This paper uses the Pagano-Hartley procedure to estimate the lag length and polynomial degree for the case of a quarterly hog supply equation. The results show that the nicely humped shapes which materialize when using the Almon lag may be caused by the failure in accounting for autocorrelation in determining lag length and polynominal degree.Livestock Production/Industries, Research Methods/ Statistical Methods,

Azzam, Azzeddine M.

Yanagida, John F.

English

Research Papers in Economics

A Cautionary Note  on Polynomial
Distributed Lag Formulations of
Supply  Response
Azzeddine  Azzam  and John F. Yanagida
This paper  uses the Pagano-Hartley  procedure to estimate  the lag length and
polynomial  degree for the case of a quarterly  hog supply equation.  The results show
that the nicely  humped shapes which  materialize when using the Almon  lag may be
caused by the failure  in accounting for autocorrelation  in determining  lag length  and
polynominal  degree.
Key  words: Almon  lag, autocorrelation,  supply response.
Supply  response  in  agricultural  commodity
markets,  especially  livestock,  has proven dif-
ficult  to  estimate  statistically  (Chavas  and
Johnson).  Often,  the  temporal  modeling
framework  does  not  correspond  to  the  pro-
duction cycle as in the case of crops. The poly-
nomial  distributed  lag method  developed by
Almon  has been  widely  used by  agricultural
economists  to  explain  livestock  supply  re-
sponse (Chen, Courtney, and Schmitz; Meilke,
Zwart, and Martin; Kulshreshtha;  Meilke). The
general form  of the Almon model  with  finite
lag length X  can be expressed as
x
(1)  Yt,=  ~  fjX-yj+ E,  t = 1,2,...,  T,
j=0
where the  pattern  of the fj's is  described  by
the pth order polynomial
P
(2)  i =  ak)
k j  =  0  , 2, ... , X,
k=O
Xt is nonstochastic and  et  (0, a
2
) for all t.
Rarely are the length of lag and polynomial
degree  known parameters.  Consequently,  the
previously mentioned studies using the Almon
The authors are,  respectively,  an assistant professor and an asso-
ciate professor, Department of  Agricultural Economics, University
of Nebraska.
Nebraska  Agricultural  Experiment  Station  Journal Article  No.
7994.
The authors thank James  Hassler, James  Schmidt,  Emilio  Pa-
goulatos,  Marcello  Pagano,  T. Dudley  Wallace,  and two  anony-
mous Journal  reviewers for  helpful comments and suggestions.
technique relied on ad hoc methods of param-
eter selection.  Although these ad hoc methods
can introduce  specification  error  (Harper;
Hendry,  Pagan, and Sargan) in the distributed
lag model by omission (inclusion)  of relevant
(irrelevant) explanatory variables, it is possible
for researchers  to  obtain  desirable  statistical
results  and  fail to detect  possible  misspecifi-
cations.
Recently,  Pagano  and  Hartley  (PH)  pro-
posed  a  two-step  procedure  for determining
the lag length, X,  and polynomial degree, p in
the  presence  and  absence  of autoregressive
errors.1 Like  alternative  parameter  selection
procedures,  the PH  method uses  the sample
data  to  obtain  information  about  X and  p.2
Unlike  these  alternative  procedures,  the  PH
method is computationally  more  efficient  by
circumventing the need for nested  sequential
hypothesis testing through the use of orthog-
onal reparameterization  of the basic  model.3
This avoids repeated calculations  for different
choices of X  and p.
The  primary  objective  of this  paper  is  to
reexamine  Meilke's  study  on  the  supply  re-
This testing procedure was originally developed  by Mizon.
2 There are numerous procedures and criteria for determing the
appropriate  lag length  and polynomial  degree.  Useful summaries
of these  procedures are found in Trivedi and Pagan; Judge et al.;
and Hendry,  Pagan, and Sargan.
3  See  Batten and Thorton (1983,  1984) for an application of the
PH method and comparisons  with alternative  procedures.
