1.1 Cointegration and common trends

Stock and Watson (1988) show that it is possible to express \(y_{1,t}\) and \(y_{2,t}\), which represent \(I(1)\) variables, as

\[\begin{eqnarray*} y_{1,t} &=& \mu_{y_{1t}} + \upsilon_{y_{1t}} \\ y_{2,t} &=& \mu_{y_{2t}} + \upsilon_{y_{2t}} \end{eqnarray*}\]

where \(\mu_{i,t}\) is the random walk component representing the trend in variable \({\bf{y}}_t\), and \(\upsilon_{i,t}\) is the stationary component. We are then able to multiply \(y_{1,t}\) by \(\beta_1\) and \(y_{2,t}\) by \(\beta_2\) to yield

\[\begin{eqnarray*} \beta_1 y_{1,t}=\beta_1 \mu_{y_{1t}} + \beta_1 \upsilon_{y_{1t}} \\ \beta_2 y_{2,t}=\beta_2 \mu_{y_{2t}} + \beta_2 \upsilon_{y_{2t}} \end{eqnarray*}\]

If these variables are \(CI(1,1)\), then a linear combination of these variables yields;

\[\begin{eqnarray*} \beta_{1}y_{1,t}+\beta_{2} y_{2,t}& = &\beta_{1}(\mu_{y_{1,t}}+\upsilon_{y_{1,t}})+\beta_{2}(\mu_{y_{2,t}}+\upsilon_{y_{2,t}})\\ & =& (\beta_{1}\mu_{y_{1,t}}+\beta_{2}\mu_{y_{2,t}})+(\beta_{1}\upsilon_{y_{1,t}}+\beta_{2}\upsilon_{y_{2,t}}) \end{eqnarray*}\]

If the errors, \((\beta_{1}\upsilon_{y_{1,t}}+\beta_{2}\upsilon_{y_{2,t}})\) are stationary, and the linear combination of the variables \(\beta_{1}y_{1,t}+\beta_{2}y_{2,t}\) are also stationary; then the stochastic trends \((\beta_{1}\mu_{y_{1,t}}+\beta_{2}\mu_{y_{2,t}})\) would need to vanish. Hence, for \(y_{1,t}\) and \(y_{1,t}\) to be \(CI(1,1)\),

\[\begin{eqnarray*} \mu_{y_{1,t}} = \frac{-\beta_{2}\mu_{y_{2,t}}}{\beta_{1}} \end{eqnarray*}\]

This implies that they must have the same stochastic trend up to the scalar \({-\beta_{2}}/{\beta_{1}}\). The essential insight that is provided by Stock and Watson (1988) is that the parameters in the cointegrating vector must purge the trend from the linear combination of the variables. Such a cointegrating vector is unique up to the scalar \({-\beta_{2}}/{\beta_{1}}\).

This analysis could also be extended to the \(n\) variable case, with the aid of the regression

\[\begin{equation*} {\bf{y}}_{t}=\mu_{t} + {\bf{u}}_{t} \end{equation*}\]

where \({\bf{y}}_{t}\) is a vector of \(\{y_{1_{t}},y_{2_{t}}, \ldots, y_{n_{t}}\}\) variables that are integrated of the first-order, and \(\mu_{t}\) is a vector of stochastic trends \(\{\mu_{1_{t}},\mu_{2_{t}}, \ldots, \mu_{n_{t}}\}\), and \({\bf{u}}_{t}\) is an \(n \times 1\) vector of irregular components.

In addition, if we can then express one trend as a linear combination of other trends it means that there exists a vector \(\beta\) such that

\[\begin{equation*} \beta_{1}\mu_{1_{t}}+\beta_{2}\mu_{2_{t}}+ \ldots +\beta_{n}\mu_{n_{t}}=0 \end{equation*}\]

where we once again multiply though by \(\beta\) to get \(\beta {\bf{y}}_{t}=\beta \mu_{t} + \beta {\bf{u}}_{t}\). Since the linear combination of all \(\beta\mu_{t}=0\), we are left with \(\beta {\bf{y}}_{t} = \beta {\bf{u}}_{t}\), where both sides are stationary.

A cointegration model may make use of the term equilibrium which refers to the existence of a long-run relationship. This type of equilibrium can only occur if there is a common stochastic trend amongst the variables (i.e. two variables share a common equilibrium path). These variables will periodically move away from the equilibrium path, but the effect of this will not be permanent (i.e. the errors are stationary). Over time, the variables return towards the equilibrium path and the residuals in the cointegrated model are then described as equilibrium errors.

