Reduced-Rank VARs And Cointegration Rank

Why the Long-Run Matrix Matters

A vector autoregression describes how several variables depend on their own past values and on the past values of the other variables. When the variables are integrated of order one, writing the model only in levels can obscure an important distinction: some combinations of the variables may be stationary even though the individual series are not. The vector error-correction form makes this distinction visible through one matrix, the long-run matrix.

The rank of that matrix determines how many independent stationary combinations can be represented in the long-run relation, provided the system satisfies the usual conditions for an I(1) cointegrated VAR. A reduced-rank factorization then separates the cointegrating combinations from the way they enter the equations for changes.

From a VAR in Levels to a VECM

Let \(y_t\) be an \(m\)-dimensional vector of time-series variables. For clarity, begin with a VAR(\(p\)) with no deterministic terms:

y_t=A_1y_{t-1}+A_2y_{t-2}+\cdots+A_py_{t-p}+u_t.

Each \(A_i\) is an \(m\times m\) coefficient matrix, and \(u_t\) is an \(m\)-vector of innovations. The following rearrangement is an algebraic identity; it does not require a particular probability distribution for \(u_t\).

Definition: The vector error-correction form of this VAR is \(\Delta y_t=\Pi y_{t-1}+\sum_{j=1}^{p-1}\Gamma_j\Delta y_{t-j}+u_t\), where \(\Pi=\sum_{i=1}^{p}A_i-I_m\) and \(\Gamma_j=-\sum_{i=j+1}^{p}A_i\). The matrix \(\Pi\) is the long-run matrix; each \(\Gamma_j\) is a short-run coefficient matrix.

Here \(I_m\) is the \(m\times m\) identity matrix, and \(\Delta y_t=y_t-y_{t-1}\). The formula shows how changes at time \(t\) depend on lagged changes as well as the previous level \(y_{t-1}\). In this tutorial the model has no deterministic terms, so that the rank interpretation can be stated without additional cases.

Deriving the Error-Correction Form

Subtract \(y_{t-1}\) from both sides of the levels VAR. Then express each lagged level \(y_{t-i}\) as \(y_{t-1}\) minus the intervening changes: \(y_{t-i}=y_{t-1}-\sum_{j=1}^{i-1}\Delta y_{t-j}\). For \(i=1\), the sum is empty. Substitution gives

\begin{aligned} \Delta y_t &=\left(\sum_{i=1}^{p}A_i-I_m\right)y_{t-1} -\sum_{i=2}^{p}A_i\sum_{j=1}^{i-1}\Delta y_{t-j}+u_t\\ &=\left(\sum_{i=1}^{p}A_i-I_m\right)y_{t-1} -\sum_{j=1}^{p-1}\left(\sum_{i=j+1}^{p}A_i\right)\Delta y_{t-j}+u_t. \end{aligned}

In the second line, terms involving the same lagged difference are collected together. For a fixed \(j\), that difference appears in the level representation for \(y_{t-i}\) exactly when \(i\geq j+1\), giving the inner sum shown. Identifying the coefficient matrices yields the stated \(\Pi\) and \(\Gamma_j\).

Worked Example: Rewriting a VAR(2)

Consider a hypothetical bivariate VAR(2) with \(A_1=\begin{pmatrix}0.6&0.3\\0.2&0.7\end{pmatrix}\) and \(A_2=\begin{pmatrix}0.1&0\\0&0.1\end{pmatrix}\). Find its VECM coefficient matrices.

For a VAR(2), \(\Pi=A_1+A_2-I_2\) and \(\Gamma_1=-A_2\). First add the lag matrices and subtract the identity:

\Pi= \begin{pmatrix}0.6&0.3\\0.2&0.7\end{pmatrix} +\begin{pmatrix}0.1&0\\0&0.1\end{pmatrix} -\begin{pmatrix}1&0\\0&1\end{pmatrix} = \begin{pmatrix}-0.3&0.3\\0.2&-0.2\end{pmatrix}.

