The Distributed Lag Regression Model Requires Estimation of (R+1)coefficients in the Case

The distributed lag regression model requires estimation of (r+1)coefficients in the case of a single explanatory variable.In your textbook example of orange juice prices and cold weather,r = 18.With additional explanatory variables,this number becomes even larger.
Consider the distributed lag regression model with a single regressor
Yt = β0 + β1Xt + β2Xt-1 + β3Xt-2 + ...+ βr+1Xt-r + ut
(a)Early econometric analysis of distributed lag regression models was interested in reducing the number of parameters by approximating the coefficients by a polynomial of a suitable degree,i.e. ,βi+1 ≈ f(i)for i = 0,1,…,r.Let f(i)be a third degree polynomial,with coefficients α0,.... ,α3.Specify the equations for β1,β2,β3,β4,and βr+1.
(b)Substitute these equations into the original distributed lag regression,and rearrange terms so that Y appears as a linear function of β0,α0,α1,α2,α3 and a transformation of the Xt,Xt-1,Xt-2,... ,Xt-r
(c)Assume that the third-degree polynomial approximation is quite accurate.Then what is the advantage of this polynomial lag technique?