Western Journal  of Agricultural  Economics, 12(1):  60-64
Copyright  1987  Western Agricultural  Economics  AssociationPolynomial Distributed  61
Table  1.  Standardized  t-Values  for Lag  Length Selection  for  Variable (PBG/PC)  with  Lag
Length Equal to 20,  19,  17,  15, 13,  11,  and 9 Quarters
~~~~~~~~Lagged  _________Lag  Length Lagged
Quarters  20  19  17  15  13  11  9
0  4.904  5.039  5.221  5.558  5.770  5.914  5.988
1  .589  .600  .641  .652  .649  .834  .865
2  .635  .488  .522  .422  .493  .391  .397
3  1.085  .987  1.011  .863  .885  .882  .925
4  1.389  1.327  1.377  1.356  1.350  1.391  1.460
5  1.309  1.214  1.221  1.367  1.416  1.482  1.544
6  .832  .756  .829  .708  .732  .831  .879
7  1.438  1.234  1.236  1.159  1.265  1.301  1.195
8  1.179  1.281  1.291  1.263  1.315  1.393  1.521
9  1.568
a 1.474a  1.520a  1.518a  1.567a  1.469a  1.408a
10  .553  .408  .412  .492  .524  .538
11  .906  .885  .920  .990  .952  .935
12  .557  .530  .563  .636  .694
13  .618  .674  .702  .637  .611
14  .578  .615  .653  .557
15  .604  .610  .561  .602
16  1.096  1.099  1.162
17  .895  .620  .592
18  .467  .276
19  .561  .774
20  1.110
Notes:  Lag length selection  process is to find the first  statistically significant period starting with the maximum  lag length and working
backwards  to the current period (i.e.,  lag = 0).  Batten and Thorton utilize  a significance level of 15% to minimize  effects from Type II
error.  The critical  t-values  for  15%  significance  used in this study range  from  1.40 to  1.47 for L = 20,  19,  17,  15,  13,  11,  and 9. aFirst significant  t-statistic.
sponse of hogs using the PH procedure to select
statistically the lag length and polynomial de-
gree.  This investigation  also examines the  ef-
fects of nonautoregressive  and autoregressive
errors in determining X  and p.
Application of the Pagano-Hartley
Procedure to Hog
Supply Response
The technical details of the PH procedure  are
developed in the appendix.  In this section,  we
apply the PH procedure to estimating lag length
and polynomial degree for the following quar-
\terly hog supply equation  (Meilke):
3
(3)  Qt  = ao  +  °aiDi 
+ a4T
(+  A  + PBG\ 
+
j=0  wh
where
pounds,  carcass  weight),  PBG is  the  average
price  of barrows  and gilts,  seven  markets ($/
hundredweight),  PC is  the  average  price  re-
ceived for corn, No. 2  yellow ($/bushel), T is
a time trend, and Di represent seasonal dummy
variables;  X and  p  are,  respectively,  the  lag
Table  2.  Standardized  t-Values  for  Polyno-
mial Degree  Selection  for L* = 9 Quarters
Absolute  Value
Polynomial Degreea  of t-Statisticsb
0  2.445
1  5.357
2  2.719
3  2.065c
4  .259
5  .924
6  .549
7  .073
8  .217
9  .352
P  .a  Polynomial degree selection process is to find the first statistically
(4)  i-= C  tOk(>J)  significant  polynomial  degree  starting  with the maximum  degree
k=O  length and working  backwards to degree  equal to zero.
b The critical t-value for  15%  significance  is  1.40.
and QS is commercial pork production (million  cFirst significant t-statistic.