2.1 Example: Two cointegrated share prices

By way of example, consider two share prices, \(P_1\) and \(P_2\), which are that cointegrated. If we then assume that the gap between the prices during the current period of time is relatively large, when compared to the long-run equilibrium values (i.e. we are currently not at a point of equilibrium). In this case the low priced share \(P_2\) must rise relative to the high priced share \(P_1\). This may be accomplished by either an increase in \(P_2\) or a decrease in \(P_1\), an increase in \(P_1\) with a larger decrease in \(P_2\), or a decrease in \(P_1\) with a smaller decrease in \(P_2\).

The regression that describes the relative movements in the two share prices could then take the form,

\[\begin{eqnarray} P_{1,t}=\beta_{1}P_{2,t}+u_{t} \tag{2.1} \end{eqnarray}\]

If the errors, \(u_t\), are stationary then they may be described by the autoregression,

\[\begin{eqnarray} u_{t}=\phi_{1} u_{t-1}+\varepsilon_{t} \;\;\; \mathsf{with } \; |\phi_{1}| < 1 \tag{2.2} \end{eqnarray}\]

Hence after writing (2.1) as, \(u_{t} = P_{1,t}-\beta_{1}P_{2,t}\), and inserting it into (2.2), we have

\[\begin{eqnarray*} P_{1,t}- \beta_{1}P_{2,t} & = &\phi_{1}(P_{1,t-1}- \beta_{1}P_{2,t-1})+\varepsilon_{t} \\ P_{1,t} & = &\beta_{1}P_{2,t}+\phi_{1}(P_{1,t-1}- \beta_{1}P_{2,t-1})+\varepsilon_{t} \end{eqnarray*}\]

Adding and subtracting \(P_{1,t-1}\) and \(P_{2,t-1}\) on both sides, provides us with,

\[\begin{eqnarray*} \Delta P_{1,t} &=& -(1-\phi_{1})(P_{1,t-1}- \beta_{1}P_{2,t-1})+ (\beta_{1}\Delta P_{2,t} + \varepsilon_{1,t}) \\ &=&\alpha(P_{1,t-1}- \beta_{1}P_{2,t-1})+\varepsilon_{1,t} \end{eqnarray*}\]

where \(\alpha=-(1-\phi_{1})\), while \(\Delta P_{2,t}\) is stationary and \(\varepsilon_{1,t} = (\beta_{1}\Delta P_{2,t} + \varepsilon_{1,t})\). Note that large persistence in the autoregressive error would imply a slow speed of adjustment. This representation is termed the error correction mechanism (ECM), which describes the manner in which the variables return to equilibria. It could also be used to illustrate how the variables are influenced by deviations from equilibrium.

If we assume that both share prices are \(CI(1,1)\) then we could write the respective error correction mechanisms as,

\[\begin{eqnarray} \nonumber \Delta P_{1} = \alpha_{1}(P_{2,{t-1}}-\beta_1 P_{1,{t-1}}) + \varepsilon_{1,_{t}}\\ \Delta P_{2} = \alpha_{2}(P_{2,{t-1}}-\beta_1 P_{1,{t-1}}) + \varepsilon_{2,_{t}} \tag{2.3} \end{eqnarray}\]

Note that the long-term equilibrium would be described by \((P_2 - \beta_1 P_1)\), which is stationary when variables are \(CI(1,1)\). If \(P_1\) is \(I(1)\) then \(\Delta P_1\) is stationary and when the variables are \(CI(1,1)\), then either (or both) of the terms \(\varepsilon_{1,_{t}}\) and \(\varepsilon_{2,_{t}}\) would need to be stationary.

As it currently stands, the expressions in (2.3) are only internally consistent if the two variables, \(P_{1,t}\) and \(P_{2,t}\) are cointegrated. To see this, we firstly note that the left-hand side and the error terms are assumed to be stationary. Therefore, if the \((P_{2,{t-1}}-\beta P_{1,{t-1}})\) term is non-stationary, then the right-hand side of equation (2.3) is non-stationary, which is internally inconsistent with the left-hand side. For the model to be internally consistent, \(P_{1,t}\) and \(P_{2,t}\) must be cointegrated.

