Søren Johansen

Statistical analysis of cointegration vectors

We consider a nonstationary vector autoregressive process which is integrated of order 1, and generated by i.i.d. Gaussian errors. We then derive the maximum likelihood estimator of the space of cointegration vectors and the likelihood ratio test of the hypothesis that it has a given number of dimensions. Further we test linear hypotheses about the cointegration vectors. The asymptotic distribution of these test statistics are found and the first is described by a natural multivariate version of the usual test for unit root in an autoregressive process, and the other is a x2 test. 1. Introduction The idea of using cointegration vectors in the study of nonstationary time series comes from the work of Granger (1981), Granger and Weiss (1983), Granger and Engle (1985), and Engle and Granger (1987). The connection with error correcting models has been investigated by a number of authors; see Davidson (1986), Stock (1987), and Johansen (1988) among others. Granger and Engle (1987) suggest estimating the cointegration relations using regression, and these estimators have been investigated by Stock (1987), Phillips (1985), Phillips and Durlauf (1986), Phillips and Park (1986a, b, 1987), Phillips and Ouliaris (1986,1987), Stock and Watson (1987), and Sims, Stock and Watson (1986). The purpose of this paper is to derive maximum likelihood estimators of the cointegration vectors for an autoregressive process with independent Gaussian errors, and to derive a likelihood ratio test for the hypothesis that there is a given number of these. A similar approach has been taken by Ahn and Reinsel (1987). This program will not only give good estimates and test statistics in the Gaussian case, but will also yield estimators and tests, the properties of which can be investigated under various other assumptions about the underlying data generating process. The reason for expecting the estimators to behave better *The simulations were carefully performed by Marc Andersen with the support of the Danish Social Science Research Council. The author is very grateful to the referee whose critique of the first version greatly helped improve the presentation.

Likelihood-Based Inference in Cointegrated Vector Autoregressive Models

This book gives a detailed mathematical and statistical analysis of the cointegrated vector autoregresive model. This model had gained popularity because it can at the same time capture the short-run dynamic properties as well as the long-run equilibrium behaviour of many non-stationary time series. It also allows relevant economic questions to be formulated in a consistent statistical framework. Part I of the book is planned so that it can be used by those who want to apply the methods without going into too much detail about the probability theory. The main emphasis is on the derivation of estimators and test statistics through a consistent use of the Guassian likelihood function. It is shown that many different models can be formulated within the framework of the autoregressive model and the interpretation of these models is discussed in detail. In particular, models involving restrictions on the cointegration vectors and the adjustment coefficients are discussed, as well as the role of the constant and linear drift. In Part II, the asymptotic theory is given the slightly more general framework of stationary linear processes with i.i.d. innovations. Some useful mathematical tools are collected in Appendix A, and a brief summary of weak convergence in given in Appendix B. The book is intended to give a relatively self-contained presentation for graduate students and researchers with a good knowledge of multivariate regression analysis and likelihood methods. The asymptotic theory requires some familiarity with the theory of weak convergence of stochastic processes. The theory is treated in detail with the purpose of giving the reader a working knowledge of the techniques involved. Many exercises are provided. The theoretical analysis is illustrated with the empirical analysis of two sets of economic data. The theory has been developed in close contract with the application and the methods have been implemented in the computer package CATS in RATS as a result of a rcollaboation with Katarina Juselius and Henrik Hansen.

Testing structural hypotheses in a multivariate cointegration analysis of the PPP and the UIP for UK

Abstract The paper develops some new tests for structural hypotheses in the framework of a multivariate error correction model with Gaussian errors. The tests are constructed by an analysis of the likelihood function and motivated by an empirical investigation of the PPP relation and the UIP relation for the United Kingdom. Three types of tests are discussed. First we consider the same linear restrictions on all cointegration relations, then we consider the hypothesis that certain relations are assumed to be cointegrating, and finally we formulate a general hypothesis that contains the previous ones. This hypothesis can be expressed by the condition that some of the cointegrating relations are subject to given linear restrictions, while others are unconstrained.

Cointegration in partial systems and the efficiency of single-equation analysis

Abstract It is shown how one can estimate cointegration relations in a partially modelled system by the method of maximum likelihood. The estimator is compared with the estimator based on the full system, and it is shown that the two estimators are identical if the conditioning variables are weakly exogenous for the cointegrating relations and their adjustment coefficients. Suggestions are made for analyzing the partial system, when there is no weak exogeneity.

Cointegration analysis in the presence of structural breaks in the deterministic trend

When analysing macroeconomic data it is often of relevance to allow for structural breaks in the statistical analysis. In particular, cointegration analysis in the presence of structural breaks could be of interest. We propose a cointegration model with piecewise linear trend and known break points. Within this model it is possible to test cointegration rank, restrictions on the cointegrating vector as well as restrictions on the slopes of the broken linear trend.

Identification of the long-run and the short-run structure an application to the ISLM model

In this paper we discuss the problem of identification in a model with cointegration. It is pointed out that there is an identification problem for both long-run parameters and short-run parameters. The identification of the equations and the cointegrating relations is achieved by linear restrictions on the parameters and a criterion for a statistical model to be identifying is given. We also define empirical identification of an estimated structure. A switching algorithm for calculating the restricted parameters is proposed. The concepts are illustrated with an empirical analysis of the ISLM model using Australian monetary data.

Some tests for parameter constancy in cointegrated VAR-models

Some methods for the evaluation of parameter constancy in cointegrated vector autoregressive (VAR) models are discussed. Two different ways of re-estimating the VAR-model are proposed; one in which all parameters are estimated recursively based upon the likelihood function for the first observations, and another in which the cointegrating relations are estimated recursively from a likelihood function, where the short-run parameters have been concentrated out. We suggest graphical procedures based on recursively estimated eigenvalues to evaluate the constancy of the long-run parameters in the model. Specifically, we look at the time paths of the eigenvalues using a new result on the asymptotic distribution of the estimated eigenvalues. Furthermore, we show that the fluctuation test by Ploberger et al. (1989) and the Lagrange multiplier (LM) type test for constancy of parameters by Nyblom (1989) can be applied to test the constancy of the long-run parameters in the cointegrated VAR-model. All results are illustrated using a model for the term structure of interest rates on US Treasury securities.

Identifying restrictions of linear equations with applications to simultaneous equations and cointegration

The identification problem for simultaneous equations is solved by the well-known rank condition which gives a necessary and sufficient condition for the parameters to be uniquely or statistically identified by linear restrictions. This paper formulates and solves another problem: Given a set of linear restrictions, which conditions should they satisfy for most parameter values to be identified? The main result of the paper is a simple algebraic condition on a set of linear restrictions that guarantees that most parameters satisfying the restrictions are uniquely identified.

A Small Sample Correction for the Test of Cointegrating Rank in the Vector Autoregressive Model

We derive an approximation to the expectation of the likelihood tatio test for cointegration in the vector autoregressive model. The expression depends on moments of functions of random walk, which are tabulated by simulation, and functions of the parameters, which are estimated. From this approximation we propose a correction factor with the purpose of improving the small sample performance of the test.

Asymptotic Inference on Cointegrating Rank in Partial Systems

The likelihood ratio test for cointegrating rank is analyzed for partial (or conditional) systems in the vector autoregressive error-correction model. Under the assumption of weak exogeneity for the cointegrating parameters, the asymptotic distributions are given and tables of critical values are provided. A discussion is given of some of the assumptions of the model, why they are needed, and how they are tested.