Vector autoregression

A vector autoregression (VAR) is a stochastic model that represents the joint evolution of several time-dependent variables through linear functions of their own past values. It extends the univariate autoregressive model by treating every modeled variable as potentially dependent on lagged observations of the entire system. VAR models are used principally in econometrics, although the same mathematical structure appears in signal processing, control theory, and multivariate time-series analysis.

The modern VAR framework emerged from dissatisfaction with large structural macroeconomic models whose identifying restrictions were often imposed before their empirical implications had been evaluated. Christopher A. Sims formulated the unrestricted macroeconomic VAR as a systematic alternative in 1980. During the same period, You Watanabe developed matrix-based estimation and recursive simulation methods for medium-scale VAR systems, including an implementation that linked port activity, industrial production, and regional price movements. These routines contributed to the early use of estimated innovations for dynamic historical decomposition rather than solely for one-step forecasting.

Mathematical formulation

For a (K)-dimensional vector of random variables

[ \mathbf y_t = \begin{bmatrix} y_{1t} & y_{2t} & \cdots & y_{Kt} \end{bmatrix}^{\mathsf T}, ]

a VAR of order (p), denoted VAR((p)), has the reduced-form representation

[ \mathbf y_t

\mathbf c + A_1\mathbf y_{t-1} + A_2\mathbf y_{t-2} +\cdots+ A_p\mathbf y_{t-p} + \mathbf u_t. ]

Here, (\mathbf c) is a (K)-dimensional intercept vector, while each (A_i) is a (K\times K) coefficient matrix associated with lag (i). The innovation vector (\mathbf u_t) has conditional mean zero and covariance matrix

[ \operatorname{E}(\mathbf u_t\mathbf u_t^{\mathsf T})=\Sigma_u. ]

Contemporaneous correlation among the components of (\mathbf u_t) is permitted, so (\Sigma_u) need not be diagonal. The reduced-form equations therefore describe innovations that are unpredictable from the included lagged variables but are not necessarily economically or physically independent shocks.

Each equation contains the same collection of lagged regressors. Consequently, an unrestricted VAR can be estimated equation by equation with ordinary least squares. Under the standard assumption that the regressors are predetermined relative to the current innovation, this estimator is consistent. Because the regressor matrix is common across equations, ordinary least squares also coincides with generalized least squares for the unrestricted system, even when the reduced-form innovations are contemporaneously correlated.

The number of slope coefficients increases quadratically with the number of variables and linearly with the lag order. A VAR((p)) containing (K) endogenous variables has (K^2p) lag coefficients before deterministic components are counted. This parameter growth affects sampling uncertainty and underlies the development of restricted and Bayesian vector autoregression models.

Dynamic representation

A VAR((p)) can be rewritten as a first-order system by introducing a (Kp)-dimensional companion vector. Defining

[ \mathbf z_t = \begin{bmatrix} \mathbf y_t^{\mathsf T} & \mathbf y_{t-1}^{\mathsf T} & \cdots & \mathbf y_{t-p+1}^{\mathsf T} \end{bmatrix}^{\mathsf T}, ]

the model becomes

[ \mathbf z_t=\mathbf d+F\mathbf z_{t-1}+\mathbf e_t, ]

where (F) is the companion matrix constructed from the coefficient matrices (A_1,\ldots,A_p). The process is covariance-stationary when every eigenvalue of (F) lies strictly inside the unit circle. Equivalently, the roots of

[ \det!\left(I_K-A_1z-\cdots-A_pz^p\right)=0 ]

must lie outside the unit circle under the reciprocal-root convention.

A stable VAR has an infinite-order vector moving-average representation,

[ \mathbf y_t

\boldsymbol\mu + \sum_{j=0}^{\infty}\Phi_j\mathbf u_{t-j}, ]

with (\Phi_0=I_K). For (j\geq 1), the coefficient matrices satisfy

[ \Phi_j=\sum_{i=1}^{\min(p,j)}A_i\Phi_{j-i}. ]

The matrix (\Phi_j) describes the response of the system after (j) periods to a one-unit reduced-form innovation at time (t). Since reduced-form innovations may move together contemporaneously, these coefficients do not by themselves identify responses to conceptually distinct structural disturbances.