Hence the two share prices would need to be cointegrated with the vector \((1, - \beta_1)^{\prime}\), when they are of the order \(CI(1,1)\). The parameters \(\alpha_1\) and \(\alpha_2\) could then be used to describe the speed of adjustment, which provides information on how changes to share prices react to past deviations from the equilibrium path. Small values of \(\alpha_i\) would then imply a relatively unresponsive relationship, where it would take a long time to return to equilibrium.

To ascertain how the long-run equilibrium maintained note that if \(P_{2,t-1}>\beta_1 P_{1,t-1},\) then the combined term \((P_{2,{t-1}}-\beta_1 P_{1,{t-1}})\) is positive. Now since, \(\Delta P_{2,t}\), depends negatively on the combined term through \((1-\phi_{1})\). As \(|\phi_{1}|<1\) by assumption, the total effect will be negative. In other words, if \(P_{2,t}\) is above its long-run equilibrium level relative to \(P_{1,t}\), the error correction mechanism will drive down \(P_{2,t}\) until the long-run equilibrium is restored. Conversely, if \(P_{1,t}\) is higher than its long-run equilibrium level relative to \(P_{2,t}\), then the error correction mechanism will drive up \(P_{2,t}\) until the long-run equilibrium is restored.

2.2 Vector Error Correction Representation

The above example of an error correction model is rather restrictive, in that we have specified an equilibrium relationship where the full adjustment in the respective variables would need to be attributed to a change that is described by the coefficient \(\beta\). The general form of the error correction mechanism allows for slightly richer dynamics interactions between the variables, which can be specified as,

\[\begin{eqnarray} \nonumber \Delta y_{1,t}&=& \gamma_{0} + \alpha_{1} \left[ y_{1,t-1}-\beta_1 y_{2,t-1} \right] + \sum_{i=1}^{K} \zeta_{1,i} \Delta y_{1,t-1} + \sum_{j=1}^{L} \zeta_{2,j} \Delta y_{2,t-1} + \varepsilon_{y_1,{t}} \\ &&\tag{2.4} \\ \nonumber \Delta y_{2,t}&=& \eta_{0} + \alpha_{2} \left[ y_{1,t-1}-\beta_1 y_{2,t-1} \right] + \sum_{i=1}^{K} \xi_{1,i} \Delta y_{2,t-1} + \sum_{j=1}^{L} \xi_{2,j} \Delta y_{1,t-1} + \varepsilon_{y_2,{t}} \\ && \tag{2.5} \end{eqnarray}\]

This representation is termed the vector error correction model (VECM), where it would be possible to show that if both \(\alpha_1\) and \(\alpha_2\) are equal to zero, then there is: no equilibrium relationship, no error-correction, and no cointegration.

Note that if \(y_{1,t}\) and \(y_{2,t}\) are \(CI(1,1)\) then all terms in either (2.4) or (2.5) are \(I(0)\) and statistical inference that makes use of standard \(t\) and \(F\) statistics would be applicable.

2.3 Autoregressive distributed lag model

An alternative one could also employ an indirect specification of an error correction mechanism, which takes the form of the autoregressive distributed lag (ARDL) model. A simple example of such an ARDL model would be,

\[\begin{eqnarray} y_{1,t}=\phi_{1} y_{1,t-1}+\phi_{2} y_{2,t} + \varepsilon_{t} \tag{2.6} \end{eqnarray}\]

As such, the ARDL looks very similar to the autoregressive models studied earlier, except for the inclusion of the \(y_{2,t}\) term on the right-hand side. To see how the ARDL relates to the error correction model, we subtract \(y_{1,t-1}\) and \(\phi_{2}y_{2,t-1}\) from both sides of the equality sign in equation (2.6), to provide

\[\begin{eqnarray*} \Delta y_{1,t}&=&\phi_{2}\Delta y_{2,t}- (1-\phi_{1})y_{1,t-1}+\phi_{2}y_{2,t-1}+\varepsilon_{t} \\ &=& \phi_{2}\Delta y_{2,t}-(1-\phi_{1})(y_{1,t-1}-\frac{\phi_{2}}{1-\phi_{1}}y_{2,t-1})+\varepsilon_{t} \end{eqnarray*}\]

We could then denote the long-term steady-state of the variables as, \(\bar{y}_{1}\) and \(\bar{y}_{2}\), where \(y_{1,t}=y_{1,t-1}=\bar{y}_{1}\). This would allow for use to describe the long-run relationships as,

\[\begin{eqnarray*} \bar{y}_{1}=\phi_{1}\bar{y}_{1}+\phi_{2}\bar{y}_{2} \end{eqnarray*}\]

which could be rearranged as,

\[\begin{eqnarray*} \bar{y}_{1}=\frac{\phi_{2}}{1-\phi_{1}}\bar{y}_{2} \end{eqnarray*}\]