The short-run matrix is \(\Gamma_1=-A_2=\begin{pmatrix}-0.1&0\\0&-0.1\end{pmatrix}\). Thus the VECM is \(\Delta y_t=\begin{pmatrix}-0.3&0.3\\0.2&-0.2\end{pmatrix}y_{t-1} -\begin{pmatrix}0.1&0\\0&0.1\end{pmatrix}\Delta y_{t-1}+u_t\). The negative sign in the displayed lagged-difference term is already included in \(\Gamma_1\); equivalently, its coefficient is the matrix with diagonal entries \(-0.1\).

Rank Factorization and Cointegrating Relations

A matrix of rank \(r\) can be written as the product of a matrix with \(r\) columns and a matrix with \(r\) rows. Applied to \(\Pi\), this gives a factorization that is especially useful for integrated variables.

Definition: If \(\operatorname{rank}(\Pi)=r\), a rank factorization is \(\Pi=\alpha\beta'\), where \(\alpha\) and \(\beta\) are \(m\times r\) matrices, each with full column rank \(r\). The columns of \(\beta\) span the space of cointegrating vectors, and \(\beta'y_{t-1}\) is the vector of the corresponding level combinations.

To see why this factorization exists, choose \(r\) linearly independent columns of \(\Pi\) and place them in an \(m\times r\) matrix \(\alpha\). Every column of \(\Pi\) is a linear combination of these basis columns, so there is an \(r\times m\) matrix \(C\) such that \(\Pi=\alpha C\). Since \(\Pi\) has rank \(r\), \(C\) has rank \(r\). Set \(\beta'=C\); then \(\Pi=\alpha\beta'\), with both factors having rank \(r\).

For an \(I(1)\) system, a nonzero vector \(b\) is a cointegrating vector when the scalar combination \(b'y_t\) is stationary even though the component series in \(y_t\) are nonstationary. Under the standard I(1) cointegrated-VAR conditions—variables are at most I(1), there are no I(2) components, and the system has the required regularity of its unit-root dynamics—the rank \(r\) of \(\Pi\) is the cointegration rank. There are \(r\) independent stationary level combinations, represented by the columns of \(\beta\), and \(m-r\) common stochastic trends.

The rank factorization is not unique. For any nonsingular \(r\times r\) matrix \(H\), define \(\beta^*=\beta H\) and \(\alpha^*=\alpha H^{-T}\). Then \(\alpha^*(\beta^*)'=\alpha H^{-T}H'\beta'=\alpha\beta'=\Pi\). The basis used to describe the cointegrating space can therefore change without changing the long-run matrix or the space itself.

Worked Example: A Rank-One Long-Run Matrix

For the bivariate long-run matrix from the VAR(2) example, compute its rank and give a rank factorization.

The determinant is \((-0.3)(-0.2)-(0.3)(0.2)=0.06-0.06=0\), so the rank is less than two. The matrix is not the zero matrix, so its rank is one. A factorization is

\Pi= \begin{pmatrix}-0.3\\0.2\end{pmatrix} \begin{pmatrix}1&-1\end{pmatrix} =\alpha\beta'.

The implied level combination is \(\beta'y_{t-1}=y_{1,t-1}-y_{2,t-1}\). If the variables form a regular I(1) system satisfying the conditions above, this is one cointegrating relation and the system has \(2-1=1\) common stochastic trend. The rank calculation itself is an algebraic result; stationarity of the combination follows from the cointegrated-system assumptions, not from the determinant calculation alone.

What the Three Rank Cases Mean

The rank of an \(m\times m\) long-run matrix can be zero, intermediate, or full. The interpretation below is for an I(1) system satisfying the regularity conditions just described; without those conditions, matrix rank alone is not a complete diagnosis of the time-series properties.

