Forecasting from Vector Error-Correction Models

Why Forecast From a VECM?

A vector error-correction model (VECM) is useful for forecasting a set of nonstationary variables when some linear combinations of them are stationary. It expresses changes in the variables using both short-run movements and deviations from long-run equilibrium. A forecast must therefore update both components as the horizon increases: the next change depends on the current disequilibrium and recent changes, while later changes depend partly on earlier forecasted changes.

The key practical point is that multistep VECM forecasts are recursive. At the first horizon, the model uses observed values. At later horizons, unknown future values are replaced by their forecasts. The resulting forecasts are conditional point forecasts: they use the information available at time $T$, the estimated coefficients, and the model’s assumption that future innovations have conditional mean zero.

Definition: For a $k$-variable system with cointegration rank $r$, a VECM with $p-1$ lagged differences can be written as $$ \Delta y_t = \alpha\beta'y_{t-1} + \sum_{i=1}^{p-1}\Gamma_i\Delta y_{t-i} + d_t+\varepsilon_t. $$ Here $y_t$ is a $k\times1$ vector of levels, $\Delta y_t=y_t-y_{t-1}$, $\beta$ contains the cointegrating vectors, $\alpha$ contains the adjustment coefficients, $\Gamma_i$ are short-run coefficient matrices, $d_t$ collects specified deterministic terms, and $\varepsilon_t$ is the innovation vector. The matrix $\alpha\beta'$ has rank $r$. A point forecast sets the conditional mean of future innovations to zero and uses estimated model parameters.

The Recursive Forecast

Let $\widehat y_{T+h\mid T}$ denote the forecast of $y_{T+h}$ made using information through time $T$. Define the forecasted change by $\widehat{\Delta y}_{T+h\mid T}=\widehat y_{T+h\mid T}-\widehat y_{T+h-1\mid T}$, with $\widehat y_{T\mid T}=y_T$. For the first forecast, every lagged level and change in the equation is observed. For later forecasts, a term dated after $T$ is replaced by its forecast.

\widehat{\Delta y}_{T+h\mid T} = \widehat\alpha\widehat\beta'\widehat y_{T+h-1\mid T} + \sum_{i=1}^{p-1}\widehat\Gamma_i \widehat{\Delta y}_{T+h-i\mid T} + \widehat d_{T+h}, \qquad \widehat y_{T+h\mid T} = \widehat y_{T+h-1\mid T} + \widehat{\Delta y}_{T+h\mid T}.

For a lagged change whose date is at or before $T$, the observed change is used; for a later date, its forecast is used. The deterministic term must be specified for each forecast date. For example, a known seasonal indicator can be carried forward according to its calendar, whereas an unknown future explanatory variable requires an assumed future value or its own forecast.

This recursion gives a conditional mean forecast when the model’s future innovations have conditional mean zero given the information available at each date. It does not require normally distributed innovations to calculate the point forecast. Forecast intervals require additional uncertainty calculations and are not the same as the point forecast.

Worked Example: Two-Step Forecast With Error Correction

Consider a hypothetical two-variable VECM with one lagged difference, no deterministic term, and estimated equation $$ \Delta y_t = \begin{pmatrix}-0.25\\0.10\end{pmatrix} (y_{1,t-1}-y_{2,t-1}) + \begin{pmatrix}0.20&0\\0&0.20\end{pmatrix} \Delta y_{t-1} +\varepsilon_t. $$ Suppose $y_T=(10,8)'$ and the observed change is $\Delta y_T=(1,0)'$. The forecasts set the future innovation’s conditional mean to zero.

