Dynamic e¤ects of shocks; Equation typology
Ragnar Nymoen
Department of Economics, UiO
25 August 2009
ECON 3410/4410: Lecture 2
Lecture 2: Overview
A simple, but still quite general, linear dynamic equation and
the autoregressive distributed lag (ADL) interpretation.
Making the concept of dynamic multiplier precise, with the
use of the ADL model
Macroeconomic examples:
The dynamic consumption function
The Phillips curve model of the supply side of the
macroeconomy.
The market for foreign exchange
All these will play distinct roles later in the course
Main reference is IDM, Ch 2.1-2.4
ECON 3410/4410: Lecture 2
A general dynamic equation
We let yt denote an endogenous variable in period t. xt is an
economic exogenous variable, "t is a random variable that
represent shocks. 0 , 1 and are the parameters of the
model:
yt =
0
+
1 xt
+
2 xt 1
+ yt
1
+ "t :
(1)
The equation is general enough to give precise meaning to
two of the concepts we introduced in the …rst lecture:
dynamic multiplier, and
the solution of a dynamic model (Lecture 3).
Later we will generalize the results that we obtain for (1) to
systems of dynamic relationships, which will be the main
application in the course.
ECON 3410/4410: Lecture 2
Di¤erence equation = ADL equation
Mathematically,
yt =
0
+
1 xt
+
2 xt 1
+ yt
1
+ "t :
(1)
is a non-homogenous linear di¤erence equation for y .
In economics we often refer to this equation by its own name:
The autoregressive, distributed lag model, ADL for short.
This is because the equation incudes both an autoregressive
term, yt 1 , and a distributed lag in the exogenous variable xt .
Correspondingly, is called the autoregressive parameter,and
1 in 2 are the two autoregressive coe¢ cients.
ECON 3410/4410: Lecture 2
Temporary shocks, and permanent changes
In lecture 1 we studied graphically and intuitively the e¤ect of
shocks, corresponding to changes in "t in (1).
Since "t is random, it is somewhat di¢ cult to envisage
permanent changes in that variable.
With the economic variable xt it is however natural to think of
about both temporary changes,and permanent shifts.
We …rst consider the response to a temporary shock, and then
the response to a permanent change.
ECON 3410/4410: Lecture 2
Dynamic multipliers— the e¤ects of a temporary shock (I)
Consider a change to xt in period t. The impact multiplier is
de…ned as the partial derivative of
yt =
0
+
1 xt
+
2 xt 1
+ yt
1
+ "t :
with respect to xt :
@yt
= 1:
@xt
The second multiplier would then be the partial derivative of
yt+1 with respect to xt . But where does yt+1 appear?
In the equation that holds for period t + 1, namely:
yt+1 =
0
+
1 xt+1
+
2 xt
+ yt + "t+1 :
Hence xt :
@yt+1
@yt
= 2+
= 2+ 1
@xt
@xt
which is called the interim multiplier for the …rst lag.
ECON 3410/4410: Lecture 2
Dynamic multipliers (II)
To …nd the response in period t + 2, which represents the
interim multiplier for the second lag, we write down the
equation for period t + 2
yt+2 =
0
+
1 xt+2
+
2 xt+1
+ yt+1 + "t+2 :
and take the derivative:
@yt+2
@xt
@yt+1
@xt
2+
=
=
2
1.
Generally, for period t + j :
@yt+j
=
@xt
@yt+j
@xt
1
=
j
1
+
j 1
2;
for j = 1; 2; ::
ECON 3410/4410: Lecture 2
Dynamic multiplier shapes
1
0<
1
2
2
3
4
< 1:
1 and 2 have same signs: Multipliers are either positive or
negative, magnitude drops o¤ smoothly after the second …rst
interim multiplier: @y@xt+2
= @y@xt+1
t
t
1 and 2 have opposite signs: Impact may be larger or
smaller than …rst interim multiplier, then magnitude drops o¤
smoothly
1 < < 0: Signs will shift, as in the cobweb model of
market equilibrium prices
= 0: Cut-o¤ after …rst interim multiplier
j j > 1. Explosive sequence of multipliers.
