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Abstract

  1. Top of page
  2. Abstract
  3. 1 INTRODUCTION
  4. 2 THE UNDERLYING VAR MODEL AND ASSUMPTIONS
  5. 3 BOUNDS TESTS FOR A LEVEL RELATIONSHIPS
  6. 4 THE ASYMPTOTIC POWER OF THE BOUNDS PROCEDURE
  7. 5 AN APPLICATION: UK EARNINGS EQUATION
  8. 6 CONCLUSIONS
  9. Acknowledgements
  10. APPENDIX A: PROOFS FOR SECTION 3
  11. APPENDIX B: PROOFS FOR SECTION 4
  12. REFERENCES

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 the significance of the lagged levels of the variables in a univariate equilibrium correction 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 I(1). Two sets of asymptotic critical values are provided: one when all regressors are purely I(1) and the other if they are all purely I(0). These two sets of critical values provide a band covering all possible classifications of the regressors into purely I(0), purely I(1) 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 alternatives are derived. The empirical relevance of the bounds procedures is demonstrated by a re-examination of the earnings equation included in the UK Treasury macroeconometric model. Copyright © 2001 John Wiley & Sons, Ltd.


1 INTRODUCTION

  1. Top of page
  2. Abstract
  3. 1 INTRODUCTION
  4. 2 THE UNDERLYING VAR MODEL AND ASSUMPTIONS
  5. 3 BOUNDS TESTS FOR A LEVEL RELATIONSHIPS
  6. 4 THE ASYMPTOTIC POWER OF THE BOUNDS PROCEDURE
  7. 5 AN APPLICATION: UK EARNINGS EQUATION
  8. 6 CONCLUSIONS
  9. Acknowledgements
  10. APPENDIX A: PROOFS FOR SECTION 3
  11. APPENDIX B: PROOFS FOR SECTION 4
  12. 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 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.)

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 F-statistic 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 non-standard 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 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 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 F-statistic) 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 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 following notation is used. The symbol [RIGHTWARDS DOUBLE ARROW] signifies ‘weak convergence in probability measure’, Im ‘an identity matrix of order m’, I(d) ‘integrated of order d’, OP (K) ‘of the same order as K in probability’ and oP (K) ‘of smaller order than K in probability’.

2 THE UNDERLYING VAR MODEL AND ASSUMPTIONS

  1. Top of page
  2. Abstract
  3. 1 INTRODUCTION
  4. 2 THE UNDERLYING VAR MODEL AND ASSUMPTIONS
  5. 3 BOUNDS TESTS FOR A LEVEL RELATIONSHIPS
  6. 4 THE ASYMPTOTIC POWER OF THE BOUNDS PROCEDURE
  7. 5 AN APPLICATION: UK EARNINGS EQUATION
  8. 6 CONCLUSIONS
  9. Acknowledgements
  10. APPENDIX A: PROOFS FOR SECTION 3
  11. APPENDIX B: PROOFS FOR SECTION 4
  12. REFERENCES

Let equation image denote a (k + 1)-vector random process. The data-generating process for equation image is the VAR model of order p (VAR(p)):

  • equation image(1)

where L is the lag operator, µ and γ are unknown (k + 1)-vectors of intercept and trend coefficients, the (k + 1, k + 1) matrix lag polynomial equation image with equation image (k + 1, k + 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 + 1)-vector error process equation image are given in Assumption 2 below. All the analysis of this paper is conducted given the initial observations Z0 ≡ (z1 − p, …, z0). We assume:

Assumption 1. The roots ofequation imageare either outside the unit circle |z| = 1 or satisfy z = 1.

Assumption 2. The vector error processequation imageis IN(0, Ω), Ωpositive definite.

Assumption 1 permits the elements of zt 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 equation image to be a conditionally mean zero and homoscedastic process; see, for example, PSS, Assumption 4.1.

We may re-express the lag polynomial Φ(L) in vector equilibrium correction model (ECM) form; i.e. Φ(L) ≡ −ΠL + Γ(L)(1 − L) in which the long-run multiplier matrix is defined by equation image, and the short-run response matrix lag polynomial equation image, equation image, i = 1, …, p − 1. Hence, the VAR(p) model (1) may be rewritten in vector ECM form as

  • equation image(2)

where Δ ≡ 1 − L is the difference operator,

  • equation image(3)

and the sum of the short-run coefficient matrices equation image. As detailed in PSS, Section 2, if γ0, the resultant constraints (3) on the trend coefficients a1 in (2) ensure that the deterministic trending behaviour of the level process equation image is invariant to the (cointegrating) rank of Π; a similar result holds for the intercept of equation image if µ0 and γ = 0. 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 yt given the k-vector xt and the past values equation image and Z0, where we have partitioned zt = (yt, xt′)′. Partitioning the error term ϵt conformably with zt = (yt′, xt′)′ as ϵt = (ϵyt, ϵxt′)′ and its variance matrix as

  • equation image

we may express ϵyt conditionally in terms of ϵxt as

  • equation image(4)

where utIN(0, ωuu), equation image and ut is independent of ϵxt. Substitution of (4) into (2) together with a similar partitioning of a0 = (ay0, ax0′)′, a1 = (ay1, ax1′)′, Π = (πy′, Πx′)′, Γ = (γy′, Γx′)′, Γi = (γyi′, Γxi′)′, i = 1, …, p − 1, provides a conditional model for Δyt in terms of zt − 1, Δxt, Δzt − 1, …; i.e. the conditional ECM

  • equation image(5)

where equation image, c0ay0ωax0, c1ay1ωax1, ψi′ ≡ γyiωΓxi, i = 1, …, p − 1, and πy.xπyω′Πx. The deterministic relations (3) are modified to

  • equation image(6)

where γy.xγyωΓx.

We now partition the long-run multiplier matrix Π conformably with zt = (yt, xt′)′ as

  • equation image

The next assumption is critical for the analysis of this paper.

Assumption 3. The k-vectorπxy = 0.

In the application of Section 6, Assumption 3 is an identifying assumption for the bargaining theory of wage determination. Under Assumption 3,

  • equation image(7)

Thus, we may regard the process equation image as long-run forcing for equation image as there is no feedback from the level of yt in (7); see Granger and Lin (1995).3 Assumption 3 restricts consideration to cases in which there exists at most one conditional level relationship between yt and xt, irrespective of the level of integration of the process equation image; see (10) below.4

Under Assumption 3, the conditional ECM (5) now becomes

  • equation image(8)

t = 1,2, …, where

  • equation image(9)

and πyx.xπyxωΠxx.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 matrixΠxxhas rank r, 0 ≤ rk.

Under Assumption 4, from (7), we may express Πxx as Πxx = αxxβxx′, where αxx and βxx 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 equation image is mutually cointegrated of order r, 0 ≤ rk. 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 πxy = 0andΠxx = 0, then xt is weakly exogenous for πyy and πyx.x = πyx in (8); see, for example, Johansen (1995, Theorem 8.1, p. 122). In the more general case where Πxx is non-zero, as πyy and πyx.x = πyxωΠxx are variation-free from the parameters in (7), xt is also weakly exogenous for the parameters of (8).

Note that under Assumption 4 the maximal cointegrating rank of the long-run 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, πyy = 0 and πyxϕΠxx = 0′ for some k-vector ϕ. Note that πyx.x = 0′ implies the latter condition. Thus, under Assumptions 1, 3 and 4, Π has rank r and is given by

  • equation image

Hence, we may express Π = αβ′ where α = (αyx′, αxx′)′ and β = (0, βxx′)′ are (k + 1, r) matrices of full column rank; cf. HJNR, p. 390. Let the columns of the (k + 1, kr + 1) matrices equation image and equation image, where equation image, equation image and α, β are respectively (k + 1)-vectors and (k + 1, kr) matrices, denote bases for the orthogonal complements of respectively α and β; in particular, equation image and equation image.

Assumption 5a. If rank(Π) = r, the matrixequation imageis full rank k − r + 1, 0 ≤ r ≤ k.

Cf. Johansen (1991, Theorem 4.1, p. 1559).