At horizon one, the equilibrium error is $10-8=2$. The error-correction contribution is $$ \begin{pmatrix}-0.25\\0.10\end{pmatrix}2 = \begin{pmatrix}-0.50\\0.20\end{pmatrix}. $$ The short-run contribution uses the observed change: $$ \begin{pmatrix}0.20&0\\0&0.20\end{pmatrix} \begin{pmatrix}1\\0\end{pmatrix} = \begin{pmatrix}0.20\\0\end{pmatrix}. $$ Therefore, $$ \widehat{\Delta y}_{T+1\mid T} = \begin{pmatrix}-0.50\\0.20\end{pmatrix} + \begin{pmatrix}0.20\\0\end{pmatrix} = \begin{pmatrix}-0.30\\0.20\end{pmatrix}, \qquad \widehat y_{T+1\mid T} = \begin{pmatrix}10\\8\end{pmatrix} + \begin{pmatrix}-0.30\\0.20\end{pmatrix} = \begin{pmatrix}9.70\\8.20\end{pmatrix}. $$

At horizon two, the forecasted equilibrium error is $9.70-8.20=1.50$. The error-correction contribution is $$ \begin{pmatrix}-0.25\\0.10\end{pmatrix}1.50 = \begin{pmatrix}-0.375\\0.150\end{pmatrix}. $$ The lagged change is now the horizon-one forecasted change, so the short-run contribution is $$ \begin{pmatrix}0.20&0\\0&0.20\end{pmatrix} \begin{pmatrix}-0.30\\0.20\end{pmatrix} = \begin{pmatrix}-0.060\\0.040\end{pmatrix}. $$ Thus $$ \widehat{\Delta y}_{T+2\mid T} = \begin{pmatrix}-0.375\\0.150\end{pmatrix} + \begin{pmatrix}-0.060\\0.040\end{pmatrix} = \begin{pmatrix}-0.435\\0.190\end{pmatrix}, \qquad \widehat y_{T+2\mid T} = \begin{pmatrix}9.70\\8.20\end{pmatrix} + \begin{pmatrix}-0.435\\0.190\end{pmatrix} = \begin{pmatrix}9.265\\8.390\end{pmatrix}. $$ The gap narrows from $2$ to $1.50$, but the average of the two variables changes from $9$ to $8.95$ and then to $8.8275$. Error correction does not generally preserve the average or any other particular common component.

What Error Correction Does Across Horizons

The error-correction term uses the forecasted level from the preceding horizon. If the system is above its long-run relation, the signs and magnitudes in $\alpha$ determine how each variable’s forecasted change responds. The short-run terms can reinforce that response, offset it, or temporarily move the variables in another direction. It is therefore not correct to infer the entire forecast path from the current equilibrium error alone.

A simple special case shows how the error can evolve. Suppose there are two variables, no lagged-change terms, and no deterministic terms. Let the cointegrating vector be $\beta=(1,-1)'$, so the error is $g_t=y_{1,t}-y_{2,t}$, and let $\alpha=(-0.20,0.10)'$. Then the forecasted change is $\alpha g_t$, and the next error is $$ \widehat g_{t+1\mid t} = \beta'\widehat y_{t+1\mid t} = \beta'y_t+\beta'\alpha g_t = (1+\beta'\alpha)g_t = 0.70g_t. $$ The coefficient $0.70$ follows because $\beta'\alpha=-0.20-0.10=-0.30$. Since its absolute value is less than one, this simplified model’s forecasted error decays geometrically. Actual VECMs also contain lagged changes and may contain deterministic terms, so their error paths need not follow this one-step geometric rule.

Worked Example: Tracking a Cointegrating Gap

Use the simplified model just described, and suppose the hypothetical current levels are $y_T=(20,15)'$. The current gap is $g_T=20-15=5$. The assumptions for the calculation are that the specified VECM has no lagged changes or deterministic terms and that future innovations have conditional mean zero.