ECON 3410/4410: Lecture 2
1.0
Price response to temporary demand shock. Static marked equilibrium model
0.5
1.0
0
5
10
15
20
15
20
15
20
Dynamic marked equilibrium model, cobweb.
0.5
0.0
-0.5
0
1.0
5
10
Dynamic market equilibrium model, habit formation.
0.5
0
5
10
As we shall see, the
dynamic responses
of a variable in a
system of equations
can be derived by
…rst …nding (by
solution) the ADL
equation for the
variable in question
Therefore, the
dynamic market
equilibrium model
of Lecture 1
illustrates some of
the possible shapes,
see left.
ECON 3410/4410: Lecture 2
Cumulated multipliers: The e¤ect of a permanent shock (I)
Assume that the shock to x in period t is permanent instead
of temporary. What will be the e¤ect on yt ; yt+1 , yt+2 , ....?
We denote these e¤ects by 0 , 1 , 2 , ...
0 , the impact multiplier, is the same as in the transitory
shock: 0 = 1 :
The second multiplier will be the …rst interim multiplier plus a
“new” impact multiplier in period t + 1:
1
=
@yt+1
@yt+1
+
=
@xt+1
@xt
1 (1
+ )+
2;
and note that this is the same as
@yt
@yt+1
+
1 =
@xt
@xt
since yt = xt = yt+1 = xt+1 = 1 , suggesting that we obtain
the e¤ects of a permanent change in xt by cumulation of the
interim multipliers.
ECON 3410/4410: Lecture 2
Cumulated multipliers: The e¤ect of a permanent shock
(II)
Applying this to the third multiplier ( 2 ) gives
2
=
=
@yt
@yt+1
@yt+2
+
+
@xt
@xt
@xt
2
(1
+
+
)
+
1
2 (1 + )
and, by induction,
j
=
1 (1
+
+ ::: +
j
)+
2 (1
+
+ ::: +
j 1
), for j = 1; 2; ::::
ECON 3410/4410: Lecture 2
Obtaining cumulated multipliers by recursion
It is useful to note that by rewriting the expression for
1
we see that
=
1 can
1 (1
+ )+
=
1
+
2
+
1
be obtained from the recursive formula:
1
For
2
1:
=
1
+
2
+
0
+
2
)+
2 (1
2:
2
=
1 (1
=
1
+
+
2
And generally:
j
=
1
+
2
+
+
f
|
j 1
1
+
+ )
2+
{z
1
1g
}
for j = 1; 2; 3; :::
ECON 3410/4410: Lecture 2
(2)
The long-run multiplier
What are the multipliers in the long-run, when j ! 1?
From
j
=
1 (1
+
+ ::: +
j
)+
2 (1
+
+ ::: +
j 1
), for j = 1; 2; ::::
we see that this depends on the magnitude of .
If the absolute value of is less than one, the two geometric
sequences in sum to 1
when j is in…nite. So
j
1
!
+
2
1
j !1
, if
1<
< 1:
We obtain the same from the recursive formula (2) by setting
j = j 1 = long run and solving for long run .
long run
=
1
1
+
2
, if
1<
< 1:
ECON 3410/4410: Lecture 2
Summary of dynamic multipliers
yt =
0
+
1 xt
+
Temporary change in x
@yt
@xt = 1
@yt+1
@yt
@xt = 2 + @xt
@yt+2
@yt+1
@xt =
@xt
..
.
@yt+j
@xt
@yt+j
@xt
=
0
1
2 xt 1
+ yt
1
+ "t :
Permanent change in xt
0 = 1
1 = 1+ 2+
0
2 = 1+ 2+
1
..