Second, if the long-run multiplier matrix Π has rank r + 1, then under Assumptions 1, 3 and 4, πyy ≠ 0 and Π may be expressed as Π = αyβy′ + αβ′, where αy = (αyy, 0′)′ and βy = (βyy, βyx′)′ 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 (αy, α) and (βy, β); in particular, α′(αy, α) = 0 and β′(βy, β) = 0.

Assumption 5b. If rank(Π) = r + 1, the matrixαΓβis full rank k − r, 0 ≤ r ≤ k.

Assumptions 1, 3, 4 and 5a and 5b permit the two polar cases for equation image. First, if equation image is a purely I(0) vector process, then Πxx, and, hence, αxx and βxx, are nonsingular. Second, if equation image is purely I(1), then Πxx = 0, and, hence, αxx and βxx are also null matrices.

Using (A.1) in Appendix A, it is easily seen that πy.x (ztµγt) = πy.xC*(L)ϵt, where {C*(L)ϵt} is a mean zero stationary process. Therefore, under Assumptions 1, 3, 4 and 5b, that is, πyy ≠ 0, it immediately follows that there exists a conditional level relationship between yt and xt defined by

  • equation image(10)

where θ0πy.xµyy, θ1πy.xγyy, θ ≡ −πyx.xyy and vt = πy.xC*(Ltyy, also a zero mean stationary process. If πyx.x = αyyβyx′ + (αyxω′αxx)βxx′ ≠ 0′, the level relationship between yt and xt is non-degenerate. Hence, from (10), ytI(0) if rank(βyx, βxx) = r and ytI(1) if rank(βyx, βxx) = r + 1. In the former case, θ is the vector of conditional long-run multipliers and, in this sense, (10) may be interpreted as a conditional long-run level relationship between yt and xt, whereas, in the latter, because the processes equation image and equation image are cointegrated, (10) represents the conditional long-run level relationship between yt and xt. Two degenerate cases arise. First, if πyy ≠ 0 and πyx.x = 0′, clearly, from (10), yt is (trend) stationary or ytI(0) whatever the value of r. Consequently, the differenced variable Δyt depends only on its own lagged level yt − 1 in the conditional ECM (8) and not on the lagged levels xt − 1 of the forcing variables. Second, if πyy = 0, that is, Assumption 5a holds, and πyx.x = (αyxωαxx)βxx′ ≠ 0′, as rank(Π) = r, πyx.x = (ϕω)′αxxβxx′ which, from the above, yields πyx.x (xtµxγxt) = πy.xC*(L)ϵt, t = 1,2, …, where µ = (µy, µx′)′ and γ = (γy, γx′)′ are partitioned conformably with zt = (yt, xt′)′. Thus, in (8), Δyt depends only on the lagged level xt − 1 through the linear combination (ϕω)′αxx of the lagged mutually cointegrating relations βxxxt − 1 for the process equation image. Consequently, ytI(1) whatever the value of r. Finally, if both πyy = 0 andπyx.x = 0′, there are no level effects in the conditional ECM (8) with no possibility of any level relationship between yt and xt, degenerate or otherwise, and, again, ytI(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 yt and xt, the emphasis in this paper is a test of the joint hypothesis πyy = 0 and πyx.x = 0′ in (8).7, 8 In contradistinction, the approach of BDM may be described in terms of (8) using Assumption 5b:

  • equation image(11)

BDM test for the exclusion of yt − 1 in (11) when r = 0, that is, βxx = 0 in (11) or Πxx = 0 in (7) and, thus, {xt} is purely I(1); cf. HJNR and PSS.9 Therefore, BDM consider the hypothesis αyy = 0 (or πyy = 0).10 More generally, when 0 < rk, BDM require the imposition of the untested subsidiary hypothesis αyxωαxx = 0′; that is, the limiting distribution of the BDM test is obtained under the joint hypothesis πyy = 0 and πyx.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)  c0 = 0 and c1 = 0. That is, µ = 0 and γ = 0. Hence, the ECM (8) becomes

    • equation image(12)
  • Case II (restricted intercepts; no trends)  c0 = −(πyy, πyx.x)µ and c1 = 0. Here, γ = 0. The ECM is

    • equation image(13)
  • Case III (unrestricted intercepts; no trends)  c0 ≠ 0 and c1 = 0. Again, γ = 0. Now, the intercept restriction c0 = −(πyy, πyx.x)µ is ignored and the ECM is

    • equation image(14)
  • Case IV (unrestricted intercepts; restricted trends)  c0 ≠ 0 and c1 = −(πyy, πyx.x)γ.

    • equation image(15)
  • Case V (unrestricted intercepts; unrestricted trends)  c00 and c10. Here, the deterministic trend restriction c1 = −(πyy, πyx.x)γ is ignored and the ECM is

    • equation image(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 (πyy, πyx.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

  1. Top of page
  2. Abstract
  3. 1 INTRODUCTION
  4. 2 THE UNDERLYING VAR MODEL AND ASSUMPTIONS
  5. 3 BOUNDS TESTS FOR A LEVEL RELATIONSHIPS
  6. 4 THE ASYMPTOTIC POWER OF THE BOUNDS PROCEDURE
  7. 5 AN APPLICATION: UK EARNINGS EQUATION
  8. 6 CONCLUSIONS
  9. Acknowledgements
  10. APPENDIX A: PROOFS FOR SECTION 3
  11. APPENDIX B: PROOFS FOR SECTION 4
  12. REFERENCES

In this section we develop bounds procedures for testing for the existence of a level relationship between yt and xt 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 yt and xt via the exclusion of the lagged level variables yt − 1 and xt − 1 in (12–16). Consequently, we define the constituent null hypotheses equation image, equation image, and alternative hypotheses equation image, equation image. Hence, the joint null hypothesis of interest in (12–16) is given by:

  • equation image(17)

and the alternative hypothesis is correspondingly stated as:

  • equation image(18)

However, as indicated in Section 2, not only does the alternative hypothesis H1 of (17) cover the case of interest in which πyy ≠ 0 and πyx.x0′ but also permits πyy ≠ 0, πyx.x = 0′ and πyy = 0 and πyx.x0′; cf. (8). That is, the possibility of degenerate level relationships between yt and xt is admitted under H1 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

  • equation image(19)

where ιT is a T-vector of ones, Δy ≡ (Δy1, …, ΔyT)′, ΔX ≡ (Δx1, …, ΔxT)′, ΔZi ≡ (Δz1 − i, …, ΔzTi)′, i = 1, …, p − 1, ψ ≡ (ω′, ψ1′, …, ψp − 1′)′, ΔZ ≡ (ΔX, ΔZ−1, …, ΔZ1 − p), equation image, τT ≡ (1, …, T)′, Z−1 ≡ (z0, …, zT−1)′, u ≡ (u1, …, uT)′ and

  • equation image

The least squares (LS) estimator of equation image is given by:

  • equation image(20)

where equation image, equation image, equation image, PιITιT (ιTιT)−1ιT′ and equation image. The Wald and the F-statistics for testing the null hypothesis H0 of (17) against the alternative hypothesis H1 of (18) are respectively:

  • equation image(21)

where equation image, m ≡ (k + 1)(p + 1) + 1 is the number of estimated coefficients and ũt, 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 F-statistic is a simple corollary and is not presented here or subsequently. Let Wkr + 1 (a) ≡ (Wu (a), Wkr (a)′)′ denote a (kr + 1)-dimensional standard Brownian motion partitioned into the scalar and (kr)-dimensional sub-vector independent standard Brownian motions Wu (a) and Wkr (a), a ∈ [0,1]. We will also require the corresponding de-meaned (kr + 1)-vector standard Brownian motion equation image, and de-meaned and de-trended (kr + 1)-vector standard Brownian motion equation image, and their respective partitioned counterparts kr + 1 (a) = (u (a), kr (a)′)′, and Ŵkr + 1 (a) = (Ŵu (a), Ŵkr (a)′)′, a ∈ [0,1].

