The Granger Representation Theorem And Error Correction

From Cointegration to Error Correction

When several economic time series each have a unit root, their levels may still move together in the long run. Cointegration describes this pattern: a particular linear combination of the series is stationary even though the individual series are not. The Granger Representation Theorem gives this fact a dynamic counterpart. Under appropriate conditions, the system can be represented in terms of stationary changes and lagged deviations from long-run equilibrium.

This distinction matters for modeling. A model written only in stationary changes can omit information about whether the variables were previously out of equilibrium. An error-correction representation keeps that information while allowing the levels themselves to contain common stochastic trends. In the equations below, deterministic terms are suppressed to focus on this connection; their placement can affect the interpretation of the long-run relation.

Definition: Let \(y_t\) be an \(n\)-vector of time series, each integrated of order one, or \(I(1)\). Its cointegration rank is the number \(r\) of linearly independent vectors \(\beta\) for which \(\beta'y_t\) is \(I(0)\). A system with \(0<r<n\) has \(r\) independent stationary long-run relations and \(n-r\) common stochastic trends.

A cointegrating vector is written as an \(n\)-vector \(\beta\); its stationary combination is \(\beta'y_t\). With rank \(r\), collect \(r\) linearly independent cointegrating vectors as columns of an \(n\times r\) matrix \(\beta\). The corresponding vector of equilibrium errors is \(\beta'y_t\), an \(r\)-vector. The matrix \(\alpha\), also \(n\times r\), describes how those errors enter the changes in the variables.

The Error-Correction Representation

For a finite-order VAR in levels,

y_t=A_1y_{t-1}+\cdots+A_py_{t-p}+\varepsilon_t,

where \(\varepsilon_t\) is a stationary innovation vector, an algebraic rearrangement gives the error-correction form

\Delta y_t=\Pi y_{t-1}+\sum_{j=1}^{p-1}\Gamma_j\Delta y_{t-j}+\varepsilon_t, \qquad \Pi=\sum_{i=1}^{p}A_i-I, \qquad \Gamma_j=-\sum_{i=j+1}^{p}A_i.

Here \(\Delta y_t=y_t-y_{t-1}\). The matrix \(\Pi\) collects the long-run information: its rank is \(r\) under the usual cointegrated \(I(1)\) setup. If \(0<r<n\), it can be factored as \(\Pi=\alpha\beta'\), where both \(\alpha\) and \(\beta\) have \(r\) linearly independent columns. Substitution gives

\Delta y_t=\alpha(\beta'y_{t-1})+ \sum_{j=1}^{p-1}\Gamma_j\Delta y_{t-j}+\varepsilon_t.

The term \(\beta'y_{t-1}\) records last period's deviations from the long-run relations; multiplying by \(\alpha\) maps those deviations into the current changes. The lagged differences describe additional short-run dynamics. The representation does not say that every variable must move toward equilibrium in every period: innovations and other lagged changes also affect its movement.

Theorem: In a regular finite-order VAR with stationary innovations, suppose the \(n\)-vector \(y_t\) is \(I(1)\), has cointegration rank \(r\) with \(0<r<n\), and has no \(I(2)\) components. Under the usual nondegeneracy and stability conditions—there are \(n-r\) non-explosive unit-root directions and the remaining dynamic roots are stable—the system has an error-correction representation with \(\Pi=\alpha\beta'\) of rank \(r\). Conversely, an error-correction system with rank \(r\) and suitable root conditions has \(r\) stationary cointegrating relations and \(n-r\) common stochastic trends. Rank by itself, without the root conditions, is not enough to guarantee this behavior.

Why the Representation Follows

The algebraic rearrangement and the cointegration result play different roles. Rearranging a VAR always produces a matrix \(\Pi\) and short-run matrices \(\Gamma_j\). The theorem's substantive claim is that, when the levels are \(I(1)\) and cointegrated under the stated conditions, the long-run matrix has reduced rank and its column and row spaces can be represented by adjustment and cointegrating vectors.

