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NPTELProject

EconometricModelling

VinodGuptaSchoolofManagement

Module25:Vector Autoregressive Model

Lecture 39: Vector Autoregressive Model

RudraP.Pradhan

VinodGuptaSchoolofManagement

IndianInstituteofTechnologyKharagpur,India

Email:rudrap@vgsom.iitkgp.ernet

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MODULE OBJECTIVE

This module attempts to explore the causal nexus between multiple variables in the time series

setting. In this section, we highlight the followings:

1. WHAT IS VECTOR AUTOREGRESSIVE (VAR)?

2. MODELING OF VAR

3. ESTIMATION OF VAR MODELS

4. INTERPRETATION OF VAR MODEL

WHAT ARE VECTOR AUTOREGRESSIVE MODELS?

Vector autoregression (VAR) is a statistical tool for capturing the linear interdependencies

among multiple time series variables. VAR models generalize the univariate autoregression (AR)

models. All the variables in a VAR are treated symmetrically and each variable has an equation

explaining its evolution based on its own lags and the lags of all the other variables in the model.

VAR modeling does not require expert knowledge, which previously had been used in structural

models with simultaneous equations. The 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 yt, which has as

the ith element yi,t the time t observation of variable yi. For example, if the ith variable is GDP,

then yi,t is the value of GDP at t. The general VAR model is as follows:

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where c is a k 1 vector of constants (intercept), Ai is a k k matrix (for every i = 1, ..., p) and et

is a k 1 vector of error terms satisfying the followings:

THEVAR REQUIREMENTS:

Endogenous variable: explained earlier

Exogenous variables: explained earlier

Autoregressive Scheme: explained earlier

Lag Length: explained earlier

Matrix: explained earlier

Order of integration

Cointegration

Granger causality

ORDER OF INTEGRATION AND COINTEGRATION OF THE VARIABLES

It is for checking the stationarity (or unit root). The concept of unit root is explained earlier. Note

that all the variables in the VAR system should be same order of integration. We have so the

following cases:

All the variables are I (0) (stationary): one is in the standard case, i.e. a VAR in level

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All the variables are I (d) (non-stationary) with d>0

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.

VAR IN MATRIX FORMAT

The VAR (p) can be written in a matrix format, which is as follows:

For simplicity, we will take two variables. The VAR (1) in two variables can be written in matrix

form as follows:

or, equivalently, as the following system of two equations

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.

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The 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 instance, the VAR (2) model

It can be re-designed as follows:

Where, I is the unity matrix. The equivalent VAR (1) form is more convenient for analytical

derivations and allows more compact statements.

THE STRUCTURAL VAR

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 t 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 t (structural shocks) satisfy the

conditions (1) - (3) in the definition above, with the particularity that all the elements off the

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main diagonal of the covariance matrix are zero. That means the structural shocks

are uncorrelated.

For instance, a two variable structural VAR (1) is as follows:

Where

That is, the variances of the structural shocks are denoted (I = 1, 2) and the

covariance is . 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.

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:

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Error terms are not correlated. The structural, economic shocks which drive the dynamics of

the economic/ finance 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 (an example of supply shock) should be related to a shift in consumers' preferences

towards a style of clothing (an example of demand shock); therefore one would expect these

factors to be statistically independent.

Variables can have a contemporaneous impact on other variables. This is a sufficient

condition especially when using low frequency data. For instance, 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.

THE REDUCED FORM VAR

The reduced from of VAR can be written as follows:

Where,

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

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contemporaneous effect on other variables in the model. However, the error terms in the reduced

VAR are composites of the structural shocks et = B01t. Thus, the occurrence of one structural

shock i,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