Theorem 3.1 (Limiting distribution of W) If Assumptions 1–4 and 5a hold, then under H0yy = 0 andπyx.x = 0of (17), as T [RIGHTWARDS ARROW] ∞, the asymptotic distribution of the Wald statistic W of (21) has the representation

  • equation image(22)

wherezrN(0, Ir) is distributed independently of the second term in (22) and

  • equation image

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 {xt}, k and r respectively. In Case IV, referring to (11), the first component in (22), zrzr ∼ χ2 (r), corresponds to testing for the exclusion of the r-dimensional stationary vector βxxxt − 1, that is, the hypothesis αyxωαxx = 0′, whereas the second term in (22), which is a non-standard Dickey–Fuller unit-root distribution, corresponds to testing for the exclusion of the (kr + 1)-dimensional I(1) vector equation image and, in Cases II and IV, the intercept and time-trend respectively or, equivalently, αyy = 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 Πxx 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.

Corollary 3.1(Limiting distribution of W if {xt} ∼ I(0)). If Assumptions 1–4 and 5a hold and r = k, that is, {xt} ∼ I(0), then under H0yy = 0 andπyx.x = 0of (17), as T [RIGHTWARDS ARROW] ∞, the asymptotic distribution of the Wald statistic W of (21) has the representation

  • equation image(23)

wherezkN(0, Ik) is distributed independently of the second term in (23) and

  • equation image

r = 0, …, k, where Cases I–V are defined in (12–16), a ∈ [0,1].

Corollary 3.2(Limiting distribution of W if {xt} ∼ I(1)). If Assumptions 1–4 and 5a hold and r = 0, that is, {xt} ∼ I(1), then under H0yy = 0 andπyx.x = 0of (17), as T [RIGHTWARDS ARROW] ∞, the asymptotic distribution of the Wald statistic W of (21) has the representation

  • equation image

whereFk + 1 (a) is defined in Theorem 3.1 for Cases I–V, a ∈ [0,1].

Table CI. Asymptotic critical value bounds for the F-statistic. Testing for the existence of a levels relationshipa
Table CI(i) Case I: No intercept and no trend
k0.1000.0500.0250.010MeanVariance
I(0)I(1)I(0)I(1)I(0)I(1)I(0)I(1)I(0)I(1)I(0)I(1)
03.003.004.204.205.475.477.177.171.161.162.322.32
12.443.283.154.113.884.924.816.021.081.541.081.73
22.173.192.723.833.224.503.885.301.051.690.701.27
32.013.102.453.632.874.163.424.841.041.770.520.99
41.903.012.263.482.623.903.074.441.031.810.410.80
51.812.932.143.342.443.712.824.211.021.840.340.67
61.752.872.043.242.323.592.664.051.021.860.290.58
71.702.831.973.182.223.492.543.911.021.880.260.51
81.662.791.913.112.153.402.453.791.021.890.230.46
91.632.751.863.052.083.332.343.681.021.900.200.41
101.602.721.822.992.023.272.263.601.021.910.190.37
Table CI(ii) Case II: Restricted intercept and no trend
k0.1000.0500.0250.010MeanVariance
I(0)I(1)I(0)I(1)I(0)I(1)I(0)I(1)I(0)I(1)I(0)I(1)
03.803.804.604.605.395.396.446.442.032.031.771.77
13.023.513.624.164.184.794.945.581.692.021.011.25
22.633.353.103.873.554.384.135.001.522.020.690.96
32.373.202.793.673.154.083.654.661.412.020.520.78
42.203.092.563.492.883.873.294.371.342.010.420.65
52.083.002.393.382.703.733.064.151.292.000.350.56
61.992.942.273.282.553.612.883.991.262.000.300.49
71.922.892.173.212.433.512.733.901.232.010.260.44
81.852.852.113.152.333.422.623.771.212.010.230.40
91.802.802.043.082.243.352.503.681.192.010.210.36
101.762.771.983.042.183.282.413.611.172.000.190.33
Table CI(iii) Case III: Unrestricted intercept and no trend
k0.1000.0500.0250.010MeanVariance
I(0)I(1)I(0)I(1)I(0)I(1)I(0)I(1)I(0)I(1)I(0)I(1)
06.586.588.218.219.809.8011.7911.793.053.057.077.07
14.044.784.945.735.776.686.847.842.032.522.282.89
23.174.143.794.854.415.525.156.361.692.351.231.77
32.723.773.234.353.694.894.295.611.512.260.821.27
42.453.522.864.013.254.493.745.061.412.210.600.98
52.263.352.623.792.964.183.414.681.342.170.480.79
62.123.232.453.612.753.993.154.431.292.140.390.66
72.033.132.323.502.603.842.964.261.262.130.330.58
81.953.062.223.392.483.702.794.101.232.120.290.51
91.882.992.143.302.373.602.653.971.212.100.250.45
101.832.942.063.242.283.502.543.861.192.090.230.41
Table CI(iv) Case IV: Unrestricted intercept and restricted trend
k0.1000.0500.0250.010MeanVariance
I(0)I(1)I(0)I(1)I(0)I(1)I(0)I(1)I(0)I(1)I(0)I(1)
05.375.376.296.297.147.148.268.263.173.172.682.68
14.054.494.685.155.305.836.106.732.452.771.411.65
23.384.023.884.614.375.164.995.852.092.570.921.20
32.973.743.384.233.804.684.305.231.872.450.670.93
42.683.533.053.973.404.363.814.921.722.370.510.76
52.493.382.813.763.114.133.504.631.622.310.420.64
62.333.252.633.622.903.943.274.391.542.270.350.55
72.223.172.503.502.763.813.074.231.482.240.310.49
82.133.092.383.412.623.702.934.061.442.220.270.44
92.053.022.303.332.523.602.793.931.402.200.240.40
101.982.972.213.252.423.522.683.841.362.180.220.36
Table CI(v) Case V: Unrestricted intercept and unrestricted trend
k0.1000.0500.0250.010MeanVariance
I(0)I(1)I(0)I(1)I(0)I(1)I(0)I(1)I(0)I(1)I(0)I(1)
  • a

    The critical values are computed via stochastic simulations using T = 1000 and 40,000 replications for the F-statistic for testing ϕ = 0 in the regression: Δyt = ϕ′zt − 1 + a′wt + ξt, t = 1, …, T, where xt = (x1t, …, xkt′) and

    • equation image

    The variables yt and xt are generated from yt = yt − 1 + ϵ1t and xt = Pxt − 1 + ϵ2t, t = 1, …, T, where y0 = 0, x0 = 0 and ϵt = (ϵ1t, ϵ2t)′ is drawn as (k + 1) independent standard normal variables. If xt is purely I(1), P = Ik whereas P = 0 if xt is purely I(0). The critical values for k = 0 correspond to the squares of the critical values of Dickey and Fuller's (1979) 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 ‘I(0)’ refer to the lower critical values bound obtained when xt is purely I(0), while the columns headed ‘I(1)’ refer to the upper bound obtained when xt is purely I(1).

09.819.8111.6411.6413.3613.3615.7315.735.335.3311.3511.35
15.596.266.567.307.468.278.749.633.173.643.333.91
24.195.064.875.855.496.596.347.522.443.091.702.23
33.474.454.015.074.525.625.176.362.082.811.081.51
43.034.063.474.573.895.074.405.721.862.640.771.14
52.753.793.124.253.474.673.935.231.722.530.590.91
62.533.592.874.003.194.383.604.901.622.450.480.75
72.383.452.693.832.984.163.344.631.542.390.400.64
82.263.342.553.682.824.023.154.431.482.350.340.56
92.163.242.433.562.673.872.974.241.432.310.300.49
102.073.162.333.462.563.762.844.101.402.280.260.44

In practice, however, it is unlikely that one would possess a priori knowledge of the rank r of Πxx; that is, the cointegration rank of the forcing variables {xt} or, more particularly, whether {xt} ∼ I(0) or {xt} ∼ I(1). Long-run analysis of (12–16) predicated on a prior determination of the cointegration rank r in (7) is prone to the possibility of a pre-test 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 {xt} ∼ I(0)) and 3.2 (r = 0 and {xt} ∼ 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 {xt} process is 0 ≤ rk.11 Hence, these two sets of critical values provide critical value bounds covering all possible classifications of {xt} into I(0), I(1) and mutually cointegrated processes. Asymptotic critical value bounds for the F-statistics 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 {xt} are purely I(0), and the upper bound values assume that {xt} are purely I(1).12

Hence, we suggest a bounds procedure to test H0yy = 0 and πyx.x = 0′ 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)). However, one could also allow for differential lag lengths on the lagged variables yti and xti in (2) to arrive at, for example, an ARDL(p, p1, …, pk) 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 short-run feedbacks from the lagged dependent variables, Δyti, i = 1, …, p, to Δxt in (7). Moreover, within the single-equation 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 Πxx = 0 or xt 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 αyxωαxx = 0′; that is, the limiting distributional results given below are also obtained under the joint hypothesis H0yy = 0 and πyx.x = 0′ of (17). BDM test αyy = 0 (or equation image) via the exclusion of yt − 1 in Cases I, III and V. For example, in Case V, they consider the t-statistic

  • equation image(24)

where equation image is defined in the line after (21), equation image, ŷ−1Pmath imagey−1, y−1 ≡ (y0, …, yT−1)′, −1Pmath imageX−1, X−1 ≡ (x0, …, xT−1)′, equation image, Pmath imagePmath imagePmath imageτT (τTPmath imageτT)−1τTPmath image, equation image and equation image.

