Structural analysis of vector error correction models with exogenous I(1) variables

Abstract

This paper generalizes the existing cointegration analysis literature in two respects. Firstly, the problem of efficient estimation of vector error correction models containing exogenous I(1) variables is examined. The asymptotic distributions of the (log-)likelihood ratio statistics for testing cointegrating rank are derived under different intercepts and trend specifications and their respective critical values are tabulated. Tests for the presence of an intercepts or linear trend in the cointegrating relations are also developed together with model misspecification tests. Secondly, efficient estimation of vector error correction models when the short-run dynamics may differ within and between equations is considered. A re-examination of the purchasing power parity and the uncovered interest rate parity hypotheses is conducted using U.K. data under the maintained assumption of exogenously given foreign and oil prices.

Introduction

This paper generalizes the analysis of cointegrated systems advanced by Johansen 1991, Johansen 1995 in two important respects. Firstly, we consider a sub-system approach in which we regard a subset of random variables which are integrated of order one (I(1)) as structurally exogenous; that is, any cointegrating vectors present do not appear in the sub-system vector error correction model (VECM) for these exogenous variables and the error terms in this sub-system are uncorrelated with those in the rest of the system.¹ This generalization is particularly relevant in the macroeconometric analysis of ‘small open’ economies where it is plausible to assume that some of the I(1) forcing variables, for example, foreign income and prices, are exogenous. Similar considerations arise in the empirical analysis of sectoral and regional models where some of the economy-wide I(1) forcing variables may also be viewed as exogenous. This extension paves the way for a more efficient multivariate analysis of economic time series for which data are typically only available over relatively short periods. Secondly, we allow constraints on the short-run dynamics in the VECM. This extension is also important in applied contexts where, due to data limitations, researchers may wish to use a priori restrictions or model selection criteria to choose the lag orders of the stationary variables in the model. As Abadir et al. (1999) demonstrate, the inclusion of irrelevant stationary terms in a VECM may result in substantial small sample estimator bias.

The plan of the paper is as follows. The basic vector autoregressive (VAR) model and other notation are set out in Section 2. The importance of an appropriate specification of deterministic terms in the VAR model is also highlighted here. In particular, unless the coefficients associated with the intercept or the linear deterministic trend are restricted to lie in the column space of the long-run multiplier matrix, the VAR model has the unsatisfactory feature that quite different deterministic behavior should be observed in the levels of the variables for differing values of the cointegrating rank. The problem of efficient conditional estimation of a VECM containing I(1) exogenous variables is addressed in Section 3. This section distinguishes between five different cases which are classified by the deterministic behavior in the levels of the underlying variables; that is, Case I: zero intercepts and linear trend coefficients, Case II: restricted intercepts and zero linear trend coefficients, Case III: unrestricted intercepts and zero linear trend coefficients, Case IV: unrestricted intercepts and restricted linear trend coefficients, Case V: unrestricted intercepts and unrestricted linear trend coefficients. These cases have been considered by Johansen (1995) when all the I(1) variables in the VAR are treated as endogenous. Section 4 weakens the distributional assumptions of earlier sections and develops tests of cointegration rank for all five cases in 4.1 Testing H, 4.2 Testing H relevant asymptotic critical values are provided in Table 6(a), Table 6(b), Table 6(c), Table 6(d), Table 6(e). Modifications to the tests of 4.1 Testing H, 4.2 Testing H necessitated in the presence of exogenous variables which are integrated of order zero are briefly addressed in Section 4.3. Tests for the absence of an intercept or a trend in the cointegrating relations are discussed in Section 4.4, and particular misspecification tests concerning the intercept and linear trend coefficients are developed in Section 4.5 together with misspecification tests for the weak exogeneity assumption. The problem of efficient conditional estimation of a VECM subject to restrictions on the short-run dynamics is considered in Section 5. The subsequent two sections consider the empirical relevance of the proposed tests. Section 6 addresses the issue of the small sample performance of the proposed trace and maximum eigenvalue tests using a limited set of Monte Carlo experiments. It compares the size and power performance of the standard Johansen procedure with the new tests that take account of exogenous I(1) variables as well as possible restrictions on the short-run coefficients. Section 7 presents an empirical re-examination of the validity of the Purchasing Power Parity (PPP) and the Uncovered Interest Parity (UIP) hypotheses using U.K. quarterly data over the period 1972(1)–1987(2) which was previously analyzed by Johansen and Juselius (1992) and Pesaran and Shin (1996). In contrast to this earlier work, foreign prices are assumed to be exogenously determined and a more satisfactory treatment of oil price changes is provided in the analysis. Section 8 concludes the paper. Proofs of results are collected in Appendix A and Appendix B describes the simulation method for the computation of the asymptotic critical values provided in Table 6(a)–(e).