Centre of Full Employment and Equity
Short-run models and Error
Correction Mechanisms
Professor Bill Mitchell
Director, Centre of Full Employment and Equity
Department of Economics
University of Newcastle
Australia
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Centre of Full Employment and Equity
Objectives
• To introduce the concept of a short-run
model in economics.
• To show how short- and long-run models
interact.
• To explain the concept of an Error
Correction Mechanism (ECM).
• To show how ECM and cointegration work
together.
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Centre of Full Employment and Equity
Long-run model review
• Economic theory is essentially static and
mostly considers equilibrium relationships.
• Equilibrium (long-run) relations are
normally in terms of levels.
• The problem is that with non-stationary
variables we are prone to finding spurious
relationships if we run regressions in levels.
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Centre of Full Employment and Equity
Figure 1 Z1, Z2 and Z4
• The Z variables were
simulated using
random walk functions
with r = 1:
y t  r y t 1  u t
• Any relation between
them is spurious and
because they contain
stochastic trends.
30
20
10
0
-10
-20
-30
60
65
70
75
Z1
80
85
Z2
90
95
00
Z4
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Centre of Full Employment and Equity
So this equation exhibits “good” econometric results but is
in fact spurious and tells us nothing at all.
The “good” is qualified b/c the DW statistic is the clue.
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Centre of Full Employment and Equity
The clue is in the residuals…
30
20
10
8
0
4
-10
0
-4
-8
-12
60
65
70
75
R e s id u a l
80
85
Ac tu a l
90
95
00
F itte d
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Centre of Full Employment and Equity
The long-run model quandary
• So how do we proceed?
• In the 1970s, the approach was to take
differences?
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Centre of Full Employment and Equity
Taking differences removes trends
3
 1 y t  y t  y t 1
2
1
0
-1
-2
-3
60
65
70
75
80
85
90
95
00
90
95
00
DZ1
3
2
1
0
-1
-2
-3
60
65
70
75
80
85
DZ2
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Centre of Full Employment and Equity
Taking differences…
• Do we still have a relationship?
• To test it we would run
Z1t = b0 + b1Z2t + b2Z4t + et
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Centre of Full Employment and Equity
Levels and differences
(3)
y t  a 0  a1 x t  a 2 z t  e t
 yt
(4)
 y t  1   a1  x t  x t 1   a 2  z t  z t 1    e t  e t 1 
 y t  a1  x t  a 2  z t  u t
u t   et
Problems ?
Question 1: What are the problems of estimating economic
relationships in difference form like Equation (4), given
that it can overcome the problem of non-stationarity in the
levels of the variables concerned?
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Centre of Full Employment and Equity
Error Correction Approach
• This approach attempts to use differenced
data to model the short-run adjustments but
also take into account and estimate long-run
information.
• Consider this long-run model:
y t   0   1 xt
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Centre of Full Employment and Equity
Equilibrium and disequilibrium
(6)
y t   0   1 xt
(6 a )
y t       xt  0
(6 b )
y t       xt  0
Question 2: What are the properties of Equation (6)? Does
it tell you about the path of adjustment for y if x changes?
Question 3: What are some of the reasons why equilibrium
may not hold in every period? In a forecasting environment
why would it be necessary to know about the nature of
disequilibrium adjustment paths?
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Centre of Full Employment and Equity
Equilibrium and disequilibrium
(6)
y t       xt
(6 a )
y t       xt  0
(6 b )
y t       xt  0
When Equation 6(b) holds we cannot observe the
relationship in Equation (6).
But we can observe the short-run, dynamic relationship that
would reduce to Equation (6) whenever equilibrium occurs.
So we need to learn a bit about the short-run models.
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Centre of Full Employment and Equity
Short-run model
• Short-run models are also called adjustment
functions or dynamic models or lagged models.
• A typical (simplified) version is the first-order
model:
y t  a 0   1 y t 1  b 1 x t  b 2 x t 1  u t
• The order is selected to “soak” up the serial
correlation (the “missing dynamics”)
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Centre of Full Employment and Equity
Properties of short-run model
(8)
y t  a 0   1 y t 1  b 1 x t  b 2 x t 1  u t
Question 4: What are the properties of Equation (8)? Tell a
story in words about the process through which the longrun relationship is re-established if x was to change in a
particular period?
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Centre of Full Employment and Equity
Properties of short-run model
(6)
y t       xt
(8)
y t  a 0   1 y t 1  b 1 x t  b 2 x t 1  u t
Question 4: Parameter b1 measures the immediate impact
of a change in x on y. It is not the long-run impact that
would occur from one equilibrium to another though.
Why not? What is the difference between b1 and 1? Can
you find an expression that links b1 and 1?
Solve the steady-state properties of (8).
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Centre of Full Employment and Equity
Solving for the steady-state…
y t       xt
(6)
(8)
y t  a 0   1 y t 1  b 1 x t  b 2 x t 1  u t
y t  y t 1  y
*
x t  x t 1  x
y (1   1 )  a 0   b 1  b 2  x
*
y 
*
a0
(1   1 )

