Abstract
In the preceding chapter, we used stochastic trends to model nonstationary behaviour of time series, i.e. the variance of the data generating process increases over time, the series exhibits persistent behaviour and its first difference is stationary. For many economic time series, such a data generating process is a sufficient approximation, so that, in the following, we only consider processes which are integrated of order one (I(1)).
Keywords
- Granger Causality
- Error Correction Model
- Vector Error Correction Model
- Stochastic Trend
- Vector Autoregressive Model
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
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References
The idea of cointegration goes back to
CLIVE W.J. GRANGER, Some Properties of Time Series Data and their Use in Econometric Model Specification, Journal of Econometrics 16 (1981), pp. 121 – 130, as well as
CLIVE W.J. GRANGER, Developments in the Study of Co-integrated Economic Variables, Oxford Bulletin of Economics and Statistics 48 (1986), pp. 213 – 228.
The first basic methodological paper about cointegration was
ROBERT F. ENGLE and CLIVE W.J. GRANGER, Co-Integration and Error Correction: Representation, Estimation, and Testing, Econometrica 55 (1987), pp. 251 – 276.
This was one of the essential papers for which C.W.J. GRANGER received the Nobel Prize in 2003. This and the following papers,
JAMES H. STOCK, Asymptotic Properties of Least-Squares Estimators of Cointegrating Vectors, Econometrica 55 (1987), pp. 1035 – 1056, and
SØREN JOHANSEN, Statistical Analysis of Cointegration Vectors, Journal of Economic Dynamics and Control 12 (1988), pp. 231 – 254
led to the large dissemination of this approach.
An introduction to estimation and testing of cointegration in single equations is given by
UWE HASSLER, Leitfaden zum Testen und Schätzen von Kointegration in W. GAAB, U. HEILEMANN and J. WOLTERS (eds), Arbeiten mit ökonometrischen Modellen, Physica-Verlag, Heidelberg 2004, pp. 88 – 155.
Special textbooks covering the econometric handling of cointegrated processes are
ANINDYA BANERJEE, JUAN J. DOLADO, JOHN W. GALBRAITH and DAVID F. HENDRY, Co-Integration, Error Correction, and the Econometric Analysis of Non-Stationary Data, Oxford University Press, Oxford 1993; or
SØREN JOHANSEN, Likelihood-based Inference in Cointegrated Vector Autoregressive Models, Oxford University Press, Oxford 1995.
Based on this strongly theoretically oriented book
KATARINA JUSELIUS, The Cointegrated VAR Model: Methodology and Applications, Oxford University Press, Oxford 2006,
shows how to apply and interpret Vector Error Correction Models. A short review of different approaches to identify cointegrating relations and to impose restrictions on them is given by
H. PETER BOSWIJK and JURGEN A. DOORNIK, Identifying, Estimating and Testing Restricted Cointegrated Systems: An Overview, Statistica Neerlandica 58 (2004), pp. 440 – 465.
The problem of spurious regressions was first tackled in a simulation study by
CLIVE W.J. GRANGER and PAUL NEWBOLD, Spurious Regressions in Econometrics, Journal of Econometrics 2 (1974), pp. 111 – 120.
The corresponding asymptotic distribution theory is presented in
PETER C.B. PHILLIPS, Understanding Spurious Regressions in Econometrics, Journal of Econometrics 33 (1986), pp. 311 – 340.
Critical values of residual based tests for cointegration in single equation models are given by
ROBERT F. ENGLE and BYUNG SAM YOO, Forecasting and Testing in Cointegrated Systems, Journal of Econometrics 35 (1987), pp. 143 – 159;
JAMES G. MACKINNON, Critical Values for Co-Integration Tests, in: R.F. ENGLE and C.W.J: GRANGER (eds.), Long-Run Economic Relationships, Oxford University Press, Oxford 1991, pp. 267 – 276.
A simple correction procedure which leads to asymptotically standard normal distributed t values in static regression equations is derived by
PENTTI SAIKKONEN, Asymptotically Efficient Estimation of Cointegration Regressions, Econometric Theory 7 (1991), pp. 1 – 21, and
JAMES H. STOCK and MARK W. WATSON, A Simple Estimator of Cointegrating Vectors in Higher Order Integrated Systems, Econometrica 61 (1993), pp. 783 – 820.
