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How do you model short-run dynamics among variables that share long-run equilibrium relationships?
Clive Granger's 1981 insight was that two individually nonstationary time series can move together in a way that keeps their linear combination stationary. He called this cointegration. If GDP and consumption are both I(1) but their ratio is I(0), then a VAR in first differences throws away the equilibrium relationship -- the levels contain information that differencing destroys. The Vector Error Correction Model (VECM) solves this by embedding the long-run equilibrium directly into a first-difference VAR as an error correction term. Granger shared the 2003 Nobel Prize with Robert Engle for this work.
Soren Johansen (1988, 1991) developed the maximum likelihood framework that made cointegration estimation practical for systems of n variables. His procedure jointly estimates the number of cointegrating relationships (the cointegrating rank r), the cointegrating vectors (the long-run equilibrium equations), and the short-run adjustment coefficients (the speed at which each variable corrects deviations from equilibrium). The Johansen trace and maximum eigenvalue tests for the cointegrating rank became the standard diagnostic in applied macroeconometrics.
The VECM representation is: Delta y_t = alpha * beta' * y_{t-1} + Gamma_1 * Delta y_{t-1} + ... + Gamma_{p-1} * Delta y_{t-p+1} + u_t. Here beta' * y_{t-1} is the r x 1 vector of cointegrating residuals (deviations from equilibrium), alpha is the n x r loading matrix (how fast each variable adjusts), and the Gamma matrices capture short-run dynamics in differences. The key structural restriction: the long-run impact matrix Pi = alpha * beta' has reduced rank r < n. This rank restriction distinguishes the VECM from a VAR in differences (r = 0) and a VAR in levels (r = n).
Central banks use VECMs to model systems where economic theory predicts long-run relationships: the Fisher equation linking nominal rates to inflation, purchasing power parity tying exchange rates to price levels, money demand equations relating money supply to income and interest rates. The Bank of England's medium-term macro model includes VECM components. The Riksbank and Norges Bank have published VECM-based analyses of the monetary transmission mechanism. In academic work, King, Plosser, Stock, and Watson (1991) used VECMs to test the permanent-transitory decomposition implied by real business cycle theory.
The input is an n x T matrix of I(1) variables observed at uniform frequency. The practitioner must determine the cointegrating rank r: how many independent long-run equilibrium relationships exist among the n variables. The Johansen trace test and maximum eigenvalue test provide statistical guidance, but economic theory often has the stronger say -- a system with n variables has at most n - 1 cointegrating relationships, and the actual number depends on how many common stochastic trends drive the data.
The model separates into two layers. The long-run layer is beta' * y_{t-1}: an r x 1 vector where each element is a linear combination of the n variables that is stationary. These are the equilibrium relationships. The short-run layer consists of the Gamma matrices on lagged differences, capturing transitory dynamics that wash out in the long run. The loading matrix alpha connects the two layers: it tells you which variables adjust when the system is out of equilibrium and how fast. A zero row in alpha means that variable does not respond to the equilibrium error -- it is weakly exogenous with respect to the long-run parameters.
Estimation uses Johansen's reduced-rank regression (concentrated MLE). First, regress Delta y_t and y_{t-1} on lagged differences to concentrate out the short-run parameters. Then solve a generalized eigenvalue problem on the residual moment matrices. The r largest eigenvalues give the cointegrating vectors (eigenvectors of beta), and the corresponding alpha is recovered by OLS. The trace test statistic is -T * sum of log(1 - lambda_hat_i) for i = r+1 to n; under the null of rank r, this follows a non-standard distribution tabulated by Johansen. Asymptotic critical values depend on the deterministic specification (constant inside or outside the cointegrating space, trend inside or outside).
The Bank of England uses VECM specifications to model the sterling exchange rate against a basket of currencies, anchored by the purchasing power parity cointegrating relationship. The error correction term measures the real exchange rate misalignment -- how far sterling has deviated from its PPP-implied level. The loading coefficient on the exchange rate equation measures how fast sterling corrects: estimates typically suggest a half-life of 3-5 years for PPP deviations, consistent with the Rogoff (1996) consensus.
