Partial recap of lecture 1
1. We defined static and dynamic economic models
Slides to Lecture 2 of Introductory Dynamic
Macroeconomics.
Linear Dynamic Models (Ch 2 of IDM)
2. We used economic intuition and graphical analysis to study the dynamics
of the cobweb model:
3. Starting from an initial situation where the marked was in a stationary
equilibrium, we studied the response of price and quantity to a one-period
shock to demand.
Ragnar Nymoen
University of Oslo, Department of Economics
4. Those responses, which we called dynamic multipliers, lasted for several
periods, which is a defining characteristic of dynamic models.
August 28, 2007
5. In a static model, there would only be response in the period of the shock
(adjustment is “immediate and complete” in static models).
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2
6. In most practical applications a dynamic model will describe economic
behaviour a lot better than a static model. A consumption function for
Norway provided an example.
7. Nevertheless statics models remain very useful since they can be seen as
“limiting cases” of dynamic models
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The ADL model (Ch 2.1 of IDM)
yt is the endogenous variable. yt−1 is the autoregressive term. The explanatory
variable x enters in the form of a distributed lag, DL,
yt = β0 + β1xt + β2xt−1 + αyt−1 + εt.
(1)
β1 and β2 are called distributed lag parameters, and α is called the autoregressive parameter.
(a) As models of actual behaviour in markets where adjustments are very
swift (markets for nominal exchange rates? Or betting?)
Dynamic multipliers with respect to a permanent change in x.
(b) As models of the stationary state attained by dynamic models in the
hypothetical situation where all responses to shocks have died out, and
where there are no new shocks. We call this model a “long-run model”.
Think of xt, xt+1, xt+2, .etc as differentiable functions of an underlying variable
h. When h changes permanently, starting in period t, we therefore have that
all derivatives ∂xt+j /∂h (j)1, 2, ..) are positive.
8. The distinction between 7a and 7b is essential, and a thorough understanding of is a one of the main goal of this course.
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Since xt is a function of h, so is yt, and the effect of yt of the change in h is
found as
∂xt
∂yt
= β1
.
∂h
∂h
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It is customary to consider “unit changes” in the explanatory variable, meaning
that we set ∂xt/∂h = 1. Hence the first current period response is written as
∂yt
= β1,
∂h
Usually, this is called the impact multiplier
By considering the equation for yt+2
yt+2 = β0 + β1xt+2 + β2xt+1 + αyt+1 + εt+2
∂y
(2)
t+2
we see that the pattern in (3) repeats itself for ∂h
, and for all later periods’
responses.
Hence, if we use the symbol notation
The response in the period after the occurrence of the shock is found by noting
that equation (1) is also valid for period t + 1:
δj =
∂yt+j
, j = 1, 2, ...
∂h
we have that
yt+1 = β0 + β1xt+1 + β2xt + αyt + εt+1,
and calculating the derivative ∂yt+1/∂h :
∂yt+1
∂x
∂xt
∂yt
= β1 t+1 + β2
+α
∂h
∂h
∂h
∂h
Again, considering a unit change, and using (2),
∂yt+1
= β1 + β2 + αβ1 = β1(1 + α) + β2
∂h
δ0 = β1
δj = β1 + β2 + αδj−1, for j = 1, 2, 3, . . .
(3)
(4)
(5)
for the responses to a permanent unit change in the explanatory variable in the
ADL model (1). The responses follow a recursive structure: once the impact
multiplier δ0 is determined, then δ1 and the later responses are determined by
repeated use of the second line in (5).
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Table 2: Dynamic multipliers of the ADL model. (Table 2.1 in IDM)
The long-run multiplier is defined by setting δj = δj−1 = δlong−run. Using
(5), δlong−run is
β + β2
δlong−run = 1
, if − 1 < α < 1.
