Lecture6VAR08

6. Vector Autoregression (VAR) and Vector Error Correction (VEC) Models:

6.1 Vector Autoregression (VAR) Model:

The vector autoregression (VAR) model is used for analyzing the interrelation of time series and the dynamic impacts of random disturbances (or innovations) on the system of variables.

Following Enders (1995), consider a simple bivariate first order VAR, i.e. VAR(1), model

y t =β10 - β12x t + α11y t-1 + α12 x t-1 + u yt(36) x t =β20 - β21y t + α21y t-1 + α22 x t-1 + u xt(37)

where it is assumed that both y t and x t are stationary; u yt and u xt are white noise with standard deviations of σy and σx, respectively; u yt and u xt are uncorrelated.

Equations (36) and (37) constitute a two variable first order VAR model. In this system y t is influenced by current and past values of x t, and x t is influenced by current and past values of y t.

Thus the VAR(1) model captures the feedback effects allowing current and past values of the variables in the system.

The coefficients β12 and β21 represent the contemporaneous effects of a unit change of x t

on y t and of y t on x t, respectively;

α12 is the effect of a unit change of x t-1 on y t,

α21 is the effect of a unit change of y t-1 on x t.

Hence y t and x t have mutually contemporaneous effects on each other in the system.

The disturbance terms u yt and u xt are shocks or innovations in y t and x t. The term u yt has an indirect contemporaneous influence on x t if β21≠0, and u xt has an indirect contemporaneous effect on y t if β12≠0.

Equations (36) and (37) represent the structural VAR model. This model uses economic theory

to describe the dynamic relationship between variables. However, appearance of the endogenous variables on both sides of the

equations complicates the estimation and inference processes. A standard VAR model can be applied to overcome the difficulties of the structural VAR model.

The standard form of VAR model for the two variable case can be written as

y t =γ10 + γ11y t-1 + γ12x t-1 + ε1t(38) x t =γ20 + γ21y t-1 + γ22x t-1 + ε2t(39) In (38) and (39), the terms ε1t and ε2t are random innovations or shocks, and they are correlated if there are contemporaneous effects of y t on x t and of x t on y t, but the terms ε1t and ε2t are uncorrelated if there are not contemporaneous effects on each other.

In the system each endogenous variable is determined by a function of the lagged values of the two endogenous variables. The OLS is the appropriate method since only lagged variables are included on the right hand side of the each equation, and also disturbances are assumed to be serially uncorrelated with constant variance.

Two questions arise about the construction of a general VAR model.

First, how can we determine the set of variables to include in a VAR model?

Second, how can we determine the appropriate lag length?

The included variables in a VAR model are selected according to the relevant economic theory. The selected variables must have economic influences on each other. In other terms, there must be causality between them. The overparameterization and loss of degrees of freedom problems must be avoided to capture the important information in the system.

The appropriate lag length must be determined by allowing a different lag length for each equation at each time and choosing the model with the lowest AIC and SBC values. The same sample period must be considered for different lag lengths. If the lag length is too small, the model will be misspecified; if it is too large, the degrees of freedom will be lost.

The VAR analysis determines the interrelationship among the economic time series rather than the parameter estimates.

The residual correlation in the VAR model reveals the interaction of the variables in the previous periods.

The main uses of the VAR model are the impulse response analysis, variance decomposition, and Granger causality tests.

An impulse response function traces the response of the endogenous variables to one standard deviation shock or change to one of the disturbance terms in the system.

A shock to a variable is transmitted to all of the endogenous variables through the dynamic structure of the VAR.

Therefore, an impulse response function shows the interaction between/among the endogenous variables sequence.

Variance decomposition analysis provides information about the dynamic behaviour of the model and the relative importance of each random disturbances or innovation in the VAR.

Variance decomposition shows the proportion

of the movements in the endogenous variable sequence as a result of its own shocks against shocks to other variables.

VAR models are used to test the causality relationship between the variables in the system.

Granger causality provides important information about the exogeneity, in other words x t is defined as an exogenous variable if the current and past values of y t do not affect x t.

In that case, all the coefficients on current and past y t are zero.

Granger noncausality shows that x t sequence is independent of both the u yt shocks and y t sequence.

6.2 Vector Error Correction (VEC) Model:

Engle and Granger (1987) point out that a linear combination of two or more nonstationary series may be stationary. The stationary combination may be interpreted as the cointegration, or equilibrium relationship between the variables. For example, reconsidering the consumption model in the previous section, if the consumption and income are cointegrated, then there exists a long run relationship between them.

However, if they are not cointegrated, then consumption might drift above or below income in the long run, implying that consumers either spend too much or increase savings.

A VEC model is a restricted VAR model. The VEC specification restricts the long run behaviour of the endogenous variables to converge to their long run equilibrium relationships and allow the short run dynamics. Consider the relationship between consumption and income in a simple EC model

ΔC t = θ1(C t-1 - λY t-1) + u1t, θ1>0 (40)

(41) ΔY t = - θ1(C t-1 - λY t-1) + u2t, θ2>0

where u1t and u2t are white noise disturbances, θ1 and θ2 represent the speed of adjustment parameters. θ1, θ2 and λ are the positive parameters.

The cointegrating term (C t-1 - λY t-1) is the error correction term since the deviation from long run equilibrium is corrected gradually through short run adjustments. C t and Y t are the two endogenous variables.

In an EC model, the short run dynamics of the variables in a system are influenced by the deviations from the long run equilibrium. For example, C t and Y t change in response to the previous period’s deviation from long run equilibrium.

In the VEC model if:

?the deviations are positive, i.e.(C t-1 - λY t-1) >0, then the level of income would rise and the level of consumption would fall, as the other things are constant. Long run equilibrium is achieved as C t-1 = λY t-1.

?C t-1 =λY t-1, then C t and Y t change only in response to u1t and u2t shocks.

?θ1 is large, then C t shows greater response to the previous period’s deviation from long run equilibrium.

?θ1 is small, then C t is unresponsive to the previous period’s deviations from equilibrium.

?θ2 =0, then Y t changes only in response to u2t, since ΔY t=u2t. Hence C t changes to eliminate any deviations from long run equilibrium.

?θ1 =0 or θ2 =0, there would not be a causality relationship between cointegrating variables.

?θ1 =0 and θ2 =0, there would not be a long run equilibrium relationship between the two variables. The VEC or cointegration models cannot be used for these variables.

The crucial point of using VEC models is the requirement of cointegration between the two variables with the cointegrating vector (1 -λ). In other words, (C t-1 - λY t-1) must be stationary.