Vector autoregression

Vector autoregression (VAR) is an econometric model used to capture the evolution and the interdependencies between multiple time series, generalizing the univariate AR models. All the variables in a VAR are treated symmetrically by including for each variable an equation explaining its evolution based on its own lags and the lags of all the other variables in the model. Based on this feature, Christopher Sims advocates the use of VAR models as a theory-free method to estimate economic relationships, thus being an alternative to the "incredible identification restrictions" in structural models [Christopher A. Sims, 1980, "Macroeconomics and Reality", Econometrica 48] .

pecification

Definition

A VAR model describes the evolution of a set of "k" variables (called "endogenous variables") over the same sample period ("t" = 1, ..., "T") as a linear function of only their past evolution. The variables are collected in a "k" × 1 vector "y_t", which has as the i^th element "y_i,t" the time "t" observation of variable "y_i". For example, if the "i"^th variable is GDP, then "y_i,t" is the value of GDP at "t".

A "(reduced) p-th order VAR", denoted "VAR(p)", is

:$y\_t\; =\; c\; +\; A\_1\; y\_\{t-1\}\; +\; A\_2\; y\_\{t-2\}\; +\; cdots\; +\; A\_p\; y\_\{t-p\}\; +\; e\_t,$

where "c" is a "k" × 1 vector of constants (intercept), "A_i" is a "k" × "k" matrix (for every "i" = 1, ..., "p") and e"_t" is a "k" × 1 vector of error terms satisfying

#$mathrm\{E\}(e\_\{t\})\; =\; 0,$ — every error term has mean zero;
#$mathrm\{E\}(e\_\{t\}e\_\{t\}\text{'})\; =\; Omega,$ — the contemporaneous covariance matrix of error terms is Ω (a "n" × "n" positive definite matrix);
#$mathrm\{E\}(e\_\{t\}e\_\{t-k\}\text{'})\; =\; 0,$ for any non-zero "k" — there is no correlation across time; in particular, no serial correlation in individual error terms.

The "l"-periods back observation "y"_"t"−l is called the "l"-th "lag" of "y". Thus, a "p"th-order VAR is also called a VAR with "p" lags.

Order of integration of the variables

Note that all the variables used have to be of the same order of integration. We have so the following cases:

*All the variables are I(0) (stationary): one is in the standard case, ie. a VAR in level
*All the variables are I(d) (non-stationary) with d>1:
**The variables are cointegrated: the error correction term has to be included in the VAR. The model becomes a Vector error correction model (VECM) which can be seen as a restricted VAR.
**The variables are not cointegrated: the variables have first to be differenced d times and one has a VAR in difference.

Concise matrix notation

One can write a VAR("p") with a concise matrix notation:

:$Y=BZ\; +U\; ,$

Details of the matrices are in a separate page.

Example

For a general example of a VAR(p) with "k" variables, please see this page.

A VAR(1) in two variables can be written in matrix form (more compact notation) as

:

or, equivalently, as the following system of two equations

:$y\_\{1,t\}\; =\; c\_\{1\}\; +\; A\_\{1,1\}y\_\{1,t-1\}\; +\; A\_\{1,2\}y\_\{2,t-1\}\; +\; e\_\{1,t\},$:$y\_\{2,t\}\; =\; c\_\{2\}\; +\; A\_\{2,1\}y\_\{1,t-1\}\; +\; A\_\{2,2\}y\_\{2,t-1\}\; +\; e\_\{2,t\}.,$

Note that there is one equation for each variable in the model. Also note that the current (time "t") observation of each variable depends on its own lags as well as on the lags 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 merely stacking the lags of the VAR("p") variable in the new VAR(1) dependent variable and appending identities to complete the number of equations.

For example, the VAR(2) model

:$y\_\{t\}=c\; +\; A\_\{1\}y\_\{t-1\}\; +\; A\_\{2\}y\_\{t-2\}\; +\; e\_\{t\}$

can be recast as the VAR(1) model

::

where "I" is the identity matrix.

The equivalent VAR(1) form is more convenient for analytical derivations and allows more compact statements.

tructural vs. reduced form

tructural VAR

A "structural VAR with p lags" is

:$B\_0\; y\_t\; =\; c\_0\; +\; B\_1\; y\_\{t-1\}\; +\; B\_2\; y\_\{t-2\}\; +\; cdots\; +\; B\_p\; y\_\{t-p\}\; +\; epsilon\_t,$

where "c"₀ is a "k" × 1 vector of constants, "B_i" is a "k" × "k" matrix (for every "i" = 0, ..., "p") and "ε"_"t" is a "k" × 1 vector of error terms. The main diagonal terms of the "B"₀ matrix (the coefficients on the "i"^th variable in the "i"^th equation) are scaled to 1.

The error terms ε"_t" ("structural shocks") satisfy the conditions (1) - (3) in the definition above, with the particularity that all the elements off the main diagonal of the covariance matrix $mathrm\{E\}(epsilon\_tepsilon\_t\text{'})\; =\; Sigma$ are zero. That is, the structural shocks are uncorrelated.

For example, a two variable structural VAR(1) is:

:

where

:

that is, the variances of the structural shocks are denoted $mathrm\{var\}(epsilon\_i)\; =\; sigma\_i^2$ ("i" = 1, 2) and the covariance is $mathrm\{cov\}(epsilon\_1,epsilon\_2)\; =\; 0$.

