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Cointegration

Cointegration
An econometric or financial model for time-series analysis designed to predict whether two or more variables are linearly related (cf. factor model). If the values of these variables are plotted over time, we would expect them to move together. However, observations on these two variables may have a trend over time or may be drifting apart and are therefore not consistent with the modelling process. Nevertheless a linear regression of y_t on x_t may appear to fit the data very well and to give high t-statistics. Cointegration is a property of a data series which implies that the time series move together.

Formally, a series is integrated order d, denoted as I(d), if after differencing it d times, it is stationary. A stationary time series is a series with a constant mean and variance over time and a covariance which depends on the time gap between the values rather than the period in time to which the observations relates. Consider a vector (x_t) consisting of n variables. These variables are said to be cointegrated if

each of the variables is integrated order I(d), and
there exists a vector z_t = α¹ X_t ∼ I(d − b), where b > 0.

The term α¹ is known as the cointegrating vector and the fact that the variables within X_t are cointegrated is denoted X_t ∼ CI(d, b). The cointegrating vector represents the long-run relationship between the variables in X_t and is therefore referred to as a long-run equilibrium relationship. Engle and Granger (1987) showed that, if we consider a vector which consists of two variables (y_t, x_t) which are CI(1, 1), then there exists an error correction model (ECM) representation of the form:0198288859.cointegration.1.tifwhere ε_t ∼ iid(0, σ²_ε). The ECM explains changes in y_t in terms of (lagged values) of changes in y_t itself, (lagged values) of changes in x_t, and the difference between y and its equilibrium value in the previous period.

A conventional approach to using cointegration (following the development of a theoretical model) is to test each variable for the order of integration I(d). If the variables are all integrated to the same order (d), then test for the existence of a cointegrating vector. If this exists, formulate the ECM representation of the model, estimate the coefficients, and perform nested significance tests to delete non-significant coefficients to derive a parsimonious equation.

Many tests exist for testing for the order of integration of a time series. A common test is the Augmented Dickey–Fuller (ADF) test. To test the null hypothesis: H₀: x_t ∼ I(1) against H₁: x_t ∼ I(0), using the ADF test one would use ordinary least squares regression to estimate:0198288859.cointegration.2.tifwhere as many Δ_{x − i} terms as necessary are added to make η_t stationary. The t-statistic from the null hypothesis that δ₂ = 0 would be compared to the critical values as given by Fuller (1976: 373, in the bottom part of table 8.5.2). If the t-statistic is closer to zero (the critical values are negative) than the critical value, then the null hypothesis cannot be rejected. Next, the restriction δ₁ = δ₂ = 0 has to be tested because stationarity requires that Δ x_t is not time trended. In order to determine this, the F-statistic would be calculated from the restriction δ₁ = δ₂ = 0 and compared with the value for ϕ₃ in Dickey and Fuller (1981: 1063, table vi). If this F-statistic is less than the critical value, the null hypothesis cannot be rejected, and, provided the null that δ₂ = 0 cannot also be rejected, then one would conclude that X_t is integrated of order 1 : I(1). When the number of variables in the X_t vector exceeds two, it is possible that multiple cointegrating vectors exist. To test different null hypotheses, each stating the number of cointegrating vectors, one might use the Johansen Maximum Likelihood procedure (Johansen 1988) as provided in many software routines using this modelling procedure. Cointegration offers an alternative approach to modelling time series data to that provided by ordinary least squares methods.

For details of the methods and significance tests, see: (D. A. Dickey and W. A. Fuller (1981), ‘The likelihood ratio statistics for autoregressive time series with a unit root’, Econometric, 49: 1057–72;) (R. F. Engle and C. W. Granger (1987), ‘Cointegration and error correction: representation, estimation and testing’, Econometrica: 50: 251–76;)(W. A. Fuller (1976), Introduction to Statistical Time Series (Wiley & Sons, New York);) (S. Johansen (1988), ‘Statistical analysis of cointegrating vectors’, Journal of Economic Dynamics and Control, 12: 231–54.)