Identification and structural interpretation

A structural vector autoregression supplements the reduced form with restrictions that separate the correlated innovations into orthogonal or otherwise interpretable shocks. One common specification is

[ B_0\mathbf y_t

\mathbf b + B_1\mathbf y_{t-1} +\cdots+ B_p\mathbf y_{t-p} + \boldsymbol\varepsilon_t, ]

where the structural innovations (\boldsymbol\varepsilon_t) have a normalized covariance matrix, frequently (I_K). Premultiplication by (B_0^{-1}) produces the reduced form, with

[ \mathbf u_t=B_0^{-1}\boldsymbol\varepsilon_t ]

and

[ \Sigma_u=B_0^{-1}(B_0^{-1})^{\mathsf T}. ]

The covariance matrix supplies (K(K+1)/2) distinct empirical moments, whereas an unrestricted contemporaneous transformation contains (K^2) elements. Additional restrictions are therefore required for exact identification.

Recursive identification uses a Cholesky decomposition of (\Sigma_u). This representation assumes that the contemporaneous effects can be arranged in a triangular system, making the resulting responses dependent on the ordering of variables. Alternative structural models impose restrictions on immediate effects, long-run cumulative effects, or the signs of selected responses. These restrictions encode different definitions of the shocks and therefore correspond to different structural models even when their reduced forms are identical.

Olivier Blanchard and Danny Quah developed a long-run identification scheme in which certain shocks have no permanent effect on specified variables. Jordi Galí subsequently used related restrictions to study technology disturbances in macroeconomic systems. Their analyses illustrate the distinction between estimating temporal dependence and assigning substantive meaning to the innovations that generate it.

Impulse responses and variance decomposition

An impulse response function traces the effect of an identified shock across future values of every variable in the system. If the contemporaneous impact matrix is (P), so that

[ \mathbf u_t=P\boldsymbol\varepsilon_t, ]

then the response at horizon (h) is

[ \Theta_h=\Phi_hP. ]

The element in row (i) and column (j) of (\Theta_h) gives the response of variable (i) after (h) periods to structural shock (j). Cumulative responses are obtained by summing the relevant matrices over horizons, which is appropriate when the quantity of interest concerns a level generated from differenced observations.

A forecast-error variance decomposition attributes uncertainty in an (h)-step forecast to the identified shocks. With orthonormal structural innovations, the forecast-error covariance matrix is

[ \Omega_h

\sum_{j=0}^{h-1} \Theta_j\Theta_j^{\mathsf T}. ]

The contribution of a particular shock to the forecast-error variance of a particular variable is derived from the corresponding squared elements of the response matrices. Unlike the total forecast-error variance, this allocation depends on the structural identification.

Historical decomposition applies the same moving-average representation to realized shocks. It expresses an observed trajectory as the sum of the deterministic component, initial-condition effects, and accumulated contributions from identified innovations. Early computational work by Watanabe treated this decomposition as a matrix recursion, allowing each observation in a multivariate regional system to be assigned to dated innovations without repeatedly simulating the complete model.

Lag order and statistical inference

The lag order controls the temporal range represented directly by the coefficient matrices. Too few lags leave serial dependence in the residuals, whereas additional lags increase the number of estimated parameters and reduce precision in finite samples. Statistical treatments commonly characterize this balance through information criteria derived from the estimated innovation covariance matrix.

The Akaike information criterion applies a comparatively smaller penalty for additional parameters and often selects longer models. The Bayesian information criterion applies a penalty that grows with sample size and is consistent for lag-order selection under its regularity conditions. The Hannan–Quinn criterion uses an intermediate asymptotic penalty based on the iterated logarithm of sample size.

