Dynamic equations and their estimation (long
version containing optional material, from slide
18).
Ragnar Nymoen
Department of Economics, UiO
29 March 2009 (updated from 23 March)
ECON 4610: Lecture 10
Overview
In economics, the typical case is that there are dynamic
e¤ects of a change in an explanatory variable on the
dependent variable.
This means that the models that we have mainly considered
so far, which are static models, are of limited relevance for the
modelling of time series data.
However, it possible to extend the regression model, and the
simultaneous equations model as well, so that they become
relevant for time series data.
Dynamic modelling is however a big …eld within econometrics,
and we are only give a partial introduction here.
Syllabus:
B: Lecture note DL
Greene: 20.1, 20.2.1, 20.3
K: 9.4
ECON 4610: Lecture 10
Notation for dynamic equations
We can re-interpret the linear regression model that we
started out with in lecture 1. Using Greene’s notation:
yt D
1
C
2 x2t
C ... C
k xKt
C "t , t D 1, 2, ..., T .
In the case of a single explanatory variable, we can set
x2t D xt , x3t D xt 1 , x4t D xt 2 and so on, hence:
yt D
1C
K
X2
i D0
2 Ci xt i
C "t , t D 1, 2, ..., T .
(1)
However, this is cumbersome, so in Ch 20, Greene changes
notation to:
p
X
yt D C
(2)
i xt i C " t , t D 1, 2, ..., T .
i D0
where it is understood that corresponds to 1 in (1), and
that there is a natural re-numbering of the slope coe¢ cients.
p denotes the lag-length of the model.
ECON 4610: Lecture 10
Lag and di¤erence operator
The lag operator L is de…ned as
Lxt D xt
1
and the following rules apply:
L2 xt
q
p
.L C L /xt
0
L xt
D L.Lxt / D xt
D xt
q
C xt
2,
p,
0
and Lq Lp xt D Lq Cp xt D xt
D xt , and L a D a if a is a constant.
The di¤erence operator is de…ned as
1q xt D xt
xt
q
D .1
Lq /xt
with the …rst di¤erence as an important special case:
1xt D xt
xt
1
D .1
L/xt
ECON 4610: Lecture 10
q p
Polynomial in the lag operator
Using the lag operator, (2) can be written as
yt D
or
yt D
p
X
C.
i
i L /xt
i D0
C "t , t D 1, 2, ..., T .
C B.L/xt C "t , t D 1, 2, ..., T .
where B.L/ is a polynomial in L:
B.L/ D
0
0L C
1
1L C
2
2 L C ... C
p
pL D
ECON 4610: Lecture 10
p
X
i D0
i
iL
The distributed lag model and its multipliers
Model (2), or the same equation with lag-operator notation, is
called the distributed lag model, which we refer to as DL
model from now on, because the e¤ects of a change in xt has
an e¤ect on y which is distributed over several periods.
Let j denote the marginal e¤ect that a change in x has on y
after j periods. The simplest example is when the change in x
occurs in period t and lasts for just one period, and then x
returns to the old level in period t C 1, then:
@yt
D 0
0 D
@xt
@yt C1
D 1
1 D
@xt
@yt Cp
D p
p D
@xt
@yt Cp C1
j D
D 0 for j > p.
@xt
ECON 4610: Lecture 10
A permanent change in x
Assume that x changes in period t and stays permanently at that
new level, then
0
1
p
j
@yt
D 0
@xt
@yt C1 @yt C1
D
C
D
@xt C1
@xt
p
X
D
i
D
D
i D0
p
0
C
1
for j > p.
In both examples, 0 is called the impact multiplier, and j , j > p
is called the equilibrium multiplier or the long-run multiplier.
The intermediate e¤ects are called the dynamic multiplies in the
case of a temporary change, and the cumulated multipliers in the
case of a permanent change.
ECON 4610: Lecture 10
Estimation of the DL-model
In principle there are no new issues:
If xt is uncorrelated with all the disturbances, then OLS
estimates of the DL-model are unbiased.
OLS may not be fully e¢ cient, and standard errors may need
to be corrected for e.g. heteroscedasticity.
If xt is uncorrelated with "t , "t C1 , ..., the DL-model includes
only pre-determined explanatory variables, and OLS is
consistent, and …nite sample biases that are small.
If xt is correlated with "t then we need to use IV to obtain
consistent estimators.
The problems associated with the estimation of DL model are
more practical and about relevance:
Why should there be an exact “cut o¤” point in the lag
distribution, and how do we know p?
The number of parameters may become large,
combined with multicollinearity, the estimates may become
imprecise.
