Vector Autoregression

A VAR treats each series in a multivariate time system as endogenous. In a VAR(p), every variable is regressed on p lagged values of itself and p lags of every other included variable, plus deterministic terms or exogenous inputs when specified. This symmetric setup lets interactions emerge from the data rather than designating one series as the sole response and the rest as fixed predictors. The fitted system can generate joint forecasts. For a hypothetical system of demand, price and inventory in a VAR(2), the demand equation uses two lagged observations of all three variables. The price equation and inventory equation do the same. This flexibility costs parameters: with K variables and p lags, each equation has Kp lag coefficients plus deterministic terms. A short time series with many variables can therefore be overfit, and lag order should be chosen with information criteria, residual checks and time-ordered forecast validation. VAR analysis assumes the dynamics are appropriately represented. A standard stable VAR is commonly used for stationary series. If variables have unit roots but share a long-run equilibrium, a vector error-correction model may preserve both short-run changes and cointegration. Differencing all series can remove that long-run relation. Check stationarity, cointegration and deterministic trends before selecting a model form. Granger causality tests whether lagged values of one variable add predictive information for another variable conditional on the system's other terms. It is a statement about temporal predictability under a model, not intervention-based causal identification. Omitted confounders, structural breaks and measurement timing can affect conclusions. VARs can also be used for impulse-response analysis and forecast error variance decomposition, but those interpretations depend on identification assumptions. Explain the model order, variables, transformations and uncertainty when reporting results; the equations alone do not reveal a causal mechanism.