Introduction to Dynamic Models.
Slide set #1 (Ch 1.1-1.6 in IDM).
Ragnar Nymoen
University of Oslo, Department of Economics
January 13, 2005
1
1
Introduction
We observe that economic agents take time to adjust their behaviour to changes
in circumstances.
Instantaneous adjustment is the exception in economics. Adjustment lag is the
rule.
Dynamic behaviour is therefore pervasive in economics.
Models with a dynamic formulation are therefore needed.
And we need a methodology for understanding and analyzing dynamic models.
2
The importance of dynamics is recognized by policy decision makers:.
Norges Bank [The Norwegian Central Bank] is typical of many central banks’
view:
A substantial share of the effects on inflation of an interest rate change
will occur within two years. Two years is therefore a reasonable time
horizon for achieving the inflation target of 2 12 per cent1
One important aim of this course is to learn enough about dynamic modeling
to be able to understand the economic meaning of a statement like this, and
to start forming an opinion about its realism (or lack thereof).
1 http://www.norges-bank.no/english/monetary
policy/in norway.html.
Similar statements can be found on the web pages of the central banks in e.g., Autralia,
New-Zealand, The United Kingdom and Sweden.
3
Dynamic models often include both flow and stock variables.
• Flow: in unit of (for example) million kroner per year
• Stock: in units of (for example) million of kroner at a particular period in
time (for example start or end of the year).
In this course an important class of stock variables will be price indices. For
example, Pt may represent the value of the Norwegian CPI in period t (a month,
a quarter or a year).
As you know, the values of P will be index numbers. The number 100 (often
1 is used instead) refers to the base period of the index. If Pt > 100 it means
that relative to the base period, prices are higher in period t.
4
Starting from a stock variable like Pt, a flow variable results from obtaining
the change, hence
the (absolute) change
xt = Pt − Pt−1,
Pt − Pt−1
yt =
,
the relative change, and
Pt−1
zt = ln Pt − ln Pt−1 the approximate relative change
are examples of flow variables. Note that:
• yt × 100 is inflation in percentage points. In this course we often stick to
the rate formulation (hence, we omit the scaling by 100)
• zt ≈ yt by the properties of the (natural) logarithmic function, see for
example the appendix of IDM, if in doubt.
5
600
CPI_Norway
CPI_UK
1500
400
1000
200
1760
1.5
1780
1800
1820
1760
0.4
Inflation_Norway
1.0
1780
1800
1820
1780
1800
1820
Inflation_UK
0.2
0.5
0.0
0.0
-0.5
-0.2
1760
1780
1800
1820
1760
Figure 1: Consumer price indices (stock variables), and their rate of change
(flow). Norway and UK
6
An typical empirical trait of stock variables are that they change gradually, as
a result of finite growth rates.
Sometimes however, stock variables jump from one value in period t to quite
another in period t + 1.
Empirically, the rate of change then becomes very large. Norwegian “price
history” at the breakdown of the union with Denmark is an example.
In economics, when stock variables change gradually, we need explicit dynamic
models to account for their evolution.
Sometimes though, stock variables can be treated theoretically as if the are
jump-variables. An example of such a theory in this course is the portfolio
model of the foreign exchange market. That model is static.
7
In the light of the pervasiveness of dynamics in real world phenomena: what is
the rationale and interpetation of static models?
1. as a first approximation to actual behaviour, when the speed of adjustment
is fast (although a dynamic model would give better understanding). The
portfolio model is an example.
2. as steady-state relationships, derived from and consistent with dynamic
models.
We will use both interpretations in our course, and need to distinguish between
them.
Developing the 2. interpretation in a key goal in the first part of the course.
8
As noted economic dynamics often arise from the combination of flow and
stock variables.
For example, the dynamic behaviour of debt (a stock) is linked to the value of
the current account (flow) in the following way
debt = − current account + last periods debt + corrections.
For example: If there is a primary account surplus for some time (and ignoring
corrections for simplicity), this will lead to a gradual reduction of debt–or an
increase in the nation’s net wealth. Conversely, a consistent current account
deficit raises a nation’s debt.
9
75
The Norwegian current account
Billion kroner
50
25
0
1980
1985
1990
1995
2000
1990
1995
2000
Norwegian net foreign debt
Billion kroner
0
-250
-500
-750
1980
1985
Figure 2: The Norwegian current account (upper panel) and net foreign debt
(lower panel). Quarterly data 1980(1)-2003(4)
10
In the lecture notes, Introductory Dynamic Macroeconomics (IDM) further
examples of dynamics are given in Chapter 1.1:
• Capital stock dynamics (economic growth)
• Dynamics of consumption and income (flows) and private household wealth.
