Deriving the Vector Error-Correction Form

Why Rewrite a VAR in Differences?

A levels VAR describes each variable using its own and other variables’ past levels. When the variables are integrated of order one and share long-run relations, however, their changes and departures from those relations are especially informative. The vector error-correction model (VECM) makes both features visible: it separates lagged changes from the lagged level information that can pull the system back toward a cointegrating relation.

The conversion is an algebraic identity. It does not, by itself, establish that the variables are cointegrated, determine the cointegration rank, or guarantee stable adjustment. Those conclusions require additional assumptions and evidence. Here the aim is to derive the identity and interpret its components.

Definition: Let $y_t$ be a $k$-dimensional vector of variables, and consider a finite-order levels VAR with $p$ lags: $$ y_t=c+A_1y_{t-1}+A_2y_{t-2}+\cdots+A_py_{t-p}+u_t. $$ Here $c$ is a $k$-vector of deterministic constants, each $A_j$ is a $k\times k$ coefficient matrix, and $u_t$ is a $k$-vector of innovations. Define $\Delta y_t=y_t-y_{t-1}$. The equivalent VECM is $$ \Delta y_t=c+\Pi y_{t-1}+\sum_{i=1}^{p-1}\Gamma_i\Delta y_{t-i}+u_t, $$ where $\Pi=\sum_{j=1}^{p}A_j-I_k$ and $\Gamma_i=-\sum_{j=i+1}^{p}A_j$. The symbol $I_k$ denotes the $k\times k$ identity matrix.

The levels VAR and the VECM contain exactly the same information when their coefficients are related by these formulas. The matrix $\Pi$ is called the long-run matrix. The $\Gamma_i$ matrices describe how lagged changes enter the current change, conditional on the lagged level term. For $p=1$, the sum over differenced lags is empty, so the VECM has no lagged differences.

Deriving the VECM Coefficients

The conversion follows by expressing each lagged level as the most recent level minus the intervening changes. For example, $y_{t-2}=y_{t-1}-\Delta y_{t-1}$, while $y_{t-3}=y_{t-1}-\Delta y_{t-1}-\Delta y_{t-2}$. In general, for $j\geq 1$,

y_{t-j}=y_{t-1}-\sum_{i=1}^{j-1}\Delta y_{t-i}.

Substitute these expressions into the levels VAR. Each lagged level contributes an $A_jy_{t-1}$ term, and each intervening change $\Delta y_{t-i}$ receives a contribution from every $A_j$ for which $j\geq i+1$. This gives the following complete rearrangement.

Replace lagged levels.
Substitute $y_{t-j}=y_{t-1}-\sum_{i=1}^{j-1}\Delta y_{t-i}$ for each lag $j$.

Collect terms in $y_{t-1}$.
The levels sum becomes $\left(\sum_{j=1}^{p}A_j\right)y_{t-1}$.

Collect terms in each lagged change.
The coefficient on $\Delta y_{t-i}$ is $-\sum_{j=i+1}^{p}A_j$, for $i=1,\ldots,p-1$.

Subtract $y_{t-1}$ from both sides.
The left side becomes $\Delta y_t$, and the coefficient on $y_{t-1}$ becomes $\Pi=\sum_{j=1}^{p}A_j-I_k$.

In symbols, the substitution yields

y_t=c+\left(\sum_{j=1}^{p}A_j\right)y_{t-1} -\sum_{i=1}^{p-1}\left(\sum_{j=i+1}^{p}A_j\right)\Delta y_{t-i}+u_t.

Subtracting $y_{t-1}$ on both sides proves the stated VECM form. In particular, the minus sign in $\Gamma_i=-\sum_{j=i+1}^{p}A_j$ is essential: differenced-lag coefficients are negative sums of the corresponding later levels-VAR matrices.

Worked Example: Converting a Two-Lag VAR

Suppose two hypothetical economic variables, each measured in index points, follow a VAR(2) with constant vector $c$ and coefficient matrices

A_1=\begin{pmatrix}0.7&0.2\\0.1&0.6\end{pmatrix}, \qquad A_2=\begin{pmatrix}0.2&-0.1\\0.3&0.1\end{pmatrix}.