Azzam and YanagidaWestern Journal  of Agricultural  Economics
Table  3.  Standardized  t-Values  for  Lag
Length  Selection  with  Correction  for  Serial
Correlation for Variable (PBG/PC)  with Lag
Length Equal to Twenty  Quarters
Absolute Value
Lagged Quarters  of t-Statistics
0  7.021
1  .653
2  .041
3  .704
4  1.084
5  1.020
6  .311
7  1.164
8  .830
9  1.470a
10  .305
11  .85
12  .599
13  .957
14  .888
15  .968
16  1.385
17  1.396
18  .003
19  .084
20  .483
Notes: See  notes in table  1.
a First significant t-statistic.
length and degree of the polynomial to be de-
termined. Equation (3)  is estimated in double-
logarithmic form for the time period  1960-83
(96 quarters).
Under  the assumption  of serially uncorre-
lated errors, we  chose a maximum  lag length
L = 20 and reparameterized equation (A.2) as
shown in the appendix.  The resulting tL_j sta-
tistics are listed in table  1. The PH procedure
chose a lag length of nine periods. Alternative
values ofL (lag lengths of 19,  17,  15,  13,  11,
and  9  periods) were  specified  to test the  sen-
sitivity of the lag length selection to the initial
choice  of L. As shown in table  1, the results
were unchanged in all cases. The standardized
t values  for polynomial  degree  selection  with
L* =  9  are  listed in  table  2.  The  procedure
yields a polynomial degree  equal to  3.
A test for serial correlation with L* = 9 and
P*  =  3  produced  a Durbin-Watson  statistic
equal  to  .328,  indicating  strong  positive  au-
tocorrelation of the residuals. This suggests that
the disturbances  should be modeled as an au-
toregressive  process.
Using the Gallant and  Goebel method and
Table  4.  Standardized  t-Values  for  Polyno-
mial Degree  Selection  with Correction for  Se-
rial Correlation (L* = 9 Quarters)
Absolute Value
Polynomial  Degreea  of t-Statistics
0  .919
1  4.566
2  4.526
3  3.977
4  .550
5  2.506
6  1.389
7  1.141
8  1.126
9  1.745b
aSee footnotes a and b, table 2.
b First significant t-statistic.
Harvey  (p. 204), we model  the autoregressive
effects  by
(1  - pB) (1 - P4B4) 
= Vt,
where B is the lag operator and P4 captures the
seasonal effects, while p, accounts for the serial
correlation  from  quarter to quarter. The esti-
mates,  Pi and p4,  are -.861 and  .010, respec-
tively. The t-values for lag length selection with
correction for serial correlation  are in table  3.
It is evident  that lag  length  selection,  in  our
particular example, is invariant when account-
ing for serial correlation.  However,  the degree
of the polynomial  changed  from  3  to  9  (see
table 4) when  adjusting for serial correlation.
The  Durbin-Watson  statistic  after  correcting
for serial correlation  is  1.936.  This result pro-
vides further support to PH's conclusion that
estimation  of polynomial  distributed  lags  is
extremely  sensitive  to the adjustment  of the
autoregressive  process.
Summary
In  this paper,  we  apply the PH procedure  to
estimate the lag length and polynomial degree
for the case of a quarterly hog supply process.
Our study closely resembles the work done by
Meilke but utilizes a different time period for
analysis  and  revised  data.  One  implication
from our results is that with L* = P* =  9,  the
Almon  procedure  is the same  as unrestricted
least squares.
These results also serve as  a warning to fu-
ture  users of the Almon technique  in supply
62  July 1987Polynomial Distributed  63
analyses.  Namely,  the  nicely humped  shapes
which materialize  when  applying  the Almon
distributed lag procedure  may well be  caused
by the  failure  in accounting  for serial  corre-
lation in determining  lag  length and  polyno-
mial  degree.  Furthermore,  the economic  im-
plication from our results is that short-run and
long-run elasticities derived from the estimat-
ed coefficients of the Almon model  should be
interpreted with caution.
[Received May 1986;final revision
received February 1987.]
Mizon, G. E.  "Inferential Procedures in Nonlinear Models:
An Application  in  a  U.K.  Industrial  Cross-Section
Study of Factor Substitution and  Returns  to Scale."
