modern eco no metric modelling - var models_11

8/7/2019 Modern Eco No Metric Modelling - VAR Models_11

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Econometric

Modelling -

Cointegration, VAR

and VECMsEdward Bahaw

March19th 2008

1

Natural Gas Institute of the Americas


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Outline

Regression equations

Spurious regressions

Modern Econometric Modelling techniques

1. Cointegration and error correction

models (ECMs)

2. VAR Vector Autoregressive Modelling

2


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The Regression Equation

tktktt

uXXX ! FFQ ...221

3

Multivariate

Linear regression


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A Regression Equation

4

X2t

X1t

ui

tt 221 FQ !

ttt u! 221 FQ


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Residuals (ut) arise as the regression line

might not pass through all the points

Ordinary least squares minimizes the

square of such residuals

Regression Equation

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Non stationary data

Time

Non - Stationary Time Series

Mean does not representthe value which the time

series approaches

Time

Mean represents thevalue which the series

approaches over time

Stationary Time Series

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Spurious Regression

If the residual term ut is non-stationary about

a mean of zero then the regression equation

would be spurious or unreliable

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IfX1t and X2t are two variables

OLS regression would give:

X1t = + 2X2t + ut,


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Spurious Regression

Residuals

Time

ut is non-stationary

ut corresponding to a spurious regression

ut

8

Mean


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Spurious Regression

Using X1t = + 2X2t + ut,

Then ut = X1t 2X2t

IfX1t and X2t are non-stationary a spurious

regression may be obtained

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Cointegration and Non-

Stationary VariablesIn the model: X1t = + 2X2t + ut,

Or ut = X1t 2X2t

If ut (error term) is stationary about a mean of

zero then cointegration exists.

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Cointegration and Equilbirum

If cointegration exists then there is a

long-run equilibrium relationship

between X1t and X2t

If ut is non-stationary then there is no

cointegration and the model does not

represent a long run equilibrium

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Error Correction Model

IfX1t orX2t are cointegrated then there must

be a short-run relationship which

specifies how the equilibrium is maintained.

This relationship is called the error

correction model (ECM)12


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This model expresses changes in the

dependent variable as a function of:

1. current changes in the independent

variables

2. the residual or error term in the previous

period

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Long run and Short Run

Models

tttt uXX RE (+!(1211

ttt u! 221 F (Long Run or

Equilibrium Equation)

(Short run

equation or ECM)

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tttt u R(+!( 1211

This specification of the error correction modelimplies that a current change in X1t (the dependent

variable) is a function of the current change in X2t

(the dependent variable) as well as Ut-1 (the error in

X1 in the previous period [t-1])

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If ut-1 is positive (an error exists)

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1111.. " ttei

In order to restore equilibrium X1

has to decrease in the following

period.

Thus X1t

is negative


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A positive ut-1 is associated with a

negative X1t

The E coefficient must therefore besignificantly negative

tttt uXX RE (+!( 1111

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Autoregressive Models

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Such models express the current value of a

variable as a function of past values

tktkttt uXXXX ! 12121111 ...FFF

K = lag length

Historical values of the variable help

determine or forecast future values


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Vector Autoregressive

(VAR) Modeling

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tktkttt uXAXAXAX ! ...2211

where Xt is a p1vector

ofp variables

All variables are

endogenous

!

pt

t

t

t

X

X

X

X

/

2

1


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VAR Formulation

In a two variable system (Yt and Zt) a 2 lagorder VAR can be expressed as follows.

tttttt

tttttt

eYYZZZ

eZZYYY

222112211

122112211

!

!

VVPP

HHJJ

!

t

t

t

t

t

t

t

t

e

e

Z

Y

Z

Y

Z

Y

2

1

2

2

22

22

1

1

11

11

PV

HJ

PV

HJ

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VAR Formulation

Using the following representations:

The system of equations can be

expressed more compactly as:

!

t

tt

Z

Y

!11

11

1

PV

HJA

!t

t

t

e

eu

2

1

ttttuXAXAX ! 2211

!

22

222

PV

HJA

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VARs

Applicable to time series data pertaining to

economic data

Perform well at forecasting

Used widely in sensitivity analysis

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VARs and Cointegration

If ut is stationary then the variables are

cointegrated.

That is there is a long run equilibrium

relationship exists among the variables.

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ttttuXAXAX ! 2211


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Vector Error Correction Model

(VECM)The VECM is a VAR in first difference

ttit

p

i

it XXX (*!(

!1

1

1'EF

where

is the matrix of cointegration vectors is the speed of adjustment parameter

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modern eco no metric modelling - var models_11

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