The model has two variables and two lags, so its VECM has one lagged difference. First,

\Pi=A_1+A_2-I_2 =\begin{pmatrix}0.9&0.1\\0.4&0.7\end{pmatrix} -\begin{pmatrix}1&0\\0&1\end{pmatrix} =\begin{pmatrix}-0.1&0.1\\0.4&-0.3\end{pmatrix}.

Next, $\Gamma_1=-A_2$, so

\Gamma_1=\begin{pmatrix}-0.2&0.1\\-0.3&-0.1\end{pmatrix}.

Therefore the equivalent VECM is

\Delta y_t=c+ \begin{pmatrix}-0.1&0.1\\0.4&-0.3\end{pmatrix}y_{t-1} +\begin{pmatrix}-0.2&0.1\\-0.3&-0.1\end{pmatrix}\Delta y_{t-1}+u_t.

The calculation verifies the required dimensions: $\Pi$ and $\Gamma_1$ are both $2\times2$, while their products with a two-variable vector are two-vectors. This is an algebraic conversion; it does not establish a cointegration rank for this particular numerical matrix.

Long-Run Matrix and Cointegrating Relations

The VECM form is useful for cointegrated variables because the long-run matrix can have reduced rank. If the variables are $I(1)$ and cointegrated with rank $r$, where $0<r<k$, the standard representation is

\Pi=\alpha\beta',

where $\beta$ is a $k\times r$ matrix whose columns span the cointegrating relations, and $\alpha$ is a $k\times r$ matrix of adjustment coefficients. The $r$ elements of $\beta'y_{t-1}$ are the lagged cointegrating errors. Thus $\Pi y_{t-1}=\alpha(\beta'y_{t-1})$: the long-run term affects current changes through the departures from those relations.

The rank cases have different implications. If $\Pi$ has rank zero, then $\Pi=0$ and there is no lagged-level correction term in this representation. If it has full rank $k$, the usual interpretation of $k$ $I(1)$ variables with fewer than $k$ cointegrating relations does not apply. For rank $r$ strictly between zero and $k$, the factorization expresses the reduced-rank long-run component. The factorization is not unique: changing the basis used for the cointegrating relations changes $\alpha$ and $\beta$, while leaving $\Pi$ unchanged.

Worked Example: Finding the Rank-One Long-Run Component

Consider the hypothetical long-run matrix

\Pi=\begin{pmatrix}-0.3&0.15\\0.2&-0.1\end{pmatrix}.

Its determinant is $(-0.3)(-0.1)-(0.15)(0.2)=0.03-0.03=0$, and the matrix is not zero. Therefore it has rank one. One factorization is

\alpha=\begin{pmatrix}-0.3\\0.2\end{pmatrix}, \qquad \beta'=\begin{pmatrix}1&-0.5\end{pmatrix}, \qquad \alpha\beta'= \begin{pmatrix}-0.3&0.15\\0.2&-0.1\end{pmatrix}.

The associated cointegrating error is $z_{t-1}=\beta'y_{t-1}=y_{1,t-1}-0.5y_{2,t-1}$. The long-run term in the VECM is

\Pi y_{t-1}= \begin{pmatrix}-0.3\\0.2\end{pmatrix}z_{t-1}.

This identifies one long-run relation and shows how its error enters both equations. It does not say that either variable moves only in response to that error: the VECM can also contain constants, innovations, and lagged differences.

Interpreting Adjustment to a Specified Relation

For a single cointegrating relation $z_{t-1}=\beta'y_{t-1}$, the contribution of error correction to the change in variable $j$ is $\alpha_j z_{t-1}$. Its sign and size are meaningful only relative to the chosen orientation and scale of $\beta$. Reversing the sign of the relation also reverses the corresponding adjustment coefficients, without changing $\Pi$. Rescaling the relation changes the numerical coefficient as well. Interpret the product $\alpha_j z_{t-1}$, and state clearly how the disequilibrium error was defined.

