### Example 19.5 Polynomial Distributed Lags by Using %PDL

This example shows the use of the %PDL macro for polynomial distributed lag models. Simulated data is generated so that Y is a linear function of six lags of X, with the lag coefficients following a quadratic polynomial. The model is estimated by using a fourth-degree polynomial, both with and without endpoint constraints. The example uses simulated data generated from the following model:

The LIST option prints the model statements added by the %PDL macro. The following statements generate simulated data as shown:

/*--------------------------------------------------------------*/
/*  Generate Simulated Data for a Linear Model with a PDL on X  */
/*        y = 10 + x(6,2) + e                                   */
/*        pdl(x) = -5.*(lg)**2 + 1.5*(lg) + 0.                  */
/*--------------------------------------------------------------*/
data pdl;
pdl2=-5.; pdl1=1.5; pdl0=0;
array zz(i) z0-z6;
do i=1 to 7;
z=i-1;
zz=pdl2*z**2 + pdl1*z + pdl0;
end;
do n=-11 to 30;
x  =10*ranuni(1234567)-5;
pdl=z0*x + z1*xl1 + z2*xl2 + z3*xl3 + z4*xl4 + z5*xl5 + z6*xl6;
e  =10*rannor(1234567);
y  =10+pdl+e;
if n>=1 then output;
xl6=xl5; xl5=xl4; xl4=xl3; xl3=xl2; xl2=xl1; xl1=x;
end;
run;

title1 'Polynomial Distributed Lag Example';
title3 'Estimation of PDL(6,4) Model-- No Endpoint Restrictions';

proc model data=pdl;
parms int;                  /* declare the intercept parameter */
%pdl( xpdl, 6, 4 )          /* declare the lag distribution */
y = int + %pdl( xpdl, x );  /* define the model equation */
fit y / list;               /* estimate the parameters */
run;


The LIST output for the model without endpoint restrictions is shown in Output 19.5.1. The first seven statements in the generated program are the polynomial expressions for lag parameters XPDL_L0 through XPDL_L6. The estimated parameters are INT, XPDL_0, XPDL_1, XPDL_2, XPDL_3, and XPDL_4.

Output 19.5.1: PROC MODEL Listing of Generated Program

 Polynomial Distributed Lag Example Estimation of PDL(6,4) Model-- No Endpoint Restrictions

The MODEL Procedure

Listing of Compiled Program Code
Stmt Line:Col Statement as Parsed
1 4615:14 XPDL_L0 = XPDL_0;
2 4615:14 XPDL_L1 = XPDL_0 + XPDL_1 + XPDL_2 + XPDL_3 + XPDL_4;
3 4615:14 XPDL_L2 = XPDL_0 + XPDL_1 * 2 + XPDL_2 * 2 ** 2 + XPDL_3 * 2 ** 3 + XPDL_4 * 2 ** 4;
4 4615:14 XPDL_L3 = XPDL_0 + XPDL_1 * 3 + XPDL_2 * 3 ** 2 + XPDL_3 * 3 ** 3 + XPDL_4 * 3 ** 4;
5 4615:14 XPDL_L4 = XPDL_0 + XPDL_1 * 4 + XPDL_2 * 4 ** 2 + XPDL_3 * 4 ** 3 + XPDL_4 * 4 ** 4;
6 4615:14 XPDL_L5 = XPDL_0 + XPDL_1 * 5 + XPDL_2 * 5 ** 2 + XPDL_3 * 5 ** 3 + XPDL_4 * 5 ** 4;
7 4615:14 XPDL_L6 = XPDL_0 + XPDL_1 * 6 + XPDL_2 * 6 ** 2 + XPDL_3 * 6 ** 3 + XPDL_4 * 6 ** 4;
8 4616:4 PRED.y = int + XPDL_L0 * x + XPDL_L1 * LAG1( x ) + XPDL_L2 * LAG2( x ) + XPDL_L3 * LAG3( x ) + XPDL_L4 * LAG4( x ) + XPDL_L5 * LAG5( x ) + XPDL_L6 * LAG6( x );
8 4616:4 RESID.y = PRED.y - ACTUAL.y;
8 4616:4 ERROR.y = PRED.y - y;
9 4615:15 ESTIMATE XPDL_L0, XPDL_L1, XPDL_L2, XPDL_L3, XPDL_L4, XPDL_L5, XPDL_L6;
10 4615:15 _est0 = XPDL_L0;
11 4615:15 _est1 = XPDL_L1;
12 4615:15 _est2 = XPDL_L2;
13 4615:15 _est3 = XPDL_L3;
14 4615:15 _est4 = XPDL_L4;
15 4615:15 _est5 = XPDL_L5;
16 4615:14 _est6 = XPDL_L6;

The FIT results for the model without endpoint restrictions are shown in Output 19.5.2.