Chapter 1.2 provides a detailed example (see below)
11
2
Static and dynamic models an example
The textbook consumption function, i.e., the relationship between real private
consumption expenditure (C) and real households’ disposable income (IN C)
is an example of a static equation
Ct = f (IN Ct), f 0 > 0.
(1)
Two examples of specified functional forms for the static consumption function:
(linear)
Ct = β0 + β1IN Ct + et,
ln Ct = β0 + β1 ln IN Ct + et, (log-linear)
(2)
If this is unfamiliar, read in IDM about the properties of these two functional
forms. For example for the interpretation of β.
Textbooks usually omit the term et in equation (2), but for applications of the
theory to actual data it is a necessary to get an intuitive grip on this disturbance
term in the static consumption function.
12
Using quarterly data for Norway, for the period 1967(1)-2002(4), we obtain by
using the method of least squares method:
ln Ĉt = 0.02 + 0.99 ln IN Ct
(3)
where the “hat” in Ĉt is used to symbolize the fitted value. Next, use (2) and
(3) to define the residual êt:
êt = ln Ct − ln Ĉt,
which is the empirical counterpart to et.
13
(4)
12.0
11.8
ln C t
11.6
11.4
11.2
11.0
11.0
11.1
11.2
11.3
11.4
11.5
11.6
11.7
11.8
11.9
12.0
ln INCt
Figure 3: The estimated static consumption function.
14
The dynamic consumption function:
ln Ct = β0 + β1 ln IN Ct + β2 ln IN Ct−1 + α ln Ct−1 + εt
(5)
is an example of a so called autoregressive distributed lag model, ADL. Estimated:
ln Ĉt = 0.04 + 0.13 ln IN Ct + 0.08 ln IN Ct−1 + 0.79 ln Ct−1
(6)
Compare residuals ε̂t and êt to judge which model is best (see graph).
Which explanatory variables contribute most to the improved fit?
ln Ct−1 itself! Illustrates that the dynamic framework is important.
The rather low values of the income elasticities (0.130 and 0.08) reflect that a
single quarterly change in income is “too little to build on”. We will see that
the results imply a much higher impact of a permanent change in income than
of a temporary rise.
15
0.15
Residuals of (2.7)
Residuals of (2.4)
0.10
0.05
0.00
−0.05
−0.10
−0.15
−0.20
−0.25
1965
1970
1975
1980
1985
1990
1995
2000
Figure 4: Residuals of the two estimated consumptions functions (3), and (6),
16
3
Dynamic multipliers
“A substantial share of the effects on inflation of an interest rate change will
occur within two years. Two years is therefore a reasonable time horizon for
achieving the inflation target of 2 12 per cent”
This may mean that the effect is building up gradually over 8 quarters and then
dies away quite quickly, but other interpretations are also possible.
In order to inform the public more fully about its view on the monetary policy
transmission mechanism (see topic 5 in our course), the Bank would have to
give a more detailed picture of the dynamic effects of a change in the interest
rate.
To make progress we need to understand fully a concept called the dynamic
multiplier. In order to explain dynamic multipliers, we first show what our
estimated consumption function implies about the dynamic effect of a change
in income (Ch 1.3 of IDM). Next, we derive the dynamic multipliers using a
general notation, see Ch 1.4 of IDM.
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3.1
Dynamic effects of income on consumption
Simplify equation (6) by setting εt = 0, hence we can drop the ˆ above Ct.
Assume that income rises by 1% in period t, so instead of IN Ct we have
IN Ct0 = INCt(1 + 0.01).
Using (6), we have
ln(Ct(1+δc,0) = 0.04+0.13 ln(IN Ct(1+0.01))+0.08 ln IN Ct−1 +0.79 ln Ct−1
where δc,0 denotes the relative increase in consumption in period t, the first
period of the income increase. Since ln(1 + δc,0) ≈ δc,0 when −1 < δc,0 < 1,
and noting that
ln Ct − 0.04 − 0.13 ln IN Ct − 0.08 ln IN Ct−1 − 0.79 ln Ct−1 = 0
we obtain δc,0 = 0.0013, meaning that in percentage terms the immediate
effect is a 0.13% rise in consumption.