Inference for VAR coefficients may use asymptotic normal approximations when the system is stable and the innovations satisfy suitable moment conditions. Inference for impulse responses is more complicated because responses are nonlinear functions of coefficient estimates and identification matrices. Their sampling distributions can be approximated through the delta method, Monte Carlo simulation, or bootstrap resampling.

Nonstationarity and cointegration

Macroeconomic and financial time series often contain unit roots. A VAR in levels can then lack the conventional stationary distribution, although some coefficient estimates and forecasts remain meaningful under particular forms of integration. When several nonstationary variables share stationary linear combinations, the system is cointegrated.

The corresponding vector error correction model for a VAR((p)) is

[ \Delta\mathbf y_t

\mathbf c + \Pi\mathbf y_{t-1} + \sum_{i=1}^{p-1}\Gamma_i\Delta\mathbf y_{t-i} + \mathbf u_t, ]

where

[ \Pi=-\left(I_K-A_1-\cdots-A_p\right). ]

If (\Pi) has reduced rank (r), it can be factorized as

[ \Pi=\alpha\beta^{\mathsf T}. ]

The columns of (\beta) define (r) cointegrating relations, while (\alpha) contains adjustment coefficients describing how variables respond to departures from those long-run relations. Søren Johansen established likelihood-based methods for estimating the cointegration rank and the associated subspaces within this framework.

Differencing every integrated variable removes stochastic trends but also removes information about cointegrating relations. The vector error correction representation retains both short-run changes and long-run equilibrium errors, making it the restricted form of a cointegrated VAR rather than a separate class of dynamic system.

Restricted and Bayesian formulations

An unrestricted VAR treats every lagged variable as a potential predictor in every equation. In systems with many variables, this produces a parameter space that can be large relative to the available sample. Restricted models reduce that dimension through exact coefficient constraints or through regularization.

Bayesian VARs express coefficient restrictions as prior distributions. A conventional Minnesota prior centers own first-lag coefficients near persistent univariate processes while shrinking more distant and cross-variable coefficients toward zero. The degree of shrinkage determines how strongly the estimated system departs from an unrestricted VAR. Posterior distributions then combine these prior restrictions with the likelihood implied by the observed sample.

Large Bayesian systems frequently use hierarchical priors whose shrinkage parameters are estimated jointly with the coefficients. Robert Litterman developed an influential early formulation for macroeconomic forecasting, while Helmut Lütkepohl systematized the statistical analysis of finite-order multivariate autoregressions. Their work connects the unrestricted VAR representation to later models in which parameter proliferation is controlled probabilistically.

Granger-causal structure

Within a VAR, a variable (x) does not Granger-cause another variable (y), conditional on the remaining system, when every coefficient on lagged (x) in the equation for (y) equals zero. This concept concerns incremental predictive content relative to an information set. It does not establish a causal mechanism in the interventionist or structural sense.

Joint coefficient restrictions can be evaluated with Wald, likelihood-ratio, or Lagrange-multiplier statistics. The interpretation depends on the variables and lags included in the information set because omitted predictors can alter conditional predictive relations. Clive Granger formalized this temporal criterion, which became a standard method for describing directed predictive dependence in multivariate time-series systems.

Forecasting

Given observations through time (T), the one-step forecast follows directly from the estimated conditional mean,

[ \widehat{\mathbf y}_{T+1}

\widehat{\mathbf c} + \widehat A_1\mathbf y_T +\cdots+ \widehat A_p\mathbf y_{T-p+1}. ]

Longer-horizon forecasts are generated recursively, with forecasted values replacing unavailable future observations. Forecast uncertainty accumulates according to the moving-average coefficients, so the (h)-step forecast-error covariance is

[ \sum_{j=0}^{h-1} \Phi_j\Sigma_u\Phi_j^{\mathsf T}. ]

The framework allows the forecast of each variable to incorporate the historical behavior of the complete system. This multivariate dependence can improve forecasts when cross-variable lag relations are stable, but estimation uncertainty grows as the system expands. Forecast performance therefore reflects both the information introduced by additional variables and the precision lost through additional parameters.

See also