Cointegration Tests and VECM

When the wholesale price of construction lumber rises, does the retail price rise by the same amount right away? In real markets, contract terms, inventories and distribution costs can make the two prices move differently in the short run. Even so, if a stable relationship between the wholesale and retail prices holds over time, the two series may have a long-run equilibrium.

The problem is that price data can keep rising or falling over time. Running a simple regression on nonstationary series like these can produce spurious regression, with high explanatory power and significant coefficients even when the real relationship is weak. The method for examining whether a long-run relationship exists in this situation is the cointegration test, and the model that analyzes how departures from the long-run relationship are corrected in the short run is the vector error correction model (VECM).

In this article we’ll use monthly price data to cover the difference between unit roots and cointegration, the Engle–Granger and Johansen tests, the cointegrating vector and the error-correction term, VECM estimation and diagnostics, and how to report the results.

Even if each of two prices moves with its own long-run stochastic trend, a linear combination representing the gap between them can move around a stable level. For example, let W_t be the wholesale price and R_t the retail price, and suppose the long-run relationship is as follows.

Here, if u_t = R_t − 12 − 1.15W_t is a stationary series, then under suitable conditions the two prices can be considered cointegrated. This does not mean the difference between the two prices is the same every month. It means that even if they temporarily move away from equilibrium, the gap stays within a limited range in the long run.

The two series in the figure above were generated separately, with no relationship to each other at all.

In a series with a unit root, by contrast, the effects of shocks persist for a long time, so the statistical characteristics of its levels can change over time.

The typical starting point of a cointegration analysis is checking whether each of the two variables is integrated of order one, that is, I(1). I(1) means a series that has a unit root in levels but is stationary in first differences. The ADF test and the KPSS test can be used to complement each other when checking this.

However, failing to reject the null hypothesis in an ADF test does not prove that a unit root exists. Because power can vary with the sample length, the trend specification and whether there are structural breaks, you should also look at the time-series plots and other diagnostics.

Prices were measured for a bundle of lumber of the same specification, in dollars. Both prices are assumed to be real prices, already adjusted for inflation.

The researcher expects that, even though the wholesale and retail prices each have their own long-run trend, a stable long-run relationship exists between them. The researcher also wants to check whether, when prices move away from the long-run equilibrium, subsequent monthly price changes tend to close the gap.

The research question is: "Do wholesale and retail lumber prices have a long-run equilibrium relationship, and when they move away from equilibrium, how fast does each price adjust?"

The following are the results of ADF tests with a constant. Whether to include a deterministic trend for the levels has to be decided separately based on the plots and the study design.

The null hypothesis of a unit root could not be rejected in levels but was rejected in first differences. So there are grounds for treating both variables as I(1) and proceeding with a cointegration analysis.

However, the fact that both prices have the same order of integration does not by itself mean they are cointegrated. Cointegration is a hypothesis about a long-run relationship that needs its own test.

First you estimate the long-run regression in levels, then you test whether the residuals from that regression have a unit root. For example, suppose we estimated the following long-run equation.

Then check whether the residual û_t = R_t − 12 − 1.15W_t is stationary. If the residuals are stationary, that is evidence of cointegration, meaning the two prices share a common stochastic trend.

Note that the critical values of the ordinary ADF test are not applied as-is to the estimated residuals. You must use critical values or p-values designed for cointegration tests, and the constant and trend specification of the regression also affects the test. In a multivariate setting where more than one cointegrating relationship is possible, the Johansen approach may be more natural.

The trace test and the maximum eigenvalue test are the standard versions. If two variables are I(1), the cointegration rank can be 0 or 1. Rank 0 means no cointegrating relationship was found, and rank 1 means there is one independent long-run relationship.

Critical values depend on how the deterministic terms are specified.

The first hypothesis is rejected but the second is not, so with this specification we can choose a cointegration rank of 1. This is consistent with there being one long-run equilibrium relationship between the wholesale and retail prices.

In general you can set up an equation with each of the two variables as the dependent variable. The most simplified retail price equation is as follows.