 b1  b 2 
(1   1 )
x
*
*
*
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Centre of Full Employment and Equity
Questions 6 to 9 …
• Question 6: Assume that 1 = 0.9, b1 = 0.6 and b2 = 0.3.
Starting from an equilibrium position, how long does it
take for y to return to that equilibrium, if x increases by a
unit and remains at that new level?
• Question 7: What is the change in y in the first period after
the shock? What is the change in y in the second period?
What is the total change?
• Question 8: How is the shift in equilibrium dependent on
the value taken by the AR parameter?
• Spreadsheet demonstration.
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Centre of Full Employment and Equity
Error Correction models
• The basic dynamic model may also suffer
from non-stationarity problems.
• We have seen the differencing is
unsatisfactory.
• Error Correction Mechanism (ECM) models
begin with the basic short-run model.
• After re-parameterising, the ECM form has
both dynamic and steady-state information
in the one equation.
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ECM questions
Question 9: See if you can perform the re-parameterisation
to get the ECM model which combines differences and
levels. It is shown below as Equation (10).
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ECM form and the steady-state
(8)
y t  a 0   1 y t 1  b 1 x t  b 2 x t 1  u t
y t  y t  1  a 0  y t  1   1 y t 1  b 1 x t  b 1 x t 1  b 1 x t 1  b 2 x t 1  u t

 b1  b 2  
a0
 y t  b 1  x t  (1   1 )  y t 1 

x t 1 
(1   1 )
(1   1 )


(10)
 y t  b 1  x t  (1   1 )  y t 1   0   0 x t 1 
Question 10: Would you say that Equation (8) and Equation
(10) are equivalent? What are the advantages of Equation (10)
relative to Equation (8)?
Centre of Full Employment and Equity
ECM form and the steady-state
(10)

 b1  b 2  
a0
 y t  b 1  x t  (1   1 )  y t 1 

x t 1 
(1   1 )
(1   1 )


(10)
 y t  b 1  x t  (1   1 )  y t 1   0   0 x t 1 
y* 
a0
(1   1 )

 b1  b 2 
(1   1 )
x
*
Question 11: Provide an interpretation of the expression in
square brackets in Equation (10). Have you already
encountered an expression like this earlier in this lecture?
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Centre of Full Employment and Equity
ECM form and the steady-state
(10)

 b1  b 2  
a0
 y t  b 1  x t  (1   1 )  y t 1 

x t 1 
(1   1 )
(1   1 )


(6  )
y t 1   0   1 x t 1  e t 1
e t 1   y t 1   0   1 x t 1 
You can see that the term in square brackets is equivalent to
the expression for disequilibrium in the steady-state.
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Centre of Full Employment and Equity
ECM form and the steady-state
(10)

 b1  b 2  
a0
 y t  b 1  x t  (1   1 )  y t 1 

x t 1 
(1   1 )
(1   1 )


(6  )
y t 1   0   1 x t 1  e t 1
e t 1   y t 1   0   1 x t 1 
Question 12: Is the term in square brackets stationary given
that it is in terms of levels? Under what conditions will it be
stationary?
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Centre of Full Employment and Equity
Cointegration and ECM model
(6)
y t   0   1 x t  et
(10)

 b1  b 2  
a0
 y t  b 1  x t  (1   1 )  y t 1 

x t 1 
(1   1 )
(1   1 )


(10 )
 y t  b 1  x   e t 1  u t
Two-step procedure for estimating the model:
• Test for cointegration in Equation (6).
• If null accepted then the residuals would be stationary.
• Estimate (10) with residuals from CI Equation (6) as
the ECM term.
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Centre of Full Employment and Equity
End of Talk
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