Capps, Oral, Jr.

Miller, Stephen E.

Wells, Gary J.

English

Research Papers in Economics

"'WP  122983  December  1983  \ 
MORE  ON  ESTIMATING  ELASTICITIES  USING 
LAGGED  ENDOGENOUS  VARIABLES 
by 
S.  E.  Miller,  O.  Capps,  and  G.  J.  Wells December  1983  ' 
MORE  ON  ESTIMATING  ELASTICITIES  USING 
LAGGED  ENDOGENOUS  VARIABLES 
by 
S.  E.  Miller,  O.  Capps,  and  G.  J.  Wells 
) 
* Capps  is  Associate  Professor of Agricultural  Economics  at"VPJ&SU, 
Blacksburg,  VA,  and  Miller and  Wells  are Associate  Professors  at 
Clemson  University. 
Working  Papers  are  not  subject  to  review within  the  Department  of 
nr-ir,rlt"r;:l!  Fr:nnomics  and  Rural  Sociology MORE  ON  ESTIMATING  ELASTICITIES  USING 
LAGGED  ENDOGENOUS  VARIABLES 
Introduction 
1  In an article appearing  in this  journal,  Bopp  and  Pendley  ,  here-
after abbreviated B-P,  have  considered  the problem  of constructing  con-
fidence  intervals  for  long-term elasticities estimated  from  logarithmic 
equations  containing lagged  endogenous  variables  as  explanatory vari-
abIes.  The  estimated  long-term elasticity for  explanatory variable  i  is 
(1) 
where  B.  is  the  estimated  regression coefficient for  explanatory vari-
1 
able i, or equivalently,  the  short-term elasticity for  explanatory vari-
A 
able i,  and  A is the  estimated coefficient for  the  lagged  endogenous 
variable.  Bopp  and  Pendley use  an estimate of the  asymptotic variance 
of  B./(I-~) to construct approximate  confidence  intervals  for  ~ ..  The 
1  1 
purpose  of this note  is to present a  simple  technique  for  constructing 
exact confidence  intervals  for  ~ ..  This  technique  is also illust rated 
1 
by application to  the data  used by B-P. 
Exact  Confidence  Intervals  for 
Long-Term Elasticities 
The  ratio  B./(I-~) is,  under  the  usual  assumptions,  a  ratio of two 
1 
normally distributed  random  variables with the numerator  having  an 
expected value of B.  and variance  aB~"  and  the  denominator having 
1  1 
expected value of  I-A and  variance  a~.  The  covariance of the numerator 
and  denominator  is  -aB~'  Although Hinkley2  has  derived  an exact  , 2 
expression  for  the  exact  cumulative  distribution function  (c.d.f.)  for 
such  a  ratio,  evaluation of that function  involves  use  of  tabulations  of 
the bivariate normal  distribution function.  Since  use  of these  tabula-
tions  usually requires  trivariate interpolation,  construction of exact 
confidence  intervals  for  ~.  using this  approach would  be  at best inc  on-
1 
venient. 
Howev~r,  a  result due  to Fieller3  allows  construction of exact  con-
fidence  intervals  for  ~.  and  requires  no  more  than solutions  t o  quad-
1 
ratic equations.  This  result makes  use  of the fact that the  ratio R = 
~  A 
B./(I-A)  is equivalent to 
1 
B.  - R(I-~) =  0  . 
1 
A I-a percent  confidence  interval for  R is  given by  those values  of R 
for which 
Prob  I-a 
(2) 
(3) 
2  A  2 
where  sBi  is  the  sample variance of Bi ;  s(I_~) is the  sample variance  of 
(1-~);  sBi,(I-~) is the  sample  covariance of Bi  and  (1-~);  and  t~/2 is 
the  squared value  of the upper a/2 percentage point of the t-distribu-
tion with appropriate degrees  of  freedom,  or equivalently the upper  a 
percentage point of the F-distribution ~ith one  degree  of  freedom  in the 
numerator  and  the appropriate  degrees  of  freedom  in the  denominator. 
For the problem at hand,  the appropriate  degrees  of  freedom  for the 
t-distribution or the  denominator of the F-distribution are  the error 
A  A 
degrees  of freedom  of the  regression used  to estimate B.  and  A.  For the 
1 
confidence  interval to be  closed,  (I-A)  must  be  significantly different 3 
from  zero.  The  exact  l,a confidence  limits  for  ~.  are  those values  of R 
~ 
for which 
(4) 
Note  that the  solution to this quadratic  equation in R  need  not,  and 
