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The PDLREG Procedure

Example 15.1: Industrial Conference Board Data

In the following example, a second-degree Almon polynomial lag model is fit to a model with a five-period lag, and dummy variables are used for quarter effects. The PDL model is estimated using capital appropriations data series for the period 1952 to 1967. The estimation model is written

               CEt = a0 + b1Q1t + b2Q2t + b3Q3t
                                   + c0CAt + c1CAt-1 + ... + c5CAt-5

where CE represents capital expenditures and CA represents capital appropriations.

   title 'National Industrial Conference Board Data';
   title2 'Quarterly Series - 1952Q1 to 1967Q4';
   
   data a;
      input ce ca @@;
      qtr = mod( _n_-1, 4 ) + 1;
      q1  = qtr=1;
      q2  = qtr=2;
      q3  = qtr=3;
   cards;
      2072 1660 2077 1926 2078 2181 2043 1897 2062 1695
      2067 1705 1964 1731 1981 2151 1914 2556 1991 3152
      2129 3763 2309 3903 2614 3912 2896 3571 3058 3199
      3309 3262 3446 3476 3466 2993 3435 2262 3183 2011
      2697 1511 2338 1631 2140 1990 2012 1993 2071 2520
      2192 2804 2240 2919 2421 3024 2639 2725 2733 2321
      2721 2131 2640 2552 2513 2234 2448 2282 2429 2533
      2516 2517 2534 2772 2494 2380 2596 2568 2572 2944
      2601 2629 2648 3133 2840 3449 2937 3764 3136 3983
      3299 4381 3514 4786 3815 4094 4093 4870 4262 5344
      4531 5433 4825 5911 5160 6109 5319 6542 5574 5785
      5749 5707 5715 5412 5637 5465 5383 5550 5467 5465
   ;
   
   proc pdlreg data=a;
      model ce = q1 q2 q3 ca(5,2) / dwprob;
   run;

The printed output produced by the PDLREG procedure is shown in Output 15.1.1. The small Durbin-Watson test indicates autoregressive errors.

Output 15.1.1: Printed Output Produced by PROC PDLREG
 
National Industrial Conference Board Data
Quarterly Series - 1952Q1 to 1967Q4

The PDLREG Procedure

Dependent Variable ce
 
Ordinary Least Squares Estimates
SSE 1205186.4 DFE 48
MSE 25108 Root MSE 158.45520
SBC 733.84921 AIC 719.797878
Regress R-Square 0.9834 Total R-Square 0.9834
Durbin-Watson 0.6157 Pr < DW <.0001
Pr > DW 1.0000    
 
Variable DF Estimate Standard Error t Value Approx
Pr > |t|
Intercept 1 210.0109 73.2524 2.87 0.0061
q1 1 -10.5515 61.0634 -0.17 0.8635
q2 1 -20.9887 59.9386 -0.35 0.7277
q3 1 -30.4337 59.9004 -0.51 0.6137
ca**0 1 0.3760 0.007318 51.38 <.0001
ca**1 1 0.1297 0.0251 5.16 <.0001
ca**2 1 0.0247 0.0593 0.42 0.6794
 
Estimate of Lag Distribution
Variable Estimate Standard Error t Value Approx
Pr > |t|

0                                    0.2444
ca(0) 0.089467 0.0360 2.49 0.0165 |***************                          |
ca(1) 0.104317 0.0109 9.56 <.0001 |*****************                        |
ca(2) 0.127237 0.0255 5.00 <.0001 |*********************                    |
ca(3) 0.158230 0.0254 6.24 <.0001 |***************************              |
ca(4) 0.197294 0.0112 17.69 <.0001 |*********************************        |
ca(5) 0.244429 0.0370 6.60 <.0001 |*****************************************|

The following statements use the REG procedure to fit the same polynomial distributed lag model. A DATA step computes lagged values of the regressor X, and RESTRICT statements are used to impose the polynomial lag distribution. Refer to Judge, Griffiths, Hill, Lutkepohl, and Lee (1985, pp 357--359) for the restricted least squares estimation of the Almon distributed lag model.

   data b;
      set a;
      ca_1 = lag( ca );
      ca_2 = lag2( ca );
      ca_3 = lag3( ca );
      ca_4 = lag4( ca );
      ca_5 = lag5( ca );
   run;
   
   proc reg data=b;
      model  ce = q1 q2 q3 ca ca_1 ca_2 ca_3 ca_4 ca_5;
      restrict   - ca + 5*ca_1 - 10*ca_2 + 10*ca_3 - 5*ca_4 +   ca_5;
      restrict     ca - 3*ca_1 +  2*ca_2 +  2*ca_3 - 3*ca_4 +   ca_5;
      restrict  -5*ca + 7*ca_1 +  4*ca_2 -  4*ca_3 - 7*ca_4 + 5*ca_5;
   run;

The REG procedure output is shown in Output 15.1.2.

Output 15.1.2: Printed Output Produced by PROC REG
 
The REG Procedure
Model: MODEL1
Dependent Variable: ce

Analysis of Variance
Source DF Sum of
Squares
Mean
Square
F Value Pr > F
Model 6 71343377 11890563 473.58 <.0001
Error 48 1205186 25108    
Corrected Total 54 72548564      
 
Root MSE 158.45520 R-Square 0.9834
Dependent Mean 3185.69091 Adj R-Sq 0.9813
Coeff Var 4.97397    
 
Parameter Estimates
Variable DF Parameter
Estimate
Standard
Error
t Value Pr > |t|
Intercept 1 210.01094 73.25236 2.87 0.0061
q1 1 -10.55151 61.06341 -0.17 0.8635
q2 1 -20.98869 59.93860 -0.35 0.7277
q3 1 -30.43374 59.90045 -0.51 0.6137
ca 1 0.08947 0.03599 2.49 0.0165
ca_1 1 0.10432 0.01091 9.56 <.0001
ca_2 1 0.12724 0.02547 5.00 <.0001
ca_3 1 0.15823 0.02537 6.24 <.0001
ca_4 1 0.19729 0.01115 17.69 <.0001
ca_5 1 0.24443 0.03704 6.60 <.0001
RESTRICT -1 623.63242 12697 0.05 0.9614*
RESTRICT -1 18933 44803 0.42 0.6772*
RESTRICT -1 10303 18422 0.56 0.5814*

* Probability computed using beta distribution.

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