ECON 4610: Lecture 10
Dealing with multicollinearity
Consider
yt D
C
0 xt
C
1 xt 1
C "t , t D 1, 2, ..., T .
where xt and xt 1 are often quite highly correlated.
However, we can transform the equation in a way that does not
a¤ect the disturbance "t , and therefore does not a¤ect the
statistical properties of the model:
yt
D
C
D
C
0 1xt
0 1xt
C.
C
0
C
1 xt 1
1 /xt 1
C "t
C "t
In this equation, the regressors have a low correlation, meaning
that if we concentrate on the estimation of the impact multiplier
0 , and the long-run multiplier 1 collinearity is not a practical
problem. This generalizes.
ECON 4610: Lecture 10
Speci…c distributed lags
In order to deal with the problem of unknown lag order p, and
that the number of variables may become large, it has been
proposed to specify the distributed lag more closely.
Two models are much referred to:
The polynomial lag distribution, or “Almon lag” after one of
its inventors.
The geometric lag distribution.
We brie‡y explain these, and then mention some arguments
for taking a slightly more general starting point for dynamic
modelling than the DL model.
ECON 4610: Lecture 10
Almon lag
It is often reasonable to assume that the dynamic multipliers …rst
increase, and reach a peak before they decline. A model which
captures that idea is:
j
D
0
C
with 1 > 0, 2 < 0 and
equation gives:
yt
D
.
C
0
0 xt
Cp
1j
2j
C
>
1
C.
0C 1C
2
1 Cp
2 /xt p
2
2.
, j D 0, 1, ..., p
Substitution into the original
2 /xt 1
C "t
C.
0
C2
1
C4
Collecting terms gives
yt
z0t
D
D
z1t
D
z2t
D
C
Xp
0 z0t
1 z1t
C
x ,
j D0 t j
Xp
j D0
Xp
j D0
C
2 z2t
C "t
jxt j ,
j 2 xt j .
ECON 4610: Lecture 10
2 /xt 2
C ...
Almon lag, restrictions
The number of parameters is reduced by p
quadratic form.
2 in the case of
We can think of this as p 2 restrictions on the unrestricted
DL model with p lags: The coe¢ cients of xt , xt 1 and xt 2
are not restricted by the assumed quadric polynomial, but the
coe¢ cients of xt 3 and all longer lags are restricted, eg.
3
D
0
C3
1
C9
2
()
3
3
2
C3
1
0
D0
which is a linear restriction on the unrestricted DL-model with
p D 3.
So the validity of the quadratic Almon lag speci…cation can be
tested by …rst estimating an unrestricted model with p set to a
high number, and then testing the p 2 restriction by a F -test.
Can generalize to cubic form for example: The number of
restriction is p 3. And p q in general (q is the order of
polynomial).
ECON 4610: Lecture 10
Almon lag, comments
In principle, the Almon lag model is attractive, but in practice
multicollinearity is often a big problem, cf the expressions for
z0t and z1t .
Hence, the variances of bj tend to be large
Another problem, is that it is di¢ cult in practice to obtain
Almon lag models that “survive” mis-speci…cation tests: For
example: Tests of residual autocorrelation often reject the null
of no autocorrelation.
The source of this problem may be the DL model itself: since
the Almon lag model is only a restricted version of that, and
suggest that a more general starting point for the speci…cation
of a dynamic model is needed.
ECON 4610: Lecture 10
Geometric lag distribution and the Koyck-transformation
If there are economic reasons to believe that the dynamic
multipliers are declining then a simple model is
j
D
j
0,
0<
< 1, j D 1, 2, ....
where it is understood that p is set p to in…nity. Hence (2)
becomes:
1
X
i
yt D C 0
xt i C "t
(3)
i D0
The Koyck-transformation: lag (3) one period and multiply
through by :
yt
1
D
C
0
1
X
i
xt
i 1
i D0
C "t
1
and subtract from (3)
yt D .1
/C
0 xt
C yt
1
C
ECON 4610: Lecture 10
t
(4)
OLS estimators and Koyck transformation
The disturbance in (4) is autocorrelated: t D "t
"t 1 .
The combination of a lagged regressor (yt 1 ) and autocorrelation
means that OLS yields inconsistent estimators. Recall that in a
model with no exogenous regressors the estimator will be biased if
X
plim
yt 1 disturbance 6D 0
t
which in this case becomes
X
plim
yt 1 "t
t
plim
X
yt
1 "t 1
t
the …rst term is zero (cf …rst lecture), but the second is not, so
inconsistency follows.
