Question

a) Explain what is meant by unrestricted Var models
b) Explain how one can estimate the unrestricted Var model
c) Explain how VECM can be estimated

Answer 1

c) Explain how VECM can be estimated

answer:- The vector autoregressive (VAR) model is a general framework used to describe the dynamic interrelationship among stationary variables. So, the first step in the time-series analysis should be to determine whether the levels of the data are stationary. If not, take the first differences between the series and try again. Usually, if the levels (or log-levels) of your time series are not stationary, the first differences will be.

If the time series is not stationary then the VAR framework needs to be modified to allow consistent estimation of the relationships among the series. The vector error correction (VEC) model is just a special case of the VAR for variables that are stationary in their differences (i.e., I(1)). The VEC can also take into account any cointegrating relationships among the variables.

Consider two time-series variables, t y and .t x Generalizing the discussion about dynamic relationships to these two interrelated variables yields a system of equations :

The equations describe a system in which each variable is a function of its own lag and the lag of the other variable in the system. In this case, the system contains two variables y and x. Together the equations constitute a system known as a vector autoregression (VAR). In this example, since the maximum lag is of order one, we have a VAR(1). If y and x are stationary, the system can be estimated using least squares applied to each equation. If y and x are not stationary in their levels but stationary in differences (i.e., I(1)), then take the difference and estimate ;

using least squares. If y and x are I(1) and cointegrated, then the system of equations is modified to allow for the cointegrating relationship between the I(1) variables. Introducing the cointegrating relationship leads to a model is known as the vector error correction (VEC) model.

ESTIMATING A VEC MODEL :

In the first example, data on the Gross Domestic Product of Australia and the U.S. are used to estimate a VEC model. We decide to use the vector error correction model because (1) the time series is not stationary in their levels but are in their differences (2) the variables are cointegrated. Our initial impressions are gained from looking at plots of the two series.

b) Explain how one can estimate the unrestricted Var model

Answer:- 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 values. ... For example, if the i th variable is GDP, then yi,t is the value of GDP at time t.

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 values. The variables are collected in a k-vector ((k × 1)-matrix) yt, which has as the i th element, yi,t, the observation at a time "t" of the i th variable. For example, if the i th variable is GDP, then yi,t is the value of GDP at time t.

a) Explain what is meant by unrestricted Var models

Vector autoregression (VAR) is a stochastic process model used to capture the linear interdependencies among multiple time series. VAR models generalize the univariate autoregressive model (AR model) by allowing for more than one evolving variable. All variables in a VAR enter the model in the same way: each variable has an equation explaining its evolution based on its own lagged values, the lagged values of the other model variables, and an error term. VAR modeling does 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 intertemporally.