A short justification clarifies why the error term is the lagged equilibrium deviation. If \(\Pi=\alpha\beta'\), then \(\Pi y_{t-1}=\alpha(\beta'y_{t-1})\). By definition, each component of \(\beta'y_t\) is stationary. Also, if \(y_t\) is \(I(1)\), its first differences and any fixed number of their lags are stationary. Thus every systematic term on the right-hand side of the error-correction equation is stationary, as is \(\Delta y_t\), when the innovations are stationary.

The converse requires more than this substitution. A rank-\(r\) matrix gives \(r\) directions in which levels enter the equation, but the system's other dynamic roots determine whether the remaining movements are stable or explosive. With the required stability conditions, the \(r\) combinations represented by \(\beta'y_t\) are stationary, while the \(n-r\) directions not pinned down by those relations can carry permanent stochastic movements. These are the common trends. In other words, cointegration restricts the long-run behavior; stability ensures that short-run dynamics do not overturn that conclusion.

Identify the integration structure.
Establish that the level variables are \(I(1)\), rather than assuming that a levels VAR automatically has this property.

Determine the long-run rank.
For cointegration rank \(r\), the long-run matrix \(\Pi\) has rank \(r\), where \(0<r<n\) in the cointegrated case considered here.

Factor the long-run matrix.
Write \(\Pi=\alpha\beta'\). The columns of \(\beta\) define stationary long-run combinations; \(\alpha\) maps their lagged deviations into current changes.

Check the dynamics.
Verify the relevant root and nondegeneracy conditions. The rank factorization alone does not establish stable transitory adjustment or the number of common stochastic trends.

Worked Interpretations

Worked Example: A Common Trend and a Stationary Spread

Suppose two hypothetical series are generated by \(x_t=x_{t-1}+\eta_t\), where \(\eta_t\) is a stationary innovation, and by stationary processes \(u_t\) and \(v_t\):

y_{1t}=x_t+u_t, \qquad y_{2t}=2x_t+v_t.

Assume the stationary processes are well behaved and that neither series' common-trend loading is canceled. Each \(y_{it}\) is then \(I(1)\), because it includes the random walk \(x_t\). Consider the candidate vector \(\beta=(1,-1/2)'\). Its combination is

\beta'y_t=y_{1t}-\tfrac12 y_{2t} =(x_t+u_t)-\tfrac12(2x_t+v_t) =u_t-\tfrac12 v_t.

The \(x_t\) terms cancel, leaving a stationary combination. Thus these two \(I(1)\) series are cointegrated with rank one, provided there is no additional independent cointegrating relation. Their levels share one stochastic trend, while their spread \(y_{1t}-\tfrac12y_{2t}\) is stationary. This construction establishes the long-run relation; it does not, by itself, specify the short-run adjustment coefficients.

Worked Example: Factoring a Long-Run Matrix

Consider a hypothetical bivariate error-correction system with no lagged differences:

\Delta y_t= \begin{pmatrix} -0.3&0.3\\ 0.1&-0.1 \end{pmatrix}y_{t-1}+\varepsilon_t.

The long-run matrix can be factored using \(\beta=(1,-1)'\) and \(\alpha=(-0.3,0.1)'\):

\alpha\beta' = \begin{pmatrix}-0.3\\0.1\end{pmatrix} \begin{pmatrix}1&-1\end{pmatrix} = \begin{pmatrix} -0.3&0.3\\ 0.1&-0.1 \end{pmatrix}.

Its rank is one, and the error-correction term is \(\beta'y_{t-1}=y_{1,t-1}-y_{2,t-1}\). If the previous-period spread is \(4\) units, the systematic long-run contribution to \(\Delta y_t\), before adding the innovation, is

\alpha(\beta'y_{t-1}) = \begin{pmatrix}-0.3\\0.1\end{pmatrix}(4) = \begin{pmatrix}-1.2\\0.4\end{pmatrix}.

