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

question 7

Essay

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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Definitions:

Thinner Walls

Structures, particularly in the context of blood vessels or chambers of the heart, having less thickness, which can impact their function and efficiency.

Vasodilation

The widening of blood vessels, which decreases blood pressure and increases blood flow to specific areas of the body.

Vessel Diameter

The width of a blood vessel, which affects blood flow and pressure, and is regulated by various physiological mechanisms.

Resistance

In a broad sense, it refers to the opposition to flow or movement, such as electrical resistance in physics or the body's resistance to disease in biology.