ECON 4610: Lecture 10
A more general approach: ADL model
The autocorrelation above was induced by the Koyck
transformation. It does not imply that all models with lagged
regressors have autocorrelated residuals.
In modern dynamic econometric modelling the DL model is no
longer seen as the right framework, instead the stochastic
di¤erence equation
p
q
X
X
yt D C
x
C
i yt i 1 C " t , t D 1, 2, ..., T . (5)
i t i
i D0
i D0
has taken over this role.
(5) is called the autoregressive distributed lag model, or ADL.
Classical disturbance properties are usually assumed with
reference to the principle that the lag orders p and q can be
chosen so that the model explains all the systematic variation
in yt . Hence residual autocorrelation is typically seen as a sign
that the model is not general enough, and the advise is to
re-specify the model to install classical disturbance properties
ECON 4610: Lecture 10
Attributes of the ADL model
Dynamic multipliersP
and the long run multiplier are well
q
de…ned (as long as i D0 i < 1).
Available from PcGive after estimation by OLS
The rationale for OLS is that yt 1 and higher order lags are
pre-determined variables as long at there is no residual
autocorrelation. Hence, with reference to lecture 1, the
estimators
Pq are consistent, and small sample bias is not large
unless i D0 i is very close to unity.
De…nes a large typology of di¤erent models, all of them are of
interest in economics.
ECON 4610: Lecture 10
A model typology
Several single equation models are encompassed by the
ADL(1,1)-model:
yt D
C
1 yt 1
C
0 xt
C
1 xt 1
C "t
(6)
where "t white-noise., E[xt " t Cj ] D E[xt 1 "t 1Cj ] D 0,
j D 0, 1, 2, .....
ADL equations can be part of a simultaneous equations system,
E[xt " t ] 6D 0 in that case, but herer we consider the case of
predetermined xt , and OLS yields consistent estimates. A selection
of models is
Static model
Model with di¤erenced data
Common-factor model
Equilibrium correction(EqCM)
ECON 4610: Lecture 10
Static model
If
1
D
1
D 0 (6) reduces to
yt D
C
0 xt
C "t .
White-noise "t requires that the autocorrelation of yt is matched
exactly by xt , thus 1 D 1 D 0. Unusual in practice. Instead, a
regression
yt D KO 0 C KO 1 xt C uO t
(7)
usually produces a residual uO t which is autocorrelated because of
the omitted variables (xt 1 and yt 1 /.
The static model is nevertheless important in econometrics,
because of interpretation as long-run equilibrium relationship.
ECON 4610: Lecture 10
Di¤erenced data
1
D 1 and
1
D
0,
yields:
1yt D
C
0 1xt
C "t .
(8)
In this case neither the expectation nor the variance of yt exists, so
yt is not covariance-stationary. In addition yt contains a
deterministic trend if 6D 0. Below, we shall see that (8)
represents a situation where yt and xt are integrated of degree 1,
but not cointegrated.
ECON 4610: Lecture 10
Common-factor model
ADL(1,1) with lag-operator notation
.1
1 L/yt
D
C.
D
C .1
0
C
1 L/xt
C "t .
(9)
Writing the equation as
.1
1 L/yt
1 L/ 0 xt
C "t
shows that the COMFAC restriction
.
0
C
1 L/
D .1
1 L/ 0 ,
or alternatively,
1
D
1 0,
must hold if the order of the dynamics is to be reduced by 1.
ECON 4610: Lecture 10
(10)
Imposing the restriction yields
yt D /.1
1 L/ C
0 xt
C "t /.1
1 L/
or
yt D
C
0 xt
C ut ,
ut D
1 ut 1
C "t
(11)
Note that ut is AR(1) and that the long run multiplier is equal to
the short run multiplier 0
yt D .1
1/ C
0 xt
0
1 xt 1
C
1 yt 1
C "t ,
(12)
so @yt /@xt D 0 and, for yt D y and xt D x, @y /@x D 0 .
Greater ‡exibility is obtained by starting from a general ADL(p,q).
By imposing m COMFAC restrictions, the order of the dynamics
can be reduced to p m and q m.
ECON 4610: Lecture 10
Equilibrium correction (EqCM)
EqCM is a 1
ADL(1,1).
yt
yt
1
1 transformation of the ADL-model. Consider a
C.
D
C.
D
1
1/yt
1
1
1/yt
C
1
C
0 xt
1 xt 1
C
0 .xt
C "t
(13)
xt
1/ C . 0
C
1 /xt 1
C
1 /xt 1
C "t ,
i.e.,
1yt D
C
0 1xt
C.
1
1/yt
1
C.
0
C "t .
