Skip to main content
Disciplines
Automotive Business IT + Informatics Construction + Real Estate Electrical Engineering + Electronics Energy + Sustainability Insurance + Risk Finance + Banking Management + Leadership Marketing + Sales Mechanical Engineering + Materials
Events
DE
EN
Books
Journals
Topic Page
Marketing
Start single access now
Access for companies
EXTENDED SEARCH
Log in
Top
Published in:
Introduction Read chapter Read first chapter

2023 | OriginalPaper | Chapter

12. Cointegration and VECMs

Author : John D. Levendis

Published in: Time Series Econometrics

Publisher: Springer International Publishing

Log in

Activate our intelligent search to find suitable subject content or patents.

search-config
loading …

Abstract

In this chapter we show how to model the long-run relationship between variables in their levels, even if they are integrated. This is possible if two or more variables are “cointegrated.” Two variables are cointegrated is the difference between them is stationary. Or, to put it loosely, they move in parallel. In this chapter we explore the concept of cointegration, error correction mechanisms, and some of the more popular tests of contegration.

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

  • über 102.000 Bücher
  • über 537 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Maschinenbau + Werkstoffe
  • Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 340 Zeitschriften

aus folgenden Fachgebieten:

  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Versicherung + Risiko




Jetzt Wissensvorsprung sichern!

inform now
previous chapter Vector Autoregressions II: Extensions
next chapter Static Panel Data Models
Footnotes
1
More precisely, two more variables which are integrated of order I(b) are cointegrated if a linear combination of them is integrated of a lower order than b.
 
2
Testing the theory of purchasing power parity is a classic use of cointegration analysis. Notable examples include Juselius et al. (1992), Corbae and Ouliaris (1988), Taylor (1988), and Kim (1990). Pedroni (2001) provides a cointegration test of PPP for panel data.
 
3
That is, ADF with zero lags.
 
4
MacKinnon (2010) repeated his Monte Carlo simulations from MacKinnon (1991) using many more replications. This allowed him to provide a more accurate third-degree response surface rather than his earlier second-degree surface.
 
5
Since it is user-written and not an official Stata command, you must install it. You can do this by typing ssc install egranger.
 
6
There are many features which recommend Johansen’s (1988) approach. For example, Gonzalo (1994) shows that Johansen’s method outperforms four rival methods—asymptotically and in small samples—at estimating cointegrating vectors. This is the case even when the errors are not normal or when the correct number of lags is unknown.
 
7
We do not consider the I(2) case in this book. A workable but incomplete solution is to difference the I(2) variables once to render them I(1) and then follow the procedures as outlined below.
 
8
The online help for the Eviews econometric software also warns against using Cases 1 and 5 (http://​www.​eviews.​com/​help/​helpintro.​html#page/​content/​coint-Johansen_​Cointegration_​Test.​html). Likewise, Zivot and Wang (2007) warn against using Case 1. Sjö (2008, p.18) calls Case 4 “the model of last resort” (since including a time in the vectors might induce stationarity) and Case 5 as “quite unrealistic and should not be considered in applied work.” Thus, we are left with Cases 2 and 3 as reasonable choices.
 
9
That is, the trend is due to drift from a random walk.
 
10
Dwyer (2014, p.6) explains that the trace statistic does not refer to the trace of \(\hat {\boldsymbol {\Pi }}\) but refers instead to the “trace of a matrix based on functions of Brownian motion.” It also shares the similarity with the trace of the matrix in that both involve the sum of terms (here the sum of the eigenvalues); more specifically, we sum \(ln(1-\lambda ) \approx \lambda \) when (\(\lambda \approx 0\)).
 
11
It is unclear to me why Stata opted not to have trace and max options.
 
12
Cointegration merely requires that a linear combination of the variables is stationary. In practical terms, this means that the two variables can be tilted up or down until their difference is stationary. Two parallel lines are stationary, regardless of the constant difference between them. Or, what we care about is the slopes that establish stationarity; econometrically, we are less concerned with the constant. Economically, the constant term seldom has practical significance.
 
13
I am indebted to David Giles and his popular “Econometrics Beat” blog for bringing this and the Toda-Yamamoto procedure to my attention. The blog piece can be found at http://​davegiles.​blogspot.​com/​2011/​10/​var-or-vecm-when-testing-for-granger.​html. Readers are encouraged to read the cited references in that blog entry, especially the work by Clarke and Mirza (2006).
 
