A univariate time series *y** _{t}* is

An *n*-dimensional time series *y** _{t}* is

Cointegration is distinguished from traditional economic equilibrium, in which a balance of forces produces stable long-term levels in the variables. Cointegrated variables are generally unstable in their levels, but exhibit mean-reverting "spreads" (generalized by the cointegrating relation) that force the variables to move around common stochastic trends. Cointegration is also distinguished from the short-term synchronies of positive covariance, which only measures the tendency to move together at each time step. Modification of the VAR model to include cointegrated variables balances the short-term dynamics of the system with long-term tendencies.

The tendency of cointegrated variables to revert to common stochastic
trends is expressed in terms of *error-correction*.
If *y** _{t}* is
an

In general, there may be multiple cointegrating relations among
the variables in *y** _{t}*,
in which case the vectors

$$\Delta {y}_{t}=C{y}_{t-1}+{\displaystyle \sum _{i=1}^{q}{B}_{i}\Delta {y}_{t-i}}+{\epsilon}_{t}.$$

If the variables in *y** _{t}* are
all

By collecting differences, a VEC(*q*) model
can be converted to a VAR(*p*) model in levels, with *p* = *q*+1:

$${y}_{t}={A}_{1}{y}_{t-1}+\mathrm{...}+{A}_{p}{y}_{t-p}+{\epsilon}_{t}.$$

Conversion between VEC(*q*)
and VAR(*p*) representations of an *n*-dimensional
system are carried out by the functions `vec2var`

and `var2vec`

using the formulas:

$$\begin{array}{l}{A}_{1}=C+{I}_{n}\text{+}{B}_{1}\hfill \\ {A}_{i}={B}_{i}-{B}_{i-1}\text{,}i=2,\mathrm{...},q\hfill \\ {A}_{p}=-{B}_{q}\hfill \end{array}\}\text{VEC(}q\text{)toVAR(}p=q+1)\text{(using}vec2var\text{)}$$

$$\begin{array}{l}C={\displaystyle \sum _{i=1}^{p}{A}_{i}-{I}_{n}}\hfill \\ {B}_{i}=-{\displaystyle \sum _{j=i+1}^{p}{A}_{j}}\hfill \end{array}\}\text{VAR(}p\text{)toVEC(}q=p-1\text{)(using}var2vec\text{)}$$

Because of the equivalence of the
two representations, a VEC model with a reduced-rank error-correction
coefficient is often called a *cointegrated VAR model*.
In particular, cointegrated VAR models can be simulated and forecast
using standard VAR techniques.

The cointegrated VAR model is often augmented with exogenous
terms *D**x*:

$$\Delta {y}_{t}=A{B}^{\prime}{y}_{t-1}+{\displaystyle \sum _{i=1}^{q}{B}_{i}\Delta {y}_{t-i}}+Dx+{\epsilon}_{t}.$$

Variables in *x* may include seasonal or interventional
dummies, or deterministic terms representing trends in the data. Since
the model is expressed in differences ∆*y** _{t}*,
constant terms in

Case | Form of AB′y_{t − 1} + Dx | Model Interpretation |

H2 | AB′y_{t − 1} | There are no intercepts or trends in the cointegrating relations and there are no trends in the data. This model is only appropriate if all series have zero mean. |

H1* | A(B′y_{t − 1} + c_{0}) | There are intercepts in the cointegrating relations and there are no trends in the data. This model is appropriate for nontrending data with nonzero mean. |

H1 | A(B′y_{t − 1}+c_{0}) + c_{1} | There are intercepts in the cointegrating relations and there
are linear trends in the data. This is a model of deterministic
cointegration, where the cointegrating relations eliminate
both stochastic and deterministic trends in the data. |

H* | A(B′y_{t − 1} + c_{0} + d_{0}t) + c_{1} | There are intercepts and linear trends in the cointegrating
relations and there are linear trends in the data. This is a model
of stochastic cointegration, where the cointegrating
relations eliminate stochastic but not deterministic trends in the
data. |

H | A(B′y_{t − 1} + c_{0} + d_{0}t) + c_{1} + d_{1}t | There are intercepts and linear trends in the cointegrating relations and there are quadratic trends in the data. Unless quadratic trends are actually present in the data, this model may produce good in-sample fits but poor out-of-sample forecasts. |

In Econometrics Toolbox™, deterministic terms outside of
the cointegrating relations, *c*_{1} and *d*_{1},
are identified by projecting constant and linear regression coefficients,
respectively, onto the orthogonal complement of *A*.

Integration and cointegration both present opportunities for transforming variables to stationarity. Integrated variables, identified by unit root and stationarity tests, can be differenced to stationarity. Cointegrated variables, identified by cointegration tests, can be combined to form new, stationary variables. In practice, it must be determined if such transformations lead to more reliable models, with variables that retain an economic interpretation.

Generalizing from the univariate case can be misleading. In the standard Box-Jenkins [15] approach to univariate ARMA modeling, stationarity is an essential assumption. Without it, the underlying distribution theory and estimation techniques become invalid. In the corresponding multivariate case, where the VAR model is unrestricted and there is no cointegration, choices are less straightforward. If the goal of a VAR analysis is to determine relationships among the original variables, differencing loses information. In this context, Sims, Stock, and Watson [97] advise against differencing, even in the presence of unit roots. If, however, the goal is to simulate an underlying data-generating process, integrated levels data can cause a number of problems. Model specification tests lose power due to an increase in the number of estimated parameters. Other tests, such as those for Granger causality, no longer have standard distributions, and become invalid. Finally, forecasts over long time horizons suffer from inconsistent estimates, due to impulse responses that do not decay. Enders [35] discusses modeling strategies.

In the presence of cointegration, simple differencing is a model misspecification, since long-term information appears in the levels. Fortunately, the cointegrated VAR model provides intermediate options, between differences and levels, by mixing them together with the cointegrating relations. Since all terms of the cointegrated VAR model are stationary, problems with unit roots are eliminated.

Cointegration modeling is often suggested, independently, by economic theory. Examples of variables that are commonly described with a cointegrated VAR model include:

Money stock, interest rates, income, and prices (common models of money demand)

Investment, income, and consumption (common models of productivity)

Consumption and long-term income expectation (Permanent Income Hypothesis)

Exchange rates and prices in foreign and domestic markets (Purchasing Power Parity)

Spot and forward currency exchange rates and interest rates (Covered Interest Rate Parity)

Interest rates of different maturities (Term Structure Expectations Hypothesis)

Interest rates and inflation (Fisher Equation)

Since these theories describe long-term equilibria among the variables, accurate estimation of cointegrated models may require large amounts of low-frequency (annual, quarterly, monthly) macroeconomic data. As a result, these models must consider the possibility of structural changes in the underlying data-generating process during the sample period.

Financial data, by contrast, is often available at high frequencies (hours, minutes, microseconds). The mean-reverting spreads of cointegrated financial series can be modeled and examined for arbitrage opportunities. For example, the Law of One Price suggests cointegration among the following groups of variables:

Prices of assets with identical cash flows

Prices of assets and dividends

Spot, future, and forward prices

Bid and ask prices

- Test for Cointegration Using the Engle-Granger Test
- Determine Cointegration Rank of VEC Model
- Estimate VEC Model Parameters Using egcitest
- Simulate and Forecast a VEC Model
- Test for Cointegration Using the Johansen Test
- Estimate VEC Model Parameters Using jcitest
- Compare Approaches to Cointegration Analysis
- Test Cointegrating Vectors
- Test Adjustment Speeds

Was this topic helpful?