Here the term in parentheses is last month’s deviation from the long-run equilibrium, which is called the error-correction term (ECT). α_R is the adjustment coefficient showing how this deviation feeds into this month’s change in the retail price.

The full VECM can also include an equation for changes in the wholesale price, and each equation includes several lagged differences and any deterministic terms needed. So a VECM is not simply a regression with one residual added; it is a system that estimates the long-run relationship and the short-run dynamics together.

If the error-correction coefficient in the retail price equation is -0.30, then, with other short-run factors held equal, when last month’s retail price was $10 above its long-run equilibrium level, an adjustment pressure of about -$3 acts on this month’s change in the retail price.

This is often described as correcting about 30% of the deviation from equilibrium in one period. In the full VECM, where both prices change at the same time, however, it does not mean the equilibrium error actually shrinks by exactly 30% every month, because adjustment of the wholesale price, short-run lag terms and new shocks are also at work.

The sign of the error-correction coefficient depends on which direction the ECT was defined in. So when interpreting results, always check how the cointegrating vector was normalized.

To keep the explanation simple, the table shows only the main coefficients.

The error-correction coefficient for the retail price was significant at -0.30. This is consistent with the interpretation that when the retail price is above the long-run equilibrium, it adjusts downward. The adjustment coefficient in the wholesale price equation, on the other hand, was not significant, so in these data there is not enough evidence that the wholesale price responds actively to deviations from equilibrium.

A coefficient of 0.45 on the previous month’s wholesale price change means that, other model conditions being equal, when last month’s wholesale price rose by $1, this month’s change in the retail price tended to be $0.45 higher on average. This lagged relationship alone does not prove a causal effect of price pass-through, however.

If the lags are too short, autocorrelation can remain in the residuals; if they are too long, degrees of freedom are used up and the estimates can become unstable. Information criteria such as AIC and BIC are usually used together with residual autocorrelation tests. Take care to check how the number of lags in the VAR in levels corresponds to the number of lagged differences in the VECM.

Also, the Johansen test and the estimation results can differ depending on whether the price series have a deterministic trend and whether a constant is allowed in the long-run equilibrium equation. So it’s important to keep the specification of the deterministic terms consistent between the testing stage and the model estimation stage.

If the cointegration test finds no evidence of a long-run relationship, you can consider models such as a VAR on differences, which are stationary.

On the other hand, if the variables have different orders of integration or include I(2) variables, applying the standard I(1) Johansen VECM as-is may not be appropriate. A different approach is needed depending on the integration properties of the variables.

Also, if the long-run relationship itself changed because of a policy change or a restructuring of the distribution system, a model that applies a single cointegrating vector to the whole period may be inappropriate. Structural break tests and analyses by subperiod should be considered.

A high correlation coefficient is not sufficient evidence of cointegration. Cointegration is about whether a particular linear combination of nonstationary series is stationary.

Check the specifications of the unit root and cointegration tests.

Test results can vary with the trend term, the number of lags and the sample period, and when there are structural breaks a simple test can lead to the wrong conclusion.

Report the normalization of the cointegrating vector together with the sign of the adjustment coefficients.

If the ECT is defined in the opposite direction, the signs of the adjustment coefficients can also flip.

Check the diagnostic results as well.

Examine residual autocorrelation, heteroskedasticity, normality, model stability and so on, and assess whether the estimated model describes the data adequately.

Do not take a long-run relationship to be a causal relationship.

Even if wholesale and retail prices are cointegrated, that fact alone cannot tell you which price causes the other. Causal claims require institutional background or an additional identification strategy.

Stating the cointegration rank and the definition of the ECT makes the results easier to reproduce and interpret.

A VECM analyzes the short-run changes of cointegrated variables and their deviations from the long-run equilibrium together in a single system.

As in the lumber wholesale and retail price example, two prices can move differently from month to month yet maintain a stable relationship in the long run. The error-correction coefficient then shows how a departure from equilibrium feeds into later price changes. Rather than checking only the significance of the tests, though, you should also examine the variables’ orders of integration, the cointegration rank, the lag specification, residual diagnostics and the possibility of structural breaks.