Vector autoregression
Vector autoregression (VAR) is a multivariate time series modeling framework that extends univariate autoregressive models to capture the dynamic interdependencies among multiple endogenous variables, where each variable is expressed as a linear function of its own past values and the past values of all other variables in the system, plus a vector of white noise errors.[1] In its basic form, a VAR(p) model for an n-dimensional time series Y_t is specified as Y_t = c + \Pi_1 Y_{t-1} + \cdots + \Pi_p Y_{t-p} + \varepsilon_t, where c is a constant vector, the \Pi_i are n × n coefficient matrices, p is the lag order, and \varepsilon_t is a multivariate white noise process with covariance matrix \Sigma.[1] This structure assumes covariance stationarity and allows for the inclusion of deterministic trends or exogenous variables as needed.[1] Developed in the late 1970s and early 1980s, VAR models were pioneered by economist Christopher A. Sims as a response to the perceived shortcomings of traditional large-scale macroeconometric models, which relied on "incredible" identifying assumptions like strict exogeneity and exclusion restrictions that often lacked empirical justification and led to unreliable policy inferences.[2] In his seminal 1980 paper, "Macroeconomics and Reality," Sims argued that these conventional approaches imposed too many theoretical priors, stifling data-driven analysis, and proposed VARs as a more atheoretical, reduced-form alternative that lets the data speak with minimal restrictions.[3] This innovation marked a paradigm shift in empirical macroeconomics, earning Sims the 2011 Nobel Prize in Economic Sciences (shared with Thomas Sargent) for advancing time series analysis in economic policy evaluation.[2] VAR models are estimated equation-by-equation using ordinary least squares (OLS), treating the system as a set of seemingly unrelated regressions, which yields efficient and consistent estimates under standard assumptions.[1] Lag length selection is typically guided by information criteria such as the Akaike Information Criterion (AIC), Bayesian Information Criterion (BIC), or Hannan-Quinn (HQ) to balance model fit and parsimony.[1] Once estimated, VARs facilitate a range of analyses, including multi-step forecasting by iteratively applying the model, impulse response functions to trace the effects of shocks on variables over time, variance decompositions to assess shock contributions to forecast error variance, and Granger causality tests to evaluate predictive relationships among variables.[4] Beyond economics, VARs have found applications in finance for modeling asset returns and volatility spillovers,[4] in neuroscience for brain network analysis,[5] and in political science for studying event interdependencies.[6] Extensions such as Bayesian VARs address overfitting in high-dimensional settings by incorporating priors, while structural VARs impose economic identifying restrictions to interpret shocks more causally.[7] Despite their flexibility, VARs require stationary data (or differencing for integrated series) and can suffer from the curse of dimensionality with many variables, prompting ongoing refinements like factor-augmented or sparse VARs.[4]Introduction
Definition and Motivation
Vector autoregression (VAR) is a multivariate time series model that captures the linear interdependencies among multiple variables evolving over time. In this framework, each variable in the system is modeled as a linear function of its own past values, the past values of all other variables in the system, and a contemporaneous error term. This approach treats all variables symmetrically as endogenous, allowing the model to flexibly represent how shocks to one variable propagate through the system to affect others without imposing restrictive assumptions about causality or exogeneity a priori.[8][1] The motivation for VAR models stems from the need to generalize univariate autoregressive processes to multivariate contexts, where economic or financial variables influence each other dynamically over time. Unlike single-variable AR models, VARs enable the examination of feedback loops and joint dynamics, such as how changes in interest rates might impact output and inflation simultaneously in macroeconomic analysis. In particular, they provide a practical tool for policy evaluation by simulating the effects of interventions without relying on overly structured theoretical models that may fail empirical tests.[8][4] VAR models were introduced by Christopher A. Sims in 1980 as an alternative to the large-scale macroeconometric models prevalent at the time, which Sims critiqued for their implausible identifying restrictions and poor forecasting performance. By emphasizing unrestricted reduced-form representations, VARs allow researchers to let the data reveal empirical regularities in variable interactions, supporting applications in forecasting economic time series and assessing policy impacts.[3]Historical Development
