Polynomial Distributed Lag Estimation

The PDLREG Procedure

Polynomial Distributed Lag Estimation

The simple finite distributed lag model is expressed in the form

$y_{t} = {\alpha} + \sum_{i=0}^p{{\beta}_{i}x_{t-i}} + {\epsilon}_{t}$

When the lag length (p) is long, severe multicollinearity can occur. Use the Almon or polynomial distributed lag model to avoid this problem, since the relatively low degree d ( ${{\le} p}$ ) polynomials can capture the true lag distribution. The lag coefficient can be written in the Almon polynomial lag

${\beta}_{i} = {\alpha}_{0}^{*} + \sum_{j=1}^d{{\alpha}_{j}^{*} i^j}$

Emerson (1968) proposed an efficient method of constructing orthogonal polynomials from the preceding polynomial equation as

${\beta}_{i} = {\alpha}_{0} + \sum_{j=1}^d{{\alpha}_{j} f_{j}(i)}$

where f_j(i) is a polynomial of degree j in the lag length i. The polynomials f_j(i) are chosen so that they are orthogonal:

$\sum_{i=1}^n{w_{i}f_{j}(i)f_{k}(i)} = \cases{ 1 & \hspace*{1em}\rm{if} j = k\space \cr 0 & \hspace*{1em}\rm{if} j {\neq} k\space }$

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:

f_j(i) = (A_ji + B_j)f_j-1(i) - C_jf_j-2(i) j = 1, ... , d

The constants A_j, B_j, and C_j are determined as follows:

$A_{j} &=& \{ \sum_{i=1}^n{w_{i}i^2f_{j-1}^2(i)} \hspace*{1em}-(\sum_{i=1}^n{w_{... ...{w_{i}i f_{j-1}^2(i)} \ C_{j} &=& A_{j}\sum_{i=1}^n{w_{i}i f_{j-1}(i)f_{j-2}(i)}$

where f_-1(i)=0 and ${f_{0}(i)=1/\sqrt{\sum_{i=1}^n{w_{i}}} }$ .

PROC PDLREG estimates the orthogonal polynomial coefficients, ${{\alpha}_{0},{ ... },{\alpha}_{d}}$ ,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 ${\hat{{\alpha}}_{0}, \hat{{\alpha}}_{1}, \hat{{\alpha}}_{2}}$ , and ${\hat{{\alpha}}_{3}}$ , 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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