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 = β₀ + β₁X_t + β₂X_t_-₁ + β₃X_t_-₂ + ... + β_r₊₁X_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., β_i₊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 β_r₊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 X_t, X_t_-₁, X_t_-₂, ..., X_t_-_r
(c)Assume that the third-degree polynomial approximation is quite accurate. Then what is the advantage of this polynomial lag technique?

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The Answer of 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 = β₀ + β₁X_t + β₂X_t_-₁ + β₃X_t_-₂ + ... + β_r₊₁X_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., β_i₊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 β_r₊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 X_t, X_t_-₁, X_t_-₂, ..., X_t_-_r (c)Assume that the third-degree polynomial approximation is quite accurate. Then what is the advantage of this polynomial lag technique?

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