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Distributed Lag Models:

In other words, the effect of an event may be distributed over several time periods. Fo~.example,  firms do not rush  to  increase investment  unless they  see a  sustained increase  in  sales.  A  one time increase  in  sales because of an  unexpectedly harsh winter or a natural calamity will not see firms increasing investment expenditure and expanding productive capacity. Similarly, investment  in  new  equipment  (Xt)  in period  t may  impact profit  (Y,)  over several periods, βoXt  in  period t, βiXi,  in  period t+l,  and so on up to βiXi,  in period t+s.  This may be expressed as

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Given that the usual Gauss-Markov conditions1  hold this model may be estimated by ordinary  least  squares. However, there are several questions/problems  that arise, some of which are particularly important. First, how do we  decide on  the  length of the  lag structure? Should  it be  finite or infinite?  Theory is usually silent on  this matter. Further, too short a lag structure may lead to inaccuracies and too long a lag structure may result in a model with very few degrees of freedom.

Secondly,  the  lagged  values of X  are  likely  to  be highly correlated resulting in imprecise coefficient estimates (large standard errors) and making inference difficult.

Thirdly, what will be  the  response  structure  or  the  lag  structure?  Usually  an assumption is made that all the coefficients will have the same sign,  i.e.,  the impact of  the  event will  be monotonic in nature.  In  other  words, investment  in  new equipment  today will  either  increase profit for the next periods or decreage profit' for the next s  periods.  Specifically, we do not allow for the case that it may decfease profit  for the next five periods and then increase  it from period  six to s.  It is also usual to assume that the impact of the event will decline in magnitude over time (the coefficients will decline in absolute magnitude to zero) although there may be a short initial period when  the absolute  magnitude of  the  coefficients  increases.  The assumption, while not always realistic,  is necessary for ease of estimation.

The need to economize on  the number of coefficients has  led to the development of different  lag  schemes, essentially  different  assumptions about the  weights (coefficients) β0,  β1,.  .  .  .βs.

De ~eeuw*  advocated  the use of an  inverted V  structure  for the weights.  In  a total lag of s periods,  the  first  half  of.  the  weights  are  taken  as  proportional  to  the increasing  series  1,2,3,.  ...  s/2,  and  the  second half  of  the  weights  are  chosen
proportional  to  the weights s/2, (s/2 -  1),. . .  ..3,2,1.

A  few years  later Almon3 suggested a more flexible approach.  He suggested (using the Weierstrass theorem) that the  actual  lag  function  could  be approximated by a polynomial  of  suitable degree4.  The estimation  of the model  can be  done by  least squares as  the Gauss Markov conditions are satisfied.

In  both  the  above approaches  the total  lag period is  finite and has  to be specified prior  to estimation. The biggest problem  in  using models with finite  lag structures is: How do we determine the  length of the  lag?  There are no easy answers to this question  and consequently the emphasis  has been on infinite  lag  models.

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