Econometrica  45(1977): 1221-42.
Pagano,  Marcello,  and  Michael J.  Hartley.  "On  Fitting
Distributed  Lag Models  Subject to  Polynomial  Re-
strictions." J. Econometrics 16(1981):171-98.
Trivedi, P. K., and A. R. Pagan.  "Polynomial Distributed
Lags: A Unified Treatment." Econ. Stud. Quart., April
1979,  pp. 37-49.
Appendix
Pagano-Hartley  Procedure
References
Almon,  S.  "The  Distributed  Lag Between  Capital  Ap-
propriations  and  Expenditures."  Econometrica
33(1965):178-96.
Batten,  D. S.,  and D.  L. Thorton.  "How Robust Are the
Policy Conclusions of the St. Louis  Equation?: Some
Further Evidence."  Review, Federal Reserve Bank of
St.  Louis  66(1984):26-32.
. "Polynomial  Distributed  Lags and  the Estima-
tion of the St. Louis  Equation."  Review,  Federal Re-
serve Bank of St. Louis  65(1983):13-25.
Chavas, Jean-Paul, and Stanley R. Johnson.  "Supply Dy-
namics: The Case of  U.S. Broilers and Turkeys." Amer.
J. Agr. Econ. 64(1982):558-64.
Chen, Dean, Richard Courtney, and Andrew Schmitz.  "A
Polynomial  Distributed  Lag  Formulation  of Milk
Production Response."  Amer. J.  Agr. Econ. 54(1972):
77-83.
Gallant,  A.  Ronald,  and J.  Jeffery  Goebel.  "Nonlinear
Regression  with  Autocorrelated  Errors."  J. Amer.
Statist. Assoc.  71(1976):961-67.
Harper, Charles P.  "Testing for the Existence of a Lagged
Relationship  within  Almon's  Method."  Rev.  Econ.
and Statist. 50(1977):204-10.
Harvey,  A. C.  The Econometric Analysis of Time Series.
New York: John Wiley & Sons,  1981.
Hendry,  D. F., A. R. Pagan,  and J. D. Sargan.  "Dynamic
Specification." Handbook  ofEconometrics, vol. 2, ed.,
Z. Griliches and M.  D.  Intriligator.  Amsterdam:  El-
sevier  Science Publishers,  1984.
Judge,  G.  G.,  W.  E.  Griffiths,  R. C.  Hill,  and T.-C.  Lee.
The Theory and Practice  of Econometrics. New York:
John Wiley & Sons,  1980.
Kulshreshtha,  S.  "An  Analysis  of the  Canadian  Cattle
Supply Using Polynomial Distributed Lags."  Can. J.
Agr.  Econ. 24(1976):1-14.
Meilke, Karl D.  "Another Look at the Hog-Corn Ratio."
Amer. J. Agr. Econ.  59(1977):216-19.
Meilke, K. D., A. C. Zwart, and L. J. Martin."North Amer-
ican  Hog  Supply:  A  Comparison of Geometric  and
Polynomial  Distributed  Lag  Models."  Can. J. Agr.
Econ. 22(1974):15-30.
The basic PH model, in matrix form, is similar to equation
(1) with the addition of a matrix of contemporaneous  ex-
planatory variables Z, i.e.,
(A.1) Y-  = ZA6  + Xxfx  + EX,
where  Z. and  X.  are,  respectively,  (T  x  m)  and  (T  x
(X +  1))  matrices  of  observations  on  the  explanatory
variables;  and  6, and  fl  are,  respectively,  (m  x  1) and
((X  +  1) x  1) vectors of parameters to be estimated.
Replacing  the unknown X  by the largest lag L, we  can
write equation  (A.1) as
(A.2)  YL =  [ZL  XL]  L  + EL  = WL  + EL,
where  WL  and VL  are  a ((T-  L)  x  (m  + L  +  1))  and
((m + L +  1) x  1) augmented matrix and vector, respec-
tively.
