similarly to the analysis following Theorem 3.1, we detail the limiting distribution of the tstatistic t in the two polar cases in which the forcing variables {x_{t}} are purely integrated of order zero and one respectively.
Corollary 3.3(Limiting distribution of tif {X_{t}} ∼ I(0)). If Assumptions 14 and 5a hold and r = k, that is, {x_{t}} ∼ I(0), then under H_{0} : π_{yy} = 0 andπ_{yx.x} = 0′ of (17), as T ∞, the asymptotic distribution of the tstatistic tof (24) has the representation
where
and Cases I, III and V are defined in (12), (14) and (16), a ∈ [0,1].
Corollary 3.4(Limiting distribution of tif {X_{t}} ∼ I(1)). If Assumptions 14 and 5a hold, γ_{xy} = 0, whereΓ_{x} = (γ_{xy}, Γ_{xx}), and r = 0, that is, {x_{t}} ∼ I(1), then under, as T ∞, the asymptotic distribution of the tstatistic tof (24) has the representation
where F_{k} (a) is defined in Theorem 3.2 for Cases I, III and V, a ∈ [0,1].
Table CII. Asymptotic critical value bounds of the tstatistic. Testing for the existence of a levels relationshipaTable CII(i): Case I: No intercept and no trend 

k  0.100  0.050  0.025  0.010  Mean  Variance 

I(0)  I(1)  I(0)  I(1)  I(0)  I(1)  I(0)  I(1)  I(0)  I(1)  I(0)  I(1) 

0  −1.62  −1.62  −1.95  −1.95  −2.24  −2.24  −2.58  −2.58  −0.42  −0.42  0.98  0.98 
1  −1.62  −2.28  −1.95  −2.60  −2.24  −2.90  −2.58  −3.22  −0.42  −0.98  0.98  1.12 
2  −1.62  −2.68  −1.95  −3.02  −2.24  −3.31  −2.58  −3.66  −0.42  −1.39  0.98  1.12 
3  −1.62  −3.00  −1.95  −3.33  −2.24  −3.64  −2.58  −3.97  −0.42  −1.71  0.98  1.09 
4  −1.62  −3.26  −1.95  −3.60  −2.24  −3.89  −2.58  −4.23  −0.42  −1.98  0.98  1.07 
5  −1.62  −3.49  −1.95  −3.83  −2.24  −4.12  −2.58  −4.44  −0.42  −2.22  0.98  1.05 
6  −1.62  −3.70  −1.95  −4.04  −2.24  −4.34  −2.58  −4.67  −0.42  −2.43  0.98  1.04 
7  −1.62  −3.90  −1.95  −4.23  −2.24  −4.54  −2.58  −4.88  −0.42  −2.63  0.98  1.04 
8  −1.62  −4.09  −1.95  −4.43  −2.24  −4.72  −2.58  −5.07  −0.42  −2.81  0.98  1.04 
9  −1.62  −4.26  −1.95  −4.61  −2.24  −4.89  −2.58  −5.25  −0.42  −2.98  0.98  1.04 
10  −1.62  −4.42  −1.95  −4.76  −2.24  −5.06  −2.58  −5.44  −0.42  −3.15  0.98  1.03 
Table CII(iii) Case III: Unrestricted intercept and no trend 

k  0.100  0.050  0.025  0.010  Mean  Variance 

I(0)  I(1)  I(0)  I(1)  I(0)  I(1)  I(0)  I(1)  I(0)  I(1)  I(0)  I(1) 