Output 19.5.2: PROC MODEL Results That Specify No Endpoint Restrictions

 Polynomial Distributed Lag Example Estimation of PDL(6,4) Model-- No Endpoint Restrictions

The MODEL Procedure

Nonlinear OLS Summary of Residual Errors
Equation DF Model DF Error SSE MSE Root MSE R-Square Adj R-Sq
y 6 18 2070.8 115.0 10.7259 0.9998 0.9998

Nonlinear OLS Parameter Estimates
Parameter Estimate Approx Std Err t Value Approx
Pr > |t|
Label
int 9.621969 2.3238 4.14 0.0006
XPDL_0 0.084374 0.7587 0.11 0.9127 PDL(XPDL,6,4) parameter for (L)**0
XPDL_1 0.749956 2.0936 0.36 0.7244 PDL(XPDL,6,4) parameter for (L)**1
XPDL_2 -4.196 1.6215 -2.59 0.0186 PDL(XPDL,6,4) parameter for (L)**2
XPDL_3 -0.21489 0.4253 -0.51 0.6195 PDL(XPDL,6,4) parameter for (L)**3
XPDL_4 0.016133 0.0353 0.46 0.6528 PDL(XPDL,6,4) parameter for (L)**4

Portions of the output produced by the following PDL model with endpoints of the model restricted to zero are presented in Output 19.5.3.

title3 'Estimation of PDL(6,4) Model-- Both Endpoint Restrictions';

proc model data=pdl ;
parms int;                  /* declare the intercept parameter */
%pdl( xpdl, 6, 4, r=both )  /* declare the lag distribution */
y = int + %pdl( xpdl, x );  /* define the model equation */
fit y /list;                /* estimate the parameters */
run;


Output 19.5.3: PROC MODEL Results Specifying Both Endpoint Restrictions

 Polynomial Distributed Lag Example Estimation of PDL(6,4) Model-- Both Endpoint Restrictions

The MODEL Procedure

Nonlinear OLS Summary of Residual Errors
Equation DF Model DF Error SSE MSE Root MSE R-Square Adj R-Sq
y 4 20 449868 22493.4 150.0 0.9596 0.9535

Nonlinear OLS Parameter Estimates
Parameter Estimate Approx Std Err t Value Approx
Pr > |t|
Label
int 17.08581 32.4032 0.53 0.6038
XPDL_2 13.88433 5.4361 2.55 0.0189 PDL(XPDL,6,4) parameter for (L)**2
XPDL_3 -9.3535 1.7602 -5.31 <.0001 PDL(XPDL,6,4) parameter for (L)**3
XPDL_4 1.032421 0.1471 7.02 <.0001 PDL(XPDL,6,4) parameter for (L)**4

Note that XPDL_0 and XPDL_1 are not shown in the estimate summary. They were used to satisfy the endpoint restrictions analytically by the generated %PDL macro code. Their values can be determined by back substitution.

To estimate the PDL model with one or more of the polynomial terms dropped, specify the largest degree of the polynomial desired with the %PDL macro and use the DROP= option in the FIT statement to remove the unwanted terms. The dropped parameters should be set to 0. The following PROC MODEL statements demonstrate estimation with a PDL of degree 2 without the 0th order term.

title3 'Estimation of PDL(6,2) Model-- With XPDL_0 Dropped';

proc model data=pdl list;
parms int;                  /* declare the intercept parameter */
%pdl( xpdl, 6, 2 )          /* declare the lag distribution */
y = int + %pdl( xpdl, x );  /* define the model equation */
xpdl_0 =0;
fit y drop=xpdl_0;          /* estimate the parameters */
run;


The results from this estimation are shown in Output 19.5.4.

Output 19.5.4: PROC MODEL Results That Specify %PDL( XPDL, 6, 2)

 Polynomial Distributed Lag Example Estimation of PDL(6,2) Model-- With XPDL_0 Dropped

The MODEL Procedure

Nonlinear OLS Summary of Residual Errors
Equation DF Model DF Error SSE MSE Root MSE R-Square Adj R-Sq
y 3 21 2114.1 100.7 10.0335 0.9998 0.9998

Nonlinear OLS Parameter Estimates
Parameter Estimate Approx Std Err t Value Approx
Pr > |t|
Label
int 9.536382 2.1685 4.40 0.0003
XPDL_1 1.883315 0.3159 5.96 <.0001 PDL(XPDL,6,2) parameter for (L)**1
XPDL_2 -5.08827 0.0656 -77.56 <.0001 PDL(XPDL,6,2) parameter for (L)**2