To establish the equilibrium relationship, we start from the
ADL-model
.1
.L//yt D 0 C .L/xt C " t
where
.L/ D
yt
1L
and .L/ D
1
.L/
1
0
C
1L
(14)
(15)
in a ADL(1,1).
.L/
"t
xt D
.L/
1
.L/
ECON 4610: Lecture 10
(16)
To establish the long-run conditional expectation, set L D 1 and
take the expectation:
yt
.1/
1
1
E[yt
K0
.1/
"t
xt D
.1/
1
.1/
K1 xt ] D 0
(17)
(18)
where
K0 D
K1 D
1
1
.1/
.1/
D
.1/
D
.
0
1
(19)
1
1
C
1/
1
ECON 4610: Lecture 10
(20)
(18) implies that the conditional long-run mean, y can be de…ned
as
y D K0 C K1 x.
(21)
Use this notation to write eq. (14) as
1yt
1yt
D
0 1xt
D
C
C.
0 1xt
1
C.
1/.yt
1
1
1/.y
yt
1 / C "t
K1 x/t
(22)
1
C "t ,
(23)
which makes the equilibrium correction mechanism explicit.
The static equilibrium is restrictive, since many series have
non-zero growth rates. The dynamic equilibrium is de…ned in such
a way that
y D KQ 0 C K 1 x
(24)
holds in a situation where 1yt D gy and 1xt D gx . In equilibrium,
gy D K1 gx
from (24).
ECON 4610: Lecture 10
(25)
Inserting in (23) gives
K 1 gx D
0
C
0 gx
C.
1
1/.y
K 1 x/
(26)
and consequently
yD
0
C.
.1
0
K1 /gx
C K x D KQ 0 C K1 x,
1/
Hence KQ 0 in (24) is
KQ 0 D
0
C.
.1
0
K1 /gx
.
1/
KQ 0 D K0 only if gx D gy D 0.
ECON 4610: Lecture 10
(27)
Generalization to ADL(p,q)
The ADL.p, q/ can be written as a EqCM in several equivalent
ways. For p D q D 4:
yt
4
X
i yt
i
i D1
D
C
4
X
i xt i
i D0
C "t ,
(28)
one possibility is to put the levels term at the fourth lag
1yt
D
C
3
X
i D1
C. .1/
0
1yt
i
1/yt
i
4
C
3
X
0
i 1xt i
i D0
C .1/xt
4
(29)
C "t
where
0
i
0
i
D
D
i
X
j
j D1
i
X
j D0
j,
1, i D 1, 2, 3,
i D 0, 1, 2, 3.
ECON 4610: Lecture 10
(30)
Alternatively,
1yt
D
C
3
X
i D1
i
C. .1/
1yt
i
1/yt
C
1
3
X
i
1xt
i D0
C .1/xt
1
i
(31)
C "t
with
i
D
0
D
i
D
4
X
j,
4
X
j,
j Di C1
i D 1, 2, 3,
(32)
0
j Di C1
i D 1, 2, 3.
ECON 4610: Lecture 10
In both cases:
K1 D
1
.1/
D
1 .1/
.1
P4
j D0
P4
and
0
0
D
0
D
j D1
j
j/
0,
but
0
i
0
i
6D
i
, i D 1, 2, 3,
6D
i
, i D 1, 2, 3.
ECON 4610: Lecture 10
(33)
Estimation
OLS for the ADL[p, q ]-model
yt
p
X
j yt j
j D1
D
C
q
X
j xt j
j D0
C "t
(34)
yields an estimator bT for the parameter vector
D [ , 1 , 2 , ... 0 , ..., k ] with the following properties ("t is
w.n.; E[Xt "s ] D 0 for all t,s):
p
T .bT
/ ! N.0, •/
tT
rFT
T !1
!
T !1
!
T !1
N.0, 1/
(35)
2 .r /
• is a constant covariance matrix, tT the “t-value” and rFT the
“F-statistic” for a r linear restrictions.
ECON 4610: Lecture 10
(35) holds as long as the roots of the characteristic polynomial
(34):
p. / D
p
1
p 1
:::
p
,
are inside the unit-circle. The proof is an application of the
Mann-Wald theorem, and is not shown here.
Finite sample bias arise because E[Yt j "t i ] 6D 0 for j i . An
important issue is how fast this bias (the Hurwicz-bias) is
disappearing.— How large is “large”?
The AR(1) process
yt D
C
1 yt 1
C "t ,
"t
N.0,
2
/, t D 1, : : : T .
from the …rst lecture illustrates several features.
ECON 4610: Lecture 10
(36)