0  −2.57  −2.57  −2.86  −2.86  −3.13  −3.13  −3.43  −3.43  −1.53  −1.53  0.72  0.71 
1  −2.57  −2.91  −2.86  −3.22  −3.13  −3.50  −3.43  −3.82  −1.53  −1.80  0.72  0.81 
2  −2.57  −3.21  −2.86  −3.53  −3.13  −3.80  −3.43  −4.10  −1.53  −2.04  0.72  0.86 
3  −2.57  −3.46  −2.86  −3.78  −3.13  −4.05  −3.43  −4.37  −1.53  −2.26  0.72  0.89 
4  −2.57  −3.66  −2.86  −3.99  −3.13  −4.26  −3.43  −4.60  −1.53  −2.47  0.72  0.91 
5  −2.57  −3.86  −2.86  −4.19  −3.13  −4.46  −3.43  −4.79  −1.53  −2.65  0.72  0.92 
6  −2.57  −4.04  −2.86  −4.38  −3.13  −4.66  −3.43  −4.99  −1.53  −2.83  0.72  0.93 
7  −2.57  −4.23  −2.86  −4.57  −3.13  −4.85  −3.43  −5.19  −1.53  −3.00  0.72  0.94 
8  −2.57  −4.40  −2.86  −4.72  −3.13  −5.02  −3.43  −5.37  −1.53  −3.16  0.72  0.96 
9  −2.57  −4.56  −2.86  −4.88  −3.13  −5.18  −3.42  −5.54  −1.53  −3.31  0.72  0.96 
10  −2.57  −4.69  −2.86  −5.03  −3.13  −5.34  −3.43  −5.68  −1.53  −3.46  0.72  0.96 
Table CII(v) Case V: Unrestricted intercept and unrestricted trend 

k  0.100  0.050  0.025  0.010  Mean  Variance 

I(0)  I(1)  I(0)  I(1)  I(0)  I(1)  I(0)  I(1)  I(0)  I(1)  I(0)  I(1) 