18
What about the second period after the shock? Note first that the estimated
model also holds for period t + 1, hence
ln(Ct+1(1 + δc,1)) = 0.04 + 0.13 ln(IN Ct+1(1 + 0.01))
+ 0.08(ln IN Ct(1 + 0.01)) + 0.79 ln(Ct(1 + δc,0),
after the shock. The relative increase in Ct in period t + 1 becomes
δc,1 = 0.0013 + 0.0008 + 0.79 × 0.0013 = 0.003125,
or 0.3%. By following the same way of reasoning, we find that the percentage
increase in consumption in period t + 2 is 0.46% (formally δc,2 × 100).
Since δc,0 measures the direct effect of a change in IN C, it is usually called the
impact multiplier, and is defined by taking the partial derivative ∂ ln Ct/∂ ln IN Ct
in equation (6). The dynamic multipliers δc,1, δc,2, ...δc,∞ are in their turn
linked by the dynamics of equation (6), namely
δc,j = 0.13δinc,j + 0.08δinc,j−1 + 0.79δc,j−1, for j = 1, 2, ....∞.
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(7)
For example, for j = 3, and setting δinc,3 = δinc,2 = 0.01, a permanent rise in
income, we obtain
δc,3 = 0.0013 + 0.0008 + 0.79 × 0.0046 = 0.005734
or 0.57% in percentage terms. The long-run multiplier : Set δc,j = δc,j−1 =
δc,long−run we obtain
0.0013 + 0.0008
δc,long−run =
= 0.01,
1 − 0.79
a 1% permanent increase in income has a 1% long-run effect on consumption.
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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
...
...
...
long-run multiplier
1.00
0.00
Table 2: Dynamic multipliers of the estimated consumption function in (6),
percentage change in consumption after a 1 percent rise in income.
21
1.00
1.10
Temporary change in income
1.05
Percentage change
Percentage change
0.75
Permanent change in income
0.50
0.25
1.00
0.95
0
20
40
60
0
20
Period
40
60
Period
1.00
Dynamic consumtion multipliers (temporary change in income)
0.75
Percentage change
Percentage change
0.10
0.05
Dynamic consumption multipliers (permanent change in income)
0.50
0.25
0
20
40
Period
60
0
20
40
Period
60
Figure 5: Temporary and permanent 1 percent changes in income with associated dynamic multipliers of the consumption function in (6).
22
The distinction between short and long-run multipliers permeates modern macroeconomics, and so is not special to the consumption function!
B&W:
Chapter 8, on money demand, Table 8.4.
Chapter 12, where short and long-run supply curves are derived.
For example, the slopes of the short-run curves in figure 12.6 correspond to the
impact multipliers of the respective models, while the vertical long-run curve
suggest that the long-run multipliers are infinite (we’ll return to this)
Norges Bank on inflation targeting –
and many, more examples.
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4
General notation of the ADL model
ADL model: yt is the endogenous variable while the xt and xt−1 make up the
distributed lag part of the model:
yt = β0 + β1xt + β2xt−1 + αyt−1 + εt.
(8)
Define xt, xt+1, xt+2, , .... as functions of a continuous variable h. When h
changes permanently, starting in period t: ∂xt/∂h > 0.
Since xt is a function of h, so is yt, and the effect of yt of the change in h is
found as
∂yt
∂xt
= β1
.
∂h
∂h
It is customary to consider “unit changes” in the explanatory variable, which
means that we let ∂xt/∂h = 1. Hence the first multiplier is
∂yt
= β1.
∂h
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(9)
The second multiplier is found by considering the equation for period t + 1,
i.e.,
yt+1 = β0 + β1xt+1 + β2xt + αyt + εt+1.
and calculating the derivative ∂yt+1/∂h :
∂yt+1
∂xt+1
∂xt
∂yt
= β1
+ β2
+α
∂h
∂h
∂h
∂h
Again, considering a unit change, and using (9),
(10)
∂yt+1
(11)
= β1 + β2 + αβ1 = β1(1 + α) + β2
∂h
The pattern in (10) repeats itself for higher order multipliers, hence: multiplier
number j + 1 is given as
δj = β1 + β2 + αδj−1, for j = 1, 2, 3, . . .
where we use the notation:
∂yt+j
δj =
, j = 1, 2, ...
∂h
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(12)
The long run multiplier is defined by setting δj = δj−1 = δlong−run. Using
(12), δlong−run is found to be
β1 + β2
δlong−run =
, if − 1 < α < 1.