Hence the relationship between \(y_1\) and \(y_2\) is described by \(\frac{\phi_{2}}{1-\phi_{1}}\), which is equivalent to the ECM that we derived earlier (although it is more general). As long as \(|\theta_{1}|<1\), it implies that \((1-\phi_{1})>0\), and the ARDL model corrects in the same manner as the error correction model that was described previously. The choice of whether to specify an error correction model, or an ARDL model where the equilibrium property is implicitly defined, is mostly a matter of convenience and interpretation.

2.4 Summary

Before we proceed with a discussion that relates to the estimation of these models, we should summarise a few important concepts. Firstly, we should note that cointegration refers to linear combinations of non-stationary variables. It is possible that there are nonlinear cointegrating relationships, but we currently don’t know how to test for this. However, we can model regime-switching cointegrating relationships using the methods discussed in Balke and Fomby (1997), as well as fractionally integrated cointegrating relationships, as discussed in Johansen and Nielsen (2012). Cointegrating vectors are unique up to a scalar, for every \(\beta_1 , \beta_2, \ldots\) there exists \(\lambda \beta_1, \lambda \beta_2, \ldots\), where \(\lambda\) is the scalar.

It is also important to reiterate that all of the variables must be integrated of the same order, where it is usually the case that a set of \(I(d)\) variables are not cointegrated. In addition, we should note that when two variables are integrated of different orders they cannot be cointegrated, however, it is possible to have multi-cointegration (i.e. \(b=2\)).

In addition, if \({\bf{y}}_{t}\) has \(n\) nonstationary components then there could be \(n-1\) linear cointegrating vectors. Therefore, if \({\bf{y}}_{t}\) has two variables then there can only be one cointegrating vector, and the number of cointegrating vectors is termed the cointegrating rank.

There is also a special case where if you have three variables and two are \(I(2)\) and one is \(I(1)\), it may be possible (but unlikely) that the two \(I(2)\) variables are \(CI(2,1)\). The remaining \(I(1)\) variable may then share a common stochastic trend with other two variables, whereupon the system will be stationary. Obviously, the chance of this occurring in practice is very small.

4.1 Vector error correction model (VECM)

In the same manner as for the single equation, we can rewrite the VAR(1) model from (4.1) as a vector error correction model (VECM)

\[\begin{eqnarray} \nonumber \Delta {\bf{y}}_{t} & = & \mu+(\Pi_{1}-I){\bf{y}}_{t-1}+{\bf{u}}_{t} \\ & = & \mu+\Pi {\bf{y}}_{t-1}+{\bf{u}}_{t} \;\;\; \mathsf{where } \; \Pi=(\Pi_{1}-I) \tag{4.2} \end{eqnarray}\]

With two variables, we saw that there could exist at most one linear stationary relationship between the variables. With \(n\) variables the number of of linear combinations of the variables in \({\bf{y}}_{t}\) that are stationary will provide us with information on the number of cointegration vectors. There are three possibilities that exist, which are considered below:

If the variables in the VAR(\(p\)) are cointegrated, then the VECM would include long-run cointegration and speed of adjustment parameters. Therefore, we can decompose \(\Pi\) as,

\[\begin{eqnarray*} \Pi=\alpha\beta^{\prime} \end{eqnarray*}\]

where \(\alpha\) and \(\ \beta\) are both dimensions \(n\times r\). We say that the matrix \(\beta\) is a matrix of cointegration parameters, so that the linear combination of \(\beta^{\prime} {\bf{y}}_{t}\) is stationary. In addition, each of the \(r\) rows in \(\beta^{\prime}{\bf{y}}_t\) is a cointegrated (long-run) relation that induces stability.