The origins of vector autoregression (VAR) models trace back to advancements in multivariate time series analysis during the 1970s, building on univariate autoregressive techniques to capture interdependencies among multiple economic variables.[1] These early efforts laid the groundwork for handling dynamic relationships in economic data without rigid theoretical restrictions. The formal introduction of VAR models to econometrics occurred in 1980, when Christopher Sims published "Macroeconomics and Reality" in Econometrica, proposing VARs as a flexible, data-driven alternative to traditional structural econometric models.[3] Sims' work represented a significant critique of the Cowles Commission approach, which relied on simultaneous equation systems with strong identifying assumptions that often lacked empirical support and failed to account for rational expectations.[3] Alongside Thomas Sargent, Sims emphasized VARs' ability to empirically assess macroeconomic relationships without imposing "incredible" a priori restrictions, influencing a shift toward more agnostic empirical methods in macroeconomics.[9] Mark Watson further advanced this framework through collaborative research on VAR estimation and inference, solidifying its role in policy analysis and forecasting.[10] In the late 1980s and 1990s, VAR models evolved from purely reduced-form specifications to structural interpretations, addressing identification challenges in Sims' original atheoretical setup. A key milestone was Olivier Blanchard and Danny Quah's 1989 paper in the American Economic Review, which introduced long-run restrictions to decompose VAR innovations into demand and supply shocks, enabling clearer causal inferences in macroeconomic disturbances. By the 1990s, structural VAR (SVAR) models gained prominence for policy counterfactuals, though debates persisted on assumption validity.[10] Post-2000, integration with Bayesian methods addressed overfitting in high-dimensional VARs; the Minnesota prior, originally developed by Thomas Doan, Robert Litterman, and Sims in 1984, was refined for shrinkage estimation, enhancing forecasting accuracy and parameter stability in large systems. This Bayesian evolution has since become standard for incorporating prior economic beliefs while maintaining empirical flexibility.[11]Model Specification
Core Components and Notation
A vector autoregression (VAR) of order p, denoted VAR(p), models the dynamic interdependencies among a set of n endogenous time series variables as a system of linear equations. The general form is given by Y_t = A_1 Y_{t-1} + A_2 Y_{t-2} + \cdots + A_p Y_{t-p} + \epsilon_t, where Y_t is an n \times 1 vector of endogenous variables at time t, each A_i (for i = 1, \dots, p) is an n \times n matrix of coefficients capturing the influence of the i-th lag of the system on the current values, p is the lag order selected based on information criteria or theoretical considerations, and \epsilon_t is an n \times 1 vector of error terms. The endogenous variables in Y_t are treated as jointly determined within the system, allowing each to respond to lagged values of all variables, including itself, which distinguishes VAR models from univariate autoregressions. Deterministic terms, such as a constant intercept vector or linear trends, are often incorporated to account for non-stochastic components; for instance, the model can be extended to Y_t = c + A_1 Y_{t-1} + \cdots + A_p Y_{t-p} + \epsilon_t, where c is an n \times 1 vector of intercepts representing the mean of each series under stationarity assumptions. Trends may be added as d t, with d an n \times 1 vector of trend coefficients, particularly for non-stationary data after differencing. The error terms \epsilon_t are assumed to be white noise, satisfying E(\epsilon_t) = 0, E(\epsilon_t \epsilon_t') = \Sigma where \Sigma is a positive definite n \times n covariance matrix, and E(\epsilon_t \epsilon_s') = 0 for t \neq s, ensuring no serial correlation across time periods. These assumptions imply that innovations are contemporaneously correlated but unpredictable from past information.Stationarity and Order of Integration
In vector autoregression (VAR) models, stationarity is defined in a multivariate context as covariance stationarity, which requires that the mean vector of the process remains constant over time, the covariance matrix is time-invariant, and the autocovariance matrices depend solely on the time lag between observations. This condition ensures that the statistical properties of the multivariate time series do not change systematically with time, allowing for reliable inference and forecasting within the VAR framework. Violation of covariance stationarity can lead to unstable parameter estimates and misleading interpretations of dynamic relationships among variables. The order of integration classifies the stationarity properties of time series variables in VAR models. A process is integrated of order zero, denoted I(0), if it is covariance stationary. In contrast, a process is integrated of order one, I(1), if it requires first differencing to become stationary, meaning the levels exhibit unit roots and non-constant variance over time. For multivariate systems, variables may share common stochastic trends, leading to cointegration, where linear combinations of I(1) variables are I(0), capturing long-run equilibrium relationships despite individual non-stationarity. Prior to estimating a VAR model, pre-estimation tests for stationarity are essential to determine the order of integration. Unit root tests, such as the Augmented Dickey-Fuller (ADF) test, are commonly applied to each variable in the system to assess the presence of unit roots under the null hypothesis of non-stationarity. The ADF test augments the basic Dickey-Fuller regression with lagged