Theorem 3.2(Limiting distribution of tmath image). If Assumptions 1-4 and 5a hold andγxy = 0, whereΓx = (γxy, Γxx), then under H0 : πyy = 0 andπyx.x = 0of (17), as T [RIGHTWARDS ARROW] ∞, the asymptotic distribution of the t-statistic tmath imageof (24) has the representation

  • equation image(25)

where

  • equation image

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 Wu (a) is replaced by the residual from an asymptotic regression of Wu (a) on the independent (kr)-vector standard Brownian motion Wkr (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 t-statistic tmath image in the two polar cases in which the forcing variables {xt} are purely integrated of order zero and one respectively.

Corollary 3.3(Limiting distribution of tmath imageif {Xt} ∼ I(0)). If Assumptions 1-4 and 5a hold and r = k, that is, {xt} ∼ I(0), then under H0 : πyy = 0 andπyx.x = 0of (17), as T [RIGHTWARDS ARROW] ∞, the asymptotic distribution of the t-statistic tmath imageof (24) has the representation

  • equation image

where

  • equation image

and Cases I, III and V are defined in (12), (14) and (16), a ∈ [0,1].

Corollary 3.4(Limiting distribution of tmath imageif {Xt} ∼ I(1)). If Assumptions 1-4 and 5a hold, γxy = 0, whereΓx = (γxy, Γxx), and r = 0, that is, {xt} ∼ I(1), then underequation image, as T [RIGHTWARDS ARROW] ∞, the asymptotic distribution of the t-statistic tmath imageof (24) has the representation

  • equation image

where Fk (a) is defined in Theorem 3.2 for Cases I, III and V, a ∈ [0,1].

Table CII. Asymptotic critical value bounds of the t-statistic. Testing for the existence of a levels relationshipa
Table CII(i): Case I: No intercept and no trend
k0.1000.0500.0250.010MeanVariance
I(0)I(1)I(0)I(1)I(0)I(1)I(0)I(1)I(0)I(1)I(0)I(1)
0−1.62−1.62−1.95−1.95−2.24−2.24−2.58−2.58−0.42−0.420.980.98
1−1.62−2.28−1.95−2.60−2.24−2.90−2.58−3.22−0.42−0.980.981.12
2−1.62−2.68−1.95−3.02−2.24−3.31−2.58−3.66−0.42−1.390.981.12
3−1.62−3.00−1.95−3.33−2.24−3.64−2.58−3.97−0.42−1.710.981.09
4−1.62−3.26−1.95−3.60−2.24−3.89−2.58−4.23−0.42−1.980.981.07
5−1.62−3.49−1.95−3.83−2.24−4.12−2.58−4.44−0.42−2.220.981.05
6−1.62−3.70−1.95−4.04−2.24−4.34−2.58−4.67−0.42−2.430.981.04
7−1.62−3.90−1.95−4.23−2.24−4.54−2.58−4.88−0.42−2.630.981.04
8−1.62−4.09−1.95−4.43−2.24−4.72−2.58−5.07−0.42−2.810.981.04
9−1.62−4.26−1.95−4.61−2.24−4.89−2.58−5.25−0.42−2.980.981.04
10−1.62−4.42−1.95−4.76−2.24−5.06−2.58−5.44−0.42−3.150.981.03
Table CII(iii) Case III: Unrestricted intercept and no trend
k0.1000.0500.0250.010MeanVariance
I(0)I(1)I(0)I(1)I(0)I(1)I(0)I(1)I(0)I(1)I(0)I(1)
0−2.57−2.57−2.86−2.86−3.13−3.13−3.43−3.43−1.53−1.530.720.71
1−2.57−2.91−2.86−3.22−3.13−3.50−3.43−3.82−1.53−1.800.720.81
2−2.57−3.21−2.86−3.53−3.13−3.80−3.43−4.10−1.53−2.040.720.86
3−2.57−3.46−2.86−3.78−3.13−4.05−3.43−4.37−1.53−2.260.720.89
4−2.57−3.66−2.86−3.99−3.13−4.26−3.43−4.60−1.53−2.470.720.91
5−2.57−3.86−2.86−4.19−3.13−4.46−3.43−4.79−1.53−2.650.720.92
6−2.57−4.04−2.86−4.38−3.13−4.66−3.43−4.99−1.53−2.830.720.93
7−2.57−4.23−2.86−4.57−3.13−4.85−3.43−5.19−1.53−3.000.720.94
8−2.57−4.40−2.86−4.72−3.13−5.02−3.43−5.37−1.53−3.160.720.96
9−2.57−4.56−2.86−4.88−3.13−5.18−3.42−5.54−1.53−3.310.720.96
10−2.57−4.69−2.86−5.03−3.13−5.34−3.43−5.68−1.53−3.460.720.96
Table CII(v) Case V: Unrestricted intercept and unrestricted trend
k0.1000.0500.0250.010MeanVariance
I(0)I(1)I(0)I(1)I(0)I(1)I(0)I(1)I(0)I(1)I(0)I(1)
  • a

    The critical values are computed via stochastic simulations using T = 1000 and 40 000 replications for the t-statistic for testing ϕ = 0 in the regression: Δyt = ϕyt − 1 + δxt − 1 + awt + ξt, t = 1, …, T, where xt = (x1t, …, xkt)′ and

    • equation image

    The variables yt and xt are generated from yt = yt − 1 + ϵ1t and xt = Pxt − 1 + ϵ2t, t = 1, …, T, where y0 = 0, x0 = 0 and ϵt = (ϵ1t, ϵ2t′)′ is drawn as (k + 1) independent standard normal variables. If xt is purely I(1), P = Ik whereas P = 0 if xt is purely I(0). The critical values for k = 0 correspond to those of Dickey and Fuller's (1979) unit root t-statistics. The columns headed ‘I(0)’ refer to the lower critical values bound obtained when xt is purely I(0), while the columns headed ‘I(1)’ refer to the upper bound obtained when xt is purely I(1).

