Like the autoregressive model, each variable has an equation modelling its evolution over time. This equation includes the variable's lagged (past) values, the lagged values of the other variables in the model, and an error term. VAR models do not require as much knowledge about the forces influencing a variable as do structural models with simultaneous equations. The only prior knowledge required is a list of variables which can be hypothesized to affect each other over time.
A VAR model describes the evolution of a set of k variables, called endogenous variables, over time. Each period of time is numbered, t = 1, ..., T. The variables are collected in a vector, yt, which is of length k. (Equivalently, this vector might be described as a (k × 1)-matrix.) The vector is modelled as a linear function of its previous value. The vector's components are referred to as yi,t, meaning the observation at time t of the i th variable. For example, if the first variable in the model measures the price of wheat over time, then y1,1998 would indicate the price of wheat in the year 1998.
VAR models are characterized by their order, which refers to the number of earlier time periods the model will use. Continuing the above example, a 5th-order VAR would model each year's wheat price as a linear combination of the last five years of wheat prices. A lag is the value of a variable in a previous time period. So in general a pth-order VAR refers to a VAR model which includes lags for the last p time periods. A pth-order VAR is denoted "VAR(p)" and sometimes called "a VAR with p lags". A pth-order VAR model is written as
The variables of the form yt−i indicate that variable's value i time periods earlier and are called the "ith lag" of yt. The variable c is a k-vector of constants serving as the intercept of the model. Ai is a time-invariant (k × k)-matrix and et is a k-vector of error terms. The error terms must satisfy three conditions:
For a general example of a VAR(p) with k variables, see General matrix notation of a VAR(p).
A VAR(1) in two variables can be written in matrix form (more compact notation) as
(in which only a single A matrix appears because this example has a maximum lag p equal to 1), or, equivalently, as the following system of two equations
Each variable in the model has one equation. The current (time t) observation of each variable depends on its own lagged values as well as on the lagged values of each other variable in the VAR.
Writing VAR(p) as VAR(1)
A VAR with p lags can always be equivalently rewritten as a VAR with only one lag by appropriately redefining the dependent variable. The transformation amounts to stacking the lags of the VAR(p) variable in the new VAR(1) dependent variable and appending identities to complete the number of equations.
The equivalent VAR(1) form is more convenient for analytical derivations and allows more compact statements.
Structural vs. reduced form
A structural VAR with p lags (sometimes abbreviated SVAR) is
where c0 is a k × 1 vector of constants, Bi is a k × k matrix (for every i = 0, ..., p) and εt is a k × 1 vector of error terms. The main diagonal terms of the B0 matrix (the coefficients on the ith variable in the ith equation) are scaled to 1.
The error terms εt (structural shocks) satisfy the conditions (1) - (3) in the definition above, with the particularity that all the elements in the off diagonal of the covariance matrix are zero. That is, the structural shocks are uncorrelated.
Writing the first equation explicitly and passing y2,t to the right hand side one obtains
Note that y2,t can have a contemporaneous effect on y1,t if B0;1,2 is not zero. This is different from the case when B0 is the identity matrix (all off-diagonal elements are zero — the case in the initial definition), when y2,t can impact directly y1,t+1 and subsequent future values, but not y1,t.
From an economic point of view, if the joint dynamics of a set of variables can be represented by a VAR model, then the structural form is a depiction of the underlying, "structural", economic relationships. Two features of the structural form make it the preferred candidate to represent the underlying relations:
1. Error terms are not correlated. The structural, economic shocks which drive the dynamics of the economic variables are assumed to be independent, which implies zero correlation between error terms as a desired property. This is helpful for separating out the effects of economically unrelated influences in the VAR. For instance, there is no reason why an oil price shock (as an example of a supply shock) should be related to a shift in consumers' preferences towards a style of clothing (as an example of a demand shock); therefore one would expect these factors to be statistically independent.
2. Variables can have a contemporaneous impact on other variables. This is a desirable feature especially when using low frequency data. For example, an indirect tax rate increase would not affect tax revenues the day the decision is announced, but one could find an effect in that quarter's data.
By premultiplying the structural VAR with the inverse of B0
one obtains the pth order reduced VAR
Note that in the reduced form all right hand side variables are predetermined at time t. As there are no time t endogenous variables on the right hand side, no variable has a direct contemporaneous effect on other variables in the model.
However, the error terms in the reduced VAR are composites of the structural shocks et = B0−1εt. Thus, the occurrence of one structural shock εi,t can potentially lead to the occurrence of shocks in all error terms ej,t, thus creating contemporaneous movement in all endogenous variables. Consequently, the covariance matrix of the reduced VAR
can have non-zero off-diagonal elements, thus allowing non-zero correlation between error terms.
Estimation of the regression parameters
Starting from the concise matrix notation (for details see this annex):
OLS estimator: for a model with a constant, k variables and p lags.
In a matrix notation, this gives:
Estimation of the estimator's covariance matrix
The covariance matrix of the parameters can be estimated as
Degrees of freedom
Vector autoregression models often involve the estimation of many parameters. For example, with seven variables and four lags, each matrix of coefficients for a given lag length is 7 by 7, and the vector of constants has 7 elements, so a total of 49×4 + 7 = 203 parameters are estimated, substantially lowering the degrees of freedom of the regression (the number of data points minus the number of parameters to be estimated). This can hurt the accuracy of the parameter estimates and hence of the forecasts given by the model.
Consider the first-order case (i.e., with only one lag), with equation of evolution
for evolving (state) vector and vector of shocks. To find, say, the effect of the j-th element of the vector of shocks upon the i-th element of the state vector 2 periods later, which is a particular impulse response, first write the above equation of evolution one period lagged:
Use this in the original equation of evolution to obtain
then repeat using the twice lagged equation of evolution, to obtain
From this, the effect of the j-th component of upon the i-th component of is the i, j element of the matrix
It can be seen from this induction process that any shock will have an effect on the elements of y infinitely far forward in time, although the effect will become smaller and smaller over time assuming that the AR process is stable — that is, that all the eigenvalues of the matrix A are less than 1 in absolute value.
An estimated VAR model can be used for forecasting, and the quality of the forecasts can be judged, in ways that are completely analogous to the methods used in univariate autoregressive modelling.
Christopher Sims has advocated VAR models, criticizing the claims and performance of earlier modeling in macroeconomiceconometrics. He recommended VAR models, which had previously appeared in time series statistics and in system identification, a statistical specialty in control theory. Sims advocated VAR models as providing a theory-free method to estimate economic relationships, thus being an alternative to the "incredible identification restrictions" in structural models. VAR models are also increasingly used in health research for automatic analyses of diary data or sensor data.
R: The package vars includes functions for VAR models. Other R packages are listed in the CRAN Task View: Time Series Analysis.
Python: The statsmodels package's tsa (time series analysis) module supports VARs. PyFlux has support for VARs and Bayesian VARs.
^van der Krieke; et al. (2016). "Temporal Dynamics of Health and Well-Being: A Crowdsourcing Approach to Momentary Assessments and Automated Generation of Personalized Feedback (2016)". Psychosomatic Medicine: 1. doi:10.1097/PSY.0000000000000378. PMID27551988.