Dynamic models. Slides for 15. January
2004 lecture
Ragnar Nymoen
University of Oslo, Department of Economics
January 13, 2004
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1
Introduction
Useful macroeconomic theories should help us understand the behaviour of
macroeconomic time series.
Examples (in class)
Formally
yt is a time series if we observe it over a sequence of time periods represented
by the subscript t., i.e., {yT , yT −1, ..., y1} if we have observations from period
1 to T . Usually, we use the simpler notation yt, t = 1, ...., T ,
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Typically, economic agents need/take time to adjust to changes in circumstances.
Instantaneous adjustment is the exception in economics. Adjustment lags the
rule.
Dynamic behaviour is pervasive in economics.
Models with a dynamic formulation therefore needed.
3
Dynamics is important in policy decisions.
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.
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Static models nevertheless have an important role in our discipline:
• as a first approximation to actual behaviour when the adjustment is fast
relative to the time period.
— static model of the exchange rate market may suffice, when observation
period is year of quarters.
— need dynamic model if monthly, daily, hourly data
• 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.
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2
Static and dynamic models
Two examples of static consumption functions (the difference in function form)
(linear)
Ct = β0 + β1IN Ct + et,
ln Ct = β0 + β1 ln IN Ct + et, (log-linear)
(1)
Textbooks usually omit the term et in equation (1), 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.
Using quarterly data for Norway, for the period 1967(1)-2002(4), we obtain by
using the method of least squares method (in PcGive):
ln Ĉt = 0.02 + 0.99 ln IN Ct
(2)
where the “hat” in Ĉt is used to symbolize the fitted value. Next, use (1) and
(2) to define the residual êt:
êt = ln Ct − ln Ĉt,
which is the empirical counterpart to et.
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(3)
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 1: The estimated static consumption function.
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A dynamic consumption function:
ln Ct = β0 + β1 ln IN Ct + β2 ln IN Ct−1 + α ln Ct−1 + εt
(4)
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
Compare, residuals ε̂t and êt to judge which model is best (see graph).
Which explanatory variables contribute most to the improved fit?
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(5)
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 2: Residuals of the two estimated consumptions functions (2), and (5),
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3
3.1
Dynamic multipliers
Income and consumption
Simplify equation (5) 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 (5), 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.
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What happens in 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).
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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 (5). The dynamic multipliers δc,1, δc,2, ...δc,∞ are in their turn
linked by the dynamics of equation (5), namely
δc,j = 0.13δinc,j + 0.08δinc,j−1 + 0.79δc,j−1, for j = 1, 2, ....∞.
(6)
For example, for j = 3, and setting δinc,3 = δinc, = 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
= 0.01,
1 − 0.79
a 1% permanent increase in income has a 1% long-run effect on consumption.
δc,long−run =
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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 (5),
percentage change in consumption after a 1 percent rise in income.
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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
0
60
20
40
Period
60
Figure 3: Temporary and permanent 1 percent changes in income with associated dynamic multipliers of the consumption function in (5).
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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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3.2
General notation
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.
(7)
Define xt, xt+1, xt+2, , .... as functions of a continuous variable h. When h
changes permanently, starting in period t:e ∂xt/∂h > 0. Since xt is a function
of h, so is yt, and the effect of yt of the change in h is founds 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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(8)
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 (8),
(9)
∂yt+1
(10)
= β1 + β2 + αβ1 = β1(1 + α) + β2
∂h
The pattern in (9) 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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(11)
The long run multiplier is defined by setting δj = δj−1 = δlong−run. Using
(11), δlong−run is found to be
β1 + β2
δlong−run =
, if − 1 < α < 1.
(12)
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 unit change in x(1)
δ0 = β1
δ1 = β1 + β2 + αδ0
δ2 = β1 + β2 + αδ1
..
δj = β1 + β2 + αδj−1
Temporary unit 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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