Rank	Long-run structure under the I(1) conditions	Interpretation
\(r=0\)	\(\Pi=0\)	No cointegrating level combinations; \(m\) common stochastic trends.
\(0<r<m\)	\(\Pi=\alpha\beta'\) with \(r\) columns in each factor	\(r\) cointegrating relations and \(m-r\) common stochastic trends.
\(r=m\)	\(\Pi\) has full rank	No common stochastic trends in the standard framework; the levels are stationary when the remaining dynamics satisfy the required stability conditions.

In the intermediate case, the VECM’s level term can be written as \(\alpha(\beta'y_{t-1})\). The vector \(\beta'y_{t-1}\) contains the deviations in the cointegrating combinations, while \(\alpha\) distributes those combinations across the equations for \(\Delta y_t\). This is the structural reason the reduced-rank form is called an error-correction representation. The exact sizes and interpretations of individual adjustment coefficients are a separate issue.

Worked Example: Two Relations in a Three-Variable System

Consider the hypothetical long-run matrix \(\Pi=\begin{pmatrix}-0.2&0.2&0\\0.1&-0.2&0.1\\0&0.2&-0.2\end{pmatrix}\). Factor it and identify the rank.

Choose \(\alpha=\begin{pmatrix}-0.2&0\\0.1&-0.1\\0&0.2\end{pmatrix}\) and \(\beta=\begin{pmatrix}1&0\\-1&1\\0&-1\end{pmatrix}\). Then \(\beta'=\begin{pmatrix}1&-1&0\\0&1&-1\end{pmatrix}\), and multiplication gives

\alpha\beta' = \begin{pmatrix} -0.2&0.2&0\\ 0.1&-0.2&0.1\\ 0&0.2&-0.2 \end{pmatrix} =\Pi.

Both factors have two linearly independent columns, so \(\operatorname{rank}(\Pi)=2\). The two level combinations are \(y_{1,t}-y_{2,t}\) and \(y_{2,t}-y_{3,t}\). If the variables satisfy the standard I(1) conditions, these are two independent cointegrating relations, leaving \(3-2=1\) common stochastic trend. The matrix arithmetic establishes the factorization and rank; it does not by itself establish that either combination is stationary.

Common Pitfall: Treating Rank as a Cointegration Test

A frequent error is to calculate that \(\Pi\) has rank one and conclude, without qualification, that a data set contains exactly one stationary linear combination. The calculation only identifies the algebraic rank of an estimated or specified matrix. The cointegration interpretation additionally requires the I(1) framework and its regularity conditions. If variables are stationary, I(2), or the system has other problematic unit-root dynamics, the same rank statement need not have the stated interpretation.

A related mistake is to assume that the columns of \(\Pi\) themselves are cointegrating vectors. In general they are not. The columns of \(\beta\) span the relevant level-combination space in the factorization \(\Pi=\alpha\beta'\); \(\alpha\) supplies the other factor. For a given \(\Pi\), the factorization and its rank must be examined rather than assigning the role of \(\beta\) directly to \(\Pi\).

Key takeaway: Rewriting the levels VAR exposes \(\Pi=\sum_i A_i-I_m\). When the system satisfies standard I(1) conditions, a rank-\(r\) factorization \(\Pi=\alpha\beta'\) represents \(r\) independent cointegrating relations and \(m-r\) common stochastic trends.

Check Your Understanding

Answer each question using the definitions and conditions in this tutorial.

For a VAR(3), write the expressions for \(\Pi\), \(\Gamma_1\), and \(\Gamma_2\).
If a four-variable I(1) system has cointegration rank two under the standard conditions, how many common stochastic trends does it have?
Why does \(\operatorname{rank}(\Pi)=1\) alone not establish that a specified linear combination of observed variables is stationary?
What does \(\beta'y_{t-1}\) represent in the factorization \(\Pi=\alpha\beta'\)?
Explain why changing \(\beta\) to \(\beta H\), for nonsingular \(H\), need not change \(\Pi\) when \(\alpha\) is transformed accordingly.