By the  Gram-Schmidt  decomposition,  WL  can  be de-
composed  into  WL  =  QLRL,  where  QL  is a ((T - L)  x
(m  + L  +  1))  matrix  with  orthonormal  columns,  and
RL  is an  upper  triangular  square  matrix of order (m  +
L + 1).  Equation (A.2) now can be rewritten  as
(A.3) YL  = QLaL + EL,
where aL = RLaL = [a  : a]'.  Since  QL is orthonormal, the
least squares estimate  of aL is dL  = [da: d]' = QLYL.  To
recover the structural parameters 
1L,  we  simply calculate
RL-  dL
By virtue of orthogonal reparameterization of equation
(A. 1),  the hypotheses  regarding the ij's in equation (A. 1)
and the aj's in equation (A.3)  are equivalent,  i.e.,
HL-i :  fL  =  OL-  1=  . · =  fL-  j
H-I i:  af =  ap-, =...  = af  -=
forj = 0,  1, 2, ... ,L,  and
=0
forj = 0,  1, 2, ... ,L.
Pagano  and Hartley  note  that in view of the orthogo-
nality of the procedure defining  the aj's, one can equiva-
lently consider  the simple  hypothesis:
(A.4)  KL  : afj = 0  forj = 0,  1, 2  ... , L.
The estimate of the lag length is equal to the index L-j,
where j is the smallest number for which KLj is rejected.
The appropriate  t-statistic for the hypothesis KL_j  is
Azzam and YanagidaWestern Journal of Agricultural Economics
tL-j =  aiLj/s,
where  s  is  the  standard  error  such  that  s2  =
Y  YL  - LAL
T-m-L-  1
The  procedure  for  selecting  the  polynomial  degree  is
analogous  to that  of the  lag length.  Denoting  L* as  the
optimal lag, we rewrite equation  (A. 1)  as
(A.5)  Y  = ZL.BL,  + XL*L*  + EL*.
Substituting for the fj's from equation (2) in the text, and
setting p = L*, we obtain
(A.6) YL*  ZL*6L*  + XL**a  + EL*
where  XL.  = XL.H, and
1  0  2  0O
1  1  . . . 1
1  2  . 2L*
H=  . .
1  L*  . . . L
*L *
By  orthogonal  reparameterization  of equation  (A.6),  we
arrive  at the  analogous  hypothesis  in  equation (A.4) re-
garding the polynomial degree.
Dropping the assumption of  serially uncorrelated errors,
we  slightly modify the PH procedure for determining the
lag length  L* and  the degree of the polynomial  P*. This
adjustment  is an  extension of the work  done by Gallant
and Goebel  on nonlinear regression  with autocorrelated
errors in the linear case. The essence of the Gallant-Goebel
method is (a) to construct  a r  matrix from the estimates
of the autocovariances  up to a lag r of the autoregressive
process,
E, +  PjEt-j =  Vt,
j=l
and (b) to find a matrix  U such that U' U = rF' .The matrix
U is used  to transform  the  original  observations  of the
regression  model. The generalized least squares (GLS) es-
timate  is  then  obtained  by  using ordinary  least  squares
(OLS).
64  July 1987

A Polynomial Distributed Lag Formulation of Milk Production Response.&quot;

Another Look at the Hog-Corn Ratio.&quot;

Distributed Lags and the Estimation of the St. Louis Equation.&quot; Review, Federal Reserve Bank of St. Louis 65(1983):13-25.

Dynamic Specification.&quot; Handbook ofEconometrics,

How Robust Are the Policy Conclusions of the St. Louis Equation?: Some Further Evidence.&quot; Review, Federal Reserve Bank of St.

Martin.&quot;North American Hog Supply: A Comparison of Geometric and Polynomial Distributed Lag Models.&quot;

A CAUTIONARY NOTE ON POLYNOMIAL DISTRIBUTED LAG FORMULATIONS OF SUPPLY RESPONSE

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