The first forecasted change is $$ \widehat{\Delta y}_{T+1\mid T} = \begin{pmatrix}-0.20\\0.10\end{pmatrix}5 = \begin{pmatrix}-1\\0.50\end{pmatrix}, $$ so $\widehat y_{T+1\mid T}=(19,15.50)'$, with gap $19-15.50=3.50$. At the next horizon, $$ \widehat{\Delta y}_{T+2\mid T} = \begin{pmatrix}-0.20\\0.10\end{pmatrix}3.50 = \begin{pmatrix}-0.70\\0.35\end{pmatrix}, $$ so $\widehat y_{T+2\mid T}=(18.30,15.85)'$, with gap $2.45$. At horizon three, $$ \widehat{\Delta y}_{T+3\mid T} = \begin{pmatrix}-0.20\\0.10\end{pmatrix}2.45 = \begin{pmatrix}-0.49\\0.245\end{pmatrix}, $$ giving $\widehat y_{T+3\mid T}=(17.81,16.095)'$, with gap $1.715$. The sequence of gaps is $5,\ 3.50,\ 2.45,\ 1.715$, each obtained by multiplying the preceding gap by $0.70$. The levels move toward one another in this example, but their average is not fixed: it is $17.50$ initially, $17.25$ at horizon one, $17.075$ at horizon two, and $16.9525$ at horizon three.

Connection to an Unrestricted Levels VAR

A VECM and a levels VAR can encode the same forecasting equation. For a VECM with one lagged difference, write $\Pi=\alpha\beta'$. Since $\Delta y_{t-1}=y_{t-1}-y_{t-2}$, the model without deterministic terms can be rearranged as $$ \Delta y_t=\Pi y_{t-1}+\Gamma_1(y_{t-1}-y_{t-2})+\varepsilon_t $$ and hence $$ y_t=(I+\Pi+\Gamma_1)y_{t-1}-\Gamma_1y_{t-2}+\varepsilon_t. $$ Thus the equivalent levels VAR(2) coefficient matrices are $A_1=I+\alpha\beta'+\Gamma_1$ and $A_2=-\Gamma_1$. When these are the same coefficients and deterministic terms, forecasting recursively from either representation must give identical results: the two equations are algebraic rearrangements of one another.

Worked Example: Checking the Equivalent Levels VAR

For the VECM in the first example, the cointegrating vector is $\beta=(1,-1)'$, and $$ \alpha\beta' = \begin{pmatrix}-0.25\\0.10\end{pmatrix} \begin{pmatrix}1&-1\end{pmatrix} = \begin{pmatrix}-0.25&0.25\\0.10&-0.10\end{pmatrix}. $$ With $\Gamma_1=\begin{pmatrix}0.20&0\\0&0.20\end{pmatrix}$, the corresponding levels VAR matrices are $$ A_1=I+\alpha\beta'+\Gamma_1 = \begin{pmatrix}0.95&0.25\\0.10&1.10\end{pmatrix}, \qquad A_2=-\Gamma_1 = \begin{pmatrix}-0.20&0\\0&-0.20\end{pmatrix}. $$ Since $\Delta y_T=(1,0)'$ and $y_T=(10,8)'$, $y_{T-1}=(9,8)'$. The levels VAR forecast is $$ \widehat y_{T+1\mid T} = A_1y_T+A_2y_{T-1} = \begin{pmatrix}11.50\\9.80\end{pmatrix} + \begin{pmatrix}-1.80\\-1.60\end{pmatrix} = \begin{pmatrix}9.70\\8.20\end{pmatrix}. $$ This matches the VECM forecast. The equality is a direct check of the representation, not a claim that separately estimated unrestricted coefficients must yield the same forecasts.

An unrestricted levels VAR estimates its lag matrices without imposing the reduced-rank structure $\Pi=\alpha\beta'$. If it is estimated from the same sample, its coefficients generally differ from those of a restricted VECM. Its forecasts can therefore differ, even when both models use the same lag order and observed data. The VECM imposes the chosen cointegrating structure, which can be valuable when that structure is appropriate; the unrestricted VAR allows more flexibility but does not impose that long-run restriction. Forecast performance is an empirical question, not something guaranteed by either parameterization.