Writing the first equation explicitly and passing "y_2,t" to the right hand side one obtains

:$y\_\{1,t\}\; =\; c\_\{0;1\}\; -\; B\_\{0;1,2\}y\_\{2,t\}\; +\; B\_\{1;1,1\}y\_\{1,t-1\}\; +\; B\_\{1;1,2\}y\_\{2,t-2\}\; +\; epsilon\_\{1,t\},$

Note that "y"_2,"t" can have a contemporaneous effect on "y_1,t" if "B"_0;1,2 is not zero. This is different from the case when "B"₀ is the identity matrix (all off-diagonal elements are zero — the case in the initial definition), when "y"_2,"t" can impact directly "y"_1,"t"+1 and subsequent future values, but not "y"_1,"t".

Because of the parameter identification problem, ordinary least squares estimation of the structural VAR would yield inconsistent parameter estimates. This problem can be overcome by rewriting the VAR in reduced form.

From an economic point of view it is considered that, 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.

Reduced VAR

By premultiplying the structural VAR with the inverse of "B"₀

: $y\_t\; =\; B\_0^\{-1\}c\_0\; +\; B\_0^\{-1\}\; B\_1\; y\_\{t-1\}\; +\; B\_0^\{-1\}\; B\_2\; y\_\{t-2\}\; +\; cdots\; +\; B\_0^\{-1\}\; B\_p\; y\_\{t-p\}\; +\; B\_0^\{-1\}epsilon\_t,$

and denoting

: $B\_\{0\}^\{-1\}\; c\_0\; =\; c,quad\; B\_\{0\}^\{-1\}B\_i\; =\; A\_\{i\}\; ext\{\; for\; \}i\; =\; 1,\; dots,\; p\; ext\{\; and\; \}B\_\{0\}^\{-1\}epsilon\_t\; =\; e\_t$

one obtains the "p"th order reduced VAR

:$y\_t\; =\; c\; +\; A\_1\; y\_\{t-1\}\; +\; A\_2\; y\_\{t-2\}\; +\; cdots\; +\; A\_p\; y\_\{t-p\}\; +\; e\_t$

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 "e"_"t" = "B"₀⁻¹"ε"_"t". Thus, the occurrence of one structural shock "ε_i,t" can potentially lead to the occurrence of shocks in all error terms "e_j,t", thus creating contemporaneous movement in all endogenous variables. Consequently, the covariance matrix of the reduced VAR

:$Omega\; =\; mathrm\{E\}(e\_t\; e\_t\text{'})\; =\; mathrm\{E\}\; (B\_0^\{-1\}\; epsilon\_t\; epsilon\_t\text{'}\; (B\_0^\{-1\})\text{'})\; =\; B\_0^\{-1\}Sigma(B\_0^\{-1\})\text{'},$

can have non-zero off-diagonal elements, thus allowing non-zero correlation between error terms.

Estimation

Estimation of the regression parameters

Starting from the concise matrix notation (for details see this annex):

:$Y=BZ\; +U\; ,$

*Ordinary least squares(OLS) estimation of each equation in the reduced VAR is both consistent and asymptotically efficient. It is furthemore equal to the maximum likelihood estimator (MLE) (Hamilton 1994, p 293).

The OLS estimator for B is given by:

:$hat\; B=\; YZ^\{\text{'}\}(ZZ^\{\text{'}\})^\{-1\}$

*Generalized least square (GLS) yields the same estimation, as was shown by Zellner (1962).

The GLS estimator for B is given by:

:$mbox\{Vec\}(hat\; B)\; =\; ((ZZ^\{\text{'}\})^\{-1\}\; Z\; otimes\; I\_\{k\})\; mbox\{Vec\}(Y)$

Where $otimes$ denotes the Kronecker product and Vec the vectorization of the matrix "Y".

Estimation of the covariance matrix of the errors

As in the standard case, the MLE estimator of the covariance matrix differs from the OLS estimator.

MLE estimator: $hat\; Sigma\; =\; frac\{1\}\{T\}\; sum\_\{t=1\}^T\; hat\; epsilon\_that\; epsilon\_\{t\}^\{\text{'}\}$

OLS estimator: $hat\; Sigma\; =\; frac\{1\}\{T-kp-1\}\; sum\_\{t=1\}^T\; hat\; epsilon\_that\; epsilon\_t^\text{'}$ for a model with a constant, "k" variables and "p" lags

In a matrix notation, this gives:

: $hat\; Sigma\; =\; frac\{1\}\{T-kp-1\}\; (Y-hat\{B\}Z)(Y-hat\{B\}Z)^\text{'}.$

Note that for the GLS estimator the covariance matrix of the errors becomes:

:

Estimation of the covariance matrix of the parameters

The covariance matrix of the parameters can be estimated as

: $hat\; mbox\{Cov\}\; (mbox\{Vec\}(hat\; B))\; =(\{ZZ\text{'}\})^\{-1\}\; otimeshat\; Sigma.,$

References

* Walter Enders, "Applied Econometric Time Series", 2nd Edition, John Wiley & Sons 2003, ISBN 0-471-23065-0
* James D. Hamilton. "Time Series Analysis". Princeton University Press. 1995.
* Helmut Lütkepohl. "New Introduction to Multiple Time Series Analysis". Springer. 2005.
* Zellner (1962) An Efficient Method of Estimating Seemingly Unrelated Regressions and Tests for Aggregation Bias. "Journal of the American Statistical Association", Vol. 57, No. 298 (Jun., 1962), pp. 348-368

Notes