(13)
1−α
Clearly, if α = 1, the expression does not make sense mathematically, since
the denominator is zero. Economically, it doesn’t 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 explained below,
the dynamics is essentially unstable also in this case meaning that the long run
multiplier is not well defined for the case of α = −1.
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Table 3: Dynamic multipliers of the general autoregressive distributed lag
model.
ADL model:
1. multiplier:
2. multiplier:
3. multiplier:
..
j+1 multiplier
long-run
notes:
yt = β0 + β1xt + β2xt−1 + αyt−1 + εt.
Permanent change in x(1)
δ0 = β1
δ1 = β1 + β2 + αδ0
δ2 = β1 + β2 + αδ1
..
δj = β1 + β2 + αδj−1
Temporary change in x(2)
δ0 = β1
δ1 = β2 + αδ0
δ2 = αδ1
..
δj = αδj−1
1 +β2
δlong−run = β1−α
0
(1) As explained in the text, ∂xt+j /∂h = 1, j = 0, 1, 2, ...
(2) ∂xt/∂h = 1, ∂xt+j /∂h = 0, j = 1, 2, 3, ...
If y and x are in logs, the multipliers are in percent.
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5
A typology of linear models
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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Table 4: A model typology.
Type
Equation
Restrictions
ADL
yt = β0 + β1xt + β2xt−1 + αyt−1 + εt. None
Static
yt = β0 + β1xt + εt.
β2 = α = 0
DL
yt = β0 + β1xt + β2xt−1 + εt
α=0
Differenced data1
∆yt = β0 + β1∆xt + εt
β2 = −β1, α = 1
ECM
∆yt = β0 + β1∆xt + (β1 + β2)xt−1
+(α − 1)yt−1 + εt
None
Homogenous ECM ∆yt = β0 + β1∆xt
+(α − 1)(yt−1 − xt−1) + εt
1
∆ is the difference operator, defined as ∆zt ≡ zt − zt−1.
29
β1 + β2 = −(α − 1)
6
Extensions and examples
In this subsection we briefly point to several important extensions of the ADL
model. Second, to help solidify the understanding of the ADL framework, we
provide additional economic examples.
6.1
Extensions
The most important extensions of (8) are:
1. Several explanatory variables
2. Longer lags
3. Systems of ADL equations
30
Two exogenous variables, x1,t and x2,t. The extension of (8) to this case is
yt = β0 + β11x1,t + β21x1,t−1
+ β12x2,t + β22x2,t−1 + αyt−1 + εt,
(14)
where βik is the coefficient of the i0th lag of the explanatory variable k.
Everything goes through as before, but two different sets of multipliers, with
respect to changes in x1 and x2.
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6.2
A few more examples
The dynamic consumption function (again)
This of course has been the prime example so far in section 3. We have
considered the log-linear specification in detail. Of course exactly the same
analysis applies to a linear functional form of the consumption function, only
that the multipliers will be in units of million kroner (at fixed prices) rather
than percentages. In section 7 the linear consumption function is combined
with the general budget equation to form a dynamic system.
In modern econometric work on the consumption function, more variables are
usually included than just income. Hence, there is usually more multipliers to
consider than just with respect to IN Ct. The most commonly found additional explanatory variables are wealth, the real interest rate and indicators of
demographic developments.
32
The Phillips curve
In Chapter 2 of IDM, and several times later in the course, we will consider the
Phillips curve:
e +ε .
πt = β0 + β11ut + β12ut−1 + β21πt+1
t
(15)
πt is the rate of inflation, 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 log.
e
denotes the expected rate of inflation one period ahead, so (15)
Finally, πt+1
is formally an ADL with 2 explanatory variables. Moreover, if
e
= τ πt−1
πt+1
(16)
(15) can be reduced to the single variable ADL equation (8),
πt = β0 + β11ut + β12ut−1 + απt−1 + εt.
with α = β21τ .
33
(17)
e
πt+1
= τ πt−1
is just one out of many hypotheses of expectations formation. Alternative
specifications give rise to different dynamic models of the rate of inflation.
Consider an explicit inflation target, π̄. In this case,we may set
e
πt+1
= π̄
(18)
As an exercise, convince yourself that equations (15) and (18) imply an equation
for inflation which is a distributed lag model (DL model).
More generally, firms and households take into consideration the possibility that
future inflation is not exactly on target. Hence they may adopt a more robust
forecasting rule, for example
e
πt+1
= (1 − τ )π̄ + τ πt−1,
0 < τ ≤ 1.
(19)
In this case, the derived dynamic equation for inflation again takes the form of
an ADL model.
34