Literature
Banerjee, A., Dolado, J. J., Galbraith, J. W., & Hendry, D. (1993). Co-integration, error correction, and the econometric analysis of non-stationary data. Oxford University Press.CrossRef
Braun, P. A., & Mittnik, S. (1993). Misspecifications in vector autoregressions and their effects on impulse responses and variance decompositions. Journal of Econometrics, 59(3), 319–341.CrossRef
Brooks, C. (2014). Introductory econometrics for finance. Cambridge University Press.CrossRef
Campos, J., Ericsson, N. R., & Hendry, D. F. (1996). Cointegration tests in the presence of structural breaks. Journal of Econometrics, 70(1), 187–220.CrossRef
Clarke, J. A., & Mirza, S. (2006). A comparison of some common methods for detecting granger noncausality. Journal of Statistical Computation and Simulation, 76(3), 207–231.CrossRef
Corbae, D., & Ouliaris, S. (1988). Cointegration and tests of purchasing power parity. The Review of Economics and Statistics, 70(3), 508–511.CrossRef
Elliott, G. (1998). On the robustness of cointegration methods when regressors almost have unit roots. Econometrica, 66(1), 149–158.CrossRef
Enders, W. (2014). Applied econometric time series (3rd ed.). Wiley & Sons.
Engle, R. F., & Granger, C. W. (1987). Co-integration and error correction: Representation, estimation, and testing. Econometrica: Journal of the Econometric Society, 55, 251–276.CrossRef
Engle, R. F., Granger, C. W. J., Hylleberg, S., & Lee, H. S. (1993). Seasonal cointegration: The Japanese consumption function. Journal of Econometrics, 55(1–2), 275–298.CrossRef
Engle, R. F., & Yoo, B. S. (1987). Forecasting and testing in co-integrated systems. Journal of Econometrics, 35(1), 143–159.CrossRef
Engle, R., & Granger, C. (1991). Long-run economic relationships: Readings in cointegration. Oxford University Press.CrossRef
Ghysels, E. & Osborn, D. R. (2001). The econometric analysis of seasonal time series. Cambridge University Press.CrossRef
Gonzalo, J. (1994). Five alternative methods of estimating long-run equilibrium relationships. Journal of Econometrics, 60(1–2), 203–233.CrossRef
Gonzalo, J., & Pitarakis, J.-Y. (1998). Specification via model selection in vector error correction models. Economics Letters, 60(3), 321–328.CrossRef
Granger, C. W. (1988). Some recent development in a concept of causality. Journal of Econometrics, 39(1–2), 199–211.CrossRef
Granger, C. W., & Newbold, P. (1974). Spurious regressions in econometrics. Journal of Econometrics, 2(2), 111–120.CrossRef
Hansen, P. R., & Johansen, S. (1998). Workbook on Cointegration. Oxford University Press on Demand.CrossRef
Harris, R., & Sollis, R. (2003). Applied Time Series Modelling and Forecasting. John Wiley & Sons.
Johansen, S. (1988). Statistical analysis of cointegration vectors. Journal of Economic Dynamics and Control, 12(2–3), 231–254.CrossRef
Johansen, S. (1991). Estimation and hypothesis testing of cointegration vectors in Gaussian vector autoregressive models. Econometrica: Journal of the Econometric Society, 59(6), 1551–1580.CrossRef
Johansen, S. (1994). The role of the constant and linear terms in cointegration analysis of nonstationary variables. Econometric Reviews, 13(2), 205–229.CrossRef
Johansen, S. (1995a). Likelihood-based inference in cointegrated vector autoregressive models. Oxford University Press.CrossRef
Johansen, S. (1995b). A statistical analysis of cointegration for I(2) variables. Econometric Theory, 11(1), 25–59.CrossRef
Juselius, K. (2006). The cointegrated VAR model: methodology and applications. Oxford University Press.CrossRef
Juselius, K. et al. (1992). Testing structural hypotheses in a multivariate cointegration analysis of the PPP and the UIP for UK. Journal of Econometrics, 53(1–3), 211–244.
Kim, Y. (1990). Purchasing power parity in the long run: a cointegration approach. Journal of Money, Credit and Banking, 22(4), 491–503.CrossRef
Lütkepohl, H. (2005). New introduction to multiple time series analysis. Springer Science & Business Media.
Lütkepohl, H., & Saikkonen, P. (1999). Order selection in testing for the cointegrating rank of a VAR process. In R. Engle, & H. White (Eds.), Cointegration, causality, and forecasting: A festschrift in honour of Clive WJ Granger (Chapter 7, pp. 168–199). Oxford: Oxford University Press
MacKinnon, J. G. (1991). Critical values for cointegration tests. In R. F. Engle, & C. W. J. Granger (Eds.), Long-run economic relationships: Readings in cointegration (Chapter 13).
MacKinnon, J. G. (2010). Critical values for cointegration tests. Technical report, Queen’s Economics Department Working Paper.
Murray, M. P. (1994). A drunk and her dog: An illustration of cointegration and error correction. The American Statistician, 48(1), 37–39.
Pedroni, P. (2001). Purchasing power parity tests in cointegrated panels. The Review of Economics and Statistics, 83(4), 727–731.CrossRef
Quintos, C. E., & Phillips, P. C. (1993). Parameter constancy in cointegrating regressions. Empirical Economics, 18(4), 675–706.CrossRef
Rao, B. B. (2007). Cointegration for the applied economist (2nd ed.). Palgrave Macmillan.
Taylor, M. P. (1988). An empirical examination of long-run purchasing power parity using cointegration techniques. Applied Economics, 20(10), 1369–1381.CrossRef
Toda, H. Y., & Yamamoto, T. (1995). Statistical inference in vector autoregressions with possibly integrated processes. Journal of Econometrics, 66(1), 225–250.CrossRef
Zivot, E., & Wang, J. (2007). Modeling financial time series with S-Plus® (Vol. 191). Springer Science & Business Media.
Metadata
Title
Cointegration and VECMs
Author
John D. Levendis
Copyright Year
2023
Book
Time Series Econometrics

Print ISBN: 978-3-031-37309-1

Electronic ISBN: 978-3-031-37310-7

Copyright Year: 2023

DOI
https://doi.org/10.1007/978-3-031-37310-7_12

Premium Partner

    Image Credits