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Slides to Lecture 3 of Introductory Dynamic
Macroeconomics.
Linear Dynamic Models (Ch 2.5-2.8 of IDM)
The error correction model (Ch 2.5 in IDM)
In lecture 2 we
• used the ADL model to make the concept of dynamic multiplier precise,
• Saw also that the simple ADL model can be generalized, and
• how macroeconomic hypotheses can be specified as ADLs.
Ragnar Nymoen
University of Oslo, Department of Economics
In this lecture, start by recalling the typology:
January 30, 2006
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Table 2: A model typology.
Type
Equation
Restrictions
And concentrate on the ECM and the Homogenous ECM. These models are
particularly useful in dynamic macroeconomics.
ADL
yt = β0 + β1xt + β2xt−1 + αyt−1 + εt. None
Static
yt = β0 + β1xt + εt.
β2 = α = 0
Random walk
yt = β0 + yt−1 + εt
β1 = β2 = 0, α = 1.
DL
yt = β0 + β1xt + β2xt−1 + εt
α=0
can be transformed to error correction form.
Differenced data1
∆yt = β0 + β1∆xt + εt
β2 = −β1, α = 1
We then give an interpretation.
ECM
∆yt = β0 + β1∆xt + (β1 + β2)xt−1
+(α − 1)yt−1 + εt
None
We start by showing how the ADL model
Homogenous ECM ∆yt = β0 + β1∆xt
+(α − 1)(yt−1 − xt−1) + εt
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The difference operator ∆is defined as ∆zt ≡ zt − zt−1.
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yt = β0 + β1xt + β2xt−1 + αyt−1 + εt.
β1 + β2 = −(α − 1)
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(1)
The ECM
Transformation of ADL to an error correction model (ECM)
∆yt = β0 + β1∆xt + (β1 + β2)xt−1 + (α − 1)yt−1 + εt
brings to the open what is implicit in the ADL, namely that the growth of yt
is explained by the growth of the explanatory variable and the past levels of xt
and yt.
Step 1: Subtract yt−1 on both sides of the ADL equation, which gives
yt − yt−1 = β0 + β1xt + β2xt−1 + (α − 1)yt−1 + εt
Step 2: Add and subtract β1xt−1 on the right hand side:
yt − yt−1 = β0 + β1(xt − xt−1) + (β1 + β2)xt−1 + (α − 1)yt−1 + εt
Step 3: Use the difference operator ∆, meaning for example ∆yt = yt − yt−1,
and write the ECM as:
∆yt = β0 + β1∆xt + (β1 + β2)xt−1 + (α − 1)yt−1 + εt
(3)
The occurrence of both a variable’s growth and its level is a defining characteristic of genuinely dynamic models. In 1929 Frisch put this down as a
definitional trait of dynamic models:
(2)
Step 1-3 is a re-parameterization: If the values y0, y1, y2, y3.... satisfy (1), they
also satisfy (2).
“A theoretical law which is such that it involves the notion of rate
of change or the notion of speed of reaction (in terms of time) is a
dynamic law. All other laws are static.”
The transformation ADL to ECM shows that Frisch was right!
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The name error correction, stems from the fact that
If yt and xt are measured in logarithms. ∆yt and ∆xt are their respective
growth rates (check the appendix to IDM if in doubt
For example
and the ECM version of the estimated ADL equation (1.18) in IDM is
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embodies adjustments of yt with respect to deviations from the long-run equilibrium relationship between y and x. To make this interpretation clear, collect
yt−1 and xt−1 inside a bracket:
½
Ct − Ct−1
Ct − Ct−1
∆ ln Ct = ln(Ct/Ct−1) = ln(1 +
)≈
.
Ct−1
Ct−1
∆ ln Ct = 0.04 + 0.13∆ ln(IN Ct) + 0.21 ln IN Ct−1 − 0.21 ln Ct−1
∆yt = β0 + β1∆xt + (β1 + β2)xt−1 + (α − 1)yt−1 + εt
(4)
¾
β + β2
∆yt = β0 + β1∆xt − (1 − α) y − 1
+ εt,
(5)
x
1−α
t−1
and assume the following long-run relationship for a situation with constant
growth rates (possibly zero) in y and x:
y ∗ = k + γx,
(6)
where y ∗ denotes the steady-state equilibrium of yt . We want equation (5) to
be consistent with a steady-state, and the slope coefficient γ must therefore
be equal to the long-run multiplier of the ADL-ECM, that is:
β + β2
γ = δlong−run ≡ 1
, −1<α<1
(7)
1−α
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Since y ∗ = k + γx, the expression inside the brackets in (5) can be rewritten
as
β + β2
y− 1
(8)
x = y − γx = y − y ∗ + k.