generally will not,  be  symmetric  about  ~ ..  Thus,  the use  of the  asymp-
~ 
totic variance  of  ~.  to  construct approximate  confidence  intervals  for 
~ 
~.  centered upon  ~ .  can be  misleading. 
~  ~ 
An  Application 
In order to  illustrate the  construction of exact  confidence inter-
vals  for  long-term elasticities,  we  make  use  of the  data  analyzed  by 
B-P.  They  estimated  a  logarithmic  equation expressing  residential elec-
trical  consumption  in Connecticut  (Qt)  as  a  function of real disposable 
income  (Yt ) ,  own  marginal price  (Pt ) ,  and  lagged  consumption  (Qt-1) 
using  annual  observations  from  1960  to  1975.  The  estimated short- and 
long-term elasticities and  their respective variances  are  summarized  in 
Table  1.  The  estimated coefficient of adjustment  (1-~)  was  0.152,  and 
the  relevant  covariance  terms  were  estimated as  follows:  Sy,(1-~) = 
-sy,~ = 0.005;  and  sp,(l_~) =  -sP,~ = 0.0008.  The  error degree  of  free-
darn  is  12.  With this  information,  exact  confidence  intervals  can be 
constructed for both the  short- and  long-term elasticities.  These  are 
also  displayed  in Table  1. 
Note  that the  exact confidence  intervals for  the  long-term elastic-
ities are much  wider  than their respective  short-term counterparts. 
Also,  the  exact  confidence  intervals  for  the  long-term elasticities are 
not  symmetric  about their respective point estimates,  a  result not Table  1.  Exact  Confidence  Intervals  for Short- and  Long-Term 
Elasticities 
Estimated Elasticitya  Exact  95%  Confidence  Intervals 
Variable  Short-term  Long-term  Short-term  Long-term 
Income  0.17  1.13  -0.10  to 0.44  -1.51  to  1. 78 
(0.1221)  (0.474) 
Price  -0.21  -1.4  -0.31  to  -0.11  -4.08  to  -0.52 
(0.0480)  (0.587) 
a.  Standard errors  of estimated elasticities are  shown  in 
parentheses.  The  standard errors  for  long-term elasticities are 
asymptotic. 
~ 
detected by the  use  of the  asymptotic variance  of  ~i in constructing 
appropriate  confidence  intervals  for  ~ ..  Although the  exact  confidence 
1. 
4 
interval for  the  short-term income  elasticity is wider  than the interval 
for  short-term own  price elasticity,  the  relative widths  of the  long-
term elasticity intervals are  reversed,  a  result agreeing with the  find-
ings  of  B-P. 
Summary  and  Conclusions 
The  purpose  of this note  is to  demonstrate  a  technique  for  con-
structing exact  confidence  intervals  for  long-term elasticities  from 
logarithmic equations  containing  lagged  endogenous  variables  as  regres-
sors.  This  technique  involves  only simple  computations  based  upon 
results  generated  in the  course  of parameter estimation. Notes 
1 Bopp,  Anthony E.,  and  Robert E.  Pendley.  "A  Note  on Estimating 
Economic  Elasticities Using  Lagged  Endogenous  Variables,"  Intermountain 
Economic  Review  (Fall  1978),  pp.  111-7. 
2Hinkley,  D.  V.  "On  the Ratio  of Two  Correlated Normal  Random 
Variables,"  Biometrika  56(1969),  pp.  635-9. 
5 
3Fieller,  E.  C.  "The  Distribution of the  Index  in a  Normal  Bivari-
ate Population,"  Biometrika  24(1932),  pp.  428-40. SOUTH  CAROLINA  AGRICULTURAL  EXPERIMENT  STATION, 
CLEMSON  UNIVERSITY,  CLEMSON,  SOUTH  CAROLINA 
W.  C.  GODLEY, ASSOCIATE DEAN  AND  DIRECTOR 
S.  C.  EXPERIMENT STATION 
lUTHER P.  ANDERSON, DEAN, COllEGE OF AGRICULTURAL SCIENCES 
The  South  Carolina  Agricultural  Experiment  Station  is  a  cooperative  program  financed  from  federal 
and  state  funds.  1\  Is  the  polley  of  the  Experiment  Stall  on  to  comply  fully  with  the  regulations  of  Title 
VI,  the  Civil  Rights  Act  of  1964.  Complaints  may  be  filed  with  the  Director,  S.  C.  Agricultural  Experi-
ment  Stalion,  Clemson  University,  Clemson,  S.  C.  29631. 
I  .. 

More on Estimating Elasticities using Lagged Endogenous Variables

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