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A MODIFIED INFORMATION CRITERION FOR COINTEGRATION TESTS BASED ON A VAR APPROXIMATION

  • Zhongjun Qu (a1) and Pierre Perron (a2)
Abstract

We consider the cointegration tests of Johansen (1988, Journal of Economic Dynamics and Control 12, 231–254; 1991, Econometrica 59, 1551–1580) 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 information criterion (AIC) or the Bayesian information criterion, 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, Econometrica 69, 1519–1554) 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.We are grateful to two referees for especially useful and constructive comments.

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Corresponding author
Address correspondence to Zhongjun Qu, Department of Economics, University of Illinois at Urbana-Champaign, 1206 S. Sixth Street, Champaign, Illinois 61820, USA; e-mail: zqu@uiuc.edu
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Ahn, S.K. & G.C. Reinsel (1990) Estimation for partially non-stationary multivariate autoregressive models. Journal of the American Statistical Association 85, 813823.

Anderson, T.W. (1951) Estimating linear restrictions on regression coefficients for multivariate normal distributions. Annals of Mathematical Statistics 22, 327351.

Cavanaugh, J.E. (1997) Unifying the derivations for the Akaike and corrected Akaike information criteria. Statistics and Probability Letters 33, 201208.

Chang, Y. & J.Y. Park (2002) On the asymptotics of ADF tests for unit roots. Econometric Reviews 21, 431447.

Chao, J.C. & P.C.B. Phillips (1999) Model selection in partially nonstationary vector autoregressive processes with reduced rank structure. Journal of Econometrics 91, 227271.

Cheung, Y.W. & K.S. Lai (1993) Finite-sample sizes of Johansen's likelihood ratio tests for cointegration. Oxford Bulletin of Economics and Statistics 55, 313328.

Engle, R.F. & C.W.J. Granger (1987) Co-integration and error correction: Representation, estimation, and testing. Econometrica 55, 251276.

Granger, C.W.J. (1981) Some properties of time series data and their use in econometric model specification. Journal of Econometrics 16, 121130.

Hurvich, C.M. & C.L. Tsai (1989) Regression and time series model selection in small samples. Biometrika 76, 297307.

Johansen, S. (1988) Statistical analysis of cointegration vectors. Journal of Economic Dynamics and Control 12, 231254.

Johansen, S. (1991) Estimation and hypothesis testing of cointegration vectors in Gaussian vector autoregressive models. Econometrica 59, 15511580.

Kuersteiner, G. (2005) Automatic inference for infinite order vector autoregressions. Econometric Theory 21, 85115.

Leeb, H. & B.M. Pötscher (2005) Model selection and inference: Facts and fiction. Econometric Theory 21, 2159.

Lewis, R. & G.C. Reinsel (1985) Prediction of multivariate time series by autoregressive model fitting. Journal of Multivariate Analysis 16, 393411.

Lütkepohl, H. & P. Saikkonen (1997) Impulse response analysis in infinite order cointegrated vector autoregressive processes. Journal of Econometrics 81, 127157.

Nabeya, S. & P. Perron (1994) Local asymptotic distribution related to the AR(1) model with dependent errors. Journal of Econometrics 62, 229264.

Ng, S. & P. Perron (1995) Unit root tests in ARMA models with data-dependent methods for the selection of the truncation lag. Journal of the American Statistical Association 90, 268281.

Ng, S. & P. Perron (2001) Lag length selection and the construction of unit root tests with good size and power. Econometrica 69, 15191554.

Ng, S. & P. Perron (2005) A note on the selection of time series models. Oxford Bulletin of Economics and Statistics 67, 115134.

Perron, P. & J.Y. Campbell (1993) A note on Johansen's cointegration procedure when trends are present. Empirical Economics 18, 777789.

Perron, P. & S. Ng (1996) Useful modifications to some unit root tests with dependent errors and their local asymptotic properties. Review of Economic Studies 63, 435463.

Phillips, P.C.B. (1996) Econometric model determination. Econometrica 64, 763812.

Phillips, P.C.B. & W. Ploberger (1994) Posterior odds testing for a unit root with data-based model selection. Econometric Theory 10, 774808.

Phillips, P.C.B. & V. Solo (1992) Asymptotics for linear processes. Annals of Statistics 20, 9711001.

Pötscher, B.M. (1989) Model selection under nonstationarity: Autoregressive models and stochastic linear regression models. Annals of Statistics 17, 12571274.

Reinsel, G.C. & S.K. Ahn (1992) Vector AR models with unit roots and reduced rank structure: Estimation, likelihood ratio test, and forecasting. Journal of Time Series Analysis 13, 353375.

Said, S.E. & D.A. Dickey (1984) Testing for unit roots in autoregressive-moving average models of unknown order. Biometrika 71, 599608.

Saikkonen, P. (1992) Estimation and testing of cointegrated systems by an autoregressive approximation. Econometric Theory 8, 127.

Saikkonen, P. & R. Luukkonen (1997) Testing cointegration in infinite order vector autoregressive processes. Journal of Econometrics 81, 93126.

Schwarz, G. (1978) Estimating the dimension of a model. Annals of Statistics 6, 461464.

Shibata, R. (1980) Asymptotically efficient selection of the order of the model for estimating parameters of a linear process. Annals of Statistics 8, 147164.

Wei, C.Z. (1992) On predictive least squares principles. Annals of Statistics 20, 142.

Yap, S.F. & G.C. Reinsel (1995) Estimation and testing for unit roots in a partially nonstationary vector autoregressive moving average model. Journal of the American Statistical Association 90, 253267.

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Econometric Theory
  • ISSN: 0266-4666
  • EISSN: 1469-4360
  • URL: /core/journals/econometric-theory
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