(6)
1−α
If α = 1, the expression does not make sense mathematically, since the denominator is zero. Economically, it does not make sense either since the long run
effect of a permanent unit change in x is an infinitely large increase in y (if
β1 + β2 > 0). The case of α = −1, may at first sight seem to be acceptable
since the denominator is 2, not zero. However, as we shall see below, the
dynamics is then unstable, so the long-run multiplier is not well defined.
ADL model:
1. multiplier:
2. multiplier:
Permanent(1) unit change in x
δ0 = β1
Temporary unit change
∂yt
∂x = β1
t
∂yt+1
∂yt
∂xt = β2 + α ∂xt
∂yt+2
∂yt+1
∂xt = α ∂xt
3. multiplier:
..
δ1 = β1 + β2 + αδ0
δ2 = β1 + β2 + αδ1
..
j+1 multiplier
δj = β1 + β2 + αδj−1
∂yt+j
∂yt+j−1
∂xt = α ∂xt
long-run(3)
1 +β2
δlong−run = β1−α
0
notes:
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yt = β0 + β1xt + β2xt−1 + αyt−1 + εt.
..
(1) As explained in the text, ∂xt+j /∂h = 1, j = 0, 1, 2, ...
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Dynamic multipliers with respect to a temporary change in x
The third column shows the multipliers in the case of a temporary shock, similar
to the one-period demand shock that we studied in Lecture 1.
In the case of a temporary change in x, there is no longer any gains in exposition
from invoking the idea that the x’s are functions of h. This is because, by
definition, a temporary change in period t affects only xt, not xt+1 or any
other x’s further ahead the future. Hence from (1) we have directly that the
impact multiplier is the partial derivative of (1) with respect to xt:
From the equation for yt+2 we obtain the interim multiplier for second lag as:
∂y
∂yt+2
= α t+1 .
∂xt
∂xt
Since we look at the case of −1 < α < 1, it is clear that the magnitude of
interim multipliers can be larger in the second period after the shock, ie. we
can have
¯
¯
¯
¯
¯ ∂y
¯
¯ ∂y ¯
¯ t+1 ¯
¯ t¯
¯
¯>¯
¯,
¯ ∂xt ¯
¯ ∂xt ¯
but
¯
¯
¯
¯
¯ ∂y
¯
¯ ∂y
¯
¯ t+j+1 ¯
¯ t+j ¯
¯
¯<¯
¯ for j = 1, 2, ...
¯ ∂xt ¯
¯ ∂xt ¯
∂yt
= β1.
∂xt
The second multiplier is the partial derivative of yt+1 with respect to xt:
In the limit, when j → ∞, the interim multipliers from a temporary shock is
zero.
∂yt+1
∂yt
= β2 + α
∂xt
∂xt
which is called the interim multiplier for the first lag.
Since a permanent and a temporary shock give rise to so different sequences
of dynamic multipliers, it is important to be precise about which kind of shock
we in mind when we talk about dynamic response and multipliers.
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1.0
10
Price response to temporary demand shock. Static marked equilibrium model
0.5
Relationship between multipliers
1.0
0
5
10
15
20
Dynamic marked equilibrium model, cobweb.
Heuristically, the effect of a permanent change in the x’s can be viewed as the
sum of the changes triggered by a temporary change in period t,see IDM for a
‘proof’.
0.5
0.0
-0.5
0
5
10
15
20
1.0
Dynamic market equilibrium model, habit formation.
0.5
0
5
10
15
20
The interim multipliers can therefore be viewed as the more fundamental of the
two types of dynamic multiplies that we have considered: If we first calculate
the effects of a temporary shock, the dynamic effects of a permanent shock can
be calculated afterwards by summation of the impact and interim multipliers.
Figure 1: Examples of interim multipliers from the single marked models in Ch
1 of IDM .
