Vector Autoregression (VAR)

VAR models treat all variables as jointly endogenous, eliminating the need for ad-hoc distinctions between dependent and independent variables. This symmetry allows the system to capture complex feedback loops where, for example, interest rates and inflation simultaneously influence each other. The model estimates an equation for each variable using the same set of lagged regressors, creating a closed dynamic system that reflects the true interconnected nature of financial markets.

The optimal number of lags (p) is critical for balancing model fit against over-parameterization. Information criteria penalize the inclusion of extra parameters to prevent overfitting:

• Akaike Information Criterion (AIC): Minimizes prediction error, often selecting longer lags.
• Bayesian Information Criterion (BIC): Imposes a heavier penalty for parameters, favoring parsimonious models.
• Hannan-Quinn Criterion (HQ): Provides a middle ground, consistent in large samples. Selecting too few lags causes omitted variable bias; too many inflates variance.

IRFs trace the dynamic effect of a one-unit orthogonalized shock to one variable's error term on all variables in the system over time. Because the contemporaneous errors are typically correlated, a Cholesky decomposition is applied to isolate structural shocks, though this imposes a recursive causal ordering that must be justified by economic theory. IRFs reveal the persistence, magnitude, and direction of spillover effects, such as how a volatility spike in one asset class propagates to others.

FEVD quantifies the proportion of the forecast error variance of each variable that is attributable to its own shocks versus shocks to other variables in the system. This analysis answers questions like 'What fraction of the uncertainty in equity returns is explained by innovations in credit spreads?' It provides a clear measure of the relative importance of different structural disturbances, helping quantitative researchers identify dominant drivers of market dynamics over specific horizons.

Standard VAR models require all variables to be covariance stationary—meaning the mean, variance, and autocovariance structure are constant over time. If variables contain unit roots (are I(1)), the model risks producing spurious regressions with artificially high R-squared values. In such cases, variables must be differenced before estimation. If the non-stationary series are cointegrated, a Vector Error Correction Model (VECM) is the appropriate specification to preserve long-run equilibrium relationships.

Within a VAR framework, Granger causality tests determine whether past values of one variable provide statistically significant information about the future of another variable, beyond the information contained in its own past. A variable X is said to Granger-cause Y if the coefficients on the lagged X terms are jointly significant in the Y equation. This is a test of predictive causality, not true structural causality, and is essential for building the causal ordering required for Cholesky decompositions.

A statistical hypothesis test that determines whether one time series is useful for forecasting another. In a VAR framework, a variable X Granger-causes Y if past values of X contain information that helps predict Y beyond the information contained in past values of Y alone.

• Tests predictive causality, not true structural causality
• Relies on the assumption that the future cannot cause the past
• Essential for variable selection and ordering in VAR models

A function that traces the dynamic effect of a one-time exogenous shock in one variable on the current and future values of all endogenous variables in a VAR system. IRFs are the primary tool for interpreting the dynamic interactions estimated by a VAR.

• A one-standard-deviation shock to variable i propagates through the system
• Requires orthogonalization (e.g., Cholesky decomposition) to isolate shocks
• Critical for understanding the persistence and transmission of market shocks

A fundamental property of a stochastic process whose unconditional joint probability distribution does not change when shifted in time. A standard VAR model requires all component time series to be covariance stationary.

• A stationary series has a constant mean, variance, and autocorrelation structure
• Non-stationary data (e.g., asset prices) must be differenced to log-returns before VAR estimation
• Failure to ensure stationarity leads to spurious regression and invalid inference

A statistical property of a collection of non-stationary time series that share a common stochastic drift. If a linear combination of integrated variables is stationary, they are cointegrated and possess a long-run equilibrium relationship.

• Pairs trading strategies are built on cointegration
• If cointegration is present, a standard VAR in differences is misspecified
• Requires the use of a Vector Error Correction Model (VECM) instead

A restricted VAR designed for use with cointegrated non-stationary series. A VECM separates the long-run equilibrium relationship (the error correction term) from the short-run dynamic adjustments.

• The error correction term measures the deviation from equilibrium in the previous period
• The speed-of-adjustment coefficient determines how quickly variables revert to equilibrium
• Prevents the loss of long-run information that occurs when differencing data for a standard VAR

A condition where an explanatory variable is correlated with the error term, violating a core assumption of classical regression. In VARs, endogeneity is explicitly modeled rather than avoided—every variable is treated as endogenous and determined by the system.

• Arises from simultaneity, omitted variables, or measurement error
• VAR solves simultaneity bias by making all variables dependent on lags
• Structural VAR (SVAR) imposes theoretical restrictions to recover structural shocks from reduced-form innovations