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1. DISTRIBUTED LAG MODELS

Lagsare present in econometrics for several reasons. Psychological inertia (habit),

permanent vs. transitory income, technical and technological reasons causing delay in

implementing the changes in capital labor compositions, institutional reasons, laborcontracts, etc.

Finite or Infinitenumber of lag terms to include in regression is one issue.Yt= +0Xt+1Xt-1+ kXt-k This has k finite lags.Some types of lags considered in the literature:

Arithmetic Lag:

If i= (k+1i) where the lags decline on a straight line from to 0.Inverted V Lag:

If i= ifor i [0, k/2] and i= (ki)for i (k/2, k], line goes up and comes down.Koyck Lag Structure (geometrically declining effect of past on current events):

k=is possible by the Koyck method. There is no theory w.r.t. when to stop lagging theregressors. Seems mostly ad hoc and estimation is subject to serious collinearity.

Koyck assumes that all coefficients are of same sign (positive) and decline geometrically

as: k= 0k

or k= 0(1)k

, where (0,1) is the rate of decline and (1) is speed ofadjustment. If we estimate we know rate of decline and speed of adjustment.

If k= 0kthen sum of coefficients of lag terms

k= 0(1++ +k+ ...) or k= 0(1)

1.

Steps in derivation of Koyck are designed to make it easy to estimate.Write the distributed lag model as

ADL(0,): Yt= +0Xt+1Xt-1+ + ut

substitute the geometric declining definition of coefficients in this model to yield:

(1) Yt= +0Xt+0Xt-1+02Xt-2+ + ut

lag the above eq. by 1 period to yield:

Yt-1= +0Xt-1+0Xt-2+02Xt-3+ + ut

multiply by to both sides of this second equation to give(2) Yt-1= +0Xt-1+0

2Xt-2+0

3Xt-3+ + ut

Now subtract (2) from (1) to yield ADL(1,1)

(3) YtYt-1= (1) + 0Xt+ (utut-1).

Error term here has and that must be recognized.

Re write (3) as

(4) Yt= (1) +Yt-1+ 0Xt+ (utut-1).Lagged dependent variable is a problem in finding reliable estimates of (4).Need to use Durbin h test since lagged dep var is present.

Even if maximum lag is infinite, the average lag for Koyck model need not be long.

What is the mean lag of Koyck model =[kk]/[k] both s are from 0 to .

Mean lag simplifies to /[1].