The implied contribution to the change in the spread is \(-1.2-0.4=-1.6\) units: the spread's systematic change is in the direction that reduces its positive deviation. This is a conditional description of the specified model, not a claim that either variable causes the other. The rank factorization identifies a candidate equilibrium relation; the theorem's dynamic conditions must still be checked before concluding that the system has the stated \(I(1)\) and stationary-transitory structure.

Worked Example: Two Cointegrating Relations in Three Series

For a hypothetical three-variable system, let the columns of \(\beta\) encode \(y_1-y_3\) and \(y_2-y_3\), and let \(\alpha=-0.2\beta\):

\beta= \begin{pmatrix}1&0\\0&1\\-1&-1\end{pmatrix}, \qquad \alpha= \begin{pmatrix}-0.2&0\\0&-0.2\\0.2&0.2\end{pmatrix}.

Multiplication gives the rank-two long-run matrix

\Pi=\alpha\beta' = \begin{pmatrix} -0.2&0&0.2\\ 0&-0.2&0.2\\ 0.2&0.2&-0.4 \end{pmatrix}.

The two equilibrium errors are \(d_1=y_1-y_3\) and \(d_2=y_2-y_3\). At the hypothetical previous-period values \(y_{t-1}=(8,5,2)'\), these are \(d_1=6\) and \(d_2=3\). The long-run contribution to the changes is

\Pi y_{t-1} = \begin{pmatrix}-1.2\\-0.6\\1.8\end{pmatrix}.

Consequently, the contribution to the first spread's change is \(-1.2-1.8=-3\) units, and the contribution to the second spread's change is \(-0.6-1.8=-2.4\) units. These calculations show how two stationary relations can enter the adjustment of three variables. In a system with suitable stable roots, rank two leaves one common stochastic trend; the matrix calculation alone does not prove those stability conditions.

What the Theorem Does—and Does Not—Establish

The theorem is useful because it gives a coherent way to represent both short-run movements and long-run restrictions. The differenced variables can be stationary and modeled with standard stationary dynamics, while the lagged equilibrium errors preserve information about the levels' long-run relationship. Omitting those errors when the variables are cointegrated can leave systematic equilibrium deviations out of the model.

The distinction between the matrices is also important. The columns of \(\beta\) span the cointegrating relations, while the columns of \(\alpha\) describe how those relations enter changes across the variables. A factorization is not unique: for any invertible \(r\times r\) matrix \(H\), \(\alpha\beta'=(\alpha H)(\beta H^{-T})'\). The particular coordinates for the \(r\) relations can therefore change without changing \(\Pi\). The long-run space and the product \(\Pi\), rather than an arbitrary normalization alone, are central to the representation.

A common mistake is to reason, “\(\Pi\) has rank \(r\), so the system must be a stable cointegrated system with \(n-r\) common trends.” This is incomplete. Rank describes the dimension of the long-run matrix, but the dynamic roots and integration assumptions matter too. A second mistake is to call the levels stationary because the error-correction terms are stationary. The levels can remain \(I(1)\); it is the specified linear combinations and the changes that are stationary under the theorem's conditions.

Key takeaway: Under the Granger Representation Theorem's integration, rank, and stability conditions, cointegration and error correction are two descriptions of the same system: stationary long-run combinations enter the dynamics of stationary changes, while the levels may retain common stochastic trends.

Check Your Understanding

Answer each question by relating the algebra to the long-run and short-run behavior.

If \(n=4\) and the cointegration rank is \(r=2\), how many independent cointegrating relations and common stochastic trends does the theorem imply under its conditions?
In \(\Delta y_t=\alpha\beta'y_{t-1}+\cdots\), which matrix defines the long-run combinations, and what role does the other matrix play?
Why does a rank-\(r\) long-run matrix not, by itself, establish stable adjustment dynamics?
Suppose two \(I(1)\) variables have a stationary difference. What does that imply about their cointegration rank, assuming there is no additional independent relation?
Why is it incorrect to infer that the individual levels are stationary merely because the error-correction term is stationary?