The Distributed Lag Regression Model Requires Estimation Of $( r + 1 )$

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
$$Y _ { t } = \beta _ { 0 } + \beta _ { 1 } X _ { t } + \beta _ { 2 } X _ { t - 1 } + \beta _ { 3 } X _ { t - 2 } + \ldots + \beta _ { r + 1 } X _ { t - r } + u _ { t }$$ (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., $\beta _ { i + 1 } \approx f ( i )$ for $i = 0,1 , \ldots , r$ . Let $f ( i )$ be a third degree polynomial, with coefficients $\alpha _ { 0 } , \ldots , \alpha _ { 3 }$ . Specify the equations for $\beta _ { 1 } , \beta _ { 2 } , \beta _ { 3 } , \beta _ { 4 }$ and $\beta _ { r + 1 }$ .