Vector Autoregression (VAR)
Vector Autoregression Model · Also known as: VAR, VAR model, vector autoregressive model, multivariate autoregression
Vector Autoregression is a multivariate time-series model in which each variable is regressed on its own lags and the lags of all other variables in the system. Originally proposed by Sims (1980) as a data-driven alternative to large structural macroeconomic models, VAR has become the standard workhorse for dynamic analysis in empirical economics and finance.
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When to use it
Use VAR when you have two or more stationary time series that are theoretically interrelated and you want to study their mutual dynamic interactions without imposing tight a-priori restrictions. It is well-suited for short-horizon forecasting of economic and financial series and for Granger-causality testing. Ensure all series are I(0) before entering the model; if variables are cointegrated, prefer the Vector Error Correction Model (VECM) to avoid misspecification. Avoid VAR when the system contains more than eight or ten variables, as the parameter count grows quadratically with the number of series and degrees of freedom evaporate rapidly.
Strengths & limitations
- Treats all variables as jointly endogenous, avoiding arbitrary exogeneity assumptions.
- Produces reliable multi-step-ahead forecasts that often outperform structural models.
- Provides a coherent framework for Granger-causality tests, impulse-response analysis, and variance decompositions.
- Each equation is estimated by simple OLS, making implementation straightforward.
- Flexible lag structure allows the data to determine the persistence of each relationship.
- Parameter proliferation: a VAR(p) with k variables requires k + pk² parameters, making large systems highly over-parameterised.
- IRFs are not uniquely identified without additional restrictions; a Cholesky decomposition or sign restrictions must be imposed, introducing researcher judgement.
- Requires all variables to be stationary; non-stationary inputs produce spurious inference unless a VECM is used instead.
- Reduced-form VAR coefficients have no direct structural interpretation on their own.
Frequently asked
How do I choose the lag length for a VAR?
Estimate the model for a range of lags (e.g. 1 to 8) and pick the lag that minimises an information criterion — AIC favours model fit and is common in forecasting contexts, while BIC imposes a heavier penalty for additional lags and is preferred when parsimony matters. Always check residual autocorrelation tests afterwards to confirm the chosen lag adequately captures the dynamics.
When should I use VECM instead of VAR?
If Johansen or Engle-Granger tests indicate that the variables share one or more cointegrating relationships, a VAR in first differences is misspecified because it discards the long-run error-correction mechanism. Use a VECM, which nests the VAR dynamics while explicitly modelling the cointegrating relationship and the speed of adjustment back to the long-run equilibrium.
What is the difference between a reduced-form VAR and a structural VAR?
A reduced-form (standard) VAR estimates each equation by OLS with no restrictions other than the lag structure, so its error terms are linear combinations of the underlying structural shocks and cannot be given a direct economic interpretation. A structural VAR (SVAR) imposes identifying restrictions — typically a Cholesky ordering, sign restrictions, or long-run restrictions — to recover the economically meaningful shocks and produce interpretable impulse responses.
How many variables can I include in a VAR?
As a practical rule, keep the number of variables small relative to sample size. A VAR(p) with k variables has k(1 + kp) parameters; with, say, k = 6 variables and p = 4 lags, that is 150 parameters — a substantial burden for any sample shorter than a few hundred observations. Bayesian VAR (BVAR) with shrinkage priors is the standard remedy when a larger system is required.
Does variable ordering matter for impulse-response analysis?
It matters for Cholesky-decomposition IRFs, where the variable listed first is assumed to respond to contemporaneous shocks from no other variable. A different ordering assigns contemporaneous effects differently and can produce noticeably different IRFs. To assess sensitivity, compare IRFs across orderings; if results are robust, the ordering is less critical. Alternatively, use sign restrictions or external instruments to identify shocks without relying on ordering.
Sources
- Sims, C. A. (1980). Macroeconomics and Reality. Econometrica, 48(1), 1–48. DOI: 10.2307/1912017 ↗
- Lütkepohl, H. (2005). New Introduction to Multiple Time Series Analysis. Springer. ISBN: 978-3540401728
How to cite this page
ScholarGate. (2026, June 3). Vector Autoregression Model. ScholarGate. https://scholargate.app/en/econometrics/vector-autoregression
Which method?
Set this method beside its closest kin and read them side by side — the library lays the books on the table; the choice is yours.
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- Structural VAREconometrics↔ compare
- Vector Error Correction ModelEconometrics↔ compare