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:

\[  \sum _{i=1}^{n}{w_{i}f_{j}(i)f_{k}(i)} = \begin{cases}  1 &  \mr {if} j = k \\ 0 &  \mr {if} j {\neq } k \end{cases}  \]

where ${w_{\mi {i} }}$ is the weighting factor, and ${n = p+1 }$. PROC PDLREG uses the equal weights (${w_{\mi {i} }=1}$) for all i. To construct the orthogonal polynomials, the following recursive relation is used:

\[  f_{j}(i) = (A_{j}i + B_{j})f_{j-1}(i) - C_{j}f_{j-2}(i) j=1,{\ldots }, d  \]

The constants ${A_{j}, B_{j}}$, and ${C_{j}}$ are determined as follows:

$\displaystyle  A_{j}  $
$\displaystyle = $
$\displaystyle  \left\{  \sum _{i=1}^{n}{w_{i}i^{2}f_{j-1}^{2}(i)} -\left(\sum _{i=1}^{n}{w_{i}i f_{j-1}^{2}(i)} \right)^{2} \right.  $
$\displaystyle  $
$\displaystyle  $
$\displaystyle  ~ ~  - \left. \left(\sum _{i=1}^{n}{w_{i}i f_{j-1}(i)f_{j-2}(i)} \right)^{2} \right\} ^{-1/2}  $
$\displaystyle B_{j}  $
$\displaystyle = $
$\displaystyle  -A_{j}\sum _{i=1}^{n}{w_{i}i f_{j-1}^{2}(i)}  $
$\displaystyle C_{j}  $
$\displaystyle = $
$\displaystyle  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, ${{\alpha }_{0},{\ldots },{\alpha }_{d}}$, 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 10 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).