Worked Example: Interpreting a Disequilibrium Error

Use the preceding factorization, and suppose that at time $t-1$ the hypothetical variables have values $y_{1,t-1}=10$ index points and $y_{2,t-1}=8$ index points. With the specified relation $z_{t-1}=y_{1,t-1}-0.5y_{2,t-1}$,

z_{t-1}=10-0.5(8)=6\text{ index points}.

The error-correction contribution to the next changes is

\alpha z_{t-1} =\begin{pmatrix}-0.3\\0.2\end{pmatrix}(6) =\begin{pmatrix}-1.8\\1.2\end{pmatrix}\text{ index points}.

Holding the constant, lagged-difference terms, and innovation fixed, the long-run term contributes $-1.8$ index points to $\Delta y_{1,t}$ and $+1.2$ index points to $\Delta y_{2,t}$. The induced contribution to the change in the cointegrating error is

\beta'\alpha z_{t-1} =\left[-0.3-0.5(0.2)\right](6) =-0.4(6)=-2.4\text{ index points}.

So, for this specified relation and these coefficients, the long-run term alone acts to reduce the positive error. This is a conditional component of the change, not a forecast of the total change: the remaining VECM terms may reinforce or offset it. Nor does this one calculation prove that the full dynamic system converges.

Assumptions, Scope, and a Common Pitfall

The algebraic identity requires a correctly specified finite-order levels VAR with consistently defined coefficient matrices and the same variable vector at every time. It does not require stationarity. To interpret a reduced-rank $\Pi$ as encoding cointegrating relations, one additionally needs the relevant integration and cointegration assumptions: in the standard case, the variables are $I(1)$, the cointegration rank is $r$, and $\Pi$ has rank $r$. Deterministic components can be specified in different ways in cointegrated models; the constant $c$ above is carried through the algebra unchanged. Its placement or restrictions may matter for economic interpretation, but they do not change the formulas for $\Pi$ and $\Gamma_i$.

Pitfall: treating $\Pi$ as a single adjustment speed. In a multivariable system, $\Pi$ is a matrix, and its elements combine information about the cointegrating relations and adjustment. Correction: when the rank is $r$, interpret the factorization $\Pi=\alpha\beta'$, including the definition and scaling of $\beta'y_{t-1}$.
Pitfall: reading a negative adjustment coefficient as automatic convergence. Its sign alone does not establish stability, especially when several variables and lagged changes interact. Correction: describe the coefficient’s conditional contribution for a specified error orientation, and assess the complete dynamic system separately.
Pitfall: omitting the minus sign in $\Gamma_i$. This changes the short-run coefficients. Correction: derive each one by collecting the contributions of $A_{i+1},\ldots,A_p$ after expressing lagged levels as sums of changes.

Key takeaway: For a levels VAR($p$), the exact VECM has $\Pi=\sum_{j=1}^{p}A_j-I_k$ and $\Gamma_i=-\sum_{j=i+1}^{p}A_j$. Under cointegration, a reduced-rank factorization $\Pi=\alpha\beta'$ separates the long-run relations from the coefficients governing how current changes respond to their errors.

Check Your Understanding

Answer each question using the VECM definitions and coefficient formulas.

For a VAR(3), which levels-VAR matrices contribute to $\Gamma_1$, and which contribute to $\Gamma_2$? Include the signs.
Why is the conversion from a levels VAR to a VECM an algebraic identity rather than evidence of cointegration?
If $\Pi$ is nonzero and has rank one in a two-variable system, what does a factorization $\Pi=\alpha\beta'$ identify?
For $z_{t-1}=\beta'y_{t-1}>0$, what does $\alpha_j z_{t-1}$ describe in equation $j$, and why is it not necessarily the total change?
Why can the sign of an adjustment coefficient not be interpreted without knowing how the cointegrating relation is defined?