Problems which might arise by neglecting the dynamic structure when using the Engle-Granger approach are shown by
ANINDYA BANERJEE, JUAN J. DOLADO, DAVID F. HENDRY and GREGOR W. SMITH, Exploring Equilibrium Relationships in Econometrics Through Static Models: Some Monte Carlo Evidence, Oxford Bulletin of Economics and Statistics 48 (1986), pp. 253 – 277,
Critical values for tests of cointegration in error correction models are given in
ANINDYA BANERJEE, JUAN J. DOLADO and RICARDO MESTRE, Error-Correction Mechanism Tests for Cointegration in a Single-Equation Framework, Journal of Time Series Analysis 19 (1998), pp. 267 – 283.
The critical values which are appropriate when the variables also include linear time trends is discussed in
UWE HASSLER, Cointegration Testing in Single Error-Correction Equations in the Presence of Linear Time Trends, Oxford Bulletin of Economics and Statistics 62 (2000), pp. 621 – 632.
Further test procedures for testing in single error correction equations are presented in
UWE HASSLER and JÜRGEN WOLTERS, Autoregressive Distributed Lag Models and Cointegration, Allgemeines Statistisches Archiv 90 (2006), pp. 59 – 74; reprinted in: O. HÜBLER and J. FROHN (eds.), Modern Econometric Analysis, Springer, Berlin 2006, pp. 57 – 72.
Critical values for trace and λmax tests proposed by SØREN JOHANSEN are given by
MICHAEL OSTERWALD-LENUM, A Note with Quantiles of the Asymptotic Distribution of the Maximum Likelihood Cointegration Rank Test Statistics, Oxford Bulletin of Economics and Statistics, 54 (1992), pp. 461 – 472.
JAMES G. MACKINNON, ALFRED A. HAUG and LEO MICHELIS, Numerical Distribution Functions of Likelihood Ratio Tests for Cointegration, Journal of Applied Econometrics 14 (1999), pp. 563 – 577
present critical values which are much more accurate than those available previously and also take into account the possibility for exogenous variables in the cointegrating relation.
Tests for hypotheses about the cointegration matrix have been developed by
SØREN JOHANSEN and KATARINA JUSELIUS, Maximum Likelihood Estimation and Inference on Cointegration – with Applications to the Demand for Money, Oxford Bulletin of Economics and Statistics, 52 (1990), pp. 169 – 210.
Compared to the Johansen approach, an alternative handling of the deterministic components in error correction models is proposed by
HELMUT LÜTKEPOHL and PENTTI SAIKKONEN, Testing for the Cointegration Rank of a VAR Process with a Time Trend, Journal of Econometrics 95 (2000), pp. 177 – 198, and
PENTTI SAIKKONEN and HELMUT LÜTKEPOHL, Trend Adjustment Prior to Testing for the Cointegration Rank of a Vector Autoregressive Process, Journal of Time Series Analysis 21 (2000), pp. 435 – 456.
This approach can be extended to modelling deterministic structural breaks in the data. See for this
PENTTI SAIKKONEN and HELMUT LÜTKEPOHL, Testing for the Cointegration Rank of a VAR Process with Structural Shifts, Journal of Business and Economic Statistics 18 (2000), pp. 451 – 464.
Tests for cointegration in the Engle-Granger framework in the presence of structural breaks are presented in
UWE HASSLER, Dickey-Fuller Cointegration Test in the Presence of Regime Shifts at Known Time, Allgemeines Statistisches Archiv 86 (2002), pp. 263 – 276.
For the analysis of structural vector error correction models see
SØREN JOHANSEN, Cointegration in Partial Systems and the Efficiency of Single- Equation Analysis, Journal of Econometrics 52 (1992), pp. 389 – 402,
H. PETER BOSWIJK, Efficient Inference on Cointegration Parameters in Structural Error Correction Models, Journal of Econometrics 69 (1995), pp. 133 – 158, as well as
NEIL R. ERICSSON, Conditional and Structural Error Correction Models, Journal of Econometrics 65 (1995), pp. 159 – 171.
