## 1 INTRODUCTION

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 ⇒ signifies ‘weak convergence in probability measure’, **I**_{m} ‘an identity matrix of order *m*’, *I*(*d*) ‘integrated of order *d*’, *O*_{P} (*K*) ‘of the same order as *K* in probability’ and *o*_{P} (*K*) ‘of smaller order than *K* in probability’.