The matrix \(\alpha\) is contains the speed of adjustment parameters, which account for the time that it takes to move back to equilibrium (i.e. the speed at which the error corrects following a movement away from equilibrium). For instance, under the maintained assumption about cointegration the bivariate case examined above, (4.2) can be written as:

\[\begin{eqnarray*} \left[ \begin{array} [c]{c} \Delta y_{1,t}\\ \Delta y_{2,t} \end{array} \right] =\left[ \begin{array} [c]{c} \mu_{1}\\ \mu_{2} \end{array} \right] +\left[ \begin{array} [c]{c} \alpha_{1}\\ \alpha_{2} \end{array} \right] \Big[\beta_{1}\mathsf{ \ }\beta_{2}\Big] \left[ \begin{array} [c]{c} y_{1,t-1}\\ y_{2,t-1} \end{array} \right] +\left[ \begin{array} [c]{c} u_{1,t}\\ u_{2,t} \end{array} \right] \end{eqnarray*}\]

which could be written as,

\[\begin{eqnarray*} \nonumber \Delta y_{1,t} & =\mu_{1}+\alpha_{1}(\beta_{1}y_{1,t-1}+\beta_{2}y_{2,t-1})+u_{1,t}\\ \Delta y_{2,t} & =\mu_{2}+\alpha_{2}(\beta_{1}y_{1,t-1}+\beta_{2}y_{2,t-1})+u_{2,t} \end{eqnarray*}\]

The cointegration relationship (or combination) \(\beta^{\prime} {\bf{y}}_{t}\) would then be given by,

\[\begin{eqnarray*} \beta^{\prime}{\bf{y}}_{t}=\beta_{1}y_{1,t}+\beta_{2}y_{2,t}\sim I(0) \end{eqnarray*}\]

Hence, the nonstationary variables in the \({\bf{y}}_{t}\) vector are cointegrated if there is a linear combination of these variables that is stable (stationary). Such a linear combination of variables could be related to economic theory and is often referred to as a long-run equilibrium relationship. The intuition is that the variables will drift together along some form of long-run equilibrium path. These variables may of course move away from this path over certain periods of time, but there will always be forces that move the variables back towards the long-run equilibrium path.

So far we have only investigated a situation with two variables, implying a maximum of one cointegration relationship. However, when \(n=3\), we can potentially have \(r=2\) equilibrium relationships. To see this, assume that we have \(n=3\) and \(r=2\). The respective matrices could then be written as,

\[\begin{eqnarray*} \nonumber \left[ \begin{array} [c]{c} \Delta y_{1,t}\\ \Delta y_{2,t}\\ \Delta y_{3,t} \end{array} \right] &=& \left[ \begin{array} [c]{c} \mu_{1}\\ \mu_{2}\\ \mu_{3} \end{array} \right] +\left[ \begin{array} [c]{cc} \alpha_{11} & \alpha_{12}\\ \alpha_{21} & \alpha_{22}\\ \alpha_{31} & \alpha_{32} \end{array} \right] \ldots \\ && \left[ \begin{array} [c]{ccc} \beta_{11} & \beta_{21} & \beta_{31}\\ \beta_{12} & \beta_{22} & \beta_{32} \end{array} \right] \left[ \begin{array} [c]{c} y_{1,t-1}\\ y_{2,t-1}\\ y_{3,t-1} \end{array} \right] +\left[ \begin{array} [c]{c} u_{1,t}\\ u_{2,t}\\ u_{3,t} \end{array} \right] \end{eqnarray*}\]

In this case the the three variables allow for two cointegration relationships between the variables, which we denote for \(\beta_{1}^{\prime} {\bf{y}}_{t-1}\) and \(\beta_{2}^{\prime} {\bf{y}}_{t-1}\). These could be expanded as,

\[\begin{eqnarray*}\nonumber \beta_{1}^{\prime}{\bf{y}}_{t} & =\beta_{11}y_{1,t}+\beta_{21}y_{2,t}+\beta_{31}y_{3,t}\sim I(0)\\ \beta_{2}^{\prime}{\bf{y}}_{t} & =\beta_{12}y_{1,t}+\beta_{22}y_{2,t}+\beta_{32}y_{3,t}\sim I(0) \end{eqnarray*}\]

Hence, we now have two linear relationships, \(\beta_{1}\) and \(\beta_{2}\), between the three variables, \(y_{1}\), \(y_{2}\) and \(y_{3}\) which are able to induce stationarity.

Such a finding for the VAR(1) model could be generalised to a VAR(\(p\)) in the following way,

\[\begin{eqnarray*} \Delta {\bf{y}}_{t}=\mu+\alpha\beta {\bf{y}}_{t-1}+\Gamma_{1}\Delta {\bf{y}}_{t-1}+\Gamma_{2}\Delta {\bf{y}}_{t-2}+ \ldots \\ \nonumber +\Gamma_{p-1}\Delta {\bf{y}}_{t-p-1} + {\bf{u}}_{t} \end{eqnarray*}\]

where we have added \(p\) lags of the vector of variables.