Estimating and Restricting VECM Parameters

From a VECM to Estimated Parameters

A vector error-correction model (VECM) separates short-run changes from deviations from long-run equilibrium. Estimating one at a specified cointegration rank means estimating both parts while requiring the long-run impact matrix to have that rank. The rank is taken as given here; the task is to estimate and interpret the parameters conditional on it.

Let \(y_t\) be a \(k\)-dimensional vector of variables that are integrated of order one, and suppose their cointegration rank is \(r\), where \(0<r<k\). For a VAR with \(p\) lags, write the VECM as

Here \(\beta\) is a \(k\times r\) matrix whose columns span the cointegrating relations, and \(\alpha\) is a \(k\times r\) matrix of adjustment coefficients. Their product \(\Pi=\alpha\beta'\) is the long-run impact matrix and has rank \(r\). The vector \(d_t\) contains the deterministic terms included under the chosen specification, such as an intercept, and \(\Psi\) contains their coefficients. The \(\Gamma_i\) matrices describe short-run effects of lagged changes.

The standard reduced-rank maximum-likelihood procedure is conditional on the chosen lag and deterministic specification. Its usual likelihood interpretation assumes Gaussian innovations with a constant positive-definite covariance matrix. The model must also be adequately specified: in particular, the innovations should not retain serial dependence, and the assumed integration and rank structure should be appropriate. These conditions matter for interpreting likelihood-ratio tests as well as for the estimated parameters.

How Reduced-Rank Estimation Works

The estimation separates the lagged differences and deterministic terms from the candidate long-run relations. First, regress \(\Delta y_t\) and \(y_{t-1}\) on the short-run controls: the included lagged differences and deterministic terms. Let \(z_{0t}\) and \(z_{1t}\) be the respective residual vectors. This residualization asks which linear combinations of the lagged levels best explain the remaining variation in current changes, after accounting for the short-run controls.

Define \(S_{ij}=T^{-1}\sum_t z_{it}z_{jt}'\), using the same effective sample of \(T\) observations for each covariance matrix. The \(r\) eigenvectors associated with the largest eigenvalues in the generalized eigenvalue problem below give the estimated cointegrating space. This is a reduced-rank problem: only \(r\) long-run relations are allowed, rather than an unrestricted \(k\times k\) matrix of long-run coefficients.

After selecting and normalizing the \(r\) eigenvectors as the columns of \(\widehat{\beta}\), the loading matrix can be estimated by regressing \(z_{0t}\) on \(\widehat{\beta}'z_{1t}\). In this residualized regression, the coefficient matrix is

The remaining short-run and deterministic coefficients are estimated consistently with the chosen specification. The residual covariance matrix is estimated from the fitted system. In practice, software implements the corresponding concentrated Gaussian likelihood; the displayed equations show the core reduced-rank calculation rather than every implementation detail.

Normalization and Interpretation

The estimated cointegrating space, not the scale or sign of each displayed vector, is the economically relevant object. For any nonsingular \(r\times r\) matrix \(Q\), define \(\beta^*=\beta Q\) and \(\alpha^*=\alpha Q^{-T}\). Then

Consequently, the fitted VECM is unchanged. For a single relation, multiplying \(\beta\) by a nonzero constant requires dividing \(\alpha\) by that constant. A common normalization sets one coefficient in each relation to one, when the selected coefficients permit this. Normalization makes a vector readable and gives its coefficients a reference scale; it does not establish that a variable is economically exogenous or that a particular coefficient is causal.

Once a normalization is fixed, a coefficient in \(\beta\) can be interpreted relative to the normalized variable. In the example, \(x_t-1.2m_t\) is stationary under the assumed model. The number 1.2 describes the estimated long-run relation in the chosen log units. A coefficient in \(\alpha\), in contrast, describes how a variable's change responds to disequilibrium. A zero row of \(\alpha\) means that the corresponding variable does not adjust through these error-correction terms; it does not, by itself, rule out short-run effects through the \(\Gamma_i\) terms.

Testing Economically Motivated Restrictions

Restrictions can encode questions about which variables adjust and about the form of the long-run equilibrium. For example, the restriction that a row of \(\alpha\) is zero tests whether the corresponding variable adjusts to disequilibrium. In cointegration terminology, this is a weak-exogeneity restriction for the long-run parameters. A restriction on \(\beta\), such as a unit long-run coefficient after normalization, tests a claim about the equilibrium relation itself.

Compare the unrestricted model with a model estimated subject to the restriction, keeping the rank, sample, lag structure, and deterministic specification the same. Under the null hypothesis that the restriction is correct, and under the regularity conditions for the likelihood-ratio result, the statistic has an asymptotic chi-square distribution. Its degrees of freedom are the number of independent restrictions, after accounting for the normalization and identification of the cointegrating space.

Here \(\ell_U\) and \(\ell_R\) are the maximized log likelihoods in the unrestricted and restricted models, respectively, and \(q\) is the appropriate number of independent restrictions. A small p-value is evidence against the restrictions under the stated model; it is not the probability that the null is true. For restrictions on \(\beta\), the restrictions must be defined on an identified cointegrating space, commonly by first stating a normalization. Counting raw equations in a restriction without considering identification can give the wrong degrees of freedom.

Common Pitfall and Practical Checks

A frequent error is to interpret the entries of \(\widehat{\alpha}\) as fixed adjustment speeds while ignoring the scale of \(\widehat{\beta}\). If a cointegrating vector is multiplied by two, its associated loading vector must be divided by two; otherwise the product \(\alpha\beta'\) changes. The correction is to check the normalization and interpret the product, or the normalized error-correction term and its matching loading, together.

Before reporting a restricted estimate or test, verify that the unrestricted and restricted fits use the same observations and specification, that the rank is held fixed, and that the restrictions are identified under the stated normalization. Report the restriction, likelihood-ratio statistic, degrees of freedom, p-value, and conclusion in economic context. A statistically rejected restriction may still be a useful benchmark, but it should not be described as supported by the data.