.
j
=
1
+
long run
2
+
=
1
j 1
1+ 2
ECON 3410/4410: Lecture 2
Note on derivation of multipliers
1
2
3
IDM we also give a di¤erent derivation of the cumulated
multipliers, in terms of di¤erentiation with respect to an
underlying continuous variable. You should have a good
understanding of at least one of the derivations.
The transitory shock, and the permanent shock, are extreme
cases. Often we would like to investigate the response to more
realistic shocks: For example. What are the e¤ects of an
increase in government spending that lasts for 1 or 3 years,
before it is reverted in order to “balance” government budgets
(as with the …nancial crisis)?
Logically, this question is the same exercise as above, but
calculation of the multipliers “by hand”, is often impractical.
Instead we use computer simulation. We obtain two solutions
(to be discussed later): One without the shock (or policy),
and the other with the shock imposed. The di¤erence
between the two solutions will give the multipliers!
ECON 3410/4410: Lecture 2
Consumption function example (I)
Consider the dynamic consumption function:
ln Ct =
0
+
1
ln INCt +
2
ln INCt
1
+
ln Ct
1
+ "t
(3)
where C denoted private consumption period t and INC denotes
disposable income. The estimated version (using Norwegian
quarterly data) is
ln Ct = 0:04 + 0:13 ln INCt + 0:08 ln INCt
1
+ 0:79 ln Ct
1
(4)
Permanent 1% change Temporary 1% change
Impact period
0:13
0:13
1. period after shock
0:31
0:18
2.period after shock
0:46
0:14
3.period after shock
0:57
0:11
:::
:::
:::
long-run multiplier
1:00
0:00
ECON 3410/4410: Lecture 2
Consumption function example (II)
1.00
1.10
Temporary change in income
Permanent change in income
0.75
1.05
0.50
1.00
0.25
0.95
0
20
40
60
Left:Transitory shock
and interim multipliers
0
20
Period
40
60
Period
1.00
Dynamic consumtion multipliers (temporary change in income)
0.10
0.75
Dynamic consumption multipliers (permanent change in income)
0.50
0.05
0.25
0
20
40
Period
60
0
20
40
Period
60
Right:Permanent
shock and cumulated
multipliers.
All changes are in
percent, because of
the log-linear
functional form.
ECON 3410/4410: Lecture 2
A typology of dynamic equations
Although
yt =
0
+
1 xt
+
2 xt 1
+ yt
1
+ "t :
(1)
is simple, it still contains several equations that appear in
macroeconomic models as special cases.
See Table 2.3 in IDM.
If we impose
2
=
= 0, we have the static model.
= 0 gives a distributed lag (DL) model. What is typical for
the dynamic multiplier of the DL model?
= 1 and
1
=
2
= 0 is called the random walk model.
In lecture 3, we will pay particular attention to the equilibrium
correction model, ECM, in the typology.
ECON 3410/4410: Lecture 2
Generalizing the basic dynamic equation
The most important extensions of
yt =
0
+
1 xt
+
2 xt 1
+ yt
1
+ "t :
(1)
are:
1
Two ore more explanatory variables.
2
Longer lags.
3
Systems of ADL equations.
We comment on 1. and 2 here, and return to 3. in the next lecture.
ECON 3410/4410: Lecture 2
Two or more explanatory variables, for example
yt =
+
0
+
12 x2;t
11 x1;t
+
+
21 x1;t 1
22 x2;t 1
+ yt
1
+ "t ;
pose no new problems. Straight-forward to obtain separate
multipliers for x1 and x2 .
Longer distributed lag–
creates no new logical problems, but care must be taken when
calculating the interim multipliers of course! But impact (the
short-run) and long-run multipliers are always easy!
Longer lags in yt (autoregressive part)—
Same as with long DLs, but in addition: 1 < < 1 is no
longer su¢ cient for the existence of long run as a …nite
number (full analysis goes beyond the this course), so in this
case we have to answer question about the long-run e¤ects
conditional on the stability of the dynamic equation.