0  −3.13  −3.13  −3.41  −3.41  −3.65  −3.66  −3.96  −3.97  −2.18  −2.18  0.57  0.57 
1  −3.13  −3.40  −3.41  −3.69  −3.65  −3.96  −3.96  −4.26  −2.18  −2.37  0.57  0.67 
2  −3.13  −3.63  −3.41  −3.95  −3.65  −4.20  −3.96  −4.53  −2.18  −2.55  0.57  0.74 
3  −3.13  −3.84  −3.41  −4.16  −3.65  −4.42  −3.96  −4.73  −2.18  −2.72  0.57  0.79 
4  −3.13  −4.04  −3.41  −4.36  −3.65  −4.62  −3.96  −4.96  −2.18  −2.89  0.57  0.82 
5  −3.13  −4.21  −3.41  −4.52  −3.65  −4.79  −3.96  −5.13  −2.18  −3.04  0.57  0.85 
6  −3.13  −4.37  −3.41  −4.69  −3.65  −4.96  −3.96  −5.31  −2.18  −3.20  0.57  0.87 
7  −3.13  −4.53  −3.41  −4.85  −3.65  −5.14  −3.96  −5.49  −2.18  −3.34  0.57  0.88 
8  −3.13  −4.68  −3.41  −5.01  −3.65  −5.30  −3.96  −5.65  −2.18  −3.49  0.57  0.90 
9  −3.13  −4.82  −3.41  −5.15  −3.65  −5.44  −3.96  −5.79  −2.18  −3.62  0.57  0.91 
10  −3.13  −4.96  −3.41  −5.29  −3.65  −5.59  −3.96  −5.94  −2.18  −3.75  0.57  0.92 
As above, it may be shown by simulation that the asymptotic critical values obtained from Corollaries 3.3 (r = k and {x_{t}} is purely I(0)) and 3.4 (r = 0 and {x_{t}} is purely I(1)) provide lower and upper bounds respectively for those corresponding to the general case considered in Theorem 3.2. Hence, a bounds procedure for testing based on these two polar cases may be implemented as described above based on the tstatistic t for the exclusion of y_{t − 1} in 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 for the tstatistic t of (24) is conducted under only, the resultant limit distribution for t depends on the nuisance parameter ω − ϕ in addition to the cointegrating rank r, where, under Assumption 5a, α_{yx} − ϕ′α_{xx} = 0′. Moreover, if Δy_{t} is allowed to Grangercause Δx_{t}, that is, γ_{xy, i} ≠ 0 for some i = 1, …, p − 1, then the limit distribution also is dependent on the nuisance parameter γ_{xy}/(γ_{yy} − ϕ′γ_{xy}); see Appendix A. Consequently, in general, where ω ≠ ϕ or γ_{xy} ≠ 0, although the tstatistic t has a welldefined limiting distribution under , the above bounds testing procedure for based on t is not asymptotically similar.14
Consequently, in the light of the consistency results for the above statistics discussed 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_{t}: test H_{0} of (17) using the bounds procedure based on the Wald or Fstatistic of (21) from Corollaries 3.1 and 3.2: (a) if H_{0} is not rejected, proceed no further; (b) if H_{0} is rejected, test using the bounds procedure based on the tstatistic t of (24) from Corollaries 3.3 and 3.4. If is false, a large value of t should result, at least asymptotically, confirming the existence of a level relationship between y_{t} and x_{t}, which, however, may be degenerate (if π_{yx.x} = 0′).
4 THE ASYMPTOTIC POWER OF THE BOUNDS PROCEDURE
 Top of page
 Abstract
 1 INTRODUCTION
 2 THE UNDERLYING VAR MODEL AND ASSUMPTIONS
 3 BOUNDS TESTS FOR A LEVEL RELATIONSHIPS
 4 THE ASYMPTOTIC POWER OF THE BOUNDS PROCEDURE
 5 AN APPLICATION: UK EARNINGS EQUATION
 6 CONCLUSIONS
 Acknowledgements
 APPENDIX A: PROOFS FOR SECTION 3
 APPENDIX B: PROOFS FOR SECTION 4
 REFERENCES
This section first demonstrates that the proposed bounds testing procedure based on the Wald statistic of (21) described in Section 3 is consistent. Second, it derives the asymptotic distribution of the Wald statistic of (21) under a sequence of local alternatives. Finally, we show that the bounds procedure based on the tstatistic of (24) is consistent.
In the discussion of the consistency of the bounds test procedure based on the Wald statistic of (21), because the rank of the longrun multiplier matrix Π may be either r or r + 1 under the alternative hypothesis of (18) where and , it is necessary to deal with these two possibilities. First, under , the rank of Π is r + 1 so Assumption 5b applies; in particular, α_{yy} ≠ 0. Second, under , the rank of Π is r so Assumption 5a applies; in this case, holds and, in particular, α_{yx} − ω′α_{xx} ≠ 0′.
Hence, combining Theorems 4.1 and 4.2, the bounds procedure of Section 3 based on the Wald statistic W (21) defines a consistent test of of (17) against of (18). This result holds irrespective of whether the forcing variables {x_{t}} are purely I(0), purely I(1) 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, π_{y.x}[ = (π_{yy}, π_{yx.x})] = (α_{yy}β_{yy},α_{yy}β_{xy}′ + (α_{yx} − ω′α_{xx})β_{xx}′). Consequently, we define the sequence of local alternatives
 (26)
Hence, under Assumption 3, defining
and recalling Π = αβ′, where (1,−ω′)α = α_{yx} − ω′α_{xx} = 0′, we have
 (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 which obeys the stochastic integral and differential equations, and , where W_{k − r + 1} (a) is a (k − r + 1)dimensional standard Brownian motion, , , together with the demeaned and demeaned and detrended counterparts and partitioned similarly, a ∈ [0,1]. See, for example, Johansen (1995, Chapter 14, pp. 201–210).
Theorem 4.3(Limiting distribution of W under H_{1T}). If Assumptions 1–4 and 5a hold, then under H_{1T}:π_{y.x} = T^{−1}α_{yy}β_{y}′ + T^{−1/2} (δ_{yx} − ω′δ_{xx})β′ of (26), as T ∞, the asymptotic distribution of the Wald statistic W of (21) has the representation
 (28)
wherez_{r} ∼ N (Q^{1/2}η, I_{r}), , η ≡ (δ_{yx} − ω′δ_{xx})′, is distributed independently of the second term in (28) and