0−3.13−3.13−3.41−3.41−3.65−3.66−3.96−3.97−2.18−2.180.570.57
1−3.13−3.40−3.41−3.69−3.65−3.96−3.96−4.26−2.18−2.370.570.67
2−3.13−3.63−3.41−3.95−3.65−4.20−3.96−4.53−2.18−2.550.570.74
3−3.13−3.84−3.41−4.16−3.65−4.42−3.96−4.73−2.18−2.720.570.79
4−3.13−4.04−3.41−4.36−3.65−4.62−3.96−4.96−2.18−2.890.570.82
5−3.13−4.21−3.41−4.52−3.65−4.79−3.96−5.13−2.18−3.040.570.85
6−3.13−4.37−3.41−4.69−3.65−4.96−3.96−5.31−2.18−3.200.570.87
7−3.13−4.53−3.41−4.85−3.65−5.14−3.96−5.49−2.18−3.340.570.88
8−3.13−4.68−3.41−5.01−3.65−5.30−3.96−5.65−2.18−3.490.570.90
9−3.13−4.82−3.41−5.15−3.65−5.44−3.96−5.79−2.18−3.620.570.91
10−3.13−4.96−3.41−5.29−3.65−5.59−3.96−5.94−2.18−3.750.570.92

As above, it may be shown by simulation that the asymptotic critical values obtained from Corollaries 3.3 (r = k and {xt} is purely I(0)) and 3.4 (r = 0 and {xt} 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 equation image based on these two polar cases may be implemented as described above based on the t-statistic tmath image for the exclusion of yt − 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 t-statistic tmath image of (24) is conducted under equation image only, the resultant limit distribution for tmath image depends on the nuisance parameter ωϕ in addition to the cointegrating rank r, where, under Assumption 5a, αyxϕαxx = 0′. Moreover, if Δyt is allowed to Granger-cause Δxt, that is, γxy, i0 for some i = 1, …, p − 1, then the limit distribution also is dependent on the nuisance parameter γxy/(γyyϕγxy); see Appendix A. Consequently, in general, where ωϕ or γxy0, although the t-statistic tmath image has a well-defined limiting distribution under equation image, the above bounds testing procedure for equation image based on tmath image 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 yt and xt: test H0 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, proceed no further; (b) if H0 is rejected, test equation image using the bounds procedure based on the t-statistic tmath image of (24) from Corollaries 3.3 and 3.4. If equation image is false, a large value of tmath image should result, at least asymptotically, confirming the existence of a level relationship between yt and xt, which, however, may be degenerate (if πyx.x = 0′).

4 THE ASYMPTOTIC POWER OF THE BOUNDS PROCEDURE

  1. Top of page
  2. Abstract
  3. 1 INTRODUCTION
  4. 2 THE UNDERLYING VAR MODEL AND ASSUMPTIONS
  5. 3 BOUNDS TESTS FOR A LEVEL RELATIONSHIPS
  6. 4 THE ASYMPTOTIC POWER OF THE BOUNDS PROCEDURE
  7. 5 AN APPLICATION: UK EARNINGS EQUATION
  8. 6 CONCLUSIONS
  9. Acknowledgements
  10. APPENDIX A: PROOFS FOR SECTION 3
  11. APPENDIX B: PROOFS FOR SECTION 4
  12. 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 t-statistic 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 long-run multiplier matrix Π may be either r or r + 1 under the alternative hypothesis equation image of (18) where equation image and equation image, it is necessary to deal with these two possibilities. First, under equation image, the rank of Π is r + 1 so Assumption 5b applies; in particular, αyy ≠ 0. Second, under equation image, the rank of Π is r so Assumption 5a applies; in this case, equation image holds and, in particular, αyxωαxx0′.

Theorem 4.1(Consistency of the Wald statistic bounds test procedure underequation image). If Assumptions 1-4 and 5b hold, then underequation imageof (18) the Wald statistic W (21) is consistent againstequation imagein Cases I–V defined in (12–16).

Theorem 4.2(Consistency of the Wald statistic bounds test procedure underequation image). If Assumptions 1–4 and 5a hold, then underequation imageof (18) andequation imageof (17) the Wald statistic W (21) is consistent againstequation imagein Cases I–V defined in (12–16).

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 equation image of (17) against equation image of (18). This result holds irrespective of whether the forcing variables {xt} 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, πy.x[ = (πyy, πyx.x)] = (αyyβyyyyβxy′ + (αyxωαxx)βxx′). Consequently, we define the sequence of local alternatives

  • equation image(26)

Hence, under Assumption 3, defining

  • equation image

and recalling Π = αβ′, where (1,−ω′)α = αyxωαxx = 0′, we have

  • equation image(27)

In order to detail the limit distribution of the Wald statistic under the sequence of local alternatives H1T of (26), it is necessary to define the (kr + 1)-dimensional Ornstein–Uhlenbeck process equation image which obeys the stochastic integral and differential equations, equation image and equation image, where Wkr + 1 (a) is a (kr + 1)-dimensional standard Brownian motion, equation image, equation image, together with the de-meaned and de-meaned and de-trended counterparts equation image and equation image partitioned similarly, a ∈ [0,1]. See, for example, Johansen (1995, Chapter 14, pp. 201–210).

Theorem 4.3(Limiting distribution of W under H1T). If Assumptions 1–4 and 5a hold, then under H1Ty.x = T−1αyyβy′ + T−1/2 (δyxωδxx)βof (26), as T [RIGHTWARDS ARROW] ∞, the asymptotic distribution of the Wald statistic W of (21) has the representation

  • equation image(28)

wherezrN (Q1/2η, Ir), equation image, η ≡ (δyxωδxx)′, is distributed independently of the second term in (28) and

  • equation image

r = 0, …, k, and Cases I–V are defined in (12–16), a ∈ [0,1].

The first component of (28) zrzr is non-central chi-square distributed with r degrees of freedom and non-centrality parameter η and corresponds to the local alternative equation image under equation image. The second term in (28) is a non-standard Dickey–Fuller unit-root distribution under the local alternative equation image and δyxωδxx = 0′. Note that under H0 of (17), that is, αyy = 0 and δyxωδxx = 0′, the limiting representation (28) reduces to (22) as should be expected.

The proof for the consistency of the bounds test procedure based on the t-statistic of (24) requires that the rank of the long-run multiplier matrix Π is r + 1 under the alternative hypothesis equation image. Hence, Assumption 5b applies; in particular, αyy ≠ 0.

Theorem 4.4(Consistency of the t-statistic bounds test procedure underequation image). If Assumptions 1–4 and 5b hold, then underequation imageof (18) the t-statistic tmath image (24) is consistent againstequation imagein Cases I, III and V defined in (12), (14) and (16).

As noted at the end of Section 3, Theorem 4.4 suggests the possibility of using tmath image to discriminate between equation image and equation image, although, if equation image is false, the bounds procedure given via Corollaries 3.3 and 3.4 is not asymptotically similar.

5 AN APPLICATION: UK EARNINGS EQUATION

  1. Top of page
  2. Abstract
  3. 1 INTRODUCTION
  4. 2 THE UNDERLYING VAR MODEL AND ASSUMPTIONS
  5. 3 BOUNDS TESTS FOR A LEVEL RELATIONSHIPS
  6. 4 THE ASYMPTOTIC POWER OF THE BOUNDS PROCEDURE
  7. 5 AN APPLICATION: UK EARNINGS EQUATION
  8. 6 CONCLUSIONS
  9. Acknowledgements
  10. APPENDIX A: PROOFS FOR SECTION 3
  11. APPENDIX B: PROOFS FOR SECTION 4
  12. REFERENCES

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' utility. Following Darby and Wren-Lewis (1993), the theoretical real wage equation underlying the Treasury's earnings equation is given by

  • equation image(29)

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 f(URt) 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 (ΔURt) is also included in the Treasury's wage equation. CSW cite two different theoretical rationales for the inclusion of ΔURt in the wage equation: the differential moderating effects of long- and short-term 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 log-linear 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

  • equation image

The asymptotic theory developed in the paper is not affected by the inclusion of such ‘one-off’ dummy variables.17 Let zt = (wt, Prodt, URt, Wedget, Uniont)′ = (wt, xt′)′. Then, using the analysis of Section 2, the conditional ECM of interest can be written as

  • equation image(30)

Under the assumption that lagged real wages, wt − 1, do not enter the sub-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 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 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 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: wt = ln(ERPRt/PYNONGt), Wedget = ln(1 + TEt) + ln(1−TDt)−ln(RPIXt/PYNONGt), URt = ln(100 × ILOUt/(ILOUt + WFEMPt)), Prodt = ln((YPROMt + 278.29 × YMFt)/(EMFt + ENMFt)), and Uniont = ln(UDENt), where ERPRt is average private sector earnings per employee (£), PYNONGt is the non-oil non-government GDP deflator, YPROMt is output in the private, non-oil, non-manufacturing, and public traded sectors at constant 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 non-manufacturing sectors (thousands), ILOUt is the International Labour Office (ILO) measure of unemployment (thousands), WFEMPt is total employment (thousands), TEt is the average employers' National Insurance contribution rate, TDt is the average direct tax rate on employment incomes, RPIXt is the Retail Price Index excluding mortgage 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 Figures 1–3.

