VARmallit

VARmallit, or VAR models, are a class of multivariate time series models used to analyze dynamic interdependencies among several endogenous variables. A VAR describes each variable as a linear function of its own past values and the past values of all other variables in the system, plus a random disturbance.

In compact form: x_t = A1 x_{t-1} + ... + Ap x_{t-p} + e_t, where x_t is a vector of variables, A_i are coefficient matrices, and e_t is a vector of innovations assumed to be white noise. This structure allows the variables to influence each other over time in a coordinated way.

Estimation and inference: The model is typically estimated by ordinary least squares for each equation, exploiting the fact that the regressors are the same across equations. Lag length p is determined using information criteria such as AIC or BIC, or via cross-validation.

Uses and interpretation: VAR models are widely used for forecasting and for analyzing the dynamic effects of shocks through impulse response functions and forecast error variance decomposition. They are common in macroeconomics to study interactions among variables such as output, inflation, interest rates, and exchange rates.

Extensions and variants: Structural VAR (SVAR) imposes contemporaneous restrictions to identify shocks. Bayesian VARs incorporate prior information to improve estimation in small samples. Factor-augmented VAR (FAVAR) uses latent factors to handle a large set of variables. When non-stationary and cointegrated, VECMs (vector error correction models) may be used.

Limitations: The approach assumes linear relationships and stable dynamics over time; results can be sensitive to the choice of lag length, variable set, and identification in SVARs. Large systems require many parameters and may suffer from overfitting. Data quality and measurement error affect results.

History: The VAR framework was popularized by Christopher Sims in 1980 as an alternative to structural econometric models.

See also: Vector autoregression, impulse response, cointegration, VECM, Bayesian VAR, FAVAR.