1−α
Using (8) in (5) we obtain
∆yt = β0 − (1 − α)k + β1∆xt − (1 − α) {y − y ∗}t−1 + εt,
As a point of detail, consider the steady-state situation with constant growth
∆xt = gx and ∆yt = gy , and zero disturbance: εt = 0 (the mean of the disturbance term). Imposing this in (9), and noting that {y − y ∗}t−1 by definition
of the steady-state equilibrium, give
− 1 < α < 1 (9)
showing that ∆yt is explained by two factors:
1. the change in the explanatory variable, ∆xt, and
2. the correction of the last period’s disequilibrium y − y ∗.
One can argue that it would be more logical to use the acronym ECM for
equilibrium correction model, since the model implies that ∆yt is correcting in
an equilibrating way. But the term error correction seems to have stuck.
gy = β0 − (1 − α)k + β1gx,
This means that k in the theoretical model y ∗ = k + γx depends on the two
growth terms.
−gy + β0 + β1gx
k=
, if − 1 < α < 1
(10)
1−α
In the case of a stationary steady-state (no growth): with gx = gy = 0, (10)
simplifies to k = β0/(1 − α).
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The homogenous ECM
The ECM shows that there are important points of correspondence between
the dynamic ADL model and a static relationship for long-run:
1. A theoretical linear relationship, y ∗ = k + γx, represents the steady-state
solution of the dynamic model (1).
2. The theoretical long-run slope coefficient γ is identical to the corresponding
long-run multiplier
3. Conversely, if we are only interested in quantifying a long-run multiplier
(rather than the whole sequence of dynamic multipliers), it can be found
by using the identity in (7).
As we have seen, the ECM is a “1-1” transformation of the ADL. It is a
reparameterization, there are no parameter restrictions involved, and therefor
no loss of information.
If the long-run slope coefficient γ is restricted to 1, we get the Homogenous
ECM, the last model in the Typology.
γ = 1 ⇐⇒ β1 + β2 = −(α − 1) as in Typology
The name refers to the property that a permanent unit increase in the exogenous variable leads to a unit long- run increase in y (as with homogeneity of
degree one of equilibrium market prices in a general equilibrium model).
γ = 1 may fail to match the properties of the data, but in our example of a
Norwegian consumption function it fits quite well. From (4) we have
∆ ln Ct = 0.04 + 0.13∆ ln(IN Ct) − 0.21 {ln C − ln IN C}t−1
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Static models reconsidered ( Ch 2.6 in IDM)
As noted in the first lecture, a static relationship has two distinct interpretations
in macroeconomics
1. As an (approximate) descriptions of dynamics
This corresponds to simplifying the ADL by setting β2 = 0 and α = 0 in
the Typology.
In Frisch terminology, this is the case of “infinitely great speed of reaction”.
2. As a long-run relationship which applies to a steady-state situation.
The validity of this interpretation only hinges on α being “less than one”.
Frisch: “static laws basically express what would happen in the long-run if
the static theory’s assumptions prevailed long enough for the phenomena
to have time to have time to react in accordance with these assumptions.”
The PPP hypothesis
It is important to be aware of the two interpretations of static models, and to
be able to distinguish between them.
Consider for example the real-exchange rate, σ
EP ∗
P
where P ∗ is the foreign price level, E is the nominal exchange rate (Kroner/Euro) and P is the domestic price level.
σ=
The purchasing power parity hypothesis, PPP, says that the real exchange rate
is constant.
But what is the time perspective of the PPP hypothesis?
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If PPP is taken as a short-run proposition, then σt = σt−1 = k, and, in the
case of exogenous Et:
ln Pt = − ln k + ln Et + ln Pt∗,
a static price equation which says that the pass-through of a currency change
on Pt is full and immediate
If on the other hand, PPP is taken a hypothesis of the long-run behaviour, we
have instead that σt = σ̄ in a steady-state situation, and
Solution and simulation of dynamic models (Ch
2.7)
The existence of a long-run multiplier, and thereby the validity of the correspondence between the ADL model and long-run relationships, depends on the
autoregressive parameter α in (1) being different from unity.
We next show that the parameter α is also crucial for the nature and type of
solution of equation (1).
∆ ln Pt = gE + π ∗
where gE and π ∗ denote the long-run constant growth rates of the nominal
exchange rates and of foreign prices. On this interpretation, PPP implies full
long-run pass-through, but PPP has no implications about the short-run pass
through of a change in Et on Pt.
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3.1
Solution of ADL equations
We loose little by considering the case of a deterministic autoregressive model,
where we abstract from xt and xt−1, as well as from the disturbance term (εt).
One way to achieve this simplification of (1) is to assume that both xt and εt
are fixed at their respective constant means:
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εt = 0 for t = 0, 1, ..., and
The solution can be stable, unstable or explosive.
xt = mx for t = 0, 1....
stable: often implicit in discussions about the effects of a changes in policy instruments, i.e., multiplier analysis. (example: interest rate effects on inflation)
We write the simplified model as
yt = β0 + Bmx + αyt−1, where B = β1 + β2.
(11)
In the following we proceed as if the coefficients β0, β1, β2 and α are known
numbers (in practice they will be estimated).
unstable: sometimes macroeconomists believe that the effects of a shock linger
on after the shock itself has gone away, often referred to as hysteresis.