ECON 3410/4410: Lecture 2
Example: The price Phillips curve
t
=
0
+
11 ut
+
12 ut 1
+
e
21 t+1
+ "t .
(5)
denotes the rate of in‡ation in period t, hence t =
ln(Pt =Pt 1 ) where Pt is an index of the price level of the
economy.
ut is the rate of unemployment, or its log.
e
t+1 denotes the expected rate of in‡ation one period ahead.
A simple hypothesis, is that expectations are based primarily
on past in‡ation, hence we set
t
e
t+1
=
t 1,
6= 0
(6)
and obtain the typical Phillips curve relationship:
t
where
=
=
0
+
11 ut
+
21
> 0 (if both
12 ut 1
21 and
+
t 1
+ "t ,
are positive).
ECON 3410/4410: Lecture 2
(7)
The Phillips curve and Okun’s law.
In textbooks, equations like (5) are often in terms of an
output-gap variable: the di¤erence between log GDP and the
trend in log GDP.
It is of some importance to understand that these alternative
speci…cations express the same theory: that a main source of
domestic in‡ation pressure are domestic product and labour
markets.
To keep it simple, we let either the output-gap or the rate of
unemployment, represent both pressures.
Reference is also often made to Okun’s law (see IAM p 320),
which is a “stylized fact” saying that there is a tight
functional relationship between ut: and the GDP output-gap
(or the rate of growth in GDP).
ECON 3410/4410: Lecture 2
Dynamic implication of in‡ation expectations (I).
e
t+1
=
t 1
is just one possible hypothesis of expectations formation.
Alternative speci…cations give rise to di¤erent dynamic models
of in‡ation.
Consider for example a completely credible in‡ation targeting
regime. In this case,we may set
e
t+1
where
=
(8)
denotes the in‡ation target.
Equations (5) and (8) imply a DL model for in‡ation.
How will in‡ation react to a temporary supply shock in the two
cases?
What about a demand shock?
ECON 3410/4410: Lecture 2
Dynamic implication of in‡ation expectations (II).
More generally, …rms and households take into consideration
the possibility that future in‡ation is not exactly on target.
Hence they may adopt a ‡exible more expectation rule, for
example
e
t+1
= (1
) +
t 1;
0<
1:
(9)
In this case, the derived dynamic equation for in‡ation again
takes the form of an ADL model.
Towards the end of the course, where we consider alternative
speci…cation of the supply side of the macroeconomic model,
we will discuss the new Keynesian Phillips curve. It takes the
form:
t
=
0
+
11 ut
+
12 ut 1
+
f
t+1
+ "t ,
Clearly this is speci…cation that is not covered by the ADL
framework, and it requires separate analysis.
ECON 3410/4410: Lecture 2
Floating exchange rate dynamics (I)
Although the long swings in a country’s exchange rate are
believed to re‡ect long-run fundament features of the real
economy (current account and net debt), a large part of the
variations in the medium term are due to:
Shocks
Changes in the risk premium
Both these re‡ect that the market for foreign exchange is
dominated by speculative motifs.
A reasonable starting point is therefore that the nominal
exchange rate (kroner/euro) Et is decreasing in the risk
premium:
ln Et =
0
it e e )
1 (it
risk-premium
+ "t ,
1
> 0;
it and it are domestic and foreign interest rates, and e e is the
expected rate of depreciation.
ECON 3410/4410: Lecture 2
(10)
Floating exchange rate dynamics (II)
If e e is constant, the model of the nominal exchange rate is
static.
However, e e 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 =
Et
1,
> 0;
the equation for ln Et can be written as
ln Et =
0
1 (it
it ) + Et
1
+ "t
(11)
with 1 > 0 and =
1 < 0.
(11) is an ADL model. The coe¢ cient of the lagged
endogenous variable is negative due to regressive anticipations.
What does this (simple) model predict about a one period
increase in the interest rate di¤erential?
ECON 3410/4410: Lecture 2