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

question 24

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
$$Y _ { t } = \beta _ { 0 } + \beta _ { 1 } X _ { t } + \beta _ { 2 } X _ { t - 1 } + \beta _ { 3 } X _ { t - 2 } + \ldots + \beta _ { r + 1 } X _ { t - r } + u _ { t }$$ (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 $\beta _ { r + 1 }$ .

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Definitions:

Standard Deviation

A statistic that measures the dispersion or variability within a dataset, indicating how spread out the numbers are.

Setters

Positions or roles in various games or activities that involve organizing or preparing something for use, specifically referring to a key player in volleyball responsible for setting the ball for attacks.

Assists

In various contexts, this term refers to the action of helping or supporting.

Days Absent

The number of days an individual is not present at their scheduled location, often used in contexts like school or work attendance records.