differences to account for serial correlation, providing critical values for inference since standard t-statistics do not apply under the unit root null. If variables are found to be I(1), differencing is used to induce stationarity, transforming the model to operate on \Delta Y_t = Y_t - Y_{t-1}, where Y_t is the vector of variables. Fitting a VAR model to non-stationary data without addressing integration issues results in spurious regressions, where apparent statistical significance arises from shared trends rather than true economic relationships. Granger and Newbold demonstrated this phenomenon through simulations of independent random walks, showing inflated R-squared values and invalid t-statistics in regressions of non-stationary series. For cointegrated systems, a VAR in levels may still be valid but requires reparameterization into a vector error correction model (VECM) to explicitly incorporate the error correction mechanism that enforces long-run equilibria while modeling short-run dynamics.Companion Matrix Representation
A vector autoregression of order p, denoted VAR(p), with n endogenous variables can be reformulated into an equivalent VAR(1) model using the companion matrix representation. This involves stacking the current and lagged values of the vector Y_t into a state vector Z_t = (Y_t', Y_{t-1}', \dots, Y_{t-p+1}')', which is an np \times 1 vector. The model then becomes Z_t = F Z_{t-1} + u_t, where F is the np \times np companion matrix composed of blocks from the original coefficient matrices A_i (i=1,\dots,p), and u_t = (\epsilon_t', 0_{n \times 1}', \dots, 0_{n \times 1}')' is the error vector with the innovation \epsilon_t in the first block and zeros elsewhere. The companion matrix F has a specific block structure that embeds the dynamics of the higher-order lags. Its first block row consists of the coefficient matrices A_1, A_2, \dots, A_p, each of size n \times n. The subsequent block rows feature identity matrices I_n along the subdiagonal to shift the lags, with zeros elsewhere: F = \begin{pmatrix} A_1 & A_2 & \cdots & A_{p-1} & A_p \\ I_n & 0 & \cdots & 0 & 0 \\ 0 & I_n & \cdots & 0 & 0 \\ \vdots & \vdots & \ddots & \vdots & \vdots \\ 0 & 0 & \cdots & I_n & 0 \end{pmatrix}. This structure ensures that the reformulation preserves the original VAR(p) dynamics while reducing it to a first-order form. The companion matrix representation offers several analytical advantages. It facilitates eigenvalue analysis of F, where the process is stable if all eigenvalues of F have modulus less than 1, equivalent to the roots of the characteristic polynomial lying outside the unit circle.Illustrative Example
To illustrate the specification and computation of a vector autoregression (VAR) model, consider a simple bivariate VAR(1) with two variables: GDP growth rate y_{1t} (in percentage points) and inflation rate y_{2t} (also in percentage points). This setup, with n=2 variables and lag order p=1, captures the interdependence between the two series through lagged values, assuming the process is stationary. The system of equations is specified as follows: \begin{align} y_{1t} &= \nu_1 + a_{11} y_{1,t-1} + a_{12} y_{2,t-1} + \epsilon_{1t}, \\ y_{2t} &= \nu_2 + a_{21} y_{1,t-1} + a_{22} y_{2,t-1} + \epsilon_{2t}, \end{align} where \boldsymbol{\nu} = \begin{pmatrix} \nu_1 \\ \nu_2 \end{pmatrix} is the intercept vector, \mathbf{A}_1 = \begin{pmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{pmatrix} is the coefficient matrix for lag 1, and \boldsymbol{\epsilon}_t = \begin{pmatrix} \epsilon_{1t} \\ \epsilon_{2t} \end{pmatrix} \sim N(\mathbf{0}, \boldsymbol{\Sigma}) with covariance matrix \boldsymbol{\Sigma} = \begin{pmatrix} 1 & 0.5 \\ 0.5 & 1 \end{pmatrix}. For this hypothetical example, inspired by standard simulated bivariate VAR(1) models, set \nu_1 = 0.5, \nu_2 = 1.0, a_{11} = 0.6, a_{12} = 0.3, a_{21} = 0.2, and a_{22} = 0.7; these coefficients ensure stationarity, as the eigenvalues of \mathbf{A}_1 lie inside the unit circle (0.4 and 0.9). The off-diagonal elements a_{12} and a_{21} reflect cross-variable dynamics: a shock to GDP growth influences future inflation, and vice versa. The population mean is \boldsymbol{\mu} = (I - \mathbf{A}_1)^{-1} \boldsymbol{\nu} \approx (7.50, 8.33)^\top. To demonstrate computation, simulate a short sample path starting from initial values y_{1,0} = 0, y_{2,0} = 0, and specific shocks to highlight propagation. Assume shocks \boldsymbol{\epsilon}_1 = (0.5, 0)^\top, \boldsymbol{\epsilon}_2 = (0, 0.3)^\top, \boldsymbol{\epsilon}_3 = (-0.2, 0.1)^\top, and \boldsymbol{\epsilon}_t = (0, 0)^\top for t \geq 4; these are drawn from the specified normal distribution for illustration. The first few periods are computed step-by-step:- At t=1: y_{11} = 0.5 + 0.6(0) + 0.3(0) + 0.5 = 1.00, y_{21} = 1.0 + 0.2(0) + 0.7(0) + 0 = 1.00.
- At t=2: y_{12} = 0.5 + 0.6(1.0) + 0.3(1.0) + 0 = 1.40, y_{22} = 1.0 + 0.2(1.0) + 0.7(1.0) + 0.3 = 2.20.
- At t=3: y_{13} = 0.5 + 0.6(1.4) + 0.3(2.2) - 0.2 = 1.80, y_{23} = 1.0 + 0.2(1.4) + 0.7(2.2) + 0.1 = 2.92.
- At t=4: y_{14} = 0.5 + 0.6(1.8) + 0.3(2.92) + 0 = 2.46, y_{24} = 1.0 + 0.2(1.8) + 0.7(2.92) + 0 = 3.40.
| Time t | GDP Growth y_{1t} | Inflation y_{2t} | Shocks (\epsilon_{1t}, \epsilon_{2t}) |
|---|---|---|---|
| 0 | 0.00 | 0.00 | - |
| 1 | 1.00 | 1.00 | (0.5, 0) |
| 2 | 1.40 | 2.20 | (0, 0.3) |
| 3 | 1.80 | 2.92 | (-0.2, 0.1) |
| 4 | 2.46 | 3.40 | (0, 0) |