Optimal power for testing potential cointegrating vectors with known parameters for nonstationarity.

Theory often specifies a particular cointegrating vector among integrated variables, and testing for a unit root in the known cointegrating vector is often required. Although it is common to simply use a univariate test for a unit root for this test, it is known that this does not take into account all available information. We show here that in such testing situations, a family of tests with optimality properties exists. We use this to characterize the extent of the loss in power from using popular methods, as well as to derive a test that works well in practice. We also characterize the extent of the losses of not imposing the cointegrating vector in the testing procedure. We apply various tests to the hypothesis positing that price forecasts from the Livingston data survey are cointegrated with prices, and find that although most tests fail to reject the presence of a unit root in forecast errors, the tests presented here strongly reject this (implausible) hypothesis.

KEY WORDS: Cointegration; Optimal test; Unit root.

1. INTRODUCTION

This article examines tests for cointegration when the researcher knows the cointegrating vector a priori and when the "[??]" variables in the cointegrating regression are known to be integrated of order one [I(1)]. In particular, we characterize a family of optimal tests for the null hypothesis of no cointegration when there is one cointegrating vector. This enables us to examine the loss in power from suboptimal methods (e.g., univariate unit root tests on the cointegrating vector) and also losses that arise from testing cointegration with estimated cointegrating vectors.

There are a number of practical reasons for our interest in this family of tests. First, in many applications a potential cointegrating vector is specified by economic theory (see Zivot 2000 for a list of examples), and researchers are confident or willing to assume that variables are I(1). The test of interest then becomes testing whether the implied cointegrating vector has a unit root (which would falsify the theory). The empirical strategy commonly followed is to simply construct the potential cointegrating vector and use a univariate test for a unit root. However, this method avoids using useful information in the original multivariate model that could lead to more powerful tests (see Zivot 2000). Although tests are available that do exploit this extra information in the problem (e.g., those in Horvath and Watson 1995; Johansen 1988, 1995; Kremers, Ericsson, and Dolado 1992; Zivot 2000), these tests do not use this information optimally. The class of tests suggested herein, identical apart from the treatment of deterministic terms to those of Elliott and Jansson (2003) for testing for unit roots with stationary covariates, do have optimality properties.

Second, the optimal family that we derive allows the power bound of such tests to be derived. This is interesting in the sense that it gives an objective for examining the loss of power in estimating rather than specifying the cointegrating vector. A quantitative understanding of this loss and how it varies with nuisance parameters of the model is important for understanding differences in empirical results. If one researcher specifies the parameters of the cointegrating vector and rejects while another estimates the vector and fails to reject, then we are more certain that this is likely due to loss of power when there are large losses in power from estimating the cointegrating vector. If the power losses were small, then we would probably conclude that the imposed parameters are in error. By deriving the results analytically, we are able to say what types of models (or, more concretely, what values for a certain nuisance parameter) are likely to be related to large or small power losses in estimating the cointegrating vector. For many values of the nuisance parameter (which is consistently estimable and is produced as a byproduct of the test proposed herein) the differences in power is large.

Section 2 presents our model and relates it to error-correction models (ECMs). Section 3 considers tests for cointegration when the cointegrating vector is known. We discuss a number of approaches that have been used in the literature and present the methods of Elliott and Jansson (2003) in the context of this problem. Section 4 presents numerical results to show the asymptotic and small sample performances of the Elliott and Jansson (2003) test relative to others in the literature. Section 5 describes an empirical application relating to the cointegration of forecasts and their outcomes.