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2
Effects of income on consumption (Ch 2.2 in IDM)
1.00
1.10
Temporary change in income
0.75
(7)
and the estimated version (using Norwegian quarterly data).
ln Ĉt = 0.04 + 0.13 ln IN Ct + 0.08 ln IN Ct−1 + 0.79 ln Ct−1
1.05
Percentage change
ln Ct = β0 + β1 ln IN Ct + β2 ln IN Ct−1 + α ln Ct−1 + εt
Percentage change
The first lecture introduced the dynamic consumption function:
0.50
0.25
(8)
Permanent change in income
1.00
0.95
0
20
40
60
0
20
Period
40
60
Period
1.00
Dynamic consumtion multipliers (temporary change in income)
Permanent 1% change Temporary 1% change
0.13
0.13
0.31
0.18
0.46
0.14
0.57
0.11
...
...
1.00
0.00
0.10
0.75
Percentage change
Percentage change
Impact period
1. period after shock
2.period after shock
3.period after shock
...
long-run multiplier
0.05
Dynamic consumption multipliers (permanent change in income)
0.50
0.25
0
20
40
Period
0
60
20
40
Period
60
Table 3: Dynamic multipliers of the estimated consumption function in (8),
percentage change in consumption after a 1 percent rise in income.
Figure 2: Temporary and permanent 1 percent changes in income with associated dynamic multipliers of the consumption function in (8).
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Table 4: A model typology.
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Type
A typology of linear models (Ch 2.3 in IDM)
The discussion at the end of the last section suggests that if the coefficient α in
the ADL model is restricted to for example 1, or to 0, quite different dynamic
behaviour of yt is implied. In fact the resulting models are special cases of
the unrestricted ADL model. For reference, this section gives a typology of
models that are encompassed by the ADL model. Some of these model we
have already mentioned, while others will appear later in the book.
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Equation
ADL
yt = β0 + β1xt + β2xt−1 + αyt−1.
Static
yt = β0 + β1xt
Random walk
yt = β0 + yt−1
DL
yt = β0 + β1xt + β2xt−1
Differenced data1
∆yt = β0 + β1∆xt + εt
ECM
∆yt = β0 + β1∆xt + (β1 + β2)xt−1
+(α − 1)yt−1
Homogenous ECM ∆yt = β0 + β1∆xt
+(α − 1)(yt−1 − xt−1)
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∆ is the difference operator, defined as ∆zt ≡ zt − zt−1.
Restrictions
None
−1 < α < 1
for stability
β2 = α = 0
β1 = β2 = 0,
α = 1.
α=0
β2 = −β1,
α=1
Same as ADL
β1 + β2
= −(α − 1)
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Extensions and more examples (Ch 2.4 in IDM)
Two exogenous variables, x1,t and x2,t. The extension of (1) to this case is
yt = β0 + β11x1,t + β21x1,t−1
+ β12x2,t + β22x2,t−1 + αyt−1 + εt,
The most important extensions of (1) are:
(9)
where βik is the coefficient of the i0th lag of the explanatory variable k.
1. Several explanatory variables
Everything goes through as before, but two different sets of multipliers, with
respect to changes in x1 and x2.
2. Longer lags
Longer lags in DL or a part of mode, makes for richer dynamics. Cumbersome to calculate multipliers by hand, but automatized in PcGive and other
programmes.
3. Systems of ADL equations, Lecture 3
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The Phillips curve
The price Phillips curve is an important macroeconomic example of a dynamic
equation:
(10) becomes a single variable ADL model:
e +ε .
πt = β0 + β11ut + β12ut−1 + β21πt+1
t
(10)
πt denotes the rate of inflation in period t, hence πt = ln(Pt/Pt−1) where Pt is
an index of the price level of the economy. ut is the rate of unemployment, or its
e
e
denotes the expected rate of inflation one period ahead. Since πt+1
is
log. πt+1
unobservable, we need to introduce an explicit assumption about expectations
formation.
πt = β0 + β11ut + β12ut−1 + απt−1 + εt,
α = β21τ > 0 (if both β21and τ are positive).
