Why Adjustment and Long-Run Restrictions Matter
Cointegration describes a system in which some combinations of nonstationary variables are stationary. A vector error-correction model (VECM) expresses changes in those variables in relation to past departures from those long-run equilibria. Its two central matrices answer different questions: the cointegrating matrix describes the equilibrium combinations, while the loading matrix describes how the variables respond when those combinations are out of equilibrium.
This distinction is economically useful. A long-run relation may connect prices, wages, and productivity, but not every variable must adjust to restore it. Testing which variables respond can help distinguish plausible directions of adjustment. Testing restrictions on the long-run relation can assess whether its coefficients match an economic implication, such as a unit long-run elasticity. These are model-based statistical conclusions; they do not by themselves establish causation.
The VECM Loading and Cointegrating Matrices
Let \(y_t\) be a \(k\)-dimensional vector of time-series variables. For a system with cointegration rank \(r\), write a VECM as
Here \(\Delta y_t=y_t-y_{t-1}\); \(\alpha\) and \(\beta\) are \(k\times r\) matrices; the columns of \(\beta\) define the \(r\) cointegrating relations; and the columns of \(\alpha\) give the corresponding adjustment loadings in each variable’s equation. The matrices \(\Gamma_i\) describe lagged short-run changes, \(d_t\) denotes the included deterministic terms, and \(\varepsilon_t\) is the system innovation. The vector of lagged disequilibria is \(\beta'y_{t-1}\), with one component for each relation.
The long-run relation is a linear combination, not necessarily an equation with a unique scale. Multiplying a column of \(\beta\) by a nonzero constant and dividing the corresponding column of \(\alpha\) by that constant leaves \(\alpha\beta'\) unchanged. More generally, the same product can be represented with different bases for the cointegrating space. Therefore, coefficient interpretations and restrictions on \(\beta\) require a stated normalization or another basis-invariant formulation.
A loading is not automatically a percentage correction. In a one-relation, two-variable system with no lagged changes or deterministic terms, the relation \(e_t=\beta'y_t\) changes according to \(\Delta e_t=\beta'\alpha e_{t-1}+\beta'\varepsilon_t\). The scalar \(\beta'\alpha\) then governs the systematic change in the disequilibrium. With multiple relations, lagged changes, or additional dynamics, a single loading entry does not by itself give the full speed or stability of adjustment.
Worked Example: Interpreting Adjustment in a Two-Variable System
Suppose \(y_t=(x_t,z_t)'\), both variables are measured in log points, and the model has one cointegrating relation \(e_t=x_t-z_t\). Let \(\beta=(1,-1)'\) and \(\alpha=(-0.30,0.10)'\). For this illustration, consider the error-correction contributions alone; omit lagged changes and deterministic terms.
The equations’ adjustment contributions are \(\Delta x_t=-0.30e_{t-1}\) and \(\Delta z_t=0.10e_{t-1}\). If the previous disequilibrium is \(e_{t-1}=0.040\), the contributions are
Thus the first variable falls by 0.012 log points and the second rises by 0.004 log points through these terms. Their combined effect on the gap is \(\Delta e_t=\Delta x_t-\Delta z_t=-0.016\), so the gap moves from 0.040 to 0.024, a reduction of 40% in this simplified one-step calculation.
The reduction follows from \(\beta'\alpha=(1,-1)(-0.30,0.10)'=-0.30-0.10=-0.40\), giving \(e_t=(1-0.40)e_{t-1}=0.60e_{t-1}\) when innovations and all other terms are set aside. The individual loadings describe each equation’s response; it is their combination through the relation that gives the gap’s change here. In a full fitted model, other terms and shocks also affect the observed change.
Estimating Adjustment Speeds
In a fitted VECM, estimation of \(\alpha\) depends on the method used to estimate the system and the cointegrating space. In a Johansen maximum-likelihood analysis, \(\alpha\) and \(\beta\) are estimated jointly subject to the selected rank. A useful way to understand a loading coefficient is to hold a cointegrating relation fixed and estimate its coefficient in the relevant change equation.
For example, if \(e_{t-1}=\beta'y_{t-1}\) is treated as known, the equation for variable \(j\) can be written as a regression of \(\Delta y_{j,t}\) on \(e_{t-1}\), lagged changes, deterministic terms, and any other specified regressors. The coefficient on \(e_{t-1}\) estimates the relevant loading. This conditional regression is illustrative; in practice the relation is estimated and its uncertainty matters. The regression must also include the short-run terms required by the model, rather than attributing their effects to the loading.
Worked Example: Estimating a Loading by Conditional Least Squares
Suppose a hypothetical six-period illustration uses a specified relation \(e_{t-1}\) and the equation \(\Delta x_t=c+\alpha_x e_{t-1}+u_t\). Assume there are no lagged-change regressors in this simplified equation. The six pairs \((e_{t-1},\Delta x_t)\), in log points, are \((-3,1.05),(-2,0.60),(-1,0.45),(1,-0.05),(2,-0.40),(3,-0.45)\).
The mean of the disequilibria is zero, and the mean change is \(1.20/6=0.20\). The least-squares slope is the sum of the products of demeaned regressor and outcome divided by the sum of squared demeaned regressor:
The intercept estimate is \(\hat{c}=0.20\). For example, the first product in the numerator is \((-3)(1.05-0.20)=-2.55\); summing the six products gives \(-7.00\). The estimated conditional equation is therefore \(\widehat{\Delta x}_t=0.20-0.250e_{t-1}\). Holding the other included terms fixed, a one-log-point larger disequilibrium is associated with a 0.250-log-point lower predicted change in \(x_t\).
These six hypothetical observations demonstrate the calculation, not a reliable inferential procedure. They do not establish that the cointegrating relation is known without error or that the model is adequately specified. In an applied analysis, use an appropriate system estimator or account for estimation uncertainty, include justified short-run dynamics, and base inference on suitable large-sample or finite-sample methods.
Weak Exogeneity and Restrictions on the Loading Matrix
A variable is weakly exogenous for inference about the long-run parameters when its conditional distribution can be used without modeling the marginal process for that variable for the purpose of estimating those parameters. In the standard cointegrated-system setup, a common testable implication is that the row of \(\alpha\) for that variable is zero. If variable \(j\) has no loading on any of the \(r\) disequilibria, it does not adjust to those long-run errors in its own change equation.
For rank \(r\), testing \(\alpha_{j\cdot}=0\) imposes \(r\) restrictions. A likelihood-ratio test compares the unrestricted system fit with the fit under this row restriction. A Wald test uses the estimated row and its covariance matrix. Under the null and standard large-sample conditions, either statistic has an approximate chi-square distribution with \(r\) degrees of freedom. The null hypothesis is that the row is zero; a small p-value is evidence against that restriction.
Worked Example: Testing Weak Exogeneity for One Variable
Suppose a hypothetical rank-two VECM estimates the loading row for a variable \(q_t\) as \(\hat{\alpha}_{q\cdot}=(0.020,-0.030)\). For a Wald illustration, suppose its estimated covariance matrix is \(0.0004I_2\), where \(I_2\) is the two-dimensional identity matrix. Test \(H_0:\alpha_{q\cdot}=(0,0)\) against the alternative that at least one loading is nonzero.
The Wald statistic is the estimated row multiplied by the inverse covariance matrix and by its transpose. Since the inverse covariance matrix is \(2500I_2\),
Under the null, the reference distribution is approximately \(\chi^2_2\), so the p-value is \(P(\chi^2_2\geq 3.250)\approx 0.197\). At a 5% significance level, this test does not reject the zero-loading restriction. The result does not prove that \(q_t\) is weakly exogenous: it says the data and this test do not provide sufficient evidence against the restriction at that level. The covariance matrix and large-sample reference are assumed for this hypothetical calculation.
Economically Meaningful Restrictions on Long-Run Relations
Restrictions on \(\beta\) translate economic claims into testable statements about long-run combinations. A restriction might set a normalized coefficient to a specified value, constrain two coefficients to be equal, or impose a known proportionality. Before testing, state the normalization and the economic meaning of the restriction. Without that information, a numerical coefficient may be difficult to interpret because the scale of a cointegrating vector is arbitrary.
A likelihood-ratio test compares the unrestricted cointegrated-system fit with the fit under the long-run restriction. A Wald test evaluates the estimated restriction relative to its estimated sampling uncertainty. Under regularity conditions, a test of \(q\) independent overidentifying restrictions has an approximate chi-square reference distribution with \(q\) degrees of freedom. The restrictions used to identify or normalize the cointegrating vectors are not additional economic restrictions to count as overidentifying restrictions.
Worked Example: Testing a Unit Long-Run Elasticity
Suppose two hypothetical log price series have a rank-one relation normalized as \(p_{1,t}-\theta p_{2,t}\), with the coefficient on \(p_{1,t}\) fixed at one. The estimated long-run coefficient is \(\hat{\theta}=1.080\), with estimated standard error \(0.030\). The economic restriction of a unit long-run elasticity is \(H_0:\theta=1\).
The Wald statistic is
For one restriction, the asymptotic reference distribution is \(\chi^2_1\), giving a p-value of approximately \(0.0077\). At the 5% level, reject the unit-elasticity restriction under the assumed model and inference conditions. The estimate indicates a long-run coefficient above one; the rejection is evidence that the coefficient differs from one, not proof of a particular economic mechanism. This example assumes that the normalization is appropriate and that the reported standard error accounts for the estimation method.
Common Mistakes and Practical Checks
- Calling a loading a universal percentage speed. A loading applies to a specified equation and relation. The evolution of the disequilibrium depends on the combination \(\beta'\alpha\), as well as on short-run dynamics and other terms.
- Interpreting a zero-loading test too broadly. A zero row concerns adjustment to the cointegrating errors and weak exogeneity for long-run parameters under the model’s conditions. It does not imply independence, no short-run response, or strict exogeneity.
- Testing an unnormalized coefficient without considering scale. A cointegrating vector can be rescaled. State the normalization, or formulate the restriction so that it does not depend on an arbitrary scale or basis.
- Treating failure to reject as proof. A large p-value means the test has not found sufficient evidence against the null at the chosen level; it does not establish that the null is true.
- Ignoring model uncertainty. Inference depends on the specified cointegration rank, deterministic terms, lag structure, and regularity conditions. A reported restriction test should be interpreted in that context.
Check Your Understanding
Answer each question using the definitions and test logic above.
- In a rank-two VECM, what does setting the loading row for one variable to zero restrict, and how many restrictions does it impose?
- Why is an individual loading coefficient not generally the percentage of disequilibrium corrected in one period?
- If a normalized long-run coefficient is estimated as 0.94 with standard error 0.02, calculate the Wald statistic for a null value of 1.
- What does a p-value of 0.20 in a weak-exogeneity test allow you to conclude, and what does it not allow you to conclude?
- Why must a long-run coefficient restriction state its normalization or be formulated to be invariant to the scale of the cointegrating vector?