r = 0, …, k, and Cases I–V are defined in (12–16), a ∈ [0,1].
The proof for the consistency of the bounds test procedure based on the tstatistic of (24) requires that the rank of the longrun multiplier matrix Π is r + 1 under the alternative hypothesis . Hence, Assumption 5b applies; in particular, α_{yy} ≠ 0.
5 AN APPLICATION: UK EARNINGS EQUATION
 Top of page
 Abstract
 1 INTRODUCTION
 2 THE UNDERLYING VAR MODEL AND ASSUMPTIONS
 3 BOUNDS TESTS FOR A LEVEL RELATIONSHIPS
 4 THE ASYMPTOTIC POWER OF THE BOUNDS PROCEDURE
 5 AN APPLICATION: UK EARNINGS EQUATION
 6 CONCLUSIONS
 Acknowledgements
 APPENDIX A: PROOFS FOR SECTION 3
 APPENDIX B: PROOFS FOR SECTION 4
 REFERENCES
Following the modelling approach described earlier, this section provides a reexamination 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' utility. Following Darby and WrenLewis (1993), the theoretical real wage equation underlying the Treasury's earnings equation is given by
 (29)
where w_{t} is the real wage, Prod_{t} is labour productivity, RR_{t} is the replacement ratio defined as the ratio of unemployment benefit to the wage rate, Union_{t} is a measure of ‘union power’, and f(UR_{t}) is the probability of a union member becoming unemployed, which is assumed to be an increasing function of the unemployment rate UR_{t}. The econometric specification is based on a loglinearized 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 longrun wedge effect on real wages is mixed and, as emphasized by CSW, whether such longrun effects are present is an empirical matter. The change in the unemployment rate (ΔUR_{t}) is also included in the Treasury's wage equation. CSW cite two different theoretical rationales for the inclusion of ΔUR_{t} in the wage equation: the differential moderating effects of long and shortterm unemployed on real wages, and the ‘insider–outsider’ theories which argue that only rising unemployment 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 loglinear VAR(p) model, augmented with appropriate deterministics such as intercepts and time trends. To ensure comparability of our results with those of the Treasury, the replacement ratio is not included in the analysis. CSW, p. 50, report that ‘... it has not proved possible to identify 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
The asymptotic theory developed in the paper is not affected by the inclusion of such ‘oneoff’ dummy variables.17 Let z_{t} = (w_{t}, Prod_{t}, UR_{t}, Wedge_{t}, Union_{t})′ = (w_{t}, x_{t}′)′. Then, using the analysis of Section 2, the conditional ECM of interest can be written as
 (30)
Under the assumption that lagged real wages, w_{t − 1}, do not enter the subVAR model for x_{t}, 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 u_{t} are serially uncorrelated. It is therefore important that the lag order p of the underlying 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 overparameterized, 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 1970q11997q4, and are seasonally adjusted (where relevant).21 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 following manner: w_{t} = ln(ERPR_{t}/PYNONG_{t}), Wedge_{t} = ln(1 + TE_{t}) + ln(1−TD_{t})−ln(RPIX_{t}/PYNONG_{t}), UR_{t} = ln(100 × ILOU_{t}/(ILOU_{t} + WFEMP_{t})), Prod_{t} = ln((YPROM_{t} + 278.29 × YMF_{t})/(EMF_{t} + ENMF_{t})), and Union_{t} = ln(UDEN_{t}), where ERPR_{t} is average private sector earnings per employee (£), PYNONG_{t} is the nonoil nongovernment GDP deflator, YPROM_{t} is output in the private, nonoil, nonmanufacturing, and public traded sectors at constant factor cost (£ million, 1990), YMF_{t} is the manufacturing output index adjusted for stock changes (1990 = 100), EMF_{t} and ENMF_{t} are respectively employment in UK manufacturing and nonmanufacturing sectors (thousands), ILOU_{t} is the International Labour Office (ILO) measure of unemployment (thousands), WFEMP_{t} is total employment (thousands), TE_{t} is the average employers' National Insurance contribution rate, TD_{t} is the average direct tax rate on employment incomes, RPIX_{t} is the Retail Price Index excluding mortgage payments, and UDEN_{t} 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 Figures 1–3.
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 I(1), or mutually cointegrated.
To determine the appropriate lag length p and whether a deterministic linear trend is required in addition to the productivity variable, we estimated the conditional 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, ΔProd_{t − 1}, ΔProd_{t−2}, …, were insignificant (either singly or jointly) in all regressions. Therefore, for the sake of parsimony and to avoid unnecessary overparameterization, we decided to reestimate the regressions without these lagged variables, but including lagged changes of all other variables. Table I gives Akaike's and Schwarz's Bayesian Information Criteria, denoted respectively by AIC and SBC, and Lagrange multiplier (LM) statistics for testing the hypothesis of no residual serial correlation against orders 1 and 4 denoted by and respectively.
Table II gives the values of the F and tstatistics 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.
Table II. F and tstatistics for testing the existence of a levels earnings equationp  With deterministic trends  Without deterministic trends 

