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:

Wage Differentials

The differences in wage rates due to the variations in the skill level, occupation, region, and other factors affecting compensation.

Diminishing Returns

The principle stating that as one factor of production is increased while others are held constant, a point will eventually be reached where additions of the factor yield progressively smaller increases in output.

Value Added

The increase in worth of a product or service as a result of a particular stage of production or as it moves through the supply chain.

Marginal Product

The increase in output resulting from a one-unit increase in the amount of a single input used, keeping all other inputs constant.