Dynamic equations and their estimation. Ragnar Nymoen Department of Economics, UiO
Dynamic equations and their estimation.
Ragnar Nymoen
Department of Economics, UiO
23 March 2009 (updated from 20 March)
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
Notation for dynamic equations
We can re-interpret the linear regression model that we started out with in lecture 1. Using Greene’s notation: y t
D
1
C
2 x
2 t
C ...
C k x
Kt
C " t
, t D 1 , 2 , ..., T .
In the case of a single explanatory variable, we can set x
2 t
D x t
, x
3 t
D x t 1
, x
4 t
D x t 2 and so on, hence: y t D
1
C
X 2 i
D
0
2 C i x t i C
" t
, t D 1 , 2 , ..., T .
However, this is cumbersome, so in Ch 20, Greene changes notation to:
(1) y t
D C i x t i
C " t
, t D 1 , 2 , ..., T .
(2) i D 0 where it is understood that corresponds to
1
that there is a natural re-numbering of the slope coe¢ cients.
p denotes the lag-length of the model.
Lag and di¤erence operator
The lag operator L is de…ned as
Lx t
D x t 1 and the following rules apply:
.
L q
L
2 x t
D L .
Lx t
/ D x t 2
, and L q
L p x t
D L q
C p x t
C L p
/ x t
D x t q
C x t p
,
L
0 x t D x t
, and L
0 a D a if a is a constant.
D x t q p
The di¤erence operator is de…ned as
1 q x t D x t x t q D
.
1 L q
/ x t with the …rst di¤erence as an important special case:
1 x t
D x t x t 1
D
.
1 L / x t
Polynomial in the lag operator
Using the lag operator, (2) can be written as
y t
D C .
X i
D
0 i
L i
/ x t
C " t
, t D 1 , 2 , ..., or y t D C B .
L / x t C
" t
, t D 1 , 2 , ..., T .
where B .
L / is a polynomial in L :
T .
B .
L /
D
0
L
0
C
1
L
1
C
2
L
2
C ...
C p
L p
D i
D
0 i
L i
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 x t an e¤ect on y which is distributed over several periods.
has
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: p j
0
1
D
D
D
D
@ y t
@ x t
D
@ y t
C
1
0
D
1
@ x t
@ y t C p
@ x t
D p
@ y t
C p
C
1
@ x t
D 0 for j > p .
A permanent change in x
Assume that x changes in period t and stays permanently at that new level, then
0
1
D
D
@ y t
@ x t
D
@ y t C 1
0
@ x t C 1
C
@ y t C 1
@ x t
D
0
C
1 j p
D i i
D
0
D p 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.
Estimation of the DL-model
In principle there are no new issues:
If x t 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 x t is uncorrelated with " t
, " t
C
1
, ..., the DL-model includes only pre-determined explanatory variables, and OLS is consistent, and …nite sample biases that are small.
If x t 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.
Dealing with multicollinearity
Consider y t
D C
0 x t
C
1 x t 1
C
" t
, t D 1 , 2 , ..., T .
where x t and x t 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: y t
D C
0
1 x t
C .
0
C
1
/ x t 1
C " t
D C
0
1 x t
C
1 x t 1
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.
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.
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
1 j C
2 j
2
, j D 0 , 1 , ..., p with
1
> 0, equation gives:
2
< 0 and
1
>
2
. Substitution into the original y t
D C
0 x t
C .
.
0
C p
1
0
C p
2
C
1
C
2
/ x t 1
C .
2
/ x t p
C
" t
Collecting terms gives
0
C 2
1
C 4
2
/ x t 2
C ...
y t
D C z
0 t
D p
X j p
D
0 z
1 t D
0
X j D 0 p z
0 t x t j
C
, jx t j
, z
2 t
D j D 0 j
2 x t j
.
1 z
1 t
C
2 z
2 t
C " t
Almon lag, restrictions
The number of parameters is reduced by p 2 in the case of quadratic form.
We can think of this as p 2 restrictions on the unrestricted
DL model with p lags: The coe¢ cients of x t
, x t 1 and x t 2 are not restricted by the assumed quadric polynomial, but the coe¢ cients of x t 3 and all longer lags are restricted, eg.
3
D
0
C
3
1
C
9
2
()
3
3
2
C
3
1 0
D
0 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).
Almon lag, comments
In principle, the Almon lag model is attractive, but in practice there multicollinearity is often a big problem, cf the expressions for z
0 t and z
1 t
.
Hence, the variances of b j 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.
Geometric lag distribution and the Koyck-transformation
If there is 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
becomes: y t
D C
0 j x t i
C " t i D 0
The Koyck-transformation: lag (3) one period and multiply
through by :
(3) y t 1
D C
0 i D 0
j x t i 1
C " t 1 y t
D .
1 / C
0 x t
C y t 1
C t
(4)
OLS estimators and Koyck transformation
The disturbance in (4) is autocorrelated:
t
D " t
" t 1
.
The combination of a lagged regressor ( y t 1
) and autocorrelation means that OLS yields inconsistent estimators. Recall that in a model with no exogenous regression the estimator will be biased if plim
X y t 1 t disturbance 6D 0 which in this case becomes plim
X t y t 1
" t plim
X y t 1
" t 1 t the …rst term is zero, but the second is not, so inconsistency follows.
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 y t
D C i x t i
C i y t i 1
C " t
, t D 1 , 2 , ..., T . (5) i D 0 i D 0 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 y t
. 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
Attributes of the ADL model
de…ned (as long as i q
D
0 i
< 1).
Available from PcGive after estimation by OLS
The rationale for OLS is that y t 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 are consistent, and small sample bias is not large q unless i D 0 i is very close to unity.
De…nes a large typology of di¤erent models, all of them are of interest in economics.