F_{IV}  F_{V}  t_{V}  F_{III}  t_{III} 


4  2.99a  2.34a  −2.26a  3.63b  −3.02b 
5  4.42c  3.96b  −2.83a  5.23c  −4.00c 
6  4.78c  3.59b  −2.44a  5.42c  −3.48b 
The various statistics in Table II should be compared with the critical value bounds provided in Tables CI and CII. First, consider the bounds Fstatistic. 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 F_{V}, Case V of (16), which ignores this constraint. Note that, if the trend coefficient c_{1} is not subject to this restriction, (30) implies a quadratic trend in the level of real wages under the null hypothesis of π_{ww} = 0 and π_{wx.x} = 0′, which is empirically 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 F_{V}, respectively.25 The test outcome depends on the choice of the lag order p. For p = 4, the hypothesis that there exists no level earnings equation is not rejected at the 0.05 level, irrespective of whether the regressors are purely I(0), purely I(1) 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 I(0), purely I(1) or mutually cointegrated.26 This finding is even more conclusive when the bounds Ftest 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 ttest to the earnings equations are less clearcut and do not allow the imposition of the trend restrictions discussed above. The 0.05 critical value bounds for t_{III} and t_{V}, when k = 4, are (−2.86, −3.99) and (−3.41, −4.36).28 Therefore, if a linear trend is included, the bounds ttest 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 existence of a levels earnings equation when a sufficiently high lag order is selected and 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 (π_{ww} = 0, π_{wx.x} = 0) it is important that the coefficients of lagged changes remain unrestricted, otherwise these tests could be subject to a pretesting problem. However, for the subsequent estimation of levels effects and shortrun 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 discussed in Pesaran and Shin (1999).29 First, the (estimated) orders of an ARDL(p, p_{1}, p_{2}, p_{3}, p_{4}) model in the five variables (w_{t}, Prod_{t}, UR_{t}, Wedge_{t}, Union_{t}) were selected by searching across the 7^{5} = 16,807 ARDL models, spanned by p = 0,1, …, 6, and p_{i} = 0,1, …, 6, i = 1, …, 4, using the AIC criterion.30 This resulted in the choice of an ARDL(6, 0, 5, 4, 5) specification with estimates of the levels relationship given by
 (31)
where v̂_{t} is the equilibrium correction term, and the standard errors are given in parenthesis. 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 −0.105 and 1.481, are also in line with the Treasury estimates of −0.09 and 1.31.31 The main difference between the two sets of estimates concerns the levels coefficient of the wedge variable. We obtain a much larger estimate, almost twice that obtained by the Treasury. Setting the levels coefficients of the Prod_{t} and Wedge_{t} 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 conditional ECM regression associated with the above level relationship is given in Table III.33 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 statistically significant, further justifying the choice of p = 6. The equilibrium correction coefficient is estimated as −0.229 (0.0586) which is reasonably large and 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 the 0.05 level which may be linked to the presence of some nonlinear 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 equationRegressor  Coefficient  Standard error  pvalue 


v̂_{t − 1}  −0.229  0.0586  N/A 
Δw_{t − 1}  −0.418  0.0974  0.000 
Δw_{t−2}  −0.328  0.1089  0.004 
Δw_{t−3}  −0.523  0.1043  0.000 
Δw_{t−4}  −0.133  0.0892  0.140 
Δw_{t−5}  −0.197  0.0807  0.017 
ΔProd_{t}  0.315  0.0954  0.001 
ΔUR_{t}  0.003  0.0083  0.683 
ΔUR_{t − 1}  0.016  0.0119  0.196 
ΔUR_{t−2}  0.003  0.0118  0.797 
ΔUR_{t−3}  0.028  0.0113  0.014 
ΔUR_{t−4}  0.027  0.0122  0.031 
ΔWedge_{t}  −0.297  0.0534  0.000 
ΔWedge_{t − 1}  −0.048  0.0592  0.417 
ΔWedge_{t−2}  −0.093  0.0569  0.105 
ΔWedge_{t−3}  −0.188  0.0560  0.001 
ΔUnion_{t}  −0.969  0.8169  0.239 
ΔUnion_{t − 1}  −2.915  0.8395  0.001 
ΔUnion_{t−2}  −0.021  0.9023  0.981 
ΔUnion_{t−3}  −0.101  0.7805  0.897 
ΔUnion_{t−4}  −1.995  0.7135  0.007 
Intercept  0.619  0.1554  0.000 
D7475_{t}  0.029  0.0063  0.000 
D7579_{t}  0.017  0.0063  0.009 
R^{2} = 0.5589, , AIC = 339.57, SBC = 302.55, 
, 
, . 