Evaluating Inflation Targeting Using a Macro Eco No Metric Model
Modern Eco No Metric Modelling - VAR Models_11
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Transcript of Modern Eco No Metric Modelling - VAR Models_11
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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
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The Regression Equation
tktktt
uXXX ! FFQ ...221
3
Multivariate
Linear regression
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A Regression Equation
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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
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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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Error Correction Model
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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Error Correction Model
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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Error Correction Model
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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Error Correction Model
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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