Money demand estimation is a canonical VECM application. The long-run demand for real money balances is a function of real income and the opportunity cost of holding money (the interest rate). The cointegrating vector recovers the income elasticity and interest semi-elasticity of money demand. The adjustment speed alpha tells the central bank how quickly money holdings return to equilibrium after a shock. The ECB's monetary analysis pillar historically relied on VECM-based money demand equations to assess whether M3 growth was consistent with price stability.
Term structure models use VECMs to capture the comovement among interest rates at different maturities. If the expectations hypothesis holds, the spread between the 10-year and 1-year rate is stationary even though both rates are I(1). The cointegrating vector is the yield spread; the error correction term captures the tendency of the spread to revert to its long-run mean. The Bank for International Settlements uses VECM-based term structure analysis for monitoring global financial conditions.
VECMs fail when the cointegrating relationship is unstable, when the sample is too short for reliable rank determination (T < 80 for systems with n > 4), or when the adjustment to equilibrium is nonlinear. Exchange rate cointegration at short horizons (quarterly data, T < 80) is notoriously fragile: the PPP relationship holds over decades but the adjustment speed is so slow that finite-sample tests cannot distinguish it from a unit root. In these cases, a BVAR with sum-of-coefficients priors may be more robust.
The n x r matrix of adjustment speeds. Column j of alpha tells you how each variable responds to a unit deviation from the j-th cointegrating relationship.
The n x r matrix whose columns are the cointegrating vectors. beta' * y_{t-1} gives the r equilibrium errors.
Pi = alpha * beta'. The n x n matrix of rank r that captures the long-run adjustment mechanism. Reduced rank is the defining structural restriction.
The n x n matrices on lagged first differences. Capture transitory dynamics that do not affect the long-run equilibrium.
Number of linearly independent cointegrating relationships. r = 0 means no cointegration (VAR in differences). r = n means all variables are stationary (VAR in levels).
The r x 1 vector beta' * y_{t-1}. Each element measures the deviation from one long-run equilibrium at time t-1.
The ordered eigenvalues from the reduced-rank regression. The r largest correspond to the cointegrating vectors; the remaining n - r are near zero under the correct rank.
n x 1 vector of reduced-form shocks. E[u_t] = 0, E[u_t u_t'] = Sigma. Same object as in the VAR framework.
Each variable in the system is integrated of order 1: nonstationary in levels but stationary in first differences. The Johansen framework extends to I(2) systems but the standard implementation assumes I(1).
If violated: Including I(0) variables inflates the apparent cointegrating rank. Including I(2) variables causes the Johansen asymptotic distributions to be incorrect. Pre-test with ADF or KPSS unit root tests.
The number of cointegrating relationships r is known or correctly estimated by the trace/max eigenvalue tests. The rank determines the dimension of the long-run equilibrium space.
If violated: Over-specifying r (too many cointegrating vectors) forces spurious stationary combinations into the equilibrium space, biasing short-run dynamics. Under-specifying r omits genuine long-run information, losing forecast accuracy at medium-to-long horizons.
The equilibrium relationship beta' * y_t is linear. No threshold cointegration, no smooth transition between regimes.
If violated: Transaction costs, policy regime switches, and other nonlinearities can make the equilibrium relationship piecewise linear or nonlinear. Linear VECM adjustment speeds will be biased if the true adjustment is asymmetric.
u_t ~ N(0, Sigma). Required for the Johansen MLE to achieve its asymptotic distribution. The trace test has some robustness to non-normality but loses power under heavy tails.
If violated: Non-normal innovations affect the size of the trace and max eigenvalue tests. Bootstrap versions of the Johansen test (Cavaliere, Rahbek, Taylor 2012) are recommended when residuals show excess kurtosis or skewness.
The cointegrating vectors beta and the adjustment speeds alpha are constant over the sample period.
If violated: A structural break in beta means the long-run equilibrium has shifted -- the old cointegrating vector is no longer valid. Johansen tests have low power against the alternative of a break in beta. Gregory-Hansen (1996) tests allow for a single break in the cointegrating relationship.
The number of lagged differences p-1 is large enough to whiten the residuals. Selected by information criteria applied to the unrestricted VAR in levels before imposing the rank restriction.
If violated: Too few lags leave serial correlation in the residuals, invalidating the asymptotic distributions. Too many lags waste degrees of freedom. Standard practice: select p for the VAR in levels, then use p-1 lagged differences in the VECM.