Topic

# Cointegration

About: Cointegration is a(n) research topic. Over the lifetime, 17130 publication(s) have been published within this topic receiving 506215 citation(s).

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TL;DR: In this paper, the authors consider a nonstationary vector autoregressive process which is integrated of order 1, and generated by i.i.d. Gaussian errors, and 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.

Abstract: 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.

15,356 citations

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TL;DR: In this paper, the estimation and testing of long-run relations in economic modeling are addressed, starting with a vector autoregressive (VAR) model, the hypothesis of cointegration is formulated as a hypothesis of reduced rank of the long run impact matrix.

Abstract: The estimation and testing of long-run relations in economic modeling are addressed. Starting with a vector autoregressive (VAR) model, the hypothesis of cointegration is formulated as the hypothesis of reduced rank of the long-run impact matrix. This is given in a simple parametric form that allows the application of the method of maximum likelihood and likelihood ratio tests. In this way, one can derive estimates and test statistics for the hypothesis of a given number of cointegration vectors, as well as estimates and tests for linear hypotheses about the cointegration vectors and their weights. The asymptotic inferences concerning the number of cointegrating vectors involve nonstandard distributions. Inference concerning linear restrictions on the cointegration vectors and their weights can be performed using the usual chi squared methods. In the case of linear restrictions on beta, a Wald test procedure is suggested. The proposed methods are illustrated by money demand data from the Danish and Finnish economies.

11,867 citations

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01 Jan 2005

9,664 citations

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19 Oct 2012

TL;DR: In this paper, the authors present the likelihood methods for the analysis of cointegration in VAR models with Gaussian errors, seasonal dummies, and constant terms, and show that the asymptotic distribution of the maximum likelihood estimator is mixed Gausssian.

Abstract: Presents the likelihood methods for the analysis of cointegration in VAR models with Gaussian errors, seasonal dummies, and constant terms. Discusses likelihood ratio tests of cointegration rank and find the asymptotic distribution of the test statistics. Shows that the asymptotic distribution of the maximum likelihood estimator is mixed Gausssian.

9,355 citations

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21 Nov 1994

TL;DR: In this article, the authors present an alternative solution method for Deterministic Processes by iteratively solving homogeneous difference equation and finding particular solutions for deterministic processes, and conclude that the proposed solution is the best solution.

Abstract: PREFACE. ABOUT THE AUTHOR. Chapter DIFFERENCE EQUATIONS . 1 Time-Series Models. 2 Difference Equations and Their Solutions. 3 Solution by Iteration. 4 An Alternative Solution Methodology. 5 The Cobweb Model. 6 Solving Homogeneous Difference Equations. 7 Finding Particular Solutions for Deterministic Processes. 8 The Method of Undetermined Coefficients. 9 Lag Operators. Summary and Conclusions. Questions and Exercises. Endnotes. Appendix 1 Imaginary Roots and de Moivre's Theorem. Appendix 2 Characteristic Roots in Higher-Order Equations. Chapter 2 STATIONARY TIME-SERIES MODELS . 1 Stochastic Difference Equation Models. 2 ARMA Models. 3 Stationarity. 4 Stationarity Restrictions for an ARMA(p, q) Model. 5 The Autocorrelation Function. 6 The Partial Autocorrelation Function. 7 Sample Autocorrelations of Stationary Series. 8 Box-Jenkins Model Selection. 9 Properties of Forecasts. 10 A Model of the Interest Rate Spread. 11 Seasonality. 12 Parameter Instability and Structural Change. Summary and Conclusions. Questions and Exercises. Endnotes. Appendix 1 Estimation of an MA(1) Process. Appendix 2 Model Selection Criteria. Chapter 3 MODELING VOLATILITY . 1 Economic Time Series The Stylized Facts. 2 ARCH Processes. 3 ARCH and GARCH Estimates of Inflation. 4 Two Examples of GARCH Models. 5 A GARCH Model of Risk. 6 The ARCH-M Model. 7 Additional Properties of GARCH Processes. 8 Maximum Likelihood Estimation of GARCH Models. 9 Other Models of Conditional Variance. 10 Estimating the NYSE International 100 Index. 11 Multivariate GARCH. Summary and Conclusions. Questions and Exercises. Endnotes. Appendix 1 Multivariate GARCH Models. Chapter 4 MODELS WITH TREND . 1 Deterministic and Stochastic Trends. 2 Removing the Trend. 3 Unit Roots and Regression Residuals. 4 The Monte Carlo Method. 5 Dickey-Fuller Tests. 6 Examples of the ADF Test. 7 Extensions of the Dickey-Fuller Test. 8 Structural Change. 9 Power and the Deterministic Regressors. 10 Tests with More Power. 11 Panel Unit Root Tests. 12 Trends and Univariate Decompositions. Summary and Conclusions. Questions and Exercises. Endnotes. Appendix 1 The Bootstrap. Chapter 5 MULTIEQUATION TIME-SERIES MODELS . 1 Intervention Analysis. 2 Transfer Function Models. 3 Estimating a Transfer Function. 4 Limits to Structural Multivariate Estimation. 5 Introduction to VAR Analysis. 6 Estimation and Identification. 7 The Impulse Response Function. 8 Testing Hypothesis. 9 Example of a Simple VAR Terrorism and Tourism in Spain. 10 Structural VARs. 11 Examples of Structural Decompositions. 12 The Blanchard and Quah Decomposition. 13 Decomposing Real and Nominal Exchange Rate Movements An Example. Summary and Conclusions. Questions and Exercises. Endnotes. Chapter 6 COINTEGRATION AND ERROR-CORRECTION MODELS . 1 Linear Combinations of Integrated Variables. 2 Cointegration and Common Trends. 3 Cointegration and Error Correction. 4 Testing for Cointegration The Engle-Granger Methodology. 5 Illustrating the Engle-Granger Methodology. 6 Cointegration and Purchasing-Power Parity. 7 Characteristic Roots, Rank, and Cointegration. 8 Hypothesis Testing. 9 Illustrating the Johansen Methodology. 10 Error-Correction and ADL Tests. 11 Comparing the Three Methods. Summary and Conclusions. Questions and Exercises. Endnotes. Appendix 1 Characteristic Roots Stability and Rank. Appendix 2 Inference on a Cointegrating Vector. Chapter 7 NONLINEAR TIME-SERIES MODELS . 1 Linear Versus Nonlinear Adjustment. 2 Simple Extensions of the ARMA Model. 3 Regime Switching Models. 4 Testing For Nonlinearity. 5 Estimates of Regime Switching Models. 6 Generalized Impulse Responses and Forecasting. 7 Unit Roots and Nonlinearity. Summary and Conclusions. Questions and Exercises. Endnotes. STATISTICAL TABLES. A. Empirical Cumulative Distributions of the tau. B. Empirical Distribution of PHI . C. Critical Values for the Engle-Granger Cointegration Test. D. Residual Based Cointegration Test with I (1) and I (2) Variables. E. Empirical Distributions of the lambda max and lambda trace Statistics. F. Critical Values for beta 1 = 0 in the Error-correction Model. G. Critical Values for Threshold Unit Roots. REFERENCES. SUBJECT INDEX.

6,261 citations