(11) holds for t = 0, 1, 2, .... t = 0 is the initial period.
When y0 is a fixed and known number there is a unique sequence of numbers
y0, y1, y2, ... which is the solution of (11).
explosive: “bubbles” in capital markets.
The condition
The solution is found by first solving (11) for y1, then for y2 and so on. This
recursive method is the same as we used when we obtained the dynamic multipliers. Follow the steps on page 50 in IDM and derive the solution
yt = (β0 + Bmx)
t−1
X
αs + αty0, t = 1, 2, ...
−1 < α < 1
(13)
is the necessary and sufficient condition for the existence of a (globally asymptotically) stable solution.
(12)
s=0
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Stable solution −1 < α < 1. Since
t−1
X
1 as t → ∞, we have:
αs → 1−α
5.10
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DFystable_ar.m8
DFystable_ar.8
s=0
(14)
1−α
where y ∗ denotes the equilibrium of yt. The solution can also be expressed
as (see page 52 of IDM).
yt = y ∗ + αt(y0 − y ∗).
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200
205
210
DFyunstable_201
200
205
210
DFyexplosive
15000
3.70
12500
(15)
showing that the solution contains a trend, and that the initial condition
exerts full influence on all solution periods. Marked contrast to stable case.
Explosive solution When α is greater than unity in absolute value the solution
is called explosive, for reasons that become obvious when you consult (12).
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5.00
DFyunstable
Unstable solution (hysteresis) When α = 1, we obtain from equation (12):
yt = (β0 + β1mx)t + y0, t = 1, 2, ...
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5.05
(β + β1mx)
y∗ = 0
10000
3.65
7500
3.60
200
5000
205
200
210
220
230
Figure 1: Panel a) and b): Two stable solutions of (11), (corresponding to
positive and negative values of α. Panel c): Two unstable solutions (corresponding to different initial conditions). Panel d). An explosive solution, See
the text for details about each case.
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3.2
ln(C), actual value
12.05
Simulation of dynamic models
Solution of estimated ADL for Norwegian consumption
12.00
Solution based on:
11.95
a) observed values of l n(INC t ) and ln(INCt−1 ) for the period 1990(1)-2001(4).
b) zero disturbance ε t for the period 1990(1)-2001(4).
c) observed values of l n(Ct−1 ) and ln(INCt−1 ) in 1989(4)
In applied macroeconomics it is not common to refer to the solution of a ADL
model. The common term is instead simulation.
11.90
Specifically, economists use dynamic simulation to denote the case where the
solution for period 1 is used to calculate the solution for period 2, and the
solution for period 2 is in its turn used to find the solution for period 3, and so
on. From what we have said above, in the case of a single ADL equation like
11.85
11.80
11.75
11.70
yt = β0 + β1xt + β2xt−1 + αyt−1 + εt,
11.65
dynamic simulation amounts to finding a solution of the model.
11.60
1990
1991
1992
1993
1994
1995
1996
1997
1998
1999
2000
2001
2002
Figure 2: Solution of the ADL consumption function for the period 1990(1)2001(4). The actual values of log(Ct) are also shown, for comparison.
Conveniently, the correspondence between solution and dynamic simulation
also holds for complex equations (with several lags and/or several explanatory
variables), as well as for the systems of dynamic equations which are used in
practice.
But use computer programs to find the solution.
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Dynamic systems (IDM ch 2.8)
It is often quite easy to bring a system of dynamic equations on a form with
two reduced form ADL equations which are of the same form that we have
considered above.
In the Introduction of ICM, and in the seminar exercises we have already seen
examples of dynamic systems: the cobweb model and the “habit model” of a
perfectly competitive commodity market.
The 2-equation dynamic system has two endogenous variables Ct and IN Ct,
while Jt and εt are exogenous. The ADL in (16) has an endogenous variable
on the right hand side.
Want to find and ADL for Ct with only exogenous and predetermined variables
on the right hand side, a so called final equation for Ct.
In this example: simply substitute IN C from (17) to give:
Ct = β̃0 + α̃Ct−1 + β̃2Jt + ε̃t
We take the Keynesian model as our first macroeconomic example. Here, it is
most convenient to use a linear specification of the consumption function.
Ct = β0 + β1IN Ct + αCt−1 + εt
(16)
(18)
β̃0 and α̃ are β0 and α divided by (1 − β1), and β̃2 = β1/(1 − β1).
(18) is the final equation for Ct in this example.
(17)
If −1 < α̃ < 1 the solution is stable. In that case, there is also a stable solution
for IN Ct, so we don’t have to derive a separate equation for IN Ct in order to
check stability of income.
where Jt denotes autonomous expenditure, and IN C is now interpreted as
GDP.
The impact multiplier of consumption with respect to autonomous expenditure
is β̃2, and the long-run multiplier is β̃2/(1 − α̃).
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together with the product market equilibrium condition
IN Ct = Ct + Jt