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The PDLREG Procedure |
The simple finite distributed lag model is expressed in the form
Emerson (1968) proposed an efficient method of constructing orthogonal polynomials from the preceding polynomial equation as
where w_{i} is the weighting factor, and n = p+1 . PROC PDLREG uses the equal weights (w_{i} = 1) for all i. To construct the orthogonal polynomials, the following recursive relation is used:
The constants A_{j}, B_{j}, and C_{j} are determined as follows:
where f_{-1}(i)=0 and .
PROC PDLREG estimates the orthogonal polynomial coefficients, ,to compute the coefficient estimate of each independent variable (X) with distributed lags. For example, if an independent variable is specified as X(9,3), a third-degree polynomial is used to specify the distributed lag coefficients. The third-degree polynomial is fit as a constant term, a linear term, a quadratic term, and a cubic term. The four terms are constructed to be orthogonal. In the output produced by the PDLREG procedure for this case, parameter estimates with names X**0, X**1, X**2, and X**3 correspond to , and , respectively. A test using the t statistic and the approximate p-value ("Approx Pr > |t|") associated with X**3 can determine whether a second-degree polynomial rather than a third-degree polynomial is appropriate. The estimates of the ten lag coefficients associated with the specification X(9,3) are labeled X(0), X(1), X(2), X(3), X(4), X(5), X(6), X(7), X(8), and X(9).
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