A simple hypothesis, is that expectations are based primarily on past inflation,
hence we set
e
= τ πt−1
πt+1
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(11)
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(12)
e
πt+1
= τ πt−1
is just one possible hypothesis of expectations formation. Alternative specifications give rise to different dynamic models of inflation.
Consider for example a completely credible inflation targeting regime. In this
case,we may set
e
πt+1
= π̄
(13)
More generally, firms and households take into consideration the possibility
that future inflation is not exactly on target. Hence they may adopt a more
expectation rule, for example
e
πt+1
= (1 − τ )π̄ + τ πt−1,
0 < τ ≤ 1.
(14)
In this case, the derived dynamic equation for inflation again takes the form of
an ADL model.
where π̄ denotes the inflation target.
Lecture 6 will consider a Phillips curve with rational expectations.
Equations (10) and (13) imply a DL model for inflation.
How will inflation react to a temporary supply shock in the two cases?
What about a demand shock?
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Exchange rate dynamics
The price of the good foreign currency is the nominal exchange rate, the price
of the foreign currency in terms of the currency of the domestic country. The
exchange rate is then the price of dollars (or euros) in kroner.
The standard open economy macro model used to treat the net supply of
currency to the central bank (essentially the negative of the net demand for
kroner) as a flow variable, primarily determined by the current account.
The modern approach takes as a premise that in modern open economies most
of the trade in foreign currency is capital movements.
The modern portfolio model approach views the net supply of foreign currency
as a stock variable. The unit of measurement for quantity is units of foreign
currency, for example euros, not euros per units of time as in the flow approach.
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In this course we will only need the basic intuition of the portfolio model: Net
supply of foreign currency depends in particular on the risk premium defined as
∗
risk premium = it − it − ee,
where it and i∗ are the domestic and foreign interest rates respectively, and ee
denotes the expected rate of depreciation.
The risk premium reflects by how much extra the investors get paid over the
expected return on euros to take the risk of investing in kroner in the spot
foreign exchange market.
Net supply of foreign currency is a rising function of the risk premium, since
investors find kroner a more profitable asset when the risk premium increases.
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In a model of a floating exchange rate regime where the net demand of foreign currency is exogenous, in the form of a fixed stock of foreign exchange
reserves, the nominal exchange rate (kroner/euro) Et is thus increasing in the
risk premium:
∗
ln Et = β0 − β1(it − it − ee) + εt, β1 > 0,
(15)
where εt as usual represents a disturbance, which by the way might be quite
large in comparison with the variability of ln Et. −β1 is called a semi-elasticity
since only the left hand side variable is in logs (see IDM appendix).
∗
ln Et = β0(CAt, CAt−1, ...CAt−j ) − β1(it − it − ee) + εt, β1 > 0
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Another source of dynamics is expectations. So far we have assumed that ee is
constant, but in fact it is likely to be highly variable, and to depend of a long
list of sentiments and also of macroeconomic variables. However, for modelling
purposes, we usually assume that the expected degree of depreciation depends
on the level of the exchange rate. If expectations are ‘regressive’:
ee = −τ Et−1, τ > 0,
the equation for ln Et can be written as
∗
ln Et = β0 − β1(it − it ) + αEt−1 + εt
(16)
with β1 > 0 and α = −β1τ < 0. (16) is an ADL model. The coefficient of the
lagged endogenous variable is negative in the portfolio model with regressive
anticipations.
What does this (simple) model predict about a one period increase in the
interest rate differential?
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When using (??) it must be understood that β0 is only invariant to the current
account over a limited time period. A current account deficit which lasts for
several months, or maybe years, will inevitably affect the stock of net supply of
currency, and we can think of this as a gradual increase in β0. In this way we
can bridge the gap between the stock approach and the older flow approach.
Formally, we can write β0 as a function of a distributed lag of past current
accounts: β0(CAt, CAt−1, ...CAt−j ), with negative partial derivatives, and
with j normally a quite large number (6-12 months for example). Hence, the
link to the current account defines a dynamic model of the exchange rate: