ECON 306
DISTRIBUTED LAG MODELS
An increase in income tax causes:
 Less disposable income means that
consumption declines.
 Profits of suppliers decline.
 The demand for productive inputs
declines.
 Profits of input providers decline.
y t  f ( xt , xt 1 , xt 2 ,.....)
SUCH MODELS ARE SAID TO BE
DYNAMIC.
THESE MODELS SPLIT INTO FINITE
AND INFINITE LAG MODELS.
FINITE LAG MODELS
E.G. A FIRM'S CAPITAL
EXPENDITURE DECISION
1.Investment approved.
2.Funds appropriated.
3.Actual expenditures are observed over
subsequent periods as plans are
finalised, labour and materials hired
and construction carried out.
If xt = capital appropriation and yt the
amount of capital expenditure then yt will
be observed over the periods t+1, t+2, …
until the project is completed.
We could say that xt affects yt, yt+1, yt+2, …
i.e. the current period's appropriation
decisions affect future capital
expenditure.
Alternatively we could say that current
period capital expenditure decisions are
affected by past capital appropriation
decisions. Also we will assume that, after
n quarters the effect of any appropriation
decision is exhausted; hence:
y t  f ( xt , xt 1 , xt 2 ,...., xt n )
THIS IS A FINITE DISTRIBUTED LAG
MODEL
In order to convert this into a statistical
model we require:
 A functional form.
 An error term.
 Some assumptions about the error term.
We assume a linear model:
y t     0 xt   1 xt 1   2 xt 2  ...   n xt n  et
t  n  1,..., T
We assume:
E (e t )  0
E (e t )   t
2
2
E (et e s )  0; t  s
Note loss of observations in such a model.
In this model α is the intercept term and
the βi are the DISTRIBUTED LAG
WEIGHTS which reflects the fact that
they measure the effect of changes in past
appropriations Δxt-i on current expected
expenditures ΔE(yi):
∂E(yt)/∂xt-i = βi
This model can be estimated by OLS but
multicollinearity may be a problem.
An alternative is a RESTRICTED
LEAST SQUARES approach which
reduces the estimator variances.
MODEL 1: THE ARITHMETIC LAG
Proposed by Fisher in 1937, under this
scheme the weights decline linearly. His
restrictions were:
 0  (n  1)
 1  n
 2  (n  1)
.
.
n  
 0
βi
n+1
The effect of a change in xt in the current
period is β0 = (n+1) γ and the amount
declines by γ in each subsequent period;
in the (n+1)th period the effect has
disappeared.
Assume a 4 period lag model:
y t     0 x t   1 xt 1   2 xt  2   3 xt 3   4 xt  4  et
t  5,..., T
y t    (5 ) x t  (4 ) xt 1  (3 ) xt  2  (2 ) x t 3  xt  4  et
y t     [5 xt  4 x t 1  3x t  2  2 xt 3  xt  4 ]  et
y t    z t  et
z t  5 xt  4 x t 1  3x t  2  2 xt 3  xt  4
ˆ i  (n  1  i )ˆ
i  1,...,0
Collinearity is no longer a problem - there
is only one RHS variable. But the
restriction may not be true. This can be
tested using an F test.
MODEL 2: POLYNOMIAL
DISTRIBUTED LAGS
Proposed by Shirley Almon in 1965. She
suggested constraining distributed lag
weights to fall on a polynomial, probably
of a low value.
With a second order polynomial, the
effect of Δxt-i on E(yt) is:
E ( yt ) / xt i   i   0   1i   2i 2
i  0,..., n
βi
i
At time t the effect of the change of a
policy variable is:
E ( y t ) / xt   0   0
This model is commonly used to evaluate
monetary and fiscal policies.
With a lag of length of 4:
0   0
1   0   1   2
 2   0  2 1  4 2
 3   0  3 1  9 2
 4   0  4 1  16 2
Substitution yields:
y t     0 xt  ( 0   1   2 ) xt 1  ( 0  2 1  4 2 ) xt  2
 ( 0  3 1  9 2 ) xt 3  ( 0  4 1  16 2 ) xt  4  et
yt     0 zt 0   1 zt1   2 zt 2  et
z t 0  x t  x t 1  xt  2  xt 3  x t  4
z t1  xt 1  2 xt  2  3x t 3  4 xt  4
z t 2  xt 1  4 xt  2  9 x t 3  16 xt  4
ˆi  ˆ0  ˆ1i  ˆ2i 2
i  0,..., n
SELECTION OF THE LENGTH OF
THE FINITE LAG
 Take the maximum length you are
willing to consider.
 Use AIC/Schwarz - minimise - on
increasingly shorter lag lengths.
 R squared and adjusted R squared are
poor discriminators.
THE INFINITE GEOMETRIC LAG
yt     0 xt  1 xt 1   2 xt  2  ....  et
yt    i 0  i xt i  et

 i   i
 1
β
i
y t     ( xt  xt 1   2 xt  2   3 x t 3 )  et
This is the INFINITE DISTRIBUTED
LAG MODEL. It has 3 parameters:
1. The intercept, α.
2. A scale factor, β.
3. Ф which controls the rate at which the
weights decline.
E ( yt ) / xt i   i  
i
β is the IMPACT MULTIPLIER.
If the change is sustained for 3 periods
then:
E ( y t  2 )       2
If the change is sustained permanently,
then the long run effect (total effect, long
run multiplier) is:
 (1     2   3  ....)   /(1   )
How can we estimate this:
 It has an infinite number of parameters.
 It is non-linear.
THE KOYCK TRANSFORMATION
Consider:
yt  yt 1
 [   ( xt  xt 1   2 xt 2   3 xt 3 .....  et ] 
 [   ( xt 1  xt 2   2 xt 3   3 xt 4 .....  et 1 )]
  (1   )  xt  (et  et 1 )
yt   (1   )  xt  (et  et 1 )  yt 1
 1   2 yt 1   3 xt  vt
where:
 1   (1   )
2 
3  
v t  (et  et 1 )
How do we estimate this?
 Lagged dependent variable.
 vt depends on et and et-1.
The OLS estimator is biased and this bias
does not disappear in large samples. It is
inconsistent.
Two stage least squares will produce
consistent estimates in large samples.
yˆ t 1  a 0  a1 xt 1