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Figure 1. (a) Real wages and labour productivity. (b) Rate of change of real wages and labour productivity

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Figure 2. The wedge and the unionization variables

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Figure 3. The unemployment rate

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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, ΔProdt − 1, ΔProdt−2, …, 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-estimate 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 equation image and equation image respectively.

Table I. Statistics for selecting the lag order of the earnings equation
pWith deterministic trendsWithout deterministic trends
AICSBCequation imageequation imageAICSBCequation imageequation image
  1. Notes: p is the lag order of the underlying VAR model for the conditional ECM (30), with zero restrictions on the coefficients of lagged changes in the productivity variable. AICpLLpsp and SBCpLLp − (sp/2)lnT denote Akaike's and Schwarz's Bayesian Information Criteria for a given lag order p, where LLp is the maximized log-likelihood value of the model, sp is the number of freely estimated coefficients and T is the sample size. equation image and equation image are LM 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.

1319.33302.1416.86*35.89*317.51301.6418.38*34.88*
2324.25301.772.1619.71*323.77302.621.9821.52*
3321.51293.740.5217.07*320.87294.431.5619.35*
4334.37301.313.48***7.79***335.37303.633.41***7.13
5335.84297.500.032.50336.49299.470.032.15
6337.06293.420.853.58337.03294.720.993.99
7336.96288.040.172.20336.85289.250.090.64

As might be expected, the lag order selected by AIC, equation image, irrespective of whether a deterministic trend term is included or not, is much larger than that selected by SBC. This latter criterion gives estimates equation image if a trend is included and equation image if not. The equation image statistics also 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.

Table II. F- and t-statistics for testing the existence of a levels earnings equation
pWith deterministic trendsWithout deterministic trends
FIVFVtVFIIItIII
  • Notes: See the notes to Table I. FIV is the F-statistic for testing πww = 0, πwx.x = 0′ and c1 = 0 in (30). FV is the F-statistic for testing πww = 0 and πwx.x = 0′ in (30). FIII is the F-statistic for testing πww = 0 and πwx.x = 0′ in (30) with c1 set equal to 0. tV and tIII are the t-ratios for testing πww = 0 in (30) with and without 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.

42.99a2.34a−2.26a3.63b−3.02b
54.42c3.96b−2.83a5.23c−4.00c
64.78c3.59b−2.44a5.42c−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 F-statistic. As argued in PSS, the statistic FIV 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, if the trend coefficient c1 is not subject to this restriction, (30) implies a quadratic trend in the level of real wages under the null hypothesis of πww = 0 and πwx.x = 0′, which is empirically implausible. The critical value bounds for the statistics FIV and FV 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 FIV and FV, 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 FV is still inconclusive, but FIV = 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 F-test is applied to the earnings equations without a linear trend. The relevant test statistic is FIII and the associated 0.05 critical value bounds are (2.86, 4.01).27 For p = 4, FIII = 3.63, and the test result is inconclusive. However, for p = 5 and 6, the values of FIII 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 tIII 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 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 (πww = 0, πwx.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 discussed in Pesaran and Shin (1999).29 First, the (estimated) orders of an ARDL(p, p1, p2, p3, p4) model in the five variables (wt, Prodt, URt, Wedget, Uniont) were selected by searching across the 75 = 16,807 ARDL models, spanned by p = 0,1, …, 6, and pi = 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

  • equation image(31)

where 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 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 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 non-normal 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 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
RegressorCoefficientStandard errorp-value
  1. Notes: The regression is based on the conditional ECM given by (30) using an ARDL(6, 0, 5, 4, 5) specification with dependent variable, Δwt estimated over 1972q1–1997q4, and the equilibrium correction term t − 1 is given in (31). R2 is the adjusted squared multiple correlation coefficient, equation image is the standard error of the regression, AIC and SBC are Akaike's and Schwarz's Bayesian Information Criteria, equation image, equation image, equation image, and equation image denote chi-squared 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).

t − 1−0.2290.0586N/A
Δwt − 1−0.4180.09740.000
Δwt−2−0.3280.10890.004
Δwt−3−0.5230.10430.000
Δwt−4−0.1330.08920.140
Δwt−5−0.1970.08070.017
ΔProdt0.3150.09540.001
ΔURt0.0030.00830.683
ΔURt − 10.0160.01190.196
ΔURt−20.0030.01180.797
ΔURt−30.0280.01130.014
ΔURt−40.0270.01220.031
ΔWedget−0.2970.05340.000
ΔWedget − 1−0.0480.05920.417
ΔWedget−2−0.0930.05690.105
ΔWedget−3−0.1880.05600.001
ΔUniont−0.9690.81690.239
ΔUniont − 1−2.9150.83950.001
ΔUniont−2−0.0210.90230.981
ΔUniont−3−0.1010.78050.897
ΔUniont−4−1.9950.71350.007
Intercept0.6190.15540.000
D7475t0.0290.00630.000
D7579t0.0170.00630.009
R2 = 0.5589, equation image, AIC = 339.57, SBC = 302.55,
equation image, equation image
equation image, equation image.

6 CONCLUSIONS

  1. Top of page
  2. Abstract
  3. 1 INTRODUCTION
  4. 2 THE UNDERLYING VAR MODEL AND ASSUMPTIONS
  5. 3 BOUNDS TESTS FOR A LEVEL RELATIONSHIPS
  6. 4 THE ASYMPTOTIC POWER OF THE BOUNDS PROCEDURE
  7. 5 AN APPLICATION: UK EARNINGS EQUATION
  8. 6 CONCLUSIONS
  9. Acknowledgements
  10. APPENDIX A: PROOFS FOR SECTION 3
  11. APPENDIX B: PROOFS FOR SECTION 4
  12. REFERENCES

Empirical analysis of level relationships has been an integral part of time series econometrics and pre-dates the recent literature on unit roots and cointegration.38 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, xt, entering the determination of the dependent variable of interest, yt, are all integrated of order 1 or more. This paper demonstrates that the problem of testing for the existence of a level relationship between yt and xt is non-standard even if all the regressors under consideration are I(0) because, under the null hypothesis of no level relationship between yt and xt, the process describing the yt process is I(1), irrespective of whether the regressors xt are purely I(0), purely I(1) or mutually cointegrated. The asymptotic theory developed in this paper provides a simple univariate framework for testing the existence of a single level relationship between yt and xt when it is not known with certainty whether the regressors are purely I(0), purely I(1) 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 yt 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 I(1) or I(0).

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 yt. 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.

Acknowledgements

  1. Top of page
  2. Abstract
  3. 1 INTRODUCTION
  4. 2 THE UNDERLYING VAR MODEL AND ASSUMPTIONS
  5. 3 BOUNDS TESTS FOR A LEVEL RELATIONSHIPS
  6. 4 THE ASYMPTOTIC POWER OF THE BOUNDS PROCEDURE
  7. 5 AN APPLICATION: UK EARNINGS EQUATION
  8. 6 CONCLUSIONS
  9. Acknowledgements
  10. APPENDIX A: PROOFS FOR SECTION 3
  11. APPENDIX B: PROOFS FOR SECTION 4
  12. REFERENCES

We are grateful to the Editor (David Hendry) and three anonymous referees for their helpful 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 and R000237334) and the Isaac Newton Trust of Trinity College, Cambridge, is gratefully acknowledged. Previous versions of this paper appeared as DAE Working Paper Series, Nos. 9622 and 9907, University of Cambridge.

