PDL.RPF

PDL.RPF demonstrates several ways to estimate polynomial distributed lags (or Almon lags or PDL’s) are the principal form of "hard" restriction used for estimating distributed lags. For the equation:

\begin{equation} {y_t} = \sum\limits_{l = {L_0}}^L {{\beta _l}} {X_{t - l}} + {u_t} \label{eq:pdl_genericdl} \end{equation}

an unconstrained third degree PDL (lags 1 to \(L\)) takes the form:

\begin{equation} \sum\limits_{k = 1}^L {{\kern 1pt} \left( {a + bk + c{k^2} + d{k^3}} \right){\kern 1pt} {\kern 1pt} {X_{t - k}}} \end{equation}

where \(a\), \(b\), \(c\) and \(d\) are the free parameters, that is, \({{\beta _l}}\) is constrained to follow a 3rd order polynomial. If you expand this sum, \(a\) is the coefficient on the sum of \({X_{t - k}}\), \(b\) on \(k{X_{t - k}}\), etc. We use ENCODE to create these four series, which we then use in the regression.

•We impose an end constraint (polynomial is zero for lag \(L+1\)) by writing the polynomial as \(\left( {k - \left( {L + 1} \right)} \right){\kern 1pt} \left( {a + bk + c{k^2}} \right)\).

•We impose near (polynomial is zero for lag –1) and far constraints using \(\left( {k - \left( {L + 1} \right)} \right){\kern 1pt} \left( {k + 1} \right){\kern 1pt} \left( {a + bk} \right)\)

The trick in handling the constrained PDL’s is to get the EWISE instruction, which creates the encoding matrix R, correct. Because lag 0 (\(k=0\)) corresponds to the first row in R (J=1 in EWISE), the roots should be at J=0 (\(k=–1\)) and J=L+2 (\(k=L+1\)).

Note that the t-statistics on individual coefficients may end up being extremely high in such a regression. This is because each tests whether its coefficient can be made zero while maintaining the shape restriction, which is often nearly impossible.

PDL's have fallen out of favor mainly because the basic distributed lag like \eqref{eq:pdl_genericdl} usually leaves very high serial correlation in the residuals. If you allow for richer dynamics by adding lagged dependent variables (the ARDL model), you will generally get results that are easier to interpret.

Full Program

open data haversample.rat
calendar(m) 1947
data(format=rats) 1947:1 2007:4 fltg ftb3
set shortrate = ftb3
set longrate = fltg
*
* Run all estimates starting in 1951:1.
*
smpl 1951:1 *
declare rect r
*
* PDL with no end constraints
*
dim r(4,25)
ewise r(i,j)=j^(i-1)
encode(results=enc) r
# shortrate{0 to 24}
linreg(unravel) longrate
# constant enc
*
* Same estimation using the PDL procedure
*
@pdl(graph) longrate
# shortrate 0 24 3
*
* PDL with far constraint
*
dim r(3,25)
ewise r(i,j)=(j-26)*j^(i-1)
encode(results=enc) r
# shortrate{0 to 24}
linreg(unravel) longrate
# constant enc
*
* Same estimation using the PDL procedure
*
@pdl(constrain=far,graph) longrate
# shortrate 0 24 3

Output

Linear Regression - Estimation by Least Squares

Dependent Variable LONGRATE

Monthly Data From 1951:01 To 2007:04

Usable Observations 676

Degrees of Freedom 671

Centered R^2 0.8796681

R-Bar^2 0.8789508

Uncentered R^2 0.9828667

Mean of Dependent Variable 6.2800887574

Std Error of Dependent Variable 2.5607685752

Standard Error of Estimate 0.8909458121

Sum of Squared Residuals 532.62935926

Regression F(4,671) 1226.3113

Significance Level of F 0.0000000

Log Likelihood -878.6343

Durbin-Watson Statistic 0.0755

Variable Coeff Std Error T-Stat Signif

************************************************************************************

1. Constant 1.750066767 0.073305774 23.87352 0.00000000

2. SHORTRATE 0.154157409 0.015507779 9.94065 0.00000000

3. SHORTRATE{1} 0.125294136 0.009521005 13.15976 0.00000000

4. SHORTRATE{2} 0.099226710 0.005332489 18.60795 0.00000000

5. SHORTRATE{3} 0.075898788 0.003812551 19.90761 0.00000000

6. SHORTRATE{4} 0.055254029 0.004734373 11.67082 0.00000000

7. SHORTRATE{5} 0.037236091 0.005966873 6.24047 0.00000000

8. SHORTRATE{6} 0.021788630 0.006728874 3.23808 0.00126265

9. SHORTRATE{7} 0.008855306 0.006936418 1.27664 0.20217113

10. SHORTRATE{8} -0.001620224 0.006666145 -0.24305 0.80803892

11. SHORTRATE{9} -0.009694302 0.006054203 -1.60125 0.10979186

12. SHORTRATE{10} -0.015423271 0.005287915 -2.91670 0.00365569

13. SHORTRATE{11} -0.018863472 0.004621830 -4.08139 0.00005016

14. SHORTRATE{12} -0.020071247 0.004351233 -4.61277 0.00000476

15. SHORTRATE{13} -0.019102939 0.004627788 -4.12788 0.00004123

16. SHORTRATE{14} -0.016014890 0.005297522 -3.02309 0.00259745

17. SHORTRATE{15} -0.010863441 0.006065075 -1.79115 0.07372030

18. SHORTRATE{16} -0.003704935 0.006676529 -0.55492 0.57913476

19. SHORTRATE{17} 0.005404285 0.006944852 0.77817 0.43674223

20. SHORTRATE{18} 0.016407879 0.006733677 2.43669 0.01508135

21. SHORTRATE{19} 0.029249503 0.005965335 4.90325 0.00000118

22. SHORTRATE{20} 0.043872816 0.004720822 9.29347 0.00000000

23. SHORTRATE{21} 0.060221475 0.003778739 15.93692 0.00000000

24. SHORTRATE{22} 0.078239139 0.005295673 14.77416 0.00000000

25. SHORTRATE{23} 0.097869465 0.009493977 10.30858 0.00000000

26. SHORTRATE{24} 0.119056111 0.015488399 7.68679 0.00000000

Linear Regression - Estimation by Weighted Least Squares