For the concept of weak exogeneity see, for example,
NEIL R. ERICSSON, Cointegration, Exogeneity, and Policy Analysis: An Overview, Journal of Policy Modeling 14 (1992), pp. 251 – 280, as well as
NEIL R. ERICSSON, DAVID F. HENDRY and GRAHAM E. MIZON, Exogeneity, Cointegration, and Economic Policy Analysis, Journal of Business and Economic Statistics 16 (1998), pp. 370 – 387.
These papers also discuss the relation between Granger causality and exogeneity. The problem of how vector error correction models with exogenous I(1) variables and restrictions with respect to the short-run dynamics can efficiently be estimated is discussed in
M. HASHEM PESARAN, YONGCHEOL SHIN and RICHARD J. SMITH, Structural Analysis of Vector Error Correction Models with Exogenous I(1)-Variables, Journal of Econometrics 97 (2000), pp. 293 – 343.
They also give the corresponding critical values of the tests for cointegration.
The problem of Granger causality in the situation of cointegrated variables is, for example, discussed in
CLIVE W.J. GRANGER and JIN-LUNG LIN, Causality in the Long Run, Econometric Theory 11 (1995), pp. 530 – 536.
Testing strategies for situations in which the question remains open whether a cointegrating relation exists or not are presented in
HIRO Y. TODA and TAKU YAMAMOTO, Statistical Inference in Vector Autoregressions with Possibly Integrated Processes, Journal of Econometrics 66 (1995), pp. 259 – 285, as well as in
JUAN J. DOLADO and HELMUT LÜTKEPOHL, Making Wald Tests Work for Cointegrated VAR Systems, Econometric Reviews 15 (1996), pp. 369 – 386.
For this, see also
HIROSHI YAMADA and HIRO Y. TODA, Inference in Possibly Integrated Vector Autoregressive Models: Some Finite Sample Evidence, Journal of Econometrics 86 (1998), pp. 55 – 95.
The possibilities and properties of predictions using error correction models are discussed in
PETER F. CHRISTOFFERSEN and FRANCIS X. DIEBOLD, Cointegration and Long- Horizon Forecasting, Journal of Business and Economic Statistics 16 (1998), pp. 450 – 458,
MICHAEL P. CLEMENTS and DAVID F. HENDRY, Forecasting with Difference- Stationary and Trend-Stationary Models, Econometrics Journal 4 (2001), pp. S1 – S19,
UWE HASSLER and JÜRGEN WOLTERS, Forecasting Money Market Rates in the Unified Germany, in: R. FRIEDMANN, L. KNÜPPEL and H. LÜTKEPOHL (eds.), Econometric Studies: A Festschrift in Honour of Joachim Frohn, Lit Verlag, Münster et al. 2001, pp. 185 – 201, as well as in
DAVID F. HENDRY and MICHAEL P. CLEMENTS, Economic Forecasting: Some Lessons from Recent Research, Economic Modelling 20 (2003), pp. 301 – 329.
Research on the German money demand is done by
JÜRGEN WOLTERS, TIMO TERÄSVIRTA and HELMUT LÜTKEPOHL, Modelling the Demand for M3 in the Unified Germany, Review of Economics and Statistics 80 (1998), pp. 399 – 409,
HELMUT LÜTKEPOHL, TIMO TERÄSVIRTA and JÜRGEN WOLTERS, Investigating Stability and Linearity of a German M1 Money Demand Function, Journal of Applied Econometrics 14 (1999), pp. 511 – 525.
HELMUT LÜTKEPOHL and JÜRGEN WOLTERS, The Transmission of German Monetary Policy in the Pre-Euro Period, Macroeconomic Dynamics 7 (2003), pp. 711 – 733.
The term structure of interest rates in the German money market is investigated by
JÜRGEN WOLTERS and UWE HASSLER, Die Zinsstruktur am deutschen Interbanken- Geldmarkt: Eine empirische Analyse für das vereinigte Deutschland, ifo Studien 44 (1998), pp. 141 – 160.
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Kirchgässner, G., Wolters, J., Hassler, U. (2013). Cointegration. In: Introduction to Modern Time Series Analysis. Springer Texts in Business and Economics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-33436-8_6
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