Worked Example: A Forecast From a Hypothetical Unrestricted VAR

For comparison, suppose an unrestricted VAR(2), estimated on the same hypothetical data, has coefficient matrices $$ \widetilde A_1= \begin{pmatrix}0.90&0.15\\0.08&1.02\end{pmatrix}, \qquad \widetilde A_2= \begin{pmatrix}-0.10&0.02\\0.01&-0.08\end{pmatrix}, $$ and no deterministic terms. Use $y_T=(10,8)'$ and $y_{T-1}=(9,8)'$, as above. Its one-step forecast is $$ \widehat y_{T+1\mid T} = \widetilde A_1y_T+\widetilde A_2y_{T-1} = \begin{pmatrix}10.20\\8.96\end{pmatrix} + \begin{pmatrix}-0.74\\-0.55\end{pmatrix} = \begin{pmatrix}9.46\\8.41\end{pmatrix}. $$ This differs from the restricted VECM forecast $(9.70,8.20)'$ because the assumed unrestricted coefficient matrices are not the algebraically equivalent matrices derived in the preceding example. For completeness, the second-step VAR forecast uses the first forecast in place of $y_{T+1}$: $$ \widehat y_{T+2\mid T} = \widetilde A_1 \begin{pmatrix}9.46\\8.41\end{pmatrix} + \widetilde A_2 \begin{pmatrix}10\\8\end{pmatrix} = \begin{pmatrix}9.7755\\9.3350\end{pmatrix} + \begin{pmatrix}-0.84\\-0.54\end{pmatrix} = \begin{pmatrix}8.9355\\8.7950\end{pmatrix}. $$ The calculation illustrates how a different estimated model produces a different recursive path; it does not establish which model forecasts better.

Common Mistakes and a Practical Check

A frequent error is to apply a coefficient matrix to the wrong lagged change. At horizon one, the short-run term uses the observed $\Delta y_T$; at horizon two, it uses $\widehat{\Delta y}_{T+1\mid T}$. Matrix multiplication must also be carried out component by component: in the first example, a diagonal matrix applied to $(1,0)'$ gives $(0.20,0)'$, not $(0.20,0.20)'$. Either mistake changes every subsequent forecast.

Another mistake is to treat the error-correction term as a rule that forces the variables to converge while holding their average fixed. The term adjusts the vector of levels according to $\alpha$ and the current cointegrating error. Unless the model’s coefficients imply preservation of a particular linear combination, that combination can change. Nor does a narrowing forecasted gap by itself prove that every variable’s forecast is moving toward a fixed level.

Record the available state.
Write down $y_T$, the observed lagged changes required by the model, and any known future deterministic terms.

Calculate the next change.
Evaluate the error-correction term and each short-run term using the appropriate observed or forecasted values.

Update the level and repeat.
Add the forecasted change to the preceding forecasted level, then use this updated forecast state at the next horizon.

Check the representation and arithmetic.
If comparing with a levels VAR, verify that its coefficient matrices are the exact algebraic transformation of the VECM before expecting identical forecasts.

Key takeaway: A VECM forecast repeatedly combines the current forecasted cointegrating error with lagged observed or forecasted changes. An exactly equivalent levels VAR gives the same forecasts; an unrestricted levels VAR generally does not impose the same cointegrating restrictions and can produce a different path.

Check Your Understanding

Use the recursion and distinctions in this tutorial to answer the following questions.

At horizon two in a VECM with one lagged difference, which change enters the short-run term: the observed $\Delta y_T$ or the horizon-one forecasted change?
For a two-variable model with $\beta=(1,-1)'$ and $\alpha=(-0.20,0.10)'$, calculate the multiplier relating the next forecasted gap to the current gap when there are no lagged-change or deterministic terms.
Why can a VECM and a levels VAR produce identical forecasts when their coefficient matrices are algebraically equivalent, but different forecasts when the levels VAR is estimated unrestricted?
Does a shrinking forecasted cointegrating gap imply that the average of the variables is fixed? Explain.
What assumption about future innovations is used to calculate the conditional point forecast?