Vector autoregression (VAR) is a statistical model used to capture the relationship between multiple quantities as they change over time.
The only prior knowledge required is a list of variables which can be hypothesized to affect each other over time.
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.
A lag is the value of a variable in a previous time period.
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.
The error terms must satisfy three conditions: The process of choosing the maximum lag p in the VAR model requires special attention because inference is dependent on correctness of the selected lag order.
The following cases are distinct: One can stack the vectors in order to write a VAR(p) as a stochastic matrix difference equation, with a concise matrix notation: 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 transformation amounts to stacking the lags of the VAR(p) variable in the new VAR(1) dependent variable and appending identities to complete the precise number of equations.
The equivalent VAR(1) form is more convenient for analytical derivations and allows more compact statements.
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
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 problem can be overcome by rewriting the VAR in reduced form.
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: By premultiplying the structural VAR with the inverse of B0 and denoting 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.
denotes the Kronecker product and Vec the vectorization of the indicated matrix.
In a matrix notation, this gives: The covariance matrix of the parameters can be estimated as[citation needed] 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
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
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.
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.
Sio Iong Ao and R. E. Caraka found that the artificial neural network can improve its performance with the addition of the hybrid vector autoregression component.