APPENDIX A: PROOFS FOR SECTION 3

  1. Top of page
  2. Abstract
  3. 1 INTRODUCTION
  4. 2 THE UNDERLYING VAR MODEL AND ASSUMPTIONS
  5. 3 BOUNDS TESTS FOR A LEVEL RELATIONSHIPS
  6. 4 THE ASYMPTOTIC POWER OF THE BOUNDS PROCEDURE
  7. 5 AN APPLICATION: UK EARNINGS EQUATION
  8. 6 CONCLUSIONS
  9. Acknowledgements
  10. APPENDIX A: PROOFS FOR SECTION 3
  11. APPENDIX B: PROOFS FOR SECTION 4
  12. REFERENCES

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 equation image has the infinite moving-average representation,

  • equation image(A1)

where the partial sum equation image, Φ(z)C(z) = C(z)Φ(z) = (1−z)Ik + 1, equation image, equation image, t = 1, 2 …; see Johansen (1991) and PSS. Note that equation image; see Johansen (1991, (4.5), p. 1559). Define the (k + 2, r) and (k + 2, kr + 1) matrices equation image and δ by

  • equation image

where equation image is a (k + 1, kr + 1) matrix whose columns are a basis for the orthogonal complement of β. Hence, equation image is a basis for k + 1. Let ξ be the (k + 2)-unit vector (1, 0′)′. Then, (β*,ξ,δ) is a basis for k + 2. It therefore follows that

  • equation image

where equation image, Bk + 1 (a) is a (k + 1)-vector Brownian motion with variance matrix Ω and [Ta] denotes the integer part of Ta, a ∈ [0,1]; see Phillips and Solo (1992, Theorem 3.15, p. 983). Also, equation image. Similarly, noting that βC = 0, we have that equation image. Hence, from Phillips and Solo (1992, Theorem 3.16, p. 983), defining equation image and equation image, it follows that

  • equation image(A2)

where BT ≡ (δ, T−1/2ξ). Similarly, defining ũPιu,

  • equation image(A3)

Cf. Johansen (1991, Lemma A.3, p. 1569) and Johansen (1995, Lemma 10.3, p. 146).

The next result follows from Phillips and Solo (1992, Theorem 3.15, p. 983); cf. Johansen (1991, Lemma A.3, p. 1569) and Johansen (1995, Lemma 10.3, p. 146) and Phillips and Durlauf (1986).

Lemma. Lemma A.1LetBT ≡ (δ, T−1/2ξ) and defineG(a) = (G1 (a)′, G2 (a))′, whereequation image, equation image, andequation image, a ∈ [0,1]. Then

  • equation image

whereequation imageandk (a) = (1 (a), k (a)′)′, a ∈ [0,1]

Proof of Theorem 3.1Under H0of (17), the Wald statistic W of (21) can be written as

  • equation image

whereequation image. Consider the matrixequation image. It follows from (A2) and Lemma A.1 that

  • equation image(A4)

Next, considerequation image. From (A3) and Lemma A.1,

  • equation image(A5)

Finally, the estimator for the error variance ωuu(defined in the line after (21)),

  • equation image(A6)

From (A4–A6) and Lemma A.1,

  • equation image(A7)

We consider each of the terms in the representation (A7) in turn. A central limit theorem allows us to state

  • equation image

Hence, the first term in (A7) converges in distribution tozrzr, a chi-square random variable with r degrees of freedom; that is,

  • equation image(A8)

From Lemma A.1, the second term in (A7) weakly converges to

  • equation image

which, asequation image, may be expressed as

  • equation image

Now, noting that under H0of (17) we may expressequation imageandequation imagewhereequation image, we define the (k − r + 1)-vector of independent de-meaned standard Brownian motions,

  • equation image

whereequation imageis independent ofk (a) andk + 1 (a) ≡ (1 (a), k (a)′)′ is partitioned according tozt = (yt, xt′)′, a ∈ [0,1]. Hence, the second term in (A7) has the following asymptotic representation:

  • equation image(A9)

Note that dW̃u (a) in (A9) may be replaced by dWu (a), a ∈ [0,1]. Combining (A8) and (A9) gives the result of Theorem 3.1.

For the remaining cases, we need only make minor modifications to the proof for Case IV. In Case I, equation imagewithequation imagea basis for ℛk + 1andBT = δ. For Case II, whereequation image, we have

  • equation image

and, consequently, we defineξas in Case IV,

  • equation image

Case III is similar to Case I as is Case V.▪

Proof of Corollary 3.1Follows immediately from Theorem 3.1 by setting r = k.▪

Proof of Corollary 3.2Follows immediately from Theorem 3.1 by setting r = 0.▪

Proof of Theorem 3.2We provide a proof for Case V which may be 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 impliesequation imagebut not necessarilyequation image; in particular, note that we may writeequation imagefor some k-vectorϕ. The t-statistic forequation imagemay be expressed as the square root of

  • equation image(A10)

whereATT−1/2 (β, T−1/2BT) andequation image. Note that only the diagonal element of the inverse in (A10) corresponding toequation imageis relevant, which implies that we only need to consider the blocksequation imageandequation imagein (A10). Therefore, using (A2) and (A3), (A10) is asymptotically equivalent to

  • equation image(A11)

whereequation image. Now,

  • equation image

where, for convenience, but without loss of generality, we have setequation imageandequation image, equation image, equation image, equation imageandequation image, equation image, a ∈ [0,1]. Hence, (A11) weakly converges to

  • equation image

Under the conditions of the theorem, ϕ = ωandequation imageand, therefore, equation imageandequation image, a ∈ [0,1].▪

Proof of Corollary 3.3Follows immediately from Theorem 3.2 by setting r = k.▪

Proof of Corollary 3.4Follows immediately from Theorem 3.2 by setting r = 0.▪

APPENDIX B: PROOFS FOR SECTION 4

  1. Top of page
  2. Abstract
  3. 1 INTRODUCTION
  4. 2 THE UNDERLYING VAR MODEL AND ASSUMPTIONS
  5. 3 BOUNDS TESTS FOR A LEVEL RELATIONSHIPS
  6. 4 THE ASYMPTOTIC POWER OF THE BOUNDS PROCEDURE
  7. 5 AN APPLICATION: UK EARNINGS EQUATION
  8. 6 CONCLUSIONS
  9. Acknowledgements
  10. APPENDIX A: PROOFS FOR SECTION 3
  11. APPENDIX B: PROOFS FOR SECTION 4
  12. REFERENCES

Proof of Theorem 4.1Again, we consider Case IV; the remaining Cases I–III and V may be dealt with similarly. Underequation image, Assumption 5b holds and, thus, Π = αyβy′ + αβwhereαy = (αyy, 0′)′ andβy = (βyy, βyx′)′; see above Assumption 5b. Under Assumptions 1–4 and 5b, the processequation imagehas the infinite moving-average representation, zt = µ + γt + Cst + C*(L)ϵt, where nowCβ[αΓβ]−1α′. We redefineβ*andδas the (k + 2, r + 1) and (k + 2, kr) matrices,

  • equation image

and

  • equation image

whereβis a (k + 1, kr) matrix whose columns are a basis for the orthogonal complement of (βy, β). Hence, (βy, β, β) is a basis for ℛk + 1and, thus, (β*, ξ, δ) a basis for ℛk + 2, where againξis the (k + 2)-unit vector (1, 0′)′. It therefore follows that

  • equation image

Also, as above, equation imageandequation image.

The Wald statistic (21) multiplied byequation imagemay be written as

  • equation image(B1)

whereλ*β* (αy, α)′(1, −ω′)′, ATT−1/2 (β*, T−1/2BT) andBT ≡ (δ, T−1/2ξ). Note that (A6) continues to hold underequation image. A similar argument to that in the Proof of Theorem 3.1 demonstrates that the first term in (B1) divided by ωuuhas the limiting representation

  • equation image(B2)

wherezr + 1N(0, Ir + 1), equation imageandequation imageis a (kr)-vector of de-meaned independent standard Brownian motions independent of the standard Brownian motion Wu (a), a ∈ [0,1]; cf. (22). Now, equation imageis mixed normal with conditional variance matrixequation image. Therefore, the second term in (B2) is unconditionally distributed as a χ2 (kr) random variable and is independent of the first term; cf. (A4). Hence, the first term in (B1) divided by ωuuhas a limiting χ2 (k + 1) distribution.

