3.3
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 additonal economic examples.
Extensions
The most important extensions of (7) are:
1. Several explanatory variables
2. Longer lags
3. Systems of ADL equations
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Two exogenous variables, x1,t and x2,t. The extension of (7) to this case is
yt = β0 + β11x1,t + β21x1,t−1+
β21x2,t + β22x2,t−1 + αyt−1 + εt,
(13)
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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Examples
Dynamic consumption functions 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.
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Wage price dynamics In section 8, and several times later in the course, we
will consider the Phillips curve
e +ε .
πt = β0 + β1,1ut + β2,1πt+1
t
(14)
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 (14)
Finally, πt+1
is formally an ADL with 2 explanatory variables. Moreover, if
e
πt+1
= τ πt−1
(15)
(14) can be reduced to the single variable ADL equation (7), with β1,1 = β1
and α = β2,1τ . An extension of that (standard) model would be
πt = β0 + β1ut + β2ut−1 + απt−1 + εt.
(16)
which is a direct example of (7), and which covers many special cases considered
in the literature.
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The transmission mechanism of monetary policy Towards the end of the
course, we will consider modern monetary policy. We then need a macroeconomic model which implies a reduced form equation between the rate of
inflation and the interest rate as the instrument of monetary policy. A Phillipscurve may be one of the equations of that macro model. In stylized form, the
reduced form equation for inflation is
πt = β0 + β1it + β2it−1 + απt−1 + εt.
(17)
which is an example of an ADL model with the rate of interest it as the
exogenous variable.
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Exhange rate dynamics Later in the course we will discuss the market for
foreign exchange. In the case of a floating exchange rate regime, the following
relationship can be derived for the nominal exchange rate (kroner/USD), Et:
∗
Et = β0 + β1(it − it − ee) + εt, β1 < 0
(18)
where i∗ is the foreign interest rate and ee denotes the expected rate of depre∗
ciation. it − it − ee is known as the risk premium. If expectations are so called
regressive:
ee = −τ Et−1, τ > 0,
the equation for Et can be rewritten as
∗
Et = β0 + β1(it − it ) + αβ1τ Et−1
with β1 < 0 and α = β1τ > 0, which we recognize as another example of an
ADL model.
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