Intertemporal Cointegration Model: A New Approach to the Lead–Lag Relationship Between Cointegrated Time Series

Abstract

We generalize the single regression-type cointegration model of Engle and Granger (Econometrica 55(2):251–276, 1987) from the lead–lag relationship perspective. A leading series is introduced into the model as an independent variable to express the long-run relationship between the leading and lagging variables if the disturbance term is stationary. The theoretical analysis shows that the estimators of the coefficients with true lead–lag intervals have stochastic characteristics equivalent to those of Engle and Granger (1987). Monte Carlo simulations suggest that inappropriate interval selection leads to seriously biased estimators and that the identification of the lead–lag intervals is successful when using the adjusted coefficient of determination. This accuracy is caused by the difference in order between the true interval and candidates, which increases the variance of disturbance proportionally. The farther the candidates are from the true interval, the more autocorrelation increases; further, it cannot be absorbed by the unit root test, which considers the autocorrelations of disturbance. This causes a severe deterioration in the power of cointegration testing. Therefore, cointegration analysis without considering the lead–lag interval may lead economists to overlook the important long-run relationship between the pair of variables.

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Data Availability Statement

All data used in the manuscript are published and downloadable from ESRI, cabinet office, Japan: http://www.esri.cao.go.jp/en/stat/di/di-e.html

Notes

  1. 1.

    Financial econometrics has also focused on this type of time series with lead–lags. Herbst et al. (1987) investigate lead–lags between stock indices and futures using cross-autocorrelations. Lo and Mackinlay (1990) and Kanas and Kouretas (2004) study lead–lags between portfolios consisting of size-sorted firms with a focus on cross-autocorrelations. Specifically, Kanas and Kouretas (2004) suggest that out-of-sample forecasting enhances predictions using an error correction model with an autoregressive distributed lag (ADL) approach. For ADL models, see Hassler and Wolters (2006).

  2. 2.

    See http://www.esri.cao.go.jp/en/stat/di/di2e.html.

  3. 3.

    We set the number of iterations as \(G = 10000\) in the Monte Carlo simulations.

  4. 4.

    For the other true case of s, the empirical distributions of \(\hat{\rho }_{s,r}\) are the same as for \(s = 5\) adopted here; hence, the relative difference between s and r does not depend on the absolute value of s.

  5. 5.

    The lags in the ADF test are selected by the AIC and we use the empirical distribution of \(\Phi _1\) in Table IV of Dickey and Fuller (1981) for the drawn samples \(\mathbf {x}_1^{N-r}\) and \(\mathbf {y}_r^N\).

  6. 6.

    These tests are based on the \(\hat{Z}_t\) statistics; the critical values are shown in Table IIa of Phillips and Ouliaris (1990). The lags in the tests are selected using the AIC.

  7. 7.

    We cannot see the black surface of \(N = 1000\) in Fig. 4. As they are almost zero, they are beneath the red one.

  8. 8.

    Precise information on the reference dates is reported on the ESRI website, http://www.esri.cao.go.jp/.

References

  1. Belke, A. (2000). Political business cycles in the German labour market? Empirical tests in the light of the Lusas-Critique. Public Choice, 104, 225–283.

    Article  Google Scholar 

  2. Bry, G., & Boschan, C. (1971). Cyclical analysis of time series: Selected procedures and computer programs. Cambridge: National Bureau of Economic Research, Inc.

    Google Scholar 

  3. Dickey, D. A., & Fuller, W. A. (1981). Likelihood ratio statistics for autoregressive time series with a unit root. Econometrica, 49(4), 1057–1072.

    Article  Google Scholar 

  4. Emerson, J. (2007). Cointegration analysis and the choice of lag length. Applied Economics Letters, 14, 143–159.

    Article  Google Scholar 

  5. Engle, R. F., & Granger, C. W. J. (1987). Co-integration and error correction: Representation, estimation, and testing. Econometrica, 55(2), 251–276.

    Article  Google Scholar 

  6. Engle, R. F., & Yoo, B. S. (1987). Forecasting and testing in co-integrated systems. Journal of Econometrics, 35, 143–159.

    Article  Google Scholar 

  7. Fuller, W. A. (1996). Introduction to statistical time series. New York, NY: Wiley.

    Google Scholar 

  8. Granger, C. W. J. (1981). Some properties of time series data and their use in econometric model specification. Journal of Econometrics, 16, 121–130.

    Article  Google Scholar 

  9. Hamilton, J. D. (1989). A new approach to the economic analysis of nonstationary time series and the business cycle. Econometrica, 57(2), 357–384.

    Article  Google Scholar 

  10. Hassler, U., & Wolters, J. (2006). Autoregressive distributed lag models and cointegration. In O. Hübler & J. Frohn (Eds.), Modern econometric analysis (pp. 57–72). Berlin, Heidelberg: Springer.

    Google Scholar 

  11. Herbst, A. F., McCormack, J. P., & West, E. N. (1987). Investigation of a lead-lag relationship between spot stock indices and their futures contracts. Journal of Futures Markets, 7(4), 373–381.

    Article  Google Scholar 

  12. Johansen, S. (1991). Estimation and hypothesis testing of cointegration vectors in Gaussian vector autoregressive models. Econometrica, 59(6), 1551–1580.

    Article  Google Scholar 

  13. Kanas, A., & Kouretas, G. P. (2004). A cointegration approach to the lead-lag effect among size-sorted equity portfolios. International Review of Economics and Finance, 14, 181–201.

    Article  Google Scholar 

  14. Lo, A. W., & Mackinlay, A. C. (1990). When are contrarian profits due to stock market overreaction? Review of Financial Studies, 3(2), 175–205.

    Article  Google Scholar 

  15. Phillips, P. C. B., & Ouliaris, S. (1990). Asymptotic properties of residual based tests for cointegration. Econometrica, 58(1), 165–193. 335–346.

    Article  Google Scholar 

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Correspondence to Takashi Oga.

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All statistical computations are executed via OxMetrics 8. Doornik, J.A. (2007), Object-Oriented Matrix Programming Using Ox, 3rd ed. London: Timberlake Consultants Press and Oxford: www.doornik.com.

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Oga, T. Intertemporal Cointegration Model: A New Approach to the Lead–Lag Relationship Between Cointegrated Time Series. J Bus Cycle Res (2021). https://doi.org/10.1007/s41549-021-00052-8

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Keywords

  • Cointegration
  • Lead–lag
  • Model selection
  • Business cycle

JEL Classification

  • C22
  • C32
  • C52