Standard tests for the cointegrating rank of a vector autoregressive (VAR) process have nonstandard limiting distributions which depend on the characteristics of intercept terms and time trends in the system. In practice these characteristics are often unknown. Therefore modified tests are proposed with limiting distributions which do not depend on the characteristics of deterministic terms under the null hypothesis. One type of tests makes use of lag augmentation, that is, a VAR process of order p + 1 is fitted when the true order is p while the tests are based on the coefficient matrices of the first p lags only. It is shown that Ø 2 limiting distributions are obtained in this way. The price for this simplicity will be reduced power, however. Therefore, we also consider LM (Lagrange multiplier) type tests. These tests are shown to have nonstandard limiting distributions which do not depend on deterministic terms and have better local power than competing LR (likelihood ratio) test...

The impact of the choice of the lag length on tests for the number of cointegration relations in a vector autoregressive (VAR) process is investigated. It is shown that the asymptotic distribution of likelihood ratio (LR) tests for the cointegrating rank remains unchanged if the true data generation process (DGP) is of finite order and a consistent model selection criterion is used for choosing the lag length. A similar result also holds if the true DGP is an infinite order VAR. In a simulation study we find that small sample power and size of LR cointegration tests strongly depend on the choice of the lag order. We thank Christian Muller and Kirstin Hubrich for helping with the computations. The Deutsche Forschungsgemeinschaft, SFB 373, provided financial support. Part of this research was done while the second author was visiting the Institute of Statistics and Econometrics at the Humboldt University in Berlin. 1 Introduction Following the invention of cointegration by Granger ...

Various types of impulse responses have been used for interpreting finite order vector autoregessive (VAR) models in the stationary as well as the nonstationary cointegrated case. In practice, finite order VAR processes are regarded as rough approximations to the actual data generation process at best. Therefore we derive some general asymptotic results for infinite order cointegrated VAR processes that are used for inference on impulse responses. The theory is based on the assumption that finite order VARs are fitted to the time series of interest although the true order may be infinite. The order of the fitted process is, however, assumed to increase with the sample size. The theoretical results are illustrated by an empirical analysis of a German money demand system. Financial support by the DFG, Sonderforschungsbereich 373, is gratefully acknowledged. 1 Introduction Finite order vector autoregressive (VAR) models have been used extensively in applied econometric studies. These...

In cointegrated vector autoregressive (VAR) analyses various nonlinear functions of the coefficients are of interest. Notable examples are impulse responses. A general theory for asymptotic inference on such functions is developed under the assumption that the actual data generation process (DGP) is of potentially infinite VAR order although finite order VAR models are fitted to the data and are used for computing the quantities of interest. In the asymptotic theory the VAR order is assumed to grow with the sample size albeit at a smaller rate. Both authors gratefully acknowledge financial support by the DFG, Sonderforschungsbereich 373. This research was partly carried out while the first author was visiting the Institute of Statistics and Econometrics at the Humboldt University and partly while the second author was a Visiting Research Fellow at the Australian National University in Canberra. We thank Kirstin Hubrich for comments. The paper was printed using funds made available b...

Introduction The last sixty years have witnessed a rapid development in the field of econometrics. In the 1940s and 50s the foundations were laid by the Cowles Commission researchers for analyzing econometric simultaneous equations models. Once the basic statistical theory was available many such models were constructed for empirical analysis. The parallel development of the computer technology in the 1950s and 60s has resulted in simultaneous equations models of increasing size in the hope that more detailed models would result in better approximations to the underlying data generation mechanisms. It turned out, however, that increasing the number of variables and equations of the models did not generally lead to improvements in the performance in terms of forecasting, for instance. In fact, in some forecast comparisons univariate time series models were found to be superior to large scale econometric models. One explanation of this failure of the latter models is their insufficient

We consider Johansen’s (1988, 1991) cointegration tests when a Vector AutoRegressive (VAR) process of order k is used to approximate a more general linear process with a possibly infinite VAR representation. Traditional methods to select the lag order, such as Akaike’s (AIC) or the Bayesian information criteria, often lead to too parsimonious a model with the implication that the cointegration tests suffer from substantial size distortions in finite samples. We extend the analysis of Ng and Perron (2001) to derive a Modified Akaike’s Information Criterion (MAIC) in this multivariate setting. The idea is to use the information specified by the null hypothesis as it relates to restrictions on the parameters of the model to keep an extra term in the penalty function of the AIC. This MAIC takes a very simple form for which this extra term is simply the likelihood ratio test for testing the null hypothesis of r against more than r cointegrating vectors. We provide theoretical analyses of its validity and of the fact that cointegration tests constructed from a VAR whose lag order is selected using the MAIC have the same limit distribution as when the order is finite and known. We also provide theoretical and simulation analyses to show how the MAIC leads to VAR approximations that yield tests with drastically improved size properties with little loss of power.