The second term in (B1) may be written as

  • equation image(B3)

and the third term as

  • equation image(B4)

asequation imageconverges in probability to a positive definite matrix. Moreover, as (1, −ω′)(αy, α) ≠ 0underequation image, the Theorem is proved.▪

Proof of Theorem 4.2A similar decomposition to (B1) for the Wald statistic (21) holds underequation imageexcept thatβ*andδare now as defined in the Proof of Theorem 3.1. Althoughequation imageholds, we haveequation image. Therefore, as in Theorem 3.2, note that we may writeequation imagefor some k-vectorϕω. Consequently, the first term divided by ωuumay be written as

  • equation image(B5)

cf. (A7). As in the Proof of Theorem 3.1, the first term of (B5) has the limiting representationzrzrwherezrN(0, Ir); cf. (22). The second term of (B5) has the limiting representation

  • equation image

whereequation image, a ∈ [0,1]; cf. Proof of Theorem 3.2. The second term of (B1) becomes

  • equation image

and the third term

  • equation image

The Theorem follows as (1, −ω′)α0underequation imageandequation image.▪

Proof of Theorem 4.3We concentrate on Case IV; the remaining Cases I–III and V are proved by a similar argument. Letequation imagedenote the process under H1Tof (26). Hence, Φ(L)(ztTµγt) = ξtT, whereξtT ≡ (ΠTΠ)[z(t − 1)Tµγ(t − 1)] + ϵtandΠTΠis given in (27). Therefore, Δ(ztT−µ−γt) = tT + C*(LξtT, C(z) = C + (1−z)C*(z) andequation image, and thus,

  • equation image(B6)

where

  • equation image

Inverting (B6) yields

  • equation image

Note that ΔξtT = (ΠTΠ)Δ[z(t − 1)Tµγ(t − 1)] + Δϵt. It therefore follows thatequation image, whereδis defined above Lemma A.1 andequation image, equation imageis an Ornstein-Uhlenbeck process andBk + 1 (a) is a (k + 1)-vector Brownian motion with variance matrixΩ, a ∈ [0,1]; cf. Johansen (1995, Theorem 14.1, p. 202).

Similarly to (A4),

  • equation image

Therefore, expression (B1) for the Wald statistic (21) multiplied byequation imageis revised to

  • equation image(B7)

The first term in (B7) may be written as

  • equation image(B8)

whereequation image. Definingη ≡ (δyxωδxx)′, consider

  • equation image(B9)

where we have made use ofequation image. Therefore, (B8) divided by ωuumay be re-expressed as

  • equation image(B9)

whereequation imageandzrN(Q1/2η, Ir).

Asequation image, equation image. Consider the second term in (B7), in particular, equation imagewhich after substitution forequation imagebecomes

  • equation image

Therefore,

  • equation image

Consider

  • equation image

where J̃u (a) = 1 (a)−ωk (a) is independent ofk (a) andk + 1 (a) ≡ (1 (a), k (a)′)′, a ∈ [0,1]. Now, equation imagesatisfies the stochastic integral and differential equations, equation imageandequation image, whereequation imageandequation image; cf. Johansen (1995, Theorem 14.4, p. 207). Note that the first element ofequation imagesatisfiesequation imageandequation image.

Therefore,

  • equation image

Hence, the second term in (B7) weakly converges to

  • equation image(B10)

whereequation image.

Combining (B9) and (B10) gives the result stated in Theorem 4.3 asequation imageunder H1Tof (26) and notingequation imagemay be replaced byequation image.▪

Proof of Theorem 4.4We consider Case V; the remaining Cases I and III may be dealt with similarly. Underequation image, from (10), ŷ−1 = −1θ + −1, whereequation imageandv−1 = (0, v1, …, vT−1)′. Therefore, equation imageandequation image.

As inAppendix A,

  • equation image

and noting thatequation image, βxxxt = T−1/2βxxµx + T−1/2βxxγxt + (0, βxx′)C*(L)ϵt. Consequently,

  • equation image

whereequation image.

Now, because T−1βxx−1−1 = OP (1), equation image, equation imageandequation image, henceequation image. Also becauseequation imageandequation image, henceequation image; cf. (A3). Hence, noting thatequation imageandequation image,

  • equation image

whereequation imageandequation image. Therefore, as T−1−1−1 = OP (1),

  • equation image(B11)

The numerator of tmath imageof (24) may be written asequation image, whereλ ≡ (βy, β)(αy, α)′(1, −ω′)′. Because T−1/2βxx−1û = OP (1) andequation image, equation image, and, asequation image, equation image. Therefore,

  • equation image

noting T−1/2−1û = OP (1). Similarly, as (1, −ω′)(αy, α) ≠ 0′, equation image, T−1λ−1−1βxx = OP (1) andequation image. Therefore,

  • equation image

noting T−1−1−1λ = OP (1). Thus,

  • equation image(B12)

Becauseequation image, combining (B11) and (B12) yields the desired result.▪

REFERENCES

  1. Top of page
  2. Abstract
  3. 1 INTRODUCTION
  4. 2 THE UNDERLYING VAR MODEL AND ASSUMPTIONS
  5. 3 BOUNDS TESTS FOR A LEVEL RELATIONSHIPS
  6. 4 THE ASYMPTOTIC POWER OF THE BOUNDS PROCEDURE
  7. 5 AN APPLICATION: UK EARNINGS EQUATION
  8. 6 CONCLUSIONS
  9. Acknowledgements
  10. APPENDIX A: PROOFS FOR SECTION 3
  11. APPENDIX B: PROOFS FOR SECTION 4
  12. REFERENCES
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  • 1

    Assumptions 5a and 5b below further restrict the maximal order of integration of equation image to unity.

  • 2

    See also Nielsen and Rahbek (1998) for an analysis of similarity issues in cointegrated systems.

  • 3

    Note that this restriction does not preclude equation image being Granger-causal/ for equation image in the short run.

  • 4

    Assumption 3 may be straightforwardly assessed via a test for the exclusion of the lagged level yt − 1 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 xt is purely I(1); that is, under the additional assumption Πxx = 0. If current and lagged values of a weakly exogenous purely I(0) vector wt are included as additional explanatory variables in (8), the lagged level vector xt − 1 should be augmented to include the cumulated sum equation image 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 < rk. 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.

  • 7

    This joint hypothesis may be justified by the application of Roy's union-intersection principle to tests of πyy = 0 in (8) given πyx.x. Let Wmath image (πyx.x) be the Wald statistic for testing πyy = 0 for a given value of πyx.x. The test maxmath imageWmath image (πyx.x) is identical to the Wald test of πyy = 0 andπyx.x = 0 in (8).

  • 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 πyx.x = 0′.

  • 9

    The BDM test is based on earlier contributions of Kremers et al. (1992), Banerjee et al. (1993), and Boswijk (1994).

  • 10

    Partitioning Γxi = (γxy, i, Γxx, i), i = 1, …, p − 1, conformably with zt = (yt, xt′)′, BDM also set γxy, i = 0, i = 1, …, p − 1, which implies γxy = 0, where Γx = (γxy, Γxx); that is, Δyt does not Granger cause Δxt.

  • 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 ≤ rk.

  • 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.

  • 13

    Although Corollary 3.3 does not require γxy = 0 and equation image is automatically satisfied under the conditions of Corollary 3.4, the simulation critical value bounds result requires γxy = 0 and equation image for 0 < r < k.

  • 14

    In principle, the asymptotic distribution of tmath image under equation image may be simulated from the limiting representation given in the Proof of Theorem 3.2 of Appendix A after substitution of consistent estimators for ϕ and equation image under equation image, where equation image. Although such estimators may be obtained straightforwardly, unfortunately, they necessitate the use of parameter estimators from the marginal ECM (7) for equation image.

  • 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 D7475t and D7579t are non-zero are only 7.6% and 19.2% respectively.

  • 18

    See Assumption 3 and the following discussion. By construction, the contemporaneous effects Δxt 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 uncorrelated with utand/ 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.

  • 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.

  • 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 FIV and FV statistics turned out to be (3.19,4.16) and (3.61,4.76), respectively, which are only marginally different from the asymptotic critical value bounds.

  • 26

    The same conclusion is also reached for p = 7.

  • 27

    See Table CI(iii).

  • 28

    See Tables CII(iii) and CII(v).

  • 29

    Note that the ARDL approach advanced in Pesaran and Shin (1999) is applicable irrespective of whether the regressors 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-distribution; see Theorem 3.2.

  • 36

    The complex roots are 0.34293 ± 0.67703i and −0.17307 ± 0.61386i, where i = √−1.

  • 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!

  • 38

    For an excellent review of this early literature, see Hendry et al. (1984).

  • 39

    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(0) and I(1) regressors.